Stability, control and application of time-delay systems 9780128149287, 9780128149294, 0128149299

Table of contents :
Cover......Page 1
Stability, Control andApplication ofTime-Delay Systems......Page 3
Copyright page......Page 4
Contributors......Page 5
Introduction......Page 9
LTI-TDS model......Page 11
Stability and spectral properties......Page 12
Algorithm steps in detail......Page 13
Continuous-time approximation......Page 16
Discrete-time approximation......Page 17
Approximation of neutral quasipolynomials......Page 18
Associated characteristic exponential polynomial approximation......Page 19
Neutral characteristic quasipolynomial approximation......Page 20
Retarded system......Page 21
Neutral system......Page 24
Conclusions......Page 27
References......Page 28
Introduction......Page 31
Modeling of series elastic actuators......Page 35
Gain design of series elastic actuators......Page 38
Critically damped controller gain design criterion......Page 39
Trade-off between torque and impedance control......Page 40
SEA impedance transfer function......Page 43
Effects of time delays and filtering......Page 44
Effect of load inertia......Page 46
Evaluation of the controller design......Page 47
Step response implementation......Page 50
Distributed operational space control of a mobile base......Page 53
Discussions and conclusion......Page 56
References......Page 57
Introduction......Page 60
Control framework......Page 62
Internal stability and string stability......Page 65
Head-to-tail transfer function......Page 67
Stability conditions......Page 70
Stability diagrams......Page 71
Numerical simulations......Page 73
Time-domain stability analysis......Page 75
Discussions......Page 78
References......Page 82
Introduction......Page 85
Problem statement......Page 87
H∞ design method......Page 91
W1,2 robustness criterion......Page 96
Numerical example......Page 100
References......Page 102
Introduction......Page 104
Notations and graph theory......Page 106
Some basic definitions of IQC......Page 107
Problem statement......Page 108
Main results......Page 109
State-feedback case......Page 110
Output-feedback case......Page 114
An illustrative example......Page 120
Conclusions......Page 122
Appendix......Page 125
References......Page 126
Introduction......Page 128
Dynamics of the rotor......Page 129
Stability analysis of the uncontrolled system......Page 131
Hovering flight: μ=0......Page 132
Horizontal flight: μ>0......Page 133
Stability analysis of the controlled system......Page 134
Hovering flight: Control without delay......Page 136
Hovering flight: Delayed control......Page 138
Stabilization of pitch divergence......Page 140
Stabilization of pitch-flap flutter......Page 141
Horizontal flight......Page 142
Concluding remarks......Page 144
References......Page 146
7The small signal stability region of power systems with time delay......Page 148
Power system and the time delay......Page 149
Power system small signal stability model with time delay......Page 150
Optimization-based boundary tracing algorithm......Page 152
Single-machine-infinite-bus system......Page 154
Time-delay impact on the power system......Page 156
Single-machine-infinite-bus system......Page 159
WSCC 3-generator-9-bus system......Page 161
References......Page 166
Introduction......Page 168
Modeling and control design with delays......Page 169
Car-following model......Page 170
Connected vehicle systems......Page 174
Robust string stability......Page 178
Uncertainties in a predecessor-follower system......Page 179
Robust connected cruise control design......Page 183
Conclusion......Page 186
References......Page 187
Introduction......Page 190
Robust stability of an uncertain system: A pseudospectral approach......Page 193
Computation of the pseudospectral abscissa......Page 197
Smoothness properties and optimization of the pseudospectral abscissa......Page 204
Numerical experiments......Page 206
Concluding remarks......Page 210
References......Page 211
Introduction......Page 213
Description of predictor feedback controllers......Page 214
Control problem without predictor......Page 215
The Smith predictor......Page 216
The modified Smith predictor......Page 218
Finite spectrum assignment......Page 220
Effect of initial conditions......Page 222
Implementation issues......Page 224
Application of observers......Page 225
Summary and conclusions......Page 226
References......Page 229
Introduction......Page 231
System model and definitions......Page 233
Relaxed L-K functional......Page 234
Stochastic admissibility criteria......Page 238
Definitions and assumptions......Page 240
Extended dissipativity criteria......Page 241
Problem formulation......Page 243
Useful lemmas......Page 244
Controller synthesis conditions......Page 246
Output-feedback control......Page 249
Extended dissipative filtering......Page 254
Conclusions......Page 256
References......Page 257
Further reading......Page 259
Introduction......Page 260
Problem formulation and preliminaries......Page 261
The introduction for fuzzy Markov jump model......Page 262
The introduction of event-triggered mechanism......Page 263
The model of reliable control......Page 264
Stability analysis and event-triggered controller design for FMJSs......Page 266
Stability analysis for FMJSs......Page 267
Event-triggered reliable controller design for FMJSs......Page 271
Numerical example......Page 274
Conclusions......Page 278
References......Page 279
Introduction......Page 281
Problem statement and preliminaries......Page 283
Mittag-Leffler stability of linear systems......Page 284
Effective algorithms for stability test......Page 285
Evaluation of the test integral......Page 286
Nyquist frequency plot......Page 287
Calculation of the rightmost characteristic roots......Page 290
Stability of a class of systems with delay-dependent coefficients......Page 291
Concluding remarks......Page 293
References......Page 294
14Sliding mode control for Markovian jumping systems with time delays......Page 296
System description......Page 297
Main results......Page 298
Augmented system and SMO formulation......Page 299
Derivation of the error dynamics......Page 301
Stability analysis of the overall closed-loop plant......Page 302
Reachability of the sliding mode surface......Page 304
The complete system synthesis algorithm......Page 306
Simulation study......Page 307
Results illustration and discussion......Page 308
Conclusions and future work......Page 310
References......Page 314
Introduction......Page 315
Preliminaries......Page 317
NPC: Observer-based output feedback case......Page 319
NPC: State feedback case......Page 320
Without delay mismatch......Page 322
With delay mismatch......Page 324
A numerical example......Page 325
State feedback case......Page 326
Observer-based output feedback case......Page 328
References......Page 329
Introduction......Page 332
Simple random walks......Page 333
Gambler's ruin......Page 335
θ=0 (τr=τp)......Page 337
θ> 0 (τr > τp)......Page 338
Comparison against computer simulations......Page 341
θ< 0 (τr < τp)......Page 342
Discussion......Page 343
Ruin probability......Page 344
Acknowledgments......Page 345
References......Page 346
Introduction......Page 347
Problem formulation......Page 349
Event generator and switching controller synthesis: The shortnetwork-induced delay case......Page 352
Event generator and switching controller synthesis: Combined the shortnetwork-induced delay and the packet dropout case......Page 355
Self-triggered control......Page 357
An example......Page 359
References......Page 362
Introduction......Page 364
Problem statement......Page 366
Convergence analysis......Page 371
Converting BMI into LMI......Page 372
Particular solution: The case of M measurements......Page 373
Classical approach......Page 374
Comparison from LMI feasibility point of view......Page 375
Illustrative examples......Page 376
Example 1......Page 377
Conclusion......Page 380
References......Page 381
Introduction......Page 384
Problem statement......Page 386
Existence conditions......Page 389
Stability conditions......Page 390
Simulation and results......Page 394
References......Page 398
Introduction......Page 401
Problem statement......Page 403
Existence conditions and gain parameterization......Page 406
Stability conditions for gain computation......Page 408
Simulation and results......Page 414
References......Page 417
Introduction......Page 419
Problem statement......Page 421
Robust sliding window observer synthesis......Page 424
Particular solution: The case of two measurements......Page 426
Discussion on the enhancement......Page 427
Filters comparison......Page 428
Comparison from computational complexity point of view......Page 429
Simulation results......Page 430
First case......Page 431
Second case......Page 432
References......Page 434
Introduction......Page 436
Definition of the fractional-order operator......Page 438
Fractional-order state space model......Page 440
Problem definition......Page 441
Identification method......Page 442
Simulation examples......Page 444
Example 1: Fractional commensurate case......Page 445
Example 2: Fractional noncommensurate example......Page 446
References......Page 454
D......Page 457
F......Page 458
I......Page 459
N......Page 460
P......Page 461
S......Page 462
T......Page 463
Z......Page 464
Back Cover......Page 465

Citation preview

Stability, Control and Application of Time-Delay Systems

Stability, Control and Application of Time-Delay Systems Edited by

Qingbin Gao Harbin Institute of Technology, Shenzhen, China

Hamid Reza Karimi Department of Mechanical Engineering, Politecnico di Milano, Milan, Italy

Butterworth-Heinemann is an imprint of Elsevier The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States © 2019 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-814928-7 For information on all Butterworth-Heinemann publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Mara Conner Acquisition Editor: Sonnini R. Yura Editorial Project Manager: Ali Afzal-Khan Production Project Manager: Nirmala Arumugam Cover Designer: Miles Hitchen Typeset by SPi Global, India

Contributors K. Abderrahim CONPRI, National Engineering School of Gabes (ENIG), University of Gabes, Gabes, Tunisia H. Souley Ali CRAN UMR 7039 CNRS, University of Lorraine, Cosnes et Romain, France Mohamed Aoun MACS Laboratory, National Engineering School of Gabes (ENIG), University of Gabes, Gabes, Tunisia Maamar Bettayeb Department of Electrical and Computer Engineering, University of Sharjah UAE and (CEIES) King Abdulaziz University, Jeddah, Saudi Arabia Francesco Borgioli Deparment of Computer Science, KU Leuven, 3001 Heverlee, Belgium Mohamed Boutayeb CRAN UMR CNRS 7039, University of Lorraine, Cosnes et Romain, France Rudy Cepeda-Gomez Institute of Automation, University of Rostock, Rostock, Germany Jie Chen School of Automation; Key Laboratory of Intelligent Control and Decision of Complex Systems, Beijing Institute of Technology, Beijing, China Tounsia Djamah Department of Control Engineering L2CSP, UMMTO, Tizi Ouzou, Algeria Chaoyu Dong School of Electrical and Information Engineering, Tianjin University, Tianjin, China Noussaiba Gasmi CRAN UMR CNRS 7039, University of Lorraine, Cosnes et Romain, France Jin I. Ge Department of Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA, United States Dávid Hajdu Department of Applied Mechanics, Budapest University of Technology and Economics and MTA-BME Lendület Human Balancing Research Group, Budapest, Hungary Karima Hammar Department of Control Engineering L2CSP, UMMTO, Tizi Ouzou, Algeria

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Contributors

Lama Hassan CRAN UMR CNRS 7039, University of Lorraine, Cosnes et Romain, France Tomohisa Imai Graduate School of Mathematics, Nagoya University, Nagoya, Japan Tamás Insperger Department of Applied Mechanics, Budapest University of Technology and Economics and MTA-BME Lendület Human Balancing Research Group, Budapest, Hungary Hongjie Jia School of Electrical and Information Engineering, Tianjin University, Tianjin, China Zhuoyu Li College of Information Science and Engineering, State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang, China Song Liang State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China Dan Ma College of Information Science and Engineering, State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang, China Wim Michiels Deparment of Computer Science, KU Leuven, 3001 Heverlee, Belgium Tamas G. Molnar Department of Applied Mechanics, Budapest University of Technology and Economics, Budapest, Hungary Toru Ohira Graduate School of Mathematics, Nagoya University, Nagoya, Japan Gábor Orosz Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI, United States and Department of Civil and Environmental Engineering, University of Michigan, Ann Arbor, MI, United States Libor Pekaˇr Faculty of Applied Informatics, Department of Automation and Control Engineering, Tomas Bata University in Zlín, Zlín, Czechia A. Sassi CRAN UMR 7039 CNRS, University of Lorraine, Cosnes et Romain, France; CONPRI, National Engineering School of Gabes (ENIG), University of Gabes, Gabes, Tunisia

Contributors

Luis Sentis Department of Aerospace Engineering and Engineering Mechanics, University of Texas, Austin, TX, United States Lei Su College of Information Science and Engineering, Northeastern University, Shenyang, China Jian Sun School of Automation; Key Laboratory of Intelligent Control and Decision of Complex Systems, Beijing Institute of Technology, Beijing, China Assem Thabet MACS Laboratory, National Engineering School of Gabes (ENIG), University of Gabes, Gabes, Tunisia Zaihua Wang State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China Fen Wu Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC, United States Hongyan Yang Harbin Institute of Technology, Harbin, China Dan Ye College of Information Science and Engineering; State Key Laboratory of Synthetical Automation of Process Industries, Northeastern University, Shenyang, China Shen Yin Harbin Institute of Technology, Harbin, China Chengzhi Yuan Department of Mechanical, Industrial and Systems Engineering, University of Rhode Island, Kingston, RI, United States M. Zasadzinski CONPRI, National Engineering School of Gabes (ENIG), University of Gabes, Gabes, Tunisia Ali Zemouche CRAN UMR CNRS 7039, University of Lorraine, Cosnes et Romain, France Baoyong Zhang School of Automation, Nanjing University of Science and Technology, Nanjing, China Linjun Zhang Ford Motor Company, Dearborn, MI, United States

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Contributors

Ye Zhao George W. Woodruff School of Mechanical Engineering, Georgia Tech, Atlanta, GA, United States Guangming Zhuang School of Mathematical Sciences,Liaocheng University, Liaocheng, China

CHAPTER

On the numerical determination of stability regions in the delay space via dominant pole estimation

1 Libor Pekaˇr

Faculty of Applied Informatics, Department of Automation and Control Engineering, Tomas Bata University in Zlín, Zlín, Czechia

Chapter outline 1 Introduction......................................................................................... 2 Preliminaries ....................................................................................... 2.1 LTI-TDS model ........................................................................ 2.2 Stability and spectral properties.................................................... 3 Numerical gridding DDS algorithm.............................................................. 3.1 Framework of the algorithm ......................................................... 3.2 Algorithm steps in detail............................................................. 3.3 Continuous-time approximation .................................................... 3.4 Discrete-time approximation ........................................................ 3.5 Approximation of neutral quasipolynomials ....................................... 4 Examples ............................................................................................ 4.1 Retarded system ...................................................................... 4.2 Neutral system ........................................................................ 5 Conclusions......................................................................................... Acknowledgments .................................................................................... References.............................................................................................

1 3 3 4 5 5 5 8 9 10 13 13 16 19 20 20

1 Introduction It is well known that linear time-invariant time-delay systems (LTI-TDSs) are typical representatives of infinite-dimensional systems [1]. It means that they own infinitely many system poles (or, characteristic values), the loci of which have many interesting and tricky features [2,3]. This fact inherently implies that analytic and control tasks related to LTI-TDSs and their spectra are generally nontrivial. Therefore, some rigorous solutions suffer from rich or hardly implementable mathematical formulations. Stability, Control and Application of Time-Delay Systems. https://doi.org/10.1016/B978-0-12-814928-7.00001-9 © 2019 Elsevier Inc. All rights reserved.

1

2

CHAPTER 1 Numerical determination of stability regions

Characteristic root loci of LTI-TDSs are closely related to system stability [4], as is the case of linear nondelayed systems; however, only the exponential stability of LTI-TDSs has a clear relation to the existence of the whole spectrum in the open left-half complex plane. In fact, these systems can be asymptotically, H∞ or boundedinput bounded-output stable even if there exist purely imaginary poles, or the rightmost spectrum asymptotically reach the imaginary axis [2,5,6]. Hence, the detection of imaginary axis crossing can be used as a tool for exponential stability analysis. The systematic searching of stabilizing regions in the delay space is called the delay-dependent stability (DDS) analysis. A couple of research results have dealt with the study of DDS. A family of indirect DDS methods utilizes LyapunovKrasovskii approaches and linear matrix inequalities; however, these approaches usually suffer from mathematical complexity, provide conservative results, and they are difficult to be implemented in practice—see, for instance, a discussion in Ref. [7]. Contrariwise, direct (or, frequency-domain) approaches adopt the above-introduced idea of the determination of purely imaginary poles that constitute the stability margin. Several techniques within this framework have been investigated so far. Namely, the cluster treatment of characteristic roots (CTCR) paradigm [8,9], the direct method [10], the Puiseux series expansion technique [11], the matrix pencil method [12], the Kronecker multiplication method [13], the use of the argument principle [14], etc. Despite numerous DDS ideas and principles, there is still a lack of relatively simple, easily programmable, and practically well-implementable DDS methods that give a sufficiently accurate estimation of stabilizing regions in the delays space. Hence, this chapter is aimed at the presentation of a gridding-based direct computational method for the determination of all multiple delays for LTI-TDSs ensuring the system (exponential) stability. The core of the proposed method lies in the iterative estimation of the rightmost (dominant) pole or a conjugate pair in every single-grid node of the discretized delay space. The delay-free dominant pole can initially be simply and exactly computed as a polynomial root. Since system pole positions are continuous with respect to delay values almost everywhere, the approximated already known dominant pole locus in the nearest grid node to the current one serves as the initial estimation for the iterative computation. The iterative estimations are computed via a polynomial approximation of the so-called characteristic quasipolynomial. Two ideas how to find the approximating polynomial are suggested. As first, the Taylor series-based expansion of a particular order in the neighborhood of the nearest dominant poles estimation is performed [15,16]. As second, the Tustin (bilinear) transformation followed by the prewarping correction is utilized to get the discrete form of the approximating characteristic polynomial [17,18]. Here, the delay (exponential) terms are subjected to a quadratic extrapolation method to get commensurate delays that are closely related to the discrete-time shifting operator. Once an imaginary axis crossing is detected, the so-called root tendency values are used to get a more accurate critical delay and the corresponding critical frequency estimations, by means of averaged (multistep) Newton’s zeropoint extrapolation principle. Finally, the set of found critical delays is eventually augmented via quadratic regression of the found points.

2 Preliminaries

This chapter, in fact, summarizes, integrates, and extends the two above-referred approaches [16,17]. Both the ideas are compared, and an extension to a very delicate family of so-called neutral systems is suggested—strong stability, the essential spectrum, and the approximation of the associated characteristic exponential polynomial are discussed as well. An example solving the stabilization of a skater on the controlled swaying bow, modeled as an unstable retarded TDS, is given to the reader. The obtained numerical results are compared to those received by the CTCR paradigm. Another academic example suggests a possibility how to modify the algorithm in the case of neutral delays. The Quasi-Polynomial mapping Rootfinder (QPmR) software tool by Vyhlídal and Zítek [19] is used to get the reference values in the examples. The rest of the chapter is organized as follows: in Section 2, the definition of an LTI-TDS along with the introduction of its exponential stability and selected spectral properties are given. Then, the reader is acquainted with an overview of the proposed DDS algorithm including its continuous- and discrete-time versions in Section 3. Moreover, the approximation of neutral quasipolynomials and its associated difference equation is suggested. Section 4 includes two numerical examples. The chapter is then concluded with a discussion of possible algorithm modifications and its summary. Notation. Re(s) and Im(s) denote, respectively, the real and imaginary parts of some s ∈ C. R, C, and N denote the set of real, complex, and natural (including zero) numbers, respectively. Rn is the n-dimensional Euclidean space. F(s | p) stands for a function of s with a parameter set p. The superscript “T” means the vector or matrix transpose, and · represents the floor function. Symbol  · s means the supreme norm.

2 Preliminaries In this preliminary section, the class of LTI-TDSs for the purpose of this contribution is introduced first; then, exponential stability is formulated and some important properties of the spectrum of LTI-TDS characteristic values are presented.

2.1 LTI-TDS model Definition 1. Let an LTI-TDS be formulated by state and output functional differential equations as x˙ (t) +

nH 

Hi x˙ (t − τH,i ) = A0 x(t) +

i=1

y(t) = Cx(t) +

nA 

i=1 n C 

Ai x(t − τA,i ) + B0 u(t) +

nB  i=1

Bi u(t − τB,i ), (1)

Ci x(t − τC,i ),

i=1

where u(t) ∈ Rm , y(t) ∈ Rl , and x(t) ∈ Rnx stand for input, output, and state variables, respectively, 0 < τ·,1 < τ·,2 < · · · < τ·,nτ ≤ L are delays, and A0 , Ai , B0 , Bi , C, Ci , Hi express real-valued matrices of compatible dimensions.

3

4

CHAPTER 1 Numerical determination of stability regions

Note that, in Definition 1, only lumped delays are considered for the simplicity. Whenever ∃i such that Hi = 0, a system is called neutral (NTDS); otherwise, it is retarded (RTDS). The characteristic function of system (1) can be constructed using the Laplace transform as ⎡ ⎛ (s) = det ⎣s ⎝I +

nH 

⎞ Hi exp(−sτH,i )⎠ − A0 −

i=1

nA  i=1

⎤ Ai exp(−sτA,i )⎦ =

n 

di (s)si , (2)

i=0

di (s) are exponential polynomials, that is, weighted sums of exponential terms exp(−s·) over R. The so-called associated characteristic exponential polynomial n˜ τ reads dn (s) = i=0 dn,i exp(−sτ˜i ), where dn,i are real coefficients and 0 = τ˜0 < τ˜1 < · · · < τ˜n˜ τ .

2.2 Stability and spectral properties Definition 2 (Gu et al. [4], Michiels and Niculescu [3], Zhang and Sun [7]). System (1) is said to be exponentially stable if there exist a > 0, μ > 0 such that x(t + θ , ϕ)s ≤ a exp(−μt)ϕs , ∀t ≥ 0 for all ϕ, where ϕ(θ ) = x(θ ) with θ ∈ [−L, 0] stands for the initial condition continuous-time function. In order to characterize exponential stability by means of the system spectrum Σ := {s : (s) = 0}, let us define the spectral abscissa as α(·) := sup ReΣ first. Then, the following statement holds. Proposition 1 (Michiels and Niculescu [3]). An RTDS is exponentially stable if and only if α(·) < 0. An NTDS is exponentially stable if and only if there exists ε > 0 such that α(·) < −ε. Whenever the rightmost system pole sk = sk (or a pair) appears exactly on the imaginary axis, one can determine the switching frequency ω = Im(sk ) and the corresponding switching delay τ k = {τ : (sk , τ ) = 0}. This pair {sk , τ k } identifies a point of the stability border in the delay space. In contrast to delay-free LTI systems, the (exponential) stability analysis and/or synthesis is, however, even more complicated for the delayed case. This is mainly due to the spectral properties of NTDSs. Let γ be the supreme of real parts of the so-called essential spectrum Σess := {s : dn (s) = 0}. Moreover, let us define γ˜ := sup{γ (τ + δτ ) : ∀δτ  < ε, ε > 0}. Now, an overview of some LTI-TDS spectral properties follows. Proposition 2 (Hale and Lunel [1], Michiels and Niculescu [3], Vanbiervliet et al. [20]). The following statements hold: (1) There are only finitely many characteristic roots sk ∈ Σ in the half-plane Re(s) > β for any 0 > β ∈ R, and these poles are isolated, for an RTDS. On the contrary, subsets of system and essential poles (sk,ess ∈ Σess ) constitute vertical strips at high frequencies that asymptotically converge to each other,

3 Numerical gridding DDS algorithm

for an NTDS, and there may be located infinitely many system poles in the half-plane Re(s) > γ . (2) For an RTDS, system poles behave continuously and smoothly with respect to system parameters and τ , where τ ∈ Rnτ > 0 represents the vector of all system delays. Whereas, the value of γ is not continuous with respect to delays for an NTDS. Moreover, function α(τ ) may be nonsmooth and/or non-Lipschitz at some points. It implies from Proposition 1 that exponential stability can be studied by the determination of the rightmost pair of poles being purely imaginary. However, because of Proposition 2, this task is rather tricky for NTDSs; in particular, it must be checked whether γ˜ > 0: in such a case, a system with α < 0 cannot be fully assumed as the stable one. In addition, the shape of the rightmost infinite vertical strip of system poles has to be taken into account as well. A possibility of abrupt changes in γ gives rise to the notion of strong stability. Definition 3 (Michiels and Niculescu [3], Michiels and Vyhlídal [21]). System (1) is said to be strongly stable if γ˜ < 0. Proposition 3. System (1) is strongly stable if and only if 1
0 sufficiently small.   (3) Apply Eq. (15) with z0 = exp(τ0 s0,ess ), compute Σ ess,z , Σ ess , and determine s1,ess as the closest pole to s0,ess . If |s1,ess − s0,ess | < ε, go to Step 4; else update s0,ess = s1,ess and go to Step 3. 



(4) Estimate γ according to Eq. (18). Set Σ ess = Σ ess ∪ {sk }∞ k=1 using Eq. (19). 

(5) Compute iteratively the spectrum Σ n of n (s, τ c ) = (s, τ c ) by means of the technique presented in Section 3.3 or 3.4 (sp is known). Only roots right from γ should be considered when updating s0 . 







(6) Set Σ = Σ ess ∪ Σ n and sc = arg max Re(Σ) (as the output).

4 Examples

Remark 4. Although it is stated in Step 5 of Algorithm 2 that roots left from γ can be neglected, it is reasonable to consider such roots with sufficiently small imaginary parts in practice. The reason is threefold. First, “low-frequency” poles have a greater impact to system dynamics compared to “high-frequency” ones. Second, the convergence Re(Σ) → γ holds for poles with high imaginary parts, as introduced in Proposition 2—therefore, real parts of the rightmost “low-frequency” poles can significantly differ from γ . Third, a finite-dimensional approximation of (s, τ c ) cannot cover “high-frequency” poles. The main contributions of the presented approach can be summarized as follows: (1) The unified concept covering continuous- and discrete-time versions of the DDS algorithm is herein presented. (2) The proposed procedure is programmable by standard software means without difficulties. (3) A technique to analyze stability in the delay space of a family of NTDSs is suggested as well, by a combination of the associated characteristic exponential polynomial approximation and the characteristic quasipolynomial approximation.

4 Examples Two illustrative numerical examples follow. The former one demonstrates a comparison of the use of techniques introduced in Sections 3.3 and 3.4 with the use of the well-established CTCR paradigm [8] when analyzing DDS for an RTDS model of a skater; whereas, the latter one intends merely to verify the approximation technique presented in Section 3.5 by an academic example.

4.1 Retarded system Zítek et al. [27] published a model of a skater on a remotely controlled swaying bow (see Fig. 1). The skater is sending remote signals to the servo that exerts power P(t) which causes a horizontal-angle deviation u(t) (system input). The horizontal asymmetry yields the angle deviation y(t) from the bow symmetry axis (system output). The model transfer function reads G(s) =

0.2 Y(s) = 2 2 exp(−(τ1 + τ2 )s), U(s) s (s − exp(−τ2 s))

(20)

where τ1 is a reaction delay of the skater, and τ2 expresses a latency of the servo, the nominal values of which are 0.3 and 0.1, respectively. Note that this nominal system is unstable with α = 0.953. Pekaˇr and Prokop [28] designed a linear thirdorder finite-dimensional controller minimizing the spectral abscissa of the nominal control system, giving rise to the following feedback characteristic quasipolynomial parameterized by τ = (τ1 , τ2 ):

13

14

CHAPTER 1 Numerical determination of stability regions

FIG. 1 A sketch of a skater on a remotely controlled swaying bow [17,27].

(s, τ ) = 0.2 · 106 exp(−(τ1 + τ2 )s)(8.223s3 + 106.523s2 + 26.247s + 5.617) + s2 (s2 − exp(−τ2 s))(s3 + 106 (0.469s2 + 0.64s + 10.56)).

(21)

An exciting feature of this feedback is that it is unstable for τ = (0, 0), while the nominal delayed case yields a stable control system. Let us now compare numerical results when using Algorithm 1 with continuous- versus discrete-time quasipolynomial approximations. The mesh grid in the delay space is set as (0, 0.01, 0.02, . . . , 0.7) × (0, 0.01, 0.02, . . . , 0.7), and the terminating precision of iterative calculations in Step 5 of Algorithm 1 let be 10−6 . Consider the technique introduced in Section 3.3 first. The overall computation duration is 1583 s (a Matlab (R2016a) file running on Intel Core i3-4000M [email protected] GHz, 4 GB RAM) with |T| = 61 (i.e., the number of found switching delays). Second, the discrete-time technique (see Section 3.4) requires determining the initial value of ts . It was found by numerical experiments that higher values of ts result in a poor approximation, while lower ones yield an unacceptable number of iterations in Step 5. We eventually set ω0 ts = 0.125. In this case, the computational time is better (1124 s) due to reduced use of symbolic math. The number of found switching pairs is 63. As a benchmark, the CTCR algorithm [8] programmed by the author takes 696 s (for a particular discretization of the pseudo-delay space to get kernel curves) with the total amount of 51 found switching pairs. Determined stable and unstable areas given by stability borders are displayed in Fig. 2—note, however, that individual curves for the three methods are graphically almost indistinguishable. Therefore, the precision of the obtained results are measured via two distinct values— first, by means of the absolute value of the spectral abscissa α(τ ) of all zeros s˜k of (s, τ ), where τ ∈ T; second, through the distance of approximated switching frequencies from those of s˜k , that is, ω = | |Im(˜sk )| − |Im(sk )| |, where sk ∈ Σ. Results for all τ ∈ T stemming from Steps 1 to 6 of Algorithm 1, and from the CTCR, are given to the reader in Figs. 3 and 4, respectively.

4 Examples

FIG. 2 Graphical representation of stable and unstable regions determined by Algorithm 1 and the CTCR paradigm for (s, τ ) given by Eq. (21). 10–5

a(t) `

10–10

10–15

10–20

Continuous-time Discrete-time CTCR

0

10

20

30

40

Index of the switching delay pair

FIG. 3 Errors in the switching delay estimates measured by |α(τ )|.

50

60

15

CHAPTER 1 Numerical determination of stability regions

10–7

10–8

10–9 Dw

16

10–10

10–11

10–12

Continuous-time Discrete-time CTCR

0

10

20

30

40

50

60

Index of the switching delay pair

FIG. 4 Errors in the switching delay estimates measured by ω.

From Fig. 3, the accuracy of the discrete-time technique is better, yet it is sacrificed to the unclear setting of ts . Note that during the algorithm, the value of ts was updated to 0.1/ω0 . It can be observed from the figure that the analytic-based CTCR paradigm gives best results; however, the herein proposed algorithm provides a sufficiently accurate data as well. Moreover, one can expect that the repetitive (i.e., multistep) use of Newton’s method yields more precise switching delay estimations. Errors in the estimates of switching frequencies are almost comparable, as can be seen from Fig. 4. Fig. 5 includes |α(τ )| computed for additional switching points obtained by standard quadratic regression of all τ ∈ T, see Step 7 of Algorithm 1. Note that the use of successive quadratic interpolation applied in Ref. [16] yields slightly better results than those obtained herein. The roots of (s, τ ) are computed using the QPmR software tool with the precision of 10−9 [19].

4.2 Neutral system The aim of this simple example is to verify the efficiency of Algorithm 2. Let the characteristic quasipolynomial be     2π (s, τ ) = s 0.5 exp(−0.9s) − 0.4 exp − s + 1 − 2 exp(−τ s) + 2 exp(−2τ s) + 0.3. 3 (22)

Notice, for the simplicity, that there is a single delay to be determined. An efficient approximation of the associated characteristic exponential polynomial

4 Examples

10–2

a(t) `

10–3

10–4 Continuous-time Discrete-time CTCR

10–5

0

100

200

300

400

500

Index of the switching delay pair

FIG. 5 Errors in additional switching delay estimates obtained using quadratic regression measured by ω.

  dn (s, ·) = 0.5 exp(−0.9s) − 0.4 exp − 2π 3 s + 1 has already been computed and presented in Ref. [16], where Steps 1–5 of Algorithm 2 have given rise to  d n (s, 0.3) = 0.5 exp(−0.9s) + (−0.0024 + 0.0031j) exp(−1.8s)

+ (−0.3996 − 0.0071j) exp(−2.1s) + (0.002 + 0.0029j) exp(−2.4s)

that has complex coefficients, which inter alia means that its zero loci are not symmetric to the real axis. Moreover, strong stability condition (3) holds also for 

d n . It is approximated that γ = −0.1073 (see Remark 3). Let us now perform Step 5 of Algorithm 2 via the technique described in 

Section 3.5.2. The dominant pole of d n (s0,ess = −0.1073 + 3.1579j) is taken as the initial estimation. Fig. 6 displays distances of the approximated dominant poles from those computed via the QPmR for particular delay values within the range τ ∈ [0, 2] (the discretization step is set to τ = 0.01). The eventual stability windows are computed as τstab ∈ (0, 0.0821) ∪ (0.2789, 0.6734)

with the corresponding switching frequencies, respectively, Im{Σ} = {3.198, 2.8478, 0.4328}.

17

18

CHAPTER 1 Numerical determination of stability regions

FIG. 6 Errors in dominant root estimates for Eq. (22) within τ ∈ [0, 2].



It is worth noting that Σ n initially includes some artificial roots that can be canceled via the rough approximation of the updated locus of sc as Re(sc ) ≤ Re(sp ) + c(τ RT(sp , τc )),

for some c ≈ 1; or by using a simple test on the value of (sk , τ ), where sk are root candidates (i.e., the true roots satisfy (sk , τ ) = 0). 

The comparison of rightmost parts of Σn and Σ n for τ = 0.2789 is given in Fig. 7. To sum up Algorithms 1 and 2 in the light of theoretical framework and numerical results, and to compare the presented concept with other known methods, the following statements can be inferred: (1) The proposed method does not require advanced mathematical knowledge; unlike some other approaches even from the family of direct DDS methods. For instance, there is no need to utilize neither complex matrix operations [29] nor integral evaluation [14]. (2) Despite its low mathematical complexity and a numeric character, the algorithm provides sufficiently accurate results that can be comparable to the well-established CTCR algorithm. (3) The complete stability image for a selected region in the delay space is obtained. Some other well-implementable methods provide only delay stability margin, that is, the minimum delay value for which the system becomes unstable, see, for example, Refs. [30,31].

5 Conclusions

Σn

50

Σn

Imaxinary axis

40

30

20

10 0 –0.8

– 0.7

– 0.5

– 0.6

– 0.4 Real axis

– 0.3

– 0.2

– 0.1

0

FIG. 7 

Spectral image of Σn versus Σn for Eq. (22) when τ = 0.2789.

(4) In this chapter, both the discrete-time and the continuous-time approximation techniques are covered in the algorithm and compared, which has not been made yet. Moreover, the complete technique for the class of NTDSs is proposed. It is worth noting that only a limited number of works deal with such systems, for instance, Ref. [14]. (5) Unfortunately, the algorithm has also some disadvantages. Especially, a region in the delay space has to be a priori determined. The algorithm natively does not give analytically precise results, sometimes the dominant pole estimation has to be reset, or the selection of base delay within the discrete-time version is not unambiguous.

5 Conclusions A numerical gridding algorithm to determine stability windows in the delay space for linear time-invariant systems with delays has been summarized in this chapter. The core of the algorithm lies in the iterative estimation of the dominant pole in the neighborhood of the closest known estimate. Two possible approximation techniques have been presented; namely, a continuous approximation by means of the Taylor expansion, and a discrete-time Tustin (bilinear) transformation including prewarping. Successive Newton’s zero-point iterative method has been used to get more accurate results. The reader has also been provided with two numerical examples on delaydependent stability of a retarded system with two delays, and a single-delay neutral system.

19

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CHAPTER 1 Numerical determination of stability regions

The main advantages of the proposed methodology rest in its theoretical, mathematical, and computational simplicity; its speed and accuracy are comparable to the well-established CTCR algorithm. On the contrary, one has to determine the region in the delay space to be examined. Moreover, sometimes the algorithm requires resetting the leading pole estimation. When using a discrete-time quasipolynomial approximation, the selection of a suitable value of the sampling period is disputable. In addition, a polynomial approximant cannot adequately describe pole loci of a neutral delay system, especially, the high-frequency modes. There can be suggested many various possibilities on how to enhance the presented methodology. For instance, the speed of the programming realization can be improved by better software and hardware tools—the use of parallel computing on graphical processing units (cards) suggests itself. As stated earlier, it is desirable to determine the region in the delay space and the sampling period more sophistically. It would also be interesting to test the accuracy of approximation polynomials of several degrees. Moreover, naturally, another polynomization techniques could be implemented. Once the sets of switching delays and frequencies in the grid are found, the additional values can be estimated by enhanced interpolation or extrapolation techniques (i.e., a more sophisticated one compared to the herein use quadratic regression). Last but not least, a multistep Newton’s zero-point extrapolation principle should lead to better eventual critical values estimations.

Acknowledgments We would like to express our gratitude to the European Regional Development Fund and to the Ministry of Education, Youth and Sports that financially supported this work under the National Sustainability Program project No. LO1303 (MSMT-7778/2014).

References [1] J.K. Hale, S.M.V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. [2] C. Bonnet, A.R. Fioravanti, J.R. Partington, Stability of neutral systems with commensurate delays and poles asymptotic to the imaginary axis, SIAM J. Control Optim. 49 (2) (2011) 498–516. [3] W. Michiels, S.-I. Niculescu, Stability and stabilization of time-delay systems: an eigenvalue-based approach, in: Advances in Design and Control, vol. 12, SIAM, Philadelphia, PA, 2007. [4] K. Gu, V.L. Kharitonov, J. Chen, Stability of Time-Delay Systems, Birkhäuser, Basel, Switzerland, 2003. [5] I. Boussaada, S.-I. Niculescu, Tracking the algebraic multiplicity of crossing imaginary roots for generic quasipolynomials: a Vandermonde-based approach, IEEE Trans. Autom. Control 61 (6) (2016) 1601–1606.

References

[6] J.R. Partington, C. Bonnet, Hinf and BIBO stabilization of delay systems of neutral type, Syst. Control Lett. 52 (3–4) (2004) 283–288. [7] X.-Y. Zhang, J.-Q. Sun, A note on the stability of linear dynamical systems with time delay, J. Vib. Control 20 (10) (2014) 1520–1527. [8] R. Sipahi, N. Olgac, A unique methodology for the stability robustness of multiple time delay systems, Syst. Control Lett. 55 (10) (2006) 819–825. [9] Q. Gao, N. Olgac, Bounds of imaginary spectra of LTI systems in the domain of two of the multiple time delays, Automatica 72 (C) (2016) 235–241. [10] K.E. Walton, J.E. Marshall, Direct method for TDS stability analysis, IEE Proc. D Control Theory Appl. 134 (2) (1987) 101–107. [11] X.-G. Li, S.-I. Niculescu, A. Çela, H.-H. Wang, T.-Y. Cai, On computing Puiseux series for multiple imaginary characteristic roots of LTI systems with commensurate delays, IEEE Trans. Autom. Control 58 (5) (2013) 1338–1343. [12] J. Ma, B. Zheng, C. Zhang, A matrix method for determining eigenvalues and stability of singular neutral delay-differential systems, J. Appl. Math. 2012 (2012) 749847. [13] J. Cao, Improved delay-dependent exponential stability criteria for time-delay system, J. Frankl. Inst. 350 (4) (2013) 790–801. [14] Q. Xu, G. Stépán, Z. Wang, Delay-dependent stability analysis by using delay-independent integral evaluation, Automatica 70 (3) (2016) 153–157. [15] L. Pekaˇr, Enhanced TDS stability analysis method via characteristic quasipolynomial polynomization, in: R. Šilhavý (Ed.), Cybernetics and Mathematics Applications in Intelligent Systems, Springer-Verlag, Berlin, Heidelberg, Germany, 2017, pp. 20–29. [16] L. Pekaˇr, R. Prokop, Direct stability-switching delays determination procedure with differential averaging, Trans. Inst. Meas. Control 40 (7) (2018) 2217–2226. [17] L. Pekaˇr, R. Matuš˚u, R. Prokop, Gridding discretization-based multiple stability switching delay search algorithm: the movement of a human being on a controlled swaying bow, PLoS ONE 12 (6) (2017). e0178950. [18] L. Pekaˇr, P. Navrátil, Polynomial approximation of quasipolynomials based on digital filter design principles, in: R. Šilhavý (Ed.), Automation Control Theory Perspectives in Intelligent Systems, Springer-Verlag, Berlin, Heidelberg, Germany, 2016, pp. 25–34. [19] T. Vyhlídal, P. Zítek, QPmR—quasi-polynomial root-finder: algorithm update and examples, in: T. Vyhlídal (Ed.), Delay Systems: From Theory to Numerics and Applications, Springer Verlag, New York, 2014, pp. 299–312. [20] T. Vanbiervliet, K. Verheyden, W. Michiels, S. Vandewalle, A nonsmooth optimization approach for the stabilization of time-delay systems, ESAIM Control Optim. Calcul. Var. 14 (3) (2008) 478–493. [21] W. Michiels, T. Vyhlídal, An eigenvalue based approach for the stabilization of linear time-delay systems of neutral type, Automatica 41 (6) (2005) 991–998. [22] L. Pekaˇr, R. Prokop, On delay (in)dependent stability for TDS, in: 2015 7th International Congress on Ultra Modern Telecommunications and Control Systems and Workshops (ICUMT), 2015. [23] V. Bobál, J. Böhm, J. Fessl, J. Macháˇcek, Digital Self-Tuning Controllers: Algorithms, Implementation and Applications, Springer-Verlag, London, 2005. [24] L.H.V. Nguyen, C. Bonnet, Hinf-stability analysis of various classes of neutral systems with commensurate delays and with chains of poles approaching the imaginary axis, in: 54th IEEE Conference on Decision and Control, Osaka, Japan, 2015, pp. 6416–6421.

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[25] L.H.V. Nguyen, A.R. Fioravanti, C. Bonnet, Analysis of neutral systems with commensurate delays and many chains of poles asymptotic to same points on the imaginary axis, IFAC Proc. Vol. 10 (1) (2012) 120–125. [26] R. Rabah, G.M. Skylar, A.V. Rezounenko, Stability analysis of neutral type systems in Hilbert space, J. Differ. Equ. 214 (2005) 391–428. [27] P. Zítek, V. Kuˇcera, T. Vyhlídal, Meromorphic observer-based pole assignment in time delay systems, Kybernetika 44 (5) (2008) 633–648. [28] L. Pekaˇr, R. Prokop, Algebraic optimal control in RMS ring: a case study, Int. J. Math. Comput. Simul. 7 (1) (2013) 59–68. [29] J.I. Mulero-Martínez, Modified Schur-Cohn criterion for stability of delayed systems, Math. Probl. Eng. 2015 (2015) 846124. [30] C. Dong, H. Jia, Q. Xu, J. Xiao, Y. Xu, P. Tu, P. Lin, X. Li, P. Wang, Time-delay stability analysis for hybrid energy storage system with hierarchical control in DC microgrids, IEEE Trans. Smart Grid 9 (6) (2018) 6633–6645. [31] H. Gündüz, C. Sönmez, S. Ayasun, Comprehensive gain and phase margins based stability analysis of micro-grid frequency control system with constant communication time delays, IET Gener. Transm. Distrib. 11 (3) (2017) 719–729.

CHAPTER

Distributed impedance control of latency-prone robotic systems with series elastic actuation

2

Ye Zhaoa , Luis Sentisb a George

W. Woodruff School of Mechanical Engineering, Georgia Tech, Atlanta, GA, United States of Aerospace Engineering and Engineering Mechanics, University of Texas, Austin, TX, United States

b Department

Chapter outline 1 Introduction......................................................................................... 23 2 Modeling of series elastic actuators ........................................................... 27 3 Gain design of series elastic actuators ........................................................ 30 3.1 Critically damped controller gain design criterion ............................... 31 3.2 Trade-off between torque and impedance control................................ 32 4 SEA impedance analysis ......................................................................... 35 4.1 SEA impedance transfer function .................................................. 35 4.2 Effects of time delays and filtering................................................. 36 4.3 Effect of load inertia ................................................................. 38 5 Experimental validation........................................................................... 39 5.1 Evaluation of the controller design ................................................. 39 5.2 Step response implementation ..................................................... 42 5.3 Distributed operational space control of a mobile base ......................... 45 6 Discussions and conclusion ..................................................................... 48 References............................................................................................. 49

1 Introduction As a result of the increasing complexity of robotic control systems, such as humancentered robots [1,2] and industrial surgical machines [3], new system architectures, especially distributed control architectures [4,5], are often being sought for communicating with and controlling the numerous device subsystems. Often, these distributed control architectures manifest themselves in a hierarchical control fashion where a centralized controller can delegate tasks to subordinate local controllers (Fig. 1). As it Stability, Control and Application of Time-Delay Systems. https://doi.org/10.1016/B978-0-12-814928-7.00002-0 © 2019 Elsevier Inc. All rights reserved.

23

24

CHAPTER 2 Robotic systems with series elastic actuation

Low-level controller N

Robot actuator N

Robot actuator N

Distributed control

(A)

(B)

Low-level controller N

Robot actuator N

(C) FIG. 1 Depiction of various control architectures. Many control systems today employ one of the control architectures above: (a) centralized control with only high-level feedback controllers (HLCs); (b) decentralized control with only low-level feedback controllers (LLCs); and (c) distributed control with both HLCs and LLCs, which is the focus of this chapter.

is known, communication between actuators and their low-level controllers can occur at high rates while communication between low- and high-level controllers occurs more slowly. The latter is further slowed down by the fact that centralized controllers tend to implement larger computational operations, for instance, to compute system models or coordinate transformations online. One concern is that feedback controllers with large delays [6,7], such as the centralized controllers mentioned earlier, are less stable than those with small delays, such as locally embedded controllers. Without the fast servo rates of embedded controllers, the gains in centralized controllers can only be raised to limited values, decreasing their robustness to external disturbances [8] and unmodeled dynamics [9]. As such, why not remove centralized controllers altogether and implement all feedback processes at the low level? Such operation might not always be possible. For instance, consider controlling the behavior of human-centered robots (i.e., highly articulated robots that interact with humans). Normally this operation is achieved by specifying the goals of some task frames such as the end effector coordinates. One established option is to create impedance controllers on those frames and transform the resulting control references to actuator commands via operational space transformations [10]. Such a strategy requires the implementation of a centralized feedback

1 Introduction

controller which can utilize global sensing data, access the state of the entire system model, and compute the necessary models and transformations for control. Because of the aforementioned larger delays on high-level controllers, does this imply that high gain control cannot be achieved in human-centered robot controllers due to stability problems? It will be shown that this may not need to be the case. But for now, this delay issue is one of the reasons why various currently existing humancentered robots cannot achieve the same level of control accuracy that it is found in high-performance industrial manipulators. More concretely, this study proposes a distributed impedance controller where only proportional (i.e., stiffness) position feedback is implemented in the high-level control process with slow servo updates. This process will experience the long latencies found in many modern centralized controllers of complex human-centered robots. At the same time, it contains global information of the model and the external sensors that can be used for operational space control. For stability reasons, our study proposes to implement the derivative (i.e., damping) position feedback part of the controller in low-level embedded actuator processes which can, therefore, achieve the desired high update rates. As it will be empirically demonstrated, the benefit of the proposed split control approach over a monolithic controller implemented at the high level is to increase control stability due to the reduced damping feedback delay. As a direct result, closed-loop actuator impedance may be increased beyond the levels possible with a monolithic high-level impedance controller. This conclusion may be leveraged on many practical systems to improve disturbance rejection by increasing gains without compromising overall controller stability. As such, these findings are expected to be immediately useful on many complex human-centered robotic systems. To demonstrate the effectiveness of the proposed methods, this study implements tests on a high-performance actuator followed by experiments on a mobile base. First, a position step response is tested on an actuator under various combinations of stiffness and damping feedback delays. The experimental results show high correlation to their corresponding simulation results. Second, the proposed distributed controller is applied to an implementation into an omnidirectional base. The results show a substantial increase in closed-loop impedance capabilities, which results in higher tracking accuracy with respect to the monolithic centralized controller counterpart approach. Series elastic actuators (SEAs) [11–14], as an emerging actuation mechanism, provide considerable advantages in compliant and safe environmental interactions, impact absorption, energy storage, and force sensing. In the control literature, adopting cascaded impedance control architectures for SEAs has attracted increasing investigations over the last few years [13,15,16]. Compared to full-state feedback control [17–19], the cascaded control performs superior when the controlled plant comprises slow dynamics and fast dynamics simultaneously. In this case, the inner fast control loop isolates the outer slow control loop from nonlinear dynamics inherent to the physical system, such as friction and stiction. Therefore, this study focuses on the cascaded control structure to simulate the distributed control structure for humanoid robots accompanied with a variety of delayed feedback loops [1,20].

25

26

CHAPTER 2 Robotic systems with series elastic actuation

This class of cascaded control structures nests feedback control loops [13,15], that is, an innertorque loop and an outer-impedance loop for the task-level control, such as Cartesian impedance control. Recently, the works in Refs. [13,16] proposed to embed a motor velocity loop inside the torque feedback loop. This velocity feedback enables to use integral gains for counteracting static errors such as drivetrain friction, while maintaining the system’s passivity. The authors in Ref. [15] extensively studied the stability, passivity, and performance for a variety of cascaded feedback control schemes incorporating position, velocity, and torque feedback loops. Robustness and effects of delay have often been studied in work regarding proportional integral derivative (PID) controller tuning. A survey of PID controllers including system plants using phase margin techniques with linear approximations is conducted in Ref. [21]. The works [22,23] study auto-tuning and adaption of PID controllers while the work [24] furthers these techniques by developing optimal design tools applied to various types of plants which include delays. The study in Ref. [25] proposed an optimal gain-scheduling method for DC motor speed control with a PI controller. In Ref. [26], a backstepping controller with time-delay estimation and nonlinear damping is considered for variable PID gain tuning under disturbances. The high volume of studies on PID tuning methods highlights the importance of this topic for robust control under disturbances. However, none of those studies considers the sensitivity discrepancy to latencies between the stiffness and damping servos as separate entities nor do they consider the decoupling of those servos into separate processes for stability purposes as it is done in this chapter. Optimal controller design methodologies are increasingly sought within the robotics and control community. Recent works in Ref. [27] devised a critically damped controller gain design criterion to accomplish high impedance for rigid actuators. However, inherent fourth-order SEA dynamics in this study make it challenging to design optimal controllers of the cascaded feedback structure. For the cascaded control, a common routine is to tune the inner loop gains first, followed by an outer-loop gain tuning. Indeed, this procedure consumes substantially handtuning efforts and lacks optimal performance guarantees. The majority of existing results rely on empirical tuning [11,15]. The work in Ref. [13] designed controller gain ranges according to a passivity criterion. However, gain parameters were highly coupled as a set of inequalities, which leaves the controller gains undetermined. In this chapter, a fourth-order gain design criterion is proposed by simultaneously solving SEA optimal impedance gains and torque gains. The “optimality” is proposed according to the phase-margin-based stability. Through this criterion, the designer only needs to specify a natural frequency parameter, and then all the impedance and torque gains are deterministically solved. A larger natural frequency represents larger impedance and torque controller gains. This dimensionality reduction and automatic solving process is not only convenient for SEA controller design but also warrants optimal performance in terms of system closed-loop stability. System passivity criteria have been extensively studied for coupled systems [17,28,29], networked control systems [30], and coordination control [31]. Among the robotics community, the authors in Ref. [13] designed passivity-based controller

2 Modeling of series elastic actuators

FIG. 2 Valkyrie robot equipped with series elastic actuators (SEAs). The top figure shows a set of high-performance NASA Valkyrie SEAs, the bottom left one shows the Valkyrie robot with SEA location annotations, and the bottom right one shows the calf and ankle structure.

gains for SEAs. However, that work only incorporates stiffness feedback, and the ignored damping feedback indeed plays a pivotal role, which will be analyzed in this study. Damping-type impedance control was investigated in Ref. [16]. However, it does not analyze the effects of time delays and filtering. Although these practical issues were tackled in Ref. [13], the time delays are so subtle that it cannot model large time delays often existing in serial communication channels. Due to the destabilizing effects of time delays, significant effort has been put forth to ensure that systems are stable, by enforcing passivity criteria [32]. In light of these discussions, the contributions of this chapter are: (i) analyze, provide control system solutions, implement, and evaluate actuators and mobile robotic systems with latency-prone distributed architectures to significantly enhance their stability and trajectory tracking capabilities; (ii) analyzing time domain controller stability of SEAs and proposing a critically damped gain selection criterion; and (iii) conducting a frequency-dependent impedance analysis of SEAs affected by time delays and filtering. We expect this study provides a promising solution of designing optimal impedance controllers for SEA-equipped humanoid robots (see Fig. 2) to achieve complex locomotion and manipulation tasks. The results presented in this chapter have been published in Refs. [27,33–35].

2 Modeling of series elastic actuators This section models an SEA constituting two nested feedback loops, that is, an outerimpedance loop and an innertorque loop. The SEA dynamics can be modeled as shown in Fig. 3. The spring torque τk is τk = k(qm − qj ),

(1)

27

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CHAPTER 2 Robotic systems with series elastic actuation

FIG. 3 SEA model. The annotated parameters are defined in Section 2. We map the motor inertia Im and motor damping bm to the joint coordinates by multiplying by the gear reduction squared.

where the spring stiffness is denoted by k. qm and qj represent motor and joint positions, respectively. As to the joint side, it is assumed that disturbance torque τdist = 0. Namely, spring torque is equal to load torque, that is, τk = Ij q¨ j + bj q˙ j ,

(2)

where Ij and bj are joint inertia and damping coefficients, respectively. Notably, this model merely models the effects of viscous friction; we leave the analysis of other types of friction for future work. Then the load plant PL (s) has PL (s) =

qj (s) 1 = 2 . τk (s) Ij s + bj s

(3)

By Eqs. (1), (2), the following transfer function can be derived: qj (s) k = 2 . qm (s) Ij s + bj s + k

(4)

We have motor torque τm = Im q¨ m + bm q˙ m + k(qm − qj ). Combining the equation above with Eq. (4) and defining the spring deflection as q = qm − qj , we establish the following mapping from the motor angle qm to q: r(s) =

Ij s2 + bj s q(s) = 2 . qm (s) Ij s + bj s + k

(5)

By Eq. (1), we can express the spring torque as τk (s) = kq(s) = kr(s)qm (s).

(6)

Given the relationship between the motor current im and the motor torque τm represented by τm (s)/im (s) = β = ηNkτ , with drivetrain efficiency η (constant for simplicity, and dynamic modeling of drivetrain losses is ignored), gear speed reduction N and motor torque constant kτ . (N represents the ratio of motor rotary velocity to actuator linear velocity. This gear ratio is achieved by using pulley reduction Np and ball screw, which is parameterized by ball screw lead lbs . Please

2 Modeling of series elastic actuators

refer to Ref. [12] for more actuator design details.) See Table 2 in Section 5 for more parameter details, the SEA plant PF (s) is represented by βr(s)k τ (s) = , PF (s) = k im (s) Im s2 + bm s + r(s)k

(7)

where Im and bm are motor inertia and damping coefficients, respectively. By Fig. 4, the closed-loop torque control plant PC is PC (s) =

PF (β −1 + C) τk (s) = . τdes (s) 1 + PF Ce−Tτ s

(8)

The torque feedback loop includes a delay term e−Tτ s and a PD compensator C = Kτ + Bτ Qτ d s (see Fig. 4), where Qτ d models a first-order low-pass filter for the torque derivative signal, Qτ d =

2π fτ d , s + 2πfτ d

(9)

where fτ d is the filter cut-off frequency. In addition, a feedforward loop is incorporated to convert the desired torque τdes to the motor current im (see Fig. 4). By Eqs. (3), (8), the following transfer function can be obtained qj (s) PF (β −1 + C) = PL P C = . τdes (s) (1 + PF Ce−Tτ s )(Ij s2 + bj s)

(10)

FIG. 4 SEA controller diagram. The innertorque controller is composed of a feedforward loop with a mapping scalar β −1 and PD torque feedback loops. The outerimpedance controller constitutes stiffness and damping feedback loops. Time delays are modeled as e−Ts . We apply first-order low-pass filters to both velocity and torque derivative feedback loops. τk represents the spring torque. The motor current input is im . The embedded torque control loop is denoted by PC , which normally has faster dynamics than the outer one.

29

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CHAPTER 2 Robotic systems with series elastic actuation

For the impedance feedback, we have the form as follows   τdes (s) = Kq qdes − e−Tqs s qj − Bq e−Tqd s Qqd sqj ,

(11)

where e−Tqs s and e−Tqd s denote the time delays of stiffness and damping feedback loops, separately. The joint velocity filter Qqd has the same format as that in Eq. (9) with a cut-off frequency fqd . Alternatively, we can also send the desired joint velocity as the input of the embedded damping loop. In that case, an extra zero will show up in the numerator of Eq. (12). Since a zero only changes transient dynamics, it does not affect system stability. Using PL and PC in Eqs. (3), (8), we obtain the SEA closed-loop transfer function PCL , PCL (s) =

qj (s) Kq PC PL   = −T qdes (s) 1 + PC PL e qd s Bq Qqd s + e−Tqs s Kq =

Kq (1 + βKτ + βBτ Qτ d s) , 4 i i=0 Di s

(12)

with the associated coefficients defined as D4 = Im Ij /k, D3 = (Ij bm + Im bj )/k + Ij βBτ Qτ d e−Tτ s ,   D2 = Ij 1 + e−Tτ s βKτ + Im + bj βBτ Qτ d e−Tτ s + βBτ Bq e−Tqd s Qqd Qτ d + bj bm /k,   D1 = bj 1 + e−Tτ s βKτ + bm + βBτ Qτ d Kq e−Tqs s + e−Tqd s (1 + βKτ )Bq Qqd , D0 = e−Tqs s (1 + βKτ )Kq .

(13)

This closed-loop transfer function is sixth order due to the existence of low-pass filters Qqd and Qτ d . Here, we formulate it in fourth-order form for the sake of clarity. Note that the numerator of Eq. (12) has a zero, induced by the torque derivative term. As to the step response, this induced zero shortens the rise time but causes an overshoot. Nevertheless, system stability is not affected since it is solely determined by the denominator’s characteristic polynomial.

3 Gain design of series elastic actuators The closed-loop transfer function derived in Eq. (12) is complex due to the cascaded impedance and torque feedback loops. This complexity makes the SEA controller design challenging. In this section, we propose a critically damped criterion to design optimal controller gains.

3 Gain design of series elastic actuators

3.1 Critically damped controller gain design criterion Impedance control gains of rigid actuators can be designed based on the wellestablished critically damped criterion of second-order systems [27]. As for high-order systems like SEAs, such a critically damped criterion is still missing. In this study, we aim at designing feedback controller gains such that the overall SEA closed-loop system behaves as two damped second-order systems [36]. To this end, we represent the fourth-order system in Eq. (12) (the time delays and filtering in Eq. (12) are ignored for problem tractability) by two second-order systems in multiplication presented as (s2 + 2ζ1 ω1 s + ω12 )(s2 + 2ζ2 ω2 s + ω22 ),

(14)

which has four design parameters ω1 , ω2 , ζ1 , and ζ2 . They will be used to design the gains Kq , Bq , Kτ , and Bτ . First, we set ζ1 = ζ2 = 1 in Eq. (14) to obtain the critically damped performance. Second, we assume ω2 = ω1 for simplicity. An optimal pole placement design is left for future work. Let us define a natural frequency fn of Eq. (14) as ω1 = ω2  ωn = 2πfn .

(15)

By comparing the denominators of Eqs. (12), (14), we obtain the nonlinear gain design criterion equations as follows: Ij bm + Im bj + Ij βBτ k = 4ωn , Im Ij k(Ij (1 + βKτ ) + Im + βBτ (bj + Bq )) + bj bm = 6ωn2 , Im Ij k(bj + Bq )(1 + βKτ ) + k(bm + βBτ Kq ) = 4ωn3 , Im Ij (1 + βKτ )kKq = ωn4 . Im Ij

(16)

These four equations with coupled gains can be solved by Matlab’s fsolve() function. Note that, representing a fourth-order system by two multiplied second-order systems in Eq. (14) maintains the properties of the fourth-order system. In our earlier method, the simplification comes from the selection of ω1 , ω2 , ζ1 , ζ2 parameters in Eq. (14). The resulting benefit is that selecting a natural frequency uniformly determines all the gains of torque and impedance controllers. This advantage avoids the commonly adopted complicated yet heuristic controller tuning procedures, like the ones in Refs. [15,36], although system dynamics in our case are restricted to specific patterns such as the critically damped one we design. Let us show an example as follows. Example 1. To validate this criterion, we test five natural frequencies. We select filter cut-off frequencies fvd = 50 Hz, fτ d = 100 Hz and time delays Tτ = Tqd = 0.5 ms, Tqs = 2 ms. These filters and delays are only used in the phasespace computation based on Eq. (12), and ignored in the critically damped selection

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Table 1 Critically damped controller gains. Frequency (Hz) fn = 12 fn = 14 fn = 16 fn = 18 fn = 20

Impedance gains (Nm/rad, Nms/rad)

Torque gains (A/Nm, As/Nm)

Kq = 65 Bq = 0.46 Kq = 83 Bq = 0.76 Kq = 103 Bq = 1.02 Kq = 124 Bq = 1.26 Kq = 148 Bq = 1.49

Kτ Bτ Kτ Bτ Kτ Bτ Kτ Bτ Kτ Bτ

= 1.18 = 0.057 = 1.80 = 0.067 = 2.56 = 0.077 = 3.45 = 0.087 = 4.48 = 0.097

Phase margin 45.1 degrees 43.2 degrees 40.0 degrees 36.5 degrees 33.2 degrees

criterion for problem tractability. The solved gains and phase margins are shown in Table 1. Noteworthily, the phase margin is computed based on the open-loop transfer function derived from PCL in Eq. (12). Increasing fn will lead to a uniform increase of all four gains. This property meets our expectation that increasing torque (or impedance) gains results in a torque (or impedance) bandwidth increase and a phase margin decrease. Note that, for simplicity, the gain design above ignores time delay, which does affect system stability. Next, we will study the effect of time delays given this gain design criterion. Since torque feedback is the inner loop, it normally suffers a smaller delay than that in the outer-impedance loop. This is why we assign Tτ = 0.5 ms in the example earlier. Notice that Tqs is chosen to be larger than Tqd since the former belongs to the outer control loop while the latter belongs to the inner control loop. The benefits of having damping feedback in the inner loop was extensively analyzed in Ref. [27]. This motivates us to implement the impedance feedback loops in a distributed pattern as shown in Fig. 4. Namely, we allocate the stiffness feedback loop at the high level while embedding the damping feedback loop at the low level for a fast servo rate. The same distributed control strategy was implemented for the rigid actuators in Ref. [27] and extended lately for the Whole-Body Operational Space Control [37,38].

3.2 Trade-off between torque and impedance control During gain tuning of the SEA-equipped bipedal robot Hume and NASA Valkyrie robot, which have similar SEA control architectures as the one in Fig. 5, a pivotal phenomenon is observed: if one increases torque controller gains or decreases impedance controller gains, the robot tends to become unstable. To reason about this observation, we propose an SEA gain scale definition as follows. Definition 1 (SEA gain scale). The gain scale of an SEA’s cascaded controller is a scaling parameter GS between adjusted gains (Kia , Bia ) and nominal gains (Kin , Bin ), i ∈ {τ , q},

3 Gain design of series elastic actuators

FIG. 5 SEA step response affected by time delays. These panels demonstrate that the larger delays that impedance feedback loops have, the worse performance that the step response has. As shown in panels (B) and (C), SEA stability has a higher sensitivity to damping delays than the stiffness counterpart. Panel (A) reveals that larger fn leads to a larger overshoot, which appears to be counterintuitive. However, by close inspection, we can observe that the largest fn in the solid magenta color already shows distortion, and its 36.4-degree phase margin is the smallest among all four cases. To study the influence of zero in Eq. (12), step responses without this zero are also simulated and represented by dashed lines in (A). By comparison, we can realize an overshoot induced by this extra zero.

GS =

K qn Kτa = , Kτn K qa

GS =

Bq Bτa = n, Bτn Bqa

(17)

where the adjusted gains denote actual gains in use while the nominal gains denote reference ones designed by the critically damped gain design criterion. It should be noted that if GS = 1, then the adjusted gains are the same as the nominal gains. For example, the controller gains in Table 1 are five sets of nominal gains. By Eq. (17), we have the following equalities: Kτa · Kqa = Kτn · Kqn ,

Bτa · Bqa = Bτn · Bqn ,

(18)

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FIG. 6 Optimality of the critically damped gain design criterion. Panel (A) samples a variety of gain scales and natural frequencies. An optimal performance is achieved by using the proposed critically damped gain design criterion. Panel (B) shows (i) a larger overshoot but slow rise time when GS > 1 and (ii) an over-damped response with distortions when GS < 1.

which maintains the same multiplicative value of nested proportional (or derivative) torque and impedance gains for the normal and adjusted conditions. An overall controller gain design procedure is shown in Algorithm 1.

Algorithm 1 Gain controller design procedure Assign system parameters sysParam in Eq. (12). Assign natural frequency fn (i.e., ω1 and ω2 by Eq. (15)), ζ1 = ζ2 ← 1. procedure ControllerSolver(fn , ζ1 , ζ2 , sysParam) Deterministically solve nominal controller gains Kqn , Bqn , Kτn , Bτn if Gain scale GS = 1 then (Kq , Bq , Kτ , Bτ ) ← (Kqn , Bqn , Kτn , Bτn ) else (Kqa , Bqa ) ← (Kqn , Bqn )/GS (Kτa , Bτa ) ← GS · (Kτn , Bτn ) (Kq , Bq , Kτ , Bτ ) ← (Kqa , Bqa , Kτa , Bτa ) end if return (Kq , Bq , Kτ , Bτ ) end procedure Assign filtering parameters fvd , fτ d and time delays Tτ , Tqs , Tqd . PM = PhaseMargin(Kq , Bq , Kτ , Bτ , fvd , fτ d , Tτ , Tqs , Tqd )

 refer to Eq. (16)

 refer to Eq. (17)

There is a trade-off between a large torque bandwidth for accurate torque tracking and a low torque bandwidth for larger achievable impedance range. The work in Ref. [29] obtained a similar observation that enlarging the inner loop controller bandwidth reduces the range of stable impedance control gains. In their experimental

4 SEA impedance analysis

validations, they do not decrease impedance gains when raising torque gains. As it is known, the product of cascaded gains grows if torque gains increase, however, this increase is not considered in their stability analysis. It is, therefore, unclear if the reduced stable impedance range is caused by enlarging the torque gains or the increased product gain due to the coupled effect of torque and impedance gains. To validate the trade-off in a more realistic manner, our method maintains a constant gain product value as shown in Eq. (18). Fig. 6A shows the sampling results for different gain scales (GS). A larger GS indicates increased torque gains with decreased impedance gains. When GS > 1, an increasing GS deteriorates the system stability (i.e., phase margin) and causes a larger oscillatory step response as shown in Fig. 6B. On the other hand, when GS < 1, a decreasing G also decreases the system stability. For instance, GS = 0.4 corresponds to a 34-degree phase margin as shown in panel (A), and accordingly a distortion appears in the step response of panel (B). We ignore delays and filtering to focus on the effects of the gain scale. The tests in Fig. 6 validate the optimal performance (i.e., maximized phase margin) of our proposed critically damped gain design criterion (i.e., GS = 1). Although GS = 1 is the optimal value for stability, changing GS to different values allows to change the impedance behavior without changing the natural frequency. Thus, we assign GS as a design parameter in Algorithm 1. In the following section, we will analyze the frequency-domain SEA impedance.

4 SEA impedance analysis Impedance control is widely used for dynamic interaction between a robot and its physically interacting environment [39]. In this section, we study SEA impedance performance in the frequency domain. In particular, we first derive the SEA impedance transfer function given the SEA controller diagram in Fig. 4, and then analyze the effects of time delays, filtering, and load inertia.

4.1 SEA impedance transfer function The SEA impedance transfer function is defined with a joint velocity q˙ j input and a joint torque τj output. Based on the zero desired joint position qdes , the SEA impedance Z(s) = τj (s)/(−sqj (s)) is formulated as follows: 4 τj (s) Nzi si = 5i=0 Z(s) = , −sqj (s) Dzi si i=0

with the numerator coefficients, Nz4 = Im Tf τ Tfv βk, Nz3 = βk(Im (Tf τ + Tfv ) + Tf τ Tfv bm ), Nz2 = Im βk + βkbm (Tf τ + Tfv ) + kkτ (Tf τ + β(Bτ + Kτ Tf τ ))(Bq e−Tqd s + Kq Tfv e−Tqs s ),

(19)

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Nz1 = bm βk + Bq kkτ (1 + Kτ β)e−Tqd s + Kq kkτ (Tfv + Tf τ + β(Bτ + Kτ (Tf τ + Tfv )))e−Tqs s , Nz0 = Kq kkτ e−Tqs s (1 + Kτ β),

and the denominator coefficients, Dz5 = Im Tf τ Tfv β, Dz4 = Im β(Tfv + Tf τ ) + Tfv Tf τ βbm , Dz3 = βIm + βbm (Tf τ + Tfv ) + Tfv kβ(Tf τ + kτ (Bτ + Kτ Tf τ )e−Tτ s ), Dz2 = β(bm + Tf τ k + kkτ (Bτ + Kτ Tf τ )e−Tτ s ) + Tfv βk(1 + Kτ kτ e−Tτ s ), Dz1 = βk(1 + Kτ kτ e−Tτ s ), Dz0 = 0.

Note that Z(s) in Eq. (19) does not incorporate the joint inertia Ij and damping bj since these parameters belong to parts of the interacting environment. Eq. (19) explicitly models time delays and filtering, which are often ignored in the literature of SEA cascaded controller architectures with PD-type controllers. Also, the SEA transfer function in Eq. (19) is complete without any approximations.

4.2 Effects of time delays and filtering The SEA impedance frequency responses are demonstrated in Fig. 7. We analyze various scenarios either with or without time delays and filtering: (i) Zi (jω) is the ideal impedance without delays and filtering; (ii) Zf (jω) is the impedance only with filtering; (iii) Zd (jω) is the impedance only with delays; and (iv) Zfd (jω) is the impedance with both delays and filtering. At low-frequency range, the SEA impedance converges to a virtual stiffness asymptote in all scenarios (when time delays are considered, we have e−Tqs jω → 1, e−Tτ jω → 1 as ω → 0) Kq kτ (β −1 + Kτ ) Nz0 = , ω→0 jω · Dz1 jω · (1 + Kτ kτ )

lim Zc (jω) = lim

ω→0

where c ∈ {i, f , d, fd}. The denominator of the final expression has a jω term, which indicates a −20 dB/dec decay rate. The low-frequency impedance Zc (jω) behaves as a constant stiffness impedance Kq /jω scaled by a constant kτ (β −1 + Kτ )/(1 + Kτ kτ ). This scaling applies to any PD-type cascaded impedance controller. Note that kτ β −1 is normally a small value. When kτ Kτ is large enough, Zc (jω) approaches Kq /(jω), that is, a pure virtual spring. This meets our intuition. As to the high-frequency range, the impedance also approaches an asymptote with a potential twist, depending on the delay and filtering conditions. First, let us start with the ideal case (i), that is, without delays and filtering. This leads to Dz5 = Dz4 = 0, and we have

SEA impedance with time delays and filtering. In panel (A), the impedance of a physical spring k/(jω) and a virtual stiffness gain controller are shown by yellow and blue-dashed lines, respectively. The ideal SEA impedance without delay and filtering is represented by a red-dashed line. At low-frequency range, SEA impedance converges to the virtual stiffness. A similar behavior was observed in Ref. [13]. At high-frequency range, it approaches another impedance asymptote. Panel (B) analyzes the filtering effect while panel (C) analyzes the time-delay effect. Accordingly, the sensitivity discrepancy of different time delays can be analyzed but not discussed here due to the space limit. These simulations have a natural frequency fn = 30 Hz, corresponding to Kq = 293.6 Nm/rad, Bq = 2.49 Nms/rad, Kτ = 11.71 A/Nm, and Bτ = 0.146 As/Nm.

4 SEA impedance analysis

FIG. 7

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lim

ω→+∞

Zi (jω) =

k(Im + kτ Bτ Bq ) Nz2 lim = , ω→+∞ jω · Dz3 jω · Im

which represents a constant stiffness-type impedance scaled from the passive spring stiffness k/(jω). The red-dashed lines in Fig. 7 illustrate this ideal SEA impedance feature. Second, we derive case (iii) only with delay, that is, Tfv = Tf τ = 0. Then, Dz5 = Dz4 = 0, and we obtain lim

ω→+∞

Zd (jω) =

k(Im + kτ Bτ Bq e−Tqd s ) Nz2 = . ω→+∞ jω · Dz3 jω · Im lim

Since the complex number e−Tqd s rotates along the unit circle, the SEA impedance will periodically twist around the passive spring stiffness at high-frequency range. This is visualizable in Fig. 7C. Third, in case (ii) only with filtering, we have Tqs = Tqd = Tτ = 0, and then obtain lim Zf (jω) =

ω→+∞

Nz4 k = , jωDz5 jω

which represents a passive spring stiffness as shown in Fig. 7B. The curve does not twist thanks to the constant limit value k/(jω). To verify the applicability of the behaviors aforementioned to different natural frequencies, we analyze the SEA impedance performance under varying natural frequencies in Fig. 8. By comparing Fig. 8A and B (or Fig. 7B and C), we conclude that time delays have a larger effect on the SEA impedance than filtering.

4.3 Effect of load inertia This section analyzes the effect of load inertia on SEA impedance performance. A second-order model of the output load Ij s + bj is added into Eq. (19), that is, Zl (jω) = Z(jω) + Ij s + bj . Since Eq. (19) becomes Z(jω) → 0 as ω → +∞, we have lim

ω→+∞

Zl (jω) =

lim (Z(jω) + Ij · jω + bj ) = Ij · jω + bj ,

ω→+∞

where Ij · jω represents a 20 dB/dec asymptote at high frequencies (see Fig. 9); the damping term bj adds a constant offset. As the equation previously shows, at high-frequency range, SEA impedance behaves as a spring-mass impedance instead of a pure spring one. In particular, this impedance is dominated by the load inertia as shown in Fig. 9. This figure simulates three scenarios with different load inertias. Different than the load mass effect studied in Ref. [13], our study has a large focus on analyzing the effect of filtering and time delays. These two factors dominate at middle-frequency range where large spikes show up in the shaded region of Fig. 9. The larger load inertia is, the smaller spike the response has.

5 Experimental validation

FIG. 8 SEA impedance with varying natural frequencies fn . First, these subfigures validate that a higher natural frequency fn results in higher SEA impedance. Panels (A) and (B) show how time delay and filtering affect SEA impedance, respectively. We use filters with fqd = 50 Hz and fτ d = 100 Hz while time delays are chosen as Tqd = Tτ = 1 ms and Tqs = 10 ms. Second, we test the cases with both filters and delays as shown in panel (C), and compare them with ideal cases with neither filter nor delays.

5 Experimental validation 5.1 Evaluation of the controller design This experiment section validates the proposed methods and criterion on our SEA testbed, parameters of which are provided in Table 2. We employ the gain design criterion proposed in Section 3 to design controller gains. Detailed stiffness and damping gains are accessible in Table 1. All of our tests have a 1 kHz sampling rate, which induces 0.5 ms effective feedback delay. To obtain larger feedback delays,

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FIG. 9 SEA impedance with varying load inertias. Three different load scenarios are illustrated. All of them use the natural frequency fn = 20 Hz, corresponding to Kq = 148 Nm/rad, Bq = 1.49 Nms/rad, Kτ = 4.48 A/Nm, and Bτ = 0.097 As/Nm. The damping term is bj = 0.1 Nms/rad. For all three scenarios, dashed lines are used to represent asymptote at low and high frequencies, respectively. Since the load inertia is modeled, the SEA impedance approaches the load inertia impedance curve Ij · jω + bj at high frequencies.

Table 2 UT SEA parameters. Parameters

Value

Parameters

Value

Spring stiffness k Motor inertia Im Motor damping bm Gear reduction N Drivetrain efficiency η Pulley reduction Np

350,000 N/m 0.225 kg m2 1.375 Nms/rad 8.3776 × 103 0.9 4

Joint pulley radius rk Joint inertia Ij Joint damping bj Ball screw lead lbs Motor torque coeff. kτ Sample rate

0.025 m 0.014 kg m2 0.1 Nms/rad 0.003 m/rev 0.0276 Nm/A 1 kHz

5 Experimental validation

a software buffering of sampling data is manually implemented. Thus, the total feedback delay has two components Td =

Ts + Te , 2

(20)

where Ts is the sampling period and Te is the extra added feedback delay. Ts is divided by 2 since the effective delay is half of the sampling period [40]. The extra feedback delay, Te , represents large round-trip communication delay between low- and highlevel architectures. The source code is public online: https://github.com/YeZhao/ series-elastic-actuation-impedance-control. Here is a video link of experimental validations: https://youtu.be/biIdlcAMPyE. In Fig. 10, a larger natural frequency produces a higher closed-loop bandwidth. Simulations match experimental results except slight discrepancies at high frequencies. To validate the trade-off between impedance gains and torque gains, we test step responses as shown in Fig. 11. The result shows that when GS > 1, a larger GS slows down the rise time and produces a larger overshoot. This observation is consistent with our theoretical analysis that SEA phase margin will be reduced by decreasing impedance gains and increasing torque gains. As for the discrepancy between simulations and experiments, a potential reason is due to the different spring location in the simulation model and the hardware. The simulation model assumes the spring to be placed between the gearbox output and the load (a.k.a., force sensing SEA) while our UT-SEA hardware places the spring between the motor housing and the chassis ground (a.k.a., reaction force sensing SEA) for compact size design. This discrepancy affects impedance characteristics only at the resonant frequency and high frequency, which is also validated by the result in Fig. 10. The reason why we choose

FIG. 10 Impedance frequency responses with different fn . At low frequencies, experimental results are matched with the simulations. Compared to simulations, the experimental data show a larger peak at the resonant frequency and a slightly larger bandwidth. The parameters are Tqs = Tqd = Tτ = 0.5 ms, fqd = 50 Hz, fτ d = 100 Hz, and GS = 1.

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FIG. 11 Step responses with different gain scales. The overshoot in the experimental results, when GS is increased, matches our simulation predictions. The parameters are Tqs = Tqd = Tτ = 1 ms, fqd = 50 Hz, fτ d = 100 Hz, and fn = 14 Hz.

a force sensing SEA model is due to being more general in the SEA literature, and more suitable for force control, and simplicity in the force measurements. For more details regarding these two mechanical designs, refer to Ref. [12]. The discrepancy between the two models is negligible in our tests since our primary target is to validate the trade-off between impedance and torque control. Torque tracking under impact dynamics is important for interactive manipulation and bipedal locomotion. By implementing an impulse test, we show the high fidelity of our torque control under external impulse disturbances. The purpose of this test is the performance of the controller under disturbances. The controller gains correspond to those of fn = 14 Hz in Table 1. As shown in Fig. 12, when a ball free falls from a 20 cm height and hits the arm with an impulse force, the SEA actuator settles down promptly and recovers after approximately 0.3 s. The recovery to the disturbance is fast and the tracking performance of the torque controller is very accurate. In the following section, we study in detail the implementation of the proposed distributed control strategy in a high-performance linear actuator and an omnidirectional mobile base.

5.2 Step response implementation The proposed controller is implemented in our linear actuator shown in Fig. 13. This actuator is equipped with a PC-104 form factor computer running Ubuntu Linux with an RTAI-patched kernel [12]. The PC communicates with the actuator using analog and quadrature signals through a custom signal conditioning board. Continuous signal time derivatives are converted to discrete form using a bilinear Tustin transform written in C. A load arm is connected to the output of the ball screw pushrod. Small displacements enable the actuator to operate in an approximately linear region of its load inertia. At the same time, the controller is simulated by

5 Experimental validation

FIG. 12 Impulse response of UT-SEA. A ball is dropped from a constant height (20 cm) and exerts an impulse force on the arm end effector. The maximum angle deviation is around 2.5 degrees. The arm recovers to its initial position within 0.3 s. Joint torque tracking is accurate. (A) Ts = 1 ms, Td = 1 ms. (B) Ts = 15 ms, Td = 1 ms. (C) Ts = 1 ms, Td = 15 ms. (D) Ts = 15 ms, Td = 15 ms.

using the closed-loop plant. Identical parameters to the real actuator are used for the simulation, thus allowing us to compare both side by side. First, a test is performed on the actuator evaluating the response to a step input on its position. The results are shown in the bottom part of Fig. 14 which shows and compares the performance of the real actuator versus the simulated closedloop controller. All the experimental tests are performed with a 1 kHz servo rate. Additional feedback delays are manually added by using a data buffer. A step input comprising desired displacements between 0.131 and 0.135 m of physical pushrod length is sent to the actuator. The main reason for constraining the experiment to a small displacement is to prevent current saturation of the motor driver. With very high stiffness, it is easy to reach the 30 A limit for step responses. If current is saturated,

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44 CHAPTER 2 Robotic systems with series elastic actuation

FIG. 13 Linear UT actuator. This linear pushrod actuator has an effective output inertia of m = 256 kg and an approximate passive damping of b = 1250 Ns/m.

5 Experimental validation

FIG. 14 Step response experiment with distributed controller. Panels (A) through (D) show various implementations on our linear actuator. Overlapped with the data plots, simulated replicas of the experiments are also shown to validate the proposed models. The experiments not only confirm the higher sensitivity of the actuator to damping than to stiffness delays but also indicate a good correlation between the real actuator and the simulations.

then the experiment will deviate from the simulation. The step response is normalized between 0 and 1 for simplicity. Various tests are performed for the same reference input with varying time delays. In particular, large and small delays are used for either or both the stiffness and damping loops. The four combinations of results are shown in the figure with delay values of 1 or 15 ms. The first thing to notice is that there is a good correlation between the real and the simulated results both for smooth and oscillatory behaviors. Small discrepancies are attributed to unmodeled static friction and the effect of unmodeled dynamics. More importantly, the experiment confirms the anticipated discrepancy in delay sensitivity between the stiffness and damping loops. Large servo delays on the stiffness servo, corresponding to panels (A) and (B) have small effects on the step response. On the other hand, large servo delays on the damping servo, corresponding to panels (C) and (D), strongly affect the stability of the controller. In fact, for panels (C) and (D), the results corresponding to fn = 12 Hz are omitted due to the actuator quickly becoming out of control. In contrast, the experiment in panel (B) can tolerate such high gains despite the large stiffness delay.

5.3 Distributed operational space control of a mobile base As a concept proof of the proposed distributed architecture on a multiaxis mobile platform, a Cartesian space feedback operational space controller (OSC) [10] is implemented on an omnidirectional mobile base. The original feedback controller was implemented as a centralized process with no distributed topology at that time.

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The mobile base is equipped with a centralized PC computer running Linux with the RTAI real-time kernel. The PC connects with three actuator processors embedded next to the wheel drivetrains via EtherCat serial communications. The embedded processors do not talk to each other. The high-level centralized PC on our robot has a roundtrip latency to the actuators of 7 ms due to process and bus communications, while the low-level embedded processors have a servo rate of 0.5 ms. Notice that 7 ms is considered too slow for stiff feedback control. To accentuate even further the effect of feedback delay on the centralized PC, an additional 15 ms delay is artificially introduced by using a data buffer. Thus, the high-level controller has a total of 22 ms feedback delay. An OSC is implemented in the mobile base using two different architectures. First, the controller is implemented as a centralized process, which will be called COSC, with all feedback processes taking place in the slow centralized processor and none in the embedded processors. In this case, the maximum stiffness gains should be severely limited due to the effect of the large latencies. Second, a distributed controller architecture is implemented inspired by the one proposed in Fig. 4 but adapted to a desired OSC, which will be called DOSC. In this version, the Cartesian stiffness feedback servo is implemented in the centralized PC in the same way than in COSC, but the Cartesian damping feedback servo is removed from the centralized process. Instead, our study implements damping feedback in joint space (i.e., proportional to the wheel velocities) on the embedded processors. A conceptual drawing of these architectures is shown in Fig. 15. The metric used for performance comparison is based on the maximum achievable Cartesian stiffness feedback gains, and the Cartesian position and velocity tracking errors. To implement the Cartesian stiffness feedback processes in both architectures, the Cartesian positions and orientations of the mobile base on the ground are computed using wheel odometry. To achieve the highest stable stiffness gains, the following procedure is followed: (1) Cartesian stiffness gains are adjusted to zero while the damping gains in either Cartesian space (COSC) or joint space (DOSC)—depending on the controller architecture—are increased until the base starts vibrating; (2) the Cartesian stiffness gains, on either architecture, are increased until the base starts vibrating or oscillating; and (3) a desired Cartesian circular trajectory is commanded to the base and the position and velocity tracking performance are recorded. Based on these experiments, DOSC was able to attain a maximum Cartesian stiffness gain of 140 N/(m kg) compared to 30 N/(m kg) for COSC. This result means that the proposed distributed control architecture allowed the Cartesian feedback process to increase the Cartesian stiffness gain (Kx in Fig. 16) by 4.7 times with respect to the centralized controller implementation. In terms of tracking performance, the results are shown in Fig. 15. Both Cartesian position and velocity tracking in DOSC are significantly more accurate. The proposed distributed architecture reduces Cartesian position root mean error between 62% and 65% while the Cartesian velocity root mean error decreases between 45% and 67%.

Omnidirectional mobile base with distributed and centralized OSC controllers. As a proof of concept, we leverage the proposed distributed architectures to our robotic mobile base demonstrating significant improvements on tracking and stability.

5 Experimental validation

FIG. 15

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CHAPTER 2 Robotic systems with series elastic actuation

FIG. 16 Detailed distributed operational space control structure. The figure above illustrates details of the distributed operational space controller used for the mobile base tracking ∗ experiment. Λ∗task and ptask are the operational space inertia matrix and gravity-based ∗ forces, respectively. Jtask is a contact consistent task Jacobin. More details about these matrices and vectors can be found in Ref. [10]. Our main contribution for this experiment lies in implementing operational space control in a distributed fashion and based on the observations performed on the previously simplified distributed controller. While the high-level operational space stiffness feedback loop suffers from large delays due to communication latencies and artificial delays (added by a data buffer), the embedded-level damping loop increases system stability. As a result, the proposed distributed architecture enables to achieve higher Cartesian stiffness gains Kx for better tracking accuracy.

6 Discussions and conclusion The motivation for this chapter has been to study the stability and performance of distributed controllers where stiffness and damping servos are implemented in distinct processors. These types of controllers will become important as computation and communications become increasingly more complex in human-centered robotic systems. The focus has been first on studying the physical performance of a simple distributed controller. Simplifying the controller allows us to explore the physical effects of time delays in greater detail. Based on this controller, we address the problem of impedance controller design and performance characterization of SEAs by incorporating time delays and filtering over a wide frequency spectrum. In particular, we proposed a critically damped controller gain selection method of the cascaded SEA control structure. By uncovering the trade-off existing between impedance gains and torque gains, we prove the optimality of our gain design criterion. We believe the critically damped gain selection criterion can be applied to many types of SEAs and robotics systems for performance analysis and optimizations.

References

To confirm the observations and analytical derivations, hardware experiments are carried out by using an actuator and a mobile base. In particular, the results have shown that decoupling stiffness servos to slower centralized processes does not significantly decrease system stability. As such, stiffness servo can be used to implement OSCs which require centralized information such as robot models and external sensors.

References [1] Y. Sakagami, R. Watanabe, C. Aoyama, S. Matsunaga, N. Higaki, K. Fujimura, The intelligent ASIMO: system overview and integration, in: IEEE/RSJ International Conference on Intelligent Robots and Systems, vol. 3, 2002, pp. 2478–2483. [2] M.A. Diftler, J.S. Mehling, M.E. Abdallah, N.A. Radford, L.B. Bridgwater, A.M. Sanders, R.S. Askew, D.M. Linn, J.D. Yamokoski, F.A. Permenter, et al., Robonaut 2-the first humanoid robot in space, in: 2011 IEEE International Conference on Robotics and Automation (ICRA), IEEE, 2011, pp. 2178–2183. [3] A.M. Okamura, Methods for haptic feedback in teleoperated robot-assisted surgery, Ind. Robot. Int. J. 31 (6) (2004) 499–508. [4] J.-Y. Kim, I.-W. Park, J. Lee, M.-S. Kim, B.-K. Cho, J.-H. Oh, System design and dynamic walking of humanoid robot KHR-2, in: Proceedings of the IEEE International Conference on Robotics and Automation, 2005, pp. 1431–1436. [5] V.M.F. Santos, F.M.T. Silva, Design and low-level control of a humanoid robot using a distributed architecture approach, J. Vib. Control 12 (12) (2006) 1431–1456. [6] H.R. Karimi, H. Gao, New delay-dependent exponential H-infinity synchronization for uncertain neural networks with mixed time delays, IEEE Trans. Syst. Man Cybern. B Cybern. 40 (1) (2010) 173–185. [7] Q. Gao, N. Olgac, Bounds of imaginary spectra of LTI systems in the domain of two of the multiple time delays, Automatica 72 (2016) 235–241. [8] L. Lu, B. Yao, A performance oriented multi-loop constrained adaptive robust tracking control of one-degree-of-freedom mechanical systems: theory and experiments, Automatica 50 (4) (2014) 1143–1150. [9] J.C. Martin, L. George, Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems, IEEE Trans. Autom. Control 26 (5) (1981) 1139. [10] O. Khatib, A unified approach to motion and force control of robot manipulators: the operational space formulation, IJRA RA-3 (1) (1987) 43–53. [11] G.A. Pratt, P. Willisson, C. Bolton, A. Hofman, Late motor processing in low-impedance robots: impedance control of series-elastic actuators, in: American Control Conference, 2004, pp. 3245–3251. [12] N. Paine, S. Oh, L. Sentis, Design and control considerations for high-performance series elastic actuators, IEEE/ASME Trans. Mechatron. 19 (3) (2014) 1080–1091. [13] H. Vallery, J. Veneman, E. Van Asseldonk, R. Ekkelenkamp, M. Buss, H. Van Der Kooij, Compliant actuation of rehabilitation robots, IEEE Robot. Autom. Mag. 15 (3) (2008) 60–69. [14] J. Lu, K. Haninger, W. Chen, M. Tomizuka, Design and torque-mode control of a cable-driven rotary series elastic actuator for subject-robot interaction, in: IEEE International Conference on Advanced Intelligent Mechatronics, 2015, pp. 158–164.

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[15] M. Mosadeghzad, G.A. Medrano-Cerda, J.A. Saglia, N.G. Tsagarakis, D.G. Caldwell, Comparison of various active impedance control approaches, modeling, implementation, passivity, stability and trade-offs, in: IEEE/ASME International Conference on Advanced Intelligent Mechatronics, 2012, pp. 342–348. [16] N.L. Tagliamonte, D. Accoto, E. Guglielmelli, Rendering viscoelasticity with series elastic actuators using cascade control, in: IEEE-RAS International Conference on Robotics and Automation, 2014, pp. 2424–2429. [17] A. Albu-Schäffer, C. Ott, G. Hirzinger, A unified passivity-based control framework for position, torque and impedance control of flexible joint robots, Int. J. Robot. Res. 26 (1) (2007) 23–39. [18] M. Hutter, C.D. Remy, M.A. Hoepflinger, R. Siegwart, Efficient and versatile locomotion with highly compliant legs, IEEE/ASME Trans. Mechatron. 18 (2) (2013) 449–458. [19] A. De Luca, B. Siciliano, L. Zollo, PD control with on-line gravity compensation for robots with elastic joints: theory and experiments, Automatica 41 (10) (2005) 1809–1819. [20] C.-L. Fok, G. Johnson, J.D. Yamokoski, A. Mok, L. Sentis, ControlIt!—a software framework for whole-body operational space control, Int. J. Hum. Robot. 13 (1) (2016). [21] C.-H. Lee, A survey of PID controller design based on gain and phase margins, Int. J. Comput. Cogn. 2 (2004) 63–100. [22] K.J. Åström, Automatic tuning and adaptation for PID controllers—a survey, Control Eng. Pract. 1 (1993) 699–714. [23] E. Poulin, A. Pomerleau, A. Desbiens, D. Hodouin, Development and evaluation of an auto-tuning and adaptive PID controller, Automatica 32 (1) (1996) 71–82. [24] O. Yaniv, M. Nagurka, Design of PID controllers satisfying gain margin and sensitivity constraints on a set of plants, Automatica 40 (1) (2004) 111–116. [25] Y. Tipsuwan, M.-Y. Chow, Gain scheduler middleware: a methodology to enable existing controllers for networked control and teleoperation—part I: networked control, IEEE Trans. Ind. Electron. 51 (6) (2004) 1218–1227. [26] J.Y. Lee, M. Jin, P.H. Chang, Variable PID gain tuning method using backstepping control with time-delay estimation and nonlinear damping, IEEE Trans. Ind. Electron. 61 (12) (2014) 6975–6985. [27] Y. Zhao, N. Paine, K.S. Kim, L. Sentis, Stability and performance limits of latency-prone distributed feedback controllers, IEEE Trans. Ind. Electron. 62 (2015) 7151–7162. [28] T. Hulin, C. Preusche, G. Hirzinger, Stability boundary for haptic rendering: influence of physical damping, in: IEEE/RSJ International Conference on Intelligent Robots and Systems, 2006, pp. 1570–1575. [29] M. Focchi, G.A. Medrano-Cerda, T. Boaventura, M. Frigerio, C. Semini, J. Buchli, D.G. Caldwell, Robot impedance control and passivity analysis with inner torque and velocity feedback loops, Control Theory Technol. 14 (2) (2016) 97–112. [30] H. Gao, T. Chen, T. Chai, Passivity and passification for networked control systems, SIAM J. Control. Optim. 46 (4) (2007) 1299–1322. [31] S. Yin, H. Yang, O. Kaynak, Coordination task triggered formation control algorithm for multiple marine vessels, IEEE Trans. Ind. Electron. 64 (6) (2017) 4984–4993. [32] J.E. Colgate, G. Schenkel, Passivity of a class of sampled-data systems: application to haptic interfaces, in: American Control Conference, vol. 3, 1994, pp. 3236–3240. [33] Y. Zhao, N. Paine, L. Sentis, Sensitivity comparison to loop latencies between damping versus stiffness feedback control action in distributed controllers, in: ASME Dynamic Systems and Control Conference, 2014.

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[34] Y. Zhao, N. Paine, S.J. Jorgensen, L. Sentis, Impedance control and performance measure of series elastic actuators, IEEE Trans. Ind. Electron. 65 (3) (2018) 2817–2827. [35] Y. Zhao, N. Paine, L. Sentis, Feedback parameter selection for impedance control of series elastic actuators, in: IEEE-RAS International Conference on Humanoid Robots, 2014, pp. 999–1006. [36] F. Petit, A. Albu-Schaffer, State feedback damping control for a multi DOF variable stiffness robot arm, in: IEEE-RAS International Conference on Robotics and Automation, 2011, pp. 5561–5567. [37] Y. Zhao, L. Sentis, Passivity of time-delayed whole-body operational space control with series elastic actuation, in: IEEE-RAS International Conference on Humanoid Robots, 2016, pp. 1290–1297. [38] Y. Zhao, A Planning and Control Framework for Humanoid Systems: Robust, Optimal, and Real-time Performance (Ph.D. thesis), The University of Texas at Austin, 2016. [39] N. Hogan, Impedance control: an approach to manipulation: part II implementation, J. Dyn. Syst. Meas. Control 107 (1) (1985) 8–16. [40] T. Hulin, C. Preusche, G. Hirzinger, Stability boundary for haptic rendering: influence of human operator, in: IEEE/RSJ International Conference on Intelligent Robots and Systems, 2008, pp. 3483–3488.

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CHAPTER

Connected cruise control with time-delayed vehicle-to-vehicle communication

3

Linjun Zhang Ford Motor Company, Dearborn, MI, United States

Chapter outline 1 Introduction......................................................................................... 53 2 Connected cruise control......................................................................... 55 2.1 Control framework .................................................................... 55 2.2 Internal stability and string stability ............................................... 58 3 Frequency-domain stability analysis ........................................................... 60 3.1 Head-to-tail transfer function ....................................................... 60 3.2 Stability conditions ................................................................... 63 3.3 Case study ............................................................................. 64 4 Time-domain stability analysis .................................................................. 68 5 Discussions and conclusions .................................................................... 71 5.1 Discussions ............................................................................ 71 5.2 Conclusions ............................................................................ 75 References............................................................................................. 75

1 Introduction Traffic congestions lead to significant wastes of time and resources, imposing adverse impacts on the sustainability of our society. Traffic congestions are usually caused by insufficient road capacities or traffic accidents. However, there are also traffic congestions caused by the inappropriate behaviors of human drivers, which can be observed in dense traffic where the decelerations of vehicles in front are amplified when propagating to the following vehicles and eventually cause a complete stop of the traffic flow. Such phenomenon is referred to as string instability [1], which occurs due to large reaction delays and limited perception capabilities of human drivers. In the recent years, a large amount of researches have been conducted to investigate the design of automatic motion control systems for vehicles by exploiting advanced sensors, in order to improve traffic efficiency and enhance vehicle safety. Stability, Control and Application of Time-Delay Systems. https://doi.org/10.1016/B978-0-12-814928-7.00003-2 © 2019 Elsevier Inc. All rights reserved.

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By applying cameras and/or range sensors (e.g., radar, LiDAR) to detect the motion of the vehicle immediately ahead, adaptive cruise control (ACC) is designed to regulate the longitudinal motion of the host vehicle. Although ACC may improve the passengers’ comfort, its improvement on traffic efficiency is limited because the applied sensors can only detect the objects within the line of sight. Recently, the wireless vehicle-to-vehicle (V2V) communication technology is employed to track the motion of vehicles beyond the line of sight. Integrating V2V communication with vehicle control systems has great potentials for increasing traffic efficiency and enhancing vehicle safety. When including V2V communication, the traffic flow becomes a connected vehicle system. In practice, the data transmission through V2V communication has time delays. For appropriately utilizing the V2V communication in the vehicle control system, the effects of time delays on the dynamics of connected vehicle systems must be investigated. In general, there are two methods to utilize V2V communication in the longitudinal motion control systems, which are cooperative adaptive cruise control (CACC) and connected cruise control (CCC). CACC is focused on controlling a platoon of vehicles where all the following vehicles are automated vehicles such that their controllers can be designed [2]. The design of CACC in the presence of timedelayed V2V communication has been widely studied. In Ref. [3], a string-stable CACC was proposed by utilizing V2V communication to access the acceleration of the vehicle immediately ahead. Considering that the persistent use of motion data received from distant vehicles may cause safety issues under certain circumstances, the author in Ref. [4] provided a control strategy for selective use of the information received from broadcasting vehicles, which can be used to enhance vehicle safety. In Ref. [5], the authors implemented CACC in heavy-duty trucks and demonstrated the string stability through experiments. There are also other experimental projects that evaluate the performance of CACC, such as California Path [6], SARTRE [7], and GCDC [8]. However, since CACC only allows the incorporation of automated vehicles, it may need a special lane dedicated for CACC implementation due to the low penetration of such vehicles in near future. CCC is designed for individual vehicles by applying V2V communication to monitor the motion of multiple broadcasting vehicles ahead [9–12], and it allows the incorporation of human-driven vehicles that may or may not broadcast information. Such flexibility reduces the difficulty for implementing CCC on the standard road. In Ref. [13], a hierarchical architecture was proposed for CCC design, where the highlevel control was designed for the desired longitudinal dynamics by incorporating the motion data of other vehicles received via V2V communication, while at the low level an adaptive control scheme was presented to regulate the vehicle to track the desired motion in the presence of uncertain vehicle dynamics. In Ref. [14], a motif-based approach was proposed for modular design of connected vehicle chains with time-delayed communication and complex connectivity topologies. The effects of time-delayed acceleration feedback and the time-delayed nonlinear dynamics on the stability of CCC were studied in Ge and Orosz [15] and Zhang and Orosz [16], respectively. In practice, packet drops in V2V communication lead to stochastic delays, and the effects of stochastic delays on the stability of CCC were studied in

2 Connected cruise control

Ref. [17]. All the aforementioned studies on CCC assume the prior knowledge about the dynamics of other vehicles, which is indeed not directly available in practice, especially when there are human-driven vehicles that do not broadcast information. To address this problem, the beyond-line-of-sight identification was proposed in Ref. [18] for estimating the dynamics of the connected vehicle system. In this chapter, fundamental methods for the design and the analysis of timedelayed CCC are introduced. In general, the characteristics of time-delayed CCC can be analyzed in the frequency domain and in the time domain. For linear timeinvariant (LTI) time-delayed CCC, the exact (i.e., necessary and sufficient) stability condition can be obtained through the frequency-domain analysis. For nonlinear time-delayed CCC, one can linearize the dynamics and apply the frequency-domain analysis to the linearized dynamics, which provides some insights in the stability of the original nonlinear system. However, the stability analysis obtained from the linearized dynamics is only valid when the state is close to the equilibrium. To ensure the stability of nonlinear time-delayed CCC, one needs to conduct the time-domain analysis to the original nonlinear time-delayed dynamics. The remainder of this chapter is arranged as follows. In Section 2, a general framework is presented for designing CCC by incorporating the time-delayed information received from V2V communication. Then, for the stability analysis of connected vehicle systems, the frequency-domain approach and the time-domain approach are introduced in Sections 3 and 4, respectively. Finally, in Section 5, we briefly discuss some open problems and also conclude this chapter.

2 Connected cruise control 2.1 Control framework Fig. 1 shows a connected vehicle chain where the CCC vehicle i at the tail monitors the positions sj and the velocities vj of vehicles j = p, . . . , i − 1 with p denoting the furthest broadcasting vehicle within the communication range of vehicle i. The symbol lj represents the length of vehicle j. And τi,j denotes the time delay for data transmission from vehicle j to vehicle i. The motion of vehicle i − 1 can be monitored by camera, range sensors (e.g., radar and LiDAR), or V2V communication, while the motion data of vehicles j = p, . . . , i−2 can only be obtained via V2V communication since they are beyond the line of sight. CCC allows the incorporation of humandriven vehicles that do not broadcast information; see vehicle i − 2 in Fig. 1. In this chapter, the state and the output of vehicle i are defined as  xi (t) =

 si (t) , vi (t)

yi (t) = vi (t).

(1)

By adopting the hierarchical controller presented in Ref. [13], one can ignore the aerodynamic drag and the rolling resistance in the physics-based longitudinal dynamics given in Ref. [19], and focus on the design of the desired longitudinal dynamics by using the double-integrator model

55

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CHAPTER 3 Time-delayed vehicle-to-vehicle communication

FIG. 1 CCC vehicle i at the tail monitors the motion of multiple broadcasting vehicles ahead. The motion of vehicle i − 1 can be monitored by using camera, range sensors, or V2V communication, but the motion of vehicles j = p, . . . , i − 2 can only be monitored via V2V communication because they are beyond the line of sight. Symbols sj , lj , and vj denote the position, length, and velocity of vehicle j, respectively. The information delays between vehicles i and j are denoted by τi,j for j = p, . . . , i − 1.



   0 1 0 x˙ i (t) = x (t) + u (t), 0 0 i 1 i   yi (t) = 0 1 xi (t),

(2)

where the acceleration ui is to be designed to regulate the state xi by incorporating the time-delayed data xp (t − τi,p ), . . . , xi−1 (t − τi,i−1 ). In general, CCC can be designed in the form ui (t) =

i−1 

γi,j fi,j (xi (t − τi,j ), xj (t − τi,j )),

(3)

j=p

where the constant γi,j is determined by the connectivity topology of information flow such that  γi,j =

1, 0,

if vehicle i uses the data received from vehicle j, otherwise.

(4)

Note that γi,i−1 = 1 always holds since CCC always monitors the motion of the vehicle immediately ahead for safety. For the subsequent studies, we assume that the connectivity topologies do not change overtime. The function fi,j is used to determine the response of vehicle i to the motion of vehicle j. Here, we use fi,j (xi (t), xj (t)) = αi,j (Vi (hi,j (t)) − vi (t)) + βi,j (Ui (vj (t)) − vi (t)),

(5)

where the quantity ⎞ ⎛ i−1  1 ⎝ hi,j = lk ⎠ sj − si − i−j k=j

(6)

2 Connected cruise control

denotes the average distance between vehicles i and j. Here, i − j represents the number of vehicles between vehicles i and j, which in practice can be obtained by applying the link-length estimator presented in Ref. [20]. The positive constants αi,j and βi,j are used to determine the response to the distance and the response to the relative speed between vehicles i and j, respectively. In Eq. (5), the range policy Vi (h) determines the relationship between the desired velocity and the distance h, and it is typically in the form ⎧ ⎪ ⎨0, Vi (h) = Fi (h), ⎪ ⎩ vmax,i ,

if h ≤ hst,i , if hst,i < h < hgo,i , if h ≥ hgo,i .

(7)

This means that for small distances h ≤ hst,i the vehicle tends to stop for safety, while for large distances h ≥ hgo,i the vehicle aims to maintain the maximum velocity determined by passengers’ preferences or traffic rules. In the middle range hst,i < h < hgo,i , the desired velocity Fi (h) monotonically increases with the distance h, satisfying Fi (hst,i ) = 0 and Fi (hgo,i ) = vmax,i . Three possible choices h − hst,i , hgo,i − hst,i    vmax,i π(h − hst,i ) · 1 − cos , Fi (h) = 2 hgo,i − hst,i     vmax,i π 2h − hgo,i − hst,i · 1 + tanh tan · Fi (h) = 2 2 hgo,i − hst,i Fi (h) = vmax,i ·

(8) (9) (10)

are shown in Fig. 2A–C, respectively. In particular, Fig. 2A shows that the linear function (8) leads to discontinuous gradients at h = hst,i and h = hgo,i , which causes discontinuous jerks. To address this problem, one may apply the nonlinear function (9) or (10). For the subsequent studies, we use the function (9). In practice, the values of hst,i , hgo,i , and vmax,i may vary for different vehicles. To simplify the calculations, we assume that all vehicles use the same parameters such that hst,i = hst = 5 m,

V

V

vmax,i

0 hst,i

(A)

hgo,i

h

V

vmax,i

0 hst,i

(B)

hgo,i

h

vmax,i

0 hst,i

hgo,i

h

(C)

FIG. 2 Range policy functions defined by Eq. (7). Panels (A), (B), and (C) correspond to formulae (8), (9), and (10), respectively.

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CHAPTER 3 Time-delayed vehicle-to-vehicle communication

hgo,i = hgo = 35 m, and vmax,i = vmax = 30 m/s for all i-s. In Eq. (3), the saturation function  v, if v ≤ vmax,i , Ui (v) = vmax,i , otherwise,

(11)

is used in case that the vehicles in front may overspeed.

2.2 Internal stability and string stability When all vehicles use the same range policy, there exists the uniform flow equilibrium. That is, all the following vehicles drive at the same speed v∗ and keep the same distance h∗ from the vehicle immediately ahead, that is, xi∗ (t) =

  ∗   si (t) s + v∗ t = i ∗ , ∗ vi (t) v

y∗i (t) = v∗ ,

(12)

for all i-s, where si is a constant offset. At the uniform flow equilibrium, we have si−1 − si − li−1 = h∗ ,

v∗ = V(h∗ ).

(13)

When vehicles use different range policies, at the equilibrium all vehicles still move at the same constant speed v∗ but they may maintain different distances from the vehicle immediately ahead. The performance of connected vehicle systems can be characterized by internal stability and string stability. Perturbations about equilibrium (12) are defined as x˜ i (t) = xi (t) − xi∗ (t) =



 s˜i (t) , v˜ i (t)

y˜ i (t) = yi (t) − y∗i (t) = v˜ i (t).

(14)

The internal stability of connected vehicle systems indicates that, if the head vehicle moves at a constant speed, the perturbations of all the following vehicles decay to zeros as time evolves, that is, x˜ i (t) → 0

as

t → ∞,

(15)

for all i-s. In practice, the internal stability is critical for vehicle safety. The internal stability and the internal instability are shown in Fig. 3A and B, respectively. The speed perturbations of the head vehicle may affect the motion of all the following vehicles. The connected vehicle system is said to be string stable if the perturbations are attenuated by each following vehicle, that is, ˜vi (t)Lp < ˜vi−1 (t)Lp

(16)

2 Connected cruise control

(A)

(B)

(C)

(D)

FIG. 3 Velocities of the head vehicle 0 and the following vehicle 1, which are displayed by the solid curve and the dashed-dotted curve, respectively. Cases of internal stability and internal instability are shown in (A) and (B), respectively. Cases of string stability and string instability are shown in (C) and (D), respectively.

for all i-s, where the Lp -norm is used to evaluate the magnitude of perturbations and it is defined as ˜v(t)Lp =

 t f t0

|˜v(t)|p dt

1/p .

(17)

In practice, a string-stable traffic flow can mitigate congestions. Comparison of string stability and string instability is demonstrated in Fig. 3C and D, respectively. We remark that the internal stability is a precondition for the string stability; otherwise, the perturbations of the head vehicle may cause the divergence of the following vehicles. For connected vehicle systems that include human-driven vehicles which may amplify disturbances, the string stability defined in Eq. (16) may not be achievable. To deal with this problem, the concept of head-to-tail string stability is proposed in Ref. [14], which requires that the disturbances caused by the head vehicle are attenuated when reaching the tail vehicle. For an (n + 1)-vehicle chain where the

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CHAPTER 3 Time-delayed vehicle-to-vehicle communication

head vehicle and the tail vehicle are indexed by 0 and n, respectively, the head-to-tail string stability is defined by ˜vn (t)Lp < ˜v0 (t)Lp .

(18)

Since the head-to-tail string stability allows some vehicles in the vehicle chain to amplify disturbances, it is particularly useful for connected vehicle systems that contain human-driven vehicles. Substituting controller (3) with Eq. (5) into system (2) leads to the closed-loop dynamics s˙i (t) = vi (t), v˙ i (t) =

i−1 

γi,j [αi,j (Vi (hi,j (t − τi,j )) − vi (t − τi,j )) + βi,j (Ui (vj (t − τi,j )) − vi (t − τi,j ))].

j=p

(19)

In the remaining part of this chapter, we study how to design the control gains αi,j and βi,j for the internal stability and the head-to-tail string stability of connected vehicle systems.

3 Frequency-domain stability analysis It is usually challenging to directly analyze the stability of nonlinear time-delayed systems. To acquire initial insights, one may linearize the dynamics and transform the linearized model to a transfer function in the frequency domain. Then, the stability can be evaluated by studying the transfer function.

3.1 Head-to-tail transfer function Linearizing the closed-loop dynamics (19) about equilibrium (12) and writing the result into the matrix form, we obtain an LTI model in the time domain x˙˜ i (t) = Ai,i x˜ i (t) +

i−1  (Ai,j x˜ i (t − τi,j ) + Bi,j x˜ j (t − τi,j )),

(20)

j=p

y˜ i (t) = C˜xi (t),

cf. Eq. (2), where x˜ i , y˜ i are defined in Eq. (14) and other matrices are given by  Ai,i =

0 0

 1 , 0

 Ai,j =

0 −ϕi,j

 0 , −κi,j

 Bi,j =

0 ϕi,j

 0 , βi,j

 C= 0

 1 ,

(21)

with ϕi,j =

αi,j dVi (h∗ ) · , i−j dh

κi,j = αi,j + βi,j

dUi (v∗ ) . dv

(22)

3 Frequency-domain stability analysis

The stability of the linearized system (20) can be evaluated by using the headto-tail transfer function (HTTF), which represents the dynamic relation between the motion of the head vehicle 0 and the motion of the tail vehicle n. Assuming zero initial perturbations and applying the Laplace transform to Eq. (20), one can get Y˜ i (s) =

i−1 

Ti,j (s)Y˜ j (s),

(23)

j=p

where s ∈ C represents the Laplace operator, Y˜ i (s) denotes the Laplace transform of the output y˜ i (t), and the link transfer function ⎛

Ti,j (s) = γi,j e−sτi,j C ⎝sI2 − Ai,i −

i−1 

⎞−1

e−sτi,j Ai,j ⎠

Bi,j E(s)

(24)

j=p

denotes the dynamic weight on the link from vehicle j to vehicle i; see the weighted graph in Fig. 4. In particular, I2 is the two-dimensional (2D) identity matrix and E(s) = [s−1 , 1]T relates the state and the output such that X˜ j (s) = E(s)Y˜ j (s), where X˜ j (s) is the Laplace transform of the state x˜ j (s). Substituting Eq. (21) into Eq. (24) yields the analytical expression of the link transfer function Ti,j (s) =

γi,j (sβi,j + ϕi,j )e−sτi,j ,  −sτi,k s2 + i−1 k=p (sκi,k + ϕi,k )e

(25)

where ϕi,j and κi,j are given in Eq. (22), which are indeed combinations of control gains αi,j and βi,j . The dynamic relation between the velocities of vehicles i and m can be described by the transfer function Gi,m (s) such that Y˜ i (s) = Gi,m (s)Y˜ m (s).

(26)

Note that the transfer function Gi,m (s) includes the dynamics of all vehicles between vehicle m and vehicle i, and it can be obtained from Eq. (23). To analyze the stability of a connected vehicle system, we need the HTTF Gn,0 (s) that characterizes how the disturbances arising from the head vehicle 0 influence the motion of the tail vehicle n. The graph theory provides a method to systematically calculate the HTTF [21]. To begin with, we construct a dynamic adjacency matrix T(s) ∈ C(n+1)×(n+1) , where the element at the ith row and the jth column is the link transfer function Ti,j (s) given in Eq. (25) for i, j = 0, . . . , n. For an (n + 1)-vehicle chain with unidirectional connections, the longest path has length n. Then, the transfer function matrix G(s) ∈ C(n+1)×(n+1) can be calculated by G(s) =

n 

(T(s))k ,

(27)

k=1

and the HTTF Gn,0 is given by the bottom-left element of matrix G(s). The graph theoretical method consumes unnecessary computations, since it also calculates the transfer functions that are not used in the stability analysis. For

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efficiency, one may apply the approach proposed in Ref. [14] that directly leads to the HTTF by utilizing the modified coupling matrix ˆ T(s) = R(T(s) + In+1 )RT ,

(28)

ˆ where R = [0n×1 , In ] with 0n×1 being an n-by-1 zero vector. Indeed, T(s) ∈ Cn×n can be obtained by deleting the first row and the last column of the matrix T(s)+In+1 . Theorem 1. The HTTF Gn,0 (s) is given by the permanent of the modified coupling matrix (28), that is, n  

ˆ Gn,0 (s) = perm(T(s)) =

Tˆ i,σi (s) =

σi ∈Sn i=1

N(s) , D(s)

(29)

where the sum is computed over all permutations of the set Sn = {1, 2, . . . , n}. The calculation of permanent in Eq. (29) is similar to the formula of determinant but without the change of signs. The proof of Theorem 1 is given in Ref. [14]. Let us consider the vehicle chain shown in Fig. 4, and calculate its HTTF by applying Eqs. (27), (29), respectively. The corresponding adjacency matrix is ⎡

0

⎢T1,0 (s) ⎢ T(s) = ⎢ ⎢T2,0 (s) ⎣T3,0 (s) T4,0 (s)

0 0 T2,1 (s) T3,1 (s) T4,1 (s)

0 0 0 T3,2 (s) T4,2 (s)

0 0 0 0 T4,3 (s)

⎤ 0 0⎥ ⎥ 0⎥ ⎥. 0⎦ 0

(30)

Applying this adjacency matrix in formula (27) and picking the bottom-left element at the last row and the first column of the resulting matrix, we obtain the HTTF G4,0 (s) = T4,0 (s) + T4,1 (s)T1,0 (s) + T4,2 (s)(T2,1 (s)T1,0 (s) + T2,0 (s)) + T4,3 (s)(T3,0 (s) + T3,1 (s)T1,0 (s) + T3,2 (s)(T2,1 (s)T1,0 (s) + T2,0 (s))).

(31)

The same result can also be obtained by applying formula (29) to the modified adjacency matrix

FIG. 4 A (4 + 1)-vehicle chain where the link transfer function Ti,j (s) acts as the dynamic weight along the link from vehicle j to vehicle i.

3 Frequency-domain stability analysis

⎡

T1,0 (s) ⎢T2,0 (s) ˆ T(s) =⎢ ⎣T3,0 (s) T4,0 (s)

1 T2,1 (s) T3,1 (s) T4,1 (s)

0 1 T3,2 (s) T4,2 (s)

⎤

0 0 1

⎥ ⎥, ⎦

(32)

T4,3 (s)

see Eq. (28) for the relationship between the matrices (30), (32).

3.2 Stability conditions Here, we introduce the conditions that ensure the internal stability and the head-totail string stability in the frequency domain. Theorem 2. For the connected vehicle system with LTI dynamics given by Eq. (20), it is internally stable if and only if the real parts of all characteristic roots are strictly negative, that is, Re(λi ) < 0 for i = 1, 2, . . .. An essential difference between the nondelayed LTI systems and the time-delayed LTI systems is that the former has finite number of characteristic roots λi ∈ C that satisfy the characteristic equation D(λi ) = 0, but the latter has infinitely many characteristic roots [22]. In practice, when there are no persistent disturbances, the negative real parts of all characteristic roots indicate that the initial perturbations decay to zeros as time evolves. When the speed of the head vehicle 0 keeps varying, such disturbances propagate along the vehicle chain and finally affect the motion of the tail vehicle n, and the corresponding amplification ratio in L2 norm is given by Gn,0  = sup |Gn,0 (jω)| = ˜yn L2 /˜y0 L2 ,

(33)

ω>0

where j2 = −1 represents the imaginary unit. Theorem 3. For the connected vehicle system with LTI dynamics given by Eq. (20), it is head-to-tail string stable in L2 norm if and only if the amplification ratio (33) is smaller than 1 (i.e., Gn,0  < 1). To obtain the exact (necessary and sufficient) conditions for choosing control gains that ensure the internal stability and the head-to-tail string stability, we apply the D-subdivision method [23] and solve the boundaries that enclose the stability regions. At the boundaries of the internal stability region, the purely imaginary characteristic roots ±jΩ (Ω ≥ 0) satisfy the characteristic equation D(±jΩ) = 0, yielding zero real part and zero imaginary part, that is, Re(D(jΩ)) = 0,

Im(D(jΩ)) = 0.

(34)

Solving these two equations for αi,j (Ω) and βi,j (Ω) with Ω > 0 leads to the boundaries for internal stability in the (βi,j , αi,j )-plane. For Ω = 0, condition (34) becomes D(0) = 0,

which leads to additional boundaries for internal stability in the (βi,j , αi,j )-plane.

(35)

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At the boundaries of the head-to-tail string stability region, we have Gn,0  = 1, which is equivalent to |Gn,0 (jωcr )| = 1,

|Gn,0 (jωcr )| = 0,

|Gn,0 (jωcr )| < 0,

(36)

where the prime represents the derivative of |Gn,0 (jω)| with respect to ω and the symbol ωcr denotes the critical frequency where the maximum occurs. Solving Eq. (36) for αi,j (ωcr ) and βi,j (ωcr ) yields the boundaries that enclose the head-to-tail string stability region in the (βi,j , αi,j )-plane. For the connected vehicle system (20), |Gn,0 (0)| = 1 and |Gn,0 (0)| = 0 always hold. Thus, one can gain the boundaries of head-to-tail string stability at ωcr = 0 by solving |Gn,0 (0)| = 0.

(37)

Plotting boundaries (34)–(37), one can construct the stability diagrams which enclose the regions for choosing control gains that ensure the internal stability and the headto-tail string stability.

3.3 Case study 3.3.1 Stability diagrams Here, we apply the frequency-domain stability analysis to a simple vehicle chain where there is only one vehicle in front of the CCC vehicle; see Fig. 5. The corresponding HTTF is (sβ1,0 + ϕ1,0 )e−sτ1,0 G1,0 (s) = 2 , s + (sκ1,0 + ϕ1,0 )e−sτ1,0

(38)

cf. Eq. (25) with i = 1 and j = 0. Applying Eq. (38) into Eqs. (34)–(37), one can obtain the regions for choosing control gains α1,0 and β1,0 that ensure the internal stability and the head-to-tail string stability. The detailed calculations can be found in Ref. [14]. To construct the stability diagram, let us consider the equilibrium distance h∗ = 20 [m], which leads to V1 (h∗ ) = 15 [m/s], and V1 (h∗ ) = maxh V (h) = π/2 [1/s]. The stability diagram corresponding to the time delay τ1,0 = 0.2 [s] is shown in Fig. 6A, where the domain for internal stability and the domain for head-to-tail string stability are shaded by light gray and dark gray, respectively. Fig. 6B displays

FIG. 5 A vehicle chain where the CCC vehicle 1 monitors the motion of the vehicle immediately ahead.

(A) Stability diagram for the connected vehicle system shown in Fig. 5 when the delay is τ1,0 = 0.2 [s]. Control gains chosen from the light gray domain can ensure internal stability, while those selected from the dark gray domain lead to both internal stability and head-to-tail string stability. (B) Frequencies Ω and ωcr correspond to the internal stability boundary and the head-to-tail string stability boundary respectively. (C) Amplification ratio curves corresponding to the control gains chosen at points C–G. (D)–(F) Leading characteristic roots when the control gains are chosen at points A–C, respectively.

3 Frequency-domain stability analysis

FIG. 6

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the frequencies Ω and ωcr where the system loses the internal stability and the head-to-tail string stability, respectively. In Fig. 6A, points A–C are marked to show how the leading characteristic roots change when the control gains are varied. Fig. 6D–F shows that, when the gains cross the internal stability boundary from the unstable domain to the stable domain, the rightmost characteristic roots move to the left-half complex plane. Points C–G are marked to demonstrate how the head-totail string stability changes with control gains. The corresponding amplification ratio curves are summarized in Fig. 6C. The gains chosen at point E ensures G1,0  < 1. When crossing the boundary for ωcr > 0 (case D), the system loses string stability for midrange frequencies (case C). On the other hand, case F shows marginal string stability at ωcr = 0, implying that string instability occurs at low frequencies when the gains are chosen at point G. The stability diagrams for different values of delay τ1,0 are summarized in Fig. 7, where the notations are the same as used in Fig. 6A. It shows that the stable domain shrinks as the value of delay increases, implying that the increase of time delays deteriorates the robustness of the system. In particular, the string-stable domain disappears when the delay is sufficiently large. For h∗ = 20 [m], the string stability cannot be achieved when the delay is larger than 0.325 [s]; see Ref. [14]. When applying the frequency-domain analysis approach to connected vehicle systems that include a large number of vehicles, the computation complexity may significantly increase. To address this problem, one can employ the motif-based approach proposed in Ref. [14] for modular design of connected vehicle systems, which remains scalable when the number of vehicles increases.

3.3.2 Numerical simulations Here, we simulate the dynamics of the vehicle chain shown in Fig. 5 by considering different combinations of control gains. In particular, we compare the gains marked by points C and E shown in Fig. 6. Moreover, to make the results more practical, we consider constraints on acceleration and decelerations such that −2 ≤ v˙ 1 (t) ≤ 2 [m/s2 ] for all t ≥ 0. To simulate the motion of the head vehicle 0, we use the experiment speed data collected in the UMTRI Safety Pilot Project [24]. The simulation results are shown in Fig. 8, where panels (A) and (B) show the results for the gains at point C and the gains at point E, respectively. According to Fig. 6A, the control gains at point C lead to string instability, while the control gains at point E make the system string stable. In the zoomed-in panel of Fig. 8A, one can observe that the string instability causes oscillations on the velocity of vehicle 1 when the speed of vehicle 0 varies. Such adverse phenomenon can be avoided by choosing control gains that ensure string stability; see the zoomed-in panel in Fig. 8B.

Stability diagrams for the vehicle chain shown in Fig. 5 for different values of delay τ1,0 as indicated. The notations are the same as used in Fig. 6A. It shows that the stable regions shrink when the time delay increases, implying that the time delay imposes adverse effects on the robustness of the system.

3 Frequency-domain stability analysis

FIG. 7

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(A)

(B)

FIG. 8 Numerical simulations. Panels (A) and (B) show the velocities v0 and v1 when the control gains are chosen at point C and E in Fig. 6, respectively. In particular, the zoomed-in panel in (A) shows that string unstable gains lead to oscillating velocities of vehicle 1, which may damage the vehicle and cause passengers’ discomfort. Such adverse effects can be prevented by choosing control gains that ensure string stability; see the zoomed-in panel in (B).

4 Time-domain stability analysis For LTI systems, the frequency-domain analysis leads to the necessary and sufficient stability condition. However, for nonlinear systems, the results obtained from the analysis of their linearized dynamics are valid only when the state is in the vicinity of the equilibrium. The study in Ref. [25] shows that, when the state is far from the equilibrium, the stability of the linearized dynamics may not guarantee the stability of the original nonlinear dynamics. To ensure the stability of nonlinear time-delayed systems, one needs to analyze their dynamics in time domain. Here, we introduce the Lyapunov-Krasovskii approach [26]. The time-domain stability analysis of the nonlinear connected vehicle systems (19) in the presence of multiple delays is presented in Ref. [16]. Here, we use the vehicle chain shown in Fig. 5 as an example to introduce the approach. The dynamic model is given in Eq. (19) with i = 1 and j = 0, yielding s˙1 (t) = v1 (t), v˙ 1 (t) = α1,0 (V1 (h1,0 (t − τ1,0 )) − v1 (t − τ1,0 )) + β1,0 (U1 (v0 (t − τ1,0 )) − v1 (t − τ1,0 )). (39)

Substituting the perturbations (14) into Eq. (39) leads to s˜˙1 (t) = v˜ 1 (t), v˜˙ 1 (t) = α1,0 (V1 (h1,0 (t − τ1,0 )) − V1 (h∗ ) − v˜ 1 (t − τ1,0 )) + β1,0 (U1 (v0 (t − τ1,0 )) − v∗ − v˜ 1 (t − τ1,0 )).

(40)

4 Time-domain stability analysis

We assume that the distance between vehicles 0 and 1 is bounded in a given range h1,0 (t) ∈ D  {h : h ≤ h ≤ h},

(41)

for all t ≥ 0, where the positive constants h and h are the lower and the upper bounds, respectively. It follows that the equilibrium distance is also in the range (i.e., h∗ ∈ D). According to the mean value theorem, there exists a variable ψ ∈ D such that V1 (h1,0 (t − τ1,0 )) − V1 (h∗ ) = V1 (ψ)(h1,0 (t − τ1,0 ) − h∗ )

= V1 (ψ)(˜s0 (t − τ1,0 ) − s˜1 (t − τ1,0 )).

(42)

Moreover, we assume that the head vehicle does not overspeed such that 0 ≤ v0 (t) ≤ vmax for all t ≥ 0. It follows that U1 (v0 (t − τ1,0 )) = v0 (t − τ1,0 ) = v∗ + v˜ 0 (t − τ1,0 ).

(43)

We substitute Eqs. (42), (43) into Eq. (40) and write the result into the matrix form, yielding x˙˜1 (t) = A1,1 x˜ 1 (t) + A1,0 (ψ)˜x1 (t − τ1,0 ) + B1,0 (ψ)˜x0 (t − τ1,0 ),

(44)

where the constant matrix A1,1 is given in Eq. (21), and the other matrices are given by 

0 A1,0 (ψ) = −α1,0 V1 (ψ)

 0 , −(α1,0 + β1,0 )



0 B1,0 (ψ) = α1,0 V1 (ψ)

0 β1,0

 .

(45)

Note that model (44) is indeed nonlinear since the matrices A1,0 (ψ) and B1,0 (ψ) depend on states nonlinearly; cf. Eq. (42). This is different from the linearized model (20) where all matrices are constant. When analyzing the internal stability of a connected vehicle system, we assume no perturbations from vehicle 0, that is, x˜ 0 (t) = 0 for all t ≥ 0. This leads to x˙˜1 (t) = A1,1 x˜ 1 (t) + A1,0 (ψ)˜x1 (t − τ1,0 ).

(46)

Based on the Newton-Leibniz formula, model (46) can be also written in the form x˙˜1 (t) = A1 (ψ)˜x1 (t) − A1,0 (ψ)

 t t−τ1,0

x˙˜1 (ξ )dξ ,

(47)

where A1 (ψ) = A1,1 + A1,0 (ψ). Then, we present a sufficient condition for the internal stability in the following theorem.

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Theorem 4. Suppose that the distance between vehicle 0 and vehicle 1 satisfies the range (41) and the speed of the head vehicle 0 is always below vmax . The connected vehicle system with dynamics given by Eq. (46) is internally stable if there exist positive matrices P, Q, and W such that the matrix ⎡ ⎢ Ω(ψ) = ⎢ ⎣

⎤

T

A1 (ψ)P+PA1 (ψ)+Q+τ1,0 AT1,1 WA1,1 ⎢ τ1,0

AT1,0 (ψ)WA1,1 −AT1,0 (ψ)P

AT1,1 WA1,0 (ψ) τ1,0 AT1,0 (ψ)WA1,0 (ψ)−Q τ1,0 02×2

−PA1,0 (ψ)⎥ ⎥ ⎥ (48) 02×2 ⎦ −W

is negative definite over the domain ψ ∈ D, where 02×2 denotes the 2D zero matrix. The proof of Theorem 4 is given in Ref. [25]. When applying Theorem 4, we first discretize the domain D and then solve the corresponding linear matrix inequalities (LMIs) for matrices P, Q, W by using the numerical solver YALMIP [27]. Note that the solution must ensure that the LMIs hold for all values in the domain D. Suppose that vehicle 0 moves at the constant speed v0 (t) ≡ v∗ = 22.5 [m/s], leading to the equilibrium distance h∗ = 25 [m]. We assume that the delay is τ1,0 = 0.2 [s] and the domain in Eq. (41) is given by D = {h: 13 ≤ h ≤ 27}. Since Theorem 4 is only a sufficient condition for the internal stability, the system may be still internally stable when the distance h1,0 is outside the range D. Then, we compare the stable domain obtained by applying Theorem 4 to the nonlinear system and the stable domain obtained through the frequency-domain analysis based on the linearized dynamics. The results are summarized in Fig. 9. In Fig. 9A, the whole shaded area marks the region of internal stability obtained by the frequency-domain analysis of the linearized dynamics, while the dark gray region is obtained via timedomain analysis of the original nonlinear dynamics. Two sets of control gains are selected as marked by points A and B. The simulation result in Fig. 9B shows that the system is indeed internally unstable although point A is within the stable region obtained by analyzing the linearized dynamics. This implies that the stability analysis based on the linearized dynamics may not hold for the original nonlinear system when the state is far from the considered equilibrium. Fig. 9C shows that, when the control gains are chosen from the stable region generated by the time-domain analysis of the original nonlinear dynamics, the perturbations decay to zero. The head-to-tail string stability of nonlinear time-delayed connected vehicle systems may not be guaranteed through analytical analysis. But the region of head-totail string stability can be approximated by applying the high-order Taylor expansion to the nonlinear dynamics [16].

5 Discussions and conclusions

FIG. 9 (A) Stability diagram for the connected vehicle chain shown in Fig. 5. The overall shaded region is obtained from the frequency-domain stability analysis of the linearized dynamics, while the dark gray region is constructed by applying Theorem 4 to the nonlinear time-delayed dynamics. (B) For the control gains selected at point A, CCC vehicle 1 loses internal stability, implying that the stability analysis of the linearized dynamics may not hold for large perturbations. (C) For control gains selected at point B, which is within the stable region obtained from the time-domain analysis of the nonlinear dynamics, the system becomes internally stable and the initial perturbations decay to zero.

5 Discussions and conclusions 5.1 Discussions For the stability analysis throughout this chapter, we assume no uncertainties about the dynamics of other vehicles, and we also assume that the connectivity topologies of connected vehicle systems do not vary in time. In practice, the dynamics of humandriven vehicles cannot be exactly known, and such uncertainties may have significant impacts on the robustness of the head-to-tail string stability. Let us consider the

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FIG. 10 A connected vehicle chain where the CCC vehicle 2 monitors the motion of vehicle 0 via V2V communication. Note that the parameters of the human-driven vehicle 1 α1,0 , β1,0 , and τ1,0 are unknown for the CCC vehicle 2.

connected vehicle system shown in Fig. 10, where the CCC vehicle 2 can obtain the motion information of vehicle 0 via V2V communication. The parameters of the human-driven vehicle 1 (i.e., α1,0 , β1,0 , and τ1,0 ) are unknown for the CCC vehicle 2. Suppose that the delays of the CCC vehicle 2 are τ2,1 = 0.5 [s] and τ2,0 = 0.2 [s]. To design the control gains for the CCC vehicle 2 to ensure the head-to-tail string stability, we assume the parameters of vehicle 1 as α1,0 = 0.6 [1/s], β1,0 = 0.7 [1/s], and τ1,0 = 0.5 [s]. Then, according to the calculations provided in Ref. [14], the gains α2,1 = 0.6 [1/s], β2,1 = 0.7 [1/s], α2,0 = 0.5 [1/s], and β2,0 = 0.6 [1/s] can lead to the head-to-tail string stability. We assume that the velocity of the head vehicle 0 has a sinusoidal disturbance such that v0 = v∗ + v˜ sin(ωt) with v∗ = 15 [m/s], v˜ = 1 [m/s], and ω = 1.8 [1/s]. When the parameters of vehicle 1 are the same as the assumed values used in the stability analysis, Fig. 11A shows that the CCC vehicle 2 can attenuate the disturbances from vehicle 0. Now, we consider the case that the parameters of vehicle 1 are different from the assumed values used in the stability analysis. In particular, we use α1,0 = 0.7 [1/s], β1,0 = 0.8 [1/s], and τ1,0 = 0.6 [s]. The corresponding simulation result is shown in Fig. 11B, which indicates that the connected vehicle system loses the head-to-tail string stability. This example shows that the uncertainties of the parameters of human-driven vehicles should be considered in the stability analysis for robustness. Considering uncertain parameters of human-driven vehicles but assuming constant and bounded uncertainties, the authors in Ref. [28] investigated the robustness of the head-to-tail string stability in frequency domain. However, in practice, the uncertainties of the dynamics of humandriven vehicles may vary in time, and how to ensure the head-to-tail string stability in the presence of time-varying uncertainties of human-driven vehicles is still an open problem. We remark that the uncertainties of human-driven vehicles do not affect the internal stability of CCC vehicles. Moreover, the connectivity topologies of connected vehicle systems may vary in time, that is, γi,j in Eq. (3) becomes a time-varying parameter instead of a constant. The variations of connectivity topologies may be caused by the packet drops in V2V communication, or they may be caused by the designed control strategies such as

5 Discussions and conclusions

(A)

(B)

FIG. 11 Velocities of the head vehicle 0 and the tail vehicle 2 in Fig. 10. The control gains of the CCC vehicle 2 are chosen based on the stability analysis that uses the assumed values for the parameters of the human-driven vehicle 1. (A) The connected vehicle system is head-to-tail string stable when the parameters of vehicle 1 are the same as the assumed values used in the stability analysis. (B) The connected vehicle system loses head-to-tail string stability when the parameters of vehicle 1 are different from the assumed values used in stability analysis.

the selective use of the information received via V2V communication [4]. Due to the variations of connectivity topologies, the closed-loop dynamics (19) becomes a switching time-delayed system s˙i (t) = vi (t), v˙ i (t) =

i−1 

γi,j (t)[αi,j (Vi (hi,j (t − τi,j )) − vi (t − τi,j )) + βi,j (Ui (vj (t − τi,j )) − vi (t − τi,j ))],

j=p

(49)

where γi,j (t) switches between 0 and 1 over time cf. Eq. (4). Since γi,i−1 = 1 always holds, if there are n broadcasting vehicles in front of the CCC vehicle i, there are 2n−1 subsystems in total for the switching system (49). We emphasize that ensuring the stability of each subsystem does not guarantee the stability of the original switching system [29]. The performance of the switching CCC is characterized by the absolute internal stability and the absolute head-to-tail string stability, where the absolute internal stability and the absolute head-to-tail string stability indicate that the internal stability and the head-to-tail string stability always hold in the presence of switching connectivity topologies, respectively. In Ref. [4], a sufficient condition is presented for the absolute internal stability of the switching system (49). Here, we apply this condition to the CCC vehicle 2 in the connected vehicle system shown in Fig. 10. Assuming τ2,0 = τ2,1 = 0.2 [s], we obtain the constraints for choosing control gains that ensure the absolute internal stability, as highlighted by the shaded region shown in Fig. 12A. Then, we choose the control gains α2,1 = 0.6 [1/s], β2,1 = 0.7

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FIG. 12 (A) The shaded region marks the constraints for choosing control gains that ensure the absolute internal stability of the CCC vehicle 2 in the presence of time-varying connectivity topologies, and the cross marker highlights the combinations of the control gains that are used in the simulation. (B) γ2,0 switches between 0 and 1 as time evolves cf. Eq. (4). Here, γ2,0 = 1 implies that the information of vehicle 0 is delivered to the CCC vehicle 2 while γ2,0 = 0 implies that the CCC vehicle 2 does not get the information of vehicle 0 due to packet drops. (C) Velocities of vehicles 0 and 2, which show the internal stability of the CCC vehicle 2.

[1/s], α2,0 = 0.5 [1/s], and β2,0 = 0.6 [1/s] such that their combination is within the absolute stability region; see the cross in Fig. 12A. Moreover, we assume that the packet drop rate is 20%, that is, the probability for γ2,0 (t) being 0 is 0.2. The values of γ2,0 (t) over time is shown in Fig. 12B, where γ2,0 = 1 implies that the information of vehicle 0 is successfully delivered to the CCC vehicle 2 while γ2,0 = 0 implies that the CCC vehicle 2 does not get the motion data of vehicle 0. Then, we simulate the switching nonlinear time-delayed dynamics (49), and the result shown in Fig. 12C indicates the internal stability of the CCC vehicle 2 in the presence of time-varying connectivity topologies. So far, how to guarantee the absolute head-totail string stability in the presence of time-varying connectivity topologies is still an open problem.

References

5.2 Conclusions In this chapter, we introduced the design of CCC that utilizes wireless V2V communication to monitor the motion of multiple vehicles ahead. The effects of time delays in V2V communication on the internal stability and the head-to-tail string stability were investigated. For the stability analysis of connected vehicle systems, the frequency-domain method and the time-domain method were both introduced. For LTI systems, the frequency-domain analysis leads to the necessary and sufficient stability conditions. However, for nonlinear systems, the frequencydomain stability analysis of the linearized dynamics only leads to the necessary conditions for the internal stability. That is, the nonlinear system is internally unstable if its linearized dynamics is internally unstable, but the internal stability of the linearized dynamics may not guarantee the internal stability of the original nonlinear system, especially when the state is far from the equilibrium. On the other hand, the time-domain stability analysis for nonlinear systems only leads to the sufficient conditions for the internal stability. That is, the nonlinear system is internally stable if the Lyapunov-Krasovskii condition is satisfied, however, failing to construct a suitable Lyapunov-Krasovskii functional does not necessarily indicate the internal instability of the nonlinear system. Therefore, for a thorough stability analysis of nonlinear time-delayed systems, both the frequency-domain analysis and the timedomain analysis are needed. The stability analysis demonstrates that the time delays in V2V communication deteriorates the robustness of the stability of the connected vehicle systems, and the head-to-tail string stability may not be achievable when the time delays are sufficiently large. The effects of the uncertainties of human-driven vehicles and the effects of time-varying connectivity topologies on the stability of connected vehicle systems are discussed via example studies. In particular, the internal stability of the CCC vehicle is immune to the uncertainties of other vehicles. The robustness of the headto-tail string stability against bounded and constant uncertainties can be studied in frequency domain. However, the conditions for ensuring the head-to-tail string stability in the presence of the time-varying uncertainties of human-driven vehicles is a direction for future research. For time-varying connectivity topologies, the system switches among multiple subsystems. We emphasize that ensuring the stability of each subsystem does not guarantee the stability of the original switching system. For switching time-delayed systems, the absolute internal stability can be guaranteed by applying the Lyapunov-Krasovskii method, however, how to ensure the absolute head-to-tail string stability in the presence of time-varying connectivity topologies is still an open problem and needs further research.

References [1] A. Kesting, M. Treiber, How reaction time, update time, and adaptation time influence the stability of traffic flow, Comput. Aided Civ. Inf. Eng. 23 (2) (2008) 125–137. [2] S. Öncü, J. Ploeg, N. van de Wouw, H. Nijmeijer, Cooperative adaptive cruise control: network-aware analysis of string stability, IEEE Trans. Intell. Transp. Syst. 15 (4) (2014) 1527–1537.

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[3] G.J.L. Naus, R.P.A. Vugts, J. Ploeg, M.J.G. van de Molengraft, M. Steinbuch, String-stable CACC design and experimental validation: a frequency-domain approach, IEEE Trans. Veh. Technol. 59 (9) (2010) 4268–4279. [4] L. Zhang, Cooperative adaptive cruise control in mixed traffic with selective use of vehicle-to-vehicle communication, IET Intell. Transp. Syst. 12 (10) (2018) 1243–1254. [5] M.R.I. Nieuwenhuijze, T. van Keulen, S. Öncü, B. Bonsen, H. Nijmeijer, Cooperative driving with a heavy-duty truck in mixed traffic: experimental results, IEEE Trans. Intell. Transp. Syst. 13 (3) (2012) 1026–1032. [6] V. Milanes, S.E. Shladover, J. Spring, C. Nowakowski, H. Kawazoe, M. Nakamura, Cooperative adaptive cruise control in real traffic situations, IEEE Trans. Intell. Transp. Syst. 15 (1) (2014) 296–305. [7] T. Robinson, E. Chan, E. Coelingh, Operating platoons on public motorways: an introduction to the SARTRE platooning programme, in: Proceedings of the 17th World Congress on Intelligent Transport Systems, 2010, pp. 1–11. [8] A. Geiger, M. Lauer, F. Moosmann, B. Ranft, H. Rapp, C. Stiller, J. Ziegler, Team AnnieWAY’s entry to the 2011 grand cooperative driving challenge, IEEE Trans. Intell. Transp. Syst. 13 (3) (2012) 1008–1017. [9] L. Zhang, G. Orosz, Designing network motifs in connected vehicle systems: delay effects and stability, in: Proceedings of the ASME Dynamic Systems and Control Conference, DSCC2013-4081, 2013, V003T42A006. [10] G. Orosz, Connected cruise control: modeling, delay effects, and nonlinear behavior, Veh. Syst. Dyn. 54 (8) (2016) 1147–1176. [11] L. Zhang, Hierarchical Design of Connected Cruise Control: Perception, Planning, and Execution (Ph.D. Dissertation), University of Michigan, Ann Arbor, 2017. [12] G. Orosz, G.I. Ge, C.R. He, S.S. Avedisov, W.B. Qin, L. Zhang, Seeing beyond the line of sight—controlling connected automated vehicles, ASME Mech. Eng. Mag. 139 (12) (2017) S8–S12. [13] L. Zhang, J. Sun, G. Orosz, Hierarchical design of connected cruise control in the presence of information delays and uncertain vehicle dynamics, IEEE Trans. Control Syst. Technol. 26 (1) (2018) 139–150. [14] L. Zhang, G. Orosz, Motif-based design for connected vehicle systems in presence of heterogeneous connectivity structures and time delays, IEEE Trans. Intell. Transp. Syst. 17 (6) (2016) 1638–1651. [15] J.I. Ge, G. Orosz, Dynamics of connected vehicle systems with delayed acceleration feedback, Transp. Res. C 46 (2014) 46–64. [16] L. Zhang, G. Orosz, Consensus and disturbance attenuation in multi-agent chains with nonlinear control and time delays, Int. J. Robust Nonlinear Control 27 (5) (2017) 781–803. [17] W.B. Qin, G. Orosz, Scalable stability analysis on large connected vehicle systems subject to stochastic communication delays, Transp. Res. C 83 (2017) 39–60. [18] L. Zhang, G. Orosz, Beyond-line-of-sight identification by using vehicle-to-vehicle communication, IEEE Trans. Intell. Transp. Syst. 19 (6) (2018) 1962–1972. [19] A.G. Ulsoy, H. Peng, M. Çakmakci, Automotive Control Systems, Cambridge University Press, 2012. [20] L. Zhang, G. Orosz, Black-box modeling of connected vehicle networks, in: Proceedings of the American Control Conference, 2016, pp. 2421–2426. [21] K.H. Rosen, Discrete Mathematics and Its Application, McGraw-Hill, 2012.

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[22] Q. Gao, N. Olgac, Bounds of imaginary spectra of LTI systems in the domain of two of the multiple time delays, Automatica 72 (2016) 235–241. [23] T. Insperger, G. Stépán, Semi-Discretization for Time-Delay Systems, Springer, 2011. [24] D. Bezzina, J. Sayer, Safety Pilot Model Deployment: Test Conductor Team Report (Report No. DOT HS 812 171), National Highway Traffic Safety Administration, Washington, DC, 2015. [25] L. Zhang, G. Orosz, Nonlinear dynamics of connected vehicle systems with communication delays, in: Proceedings of the American Control Conference, 2015, pp. 2759–2764. [26] M. Krstic, Delay Compensation for Nonlinear, Adaptive and PDE Systems, Birkhäuser, New York, USA, 2009. [27] J. Löfberg, YALMIP: a toolbox for modeling and optimization in MATLAB, in: IEEE International Symposium on Computer Aided Control Systems Design, 2004, pp. 284–289. [28] D. Hajdu, L. Zhang, T. Insperger, G. Orosz, Robust stability analysis for connected vehicle systems, in: Proceedings of the 13th IFAC Workshop on Time Delay Systems, 2016, pp. 165–170. [29] M.S. Branicky, Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE Trans. Autom. Control 43 (4) (1998) 475–482.

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CHAPTER

Delay-dependent unknown input observer for nonlinear time-delay systems with both H∞ and W 1,2 optimality criteria

4

Enhanced LMI conditions Lama Hassan, Ali Zemouche, Mohamed Boutayeb CRAN UMR CNRS 7039, University of Lorraine, Cosnes et Romain, France

Chapter outline 1 2 3 4

Introduction......................................................................................... Problem statement................................................................................. H∞ design method ............................................................................... W 1,2 robustness analysis ....................................................................... 4.1 Sobolev space and Sobolev norms ................................................. 4.2 W 1,2 robustness criterion ........................................................... 5 Numerical example................................................................................ 6 Conclusion .......................................................................................... References.............................................................................................

79 81 85 90 90 90 94 96 96

1 Introduction Time-delay systems have been the subject of considerable research activities over the years [1–3]. Different techniques were derived to deal with nonlinear time-delay systems with disturbances starting by the extended Kalman filter. In this methods additional terms in the equation of Riccati are introduced [4,5]. An alternative approach which can be viewed as an extension of the former is the robust H∞ filtering [6,7]. More recently, sliding mode observer techniques is considered to guarantee the stability of systems with Lipschitz nonlinearity and bounded uncertain terms [8]. On the other hand, motivated by several engineering applications such as the estimation of the disturbance in control systems, estimation of the transmitted signal in communication systems, etc., considerable attention has been paid to estimate the Stability, Control and Application of Time-Delay Systems. https://doi.org/10.1016/B978-0-12-814928-7.00004-4 © 2019 Elsevier Inc. All rights reserved.

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CHAPTER 4 Time-delay systems with both H∞ and W 1,2 optimality criteria

system state of a dynamic system subject to unknown input excitations. In existing literature, an unknown input observer (UIO) is an observer which assumes no a priori knowledge of the input. One of the first ideas dealing with UIO for a class of Lipschitz is presented in Ref. [9], in which sufficient conditions in the form of linear matrix inequalities (LMIs) and linear matrix equalities (LMEs) were given for the existence of the proposed UIOs. Whereas in Ref. [10] sufficient existence condition in terms of LMI only was presented. The Lipschitz UIO design problem is proven to be equivalent to an H∞ control problem in Ref. [11]. According to Ref. [12], the unknown inputs could be estimated even if it is impossible to estimate the entire state vector of the system. Another interesting idea is presented in Ref. [13], where the author transformed the system into an extended block triangular observable form suitable for the design of finite-time observers. The proposed method gives less restrictive conditions by comparison to the existing nonlinear UIO design procedures. A different approach proposed in Ref. [14], where the delayed term is decomposed into matched and mismatched portion and the uncertainties are treated as an unknown input. In the problem of UIO synthesis, the presence of the disturbances makes the designing of an observer a difficult task, due to the presence of the disturbance’s derivatives (if exist), and it might be more interesting to explore different approaches than that of classical H∞ , which makes it necessary to introduce and work in the socalled Sobolev spaces. Sobolev space was exploited in Ref. [15] for obtaining local input-output stability results. Moreover, a “local W-stability” was defined and it relationship with the asymptotic stability was studied. What makes this solution interesting is that all the nice properties of L2 are still satisfied by Sobolev space W1,2 . In this chapter, we tackle the problem of UIO design for continuous nonlinear time-delay systems with disturbances and propose two approaches to deal with the state estimation and input construction. Stability analysis is ensured by the use of a Lyapunov-Krasovskii functional together with the LMI approach. The first method proposes a Lyapunov-Krasovskii functional depending on the disturbances to avoid the presence of the disturbance’s derivatives. The second method introduces a new criterion of robustness based on the Sobolev norm in place of the Lebesgue space L2 . This proposed method requires the differentiability of the disturbances. In both cases, the stability is studied dependently on the delay, which turned the problem nonconvex. But, due the use of Young’s inequality in a judicious manner, two sufficient conditions in the form of LMI ensuring the stability of the system and guaranteeing a minimal attenuation level in the H∞ and Sobolev sense are provided, respectively. This chapter is organized as follows: In Sections 2 and 3, a new observer design method with H∞ performance is proposed. In Section 4, a new criterion of robustness is proposed based on the Sobolev norms and Sobolev spaces. Finally, Section 5 presents a numerical example to compare between the proposed methods. Notation. Throughout this chapter, we will use the following notations: • • •

 ·  is the Euclidean norm. () is used for the blocks induced by symmetry. Ir represents the identity matrix of dimension r.

2 Problem statement

•

 ∞ 1/p The notation xLrp = 0 x(t)p dt . is the Lrp norm of the vector x ∈ Rr r with p ≥ 1. The set Lp is defined by Lrp = {x ∈ Rr : xLrp +∞}.

•

For a square matrix S, S > 0 (S < 0) means that this matrix is positive definite (negative definite). ith  • es (i) = (0, . . . , 0, 1 , 0, . . . , 0)T ∈ Rs , s ≥ 1 is a vector of the canonical basis    s components

• •

of Rs . d = e (l)eT (m). Hij = eq (i)eTsi (j) and Hlm q rl The set Co(x, y) is the convex hull of the set {x, y}, that is Co(x, y) = {λx + (1 − λ)y, λ ∈ [0, 1]}.

•

The notation zd (t) means z(t − d(t)).

2 Problem statement Let us consider the following class of continuous nonlinear systems with delayed states and inputs: x˙ (t) = Ax(t) + Ad xd (t) + Au u(t) + Adu ud (t) + Bf (x(t), u(t), xd (t), ud (t)) + Eω ω(t) y(t) = Cx(t) + Du(t) + Dω ω(t),

(1)

where x ∈ Rn is the state vector, u ∈ Rm is the unknown input vector, ω ∈ Lr2 is the vector of bounded disturbances, and y ∈ Rp is the output. A, Ad , Au , Adu , C, D, Eω , and Dω are constant matrices of adequate dimensions. The delay is assumed to be ˙ with d˙ ≤ η < 1. time varying and bounded with a bounded derivative d, Our goal is to design an observer capable of estimating both the state x and the unknown input u of the system (1) in the presence of the disturbances ω and keeping the dynamics of the estimation error stable. In other words, we want to estimate robustly asymptotically both the state x and the unknown input u. In this chapter, we will investigate two types of robustness analysis, the H∞ and W1,2 , respectively. Before presenting the UIO model needed to achieve our purpose, we start by introducing some necessary notations:



E = In 0 , C = C D ,



A = A Au , Ad = Ad Adu ,

ξ T (t) = xT uT .

(2a) (2b) (2c)

81

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CHAPTER 4 Time-delay systems with both H∞ and W 1,2 optimality criteria

Hence, the dynamics (1) can be rewritten under the following descriptor form:  ˙ Eξ (t) = Aξ(t) + Ad ξd (t) + Bf (ξ(t), ξd (t)) + Eω ω(t), y(t) = Cξ(t) + Dω ω(t).

(3)

Without loss of generality f can always be written under the following detailed form: f (ξ(t), ξd (t)) =

i=q 

eq (i)fi (v(t), w(t))

(4)

i=1

with v(t) = Hi ξ(t), w(t) = Hid ξd (t), and

0 , Hi ∈ Rsi ×(n+m) , Fi

d 0 Hi , Hid ∈ Rri ×(n+m) . Hid = 0 Fid

Hi =

Hi 0

(5) (6)

The class of systems investigated in this chapter is defined by the following assumptions: • The matrix D is a full column rank. • The nonlinear function f : Rn+m × Rn+m −→ Rq is assumed to have uniformly bounded partial derivatives, that is: ∂fi (ζ , w) ≤ bij , ∀ζ ∈ Rsi , ∀w ∈ Rri ∂ζj ∂fi (v, ζ ) ≤ bdij , ∀ζ ∈ Rri , ∀v ∈ Rsi . adij ≤ ∂ζj aij ≤

(7) (8)

Remark 1. We assume, without loss of generality, that f satisfies Eqs. (7), (8) with aij = 0 and adlm = 0 for all i, l = 1, . . . , q, j = 1, . . . , s, and m = 1, . . . , r, where s = max1≤i≤q (si ) and r = max1≤i≤q (ri ) [16, Remark 2.3]. In order to estimate both the state x and the unknown input u, we propose the following structure of UIO: ⎧ ⎪ z˙(t) = Π1 z(t) + Π1d zd (t) + Π2 y(t) + Π2d yd (t) ⎪ ⎪ ⎪ i=q ⎨  Beq (i)fi (Hi ξˆ (t), Hid ξˆd (t)), +T ⎪ ⎪ ⎪ i=1 ⎪ ⎩ˆ ξ (t) = z(t) + Sy(t),  −1 T T

E E E T S = where . C C C

(9)

(10)

2 Problem statement

Notice that the assumption that D is of full column rank is of importance in order to have a solution to Eq. (10). By construction of T and S, we have TE + SC = In+m ,

(11)

since y(t) = Cξ(t) + Dω ω(t), we can write the estimation error e as follows: e(t) = ξˆ (t) − ξ(t) = z(t) + SCξ(t) + SDω ω(t) − ξ(t) = z(t) + SCξ(t) + SDω ω(t) − (TE + SC)ξ(t) = z(t) − TEξ(t) + SDω ω(t).

(12)

The presence of the derivative of ω when deriving e makes it difficult to ensure the asymptotic convergence by a classical quadratic Lyapunov function. In order to workaround this obstacle, and since z and Eξ(t) are differentiable, we have     ˙ ˙ e(t) − SDω ω(t) = z˙(t) − T Eξ (t).

(13)

By exploiting Eqs. (3), (9), we obtain    ˙ e(t) − SDω ω(t) = Π1 e(t) + Π1d ed (t) + Sω ω(t) + (Π1 + (Π2 − Π1 S)C − TA)ξ(t) + (Π1d + (Π2d − Π1d S)C − TAd )ξd (t) +

i=q 

TBeq (i)δfi ,

(14)

i=1

where

ωT (t) = ωT (t) ωdT (t) ,  Sω = (Π2 − Π1 S)Dω − TEω

(Π2d − Π1d S)Dω



(15) (16)

and δfi = fi (ˆv(t), w(t)) ˆ − fi (v(t), w(t)). Then, the H∞ filtering design is to determine the matrices Π1 , Π1d , Π2 , and Π2d so that lim e(t) = 0,

t→∞

eLn+m ≤ λωLs

2

2

for ω(·) ≡ 0 ∀ω(·)  ≡ 0;

(17)

e(0) = 0.

(18)

Since Eq. (14) depends on ω, then we need to rewrite Eq. (18) with respect to ω. In fact, it is easy to show that ω2 2s = ω2Ls + |ωd 2Ls = 2ω2Ls + L2

2

2

2

 0 −d

ωT (s)ω(s)ds.

83

84

CHAPTER 4 Time-delay systems with both H∞ and W 1,2 optimality criteria

Hence, inequality (18) becomes eLn+m 2

 1/2  0 λ 2 T ≤ √ ω 2s − ω (s)ω(s)ds L2 2 −d

∀ω(·)  ≡ 0; e(0) = 0.

Assuming for simplicity that ω(t) = 0, ∀t ∈ [−d, 0]. Then, which leads to the following inequality: λ eLn+m ≤ √ ωL2s 2 2 2

0

−d

(19)

ωT (s)ω(s)ds = 0,

∀ω(t)  ≡ 0, t ≥ 0; e(0) = 0.

(20)

Consequently, the H∞ filtering design problem can be redefined to having lim e(t) = 0

for ω(·) ≡ 0,

t→∞

λ eLn+m ≤ √ ωL2s 2 2 2

∀ω(t)  ≡ 0, t ≥ 0; e(0) = 0.

(21) (22)

Using the mean value theorem [17] as presented in Ref. [18], we deduce that there exist z ∈ Co(v, vˆ ), zd ∈ Co(w, w) ˆ so that δfi =

j=s i

hij eTsi (j)Hi e(t) +

j=1

j=r i

hdij eTri (j)Hid ed (t),

(23)

j=1

where ∂fi (zi (t), w(t)), ∂vj ∂fi (v(t), zdi (t)). hdij (zdi ) = ∂wj hij (zi ) =

(24) (25)

Thus, Eq. (14) becomes    ˙ e(t) − SDω ω(t) = Π1 e(t) + Π1d e(t − d) + Sω ω(t) + (Π1 + (Π2 − Π1 S)C − TA)ξ(t) + (Π1d + (Π2d − Π1d S)C − TAd )ξd (t) +

i=q j=s  i

TBeq (i)hij eTsi (j)Hi e(t)

i=1 j=1

+

i=q j=r  i

TBeq (i)hdij eTri (j)Hid ed (t).

(26)

i=1 j=1

Now that we have solved the problem of the presence of the derivative of ω, we can propose an H∞ criterion by choosing a Lyapunov functional that contains a particular quadratic term in e(t) − SDω ω(t). Thanks to this term we will provide, in the

3 H∞ design method

following section, a tractable LMI condition that ensure the asymptotic convergence of the UIO in the H∞ context.

3 H∞ design method In this section, we propose an observer synthesis method expressed in terms of LMI using the H∞ criterion. The problem of H∞ filtering design (21), (22) can be reduced to finding a Lyapunov functional V(t) such that ˙ + eT (t)e(t) − ϑ(t) = V(t)

λ2 T ω ω < 0. 2

(27)

In the following theorem, we state a sufficient condition that ensures inequality (27) and thus solve our H∞ problem. Theorem 1. For a prescribed λ > 0, the H∞ filtering design problem corresponding to system (1) and observer (9) is solvable, with an H∞ performance level less than λ, if the conditions below are fulfilled: 1. There exist matrices P = PT > 0, Q = QT > 0, R and Rd of adequate dimensions so that LMI (28) holds with ⎡

Γ11

Ad + Z

Γ12

M + PTΣ

PTΣd

0

AT

⎢ () −(1 − η)Q − Z −(PT Ad − RT C )T G − F 0 N 0 ATd d ⎢ ⎢ 2 λ ⎢ () −G T PTΣ −G T PTΣd 0 SωT P () Θ − X − 2 I2r ⎢ T ⎢ () () () −Υ 0 0 (PTΣ) ⎢ ⎢ () T () () () −Υ 0 (PTΣ ) d d ⎢ 1 ⎢ () Z 0 () () () () − 2 ⎢ d ⎣ () () () () () () −P ()

()

()

()

()

()

()

0 0 0 0 0 Z

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ < 0. ⎥ ⎥ ⎥ ⎥ ⎦

0 − 1 P

(28) Γ11 = A + AT + Q − Z + In+m , A = PTA − RT C, Ad = PTAd − RTd C,  T (P, R) Θ1,1 (P, R) + Θ1,1 Θ=− DTω Rd SDω

DTω ST RTd Dω 0r×r

 ,

Θ1,1 = DTω ST (RT Dω − PTEω ),   T TT P DTω (R − ST (PTA − RT C) + ST ) − Eω T , Γ12 = DTω (Rd − ST ) ⎡ ⎤

⎢ ⎥ M = M1 · · · Mq , where Mi = ⎣HiT · · · HiT ⎦ ,    si times

85

86

CHAPTER 4 Time-delay systems with both H∞ and W 1,2 optimality criteria

⎡



···

N = N1

Σ = B H11  d Σd = B H11

Nq ,

where

···

H1s1

H21

···

d H1r

d H21

1

⎤

⎢ ⎥ Ni = ⎣(Hid )T · · · (Hid )T ⎦ ,    ···



ri times

Hqsq ,  d · · · Hqr , q

  Υ = diag β11 Is1 , . . . , β1s1 Is1 , β21 Is2 , . . . , βqsq Isq ,  d I , . . . , βd I , βd I , . . . , βd I Υd = diag β11 r1 qrq rq , 1r1 r1 21 r2 βij =

2 , bij

2 βijd = d . bij

2. The matrices Π1 , Π1d , Π2 , and Π2d are given by Π1 = TA − P−1 RT C,

(29)

Π1d = TAd − P−1 RTd C,

(30)

Π2 = P−1 RT + Π1 S,

(31)

Π2d = P−1 RTd + Π1d S.

(32)

Proof. The mainidea relies in choosing an appropriate Lyapunov-Krasovskii functional capable of removing the quadratic term in ω: ˙ V(t) = (e(t) − SDω ω(t))T P(e(t) − SDω ω(t))  t eT (θ)Qe(θ)dθ + t−d(t)

+d

" !  "  0  t !  ˙  T ˙ e − SDω ω (s)Z e − SDω ω (s)dsdθ. −d t+θ

(33)

Calculating the derivative of the Lyapunov-Krasovskii functional along the trajectories of Eq. (26), we obtain !  " ˙ V˙ = 2(e(t) − SDω ω(t))T P e(t) − SDω ω(t) ˙ T (t)Qed (t) + eT (t)Qe(t) − (1 − d)e d +d −d

2

!  "T !  " ˙ ˙ e(t) − SDω ω(t) Z e(t) − SDω ω(t)

" !  "  t !  ˙  T ˙ e − SDω ω (s)Z e − SDω ω (s)ds. t−d

(34)

3 H∞ design method

Using the Jensen’s inequality, we can write −d

" !  "  t !  ˙  T ˙ e − SDω ω (s)Z e − SDω ω (s)ds t−d

≤

"T ! t !  " " ! t !  " ˙ ˙ e − SDω ω (s)ds Z e − SDω ω (s)ds

=

t−d

e − SDω ω ed − SDω ωd

T

−Z ()

Z −Z



t−d

e − SDω ω . ed − SDω ωd

(35)

Notice that the right-hand side of Eq. (35) can be written as ⎡ ⎤T ⎡ e −Z ⎣ed ⎦ ⎣ () ω ()

Z −Z ()

⎤⎡ ⎤ F e −F ⎦ ⎣ed ⎦ , ω −X

(36)

with

F = SDω −SDω , (SDw )T Z(SDw ) X = ()

−(SDw )T Z(SDw ) . (SDw )T Z(SDw )

(37) (38)

Using the matrices P and R provided by the LMI (28) and Eqs. (26), (29)–(32), and by defining the matrix G as G = SDω

0(n+m)×r ,

(39)

we obtain    ˙ e(t) − Gω(t) = (TA − P−1 RT C)e(t) + Sω ω(t) + (TAd − P−1 RTd C)ed (t) +

i=q j=s  i

TBHij ζij

i=1 j=1

+

i=q j=r  i

TBHijd ζijd

(40)

i=1 j=1

where

ζij = hij Hi e(t),

ζijd = hdij Hid ed (t).

(41)

 P−1 RTd Dω .

(42)

In addition, the matrix Sω becomes  Sω = P−1 RT Dω − TEω

87

88

CHAPTER 4 Time-delay systems with both H∞ and W 1,2 optimality criteria

From Eqs. (7), (8), (41), we have i=q j=s  i

eT HiT ζij −

i=1 j=1 i=q j=r  i

i=q j=s  i 1 ζ T ζij ≥ 0, bij ij

(43)

i=1 j=1

eTd (Hid )T ζijd −

i=1 j=1

i=q j=r  i 1

bdij

i=1 j=1

(ζijd )T ζijd ≥ 0.

(44)

consequently, the left-hand side of Eq. (27) verifies ϑ ≤ϑ +2

i=q j=s  i

eT HiT ζij − 2

i=1 j=1

+2

i=q j=r  i

i=q j=s  i 1 ζ T ζij bij ij i=1 j=1

eTd (Hid )T ζijd − 2

i=1 j=1

i=q j=r  i 1 i=1 j=1

bdij

(ζijd )T ζijd .

(45)

Hence, from Eqs. (34)–(38), (43)–(45), we get ⎡

⎤T ⎡ e(t) Γ11 ⎢ ed (t) ⎥ ⎢ () ⎢ ⎥ ⎢ ⎥ ⎢ ϑ ≤⎢ ⎢ ω(t) ⎥ ⎢ () ⎣ ζ (t) ⎦ ⎣ () () ζ d (t) 

Γ12 Γ22 () () ()

Γ13 Γ23 Γ33 () () 

Γ14 0 Γ34 Γ44 ()

⎤⎡ ⎤ e(t) Γ15 ⎢ ⎥ Γ25 ⎥ ⎥ ⎢ ed (t) ⎥ ⎢ ω(t) ⎥ Γ35 ⎥ ⎥⎢ ⎥ 0 ⎦ ⎣ ζ (t) ⎦ Γ55 ζ d (t) 

Γ

!  "T !  " ˙ ˙ + d e(t) − SDω ω(t) Z e(t) − SDω ω(t) ,

(46)

Γ11 = (PTA − RT C)T + (PTA − RT C) + Q − Z + In+m ,

(47)

Γ12 = (PTAd − RTd C) + Z, Γ13 = −(PTA − RT C)T G + F + PSω ,

(48)

Γ14 = M + PTΣ,

(50)

2

where

(49)

Γ15 = PTΣd ,

(51)

Γ22 = −(1 − η)Q − Z,

(52)

Γ23 = −(PTAd − RTd C)T G − F,

(53)

Γ25 = N , T PG − X − Γ33 = −G T PSω − Sω

Γ34 = −G T PTΣ,

(54) λ2 2

I2r ,

(55) (56)

Γ35 = −G T PTΣd ,

(57)

Γ44 = −Υ ,

(58)

Γ55 = −Υd ,

(59)

3 H∞ design method

  T T T T T · · · ζ1s ζ21 · · · ζqs ζ = ζ11 , q 1   d )T · · · (ζ d )T (ζ d )T · · · (ζ d )T T . ζ d = (ζ11 qrq 1r 21 1

(60) (61)

By applying Schur Lemma on Eq. (47), the inequality ϑ < 0 is equivalent to the following: ⎡

⎤⎤ (TA − P−1 RT C)T Z T −1 T ⎢(TAd − P R C) Z ⎥⎥ d ⎢ ⎥⎥ TZ ⎢ ⎥⎥ Sω ⎢ ⎥⎥ ⎢ ⎥⎥ TZ (TΣ) ⎢ ⎥⎥ < 0. ⎣ ⎦⎥ T (TΣd ) Z ⎥ ⎥ 0 ⎦ 1 − 2Z ⎡

⎢ ⎢ ⎢ ⎢Γ ⎢ Ω=⎢ ⎢ ⎢ ⎢ ⎣ ()

(62)

d

The previous condition (62) contains some bilinear terms. In order to treat these terms, we propose to use the Young’s inequality. So, the left-hand side of the inequality can be decomposed as follows: 

Γ Ω = ()



0

T T − 12 Z + X Y + Y X

(63)

d

with ⎡

⎡ ⎤ 0 ⎢0⎥ ⎢ ⎥ ⎢0⎥ ⎥ YT = ⎢ ⎢0⎥ . ⎢ ⎥ ⎣0⎦ Z

⎤ (TA − P−1 RT C)T T −1 T ⎢(TAd − P R C) ⎥ d ⎢ ⎥ T ⎢ ⎥ Sω T ⎥, X =⎢ ⎢ ⎥ T (TΣ) ⎢ ⎥ ⎣ ⎦ T (TΣd ) 0

(64)

In this case, and for any symmetric and positive definite matrix Π , we have  Γ Ω ≤ ()  Γ = ()

0



1 T T −1 − 12 Z +  X ΠX + Y Π Y d 

−1

0 T

0 ΠX T −Π − . X Π Y 1 − 2Z Y () − 1 Π

(65)

d

Using Schur Lemma once again, we deduce that the condition Ω < 0 is equivalent to having ⎡

Γ ⎢ () ⎢ ⎢ ⎣

0



− 12 Z d ()

Π ()

⎥ ⎥

⎥ < 0. 0 ⎦ 1Π 

ΠX Y

−

⎤



(66)

89

90

CHAPTER 4 Time-delay systems with both H∞ and W 1,2 optimality criteria

As a result, we deduce that in order to satisfy condition (27), inequality (66) should be verified, which is equivalent to the LMI (28) by choosing Π = P, thus the H∞ criterion is guaranteed. In the following section, we propose an alternative approach based on the Sobolev norms, namely the W 1,2 approach. The use of this criterion requires the differentiability of the disturbances.

4 W 1,2 robustness analysis In this section, we will introduce a different criterion based on the Sobolev norm [15,19,20], but first let us start by introducing the Sobolev spaces and Sobolev norms.

4.1 Sobolev space and Sobolev norms •

Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp norms of the function itself as well as its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, thus a Banach space. # k,p Wr ([0, +∞]) =

•

z : [0, +∞] → Rr

such that

$ ∂ iz r ∈ Lp ([0, +∞]), i = 0, . . . , k . ∂ti (67) k,p

Sobolev norm: The Sobolev space defined aforementioned Wr ([0, +∞]) admits a natural norm defined as follows: ⎡ ⎛' ' i=k ' i '  ∂ z' ⎝' zrk,p = ⎣ ' i' ' ∂t ' i=0

⎞p ⎤1/p ⎠ ⎦

Lrp

⎛ ' ' ⎞1/p i=k  +∞ ' i 'p  '∂ z' ⎠ =⎝ . ' i ' dt ' ∂t ' 0

(68)

i=0

k,p

The space Wr ([0, +∞]) equipped with the norm  · rk,p is a Banach space.

4.2 W 1,2 robustness criterion We assume that ω ∈ Wr1,2 . Then, following the same developments as in the previous section, the criterion can be stated as follows. Given system (1), observer (9), and disturbances ω, the W 1,2 filtering design problem is to determine the matrices Π1 , Π1d , Π2 , and Π2d so that

4 W 1,2 robustness analysis

lim e(t) = 0

t→∞

λ1,2 er1,2 ≤ √ ωr1,2 2

for

ω(·) ≡ 0,

∀ω(t)  ≡ 0,

(69)

t ≥ 0; e(0) = 0.

(70)

In fact, we have (ωr1,2 )2 = 2(ωr1,2 )2 +

 0 −d

ωT (s)ω(s)ds +

 0 −d

ω˙ T (s)ω(s)ds. ˙

Then, it suffices to suppose, to simplify the notation, that ω(t) = 0, ∀t ∈ [−d, 0] to get criterion (70). Indeed, in this case, we have  0 −d

ωT (s)ω(s)ds =

 0 −d

ω˙ T (s)ω(s)ds ˙ = 0.

Now that the criterion is well defined, then following the Lyapunov theory, the problem of W 1,2 filtering design can be reduced to finding a Lyapunov function V(·) such that λ21,2 λ21,2 T ϑ(t) = V˙ + eT e + e˙ T e˙ − ωT ω − ω˙ ω˙ < 0. 2 2

(71)

In the following theorem, we state the conditions needed to ensure the W-stability of the estimation error. Theorem 2. For a prescribed λP > 0, the W 1,2 filtering design problem corresponding to system (1) and observer (9) is solvable, with a W 1,2 performance level less than λP , if the conditions below are fulfilled: 1. There exist matrices P = PT > 0, Q = QT > 0, R, and Rd of adequate dimensions so that the LMI (72) holds. ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

M + PTΣ PTΣd

Γ (1, 1)

Γ (1, 2)

Γ (1, 3)

PG

()

−(1 − η)Q − Z

0

0

0

() ()

() ()

−

λ2P

2 I2r

()

0

(PT A − RT C )T

0

N

0

(PT Ad − RTd C )T

T Dω R − EωT T T P DTω Rd

0 0

0

0

0

0

λ2 − 2P I2r

0

0

0

GT P

0

0 0

(TΣ)T P (TΣd )T P

0 0

0

Z˜

() ()

() ()

() ()

() ()

−Υ ()

0 −Υd

()

()

()

()

()

()

()

()

()

()

()

()

()

−P

0

()

()

()

()

()

()

()

()

− 1 P

−

1 ˜ 2Z d

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ < 0. ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (72)

Γ (1, 1) = PTA − RT C + (PTA − RT C)T + Q − Z + In+m , Γ (1, 2) = PTAd − RTd C + Z,

Γ (1, 3) = RT Dω − PTEω RTd Dω .

91

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CHAPTER 4 Time-delay systems with both H∞ and W 1,2 optimality criteria

2. The matrices Π1 , Π1d , Π2 , and Π2d are given by Π1 = TA − P−1 RT C,

(73)

Π1d = TAd − P−1 RTd C, Π2 = P−1 RT + Π1 S, Π2d = P−1 RTd + Π1d S.

(74) (75) (76)

Proof. Let us consider the following classical Lyapunov-Krasovskii functional: V(t) = e(t)T Pe(t) +

 t t−d(t)

eT (θ)Qe(θ)dθ + d

 0  t −d t+θ

e˙ T (s)Z e˙ (s)dsdθ.

(77)

2

By taking into consideration the additive term d e˙ T (t)Z e˙ (t) and following the same steps of the previous method, of H∞ , we deduce that ⎤T ⎤ ⎡ e(t) e(t) ⎢ ed (t) ⎥ ⎢ ed (t) ⎥ ⎥ ⎥ ⎢ ⎢ ⎥ ⎢ ω(t) ⎥ ⎢ ⎥ (Q1 − Q2 Q−1 QT ) ⎢ ω(t) ⎥ , ϑP (t) ≤ ⎢ 2 ⎢ ω(t) ⎢ ω(t) ˙ ⎥ ˙ ⎥ 3 ⎥ ⎥ ⎢ ⎢ ⎣ ζ (t) ⎦ ⎣ ζ (t) ⎦ ⎡

ζ d (t)

(78)

ζ d (t)

where Q1 , Q2 , and Q3 are matrices of appropriate dimensions such that ⎤ Γ11 PSω PG Γ14 Γ15 ⎢ () Γ22 0 0 0 N ⎥ ⎥ ⎢ 2 ⎥ ⎢ λ ⎥ ⎢ () () − P I2r 0 0 0 ⎥, 2 Q1 = ⎢ ⎥ ⎢ λ2P ⎥ ⎢ () () I 0 0 () − 2r 2 ⎥ ⎢ ⎣ () () () () −Υ 0 ⎦ () () () () () −Υd ⎡ ! "⎤ 1 −1 T T ⎢ (TA − P R C) Z + d2 In+m ⎥ ⎢ ! "⎥ ⎢ ⎥ ⎢(TAd − P−1 RT C)T Z + 1 In+m ⎥ 2 ⎢ ⎥ d d ⎢ ⎥ ! " ⎢ ⎥ 1 T ⎢ ⎥ Z + S I n+m ω 2 ⎢ ⎥ d ⎥, " ! Q2 = ⎢ ⎢ ⎥ T Z+ 1 I ⎢ ⎥ G 2 n+m ⎢ ⎥ d ⎢ ⎥ ! " ⎢ ⎥ T Z+ 1 I ⎢ ⎥ (TΣ) n+m 2 ⎢ ⎥ d ⎢ ⎥ ! " ⎣ ⎦ 1 T (TΣd ) Z + 2 In+m ⎡

1 Q3 = − 2 d

!

"

1 Z + 2 In+m . d

(79)

(80)

d

(81)

4 W 1,2 robustness analysis

T Using Schur Lemma, we deduce that the inequality Q1 −Q2 Q−1 3 Q2 < 0 is equivalent to the one below

⎡ Ω1 = ⎣

Q1 ()

⎤ ! 0 " ⎦ + X T Y1 + Y T X 1 < 0 1 1 − 12 Z + 12 In+m d

(82)

d

with ⎡

⎡

⎤ (TA − P−1 RT C)T ⎢(TA − P−1 RT C)T ⎥ ⎢ ⎥ d d ⎢ ⎥ T Sω ⎢ ⎥ ⎢ ⎥ T T X1 = ⎢ ⎥, G ⎢ ⎥ T ⎢ ⎥ (TΣ) ⎢ ⎥ T ⎣ ⎦ (TΣd ) 0

⎤

0 0 0 0 0 0

⎢ ⎢ ⎢ ⎢ ⎢ T Y1 = ⎢ ⎢ ⎢ ⎢ ⎢! ⎣

Z + 12 In+m

⎥ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ "⎥ ⎦

(83)

d

For any symmetric and positive definite matrix Π1 and by applying the inequality of Young, we have ⎡ Ω1 ≤ ⎣

Γ ()

⎤ ! 0 " ⎦ − 12 Z + 12 In+m d

− X T Π1



YT

d

−Π1 ()

0 − 1 Π1

Π1 X . Y

−1

(84)

According to the Schur Lemma, Ω1 < 0 is equivalent to the following inequality: ⎡⎡

Q1

⎢⎣ ⎢ () ⎢ ⎢ ⎣

⎤ ! 0 " ⎦ − 12 Z + 12 In+m d

d

()

T X 1 Π1

−Π1 ()

Y1T



0 1 −  Π1

⎤ ⎥ ⎥ ⎥

⎥ < 0. ⎦

(85)

 By defining the matrix Z˜ = Z + 12 In+m and choosing Π1 = P, we get LMI (72) d which means criterion (71) is verified. This ends the proof of Theorem 2. Remark 2. The W 1,2 stability requires the differentiability of the disturbances. Remark 3. For differentiable disturbances, both H∞ and W 1,2 methods can be applied if the rank condition on matrix D is satisfied. In addition, if the disturbances are differentiable and the rank condition is not verified, W 1,2 approach can still be used if the rank condition is satisfied using pseudo-measurements. On the other hand, in some cases even if the rank condition holds, the W1,2 approach may be more robust from the H∞ method as we will see in the undermentioned example.

93

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CHAPTER 4 Time-delay systems with both H∞ and W 1,2 optimality criteria

5 Numerical example The considered system is a nonlinear time-delay system on the form:

−10 0 1 x(t) + u(t) + Eω ω(t) 0 −10 1



0.2 −0.01 tanh(x1 (t)) + tanh(x2 (t)) −0.5 0.45



−0.6 −0.04 tanh(x1 (t − d)) , + −0.08 −1.6 tanh(x2 (t − d))

y(t) = 1 0 x(t) + u(t) + Dω ω(t).

x˙ (t) =

(86)

The unknown input u is a sinusoidal signal of the form u(t) = sin(0.5t). The previous equations can be transformed under form (3) with

1 E= 0 H1 = H1d C= 1

−10 0 1 A= , Ad = 02×3 , 0 −10 1



1 0 0 0 1 0 = , H2 = H2d = , 0 0 0 0 0 0

1 0 1 , Eω = , Dω = 0.1. 1 0 1

0 , 0



Moreover, the disturbance vector ω is a sinusoidal signal ω(t) = 0.2 sin(0.2t),

which is added on two finite intervals of time I = 2 5 ∪ 10 20 , to show simultaneously the robustness and the asymptotic of the proposed observer convergence to zero, respectively, with and without disturbances. In other words, the disturbance is ωk χk , where χk is defined by # χk =

1 if k ∈ I 0 otherwise.

The bounds of the partial derivatives of f are aij = 0, adij = 0,

bij = 1, bdij = 1,

i, j = 1, 2.

According to Remark 1, our system does fulfill the required condition. By taking the initial conditions ξ0 = [0.1, −0.2, 0], ξˆ0 = [0.5, 1, −1] and for  = 10, we obtain the following solutions: •

H∞ method:

⎡

Π1

Π1d

−33.73 = ⎣ 29.02 33.22 ⎡ 37.46 = ⎣−45.21 −37.68

⎡ ⎤ ⎤ −22.73 1 30.02 ⎦ , Π2 = ⎣ 1 ⎦ , 22.22 −1 ⎤ 37.46 −45.21⎦ , Π2 = 03×1 , −37.68

0 −10 0 0 0 0

and the optimal value of the disturbance attenuation level is λ = 0.1972.

5 Numerical example

1 x1(t) x^1(t) with H∞criterion x^ (t) with W criterion

0.8 0.6

1

1,2

Magnitude

0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1

(A)

0

2

4

6

8

10 12 Time (t)

14

16

18

20

1.5 x2(t) x^2(t) with H∞criterion x^ (t) with W criterion

1

2

1,2

Magnitude

0.5

0

−0.5

−1

−1.5

(B)

0

2

4

6

8

12 10 Time (t)

14

16

18

20

Unknown input u Estimated input with H ∞ criterion

1.5

Estimated input with W1,2 criterion

Magnitude

1

0.5

0

−0.5

−1

(C)

0

2

4

FIG. 1 x1 , x2 and u and their estimates.

6

8

10 12 Time (t)

14

16

18

20

95

96

CHAPTER 4 Time-delay systems with both H∞ and W 1,2 optimality criteria

•

W 1,2 method: ⎡

Π1

Π1d

⎡ ⎤ ⎤ −18.97 0 −7.97 1 = ⎣ −8.86 −10 −7.86⎦ , Π2 = ⎣ 1 ⎦ , 17.73 0 6.73 −1 ⎡ ⎤ −0.28 0 −0.28 = ⎣−2.14 0 −2.14⎦ , Π2d = 03×1 , 0.38 0 0.38

and the optimal value of the disturbance attenuation level is λ1,2 = 1.2520. The simulation results are shown in Fig. 1. By analyzing the results of both methods, we notice that the W1,2 is more robust than H∞ . In addition, it is worth mentioning that the disturbance attenuation levels λ and λ1,2 do not have the same sense.

6 Conclusion We presented in this chapter two new UIO design methods for a class of nonlinear time-delay systems. The nonlinearity of the considered system is assumed to be Lipschitz with respect to its arguments. By use of the mean value theory and a particular Lyapunov-Krasovskii functional, new sufficient LMI conditions were proposed to insure the H∞ and W 1,2 robustness of the proposed observers. Finally, a numerical example is given to illustrate the effectiveness of the proposed theories.

References [1] J.P. Richard, Time-delay systems: an overview of some recent advances and open problems, Automatica 39 (10) (2003) 1667–1694. [2] V.L. Kharitonov, Robust stability analysis of time-delay systems: a survey, Annu. Rev. Control 23 (1999) 185–196. [3] O. Sename, New trends in design of observers for time-delay systems, Kybernetika 37 (4) (2001) 427–458. [4] T. Raff, F. Allgöwer, An EKF-based observer for nonlinear time-delay systems, in: Proceedings of the 2006 American Control Conference, Minneapolis, MN, USA, 2006. [5] O. Hugues-Salas, K. Shore, An extended Kalman filtering approach to nonlinear time-delay systems: application to chaotic secure communications, IEEE Trans. Circ. Syst. I 57 (9) (2010) 2520–2530. [6] O. Sename, Is a mixed design of observer-controllers for time-delay systems interesting?, Asian J. Control 9 (2) (2007) 180–189. [7] W.H. Chen, W.X. Zheng, Delay-dependent robust stabilization for uncertain neutral systems with distributed delays, Automatica 43 (1) (2007) 95–104. [8] A.J. Koshkouei, K.J. Burnham, Discontinuous observers for nonlinear time-delay systems, Int. J. Syst. Sci. 40 (4) (2009) 383–392.

References

[9] E.E. Yaz, A. Azemi, Actuator fault detection and isolation in nonlinear systems using LMI’s and LME’s, in: Proceedings of the American Control Conference, Philadelphia, PA, USA, 1998, pp. 1590–1594. [10] W. Chen, M. Saif, Unknown input observer design for a class of nonlinear systems: an LMI approach, in: Proceedings of the American Control Conference, Minneapolis, MN, USA, 2006, pp. 834–838. [11] A.M. Pertew, H.J. Marquez, Q. Zhao, H∞ synthesis of unknown input observers for non-linear Lipschitz systems, Int. J. Control 78 (15) (2005) 1155–1165. [12] F. Bejarano, L. Fridman, A. Poznyak, Unknown input and state estimation for unobservable systems, SIAM J. Control. Optim. 48 (2) (2009) 1155–1178. [13] J.P. Barbot, D. Boutat, T. Floquetc, An observation algorithm for nonlinear systems with unknown inputs, Automatica 45 (8) (2009) 1970–1974. [14] H. Trinh, M. Aldeen, S. Nahavandi, An observer design procedure for a class of nonlinear time-delay systems, Comput. Electr. Eng. 30 (1) (2004) 61–71. [15] H. Bourles, F. Colledani, W-stability and local input-output stability results, IEEE Trans. Autom. Control 40 (6) (1995) 1102–1108. [16] L. Hassan, A. Zemouche, M. Boutayeb, Robust unknown input observers for nonlinear time-delay systems, SIAM J. Control Optim. 51 (4) (2013) 2735–2752. [17] P. Sahoo, T. Riedel, Mean Value Theorems and Functional Equations, World Scientific, Berlin, Heidelberg, 1998. [18] A. Zemouche, M. Boutayeb, G. Bara, Observers for a class of Lipschitz systems with extension to H∞ performance analysis, Syst. Control Lett. 57 (18) (2008) 18–27. [19] A. Zemouche, M. Boutayeb, A new observer design method for a class of Lipschitz nonlinear discrete-time systems with time-delay. Extension to H∞ performance analysis, in: Proceedings of the 46th IEEE Conference on Decision & Control, New Orleans, LA, USA, 2007, pp. 414–419. [20] A. Alessandri, Input-output stability for optimal estimation problems, Int. Math. Forum 2 (13) (2007) 593–617.

97

CHAPTER

H∞ consensus synthesis of

multiagent systems with nonuniform time-varying input delays: A dynamic IQC approach

5

Chengzhi Yuana , Fen Wub a Department

of Mechanical, Industrial and Systems Engineering, University of Rhode Island, Kingston, RI, United States b Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC, United States

Chapter outline 1 Introduction....................................................................................... 99 2 Preliminaries and problem statement ........................................................ 101 2.1 Notations and graph theory......................................................... 101 2.2 Some basic definitions of IQC ..................................................... 102 2.3 Problem statement .................................................................. 103 3 Main results ...................................................................................... 104 3.1 State-feedback case ................................................................. 105 3.2 Output-feedback case ............................................................... 109 4 An illustrative example ......................................................................... 115 5 Conclusions....................................................................................... 117 Appendix ............................................................................................. 120 References........................................................................................... 121

1 Introduction The problem of consensus control is one of the most important and fundamental issues for multiagent systems (MASs), it specifies the control objective of driving all the agents’ states to reach a certain common agreement. Many practical multiagent coordination tasks can be formulated under the consensus framework, such as rendezvous, flocking, formation control, distributed estimation/filtering, and so on. The past decades have witnessed fruitful research achievements in the field Stability, Control and Application of Time-Delay Systems. https://doi.org/10.1016/B978-0-12-814928-7.00005-6 © 2019 Elsevier Inc. All rights reserved.

99

100

CHAPTER 5 H∞ consensus synthesis of multiagent systems

(see Refs. [1–5] and the references therein). Starting from the seminal work [6], early studies of consensus control mainly focused on first- and second-order MASs [7–9]. These results were later on extended to general networks with linear and nonlinear agent dynamics [10–13]. More recently, the research efforts have been shifting to the multiagent consensus problem subject to various physical constraints and system limitations, such as heterogeneity, modeling uncertainties, dynamically changing interconnection topologies, communication/actuation nonlinearities, and so on. In particular, time delay reflects an inherent feature in practical MASs due to limited capability of sensing, communication, and computation. Knowing that time delay might degrade the system performance or even destroy the system stability, studies have been conducted to investigate stability and performance properties of time-delay MASs [14–16]. In general, current approaches in this field can be classified into two major categories: the frequency-domain approach and the Lyapunov functional approach. The frequency-domain approach typically attempts to determine whether all roots of the characteristic equation lie in the left half-plane, which is only applicable to constant time-delay systems. The core of the Lyapunov functional approach lies in constructing a nontraditional energy-storage function/functional for time-delay closed-loop systems. This approach has been prevailing in the time-delay system analysis and synthesis literature for a long time, and has been demonstrated to be successful for single time-delay systems [17–19]. Nevertheless, due to the complexity of the network interconnected structure, extension of these results to time-delay MASs is nontrivial. Existing distributed delay control schemes are largely limited to networks with constant time delays or uniform time-varying delays, which is deemed too unrealistic. On the other hand, consensus control of MASs with general heterogeneous agent dynamics is another challenging problem in the field. It is worth mentioning the scheme of cooperative output regulation recently proposed by Su and Huang [12], where the classical output regulation technique [20,21] was successfully extended for output synchronization of heterogeneous MASs. However, this technique could be conservative or even infeasible for some consensus control problems, because it is typically required that each agent’s control input has at least the same dimension as of its state in order to guarantee solvability of the associated output regulator equations [22]. Thus, the problem of consensus control of MASs with simultaneous heterogeneity and nonuniform time-varying delays is yet to be adequately addressed, which motivates our current work. In this chapter, we will propose a novel leader-following consensus control approach for weakly heterogeneous MASs with nonuniform time-varying input delays. The agents’ dynamics are considered to be weakly heterogeneous as all the system matrices except the matrix “A” are nonidentical among different agents. The distributed delay stability and stabilization issues will be tackled under the dynamic integral quadratic constraints (IQCs) framework from robust control theory [23]. IQC, as a powerful paradigm in modeling a large number of system nonlinearities and uncertainties, has been demonstrated to be successful for both robustness analysis and robust control designs [23,24]. In particular, for time-delay control systems, its advantages over existing delay control techniques lie in: more accurate performance analysis [25,26] and better flexibility in delay control synthesis [27,28].

2 Preliminaries and problem statement

However, virtually all of the existing results pertain to single time-delay control systems, the power of the dynamic IQC techniques has not been fully exploited for MASs. Few attempts [13,29] of employing dynamic IQCs for interconnected networks have been made recently though, they were mainly focusing on the stability analysis issue, while the distributed delay control problem remains unsolved to date. The contributions of this chapter are in twofold: (i) new distributed delay control protocols with a novel exact-memory structure are proposed under the dynamic IQC framework for consensus control of weakly heterogeneous MASs subject to nonuniform time-varying input delays and local, external disturbance and (ii) the overall distributed consensus control problem is decomposed into N independent H∞ stabilization subproblems, and the associated solvability conditions for both state- and output-feedback controls are formulated as linear matrix inequalities (LMIs), which significantly facilitates the distributed control synthesis via convex optimization. The rest of this chapter is organized as follows. Some preliminary results on graph theory and some basic definitions of IQCs, together with the problem statement, are all given in Section 2. The main results of new IQC-based distributed protocols for both state- and output-feedback cases are presented in Section 3. Section 4 provides a numerical example to illustrate the proposed approach, and the conclusions are drawn in Section 5.

2 Preliminaries and problem statement 2.1 Notations and graph theory Throughout the chapter, we use R and C to stand for the set of real and complex numbers, respectively. R+ stands for the set of positive real numbers. Rm×n (Cm×n ) is the set of real (complex) m × n matrices, and Rn (Cn ) represents the set of real (complex) n × 1 vectors. In and 1n denote the n × n identity matrix and an n-dimensional column vector with all elements being 1, respectively. Sn and Sn+ are used to denote the sets of real symmetric n × n matrices and positive definite matrices, respectively. A block diagonal matrix with matrices X1 , X2 , . . . , Xp on its main diagonal is denoted by diag{X1 , X2 , . . . , Xp }. The notation A ⊗ B represents the Kronecker product of matrices A and B. For a series of column vectors x1 , . . . , xn , col{x1 , . . . , xn } stands for a column vector by stacking them together. Furthermore, we use the symbol  in LMIs to denote entries that follow from symmetry. For two integers k1 < k2 , we denote I[k1 , k2 ] = {k1 , k1 + 1, . . . , k2 }. For s ∈ C, s¯ denotes the complex conjugate of s. For a matrix M ∈ Cm×n , M T denotes its transpose and M ∗ denotes the complex conjugate transpose. The Hermitian operator He{·} is defined as He{M} = M + M T for real matrices. RL∞ denotes the set of rational functions with real coefficients that are proper and have no poles on the imaginary axis. RH∞ is the subset of functions in RL∞ that are analytic in the closed righthalf of the complex plane. RLm×n and RHm×n denote the sets of m × n matrices ∞ ∞ whose elements are in RL∞ and RH∞ , respectively. The para-Hermitian conjugate ∼ ∼ ∗ n of G ∈ RLm×n ∞ , denoted as G , is defined by G (s) := G(−¯s) . For x ∈ C , its

101

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CHAPTER 5 H∞ consensus synthesis of multiagent systems

n is the space of functions u: [0, ∞) → Rn norm is defined as x := (x∗ x)1/2 . L2+  ∞ T 1/2 n , u denotes the satisfying u2 := < ∞. Given u ∈ L2+ T 0 u (t)u(t)dt truncated function uT (t) = u(t) for t ≤ T and uT (t) = 0 otherwise. The extended space, denoted as L2e+ , is the set of functions u such that uT ∈ L2+ for all T ≥ 0. For a square matrix A, λi (A) denotes its ith eigenvalue, and Re(λi (A)) represents the real part of this eigenvalue accordingly. In graph theory, a digraph G = (V, E) consists of a finite set of nodes V = {1, 2, . . . , N} and an edge set E ⊆ V × V. An edge of E from node i to node j is denoted by (i, j), where the nodes i and j are called the parent node and the child node of each other, and the node i is also called a neighbor of the node j. Let Ni denote the subset of V which consists of all the neighbors of the node i. If the digraph G contains a sequence of edges of the form (i1 , i2 ), (i2 , i3 ), . . . , (ik , ik+1 ), then the set {(i1 , i2 ), (i2 , i3 ), . . . , (ik , ik+1 )} is called a path of G from node i1 to node ik+1 and node ik+1 is said to be reachable from node i1 . If i1 = ik+1 , the path is called a cycle. A directed tree is a digraph in which every node has exactly one parent except for one node, called the root, which has no parent and from which every other node is reachable. A digraph Gs = (Vs , Es ) is a subgraph of G = (V, E) if Vs ⊆ V and Es ⊆ E ∩ (Vs × Vs ). A subgraph Gs = (Vs , Es ) of the digraph G = (V, E) is called a directed spanning tree of G if Gs is a directed tree and Vs = V. The diagraph G = (V, E) contains a directed spanning tree if a directed spanning tree is a subgraph of G. Note that the directed graph G contains a directed spanning tree if and only if G has at least one node which can reach every other node. The weighted adjacency matrix of a digraph G is a nonnegative matrix A = [aij ] ∈ RN×N , where aii = 0 and aij > 0 ⇒ (j, i) ∈ E. The Laplacian of a digraph G is denoted by L = [lij ] ∈ RN×N , where  lii = N j=1 aij and lij = −aij if i = j. On the other hand, given a matrix A = [aij ] ∈ RN×N satisfying aii = 0, i ∈ I[1, N] and aij ≥ 0, i, j ∈ I[1, N], we can always define a digraph G such that A is the weighted adjacency matrix of the digraph G. We call G the digraph of A. It is known that at least one eigenvalue of L is at the origin and all nonzero eigenvalues of L have positive real parts. Moreover, according to the Lemma 3.3 of [30], L has one eigenvalue at the origin and all other (N − 1) eigenvalues with positive real parts if and only if the digraph G contains a directed spanning tree.

2.2 Some basic definitions of IQC Some basic definitions of dynamic IQCs are first recalled from the robust control theory, which will be useful for the subsequent developments. (m +m )×(m1 +m2 ) Definition 1 (Seiler [31]). Let Π ∈ RL∞ 1 2 be a proper, rational function, called a “multiplier,” such that Π = Ψ ∼ WΨ with W ∈ Rnz ×nz and Ψ ∈ n ×(m +m ) m1 m2 RH∞z 1 2 . Then, two signals q ∈ L2e+ and p ∈ L2e+ satisfy the IQC defined by the multiplier Π , and (Ψ , W) is a hard IQC factorization of Π if the following inequality holds for all T ≥ 0:  T 0

zT (t)Wz(t)dt ≥ 0

(1)

2 Preliminaries and problem statement

where z ∈ Rnz denotes the filtered   output of Ψ driven by inputs (q, p) with zero initial q m1 → conditions, that is, z = Ψ . Moreover, a bounded, causal operator S: L2e+ p m2 m1 satisfies the IQC defined by Π if condition (1) holds for all q ∈ L2e+ , p = S(q) L2e+ and all T ≥ 0. Note that the factorization of IQC multiplier Π = Ψ ∼ WΨ is not unique but can be computed with state-space methods [31]. Furthermore, it has been demonstrated that a broad class of IQC multipliers possess a hard factorization [23]. More discussions about the hard IQCs as defined earlier can be found in Ref. [31]. The concept of hard IQC, together with the following factorization definition and lemma, is instrumental for using IQCs within the dissipation inequality framework. Definition 2 (Seiler [31]). (Ψ , W) is called a Jm1 ,m2-spectral factorization of Π =  I 0 (m +m )×(m +m ) 1 2 Π ∼ ∈ RL∞ 1 2 if Π = Ψ ∼ WΨ , W = m1 , and Ψ , Ψ −1 ∈ 0 −Im2 (m +m )×(m1 +m2 ) . RH∞ 1 2 Note that with a Jm1 ,m2 -spectral factorization (Ψ , W), Ψ is always square, stable, and minimum phase [31]. (m1 +m2 )×(m1 +m2 ) ∼ be partitioned as Lemma 1 (Seiler   [31]). Let Π = Π ∈ RL∞ Π11 Π12 m1 ×m1 m2 ×m2 and Π22 ∈ RL∞ . Assuming that Π = , where Π11 ∈ RL∞ ∼ Π Π12 22 Π11 (jω) > 0 and Π22 (jω) < 0 for all ω ∈ R ∪ {∞}. Then, Π has a Jm1 ,m2 -spectral factorization (Ψ , W), which is also a hard factorization of Π . IQCs, as a powerful tool for modeling a large variety of nonlinearities, have been demonstrated to be successful for robust stability analysis of various dynamical systems (see, e.g., Refs. [23,25,31]). A thorough review on this topic is out of the scope of this chapter, we refer interested readers to the aforementioned references for more detailed discussions.

2.3 Problem statement In this chapter, we consider a MAS consisting of N agents with weakly heterogeneous dynamics subject to local, external disturbances, and time-varying input delays: x˙ i (t) = Axi (t) + B1,i di (t) + B2,i Di (ui (t)) , yi (t) = Ci xi (t) + Di di (t)

∀i ∈ I[1, N],

(2)

where xi ∈ Rn is the state, yi ∈ Rny,i is the measurement output, di ∈ Rnd,i is the external disturbance, and ui ∈ Rnu,i is the control input for the ith agent. The operator Di (ui (t)) := ui (t−τi (t)) denotes the time-varying delay function with τi (t) specifying the delay amount at time instant t. It is assumed that the nonuniform time-varying input delay τi ’s for all i ∈ I[1, N] satisfy τi ∈ [0, τ¯i ] and τ˙i (t) ≤ ri for all t ≥ 0, where τ¯i ≥ 0 and ri ≥ 0 denote the maximum delay amount and its variation bound, respectively. Note that when ri = 0, all the results developed in this chapter will degenerate to a special case of nonuniform constant time delays. Furthermore, for

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CHAPTER 5 H∞ consensus synthesis of multiagent systems

− t ∈ [−τ¯i , 0], we define ui (t) = u− i (t) for all i ∈ I[1, N] with ui (t) being the initial delay function. The agents are said to have weakly heterogeneous dynamics as all the system matrices except A of Eq. (2) are nonidentical among different agents. Typically, it is assumed that (A, B2,i ) is stabilizable, and (A, Ci ) is detectable for each i ∈ I[1, N]. In this chapter, we are interested in the leader-following consensus problem for the MAS (Eq. 2). Specifically, we consider a (virtual) leader governed by the following linear dynamics:

w(t) ˙ = Aw(t),

(3)

where w ∈ Rn is the state of the leader. The leader-following consensus problem is to seek ui for all i ∈ I[1, N] using local information to render all N agents to follow the leader such that lim ei (t) := xi (t) − w(t) = 0,

t→∞

∀i ∈ I[1, N]

(4)

for any initial conditions xi (0) (i ∈ I[1, N]) and w(0). By labeling the leader with i = 0, we assume that the underlying interconnection graph G for the network consisting of Eqs. (2), (3) is fixed, and contains a directed spanning three with the leader as its root. As such, we can define the adjacent matrix A associated with the graph G as A = [aij ] (i, j ∈ I[0, N]), where for i ∈ I[1, N], ai0 > 0 if and only if the ith agent has access to the leader’s state information w, and all the other elements of A are arbitrary nonnegative numbers satisfying aii = 0 for all i ∈ I[0, N]. Accounting for the effects from both external disturbance and nonuniform control input delays, the objective of this chapter is to design a distributed protocol for Eq. (2), such that the overall networked system achieves consensus with optimal H∞ disturbance attenuation performance, that is, minimizing γ > 0 such that ei 2 < γ di 2 for all i ∈ I[1, N].

3 Main results Prior to presenting the controller structure, a model transformation will be first performed on each agent’s dynamics (2) so as to single out the input delay nonlinearity from the nominal linear dynamics. This will result in a new model in a linear fractional transformation (LFT) form. Specifically, define a delay difference signal pi (t) = Si (ui (t)) := ui (t − τi (t)) − ui (t) for each i ∈ I[1, N], then we are able to rewrite the input-delayed model (2) as the following LFT system: x˙ i (t) = Axi (t) + B1,i di (t) + B2,i pi (t) + B2,i ui (t), yi (t) = Ci xi (t) + Di di (t), pi (t) = Si (ui (t)),

∀i ∈ I[1, N].

We have the following assumption regarding the delay operator Si (·).

(5)

3 Main results

Assumption 1. For each i ∈ I[1, N], Si (·) satisfies a collection of IQCs defined 2nu,i ×2nu,i k mi i , where the multipliers {Πik }m by {Π i }k=1 ∈ RL k=1 can be partitioned as

∞ k k Π12,i Π11,i k of dimension nu,i × nu,i . Each multiplier satisfies with Π11,i k k ∼ (Π12,i ) Π22,i k (jω) > 0 and Π k (jω) < 0 for all ω ∈ R ∪ {∞}. Furthermore, for each Π11,i 22,i k ∈ I[1, mi ], Πik has a Jnu,i ,nu,i -spectral factorization (Ψik , Wik ) in the form of Ψik =  k    k Ψ11,i Ψ12,i Inu,i 0 k and Wi = . 0 −Inu,i 0 Inu,i Assumption 1 does not cause any loss of generality. We follow the same discussions in Refs. [27,32,33] for single time-delay systems. i Under Assumption 1, given any collection of IQC multipliers {Πik }m k=1 for the delay operator Si (·), one can construct a Jnu,i ,nu,i -spectral factorization for each Πik as (Ψik , Wik ) by using the methods of Refs. [31,34]. Then, the state-space realization of i the associated system {Ψik }m k=1 can be described in the following linear time-invariant (LTI) form:



x˙ ψ,i zki

=

Aψ,i

Bψ1,i

k Cψ,i

Dkψ1,i

⎡x

⎤

ψ,i ⎣ ui ⎦ , k Dψ2,i pi

Bψ2,i

(6)

i where xψ,i ∈ Rnψ,i denotes the state vector of the operator {Ψik }m k=1 with xψ,i (0) = 0, zki ∈ Rnz,i for all k ∈ I[1, mi ] with nz,i = 2nu,i are the operator outputs. In particular, Assumption 1 renders the following structure for the output matrices for all k ∈ I[1, mi ] and i ∈ I[1, N]:

k C¯ ψ,i , 0

k = Cψ,i

¯k D ψ1,i , 0

Dkψ1,i =

¯k D ψ2,i , Inu,i

Dkψ2,i =

(7)

k ,D ¯ k , and D ¯ k of compatible dimensions. with C¯ ψ,i ψ1,i ψ2,i Based on this system setup, the following sections will present the new distributed control laws under the state- and output-feedback control scenarios, respectively.

3.1 State-feedback case For state-feedback control, we construct the following distributed protocol for the MAS (5) with Eq. (3): ⎛ w˙ˆ i (t) = Aw ˆ i (t) + β ⎝

N  j=1

⎞ aij (wˆ j (t) − w ˆ i (t)) + ai0 (w(t) − wˆ i (t))⎠ ,

∀i ∈ I[1, N],

(8)

ui (t) = K1,i (wˆ i (t) − xi (t)) + K2,i xψ,i (t) + K3,i pi (t)

where w ˆ i ∈ Rn is the controller state for agent i. β ∈ R+ and K1,i , K2,i , K3,i are controller coefficients for design. To understand the structure of this new distributed

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protocol, the first equation of Eq. (8) is essentially serving as a distributed observer to estimate the leader’s state w. Since not all of the follower agents have access to the leader’s state information, cooperative estimation is necessary. To this end, relative estimation in the form of w ˆj − w ˆ i will be shared among neighboring agents. On the other hand, the local control input ui is generated by feeding back the local agent state xi , the estimated state w ˆ i , the associated IQC-induced system state xψ,i , and the delay difference signal pi . This type of controller structure is motivated from the exact-memory delay control mechanism of Refs. [27,32] for single time-delay systems, which has been demonstrated to be advantageous over other existing delay control techniques. It should be pointed out that xψ,i (t) can be readily computed online using Eq. (6), and pi (t) is accessible using similar methods from Ref. [27] for each agent. Furthermore, different from most existing distributed control methods, no physical plant information (e.g., agent plant state/output signals) is exchanged among neighboring agents. This important property will not only ease distributed control implementation, but also facilitate formulation of associated distributed control synthesis problems into tractable LMI conditions, which will be clarified in the sequel. Remark 1. Note that the control law (8) contains an algebraic loop due to the delay difference term K3,i pi (t). In order to guarantee the control law is implementable, this algebraic loop is required to be well posed, that is, there exists a unique solution ui (t) at any time instant. In fact, this requirement can be translated as the nonsingularity of matrix Inu,i + K3,i . This issue will be discussed and addressed later using the LMI technique. In order to analyze the leader-following consensus problem, we denote the state error between the agent i and the leader as ei = xi − w, and the estimation error as w˜ i = w ˆ i − w. Then, by connecting the controller (8) to the agent’s LFT model (5), and adsorbing the associated IQC dynamics (6), we are able to formulate the local close-loop system in terms of error signals for agent i as ⎛ ˜ i (t) + β ⎝ w˙˜ i (t) = Aw

N 

⎞ aij (w ˜ j (t) − w˜ i (t)) − ai0 w ˜ i (t)⎠ ,

j=1

e˙ i (t) = (A − B2,i K1,i )ei (t) + B2,i K2,i xψ,i (t) + B2,i K1,i w˜ i (t) + (B2,i + B2,i K3,i )pi (t) + B1,i di (t), x˙ ψ,i (t) = −Bψ1,i K1,i ei (t) + (Aψ,i + Bψ1,i K2,i )xψ,i (t)

(9)

+ Bψ1,i K1,i w˜ i (t) + (Bψ2,i + Bψ1,i K3,i )pi (t), k k + Dk K )x (t) zi (t) = −Dkψ1,i K1,i ei (t) + (Cψ,i ψ1,i 2,i ψ,i k k + Dψ1,i K1,i w ˜ i (t) + (Dψ2,i + Dkψ1,i K3,i )pi (t), pi (t) = Si (ui (t)),

∀k ∈ I[1, mi ], i ∈ I[1, N].

The following theorem presents the synthesis conditions for the state-feedback distributed protocol (8) that renders consensus for the overall network with guaranteed H∞ controlled performance.

3 Main results

Theorem 1. Consider the MAS (2), (3) with Assumption 1. If there exist a n+n positive-definite matrices Qi ∈ S+ ψ,i , rectangular matrices Kˆ 12,i ∈ Rnu,i ×(n+nψ,i ) , Kˆ 3,i ∈ Rnu,i ×nu,i , and positive scalars β, ηˆ i , ηˆ ik , γ ∈ R+ for all k ∈ I[1, mi ] and i ∈ I[1, N] such that the following conditions hold:  A He ⎢ 0 ⎢  ⎢ T ⎢ ηˆ i B2,i ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡

  B2,i ˆ Q +    K Bψ1,i 12,i Aψ,i i     mi k T BT B T BTψ2,i + Kˆ 3,i   2,i ψ1,i k=1 (ηˆ i − 2ηˆ i )Inu,i   BT1,i 0 0 −γ Ind,i    In 0 Qi 0 0 −γ In 0





Υˆ51,i

Υˆ52,i 



0

Υˆ51,i

 ⎢ ⎢ Υˆ52,i = ⎢ ⎣

0

  −Υˆ55,i

(10)

 > 0, ηˆ i Inu,i

Re(λj (A) − βλi (H)) < 0,

⎡

0



⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ < 0, ⎥ ⎥ ⎥ ⎥ ⎦



ηˆ i Inu,i Kˆ 3,i

where H = [hij ] ∈ RN×N with hii =

0

⎤ 

N

j=0 aij

(11)

∀j ∈ I[1, n],

(12)

and hij = −aij for any i = j, and

 1 ¯ 1 Kˆ 12,i C¯ ψ,i Qi + D ψ1,i ..  . m ¯ mi Kˆ 12,i C¯ ψ,ii Qi + D ψ1,i

⎤ ¯ 1 Kˆ 3,i ¯1 +D ηˆ i D ψ2,i ψ1,i ⎥ ⎥ .. ⎥, . ⎦ m m i i ¯ ¯ ηˆ i D +D Kˆ 3,i ψ2,i

ψ1,i

m Υˆ55,i = diag{ηˆ i1 Inu,i , . . . , ηˆ i i Inu,i }.

Then the state-feedback distributed protocol (8) renders the MAS (2), (3) consensus with an H level γ . Moreover, the controller coefficient matrices are  ∞ performance  −1 ˆ given by −K1,i K2,i = Kˆ 12,i Q−1 i and K3,i = ηˆ i K3,i , and the resulting distributed control law (8) is well posed. Proof. By examining the local closed-loop system (9), it is clearly seen that only the observer dynamics (first equation of Eq. 9) involves coupling among different agents, and this dynamics is autonomous without being affected by any other signals within the same system. This important property allows us to separate the overall stability analysis into two parts: the distributed observer and the rest. As such, by ˜ = col{w introducing w ˜ 1, w ˜ 2, . . . , w ˜ N }, we have ˙˜ ˜ w(t) = ((IN ⊗ A) − β(H ⊗ In ))w(t).

(13)

Then, based on condition (12), it is ensured that (IN ⊗ A) − β(H ⊗ In ) is Hurwitz, which further implies w ˜ i → 0 exponentially as t → ∞. Since the rest part of dynamics (9) involves only local information from the agent itself, and in light of the definitions of ei and the consensus performance in Eq. (4),

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CHAPTER 5 H∞ consensus synthesis of multiagent systems

exponential stability of w ˜ i will reduce the overall network’s consensus problem to the H∞ stabilization problems of N independent LFT systems in the form of e˙ i (t) = (A − B2,i K1,i )ei (t) + B2,i K2,i xψ,i (t) + (B2,i + B2,i K3,i )pi (t) + B1,i di (t), x˙ ψ,i (t) = −Bψ1,i K1,i ei (t) + (Aψ,i + Bψ1,i K2,i )xψ,i (t) + (Bψ2,i + Bψ1,i K3,i )pi (t), k + Dk K )x (t) + (Dk k zki (t) = −Dkψ1,i K1,i ei (t) + (Cψ,i ψ1,i 2,i ψ,i ψ2,i + Dψ1,i K3,i )pi (t),

pi (t) = Si (ui (t)),

(14)

∀k ∈ I[1, mi ], i ∈ I[1, N].

According to Lemma 2 in “Appendix” section, for each i ∈ I[1, N], the previous system with time-varying input delays and external disturbance is stable and achieves an L2 gain less than γ (i.e., ei 2 < γ di 2 ) if there exist positive scaling factors ηik n+n for all k ∈ I[1, mi ] and a matrix Pi ∈ S+ ψ,i such that ⎡

⎤

⎡ T ⎤ ⎥ 1 Ccl2,i   ⎥ ⎣ 0 ⎦ Ccl2,i 0 0 0  ⎦+ γ 0 0 −γ Ind,i ⎡ k ⎤ (Ccl1,i )T mi    ⎢ k ⎥ k k Dkcl11,i Dkcl12,i < 0, ηi ⎣(Dcl11,i )T ⎦ Wik Ccl1,i + k=1 (Dkcl12,i )T

He{PAcl,i }

⎢ T ⎢ B P ⎣ cl1,i i T Bcl2,i Pi





∀i ∈ I[1, N]

(15)

where ⎡

⎤ Bcl1,i Bcl2,i Dcl11,i Dcl12,i ⎦ 0 0 ⎡         B2,i  B2,i B2,i A 0 −K K + + K 1,i 2,i ⎢ 0 Aψ,i Bψ1,i Bψ2,i Bψ1,i 3,i ⎢   =⎢   ⎢ 0 Ck k Dkψ2,i + Dkψ1,i K3,i ⎣ ψ,i +Dψ1,i −K1,i K2,i In 0 0   0 I Wik = nu,i , ∀k ∈ I[1, mi ], i ∈ I[1, N]. 0 −Inu,i

Acl,i ⎣Ccl1,i Ccl2,i

⎤ B1,i 0 ⎥ ⎥ ⎥, ⎥ 0 ⎦ 0



Under Assumption 1, we substitute the matrices of Eq. (7) into Eq. (15) to yield the following condition: ⎡

He{Pi Acl,i }

⎢ T ⎢ Bcl1,i Pi ⎢ ⎢ T ⎢ Bcl2,i Pi ⎢ ⎢ C ⎣ cl2,i Υ51,i

−

mi



k k=1 ηi Inu,i









0

−γ Ind,i



0

0

−γ In

Υ52,i

0

0



⎤

⎥  ⎥ ⎥ ⎥  ⎥ < 0, ⎥  ⎥ ⎦

Υ55,i

∀i ∈ I[1, N]

(16)

3 Main results

where ⎡ 1 ⎤ C¯ cl1,i ⎢ . ⎥ ⎥ Υ51,i = ⎢ ⎣ .. ⎦ , m i C¯ cl1,i

Then, denote ηˆ ik =

1 ηik

⎡ 1 ⎤ ¯ D cl11,i ⎢ . ⎥ ⎥ Υ52,i = ⎢ ⎣ .. ⎦ , m ¯ i D cl11,i

 1 Υ55,i = diag Inu,i , . . . , mi Inu,i . ηi ηi1 

1

and use the fact that −(ηˆ ik )−1 ≤ −ηˆ i−2 (2ηˆ i − ηˆ ik ) for all

k ∈ I[1, mi ] and any positive scalar ηˆ i , it suffices to prove ⎡

He{Pi Acl,i }

⎢ T ⎢ B ⎢ cl1,i Pi ⎢ T ⎢ Bcl2,i Pi ⎢ ⎢ C cl2,i ⎣ Υ51,i

−

mi











0

−γ Ind,i



0

0

−γ In

Υ52,i

0

0

−2 k k=1 ηˆ i (ηˆ i − 2ηˆ i )Inu,i



⎤

⎥  ⎥ ⎥ ⎥  ⎥ < 0, ⎥  ⎥ ⎦

∀i ∈ I[1, N]

Υˆ55,i

(17)

with Υˆ55,i = diag{ηˆ i1 Inu,i , . . . , ηˆ imi Inu,i }. Consequently, let Qi = P−1 for all i ∈ i I[1, N], and multiplying matrix diag{Qi , ηˆ i Inu,i , Ind,i , In , Imi nu,i } to the right and its transpose from the left  of inequality (17), we arrive at condition (10) by denoting Kˆ 12,i = −K1,i K2,i and Kˆ 3,i = ηˆ i K3,i . Condition (11) is used to guarantee the resulting control law is well posed. Specifically, it implies via Schur complement T ηˆ −1 K ˆ 3,i − ηˆ i Inu,i < 0, which is equivalent to K T K3,i < Inu,i . This ensures that Kˆ 3,i i 3,i invertibility of Inu,i + K3,i , in turn guaranteeing implementability of Eq. (8) as discussed in Remark 1. This ends the proof. Remark 2. Note that the time-delay bounds τ¯i and ri for all i ∈ I[1, N] are not explicitly presented in the synthesis conditions (10)–(12) though, this information is essentially encapsulated by the IQC dynamics (6) and hidden in the synthesis conditions in terms of the associated IQC system matrices. As an important contribution of this work, it is noted that with the proposed distributed protocol structure: (i) the overall consensus problem is decomposed into N independent H∞ control subproblems and (ii) each subproblem is formulated in terms of LMIs, which are convex on all design variables. As a result, the optimal distributed consensus control solution that renders a minimal L2 gain γ can be obtained by solving the following N independent convex optimization problems: min

Qi ,ηˆ i ,ηˆ ik ,Kˆ 12,i ,Kˆ 3,i ,∀k∈I[1,mi ],i∈I[1,N]

s.t.

γ (18)

Eqs. (10)–(12).

3.2 Output-feedback case In this section, we will further develop an output-feedback distributed protocol for leader-following consensus control. In this case, the measurement output yi will

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CHAPTER 5 H∞ consensus synthesis of multiagent systems

replace the role of the state xi for feedback control use. To this end, we construct the following output-feedback distributed protocol for MAS (5) with Eq. (3): ⎛ w˙ˆ i (t) = Awˆ i (t) + β ⎝

N 

⎞ aij (wˆ j (t) − w ˆ i (t)) + ai0 (w(t) − wˆ i (t))⎠

j=1

,

ˆ i (t) − yi (t)) + Bc2,i pi (t) x˙ c,i (t) = Ac,i xc,i (t) + Bc0,i xψ,i (t) + Bc1,i (Ci w

∀i ∈ I[1, N],

ˆ i (t) − yi (t)) + Dc2,i pi (t) ui (t) = Cc,i xc,i (t) + Dc0,i xψ,i (t) + Dc1,i (Ci w (19)

where xc,i ∈ Rn together with w ˆ i constitutes the output-feedback controller state. The first equation of Eq. (19) serving as a distributed observer has the same structure as in the state-feedback case. Ac,i , Bc0,i , Bc1,i , Cc,i , Dc0,i , Dc1,i , Dc2,i , and β of compatible dimensions are free variables subject to design. The corresponding closed-loop system for agent i under such an output-feedback control law can be obtained by connecting Eqs. (19)–(5) with Eq. (6), that is, ⎛ w˙˜ i (t) = Aw˜ i (t) + β ⎝

N 

⎞ aij (w ˜ j (t) − w ˜ i (t)) − ai0 w ˜ i (t)⎠ ,

j=1

⎡

⎤ ⎡ Acl,i x˙ cl,i (t) k ⎣ z (t) ⎦ = ⎣Ck cl1,i i ei (t) Ccl2,i pi (t) = Si (ui (t)),

Bcl1,i Dkcl11,i 0

Bcl2,i Dkcl12,i 0

⎤ ⎡ ⎤ Bcl3,i ⎢xcl,i (t)⎥ pi (t) ⎥ Dkcl13,i ⎦ ⎢ ⎣ di (t) ⎦ , 0 w ˜ (t)

(20)

i

∀k ∈ I[1, mi ], i ∈ I[1, N],

where xcl,i = col{ei , xψ,i , xc,i }, w ˜ i = wˆ i − w, and ei = xi − w for all k ∈ I[1, mi ], i ∈ I[1, N]. The closed-loop system matrices are specified as follows: ⎡

     ⎤  Ci 0 B2,i  B2,i −Dc1,i Dc0,i Cc,i ⎥ 0 Inψ,i Bψ1,i Bψ1,i Aψ,i ⎥,   ⎦   Ci 0 −Bc1,i Bc0,i Ac,i 0 Inψ,i       ⎡ ⎡ ⎤ ⎤ B2,i B2,i B2,i B1,i − + D D D i Bψ1,i c2,i ⎦ , Bcl2,i = ⎣ 0 Bψ1,i c1,i ⎦ , = ⎣ Bψ2,i Bc2,i −Bc1,i Di  ⎡ ⎤ B2,i D C = ⎣ Bψ1,i c1,i i ⎦ , Bc1,i Ci        Ci 0 k Dkψ1,i Cc,i , + Dkψ1,i −Dc1,i Dc0,i = 0 Cψ,i 0 Inψ,i    0 , = In 0

A ⎢ 0 ⎢ Acl,i = ⎣

Bcl1,i

Bcl3,i k Ccl1,i

Ccl2,i

0





+

Dkcl11,i = Dkψ2,i + Dkψ1,i Dc2,i ,

Dkcl12,i = −Dkψ1,i Dc1,i Di ,

Dkcl13,i = Dkψ1,i Dc1,i Ci .

(21)

3 Main results

The following theorem provides the control synthesis conditions for outputfeedback distributed consensus control of the input-delayed MAS (2) with Eq. (3) under the IQC and dissipation inequality framework. Theorem 2. Consider the MAS (2), (3) with Assumption 1. If there exist n+n positive-definite matrices Ri ∈ S+ ψ,i , S1,i ∈ Sn+ , rectangular matrices S2,i ∈   Rn×nψ,i , Aˆ c,i ∈ Rn×nψ,i , Bˆ c0,i Bˆ c1,i Bˆ c2,i ∈ Rn×(n+ny,i +nu,i ) , Cˆ c,i ∈ Rnu,i ×nψ,i ,   ˆ c1,i D ˆ c2,i ∈ Rnu,i ×(n+ny,i +nu,i ) , and positive scalars β, ηˆ i , ηˆ ik , γ ∈ R+ for ˆ c0,i D D all k ∈ I[1, mi ] and i ∈ I[1, N] such that condition (12) and the following conditions hold:

⎡

⎤     B2,i  A 0 ˆ c0,i Cˆ c,i      Ri + D Bψ1,i 0 Aψ,i ⎢ ⎥ ⎢ ˆ ˆ   T  T ˆT  T T  ⎥     ⎥ ⎢ Bc0,i Ac,i + A 0 + Ci Dc1,i B2,i Bψ1,i He{S1,i A + Bˆ c1,i Ci } ⎢ ⎥     ⎢ ⎥ mi ˆT BT2,i BTψ1,i ⎢ (ηˆ ik − 2ηˆ i )Inu,i    ⎥< 0, Bˆ Tc2,i ηˆ i BT2,i BTψ2,i + D c2,i k=1 ⎢ ⎥     ⎢ ⎥ ˆT BT2,i BTψ1,i ⎢ BT1,i 0 + DTi D BT1,i S1,i + DTi Bˆ Tc1,i 0 −γ Ind,i   ⎥ c1,i ⎢ ⎥   ⎣ ⎦ In 0 0 −γ In  In 0 Ri ˆ ˆ ˆ ˆ ˆ 0 −Υ66,i Υ62,i Υ63,i Υ64,i Υ61,i 

He

(22)

    Ri  > 0, In 0 S1,i    ηˆ i Inu,i ˆ c2,i ηˆ i Inu,i > 0, D

where

⎡ 

0

Υˆ61,i

 ⎢ ⎢ Υˆ62,i = ⎢ ⎣

0

⎡



Υˆ63,i

  1 ¯1 ˆ C¯ ψ,i Ri + D ψ1,i Dc0,i .. .   m ¯ mi D ˆ c0,i C¯ ψ,ii Ri + D ψ1,i

¯1 D ¯1 +D ˆ ηˆ i D ψ2,i ψ1,i c2,i  ⎢ .. Υˆ64,i = ⎢ ⎣ . ¯ mi D ¯ mi + D ˆ ηˆ i D ψ2,i ψ1,i c2,i

(23) (24)

Cˆ c,i Cˆ c,i





⎤ ¯1 D ˆ c1,i Ci D ψ1,i ⎥ ⎥ .. ⎥, . ⎦ m i ¯ ˆ D ψ1,i Dc1,i Ci

⎤ ¯1 D ˆ D ψ1,i c1,i Di ⎥ .. ⎥, ⎦ . ¯ mi D ˆ c1,i Di D ψ1,i

m Υˆ66,i = diag{ηˆ i1 Inu,i , . . . , ηˆ i i Inu,i }.

Then, the output-feedback distributed protocol (19) renders the MAS (2), (3) consensus with an H∞ performance level γ . Moreover, the controller coefficient matrices can be constructed through the following algorithm for all i ∈ I[1, N]:

  R1,i R2,i R` 1,i R` 2,i −1 • Partition matrices Ri = T , Ri = ` T with R1,i , R` 1,i ∈ Sn+ , R2,i R3,i R R` 3,i 2,i

n R2,i , R` 2,i ∈ Rn×nψ,i , R3,i , R` 3,i ∈ S+ψ,i and let

S3,i = R` 3,i + (R` 2,i − S2,i )T (S1,i − R` 1,i )−1 (R` 2,i − S2,i ).

111

112

CHAPTER 5 H∞ consensus synthesis of multiagent systems



 S1,i S2,i Then, we have Si := T > 0. Note that the matrix S1,i − R` 1,i is S2,i S3,i invertible [35]. T • Solve Ni ∈ R(n+nψ,i )×n through the factorization Si − R−1 i = NI Qi Ni , where n Qi ∈ S+ , and define Mi := −Ri Ni Qi so that Mi , Ni satisfy the identity   M1,i T Si Ri + Ni Mi = In+nψ,i . Furthermore, we partition Mi , Ni as Mi = , M2,i   N Ni = 1,i so that M1,i , N1,i ∈ Rn×n are invertible and M2,i , N2,i ∈ Rnψ,i ×n . N2,i • Compute the controller matrices Ac,i , Bc0,i , Bc1,i , Bc2,i , Cc,i , Dc0,i , Dc1,i , Dc2,i and the scaling factors ηik for all k ∈ I[1, Ni ] and i ∈ I[1, N] via



Dc0,i



Bc0,i

ˆ c1,i , Dc2,i = ηˆ −1 D ˆ c2,i , ηk = (ηˆ k )−1 , ∀k ∈ I[1, mi ] Dc1,i = −D i i i       B −1 ˆ 2,i D , Bc1,i = −N1,i Bc1,i + S1,i S2,i c1,i Bψ1,i         B2,i B2,i −1 ηˆ i−1 Bˆ c2,i − S1,i S2,i − Dc2,i , Bc2,i = N1,i Bψ2,i Bψ1,i  !     ˆ c0,i Cˆ c,i + Dc1,i Ci 0 Ri Ωi−1 , Cc,i = D        A 0 −1 Ac,i = N1,i R Bˆ c0,i Aˆ c,i − S1,i S2,i 0 Aψ,i i        B2,i  Dc0,i Cc,i Ωi − Dc1 Ci 0 Ri + Bψ1,i  "  +N1,i Bc1,i Ci 0 Ri Ωi−1 ,

 where Ωi :=

RT2,i T M1,i

(25)

 R3,i T . M2,i

Proof. The proof can be completed by following the same line of Theorem 1 combined with similar idea from Ref. [27] for single time-delay systems. Specifically, following the same discussions in the proof of Theorem 1, condition (12) guarantees exponential convergence of w ˜ i → 0, and the overall consensus problem boils down to the H∞ stabilization problems of N independent LFT systems in the form of ⎡

⎤ ⎡ Acl,i x˙ cl,i (t) k ⎣ z (t) ⎦ = ⎣Ck cl1,i i ei (t) Ccl2,i pi (t) = Si (ui (t)),

Bcl1,i Dkcl11,i 0

⎤ ⎤⎡ Bcl2,i xcl,i (t) k Dcl12,i ⎦ ⎣ pi (t) ⎦ , di (t) 0

(26)

∀k ∈ I[1, mi ], i ∈ I[1, N],

with the system matrices defined by Eq. (21). Thus, for each i ∈ I[1, N], L2 stability of the previous time-delay system is achieved if there exist positive scalar factors 2n+n ηik for all k ∈ I[1, mi ] and a matrix Pi ∈ S+ ψ,i such that condition (15) holds

3 Main results

for the closed-loop system (26) with Eq. (21). Through similar manipulation as for Theorem 1, we will arrive at the following sufficient condition for Eq. (15): ⎡

He{Pi Acl,i }

⎢ T ⎢ B ⎢ cl1,i Pi ⎢ T ⎢ Bcl2,i Pi ⎢ ⎢ C cl2,i ⎣ Υ51,i

−











0

−γ Ind,i



0

0

−γ In

Υ52,i

Υ53,i

0

mi

−2 k k=1 ηˆ i (ηˆ i − 2ηˆ i )Inu,i



⎤

⎥  ⎥ ⎥ ⎥  ⎥ < 0, ⎥  ⎥ ⎦

∀i ∈ I[1, N]

Υˆ55,i

(27)

with Υˆ55,i = diag{ηˆ i1 Inu,i , . . . , ηˆ imi Inu,i } and ⎡ 1 ⎤ C¯ cl1,i ⎢ . ⎥ ⎥ Υ51,i = ⎢ ⎣ .. ⎦ , mi ¯ Ccl1,i

⎡ 1 ⎤ ¯ D cl11,i ⎢ . ⎥ ⎥ Υ52,i = ⎢ ⎣ .. ⎦ , mi ¯ Dcl11,i

⎡ 1 ⎤ ¯ D cl12,i ⎢ . ⎥ ⎥ Υ53,i = ⎢ ⎣ .. ⎦ . mi ¯ Dcl12,i

To convert the earlier analysis condition into convex LMIs on all the design variables, we partition Pi =

n+nψ,i

with Si ∈ S+

Si NiT

⎡



S1,i ⎢ T S := ⎢ ⎣ 2,i Xi−1 T N1,i Ni

S2,i

N1,i

⎤

⎥ N2,i ⎥ ⎦ Xi−1

S3,i T N2,i

and Xi−1 ∈ Sn+ . Then define

⎡ R Z1,i = ⎣ i MiT

⎡  ⎤ R1,i In ⎢ T ⎢ ⎦ 0 := ⎣ R2,i 0 MT

1,i

R2,i R3,i T M2,i

⎤

⎡

⎥ 0⎥, ⎦ 0

⎢In+nψ,i Z2,i = ⎣ 0

In

⎤ S1,i T ⎥ S2,i ⎦ T N1,i



such that Pi Z1,i = Z2,i and Mi NiT = In+nψ,i − Ri Si , which implies Xi−1 MiT = −NiT Ri .  ⎤ ⎡ In Ri T PZ ⎣ 0 ⎦ > 0, = Based on condition (23), it can be verified that Z1,i i 1,i   In 0 S1,i in turn, Pi > 0 as Z1,i is nonsingular. Then, by multiplying matrix diag{Z1,i , ηˆ i Inu,i , Ind,i , In , Imi nu,i } to the right and its transpose from the left on both sides of inequality (27), we obtain the following results: T P A Z = ZT A Z Z1,i i cl,i 1,i 2,i cl,i 1,i ⎡    B2,i  ˆ A 0 Ri + D ⎢ Bψ1,i  c0,i = ⎣ 0 Aψ,i  Bˆ c0,i Aˆ c,i

Cˆ c,i



 ⎤    B2,i ˆ A + D C Bψ1,i c1,i i ⎥ 0 ⎦, S1,i A + Bˆ c1,i Ci

113

114

CHAPTER 5 H∞ consensus synthesis of multiagent systems

ηˆ i BTcl1,i Pi Z1,i = ηˆ i BTcl1,i Z2,i     ˆ T BT = ηˆ i BT2,i BTψ2,i + D 2,i c2,i BTcl2,i Pi Z1,i = BTcl2,i Z2,i    ˆ T BT2,i = BT1,i 0 + DTi D c1,i    Ccl2,i Z1,i = In 0 Ri In ,

BTψ1,i

BTψ1,i





 Bˆ Tc2,i ,

 BT1,i S1,i + DTi Bˆ Tc1,i ,

and for all k ∈ I[1, mi ] and i ∈ I[1, N], k Z = C¯ cl1,i 1,i

 0

  k ¯k ˆ C¯ ψ,i Ri + D ψ1,i Dc0,i

Cˆ c,i



 ¯k D ˆ D ψ1,i c1,i Ci ,

¯k ˆ ¯k ¯k ηˆ i D cl11,i = ηˆ i Dψ2,i + Dψ1,i Dc2,i ,

where 

        B2,i A 0 Ri + (−Dc1,i Ci 0 Ri + Dc0,i Cc,i Ωi ) Bψ1,i 0 Aψ,i     − N1,i Bc1,i Ci 0 Ri + N1,i Bc0,i Ac,i Ωi ,     B2,i D − N1,i Bc1,i , Bˆ c1,i = − S1,i S2,i Bψ1,i c1,i         ˆBc2,i = ηˆ i S1,i S2,i B2,i + ηˆ i S1,i S2,i B2,i Dc2,i + ηˆ i N1,i Bc2,i , Bψ2,i Bψ1,i      Cˆ c,i = −Dc1,i Ci 0 Ri + Dc0,i Cc,i Ωi ,

Bˆ c0,i Aˆ c,i

 ˆ c0,i D



  = S1,i S2,i

ˆ c1,i = −Dc1,i , D



ˆ c2,i = ηˆ i Dc2,i . D (28)

Consequently, we recover condition (22) by denoting ηˆ ik = (ηik )−1 for all k ∈ I[1, mi ] and i ∈ I[1, N]. The controller formula (25) can be further verified by inverting the relations in Eq. (28). The well posedness of controller (19) can be proved similarly as for Theorem 1 based on condition (24). Similar to the case of state-feedback control, the synthesis conditions (22)–(24) for the output-feedback control are expressed in terms of LMIs. An optimal control solution that renders the input-delayed MAS consensus with a minimal L2 gain can be obtained by solving the following convex optimization problem: min

ˆ c2,i ,ηˆ i ,ηˆ k , ∀k∈I[1,mi ],i∈I[1,N] ˆ c0,i ,D ˆ c1,i D Ri ,S1,i ,S2,i ,Aˆ c,i ,Bˆ c0,i ,Bˆ c1,i ,Bˆ c2,i ,Cˆ c,i ,D i

s.t.

γ (29)

Eqs. (22)–(24).

After obtaining a feasible solution to the earlier optimization problem, the outputfeedback distributed controller can be subsequently constructed using Eq. (25).

4 An illustrative example

4 An illustrative example In this section, a numerical example will be used to illustrate the design procedure and demonstrate the effectiveness of the proposed distributed control approach under both state- and output-feedback control scenarios. Specifically, we consider an inputdelayed MAS described by ⎡

0 −3 ⎢3 0 x˙ i (t) = ⎢ ⎣0 1 0 0  1 0 1 yi (t) = 0 1 0

0 0 0 3

⎡ 0.1 ⎤ ⎡ ⎤ ⎤ −1 0.2i i ⎢ 0.1 ⎥ ⎢ ⎥ 0⎥ ⎥ xi (t) + ⎢ i ⎥ di (t) + ⎢ −i ⎥ Di (ui (t)) ⎣ ⎣ ⎦ ⎦ 0 0.2i⎦ 0.1 , 0 0.5i 0.1    0 0.1 xi (t) + d (t) 1 0.1 i

∀i ∈ I[1, 3].

(30)

The leader dynamics is chosen as ⎡

0 ⎢3 ⎢ w(t) ˙ =⎣ 0 0

−3 0 1 0

0 0 0 3

⎤ −1 0⎥ ⎥ w(t), 0⎦ 0

such that it generates a reference signal w(t) with multiple frequencies at 2.80 and 1.07 rad/s for consensus tracking control. Furthermore, it can be easily verified that Re(λi (A)) = 0, ∀i ∈ I[1, 4], which implies that condition (12) can be satisfied for any positive β > 0. Thus, we set β = 5 in this example. The underlying interconnection network graph G is depicted in Fig. 1, and the associated Laplacian matrix is given by ⎡

0 ⎢−1 L=⎢ ⎣0 0

0 2 −1 0

0 0 1 −1

⎤ 0 −1⎥ ⎥. 0⎦ 1

The nonuniform time-varying input delay functions τi (t) are assumed to satisfy τi ∈ [0, τ¯i ] with τ¯1 = 0.2 s, τ¯2 = τ¯3 = 0.1 s, and τ˙i (t) ≤ ri < 1 with r1 = 0.5, r2 = 0.2, r3 = 0.3. As such, for IQC-based delay control synthesis, we select the following dynamic IQC multipliers from Ref. [25] to characterize the associated time-delay nonlinear operator Si (ui (t)), that is,

FIG. 1 Network graph (node 0 is the leader).

115

116

CHAPTER 5 H∞ consensus synthesis of multiagent systems

 Π1 (s) =

|φ(s)|2 0

 0 , −1

 Π2 (s) =

|ϕ(s)|2 0

 0 , −1

(31)



   τ¯ 2 s2 + c1 τ¯ s τ¯ 2 s2 + c2 τ¯ s where φ(s) = k1 + , ϕ(s) = k2 + δ, τ¯ 2 s2 +√ a1 τ¯ s + k1 c1 τ¯ 2 s2 + a2 τ¯ s + b2 1 with k1 = 1 + √ , a1 = 2k1 c1 , c1 is any positive real number such that c1 < 2k1 , 1−r # √ √ √ 8 k2 = 2−r , a2 = 6.5 + 2b2 , b2 = 50, c2 = 12.5, and , δ are two arbitrarily

small positive numbers. In this example, we select c1 = 10,  = 10−7 , δ = 0.001. It is easy to see that the selected multipliers {Πk }2k=1 satisfy Assumption 1. Thus, applying the Jnu,i ,nu,i -spectral factorization [24,31] to the earlier multipliers, we obtain ⎡

k1 (c1 − a1 ) s − k12 c1 /τ¯ 2 ⎢ τ¯ ⎢ + k1 +  Ψ1 (s) = ⎢ 2 a1 ⎣ s + s + k1 c1 /τ¯ 2 τ¯ 0 ⎡ k2 (c2 − a2 ) s − k2 b2 /τ¯ 2 ⎢ τ¯ ⎢ + k2 + δ a2 Ψ2 (s) = ⎢ ⎣ s2 + s + b2 /τ¯ 2 τ¯ 0

⎤ ⎥ 0⎥ , ⎥ ⎦ 1 ⎤ ⎥ 0⎥ . ⎥ ⎦ 1

(32)

These transfer function matrices can be readily transformed into their respective state-space representations. For illustration purpose, we use different IQCs to characterize nonuniform time-varying input delay nonlinear operators. Specifically, for agents 1 and 2, Ψ1 (s) and Ψ2 (s) will be employed to characterize S1 (u1 (t)) and S2 (u2 (t)), respectively. While for agent 3, both Ψ1 (s) and Ψ2 (s) will be utilized for S3 (u3 (t)). As a result, the associated IQC-induced dynamics for each agent can be expressed in the form of Eq. (6) with the system matrices given by ⎧ ⎡ ⎤   ⎪ 0 1 ⎪ 0 ⎪ ⎪ ⎣ ⎦ a c k A = = , Bψ2,1 = 0, , B ⎪ ψ1,1 ⎨ ψ,1 1 − 1 21 − 1 τ ¯ τ¯   2 ⎪ ⎪ ⎪ c k ⎪ k1 (c1 − a1 ) , D ¯ ψ1,1 = k1 + , D ¯ ψ2,1 = 0, ⎪ ⎩ C¯ ψ,1 = − 1 1 τ¯ τ¯ 2 ⎡ ⎤ ⎧   0 1 ⎪ ⎪ 0 ⎪ ⎣ ⎦ b a ⎪ A = = , Bψ2,2 = 0, , B 2 2 ψ1,2 ⎨ ψ,2 1 − 2 − τ¯ τ¯   ⎪ ⎪ ⎪ ⎪ ¯ ψ1,2 = k2 + δ, D ¯ ψ2,2 = 0, ⎩ C¯ ψ,2 = − b2 k2 k2 (c2 − a2 ) , D τ¯ τ¯ 2 ⎧  T ⎪ A = diag{Aψ,1 , Aψ,2 }, Bψ1,3 = 0 1 0 1 , Bψ2,3 = 0, ⎪ ⎪ ⎨ ψ,3   1 = C ¯ ψ,1 0 0 , D ¯ ¯1 ¯1 C¯ ψ,3 ψ1,3 = Dψ1,1 , Dψ2,3 = 0, ⎪ ⎪   ⎪ ⎩ C¯ 2 = 0 0 C¯ ¯2 ¯ ¯2 ψ,2 , D ψ,3 ψ1,3 = Dψ1,2 , Dψ2,3 = 0.

5 Conclusions

Based on this setup, we first consider the state-feedback case with yi (t) ≡ xi (t) for all i ∈ I[1, 3]. By solving the corresponding LMI optimization problem (18), it yields the minimized L2 gain γ = 0.53 with the controller gains given by 

K1,1

K3,1

K2,2

−3.4575  K3,2

 = 2.7835

 K1,2



= 2.6196

 K1,3





K2,1

K2,3

0.2747  K3,3

5.5077 4.0958

−1.5251 0.3072

−13.4572

418.2070

−4.5609

12.3869

 −0.3879 ,

 −0.2054 ,

 = 1.4144 −0.0027 2.2381 0.0935 5.3937 −0.1868 327.9219 8.7521 −0.1918 .

Similarly for the output-feedback case, solving the LMI optimization problem (29) yields an optimal control solution with the minimized L2 gain γ = 1.2623, which is expectably larger than the state-feedback case. Afterwards, formula (25) can be used to recover the output-feedback controller coefficients. Finally, time-domain simulations are conducted by using the synthesized distributed protocols under the initial conditions of w(0) = [2, −1, 1, 1]T , x1 (0) = [−4, 4, 6, 3]T , x2 (0) = [4, 5, −3, 2]T , x3 (0) = [3, −5, 3, 1]T and all zero initial conditions for the controller states. The delay functions are set to be τi (t) = (τ¯i − ri ) + ri sin(t). The simulation results of each agent’s state trajectories under for the state- and output-feedback cases are plotted in Figs. 2 and 3, respectively. It is seen that perfect consensus performance is indeed achievable under both cases in the presence of nonuniform time-varying input delays. In particular, as indicated by the minimized L2 gain γ values, the statefeedback protocol renders noticeably better consensus performance in terms of faster consensus convergence speed compared to the output-feedback case. Remark 3. Comprehensive comparisons between the IQC-based delay control design approach (adopted in this chapter) and conventional Lyapunov functionalbased approaches (prevailing in current delay control literature) via extensive simulation studies can be found in Ref. [27], which will not be repeated in this chapter for saving space. We refer interested readers to this reference for more details.

5 Conclusions In this chapter, a novel dynamic IQC-based approach has been proposed to address the leader-following consensus problem for MASs with weakly heterogeneous agent dynamics and nonuniform time-varying input delays. Specifically, based on the IQC and dissipation inequality mechanisms, each agent’s time-delay dynamics is first transformed to an equivalent LFT form, such that different dynamic IQCs can be employed to characterize the nonuniform time-varying delay nonlinearities. A novel distributed observer-based exact-memory delay control protocol has been proposed to achieve consensus and optimal H∞ disturbance attenuation performance. Under the proposed scheme, the overall distributed consensus control problem can be

117

118

15 Leader Agent 1 Agent 2 Agent 3

5

Leader Agent 1 Agent 2 Agent 3

10 5

0

0 −5

−5

−10 −10

0

5

10

(A)

15 Time (s)

20

25

30

−15

0

5

10

(B)

15 Time (s)

20

25

30

20

10 Leader Agent 1 Agent 2 Agent 3

5

Leader Agent 1 Agent 2 Agent 3

15 10 5

0

0 −5 −5 −10

0

(C)

5

10

15 Time (s)

20

25

30

−10

0

(D)

5

10

15 Time (s)

20

25

FIG. 2 Simulation results for state-feedback case. (A) First state xi,1 (t); (B) second state xi,2 (t); (C) third state xi,3 (t); (D) fourth state xi,4 (t).

30

CHAPTER 5 H∞ consensus synthesis of multiagent systems

10

10

10 Leader Agent 1 Agent 2 Agent 3

5

Leader Agent 1 Agent 2 Agent 3

5 0

0 −5 −5

−10

−10

0

5

10

(A)

15 Time (s)

20

25

30

−15

0

5

10

(B)

8

15 Time (s)

20

25

30

20 Leader Agent 1 Agent 2 Agent 3

6 4

Leader Agent 1 Agent 2 Agent 3

15 10

2 5 0 0

−2

−5

−4 −6

0

10

15 Time (s)

20

25

30

−10

0

(D)

5

10

15 Time (s)

20

25

FIG. 3 Simulation results for output-feedback case. (A) First state xi,1 (t); (B) second state xi,2 (t); (C) third state xi,3 (t); (D) fourth state xi,4 (t).

30

5 Conclusions

(C)

5

119

120

CHAPTER 5 H∞ consensus synthesis of multiagent systems

decomposed into N independent H∞ stabilization subproblems, and the associated distributed control synthesis conditions for both state- and output-feedback cases are formulated in terms of LMIs, which can be solved effectively via convex optimization. The effectiveness of the proposed approach has been demonstrated through simulation studies. Two important directions for future research along the proposed IQC-based approach include (i) extension to more general MASs with completely heterogeneous agent dynamics and (ii) consideration of other typical types of time-delay effects (e.g., communication/output delays).

Appendix Lemma 2. Consider the following system: x˙ cl (t) = Acl xcl (t) + Bcl1 p(t) + Bcl2 d(t), k x (t) + Dk p(t) + Dk d(t), zk (t) = Ccl1 cl cl11 cl12

∀k ∈ I[1, m],

e(t) = Ccl2 xcl (t) + Dcl21 p(t) + Dcl22 d(t),

(33)

p(t) = S(q(t)),

with xcl ∈ Rn×n . Assume that the nonlinear operator S(·) satisfies multiple IQCs n defined by {(Ψ k , W k )}m k=1 , and if there exist a positive-definite matrix P ∈ S+ , k positive scalars η , γ ∈ R+ for all k ∈ I[1, m] such that ⎡

He{PAcl }





⎤

⎡

T Ccl2

⎤

  1⎢ T ⎥ ⎥ Ccl2 Dcl21 Dcl22  ⎥ D ⎦+ ⎢ γ ⎣ cl21 ⎦ 0 −γ I DTcl22 ⎡ k T⎤ (Ccl1 ) m    ⎢ ⎥ k + ηk ⎣(Dkcl11 )T ⎦ W k Ccl1 Dkcl11 Dkcl12 < 0.

⎢ BT P ⎣ cl1 BTcl2 P

k=1

0

(34)

(Dkcl12 )T

Then, system (33) is stable and achieves an L2 gain less than γ , that is, e2 < γ d2 . T Px for the system (33), then Proof. Define the Lyapunov function V(xcl ) = xcl cl we have T P(A x + B p + B d) V˙ = (Acl xcl + Bcl1 p + Bcl2 d)T Pxcl + xcl cl cl cl1 cl2 ⎡ ⎤ ⎡ ⎤ He{PAcl }   ⎥ xcl ⎢ T  T B P 0 ⎥ ⎣ p ⎦ . = xcl pT dT ⎢ ⎣ cl1 ⎦ d 0 0 BT P cl2

T Then, consider condition (34), multiply vector [xcl condition and its transpose to the left, we obtain

pT

dT ]T from the right of this

References

V˙ +

m  k=1

ηk (zk )T W k zk +

1 T e e − γ dT d < 0. γ

Integrating both sides of the earlier inequality from t = 0 to t = T ≥ 0 with zero initial conditions, we obtain V(xcl (T)) +

m  k=1

ηk

 T 0

(zk )T (t)W k zk (t)dt < γ

 T 0

dT (t)d(t)dt −

 1 T T e (t)e(t)dt. γ 0

According to Definition 1 with the IQC constraint (1), and based on the nonnegativity of the Lyapunov function V, it can be concluded that   T 1 T T e (t)e(t)dt < γ dT (t)d(t)dt, γ 0 0

which implies that system (33) is stable and achieves e2 < γ d2 .

References [1] W. Ren, R.W. Beard, E.M. Atkins, Information consensus in multivehicle cooperative control, IEEE Control Syst. Mag. 27 (2) (2007) 71–82. [2] W. Ren, R.W. Beard, Distributed Consensus in Multi-Vehicle Cooperative Control, Springer-Verlag, London, 2008. [3] M. Mesbahi, M. Egerstedt, Graph Theoretic Methods for Multiagent Networks, Princeton University, Princeton, NJ, 2010. [4] H. Bai, M. Arcak, J. Wen, Cooperative Control Design: A Systematic, Passivity-Based Approach, Springer-Verlag, New York, 2011. [5] Y. Cao, W. Yu, W. Ren, G. Chen, An overview of recent progress in the study of distributed multi-agent coordination, IEEE Trans. Ind. Inf. 9 (1) (2013) 427–438. [6] B.P. Bertsekas, J.N. Tsitsiklis, Parallel and Distributed Computation: Numerical Methods, Prentice Hall, Englewood Cliffs, NJ, 1989. [7] W. Ren, On consensus algorithms for double-integrator dynamics, IEEE Trans. Autom. Control 53 (6) (2008) 1503–1509. [8] X. Liu, W. Lu, T. Chen, Consensus of multi-agent systems with unbounded time-varying delays, IEEE Trans. Autom. Control 55 (10) (2010) 2396–2401. [9] S. Li, H. Du, X. Lin, Finite-time consensus algorithm for multi-agent systems with double-integrator dynamics, Automatica 47 (8) (2011) 1706–1712. [10] L. Scardovi, R. Sepulchre, Synchronization in networks of identical linear systems, Automatica 45 (11) (2009) 2557–2562. [11] J. Lunze, Synchronization of heterogeneous agents, IEEE Trans. Autom. Control 57 (11) (2012) 2885–2890. [12] Y. Su, J. Huang, Cooperative output regulation of linear multi-agent systems, IEEE Trans. Autom. Control 57 (4) (2012) 1062–1066. [13] S.Z. Khong, E. Lovisari, A. Rantzer, A unifying framework for robust synchronization of heterogeneous networks via integral quadratic constraints, IEEE Trans. Autom. Control 61 (2016) 1297–1309.

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[14] E. Nuno, R. Ortega, L. Basanez, D. Hill, Synchronization of networks of nonidentical Euler-Lagrange systems with uncertain parameters and communication delays, IEEE Trans. Autom. Control 56 (4) (2011) 935–941. [15] K. You, L. Xie, Network topology and communication data rate for consensusability of discrete-time multi-agent systems, IEEE Trans. Autom. Control 56 (10) (2011) 2262–2275. [16] C. Yuan, Leader-following consensus control of general linear multi-agent systems with diverse time-varying input delays, J. Dyn. Syst. Meas. Control. 140 (6) (2018) 061010. [17] E. Fridman, Introduction to Time-Delay Systems: Analysis and Control, Birkhauser, 2014. [18] Y.S. Lee, Y.S. Moon, W.H. Kwon, P.G. Park, Delay-dependent robust H∞ control for uncertain systems with a state-delay, Automatica 40 (2004) 65–72. [19] S. Niculescu, K. Gu, Advances in Time-Delay Systems, Springer, 2004. [20] J. Huang, W.J. Rugh, On a nonlinear multivariable servomechanism problem, Automatica 26 (1990) 963–972. [21] A. Saberi, A.A. Stoorvogel, P. Sannuti, Control of Linear System with Regulation and Input Constraints, Springer, Berlin, 1999. [22] Y. Su, J. Huang, Cooperative output regulation with application to multi-agent consensus under switching network, IEEE Trans. Cybern. 42 (3) (2012) 864–875. [23] A. Megretski, A. Rantzer, System analysis via integral quadratic constraints, IEEE Trans. Autom. Control 42 (1997) 819–830. [24] S. Wang, H. Pfifer, P. Seiler, Robust synthesis for linear parameter varying systems using integral quadratic constraints, Automatica 68 (2016) 111–118. [25] C. Kao, A. Rantzer, Stability analysis of systems with uncertain time-varying delays, Automatica 43 (2007) 959–970. [26] C. Kao, On stability of discrete-time LTI systems with varying time delays, IEEE Trans. Autom. Control 57 (5) (2012) 1243–1248. [27] C. Yuan, F. Wu, Exact-memory and memoryless control of linear systems with time– varying input delay using dynamic IQCs, Automatica 77 (2017) 246–253. [28] C. Yuan, F. Wu, H∞ state-feedback control of linear systems with time-varying input delays, in: Proc. IEEE on Decision and Control, Las Vegas, NV, 2016, pp. 586–591. [29] A. Eichler, H. Werner, Improved IQC description to analyze interconnected systems with time-varying time-delays, in: Proc. American Contr. Conf., 2015, pp. 5402–5407. [30] W. Ren, R.W. Beard, Consensus seeking in multi-agent systems under dynamically changing interaction topologies, IEEE Trans. Autom. Control 50 (5) (2005) 655–661. [31] P. Seiler, Stability analysis with dissipation inequalities and integral quadratic constraints, IEEE Trans. Autom. Control 60 (6) (2015) 1704–1709. [32] C. Yuan, F. Wu, Dynamic IQC-based control of uncertain LFT systems with time-varying state delay, IEEE Trans. Cybern. 46 (12) (2016) 3320–3329. [33] C. Yuan, F. Wu, Delay scheduled impulsive control for networked control systems, IEEE Transactions on Control of Network Systems 4 (3) (2017) 587–597. [34] H. Pfifer, P. Seiler, Integral quadratic constraints for delayed nonlinear and parameter– varying systems, Automatica 56 (2015) 36–43. [35] C. Yuan, F. Wu, Hybrid control for switched linear systems with average dwell time, IEEE Trans. Autom. Control 60 (1) (2015) 240–245.

CHAPTER

Delayed feedback control of pitch-flap instabilities in helicopter rotors

6

Rudy Cepeda-Gomez Institute of Automation, University of Rostock, Rostock, Germany

Chapter outline 1 Introduction....................................................................................... 123 2 Dynamics of the rotor ........................................................................... 124 3 Stability analysis of the uncontrolled system ............................................... 126 3.1 Hovering flight: μ = 0............................................................... 127 3.2 Horizontal flight: μ > 0 ............................................................. 128 4 Stability analysis of the controlled system .................................................. 129 4.1 Hovering flight: Control without delay ............................................ 131 4.2 Hovering flight: Delayed control ................................................... 133 4.3 Horizontal flight ...................................................................... 137 5 Concluding remarks ............................................................................. 139 References........................................................................................... 141

1 Introduction The main rotor is the principal source of vibrations in a helicopter and the reduction of this vibration is important to increase the life of the components of the airframe as well as the comfort of passengers and crew. A considerable research effort has been, therefore, devoted to the study of active control systems for helicopter rotors, in order to guarantee the stability of its motion. Most works on this topic fall in one of two categories: higher harmonic control (HHC) or individual blade control (IBC). Works in the HHC category, like Refs. [1–3] consider the control forces to be applied to the swashplate by means of actuators. The IBC category [4–8] considers that each blade can be actuated independently. The main difference between the two approaches is the number of available degrees of freedom for the control action. Many control algorithms, nevertheless, can be implemented with either actuation technique. Friedman [9] compared several different control methods in both categories. A more recent survey of the developments in these topics is the work [10]. Stability, Control and Application of Time-Delay Systems. https://doi.org/10.1016/B978-0-12-814928-7.00006-8 © 2019 Elsevier Inc. All rights reserved.

123

124

CHAPTER 6 Pitch-flap instabilities in helicopter rotors

Faragher [11,12] introduced a control law for the prevention of pitch-flap flutter which can be implemented in both HHC and IBC frameworks. This control logic uses delayed feedback to improve the stability of the motion of the helicopter rotor. The inspiration for this control strategy was the previous use of delayed feedback for the stabilization of periodic motion [13] and the suppression of vibrations by means of delayed resonators [14]. The original control action proposed by Faragher [11], with a fixed delay equal to the period of the motion, does not increase the stability region of the system. On the contrary, it creates instabilities for parametric combinations which were stable before its introduction [15]. Furthermore, this logic uses both the delayed and the nondelayed states in the feedback law, a proposition which does not make much sense from a practical point of view, considering that in reality it is not possible to process the state without incurring in some delays. We revisit the idea of using delayed feedback to increase the pitch-flap stability region in the parametric domain for a helicopter rotor. A control action using only delayed feedback, which uses the principle of delay scheduling [16], is applied to a helicopter rotor model and analyzed in the parametric domain. The effect of this logic is studied for both hovering and horizontal flight. Using results obtained from numerical analysis techniques, we show the advantages and disadvantages of the proposed logic. For the case of hovering flight, in which the dynamics of the rotor are described by a linear time-invariant (LTI) model, the cluster treatment of characteristic roots (CTCR) paradigm [17,18] is used to analyze the stability of the system with respect to the delays. In horizontal flight, a rotor is described by a linear time-periodic (LTP) mathematical model, and we study its stability with respect to the delay using the semidiscretization method [19,20].

2 Dynamics of the rotor A helicopter rotor blade is usually mounted on a set of hinges which allow three angular degrees of freedom. A typical arrangement of the hinges is shown in Fig. 1. The flapping motion is defined as an up and down rotation, through an angle β, in a plane which contains both the blade and the shaft. This angle is considered positive when the blade flaps upwards. A flapping blade rotating at high speeds is subject to large Coriolis moments in the plane of rotation, and a hinge is introduced to alleviate these moments, allowing the motion of the blade in the same plane of rotation. This is denoted as lead-lag motion, and the lagging angle is denoted by ξ . The pitching or feathering motion, denoted by θ, is a rotation about an axis parallel to the blade span. The angle is considered positive when the leading edge of the blade moves upwards. The pitch-flap flutter is an unstable behavior produced by the coupling of the pitching and flapping motions of the rotor. It can lead to very high oscillatory loads in the pitch control circuit. Stammers [21] presented a detailed model to describe the coupling between θ and β for an uncontrolled rotor moving forward with a horizontal speed Vf . The basic assumptions made to obtain this model are that no lagging motion occurs, that the flapping hinge is horizontal at the blade root and

2 Dynamics of the rotor

FIG. 1 Typical hinge arrangement and degrees of freedom of a helicopter rotor blade. Wikimedia Commons.

on the axis of rotation, that the flexural axis of the nonrotating blade lies along the quarter-chord line, and that the blade is of constant chord and symmetrical crosssection. Stammer’s model has the form       θ¨ (ψ) θ˙ (ψ) θ(ψ) M ¨ + C(ψ) ˙ + K(ψ) =0 β(ψ) β(ψ) β(ψ)

(1)

with the inertia M, damping C, and stiffness K matrices defined as  1 M= 0 ⎡

r σ k2



, 1  nmθ˙ 1 + 43 μ sin(ψ) 2

C(ψ) = ⎣ 8k ⎡

0 ν12

⎤ 0

 ⎦ , nlz˙ 1 + 43 μ sin(ψ) r σ k2

⎤

⎦. K(ψ) = ⎣  4 nlθ 5 + 2μ sin(ψ) + 23 μ2 (1 − cos(2ψ)) 1 + 43 nlz˙ μ cos(ψ) + nlz˙ μ2 sin(2ψ) (2)

As it is customary in helicopter dynamics, the independent variable in Eqs. (1), (2) is not the time t, but the azimuth angle ψ = Ωt. Here, Ω is the angular speed

125

126

CHAPTER 6 Pitch-flap instabilities in helicopter rotors

Table 1 Parameters of the dynamic model. r k n mθ˙ lθ lz˙

Description

Value

Blade aspect ratio Nondimensional radius of gyration of chordwise element about its flexural axis 1 8 Lock number Aerodynamic derivative coefficient Aerodynamic derivative coefficient Aerodynamic derivative coefficient

14.14 √

0.05

1.6667 0.5 0.5 0.5

of the rotor shaft, which is assumed constant during the operation of the aircraft. In Eq. (1), a dot over a variable denotes its derivative with respect to ψ. The parameter μ in Eq. (2) is known as the tip speed ratio, and it corresponds to the ratio of the horizontal speed of helicopter to that of the tip of an advancing blade μ=

Vf , ΩR

(3)

with R being the blade span. For a helicopter in hover or vertical flight μ = 0, whereas μ > 0 for any horizontal flight condition. Among the parameters presented in Eq. (2), two are easily varied and thus the stability is studied with respect to them. These parameters are σ , which is the distance of the center of gravity (c.g.) of the blade aft from the center of pressure, expressed as a fraction of the chord; and the nonrotating torsional natural frequency ν1 . The other parameters, described in Table 1, are taken as fixed. Notice that the ν1 is given as a fraction of Ω. Table 1 also shows the numeric values used for the fixed parameters in the numerical analysis sections. These values are taken from Ref. [11]. By means of two sticks in the cockpit, the pilot can exert a control moment which directly affects the feathering motion of the rotor. The uncontrolled model (1) is then expanded using an input signal u(ψ), which corresponds to the motion of the lower end of an actuation rod. Following again Ref. [11], the controlled model becomes        2 θ¨ (ψ) θ˙ (ψ) θ(ψ) ν M ¨ + C(ψ) ˙ + K(ψ) = Bu(ψ) = 1 u(ψ). β(ψ) β(ψ) β(ψ) 0

3 Stability analysis of the uncontrolled system With the state vector x(ψ) = θ space as the fourth-order system

β

θ˙

(4)

T β˙ , Eq. (1) can be represented in state

3 Stability analysis of the uncontrolled system

x˙ (ψ) = Au (ψ)x(ψ)  022 = −M −1 K(ψ)

 I22 x(ψ), −M −1 C(ψ)

(5)

with 022 being a 2 × 2 matrix with all elements equal to zero and I22 the identity matrix.

3.1 Hovering flight: μ = 0 For hovering or vertical flight, the advance ratio of the helicopter is equal to zero. Under this condition, the damping and stiffness matrices in Eq. (2) lose their dependence on ψ and the LTP dynamic model (5) becomes the LTI system  I22 x(ψ), −M −1 Ch

 x˙ (ψ) = Auh x(ψ) =

with

 nm Ch =

θ˙ 8k2

0

022 −M −1 Kh

 0 , nlz˙

 Kh =

ν12

4 nl 5 θ

r σ k2

1

(6)

 .

If any eigenvalues of the matrix Auh have positive real parts, their locations define the type of instability that appears. When Auh has a purely real unstable root, the unstable mode is known as pitch divergence and it is evidenced by a steady increase in the pitch and flap angles. When the unstable roots are complex conjugate, the oscillatory instability known as pitch-flap flutter appears. The boundary at which the transition from stability to pitch divergence occurs is given by the presence of a characteristic root at the origin of the complex plane. A sufficient condition for Auh to have an eigenvalue at the origin is that the determinant of the stiffness matrix Kh is equal to zero   ν2  |Kh | =  4 1  nlθ 5



r σ  k2  = 0,

1 

(7)

from where we obtain ν12 =

4r σ nlθ . 5k2

(8)

To establish the boundary between stability and flutter, it is necessary to find a conjugated pair of purely imaginary characteristic roots at a frequency ωf . This is achieved by solving the characteristic equation |jωf I44 − Auh | = 0, leading to the following two equations in terms of the crossing frequency:   m˙ m˙ ω2 − 2θ ν12 = 1 + 2θ 8k lz˙ 8k lz˙ 5mθ˙ (ω4 + (lz2˙ n2 − 2)ω2 + 1) σ = . 32(r nlz˙ lθ )(ω2 − 1)

(9a) (9b)

127

CHAPTER 6 Pitch-flap instabilities in helicopter rotors

20

15

n12

128

II

10 III 5 I 0

0

0.02

0.04 s

0.06

0.08

FIG. 2 Pitch-flap stability boundaries for an uncontrolled system in hovering flight. The not-shaded region represents stable parametric combinations. The shaded regions indicate pitch divergence (I), pitch-flap flutter (II), and both (III) unstable behaviors.

For the parametric values given in Table 1, the two stability boundaries are presented in Fig. 2. The dashed line depicts the boundary at which an unstable pole introduces pitch divergence, whereas the solid line indicates a change in stability to the pitch-flap flutter. These two lines divide the parametric space in four zones. The region without shading corresponds to stable parametric selections. The lightshaded zone at the bottom of the plot (zone I) indicates the presence of a pitch divergence mode. The darker-shaded region at the top right (zone II) indicates pitchflap flutter behavior, and the darkest region to the right (zone III) corresponds to the worst possible conditions, when both unstable modes are present.

3.2 Horizontal flight: μ > 0 Stammers [21] has shown that forward flight is detrimental for the pitch divergence and beneficial for the flutter. This means that for a given value of ν12 , divergence appears for values of σ smaller than in the hovering case. The flutter boundary, on the contrary, moves to the right in the (σ , ν12 ) plane when μ > 0. These results are based on an approximate analysis made necessary by the particularities of the system. For μ > 0, the system (5) is linear time periodic, and its stability analysis requires the application of the Floquet theory [22]. For the system x˙ (ψ) = Au (ψ)x(ψ), with Au (ψ + T) = Au (ψ) and an initial condition x(0), a solution of the form x(ψ) = Φ(ψ)x(0) exists. The fundamental matrix Φ(ψ) can be factorized into the product of a periodic and a constant matrix Φ(ψ) = P(ψ)eBψ , where P(ψ) = P(ψ + T) and P(0) = I. The matrix Φ(T) = eBT is known as the monodromy matrix of the system, and its eigenvalues λi are known as the characteristic multipliers of the

4 Stability analysis of the controlled system

system. The necessary and sufficient stability condition for an LTP system is that all its characteristic multipliers must lie inside the unit circle, that is, |λi | < 1. The main difficulty in the analysis of LTP systems is that, in general, the monodromy matrix cannot be determined in closed form. There are several methods which allow an approximation of Φ(T). The approach used in this work consists in d the numerical solution of the differential equation dψ Φ(ψ) = Au (ψ)Φ(ψ) in the interval [0, 2π ], subject to the initial condition Φ(0) = I44 . This allows a numerical determination of the characteristic multipliers of the system for each (σ , ν12 ) selection given a certain value of μ. By observing the eigenvalues of Φ, we determine the stability of each point. The transition from stability to divergence occurs when a real characteristic multiplier crosses the unit circle at λ = 1. When a pair of complex conjugate eigenvalues cross the unit circle, flutter appears. These two cases are similar to what was analyzed in the previous section. An extra stability transition can occur in the case of forward flight, and it appears when a real characteristic multiplier crosses the unit circle at λ = −1. This case is known as a period doubling bifurcation, which results in flutter characterized by oscillations at frequencies which are half-integer multiples of the main rotor angular frequency Ω. This situation corresponds to the transition boundaries labeled as P = .5 in Ref. [21]. Fig. 3 shows the stability regions of the system for different values of μ.

4 Stability analysis of the controlled system Faragher [11,12] presented a method to increase the size of the stable region in Fig. 2 by means of delayed feedback. Based on the work of Pyragas [13], the control law ˙ ˙ − τ )), u(ψ) = a(β(ψ) − β(ψ − τ )) + b(β(ψ) − β(ψ

(10)

was introduced, with τ = 2π . The idea behind Eq. (10) is to use as error signal the difference between the current state and its value one period earlier. the claim in Refs. [11,12] is that making the delay exactly equal to the period of the motion greatly increases the region of asymptotic stability of the uncontrolled system. It was already shown [15] that this control law does not work as intended and that this particular selection of the delay value actually introduces instability into the system. Furthermore, given that any practical feedback loops always have delays due to noninstantaneous sensing and processing activities, this control law is not practical ˙ because the actual state (β(ψ), β(ψ)) cannot be processed instantaneously. We propose the control law ˙ − τ ), u(ψ) = −aβ(ψ − τ ) − bβ(ψ

(11)

which uses only delayed state feedback. Here, a and b are control gains to be set by the user. Under Eq. (11), the controlled system (4) can be represented in the state space as x˙ (ψ) = Au (ψ)x(ψ) + Ad x(ψ − τ ),

(12)

129

12

12

10

10

n12

14

n12

14

8

8

6

6

4

4

2

2

0

0

0.01

0.02

0.03

0.04

(A)

s

0.05

0.06

0.07

0.08

0

0

0.01

0.02

0.03

(B)

0.04

s

0.05

0.06

0.07

0.08

16 14 12

n12

10 8 6 4 2 0

(C)

0

0.01

0.02

0.03

0.04

s

0.05

0.06

0.07

0.08

FIG. 3 Pitch-flap stability boundaries for an uncontrolled system in horizontal flight for different values of the tip speed ratio μ. The solid line is the flutter boundary, the dashed line is the pitch divergence boundary, the dotted line is the P = .5 flutter boundary. (A) μ = 0.1; (B) μ = 0.2; (C) μ = 0.3.

CHAPTER 6 Pitch-flap instabilities in helicopter rotors

16

130

16

4 Stability analysis of the controlled system

with

⎡

0 ⎢0 Ad = ⎢ ⎣0 0

0 0 −ν12 a 0

0 0 0 0

⎤ 0 0 ⎥ ⎥. −ν12 b⎦ 0

The aim of the feedback law (11) is to improve the stability of the system, that is, to increase the stable operating zone of Fig. 2, by stabilizing parametric combinations which normally result in unstable operation. In the following paragraphs, we analyze the effect that each one of the control gains and the delay have in the stability boundaries of the system.

4.1 Hovering flight: Control without delay Let us first consider the case μ = 0. In this case, Eq. (12) becomes the time invariant model x˙ (ψ) = Auh x(ψ) + Ad x(ψ − τ ).

(13)

Since Eq. (13) is a retarded system, making it stable for τ = 0 will also guarantee its stability for small values of the delay. With this premise in mind, we would like to select the control gains a and b such that the largest possible stable operating region for τ = 0 is achieved. Following an analysis similar to the one presented in the previous section, we can determine the boundary between stability and pitch divergence for Eq. (4) by equating to zero the determinant of the equivalent stiffness matrix   ν2  |Kch | =  4 1  nlθ 5



r  σ + ν 2 a 1  = 0, k2

1



(14)

from where we get 4r σ nlθ . ν12 = 2 k (5 − 4anlθ )

(15)

When comparing Eqs. (15) to (8), we notice that selecting positive values for a makes the slope of the stability boundary to increase, reducing the stable operating region in the parametric domain (σ , ν12 ). Selecting a < 0, on the contrary, reduces the slope of the stability line leading to the desired result. In order to find the boundary between stability and flutter, we look for solutions to the characteristic equation |jωf I44 − Auh − Ad | = 0. This leads to the equations ν12 =

1 (5lz˙ − 4blθ )

   5m ˙ 5m ˙ 5lz˙ + 2θ ω2 − 2θ , 8k 8k

(16a)

σ = {5(32bk2 lθ + 5mθ˙ )ω4 + 5[(32alz˙ lθ n − 32blθ )k2 + (−4blz˙ lθ mθ˙ + 5lz˙ 2 mθ˙ )n2 + 4anlθ mθ˙ − 10mθ˙ ]ω2 − 4anlθ mθ˙ + 5mθ˙ }/{32(ω2 − 1)(5lz˙ − 4blθ )rnlθ }.

(16b)

131

CHAPTER 6 Pitch-flap instabilities in helicopter rotors

20

20 a = −1 a = −0.5 a=0 a = 0.5 a =1

10

10

5

0

(A)

b = −1 b = −0.5 b=0 b = 0.5 b =1

15

n12

2

15

n1

5

0

0.02

0.04

s

0.06

0.08

0.1

0

(B)

0

0.02

0.04

s

0.06

0.08

0.1

FIG. 4 Effect of the control gains in the flutter stability boundary. (A) Boundary for different values of a keeping b = 0. (B) Boundary for different values of b keeping a = 0. 20

15

II n12

132

10

III

5

0

I 0

0.02

0.04 s

0.06

0.08

FIG. 5 Pitch-flap stability boundaries for a controlled system in hovering flight with a = −0.25 and b = 1. The thin lines represent the boundaries for the uncontrolled system, for comparison. The not-shaded region represents stable parametric combinations of the controlled system. The shaded regions indicate pitch divergence (I), pitch-flap flutter (II), and both (III) unstable behaviors.

The effect of the control parameters on the flutter boundary is difficult to appreciate from Eq. (16). Fig. 4 shows how, for the parametric values used in this study, increasing either parameter while keeping the other constant moves the stability line to the right. Since increasing a reduces the robustness against pitch divergence, a compromise between the two control actions should be reached. By selecting a = −0.25 and b = 1, we obtained an increase in the stable operating region, as shown in Fig. 5.

[rad]

4 Stability analysis of the controlled system

0.1 0 −0.1 −0.2 −0.3

0

5

0

5

b

q

25

30

25

30

[rad]

0.01 0 −0.01

10

15 ψ [rad]

20

FIG. 6 Simulation results for (σ , ν12 ) = (0.05, 9) without control (top) and using the control law (11) with (a, b) = (−0.25, 1) and τ = 0 (bottom).

To show the effect of the control action, we simulated the response of the system to a perturbation 0.01 radians in the flap angle β, considering the operating point (σ , ν12 ) = (0.05, 9). Fig. 6 shows the results of the simulations for both the uncontrolled (top) and controlled (bottom) cases. It can be appreciated that the control action removes the unstable behavior as expected.

4.2 Hovering flight: Delayed control We just showed that a proper selection of a and b may increase the stable operation region of the system when the feedback loop does not present a delay, but there are always delays in practical applications due to the time required for sensing and processing. When these delays are small, that is, a small fraction of the rotational period of the rotor, we can expect the system to preserve its stability. Larger delays would be expected to be detrimental for the system stability. However, in this section, we show how introducing a larger delay, of around one-tenth of the revolution period, can lead to an increased the robustness of the system against pitch-flap flutter. The stability analysis of a linear time-invariant, time-delay system (LTI-TDS) like Eq. (13) can be done following the two propositions of the CTCR presented in Ref. [17]. These two propositions state the following: •

Although an LTI-TDS has infinitely many characteristic roots, which change as the delay does, there is only a finite number of possible purely imaginary characteristic roots, which appear at periodically spaced values of the delay. That is, the number of elements of the set of imaginary characteristic roots of the system s = jωci , i = 1, 2, . . . , m, is always finite and upper bounded: m ≤ n2 ,

133

134

CHAPTER 6 Pitch-flap instabilities in helicopter rotors

where n is the order of the system [23]. Furthermore, if s = jωc is a characteristic root of the system for τ = τc , the same imaginary root appears when τ = τc + 2kπ/ωc , k = 1, 2, 3 . . .. • When the delay increases, an imaginary characteristic root can move to the left or to the right of the complex plane, stabilizing or destabilizing the system. This direction of crossing is known as the root tendency (RT). The RT is invariant for roots created by the periodicity. That is, if by increasing the delay from τc to τc + ε (0 < ε  1) the root at s = jωc moves to the left, it does the same when the delay increases from τ = τc + 2π/ωc to τ = τc + 2π/ωc + ε. According to these ideas, to determine the values of the delay for which the system is stable we must find the purely imaginary characteristic roots of system (12). For this, we form the characteristic equation det(sI44 − Auh − Ad e−τ s ) = 0.

(17)

Since Ad has rank 1, this equation has the form P(s) + Q(s)e−τ s = 0.

(18)

By replacing s = jω in Eq. (18), and reorganizing we obtain P(ω) = e−jωτ . Q(ω)

(19)

After equating the magnitudes of both sides of Eq. (19), we arrive to an expression in terms of ω |P(ω)|2 − |Q(ω)|2 = 0.

(20)

For the particular case at hand, Eq. (20) is an eighth-degree polynomial in ω, with nonzero coefficients only for the even exponents of ω. Introducing a new variable η = ω2 , we get an expression of the form η4 + c3 η3 + c2 η2 + c1 η + c0 = 0.

(21)

The solutions of Eq. (21) with η ≥ 0 correspond to valid crossing frequencies ω = √ ± η for the LTI-TDS system (12). This equation shows that there can be up to four crossing points at the imaginary axis. To find the delay at which such crossing frequencies occur, we equate the phase angles of both members of Eq. (19) and solve for τ τ=

1  ( Q(ω) −  P(ω) + π ). ω

(22)

The explicit versions of Eqs. (21), (22) are very long, and thus left outside of this document. To find whether an imaginary root crosses to the left or to the right of the complex plane, the RT for each ω and one of its corresponding τ is found using its definition

4 Stability analysis of the controlled system

   ds . RT = signum  dτ

(23)

The root sensitivity ds/dτ is found from Eq. (17) using implicit differentiation. When RT = 1, the root crosses to the right-half of the complex plane, which introduces instability. If RT = −1, on the other hand, the root moves to the stable left-half of the complex plane. With the help of Eqs. (21)–(23), we can determine for which values of the delay the system is stable given a certain parametric selection. The procedure is simple. After finding the crossing frequencies and the delays that generate them, we create a table of stability transition points. Starting with the number of unstable characteristic roots of the nondelayed system (which is equivalent to the number of eigenvalues of Auh with positive real part), we increase τ until a crossing delay is reached. If RT = 1 for that point, we increase the number of unstable roots by 2, or decrease it if RT = −1. Those intervals of τ for which the number of unstable roots is zero are the delays that guarantee stable operation.

4.2.1 Stabilization of pitch divergence Faragher [11] states that a system that presents pitch divergence cannot be stabilized by the control action proposed by them. This is also the case for the control action (13) as shown below. When the helicopter rotor presents pitch divergence, the nondelayed case has a real characteristic root. For this root to become stable, a stabilizing transition at the origin of the complex plane must take place. That is, we need a crossing frequency ω = 0 for which RT = −1. For Eq. (21) to have a root at ω = 0, the independent term c0 must be zero. Considering the explicit form of this coefficient, the condition for this to happen is c0 =

16n2 r2 σ 2 lθ2 25k4

−

8nrσ ν12 lθ 5k2

+ ν14 −

16a2 n2 ν14 lθ2 25

= 0.

(24)

This equation shows that selecting a=±

5ν12 k2 − 4nrσ lθ 4k2 nν12 lθ

.

(25)

By selecting the positive value indicated in Eq. (25) and replacing it into the characteristic equation, we observe that s = 0 becomes a solution of Eq. (17) for any value of the delay. This indicates that the system will present marginally stable behavior, integrating any disturbance, which is unacceptable. Selecting the negative value creates a crossing at ω = 0 which can happen only at τ = ∞ according to Eq. (22). Therefore, no finite delay can stabilize the pitch divergence even if the control gain is selected as indicated by Eq. (25).

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CHAPTER 6 Pitch-flap instabilities in helicopter rotors

4.2.2 Stabilization of pitch-flap flutter In order to show how the introduction of the delay may be beneficial, we consider first a parametric combination which is unstable for τ = 0. From Fig. 5, we appreciate that (σ = 0.07, ν12 = 12) is within the flutter region when the control gains are selected as (a = −0.25, b = 1). Following the CTCR steps described at the beginning of the section, we obtain for this case three crossing frequencies for which the delays and root tendencies are presented in Table 2. Notice that under this parametric combination, the system is unstable for τ = 0. However, at τ = 0.0518, a crossing in stabilizing direction (RT = −1) occurs. The next stability change happens at τ = 0.9019 when two roots cross to the right-half of the complex plane. This indicates that in the interval 0.0518 < τ < 0.9019 all the characteristic roots of the system are in the left-half of the complex plane, that is, it does not exhibit flutter behavior. After performing a similar analysis in a wider area of the (σ , ν12 ) domain, we arrive to Fig. 7, which presents the maximum and minimum limits for stability with respect to the delay for the numeric values stated in Table 1. Those parametric combinations which have a negative minimum delay are stable for τ = 0 and lose the stability when the delay reaches the upper bound. For the cases beyond the stability Table 2 Crossing frequencies, delays, and RTs for a system with σ = 0.07, ν12 = 13, a = −0.25, b = 1. Frequency

First delay

RT

1.1793 2.2603 0.1753

0.0518 0.9019 2.4253

−1 1 1

1

0.5 t

136

0

16

14

12 n21

10

8

0.05

0.06

0.07

0.08

s

FIG. 7 Minimum and maximum stability bounds in the domain of the delay. The grid is the τ = 0 plane. The thick line is the pitch flutter stability boundary for the nondelayed case.

4 Stability analysis of the controlled system

0.04 [rad]

0.02 0 −0.02 −0.04

0

5

q10

15 b

20

0

5

10 ψ [rad]

15

20

[rad]

0.01 0 −0.01

FIG. 8 Simulation results for (σ , ν12 ) = (0.07, 12) and (a, b) = (−0.25, 1) without delay (top) and using a time delay τ = 0.5 (bottom).

boundary for the nondelayed case (marked by the black thick line in Fig. 7), a positive finite delay within the minimum and maximum removes the pitch flutter mode. Fig. 8 shows an example of the stabilizing effect of the delay for a system with σ = 0.07 and ν12 = 12 using the control gains a = −0.25 and b = 1. In the top plot of the figure, we appreciate the time evolution of the states when the delay is equal to 0, whereas the bottom plot shows the results when τ = 0.5. It is important to highlight that Fig. 7 does not include information regarding stability of the pitch divergence mode. There is a region in the (σ , ν12 ) plane for which the introduction of the delay removes the pitch-flap flutter but the system nevertheless still presents pitch divergence. The parametric selection σ = 0.07 and ν12 = 10, for example, belongs to this region.

4.3 Horizontal flight For μ > 0, the complete LTP model (12) has to be studied. If τ = 0, the system reduces itself to x˙ (ψ) = (Au (ψ) + Ad )x(ψ). Using the same method as in the case of uncontrolled forward flight, the stability regions for this case are constructed and displayed in Fig. 9. Comparing the plots of Fig. 9 with those of Fig. 3, we observe that the proposed control action worsens the stability region for the case of horizontal flight. Particularly, the stabilizing effect that the horizontal flight has on the flutter boundary, which moves to the right of the plots in Fig. 3 as μ increases, is lost when the control action is present with τ = 0. In the case of the pitch divergence, the destabilizing effect is worsened. This would definitively speak against the deployment of this control technique in a real application.

137

14

12

12

10

10

8

8

n12

n12

14

6

6

4

4 2

2

0

(A)

0 0

0.01

0.02

0.03

0.04

s

0.05

0.06

0.07

0.08

0.03

0.04

0

0.01

0.02

0.03

(B)

0.04

s

0.05

0.06

0.07

0.08

16 14 12

n12

10 8 6 4 2 0

(C)

0

0.01

0.02

s

0.05

0.06

0.07

0.08

FIG. 9 Pitch-flap stability boundaries for a controlled system in horizontal flight for different values of the tip speed ratio μ. The solid line is the flutter boundary, the dashed line is the pitch divergence boundary, and the dotted line is the P = .5 flutter boundary. (A) μ = 0.1; (B) μ = 0.2; (C) μ = 0.3.

CHAPTER 6 Pitch-flap instabilities in helicopter rotors

16

138

16

5 Concluding remarks

In order to observe the effect of the delayed control action on a rotor in horizontal flight, the full LTP-TD model (12) has to be analyzed. Just like in the case of uncontrolled horizontal flight, the stability of the system is determined by its characteristic multipliers, that is, by the eigenvalues of its monodromy matrix. The numerical complexity of the task appears from the fact that the time delay makes these systems infinite dimensional, which means that we would need to study an infinite number of multipliers. To overcome this difficulty, Inperger and Stépán introduced the semidiscretization methodology [19,20]. This method allows the generation of an approximation to the monodromy matrix of the system. The basic idea of the method is to discretize the system with respect to the time variable in such a way that the system matrices are assumed constant during the discretization intervals. Each subsystem can be solved then as a standard ODE, and the monodromy matrix can be approximated using the product of the individual fundamental matrices. The precision and the computational load of the method can be controlled by means of the so-called period resolution, which defines the number of intervals in which a period will be divided. This method could be consider similar to the integration used for the system without delays, but it contemplates approximations for cases of distributed and nonconstant delays. The results of applying the semidiscretization method to the system under consideration in this work can be observed in Fig. 10. The plots in Fig. 10 show how the presence of a delay improves the situation with respect to the case without delay, particularly for small tip speed ratios. As the horizontal speed increases, however, the flutter boundary moves considerably to the left, drastically reducing the stability region. This stresses the point of the delay being somehow beneficial for the system. It is important to notice here that the parameters of the control action and the time delay were selected to optimize the stability region for the case of hovering flight. This optimization was possible thanks to the explicit results provided by the CTCR methodology for LTI-TDS. It may be feasible to obtain more positive results if the optimization of the control parameters is done considering directly the model of the helicopter in horizontal flight. A complete understanding of the effect of a control action based on delayed feedback is difficult to extract due to the numerical (instead of analytic) nature of the tools available for this analysis and their high computational cost.

5 Concluding remarks In this document, we studied the problem of stabilization of pitch-flap motion of helicopter rotors using delayed feedback. A control law based on delayed feedback was proposed and its effected analyzed. For the case of hovering flight, in which the system is an LTI system with delays, the CTCR methodology was used to analyze the stability. For the case of horizontal flight, in which an LTP system has to be studied, the semidiscretization method was deployed.

139

14

12

12

10

10

n12

n12

14

8

8

6

6

4

4

2

2

(A)

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

s

(B)

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

s

16 14 12

n12

10 8 6 4 2

(C)

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

s

FIG. 10 Pitch-flap stability boundaries for a controlled system in horizontal flight for different values of the tip speed ratio μ and a time delay of τ = 0.5 rad. The dashed line is the pitch divergence boundary and the dotted line is the P = .5 flutter boundary. (A) μ = 0.1; (B) μ = 0.2; (C) μ = 0.3.

CHAPTER 6 Pitch-flap instabilities in helicopter rotors

16

140

16

References

It was observed that, although the delayed feedback could bring considerable improvements in the case of hovering flight, it does not seem the case for the case of horizontal flight. For the parametric selections considered in this chapter, the control action limited the region of stability of the system, making it more prone to pitch divergence and pitch-flap flutter. As a parallel result of this study, we could highlight the need of developing new tools for the stability analysis of LTP systems with time delays. The semidiscretization method allows to find results with relative good precision, but its computational cost is relatively high, and it does not provide a full stability picture of the system in the domain of the delay.

References [1] S. Hall, N. Wereley, Control algorithms for helicopter vibration reduction, J. Guid. Control Dyn. 16 (4) (1992) 793–797. [2] R. Mura, M. Lovera, An lpv/h∞ framework for the active control of vibrations in helicopters, in: Proceedings of the IEEE Conference on Control Applications, Antibes, France, 2014, pp. 427–432. [3] R. Mura, A. Esfahani, M. Lovera, Robust harmonic control for helicopter vibration attenuation, in: Proceedings of the American Control Conference, Portland, Oregon, USA, 2014, pp. 3850–3855. [4] S. Bittanti, F.A. Cuzzola, Periodic active control of vibrations in helicopter: a gain-scheduled multi-objective approach, Control Eng. Pract. 10 (2002) 1043–1057. [5] P. Arcara, S. Bittanti, M. Lovera, Periodic control of helicopter rotors for attenuation of vibrations in forward flight, IEEE Trans. Control Syst. Technol. 8 (2000) 883–894. [6] F. King, J. Maurice, W. Fichter, O. Dieterich, P. Konstanzer, In-flight rotorblade tracking control for helicopters using active trailing-edge flaps, J. Guid. Control Dyn. 37 (2) (2014) 633–643. [7] J. Shen, I. Chopra, Swashplateless helicopter rotors with trailing-edge flaps, J. Aircraft 41 (2) (2004) 208–214. [8] J. Shen, M. Yang, I. Chopra, Swashplateless helicopter rotor with trailing-edge flaps for flight and vibration con, J. Aircr. 43 (2) (2006) 346–352. [9] P. Friedmann, T. Millott, Vibration reduction in rotor using active control: a comparison of various approaches, J. Guid. Control Dyn. 18 (4) (1995) 664–673. [10] P. Friedmann, On-blade control of rotor vibration, noise and performance: just around the corner?, J. Am. Helicopter Soc. 59 (2014) 1–37. [11] J. Faragher, Stability Improvement of Periodic Motion of Helicopter Rotor Blades (Ph.D. thesis), Department of Mechanical Engineering, The University of Melbourne, Melbourne, Australia, 1996. [12] J. Krodkiewski, J. Faragher, Stabilization of motion of helicopter rotor blades using delayed feedback—modeling, computer simulation and experimental verification, J. Sound Vib. 234 (4) (2000) 591–610. [13] V. Pyragas, Continuous control of chaos by self-controlling feedback, Phys. Lett. A 170 (1992) 421–428. [14] N. Olgac, B. Holm-Hansen, A novel active vibration absorption technique: the delayed resonator, J. Sound Vib. 176 (1) (1994) 93–104.

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[15] R. Cepeda-Gomez, Control of pitch-flap instabilities in helicopter rotors using delayed feedback, in: IFAC-PapersOnLine, vol. 49, Istanbul, Turkey, 2016, pp. 82–87. [16] N. Olgac, A. Ergenc, R. Sipahi, Delay scheduling: a new concept for stabilization in time delay systems, J. Vib. Control 11 (9) (2005) 1159–1172. [17] N. Olgac, R. Sipahi, An exact method for the stability analysis of time-delayed linear time invariant systems, IEEE Trans. Autom. Control 47 (5) (2002) 793–797. [18] N. Olgac, R. Sipahi, Complete stability robustness of third order LTI multiple time delay systems, Automatica 41 (8) (2005) 1413–1422. [19] T. Insperger, G. Stépán, Semi-discretization method for delayed systems, Int. J. Numer. Methods Eng. 55 (2002) 503–518. [20] T. Insperger, G. Stépán, Semi-Discretization for Time-Delay Systems, Springer, 2011. [21] C. Stammers, The flutter of a helicopter rotor blade in forward flight, Aeronaut. Q. 21 (1970) 18–48. [22] A. Halanay, Differential Equations: Stability Oscillations, Time Lags, Academic Press, London, 1966. [23] A.F. Ergenc, N. Olgac, H. Fazelinia, Extended Kronecker summation for cluster treatment of LTI systems with multiple delays, SIAM J. Control Optim. 46 (1) (2007) 143–155.

CHAPTER

The small signal stability region of power systems with time delay

7

Hongjie Jia, Chaoyu Dong School of Electrical and Information Engineering, Tianjin University, Tianjin, China

Chapter outline 1 2 3 4

Power system and the time delay ............................................................. Power system small signal stability model with time delay .............................. Optimization-based boundary tracing algorithm............................................ Time-delay and the eigenvalue trajectory ................................................... 4.1 Single-machine-infinite-bus system............................................... 4.2 Time-delay impact on the power system.......................................... 5 Time-delay and the small signal stability region ........................................... 5.1 Single-machine-infinite-bus system............................................... 5.2 WSCC 3-generator-9-bus system .................................................. 6 Conclusions....................................................................................... References...........................................................................................

144 145 147 149 149 151 154 154 156 161 161

With the rapid development of phase measurement units (PMUs) and wide area measurement system (WAMS), the stability during the coordination in the power system has attracted more and more attention. Since there exist obvious time delays during the WAMS measurements, it is important to properly evaluate the impact of time delays on the power system stability and the controller design. In this chapter, the time-delay small signal stability of the power system is discussed. The main contributions of this work are: (1) the time-delay model and boundary structure of small signal stability region (SSSR) are revealed for power systems; (2) to trace the eigenvalue variations and determine the boundaries of the SSSR, an optimization-based algorithm is developed for the power system considering the time delays; (3) on the basis of the proposed algorithm, the time-delay impacts on the eigenvalue trajectories and the SSSR are visualized through the representative singlemachine-infinite-bus system and the WSCC 3-generator-9-bus system; (4) besides the visualized time-delay impact, an index of Err is introduced to quantitatively assess the influence caused by the time delay; and (5) it is found that time delay leads to complex influence on the small signal stability of the power system. Both negative and positive effects are detected during the investigation. Stability, Control and Application of Time-Delay Systems. https://doi.org/10.1016/B978-0-12-814928-7.00007-X © 2019 Elsevier Inc. All rights reserved.

143

144

CHAPTER 7 Stability region of power systems with time delay

1 Power system and the time delay There exist various time delays during the measurements in power systems. As the traditional power system stability control strategies only employ the local information, the measurement delays were usually very small (less than 10 ms), which makes time delays ignorable in the previous stability analyses and the controller designs [1]. With the fast deployment of PMUs and WAMS, the global coordination becomes an available choice to enhance the performance of power system with the remote information provided by WAMS/PMU. However, the time delays during the wide area measurements become obvious and larger, which might affect or even disable the previous analyses and control strategies. Hence, it is very necessary to evaluate the impact of time delays on power system stability [2–4]. In the past, researches on the time delays in the power system mainly focused on the following aspects: (1) To evaluate the time-delay influences on controller designs. For example, Kamwa [5], Wu [6], Jiang [7], and Saad [8] discussed the impacts of time delay in the design of the power system stabilizer (PSS), thyristor-controlled series compensation (TCSC), and static var compensator (SVC), respectively. (2) To analyze the cause of the time delay in power system measurement. They were helpful to find appropriate methods to reduce the negative effects of time delays [9,10]. (3) To reduce and control power system periodic or chaotic oscillations through time-delayed feedback [11,12]. However, there were few reports about the time-delay impact on the small signal stability of the power system. As the power system normally operates in equilibrium points, the small signal stability analysis is widely utilized to investigate the power system stability. SSSR is defined as a set in the parameter space that consists of all the stable operating points under a given condition [13–17]. When time delay is ignored, power system dynamics can be described by following differential algebraic equations (DAEs): x˙ = F(x, y, p)

(1a)

0 = G(x, y, p)

(1b)

where x ∈ Rn , y ∈ Rm , and p ∈ Rp are vectors of the state variables, algebraic variables, and bifurcation variables, respectively. For a given p, the system equilibrium point is the solution of the following equations: 0 = F(x, y)|p

(2a)

0 = G(x, y)|p

(2b)

Linearizing Eq. (1) at an equilibrium (x0 , y0 ), we can get the following incremental DAEs:

2 Power system small signal stability model with time delay

where A(p) =



∂F  ∂x p ,

˙x = A(p) · x + B(p) · y

(3a)

0 = C(p) · x + D(p) · y

(3b)

B(p) =



∂F  ∂y p ,

C(p) =



∂G  ∂x p ,

D(p) =



∂G  ∂y p .

If matrix D is

nonsingular, Eq. (3) can be reduced into an ordinary differential equation as follows: ˙x =  A(p) · x

(4)

where  A(p) = A(p) − B(p) · D−1 (p) · C(p). According to the Lyapunov stability theory, if all the eigenvalues of the matrix  A(p) have negative real parts, the equilibrium point (x0 , y0 ) is small signal stable. When p changes and there is a real eigenvalue crossing the imaginary axis, the saddle-node bifurcation (SNB) will occur. If there is a pair of conjugated eigenvalues crossing the imaginary axis, the Hopf bifurcation (HB) will happen. And, if matrix D turns singular, the singularity-induced bifurcation (SIB) will appear. All eigenvalues of the matrix  A(p) are denoted as follows: λ = [λ1 , λ2 , . . . , λn ]T

(5)

The power system SSSR can then be defined ΩSSSR = {p | (λi ) < 0, ∀λi ∈ λ and det(D(p))  = 0}

(6)

The boundary of the SSSR is composed of the curves of SNB, HB, and SIB as follows: ∂ΩSSSR = {SNB ∪ SIB ∪ HB}

(7)

It can be found that the small signal stability boundary without the time delay is the closure of SNB, HB, and SIB.

2 Power system small signal stability model with time delay When the time delay during the remote control is considered, power system dynamics should be described by time-delay DAE model as follows: ⎧ x˙ = f (x, xτ1 , xτ2 , . . . , xτk , y, yτ1 , yτ2 , . . . , xτk , p) ⎪ ⎪ ⎪ ⎪ 0 = g(x, y, p) ⎪ ⎪ ⎪ ⎨0 = g(xτ1 , yτ1 , p) 0 = g(xτ2 , yτ2 , p) ⎪ ⎪ ⎪ ⎪ .. .. ⎪ ⎪ ⎪ ⎩. . 0 = g(xτk , yτk , p)

(8)

145

146

CHAPTER 7 Stability region of power systems with time delay

where x ∈ Rn is the vector of state variables. xτ1 = x(t − τ1 ), xτ2 = x(t − τ2 ), . . . , xτ3 = x(t − τ3 ) are vectors of the time-delay variables. τ1 , τ2 , . . . , τk are the time delays. It can be verified that the equilibrium point of Eq. (8) is also given by the solution of Eq. (2). Linearizing equation (8) at an equilibrium point (x0 , y0 ), the following incremental DAE can be derived: ⎧ ˙x = A0 x + A1 xτ1 + A2 xτ2 + · · · + Ak xτk ⎪ ⎪ ⎪ ⎪ +B0 y + B1 yτ1 + B2 yτ2 + · · · + Bk yτk ⎪ ⎪ ⎪ ⎪ 0 = C0 x + D0 y ⎪ ⎨ 0 = C1 xτ1 + D1 yτ1 ⎪ ⎪ ⎪ 0 = C2 xτ2 + D2 yτ2 ⎪ ⎪ .. .. ⎪ ⎪ ⎪ ⎪ ⎩ . . 0 = Ck xτk + Dk yτk

(9)

      ∂f  ∂f  ∂g  ∂g  ∂f  ∂f  where A0 = ∂x  , B0 = ∂y  , C0 = ∂x  , D0 = ∂y  , Ai = ∂xτ  , Bi = ∂yτ  , i p p p p p i p   ∂g  ∂g  Ci = ∂xτ  , Di = ∂y  , (i = 1, 2, . . . , k) are the Jacobian matrices related to the i

p

τi

p

state variables and the time-delay variables. Assumed matrices D0 , D1 , D2 , . . . , Dk are nonsingular, Eq. (9) can be further simplified ˙x =  A0 x +  A1 xτ1 +  A2 xτ2 + · · · +  Ak xτk

(10)

where  Ai = Ai − Bi · D−1 i · Ci , i = 0, 1, . . . , k. On the basis of Eq. (10), the system characteristic equation is formed ⎛ Γ (λ) = det ⎝λ · I −  A0 −

k

⎞  Ai · e−λ·τi ⎠ = 0

(11)

i=1

Solving Eq. (11), all eigenvalues of the time-delay system can be obtained at the point (x0 , y0 ): λτ = (λτ1 , λτ2 , . . . , λτn ). If max(real(λτi )) < 0, ∀λτi ∈ λτ , the system is small signal stable. Otherwise, if there is at least one eigenvalue with the positive real part, the system is unstable. Similar to Eq. (6), the SSSR with the time delay can be defined as follows: τ ΩSSSR = {p | real(λτi (p)) < 0 and det(Dj (p))  = 0}

(12)

where i = 1, 2, . . . , n and j = 1, 2, . . . , k. The boundary of the SSSR defined by Eq. (12) is also composed of the curves of SNB, HB, and SIB. Theorem 1. Provide that x0 is an equilibrium point of the system (1) and λ = [λ1 , λ2 , . . . , λn ]T is the corresponding eigenvalues. Re = [Re1 , Re2 , . . . , Ren ] is the eigenvalue real parts, where Rei = real(λi ), i = 1, 2, . . . , n. (1) If all real parts of the system eigenvalues are negative, system (1) is small signal stable at point x0 .

3 Optimization-based boundary tracing algorithm

(2) If there exists at least one eigenvalue with a positive real part, that is, Rej > 0 (1 ≤ j ≤ n), system (1) is small signal unstable at that point. With the consideration of the time delay in the power system, transcendental items have been involved in the characteristic equation of the small signal stability model. As an uncertain and unstable factor, the time-delay impact on the power system needs to be carefully investigated.

3 Optimization-based boundary tracing algorithm For the linear small signal model, the system eigenvalues represent the dynamic features of the power system. During the power system analysis, the tracking of eigenvalue trajectories becomes an effective method to detect the system variation. Avoiding the unstable states crossing the imaginary axis, the SSSR of the power system can be formed by the corresponding system parameters. In order to investigate the time-delay impact on the small signal stability of the power system, an optimization-based boundary tracing algorithm [1,3,13,15,17] is proposed to solve the system eigenvalues, which provides an effective approach forming the corresponding SSSR. Defining a matrix function of the parameter λ as follows: M(λ) =  A0 +

k

 Ai · e−λ·τi ∈ Rn×n

(13)

i=1

If λ∗ satisfies the following equation: Γ (λ∗ ) = det(λ∗ · I − M(λ∗ )) = 0

(14)

An eigenvalue of the matrix M(λ∗ ) can be found. Also, it is a solution of Eq. (11) (i.e., it is an eigenvalue of Eq. (8)). Provided that all eigenvalues of are ranked Λ∗ = (λ∗1 , λ∗2 , . . . λ∗n−1 , λ∗ )

(15)

 λ∗ is a value close to λ∗ , which can be expressed as ∗

 λ = λ∗ + λ

(16)

with λ → 0. The following result can then be derived: |Γ ( λ∗ )| = | det( λ∗ · I − M( λ∗ ))| ∗ ≈ | det( λ · I − M(λ∗ ))|    n−1   ∗ ∗   λ∗ − λ∗ )| (λ − λi ) · |( =   i=1 = K( λ∗ ) · |λ|

(17)

147

148

CHAPTER 7 Stability region of power systems with time delay

FIG. 1 Algorithm for tracing the boundary of the small signal stability region.

   ∗ − λ∗ ) and λ =   where K( λ∗ ) :=  n−1 ( λ λ∗ − λ∗ . It is found that although λ i=1 i  will turn small when  λ∗ → λ∗ . However, since the value of K( λ∗ ) may be very large, ∗ |Γ ( λ )| might not be very small. The tracing algorithm is derived based on the “predictor-corrector” framework as shown in Fig. 1. The following optimization problem is used as the algorithm’s corrector: min

C(x, y, τ , p, λ)

s.t. 0 = f (x, y, τ , p, λ)

(18a) (18b)

0 = g(x, y, τ , p, λ)

(18c)

0 ≤ B(x, y, τ , p, λ)

(18d)

where p is the vector of the bifurcation parameter. λ is the key eigenvalue of the time-delay system. Total generator cost is used as the objective function. Eqs. (18b), (18c) are used to guarantee that the optimal solution is an equilibrium point of the time-delay system. Eq. (18d) includes all the system constraints, such as limits of bus voltage, limits of generator output, the constraint of small signal stability boundaries, etc. Theoretically, Eq. (11) should be included in Eq. (18d). However, since it is very difficult to directly solve, it is replaced by the following unconstraint optimization procedure: min

|Γ (λ)|

(19)

Implementation details of the developed algorithm are illustrated as follows: Step 1. Initially, a point X 1 is determined on the boundary of the SSSR with k = 1. Step 2. The next point X k+1 on the boundary is predicted by the following equation: Xk+1 = σ k ·

∂X + Xk ∂p

(20)

4 Time-delay and the eigenvalue trajectory

where X = [x, y, p, λ] and parameter σ k is set as follows: σk =

 0 h

k=1 k>1

(21)

h is the step size of the predictor. Step 3. The optimization problem in Eq. (19) is solved to guarantee the solution is on the boundary of the region. Step 4. If Step 3 converges, jump to Step 6, otherwise, go to Step 5. Step 5. If h < hmin , jump to Step 7. Otherwise, the step size h is reduced, then jump to Step 2 and try again. Step 6. If the end condition hits, turn to Step 7. Otherwise set k = k + 1, pk = pk−1 + p(σ k ) and jump to Step 2. Step 7. The program terminates and results are saved. Through these seven steps, the eigenvalue of the small signal stability model can be calculated.

4 Time-delay and the eigenvalue trajectory To visualize the time-delay impact on the eigenvalue trajectories as well as the verification of the optimization-based algorithm, the single-machine-infinite-bus system [1–4] is employed as an example in this section.

4.1 Single-machine-infinite-bus system The illustration of the time-delay impact is based on a single-machine-infinite-bus system as shown in Fig. 2. The system model can be described as follows δ˙ = ωB · ω

(22)

M ω˙ = −Dω + (Pm − PG )  T E˙ = −E + (xd − x )Id + Efd

(23)

TA E˙ fd = −KA (VG − Vref ) − (Efd − Efd0 )

(25)

d0

FIG. 2 A single-machine-infinite-bus power system.

d

(24)

149

150

CHAPTER 7 Stability region of power systems with time delay

where δ is the generator angle; ω is the generator speed and ωB is the base speed; E is the generator voltage behind the transient reactance; Efd is the exciter output voltage, and Efd0 is the reference; TA and KA are the time constant and gain of the exciter; Pm is the mechanical power; D is the generator damping factor;  is the open-loop time constant of the armature winding; and Td0

V0 is the infinite bus voltage. The electric power, d-axis current, and terminal voltage of the generator can be expressed as following equations: PG = VG = Id =

E · V0 · sin δ xe + xd  (xd + xe E cos δ)2 + (xe E sin δ)2 xe + xd

E − V0 · cos δ xe + xd

(26)

(27) (28)

Substituting Eqs. (26)–(28) into Eqs. (22)–(25), we can get the following ordinary differential equation: x˙ = f (x, p)

(29)

where x = [δ, ω, E , Efd ]T are the system state variables and p is the system bifurcation parameter. In modern power systems, the exciter inputs can be taken from remote buses [1]. In such scenarios, there may introduce time delays in the measuring data. In this chapter, we simply assume that there is a delay in the measurement of VG . Hence, Eq. (25) can be rewritten as follows: TA E˙ fd = −KA (VG (t − τ ) − Vref ) − (Efd − Efd0 )

(30)

where τ is the time delay of VG . Then, Eq. (29) could be described as the following time-delay differential equation: x˙ = f (x, xτ , p)

(31)

4 Time-delay and the eigenvalue trajectory

4.2 Time-delay impact on the power system For the investigation of the time-delay impact, Table 1 gives the parameter values of the single-machine-infinite-bus system. The generator mechanical power Pm is selected as the bifurcation parameter in the following studies. When the time delay is ignored, Fig. 3 plots the curves of system eigenvalue (eigenvalue loci) changing with Pm . Since there are two pair conjugated eigenvalue loci and they are mirrored by the real axis, only the branches upside the real axis (i.e., with positive imaginary parts) are plotted in the figure. The arrows on the loci point out the direction with the increased Pm . 1 moves to the left and locus  2 moves to the right It can be found that locus  2 crosses the imaginary axis, when Pm increases. At about Pm = 1.1056, locus  which leads to the HB at that point. Therefore, the critical eigenvalue is related to 2 instead of locus . 1 locus  When time delay τ changes in the range of 0.01–0.09 s, the system eigenvalue loci with Pm increasing are plotted in Fig. 4. In the diagrams, the solid lines are the loci without considering the delay, while the dashed lines are the loci with the delay. It can be seen that the time-delay influence becomes significantly obvious with the increased time delay. When τ is smaller than 0.02 s, that is, in Fig. 4A 1 and locus  2 is small. Once the delay value and B, the difference between locus  Table 1 Parameter values of the single-machine-infinite-bus system. M

D

xd

xd

 Td0

xe

10.0 Efd0

1.0 Vref

1.0 KA

0.4 V0

10.0 ωB

0.5

2.0

1.05

190.0

1.0

377.0

FIG. 3 System eigenvalue loci when the time delay is not considered.

151

152

CHAPTER 7 Stability region of power systems with time delay

(A)

(B)

(C)

(D)

(E)

(F)

FIG. 4 System eigenvalue loci when τ has different values. (A) τ = 0.01 s, (B) τ = 0.02 s, (C) τ = 0.024 s, (D) τ = 0.025 s, (E) τ = 0.04 s, (F) τ = 0.09 s.

exceeds 0.024 s, the trace shapes are even changed due to the growing time delay in Fig. 4C and D. Moreover, the whole loci are completed distorted in the last two figures because of the time delay. To quantitatively investigate the time-delay impact, the following error index is defined comparing the loci differences with and without the time delay: Err =

 (τ − 0 )2 + ( τ − 0 )2

(32)

where τ and τ are the real and imaginary parts of the eigenvalue when the time delay is considered; and 0 and 0 are the real and imaginary parts of the eigenvalue when the time delay is ignored.

4 Time-delay and the eigenvalue trajectory

From the definition of Eq. (32), we can find that the index of Err provides the absolute error between two eigenvalues with and without considering the delay. The corresponding locus Errs in Fig. 4 are plotted in Fig. 5A and B, which classifies the Err curve into two types based on the variation trends. According to Figs. 4 and 5, the following phenomena are observed: (1) When τ is smaller, its influence is smaller, and vice versa. For an example, when τ = 0.01 s (10 ms), the maximum error between two critical eigenvalues with and without considering the delay is less than 0.1. Thus, when τ is less than 0.01 s, it can be ignored in small signal stability analysis. (2) When τ turns larger, the maximum error increases rapidly. The time delay significantly influences the small signal stability analysis at that time. For an example, when the time delay is not considered, an HB occurs at Pm = 1.1056 and the system critical eigenvalues are λ = ±5.0779i. When τ = 0.01 s, the HB happens at Pm = 1.1188 with the critical eigenvalues λ = ±4.9254i. When τ = 0.024 s, the Pm and critical eigenvalues at the HB are Pm = 1.1275 and λ = ±4.6594. It can be observed that the maximum errors between the critical eigenvalues turn larger and larger when time delay increases as shown in Fig. 5. (3) The system critical eigenvalue is exchanged between τ = 0.024 s and τ = 0.025 s. When τ is less than 0.024 s, the system critical eigenvalue is 2 However, when τ is larger than 0.025 s, the system corresponding to locus . 2 to locus . 1 The system oscillation critical eigenvalue is changed from locus  mode is also changed at that time. We have observed that the system critical eigenvalue is exchanged between τ = 0.024 s and τ = 0.025 s from Fig. 4. Therefore, there must exist a point at which 1 and  2 coalesce together. Based on the further calculation with the proposed locus  method, it is found that the intersection occurs at τ = 0.024839 s, which is shown in Fig. 6. The crossing point in the complex plane is (τ , τ ) = (−0.35628, 4.67919). After τ > 0.024839 s, the system critical eigenvalue changes from eigenvalue locus 2 to eigenvalue locus , 1 which can be observed from Fig. 4C–F. 

(A)

(B)

FIG. 5 Differences of the critical eigenvalue loci when τ has different values. (A) τ = 0.01, 0.02, 0.024 s. (B) τ = 0.025, 0.04, 0.09 s.

153

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CHAPTER 7 Stability region of power systems with time delay

FIG. 6 System eigenvalue curves intersect at τ = 0.024839 s.

The intersection of the eigenvalue loci in Fig. 6 is similar to the scenario of the SIB; that is, a pair of conjugated eigenvalue loci coalesces at a point in the complex plane. When an SIB occurs, the conjugated eigenvalue loci coalesce at a point on the real axis. After coalescence, they turn into two real eigenvalues so that a system oscillation mode is annihilated. However, the critical eigenvalues do not disappear after they coalesce as shown in Fig. 6. The only changing caused by the time delay is their moving directions. According to Fig. 6, it can be seen that the time delay has an obvious influence on power system small signal stability. Especially when the delay is large, it can absolutely change the system stability characteristics, such as the oscillation mode and the system critical eigenvalues. Therefore, in power system wide area controller design and power system stability studies, it is necessary to appropriately consider the uncertainties induced by the time delay.

5 Time-delay and the small signal stability region On the basis of the single-machine-infinite-bus system and the WSCC 3-generator9-bus system [13–17], the change of the SSSR caused by the time delay is compared and analyzed.

5.1 Single-machine-infinite-bus system According to the optimization-based boundary tracing algorithm of seven steps, the small signal region of the single-machine-infinite-bus system is solved and studied. Fig. 7 plots the SSSR when the time delay is ignored. It can be seen that the SSSR (gray area) is surrounded by the HB curve (solid line) and SNB curve (dashed line). Since the solid line turns tangent to the dashed line at points A and B, the SSSR can be separated into two parts.

5 Time-delay and the small signal stability region

FIG. 7 Small signal stability region and the boundaries without time delay.

FIG. 8 Boundaries of some small signal stability regions with different time delays.

In order to demonstrate the time-delay impact on the SSSR, Fig. 8 plots the small signal stability boundaries when τ = 0, 20, 40, 60, 80, and 100 ms. Fig. 9 gives a zoom-in result of the gray area in Fig. 8. According to Figs. 8 and 9, the following conclusions can be drawn: (1) Time delay notably affects the SSSR especially when τ ≥ 20 ms. (2) When Pm and τ are both smaller, the increasing of τ leads to the region’s bottom moving down so as to increase the area of the SSSR in this direction. One example is the case of τ = 20 ms in Figs. 8 and 9. However, the left boundary of the SSSR moves to the left when τ increases, which causes the area of SSSR decreases significantly.

155

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CHAPTER 7 Stability region of power systems with time delay

FIG. 9 Zoom-in result of the gray area in Fig. 8.

(3) The boundary of the SSSR may cross itself when the time delay is large, for example, the scenarios of 80 and 100 ms in Fig. 9. This phenomenon may lead to the topological characteristics of the SSSR even more complicated.

5.2 WSCC 3-generator-9-bus system Similar to the single-machine-infinite-bus system, the case of WSCC 3-generator-9bus system is demonstrated and shown in Fig. 10. Generator 1 is considered as an infinite bus. Generators 2 and 3 are modeled by the following differential equations: δ˙i = ωi − ωs  − X  · I )I − (E + X  · I )I − D(ω − ω ) 2H · ω˙ i = Pmi − (Eqi s i qi qi di di di qi di     ˙ T · E qi = −E − (Xdi − X ) · Idi + Efdi qi doi di    )·I  ˙ Tqoi · E di = −Edi + (Xqi − Xqi qi

TAi · E˙ fdi = −Efdi + KAi (Vrefi − VGi )

(33a) (33b) (33c) (33d) (33e)

where i = 2, 3. Parameters used in this chapter are given in Table 2. Pm2 and Pm3 are selected as the bifurcation parameters. System load level ρ is fixed to 2.111 pu in the following studies. It is assumed that there exists a time delay during the measurement of the terminal voltage VG2 . • When time delay is not considered

5 Time-delay and the small signal stability region

FIG. 10 WSCC 3-generator-9-bus system.

Table 2 Parameters of the 3-generator-9-bus system. ωB 377 D2

Xd2 0.8958 TA2

 Xd2 0.1198 KA2

 Xq2 0.8645 Vref2

 Td02 0.1969 Xd3

 Xq02 6.0  Xd3

H2 0.54 Xq3

6.4  Xq3

0.05  Td03

0.02  Tq03

50.0 H3

1.1223 D3

0.90 TA3

0.10 KA3

0.85 Vref3

0.10 PL5

8.00 QL5 0.5

0.25 PL7 1.0

3.01 QL7 0.5

0.05 PL9 1.0

0.02 QL9 0.5

50.0

1.1223

1.0

When time delay is not considered, Fig. 11 plots the system SSSR in (Pm2 , Pm3 ) space with ρ = 2.111 pu. It is obvious that the region is a closed area surrounded by 1 , 4 which forms four vertexes in Fig. 11. four HB curves – •

When time delay is considered

Fixing load level ρ = 2.111 pu, let us change the value of τ and trace the corresponding SSSR with considering the time delay. Fig. 12 describes the region deformations when τ changes between 10 and 280 ms. In the diagrams, the area surrounded by the solid lines is the SSSR when the time delay is not considered, while the area surrounded by the dashed lines is the SSSR when the time delay is considered. It can be found that the time-delay impact on the SSSR is very complicated. Two interesting phenomena are detected in Fig. 12. (1) When τ is small, its influence on the SSSR is also negligible. For example, in the WSCC system, when τ is less than 10 ms, it can be ignored with little error.

157

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CHAPTER 7 Stability region of power systems with time delay

FIG. 11 Small signal stability region without the time delay.

Just as we know, traditional power system controllers mainly use the local data. The time delay for the local data is usually less than 10 ms, which is reasonable to ignore such delay during the traditional controller designs. However, the coordinated controllers of the power system are based on the WAMS information. The time delay during the transmission of WAMS data is usually much larger than 10 ms. Therefore, the time delay cannot be ignored. (2) The time delay can bring both negative and positive influence to the power system small signal stability. One example is shown in Fig. 12. When τ is less than 130 ms, the region’s boundaries all move to the interior. The stability of the power system turns worse. The time delay brings a negative influence on power system small signal stability. However, when τ is larger than 150 ms, the right and bottom boundaries move to the exterior. Power system stability turns better in these two directions. It can be seen that the mechanism of the time-delay influence to power system small signal stability is complicated, which needs more studies in the future. In the past, people mainly concerned about the negative influence of the time delay in the design of the power system controller. Methods were usually used to reduce the time lags. However, based on the previous studies, it is revealed that time delay could introduce positive effects to the power system stability in some scenarios. If the positive and negative effects of the time delay could be simultaneously considered, various coordination control strategies can benefit the power system more in the future.

(D)

(B)

(C)

(E)

(F)

FIG. 12 Small signal stability regions with different time delays. (A) τ = 10 ms. (B) τ = 40 ms. (C) τ = 70 ms. (D) τ = 100 ms. (E) τ = 130 ms. (F) τ = 160 ms. (Continued)

5 Time-delay and the small signal stability region

(A)

159

160

(H)

FIG. 12, CONT’D (G) τ = 190 ms. (H) τ = 220 ms. (I) τ = 250 ms.

(I)

CHAPTER 7 Stability region of power systems with time delay

(G)

References

6 Conclusions This chapter studies the time-delay influence on the small signal stability of the power system. The time-delay model and boundary structure of SSSR are investigated for power systems. An optimization-based algorithm is then proposed to trace the eigenvalue trajectory effectively, which determines the corresponding SSSR. Applying the developed algorithm, it can be found that time delay has little impact on the small signal stability when its value is small. However, the influence would become notable with the rise of the delay value. In the meantime, the time delay in the power system is detected inducing both negative and positive effects on the system small signal stability. These two effects should be properly studied and utilized during the design of wide area stability controllers for the power system.

References [1] H.J. Jia, X.D. Yu, Y.X. Yu, C.S. Wang, Power system small signal stability region with time delay, Int. J. Electr. Power Energy Syst. 30 (1) (2008) 16–22. [2] C.Y. Dong, H.J. Jia, T. Jiang, L.Q. Bai, Q.R. Hu, L. Wang, Y.L. Jiang, Effective method to determine time-delay stability margin and its application to power systems, IET Gener. Transm. Distrib. 11 (7) (2017) 1661–1670. [3] H.J. Jia, X.D. Yu, Method of determining power system delay margins with considering two practical constraints, Autom. Electr. Power. Syst. 32 (1) (2008) 6–10. [4] H.J. Jia, G.Y. Na, S.T. Lee, P. Zhang, Study on the impact of time delay to power system small signal stability, IEEE MELECON (2006) 1011–1014. [5] I. Kamwa, R. Grondin, Y. Hebert, Wide-area measurement based stabilizing control of large power systems—a decentralized/hierarchical approach, IEEE Trans. Power. Syst. 16 (1) (2001) 136–153. [6] H.X. Wu, K.S. Tsakalis, G.T. Heydt, Evaluation of time delay effects to wide-area power system stabilizer design, IEEE Trans. Power. Syst. 19 (4) (2004) 1935–1941. [7] Q.Y. Jiang, Z.Y. Zou, Y.J. Cao, Wide-area TCSC controller design in consideration of feedback signals time delays, in: IEEE PES, 2005, pp. 1676–1680. [8] M.S. Saad, M.A. Hassouneh, E.H. Abed, A.A. Edris, Delaying instability and voltage collapse in power systems using SVCs with washout filter-aided feedback, in: Proc. American Control Conference, 2005, pp. 4357–4362. [9] J. Luque, J.I. Escudero, F. Perez, Analytic model of the measurement errors caused by communications delay, IEEE Trans. Power Delivery 17 (2) (2002) 334–337. [10] Z.X. Hu, X.R. Xie, L.Y. Tong, Characteristic analysis and polynomial fitting based compensation of the time delays in wide-area damping control system, Autom. Electr. Power Syst. 29 (20) (2005) 29–34. [11] Q. Zhang, B.H. Wang, Controlling power system chaotic oscillation by time-delayed feedback, Power Syst. Tech. 28 (7) (2004) 23–26. [12] H.K. Chen, T.N. Lin, J.H. Chen, Dynamic analysis, controlling chaos and chaotification of a SMIB power system, Chaos Solitons Fractals 24 (5) (2005) 1307–1315. [13] H.J. Jia, X.D. Yu, X.D. Cao, Impact of the exciter voltage limit to small signal stability region of a three-bus power system, Int. J. Electr. Power Energy Syst. 33 (1) (2011) 1598–1607.

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[14] Y.L. Jiang, T. Jiang, H.J. Jia, C.Y. Dong, A novel LMI criterion for power system stability with multiple time-delays, Sci. China Tech. Sci. 57 (7) (2014) 1392–1400. [15] H.J. Jia, X.X. Xie, X.D. Yu, Power system small signal stability region with time delay considered, Autom. Electr. Power Syst. 30 (21) (2008) 1–5. [16] H.Y. An, H.J. Jia, X.D. Yu, On an improved delay-dependent robust stability criterion and application to power system stability analysis with time delays, in: Proc. Power Syst. Tech., 2010, pp. 1–6. [17] H.J. Jia, T.T. Song, X.D. Yu, A practical curve tracing method of delay stability margin of power system, Autom. Electr. Power Syst. 33 (4) (2009) 1–5.

CHAPTER

Robust stability of connected cruise controllers

8

Dávid Hajdua,c , Jin I. Geb , Tamás Inspergera,c , Gábor Oroszd a Department

of Applied Mechanics, Budapest University of Technology and Economics, Budapest, Hungary b Department of Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA, United States c MTA-BME Lendület Human Balancing Research Group, Budapest, Hungary d Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI, United States, and Department of Civil and Environmental Engineering, University of Michigan, Ann Arbor, MI, United States

Chapter outline 1 Introduction....................................................................................... 163 2 Modeling and control design with delays ................................................... 164 2.1 Car-following model ................................................................. 165 2.2 Connected vehicle systems ......................................................... 169 3 Robust string stability........................................................................... 173 3.1 Uncertainties in a predecessor-follower system ................................. 174 3.2 Robust connected cruise control design ......................................... 178 4 Conclusion ........................................................................................ 181 Acknowledgment.................................................................................... 182 References........................................................................................... 182

1 Introduction Over the past few decades, passenger vehicles have been equipped with more and more automation features in order to improve active safety, passenger comfort, and traffic throughput of the road transportation system. In particular, adaptive cruise control (ACC) was invented to alleviate human drivers from the constant burden of speed control [1]. While the influence of ACC is yet to be observed in real traffic due to its low penetration rate, theoretical studies have found that ACC-equipped vehicles may have limited benefits on traffic flow [2,3]. In particular, very high penetration of ACC vehicles is required to suppress speed fluctuations propagating through vehicle strings, as each ACC vehicle only responds to its immediate predecessor [4]. Stability, Control and Application of Time-Delay Systems. https://doi.org/10.1016/B978-0-12-814928-7.00008-1 © 2019 Elsevier Inc. All rights reserved.

163

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In order to overcome such limitations, cooperative adaptive cruise control (CACC) was proposed, where a platoon of ACC vehicles utilizes vehicle-to-everything (V2X) communication to coordinate their behavior [5–8]. CACC has been shown to have the potential to improve fuel economy and traffic efficiency in both theoretical and experimental studies [9–12]. However, the application of CACC in the early stages of driving automation may be significantly limited by the requirement that all vehicles in a CACC platoon must be automated aside from having V2X communication devices [13,14]. In particular, as mentioned in Ref. [15], “at low market penetrations, . . . the probability of consecutive vehicles being equipped is negligible.” Given the relatively low cost of V2X devices compared with driving automation, it is desirable to exploit the benefits of V2X without being restricted by the penetration rate of automation. Thus, we need to consider a connected automated vehicle design that is able to utilize V2X information sent from human-driven vehicles ahead. For the longitudinal control of such a connected automated vehicle design, we proposed the concept of connected cruise control (CCC) that exploits ad hoc V2X communication from multiple human-driven vehicles ahead [16]. Several theoretical studies have shown that connected cruise control is able to significantly improve active safety and fuel economy of the connected automated vehicle by providing head-to-tail string stability [17–20]. Since connected automated vehicle design relies on models of preceding vehicles, uncertainties in the models need to be considered to guarantee robust performance of the connected vehicle system. In Refs. [21–23], uncertainties of vehicle parameters in an automated platoon was considered and robust controllers were synthesized using the H∞ framework. Some other methods were also used in Refs. [24–27] to discuss the effects of unmodeled dynamics, stochastic communication delay, and measurement noise. However, a systematic method is needed to guarantee robust string stability against uncertain parameters of the human drivers ahead, such as their reaction time delays and feedback gains. In particular, uncertainties in the time delays should be taken into account without using overly conservative approximations. Moreover, such analysis should allow flexible connectivity topology and scale well as the number of vehicles connected via V2X communication increases. Therefore, in this chapter, we use structured singular value analysis [28,29] to provide tight bounds which allow connected cruise controllers to be head-to-tail string stable, despite uncertainties in human car-following behavior. We demonstrate through case studies how this robust string stability may improve the performance of a connected automated vehicle among human-driven vehicles.

2 Modeling and control design with delays In this section, we first model the car-following behavior of human-driven vehicles in nonemergency situations; see Fig. 1A. While many existing car-following models can be used to describe the longitudinal behavior of human-driven vehicles, the optimal velocity model has a very simple mathematical form and provides great physical intuitions. Thus, we choose this to model human-driven vehicles and also use it as a

2 Modeling and control design with delays

(A)

(B)

(C) FIG. 1 Two-vehicle configuration: (A) car-following model; (B) range policy function Vi (hi ) of vehicle i; and (C) block diagram of the transfer function Ti+1,i (s) of vehicle i.

basis for connected vehicle controller. For both human car-following behavior and the connected cruise controller, we introduce the notion of plant and string stability and calculate the nominal stability regions without considering parameter uncertainty.

2.1 Car-following model In this section, we introduce the model to describe the dynamics of a predecessorfollower pair. When neglecting the rolling resistance and air drag the longitudinal dynamics of the following vehicle i can be described by h˙ i (t) = vi+1 (t) − vi (t),

(1)

v˙ i (t) = ai (t), 1 a˙ i (t) = (ui (t) − ai (t)), ξi

(2) (3)

where vi+1 is the velocity of the preceding vehicle, hi , vi , and ai are the headway, velocity, and acceleration of the following vehicle i, and ui is its acceleration command. The powertrain dynamics is modeled through the actuator lag ξi [16,30]. Since the follower only uses motion information from the immediate predecessor, the acceleration command can be described by ui (t) = αi (Vi (hi (t − τi )) − vi (t − τi )) + βi (vi+1 (t − τi ) − vi (t − τi )),

(4)

where τi is the reaction time delay of a human driver and αi and βi are the control gains. Moreover, Vi (hi ) is the range policy function that describes the desired velocity based on headway. Here, we consider ⎧ ⎨0 Vi (hi ) = κi (hi − hi,st ) ⎩ vmax

if if if

hi ≤ hi,st , hi,st < hi < hi,go , hi ≥ hi,go ,

(5)

165

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CHAPTER 8 Robust stability of connected cruise controllers

see Fig. 1B. That is, the desired velocity is zero for small headways (hi ≤ hi,st ) and equal to the speed limit vmax for large headways (hi ≥ hi,go ). Between these, the desired velocity increases with the headway linearly, with gradient κi . Note that when hi,st = 0 [m], 1/κi is often referred to as the time headway. Many other range policies may be chosen, but the qualitative dynamics remains similar if the earlier characteristics are kept. Let us assume that the leader is traveling with the constant speed vi+1 (t) ≡ v∗ , then the follower admits the equilibrium hi (t) ≡ h∗i , vi (t) ≡ v∗ , ai (t) ≡ 0. We define the perturbations about the equilibrium as h˜ i (t) = hi (t) − h∗i ,

v˜ i (t) = vi (t) − v∗ ,

a˜ i (t) = ai (t),

(6)

that results in h˙˜ i (t) = v˜ i+1 (t) − v˜ i (t), v˙˜ i (t) = a˜ i (t), 1 a˙˜ i (t) = (αi (κi h˜ i (t − τi ) − v˜ i (t − τi )) + βi (˜vi+1 (t − τi ) − v˜ i (t − τi )) − a˜ i (t)). ξi

(7)

Taking the Laplace transform of Eq. (7) with zero initial conditions, one may derive the transfer function Ti+1,i (s) =

V˜ i (s) V˜ i+1 (s)

=

(αi κi + βi s)e−sτi , ξi s3 + s2 + (αi κi + (αi + βi )s)e−sτi

(8)

where s is the Laplace variable and V˜ i (s) and V˜ i+1 (s) denote the Laplace transforms of v˜ i (t) and v˜ i+1 (t). The block diagram is presented in Fig. 1C. In the rest of Section 2.1, we assume nominal parameter values in Eq. (8) and discuss their influence in human car-following behaviors. By abuse of notation, we drop the indices i of parameters and refer to them as κ, α, β, τ , and ξ . At the linear level, the system is plant stable, that is, approaches the equilibrium asymptotically, if all the infinitely many poles of Eq. (8), that is, the roots of the characteristic function D(s) = ξ s3 + s2 + (ακ + (α + β)s)e−sτ

(9)

are located on the left-half complex plane. When a real pole crosses the imaginary axis, substituting s = 0 into the characteristic equation D(s) = 0 yields the stability boundary α = 0.

(10)

On the other hand, when a complex conjugate pair of poles crosses the imaginary axis, substituting s = iΩ, Ω > 0 into D(s) = 0 and separating the real and imaginary parts result in the stability boundary

2 Modeling and control design with delays

= 0 [s] = 0.6 [s]

= 0.2 [s] = 0.4 [s]

3.0

Unstable

2.5 [1/s]

= 0.4 [s] = 0.2 [s]

= 0.5 [s] = 0.1 [s]

= 0.6 [s] = 0 [s]

Plant stable

2.0

String stable

1.5 1.0

P+

0.5 0

-1

P 0 1 [1/s]

P+ _

2 -1

P 0 1 [1/s]

P+

P+ _

2

-1

P 0 1 [1/s]

_

P+ _

2 -1

P 0 1 2 -1 [1/s]

P 0

1 [1/s]

_

2

FIG. 2 Plant and string stability charts for different delays and lags, while fixing τ + ξ = 0.6 [s], κ = 0.6 [1/s].

Ω2 (cos(Ωτ ) − ξ Ω sin(Ωτ )), κ β = Ω(ξ Ω cos(Ωτ ) + sin(Ωτ )) − α. α=

(11) (12)

Stability diagrams are presented in Fig. 2, where the thick red curves (or gray in grayscale print) are the nominal plant stability boundaries and the shading indicates the domain of the plant-stable control gains. The sum of the time delay and lag is kept τ + ξ = 0.6 [s], while having κ = 0.6 [1/s]. Comparison of the different panels reveals that the plant-stable domain decreases as the delay τ increases. To ensure string stability, that is, the attenuation of velocity perturbations between the leader and the follower at the linear level, we consider sinusoidal excitation amp v˜ i+1 (t) = vi+1 sin(ωt), which (assuming plant stability) leads to the steady-state amp amp amp sin(ωt + ψ), where vi /vi+1 = |Ti+1,i (iω)| and ψ = response v˜ ss i (t) = vi  Ti+1,i (iω). Requiring |Ti+1,i (iω)| < 1 for all ω > 0 ensures attenuation of sinusoidal signals and, as superposition holds for linear systems, for the linear combination of those signals. This condition may be rewritten as ω2 P(ω) > 0 where P(ω) = α 2 + 2αβ + ω2 + ξ 2 ω4 − 2(ακ + (α + β)ξ ω2 ) cos(ωτ ) − 2(α + β − ακξ )ω sin(ωτ ).

(13)

The stability boundaries can be identified corresponding to the minima of P becoming negative at ωˆ > 0 that is defined by P(ω) ˆ = 0, ∂ P(ω) ˆ = 0, ∂ω

(14) (15)

167

168

CHAPTER 8 Robust stability of connected cruise controllers

2

while satisfying ∂∂ 2 ωP (ω) ˆ > 0. Solving Eqs. (14), (15) for α and β one may obtain the string stability boundaries parameterized by ωˆ as α = f1 ± β=

 f12 + 2f0 ,

(16)

ακ(ξ τ ωˆ cos(ωτ ˆ ) + (ξ + τ ) sin(ωτ ˆ )) + ω(1 ˆ + 2ξ 2 ωˆ 2 ) − α, 2 (2ξ + τ )ωˆ cos(ωτ ˆ ) + (1 − ξ τ ωˆ ) sin(ωτ ˆ )

(17)

where f0 =

ωˆ 2 ((1 + (3ξ + τ )ξ ωˆ 2 + ξ 3 τ ωˆ 4 ) sin(ωτ ˆ ) − ω(τ ˆ − ξ 2 (2ξ − τ )ωˆ 2 ) cos(ωτ ˆ )) , 2(2ξ(κτ − 1) − τ )ωˆ cos(ωτ ˆ ) + 2(2κ(ξ + τ ) + ξ τ ωˆ 2 − 1) sin(ωτ ˆ )

(18)

f1 =

2ω(κ(ξ ˆ + τ ) − 1 + (κτ − 2)ξ 2 ωˆ 2 + κi ξ cos(2ωτ ˆ )) + κ(1 − ξ 2 ωˆ 2 ) sin(2ωτ ˆ ) . 2 2(2ξ(κτ − 1) − τ )ωˆ cos(ωτ ˆ ) + 2(2κ(ξ + τ ) + ξ τ ωˆ − 1) sin(ωτ ˆ )

(19)

For ωˆ = 0, the equalities |Ti+1,i (0)| = 1 and ∂ 2 |Ti+1,i | (0) ∂ω2

∂|Ti+1,i | ∂ω (0)

= 0 always hold. Thus, for

< 0 which is equivalent to P(0) = α(α + 2β − string stability, we need 2κ) > 0. This gives the stability boundaries α = 0,

(20)

α = 2(κ − β).

(21)

The nominal stability boundaries (16)–(21) are presented in Fig. 2 as black curves enclosing the string stable domain (shaded dark gray). Here, the dashed straight lines indicate the plant and string stability boundaries corresponding to Ω = 0 and ωˆ = 0, respectively. Note that the nominal string stable region grows and becomes open from above as τ decreases, indicating that the information delay τ has more significant influence on the string stability than the actuation lag ξ . Also, note that the string stable domain shrinks when the delay τ increases. In particular, there exist two anchor points P+ and P− corresponding to ωˆ = 0,  1 − 4κ(ξ + τ ) + 2κ 2 τ (2ξ + τ ) 4κ(ξ + τ ) − 2 , , 2τ (κτ − 1) + ξ(4κτ − 2) 2τ (κτ − 1) + ξ(4κτ − 2)  1 , (α − , β − ) = 0, 2(ξ + τ ) 

(α + , β + ) =

(22) (23)

which can be found by applying the L’Hospital rule to Eqs. (16)–(19). When increasing τ + ξ the anchor points move toward each other and when they meet, the string stable domain disappears. Using Eqs. (22), (23), we find the critical sum of the delay and lag is τcr + ξcr =

1 , 2κ

and for larger values the string stable domain disappears.

(24)

2 Modeling and control design with delays

... n+1 .

n.

2.

1

.

FIG. 3 A connected vehicle system resulting from a connected automated vehicle using V2X information from n human-driven vehicles ahead.

2.2 Connected vehicle systems We consider a heterogeneous chain of vehicles where all vehicles are equipped with V2X communication devices and some are capable of automated driving, as shown in Fig. 3. When an automated vehicle receives motion information broadcasted from a few vehicles ahead, it may choose to use the information in its motion control (see the dashed arrows), and thus, it becomes a connected automated vehicle. Such a V2Xbased controller then defines a connected vehicle system consisting of the connected automated vehicle and the preceding vehicles whose motion signals are used by the connected automated vehicle. Inside this connected vehicle system, we denote the connected automated vehicle as vehicle 0, and the preceding vehicles as vehicles 1, . . . , n. Note that we assume a connected automated vehicle does not “look beyond” another connected automated vehicle. For example, in Fig. 3, vehicle 0 does not include the V2X signals from vehicles farther ahead than vehicle n in its controller. This assumption greatly simplifies the topology of connected vehicle systems and eliminates intersections of links that are typically detrimental for the performance of the system [17,31]. The connected cruise controller for the connected automated vehicle (i = 0) is assumed in the form of u0 (t) =A1,0 (V0 (h0 (t − σ1,0 )) − v0 (t − σ1,0 )) +

n

Bj,0 (vj (t − σj,0 ) − v0 (t − σj,0 )), (25)

j=1

where the control gains Aj,0 and Bj,0 and communication delay σj,0 correspond to the links between vehicle j and the connected automated vehicle 0. Here, the range policy function V0 (h0 ) is defined as in Eq. (5), and its gradient is denoted by κ0 for h0,st ≤ h0 ≤ h0,go . Unlike many CACC algorithms, in the connected vehicle system shown in Fig. 3, the preceding vehicles 1, . . . , n are not required to cooperate with the connected automated vehicle. Moreover, aside from broadcasting their motion information through V2X communication, no automation of these vehicles is required. Correspondingly,

169

170

CHAPTER 8 Robust stability of connected cruise controllers

the feedback gains and delay times in Eq. (4) cannot be tuned for the connected automated vehicle design. However, the connected automated vehicle 0 may fully exploit V2X signals from vehicles 1, . . . , n with no constraint on the connectivity topology. Note that when n = 1, the connected automated vehicle only uses motion information from its immediate predecessor, and Eq. (25) gracefully degrades to the same control structure as human-driven vehicles [32] or automated vehicles without connectivity [4]. Similarly as in Section 2.1, here we consider the nominal stability of the connected vehicle system (1)–(4), (25) around the equilibrium, where the vehicles travel with the same constant speed vi (t) = v∗ , ai (t) = 0 and their corresponding headways are constant hi (t) = h∗i such that Vi (h∗i ) = v∗ . Linearization of Eqs. (1)–(4) including Eq. (25) about the equilibrium (h∗i , v∗ , 0) gives h˙˜ 0 (t) = v˜ 1 (t) − v˜ 0 (t), v˙˜ 0 (t) = a˜ 0 (t), ⎛ 1 ⎝A1,0 (κ0 h˜ 0 (t − σ1,0 ) − v˜ 0 (t − σ1,0 )) a˜˙ 0 (t) = ξ0 +

n

⎞

Bi,0 (˜vi (t − σi,0 ) − v˜ 0 (t − σi,0 )) − a˜ 0 (t)⎠

(26)

i=1

for the connected automated vehicle and h˙˜ i (t) = v˜ i+1 (t) − v˜ i (t), v˙˜ i (t) = a˜ i (t), 1 a˙˜ i (t) = (αi (κi h˜ i (t − τi ) − v˜ i (t − τi )) + βi (˜vi+1 (t − τi ) − v˜ i (t − τi )) − a˜ i (t)) ξi

(27)

for the human-driven vehicles i = 1, . . . , n. We assume that the connected vehicle system (26), (27) is plant stable, that is, when the input perturbation v˜ n+1 (t) ≡ 0, the perturbations h˜ i , v˜ i , a˜ i of the preceding vehicles and h˜ 0 , v˜ 0 , a˜ 0 of the connected automated vehicle tend to zero regardless of the initial conditions. Instead, we focus on how the connected automated vehicle responds to speed perturbations propagating through the system. When the speed fluctuation v˜ 0 of the connected automated vehicle has smaller amplitude than the input v˜ n , we call the connected automated vehicle design head-to-tail string stable. The notion of string stability between two consecutive vehicles was previously used to explain the amplification of speed perturbations along a chain of vehicles without connectivity [33]. However, by considering head-to-tail string stability, we allow speed perturbations to be amplified among the uncontrollable vehicles 1, . . . , n, and we focus on how the connected automated vehicle attenuates the perturbations.

2 Modeling and control design with delays

Being head-to-tail string stable not only enables a connected automated vehicle to enjoy better active safety, energy efficiency, and passenger comfort, but also it can help to mitigate traffic waves [17]. We assume zero initial conditions for Eqs. (26), (27) and obtain V˜ 0 (s) =

n

Ti,0 (s)V˜ i (s),

(28)

i=1

V˜ i (s) = Ti+1,i (s)V˜ i+1 (s),

where V˜ 0 (s) and V˜ i (s) denote the Laplace transforms of v˜ 0 (t) and v˜ i (t), and the link transfer functions are T1,0 (s) = Ti,0 (s) = Ti+1,i (s) =

(A1,0 κ0 + B1,0 s)e−sσ1,0 ,  ξ0 s3 + s2 + A1,0 (κ0 + s)e−sσ1,0 + nl=1 Bl,0 se−sσl,0 Bi,0 se−sσi,0

ξ0 s3 + s2 + A1,0 (κ0 + s)e−sσ1,0 +

n

(αi κi + βi s)e−sτi . ξi s3 + s2 + (αi κi + (αi + βi )s)e−sτi

l=1 Bl,0 se

(29)

−sσl,0 ,

(30) (31)

Thus, the head-to-tail transfer function of the connected vehicle system is Gn,0 (s) =

V˜ 0 (s) = det(T(s)), V˜ n (s)

(32)

where the transfer function matrix is given by ⎡

T1,0 (s) ⎢ T2,0 (s) ⎢ ⎢ T3,0 (s) ⎢ T(s) = ⎢ .. ⎢ . ⎢ ⎣T n−1,0 (s) Tn,0 (s)

−1 T2,1 (s) 0 .. . 0 0

0 −1 T3,2 (s) .. . 0 0

··· ··· ··· .. . ··· ···

0 0 0 .. . Tn−1,n−2 (s) 0

⎤

0 0 0 .. . −1

⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦

(33)

Tn,n−1 (s)

see Ref. [17] for the proof. The criterion for head-to-tail string stability at the linear level is guaranteed if the perturbations are attenuated for any frequency, that is, |det(T(iω))| < 1,

∀ω > 0,

(34)

where we substituted s = iω. In order to facilitate robustness analysis in the following section, we rewrite Eq. (34) as 1 − det(T(iω))δ c  = 0,

∀ω > 0,

(35)

where δ c is an arbitrary complex number inside the unit circle in the complex plane, that is, δ c ∈ C, |δ c | < 1.

171

172

CHAPTER 8 Robust stability of connected cruise controllers

(A)

(B) FIG. 4 A four-vehicle configuration: (A) connected vehicle system with the information flow indicated by the dashed arrows and (B) the corresponding block diagram showing the propagation of speed perturbations V˜ i (s), for i = 3, 2, 1, 0.

To illustrate the head-to-tail string stability, here we consider a connected automated vehicle using motion information from three preceding vehicles, as shown in Fig. 4A. The transfer function matrix for this connected vehicle system is ⎡

T1,0 (s) T(s) = ⎣T2,0 (s) T3,0 (s)

−1 T2,1 (s) 0

⎤ 0 −1 ⎦ , T3,2 (s)

(36)

where the elements T1,0 (s), T2,0 (s), T3,0 (s), T2,1 (s), and T3,2 (s) are given by Eqs. (29)–(31), while Eq. (32) results in the head-to-tail transfer function G3,0 (s) = det(T(s)) = T3,0 (s) + T3,2 (s)T2,0 (s) + T3,2 (s)T2,1 (s)T1,0 (s).

(37)

The flow of information is illustrated on a schematic block diagram in Fig. 4B. We consider the case when the preceding vehicles i = 1, 2 have parameters αi = 0.25 [1/s], βi = 0.5 [1/s], κi = 0.8 [1/s], τi = 0.3 [s], ξi = 0.5 [s], while the connected automated vehicle has σ1,0 = σ2,0 = σ3,0 = σ = 0.1 [s], κ0 = 0.6 [1/s], ξ0 = 0.5 [s] and the design parameters are chosen as A1,0 = 0.4 [1/s], B1,0 = 0.2 [1/s], B2,0 = 0.4 [1/s], and B3,0 = 0.4 [1/s]. In Fig. 5A, we plot the head-to-tail transfer function |G3,0 (iω)| of the connected automated vehicle (solid gray curve) and the link transfer function |T3,2 (iω)| that describes how vehicle 2 responds to the motion of vehicle 3 (dotted purple curve). Here this is equal to |T2,1 (iω)| as vehicles 2 and 1 have the same parameters. While the magnitude of the head-to-tail transfer function stays below 1, the link transfer functions of vehicles 2 and 1 reach beyond 1 for low frequencies. This indicates that

3 Robust string stability

(A)

(B)

FIG. 5 (A) Example transfer functions |T3,2 (iω)| = |T2,1 (iω)|, |G3,0 (iω)| and (B) corresponding simulations for ω = 0.6 [rad/s].

speed perturbations at low frequency are amplified by vehicles 2 and 1 but eventually are suppressed by the connected automated vehicle. This observation is supported by amp a simulation shown in Fig. 5B, where the speed input v3 (t) = v∗ + v3 sin(ωt) with amp v∗ = 15 [m/s], v3 = 5 [m/s], ω = 0.6 [rad/s] is plotted as dashed brown curve. The stationary time profiles for vehicles 2 and 1 are plotted by dotted purple and pointdotted green curves, respectively. The color code corresponds to the vehicle colors in Fig. 4A. In Fig. 5B, one may notice the difference in the equilibrium headway, that is, h∗0 = 30 [m] and h∗1 = h∗2 = 23.75 [m]. This is caused by the differences in the slopes of the range policy functions Vi (hi ). In particular, while hi,st = 5 [m] for i = 0, 1, 2, we have κ0 = 0.6 [1/s] and κ1 = κ2 = 0.8 [1/s]; see Eq. (5) and Fig. 1B. This highlights that the trajectories in Fig. 5B strongly depend on the parameters of the preceding vehicles. The same control parameters used in Fig. 5 may behave poorly with a different set of parameters κi , αi , βi , τi , and ξi .

3 Robust string stability Since a connected automated vehicle may not know the dynamics of the preceding vehicles 1, . . . , n accurately, the V2X-based controller should be robust against their parameter uncertainties (aside from the model uncertainties of the connected automated vehicle itself). In this section, we assume additive perturbations κ˜ i , α˜ i , β˜i , τ˜i , and ξ˜i in the parameters of human-driven vehicles. To ensure good performance under these parameter changes, we apply robust control design.

173

174

CHAPTER 8 Robust stability of connected cruise controllers

3.1 Uncertainties in a predecessor-follower system As an illustration of the robust string stability, we consider vehicle i that only uses information from one vehicle ahead. Here, we set i = 0 without loss of generality (see Fig. 1) and drop the index i of the parameters κ, α, β, τ , and ξ in this section. Thus, we have the input v˜ i+1 (t), the output v˜ i (t), and the nominal head-to-tail transfer function is given by Gi+1,i (s) = Ti+1,i (s) =

(ακ + βs)e−sτ ξ s3 + s2 + (ακ + (α + β)s)e−sτ

,

(38)

see Eq. (8). Uncertainties in the plant may appear in different forms. Without any restrictions, let us assume that every parameter in the model is uncertain. While additive ˜ ξ˜ , and κ˜ result in additive uncertainty terms, an additive delay uncertainties α, ˜ β, uncertainty τ˜ will result in a multiplicative exponential term e−sτ˜ in Eq. (38), that is, ˜ i+1,i (s) = Gi+1,i (s) + G

−s(τ +τ˜ ) ˜ ((κ + κ)(α ˜ + α) ˜ + (β + β)s)e

, −s(τ +τ˜ ) ˜ (ξ + ξ˜ )s3 + s2 + ((κ + κ)(α ˜ + α) ˜ + (α + α˜ + β + β)s)e (39)

˜ i+1,i (s) represents only the uncertainty. In spite of the fact that G ˜ i+1,i (s) can where G be expressed algebraically, the question is how to formulate the uncertain model that is the most suitable for robust analysis. In order to formulate the uncertainties in a general way, we must separate the uncertainty from the nominal model. We use the Rekasius substitution to handle uncertainty in the delay, such that e−sτ˜ =

˜ 1 − sϑ(s) , ˜ 1 + sϑ(s)

(40)

where we restrict ourselves to s = iω and therefore, we have ˜ ϑ(iω) =

ωτ˜ 1 tan , ω 2

(41)

see Ref. [34]. This substitution is exact and suitable for the robust analysis in the region 0 ≤ ω < π/τ˜ , since it covers the same domain in the complex plane as e−iωτ˜ . Similar attempt with higher-order approximation is given in Ref. [35]. By taking into account the uncertain parameters (with the Rekasius substitution), the block diagram shown in Fig. 6 can be drawn. Based on this graphical representation, one can get the following system of equations for the inputs u = [u1 , u2 , u3 , u4 , u5 ] , w = V˜ i+1 (s) and outputs y = [y1 , y2 , y3 , y4 , y5 ] , z = V˜ i (s) in the form 1 y1 = (w − z) , s 1 y2 = (w − z) κ + u1 − z, s

(42) (43)

3 Robust string stability

FIG. 6 Block diagram of the car-following model with parametric uncertainty.

y3 = (w − z),  1 (w − z) κ + u1 − z α + u2 + u3 + (w − z)β e−sτ − u4 s, y4 = s  1 1 y4 − u4 − u5 − y5 , y5 = s s ξ 11 z = y5 . ss

(44) (45) (46) (47)

This can be formulated as     y u = m(s) , z w u = δ(s)y,

(48) (49)

where m(s) is the interconnection matrix and δ(s) is the uncertainty matrix. In the present case, these matrices are written as  m(s) =

m1,1 (s) m2,1 (s)

 m1,2 (s) , m2,2 (s)

˜ ϑ(s), ˜ δ(s) = diag[κ, ˜ α, ˜ β, ξ˜ ],

(50) (51)

where m1,1 (s)

⎡

⎤

αe−sτ e−sτ e−sτ 2 1 2 + βse−sτ −(κ + s)e−sτ −sτ + s (κ + s)e 2(κ + s) s + κ ⎥ 1 ⎢ ⎥ −αse−sτ −se−sτ se−sτ 2s s = ⎢ ⎥, D(s) ⎣ αs3 (1 + ξ s)e−sτ s3 (1 + ξ s)e−sτ s3 (1 + ξ s)e−sτ se−sτ c(s) − s2 (1 + ξ s) e−sτ sc(s)⎦ e−sτ s3 e−sτ s3 −2s3 −s3 e−sτ s3 α

⎢ξ s3

⎡ m1,2 (s) =

1 D(s)

s + αe−sτ + s2 ξ κs − βse−sτ + s2 κξ

⎤

⎢ ⎥ ⎢ ⎥ 3 + s2 + αse−sτ ⎢ ⎥, ξ s ⎢ ⎥ ⎣(καs2 + βs3 )e−sτ (1 + ξ s)⎦ e−sτ s2 (sβ + ακ)

(52)

(53)

175

176

CHAPTER 8 Robust stability of connected cruise controllers

FIG. 7 m − δ uncertain interconnection structure.

1  −sτ αse se−sτ D(s) 1 (κα + βs)e−sτ , m2,2 (s) = D(s) m2,1 (s) =

se−sτ

−2s

 −s ,

(54) (55)

moreover D(s) is the characteristic function defined in Eq. (9) and c(s) = s(α + β) + ακ is introduced for convenience. This is called the m − δ uncertain interconnection structure that can be graphically represented by the block diagram in Fig. 7; see Ref. [28]. The transfer function between the input w and output z involves the uncertainty matrix δ(s). The solution can be expressed using the upper linear fractional transformation Fu (upper LFT) as Fu (m(s), δ(s)) := m2,2 (s) + m2,1 (s)δ(s)(I − m1,1 (s)δ(s))−1 m1,2 (s)       Gi+1,i (s)

=

˜ i+1,i (s) G

˜ 1+sϑ(s)

−sτ 1−sϑ(s) ˜ ((κ + κ)(α ˜ + α) ˜ + (β + β)s)e ˜ ˜ 1+sϑ(s)

−sτ 1−sϑ(s) ˜ (ξ + ξ˜ )s3 + s2 + ((κ + κ)(α ˜ + α) ˜ + (α + α˜ + β + β)s)e ˜

,

(56)

under the condition that det(I − m1,1 (s)δ(s))  = 0.

(57)

One can show that this condition results the perturbed characteristic equation, and therefore, a necessary condition for robust plant stability. Note that Eq. (56) is equivalent to Eq. (39) including the Rekasius substitution. Recall that the string stability criterion (35), similarly, the perturbed transfer function (56), needs to satisfy 1 − Fu (m(iω), δ(iω))δ c  = 0,

∀ω > 0

(58)

and for any complex number δ c ∈ C, |δ c | < 1. Using the Schur formula (see Ref. [36]), we rewrite Eqs. (57), (58) for s = iω as

3 Robust string stability

 det

I 0

  0 m1,1 (iω) − m2,1 (iω) 1

 m1,2 (iω) δ(iω) m2,2 (iω) 0

0 δc

  = 0.

(59)

In Eq. (59), the uncertainty matrix is not normalized. Let us introduce the weights ρi (s) and normalized uncertainties δir ∈ R, |δir | < 1, such that κ˜ = ρ1 δ1r ,

β˜ = ρ3 δ3r ,

α˜ = ρ2 δ2r ,

˜ ϑ(s) = ρ4 (s)δ4r ,

ξ˜ = ρ5 δ5r ,

(60)

where δi represents a real parameter uncertainty. Then, Eq. (59) can be rewritten in a normalized form as ˆ  = 0, ˆ det(I − m(iω) δ)

(61)

where  ˆ m(iω) =

m1,1 (iω)r(iω) m2,1 (iω)r(iω)

 m1,2 (iω) , m2,2 (iω)

(62)

r(iω) = diag[ρ1 , ρ2 , ρ3 , ρ4 (iω), ρ5 ],

(63)

δˆ = diag[δ1r , δ2r , δ3r , δ4r , δ5r , δ c ].

(64)

Eq. (63) emphasizes that the weight of each parameter is constant, except for the ˜ which varies with the frequency according to the Rekasius time delay (ρ4 = ϑ), substitution. In order to quantify the robustness of the system, we use the structured singular ˆ value (μ) analysis introduced by Doyle [28]. We define the μ-value of m(iω) as the ˆ when Eq. (61) fails at frequency ω, that is, inverse of the smallest σ¯ (δ)  ˆ det(I − m(iω) ˆ = 0} ˆ μ(ω) = min{σ¯ (δ): δ) δˆ

−1 ,

(65)

ˆ denotes the largest singular value of δ. ˆ When Eq. (61) holds regardless where σ¯ (δ) ˆ of the perturbation values in δ, we have μ(ω) = ∞, while if Eq. (61) is not satisfied for any perturbation structure, then μ(ω) = 0. As μ(ω) increases, a ˆ and results in ˆ smaller perturbation value in δˆ may lead to a singular (I − m(iω) δ) string instability. Therefore, the condition for robust string stability against bounded parameter variation is μ(ω) < 1,

∀ω > 0,

(66)

similarly to the head-to-tail string stability Eq. (34). Note that μ cannot be computed directly using Eq. (65), since the optimization problem is not convex in general and may have multiple local extrema [37]. However, if we are interested in finding upper and/or lower bounds, several alternative formulations have been developed [28,37–39]. In this work, we use the mussv function in MATLAB μ-Analysis and Synthesis Toolbox to obtain the upper and lower bounds of μ [40].

177

CHAPTER 8 Robust stability of connected cruise controllers

2.0

1.2

= 0.4 [s] = 0.2 [s]

1.5

3%

Uncert ainty 6% Uncertainty

1.0 0.5

9%

(A)

0.6 0.4

1%2 12%

0.2 0.4 0.6 0.8 1.0 [1/s]

15% 15% 12% 9% 9% 6% 3% 0%

1.0

0.8

15% 15% 0 0

1.1

1.0

( )

% 0% 0

[1/s]

178

0.2 0 0

0.7 0.7

Upper Lower |G 1,0 (i )|

1

2

(B)

3 [rad/s]

Uncertainty

4

5

1.8

6

FIG. 8 (A) Robust stability charts in the plane (β, α) for κ = 0.6 [1/s], τ = 0.2 [s], ξ = 0.4 [s] with uncertainties 0%, 3%, 6%, 9%, 12%, and 15% in κ, ξ , and τ (0% in α and β). Black curves indicate the nominal string stable boundary (0% uncertainty) and dark gray curves indicate the robust string stable boundaries. (B) The nominal transfer function |Gi+1,i (iω)| curve (thick black) and μ(ω) curves (thin gray ) for parameter point (β, α) = (0.6, 0.7) [1/s] marked by a black dot in panel (A).

As an example, a robust string stability diagram is presented in Fig. 8. To demonstrate the sensitivity of the string stable domain, we chose the parameters κ = 0.6 [1/s], τ = 0.2 [s], ξ = 0.4 [s], as shown in the middle panel in Fig. 2. Panel (A) shows the contraction of the robust string stable region as the uncertainty increases from 0% to 15% in parameters κ, τ , and ξ , while panel (B) plots the μbounds corresponding to control gains α = 0.6 [1/s] and β = 0.7 [1/s] marked as a black dot. Note that when the uncertainty in the parameters are set to zero, that is, r(iω) = 0, then the robust string stable condition (61) reduces to the nominal string stable condition (58), which is satisfied only if δ c = Gi+1,i (iω)−1 . In this case one obtains μ(ω) = |Gi+1,i (iω)| by definition from Eq. (65).

3.2 Robust connected cruise control design In order to demonstrate the applicability of the method developed here, we present a case study for the connected vehicle system consisting of a connected automated vehicle and three human-driven cars with uncertainty, that is, n = 3 in Eq. (27) as shown in Fig. 9A. The schematic block diagram with uncertainties is presented in Fig. 9B, which is the extension of Fig. 4B. While the nominal transfer function matrix is given in Eq. (36), we assume each parameter in vehicles 2 and 1 have certain levels of uncertainty and compute the robust string stable regions in the (B2,0 , B3,0 )-plane for different values of A1,0 and B1,0 . In order to construct a connected uncertainty structure, we need to include the uncertain model for each vehicle that might be different. Let us use upper indices

3 Robust string stability

(A)

(B) FIG. 9 Example configuration with uncertainties: (A) connectivity topology and (B) block diagram.

(3, 2) and (2, 1) to denote the links between vehicles 3–2 and 2–1. The corresponding system of equations for each vehicle read (2,1)

(2,1)

y1 = m1,1 (s)u1 + m1,2 (s)w1 ,

(67)

(3,2) (3,2) y2 = m1,1 (s)u2 + m1,2 (s)w2 , (2,1) (2,1) z1 = m2,1 (s)u1 + m2,2 (s)w1 , (3,2) (3,2) z2 = m2,1 (s)u2 + m2,2 (s)w2 ,

(68)

z = T1,0 (s)z1 + T2,0 (s)z2 + T3,0 (s)w2 , w1 = z2 , w2 = w.

(69) (70) (71) (72) (73)

Note that here we have w = V˜ 3 (s) and z = V˜ 0 (s). After simplifications with (2,1) (3,2) m2,2 (s) = T2,1 (s) and m2,2 (s) = T3,2 (s), the solution for Eqs. (67)–(73) directly yields ⎡ ⎤ ⎡ ⎤ y1 u1 ⎣y2 ⎦ = M(s) ⎣u2 ⎦ , z w

(74)

179

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CHAPTER 8 Robust stability of connected cruise controllers

where the interconnection structure is given by ⎡

m1,1 (s)

(2,1)

m1,2 (s)m2,1 (s)

m12 (s)T3,2 (s)

0

m1,1 (s)

m1,2 (s)

m2,1 (s)T1,0 (s)

m2,1 (s)(T2,0 (s) + T1,0 (s)T2,1 (s))

G3,0 (s)

⎢ M(s) = ⎢ ⎣

(2,1)

(2,1)

(3,2)

(2,1)

(3,2)

(3,2)

(3,2)

⎤ ⎥ ⎥ , (75) ⎦

and the weight matrix of the corresponding uncertainties reads R(s) =

 (2,1) (s) r 0

0



. r(3,2) (s)

(76)

With some generalizations, one can show that the structure of equations remains the same, that is, ˆ ˆ  = 0, det(I − M(iω) )

(77)

where  M1,1 (iω)R(iω) ˆ M(iω) = M2,1 (iω)R(iω)

 M1,2 (iω) , M2,2 (iω)

ˆ = diag[δ r , . . . , δ r , δ c ].  1 10

(78) (79)

Here, 10 corresponds to the 5 + 5 independent parameters of vehicles 1 and 2. Again, the computation of μ-values are performed by the MATLAB toolbox using the mussv function. The results are presented in Fig. 10A, where we assumed that each parameter of each vehicle is perturbed by the same percentage of their nominal value, that is, αi , βi , κi , τi , and ξi has identical relative uncertainty. The nominal human driver parameters are κi = 0.8 [1/s], αi = 0.25 [1/s], βi = 0.5 [1/s], τi = 0.3 [s], and ξi = 0.5 [s] (same for both vehicles for simplicity), while the fixed parameters of the connected automated vehicle are κ0 = 0.6 [1/s], σ = σ1,0 = σ2,0 = σ3,0 = 0.1 [s], and ξ0 = 0.5 [s]. The same set of parameters were used in the simulations in Section 2.2, Fig. 5. In this configuration, human-driven vehicles are string unstable, but head-to-tail string stability can be guaranteed by appropriate selection of the gains of the connected automated vehicle. The subplots in Fig. 10A show how the uncertain parameters (10 in total) affect the robust stable domain of control parameters (A1,0 , B1,0 , B2,0 , B3,0 ). One of the most robust parameter combination for realizable gain combinations is located at (0.4, 0.2, 0.3, 0.3) [1/s]. For this point, some μ(ω) curves are presented in Fig. 10B for different uncertainty levels. The curves show that the upper-lower bounds are sufficiently tight, therefore, the robust boundaries obtained by the numerical method are close to the real robust boundaries. The μ-curves also show that at least 20% robustness can be guaranteed with the selected control gains.

4 Conclusion

B 3,0 [1/s]

B 1,0 = 0.1 [1/s]

A1,0 = 0.1 [1/s]

A1,0 = 0.2 [1/s]

A1,0 =0.3 [1/s]

A1,0 = 0.4 [1/s]

2 20 30

1

0

10

10

20

0

20

20

10

30

10

0 0

0

B 3,0 [1/s]

B 1,0 = 0.2 [1/s]

2

20

1

0

10

30

10

20

0

30

10

20

20

30 0

10

(0.3,0.3) (0.4,0.4)

0

0

B 3,0 [1/s]

B 1,0 = 0.3 [1/s]

2

20

1

0

10

10

30 20

30

0

20

10

30

20

10

0 0

0

B 3,0 [1/s]

B 1,0 = 0.4 [1/s]

2

20

1

0

10

20

10

20

10

0 0

0 0

(A)

2

4

B 2,0 [1/s]

0

2

4

0

2

B 2,0 [1/s]

4

B 2,0 [1/s]

1.4

0

2

1.0 0.8 0.6

4

B 2,0 [1/s]

30% 25% 20% 15% 10% 5% 0%

1.2

( )

30

10

20

0

Upper Lower |G3,0(i )|

0.4

Uncertainty

0.2 0

0

(B)

0.5

1.0

1.5

2.0

2.5

3.0

[rad/s]

FIG. 10 (A) Robust string stability charts in the (B2,0 , B3,0 ) plane for different values of A1,0 and B1,0 , when using uncertainties 10−20−30%. (B) μ-curves for different levels of uncertainty at parameter point (A10 , B10 , B20 , B30 ) = (0.4, 0.2, 0.3, 0.3) [1/s].

4 Conclusion In this chapter, we applied the structured singular value analysis to investigate the influences of uncertain human car-following parameters on string stability of

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connected cruise controllers. In particular, the uncertain time delays were handled using the Rekasius substitution, so that the robust bounds on head-to-tail string stability remained tight. We demonstrated through case studies that these robustness results could be used to design connected automated vehicles that reject traffic perturbations well and improves performance of traffic flow despite uncertain human car-following behavior.

Acknowledgment This work is supported by the ÚNKP-17-3-I. The research reported in this paper was supported by the Higher Education Excellence Program of the Ministry of Human Capacities in the frame of Artificial intelligence research area of Budapest University of Technology and Economics (BME FIKP-MI). New National Excellence Program of the Ministry of Human Capacities.

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CHAPTER

A design approach for structured controllers for uncertain delay systems grounded in the real structured pseudospectra framework

9

Francesco Borgioli, Wim Michiels Deparment of Computer Science, KU Leuven, 3001 Heverlee, Belgium

Chapter outline 1 Introduction....................................................................................... 2 Robust stability of an uncertain system: A pseudospectral approach .................. 3 Computation of the pseudospectral abscissa ............................................... 4 Smoothness properties and optimization of the pseudospectral abscissa ............. 5 Numerical experiments ......................................................................... 6 Concluding remarks ............................................................................. Acknowledgments .................................................................................. References...........................................................................................

185 188 192 199 201 205 206 206

1 Introduction The stabilization of linear time-delay systems is a topic of major concern in control systems theory and a substantial amount of results have contributed to develop and partially solve this problem; without being exhaustive, possible approaches for stability analysis and stabilization are the use of Lyapunov-Krasovskii functionals and linear matrix inequality (LMI) conditions (see, for instance, Refs. [1–3]), the direct eigenvalue optimization approach [4], the continuous pole placement method [5], the use of prediction-based controllers (see Ref. [6] and references therein [7,8]). We refer the reader interested in a general overview on the many different stabilization methods to the monographs [9–11]. In this work, we consider problems where the closed-loop system takes the form Stability, Control and Application of Time-Delay Systems. https://doi.org/10.1016/B978-0-12-814928-7.00009-3 © 2019 Elsevier Inc. All rights reserved.

185

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E˙x(t) = A0 (p)x(t) +

m 

Ai (p)x(t − τi (p)),

(1)

i=1

where E ∈ Rn×n is allowed to be a singular matrix (see Section 2 for a detailed description of the assumptions), x(t) ∈ Rn is the state variable, A0 (p), . . . , Am (p) ∈ Rn×n , 0 < τ1 (p) < · · · < τm (p) are the delay terms, and p ∈ Rnp is a vector of design or controller parameters. Note that this general form also allows to address problems where delays are used as controller parameters. Most of the times, a controller is implemented in order to make a system asymptotically stable; using a frequency approach, this means to push all the roots of its associated characteristic equation to the open left-half of the complex plane, or equivalently, to have a negative spectral abscissa, that is, the real part of the rightmost eigenvalue in the spectrum. Minimizing the spectral abscissa w.r.t. some control or design parameters, as done in Ref. [4], is, therefore, a simple and effective method to stabilize a time-delay system. However, the spectral abscissa is not a good measure for the stability robustness: in real-life applications, uncertainties on system parameters are always present and potentially affect the system’s stability; in particular, they affect the spectrum of the system, generating a shift of its eigenvalues in the complex plane. In real applications, we can assume to have an upper bound ε on the size of the uncertainties affecting our system of interest: the ε-pseudospectrum is then defined as the region in the complex plane to which eigenvalues can be moved when the system is subject to some ε-bounded uncertainties; the ε-pseudospectral abscissa is then the supremum of the real parts of the points in the ε-pseudospectrum, and it constitutes a bound on the asymptotic growth rate of the solution of an uncertain system, which is uniform over all ε-bounded uncertainties. As a consequence, a negative pseudospectral abscissa guarantees the robustness of the asymptotic stability of system (1) against any ε-bounded uncertainty on the model; in this chapter, we propose a method to robustly stabilize system (1) via the minimization of its pseudospectral abscissa w.r.t. a fixed set of parameters p. In the literature, important distinctions are made regarding the perturbations of the system matrices: perturbations can be considered either complex valued or real valued, where the latter ones are certainly more realistic from an application point of view. Perturbations can also be distinguished between structured and unstructured, where the former ones allow the perturbation of any specific submatrix, for example, a single coefficient or a block. Among the many algorithms for the computation of the pseudospectral abscissa, we refer the reader to Refs. [12,13]; the former exploits unstructured real-valued perturbations of a standard eigenvalue problem, whereas the latter applies to nonlinear eigenvalue problems (e.g., polynomial and delay eigenvalue problems [DEPs]) whose matrices are perturbed by unstructured complex-valued matrices. In Ref. [14], we inherited the real-valued feature from the former and the nonlinearity structure from the latter: moreover, we simultaneously considered real-valued uncertainties on the delay terms and real-valued structured uncertainties on the system matrices. In addition to this method, as main contribution

1 Introduction

in this chapter, we take into account the potential dependency between the uncertainties perturbing the nominal matrices Ai : this has a major relevance for those uncertain system parameters which are present in more than one matrix Ai and will be motivated by an application from machine tool vibrations. An original, distinctive feature of the overall approach in this chapter is that robustness of stability, expressed in terms of negative pseudospectral abscissa, is assessed by solving an eigenvalue optimization problem where the real part of the rightmost eigenvalue is maximized as a function of the allowable perturbations. First, this setting allows an exact description of a broad class of uncertainties. As a matter of fact, the special structure of the delay system, the property that uncertain parameters are real valued, possible interdependencies between perturbations, and a nonlinear dependence on uncertain data (uncertainties affecting delay values too) can be easily exploited. Particularly, the latter is very difficult within the standard approaches in robust control, which are grounded in structured singular value characterizations (see, for instance, Refs. [15,16]). Second, due to use of rightmost eigenvalue computations and the exploitation of the structure on the uncertainty, the obtained robust stability conditions are necessary and sufficient, in contrast to many approaches grounded in Lyapunov’s second method [17] and resulting in potentially very conservative criteria, for example, LMI-based ones (see Refs. [18,19] for a tutorial). Third, we adopt a system description in terms of delay differential algebraic equations (DDAEs) (1), which allow to systematically model interconnections of subsystems (including interconnections between systems and controllers). Furthermore, in this chapter, we make the leap from robust stability analysis to the design of robustly stabilizing controllers. The distinctive feature of the adopted approach is that this is done by minimizing the pseudospectral abscissa (in such a way that the optimization problem behind the overall approach can be seen as a min-max (saddle point) problem); since the pseudospectral abscissa is by definition a continuous but only almost everywhere differentiable function, the optimization is performed using the hybrid algorithm for nonsmooth optimization (HANSO) method introduced in Ref. [20] (see website indicated in Ref. [21] for a detailed description). An additional advantage is that structured controllers can be designed, for example, low-order controllers that are easy to implement (note that within a Lyapunov framework the design of reduced order controllers would give to complicated bilinear matrix inequalities). Some preliminary results in this direction, without taking into account interdependencies between perturbations, are reported in Ref. [22]. The chapter is structured as follows: in Section 2, we formally introduce the uncertainties on system (1), the corresponding DEP and the pseudospectral approach used to investigate the stability robustness; in Section 3, we review and extend the method to compute the pseudospectral abscissa introduced in Ref. [14]; in Section 4, we briefly describe the smoothness properties of the pseudospectral abscissa and compute the gradient of the pseudospectral abscissa w.r.t. design or controller parameters employed as optimization variables; in Section 5, we show the applicability of this method to linear delay systems with static or dynamic controllers and present some numerical experiments; and conclusions are finally presented in Section 6.

187

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2 Robust stability of an uncertain system: A pseudospectral approach In Sections 2 and 3, we model the uncertainty on system (1), define and present an algorithm for the pseudospectral abscissa computation. Since we consider the set of controller parameters p as fixed, in these two sections we suppress the dependence of Ai on p in the notations (for the optimization of the pseudospectral abscissa as a function of p, we refer to Section 4). We consider system (1) and we want to include real-valued uncertainties on the matrices Ai and on the delay terms τi ; uncertainties affecting matrices Ai are also allowed to be structured, and we admit an interdependency between them. Therefore, we assume the existence of K independent uncertainties δAj , where each δAj can perturb one or more nominal matrix Ai , and the existence of m uncertainties δτi . Introducing these uncertainties in system (1), we have ⎛ E˙x(t) = ⎝A0 +

K 

⎞ Bi,j δAj Ci,j ⎠ x(t) +

j=1

m 

⎛ ⎝Ai +

i=1

K 

⎞ Bi,j δAj Ci,j ⎠ x (t − (τi + δτi )) , (2)

j=1

where δAj ∈ Rpj ×qj , δτi ∈ R and is such that |δτi | < τi ; each nominal matrix Ai can be perturbed by one or more uncertainties δAj , whose structures (and weights) are defined by auxiliary matrices Bi,j and Ci,j . Matrices Bi,j and Ci,j are real valued and their dimensions depend on the dimension of the associated uncertainty δAj : as a consequence, for each δAj ∈ Rpj ×qj , we have Bi,j ∈ Rn×pj and Ci,j ∈ Rqj ×n for all i = 0, . . . , m, j = 1, . . . , K. Obviously, if matrix Ai is not affected by uncertainty δAj , we assume Bi,j = 0 or Ci,j = 0. In the following, with the purpose  of simplifying the  notation, we will indicate with R∗ = Rp1 ×q1 × · · · × RpK ×qK the domain of the tuple of uncertainties (δA1 , . . . , δAK ). We here introduce the DEP associated with the system of perturbed Eq. (2) ⎛ M(λ)y := ⎝λE −

m  i=0

⎛ ⎝Ai +

K 

⎞

⎞

Bi,j δAj Ci,j ⎠ e−λ(τi +δτi ) ⎠ y = 0,

(3)

j=1

where λ ∈ C and y ∈ Cn . In addition, with the purpose of a simpler notation, we make the system more compact by adopting τ0 = 0, δτ0 = 0. The general DDAE models (1), (2) may arise as the feedback interconnection of a plant model with a controller with a fixed order or structure, where p in Eq. (1) is the parameterization of this controller. We here give two examples to explain the definition of the uncertainties and the benefits derived from a DDAE formulation; the same examples will be considered again in Section 5. In the first one, we show a classic case of an uncertain system with static feedback controller that can be recast using DDAEs; in the second one, we consider a model for a rotating cutting machine affected by uncertainties.

2 Robust stability of an uncertain system: A pseudospectral approach

Example 1. Consider the system with uncertainties

x˙ (t) = (A + δA)x(t) + (B + δB)u(t − (τ + δτ )) y(t) = (C + δC)x(t),

(4)

with static controller u(t) = Ky(t).

(5)

Defining the new state variable ξ := (x, y, u)T , the system can be recast as follows: ⎞

⎛ ⎡

⎤ I 0 0 ⎣ 0 0 0 ⎦ ξ˙ (t) 0 0 0    := E

⎜⎡ ⎤ ⎡ ⎤T ⎟ ⎤ ⎡ ⎤ ⎡ ⎤T ⎡ ⎜ A 0 0 I 0 I I ⎟ ⎟ ⎜ ⎜ ⎣ ⎣ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎦ I + 0 δA 0 + = ⎜ C −I 0 δC 0⎦ ⎟ ⎟ ξ (t) ⎜ 0 K −I 0 0 0 0 ⎟ ⎝        ⎠ := A0 := B δA C := B δA C 0,1 1 0,1 ⎛ ⎞ 0,2 2 0,2 ⎜⎡ ⎤ ⎡ ⎤ ⎡ ⎤T ⎟ ⎜ 0 0 B ⎟ I 0 ⎜ ⎟ ⎜ ⎟ ξ (t − (τ + δτ )). ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ + ⎜ 0 0 0 + 0 δB 0 ⎟ ⎜ 0 0 0 ⎟ 0 I ⎝     ⎠ := A1

:= B1,3 δA3 C1,3

In the last equation, we reformulated the original problem in form of DDAEs with perturbations as in Eq. (2), where p in Eq. (1) is the vectorization of the controller parameters in K. Note that the elimination of input u would lead to a nonlinear dependence of the system matrices on the uncertainty: as a matter of fact, if one substitutes control law (5) in the system of DDAE (4), then the representation (2) is not valid anymore, since the uncertainty δB cannot be decoupled from the controller parameters in K; the adopted reformulation in a DDAE framework then allows us to solve this problem. More in general, thanks to the introduction of a slack variable, we are able to handle any polynomial dependence on the uncertainties within a DDAE framework. Furthermore, original perturbations δA, δC are treated as two independent perturbations (renamed δA1 and δA2 , respectively) affecting matrix A0 of the new system of DDAE; the original δB is another independent perturbation affecting matrix A1 of the new system. Example 2. We consider a simple model for a turning process as presented in Refs. [23,24] x¨ (t) + 2ξ ω˙x(t) + ω2 x(t) =

k (x(t − τ ) − x(t)) , m

whose corresponding DEP reads   λE − A0 − A1 e−λτ y = 0,

(6)

189

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CHAPTER 9 Real structured pseudospectra framework

with  E=

1 0

0 1



 ,

A0 =

0 −ω2 − mk

1 −2ξ ω



 A1 =

,

0 k m

0 0

 ,

(7)

where ω is the natural frequency, ξ is the damping ratio, m is the modal mass, and k is the cutting force coefficient. Considering system (1), we then have τ1 = τ . Let us now assume that parameters k and ξ are, respectively, affected by uncertainties δk = δA1 and δξ = δA2 , and that delay term τ1 also admits a perturbation δτ1 . Then, we obtain the perturbed DEP     M(λ) = λE − A0 + B0,1 δkC0,1 + B0,2 δξ C0,2 − A1 + B1,1 δkC1,1 e−λ(τ1 +δτ1 ) ,

(8)

where 

 T



0 1 B0,1 = −B1,1 = 1 , C0,1 = C1,1 = 0 −m





0 , B0,2 = , C0,2 = −2ω



0 1

T .

In this case, perturbation δk = δA1 thus affects both nominal matrices A0 and A1 . The algorithm presented here to compute the pseudospectral abscissa, which generalizes the one introduced in Ref. [22], enables us to impose that the perturbation δA1 affecting both nominal matrices A0 and A1 stems from the same uncertain parameter k. Now, we need to define an overall scalar measure of the uncertainties affecting our system of interest. We first define the set of all the uncertainties as  := (A, τ ) = (δA1 , . . . , δAK , δτ1 , . . . , δτm ) , and then, after introducing weights vi ∈ R+ ∪ {+∞}, we define the global norm    :=  δA1 F

...

δAK F

v1 |δτ1 |

...

vm |δτm |

T   

∞

,

(9)

where ·F indicates the matrix Frobenius norm. Given this definition, a set of perturbations  is ε-bounded if and only if   δAj  ≤ ε, F

|δτi | ≤

ε , vi

j = 1, . . . , K, i = 1, . . . , m.

Moreover, if a weight vi = +∞ then the corresponding delay term τi is assumed free from uncertainty. It is worth to highlight that definition (9) allows bounds on different perturbations to be specified independently on each other; in this sense, it is possible to treat the case where two or more independent uncertainties are simultaneously set equal to their maximum allowed value. Equivalently, the space of ε-bounded perturbations can be represented as an hypercube rather than a hypersphere.

2 Robust stability of an uncertain system: A pseudospectral approach

At this point, we can define the real-valued, structured ε-pseudospectrum as the following set:   

Λε :=

λ ∈ C : det M(λ) = 0 ,

(10)

 ∈ R∗ × R m glob ≤ ε

and the ε-pseudospectral abscissa function αε as αε := sup{R(λ): λ ∈ Λε }. In this chapter, we will always deal with DDAEs with retarded dynamics, which means in particular that the spectral abscissa of a DEP is always continuous with respect to small variations in the system matrices and in the delay terms; as a consequence, in this case the pseudospectral abscissa can be simply reduced to a maximum function. We conclude the section presenting the adopted assumptions that guarantee our perturbed DEP (3) to have retarded dynamics. Considering system (1), let rank(E) = n − ν and let U, V ∈ Rn×ν be, respectively, a (minimal) basis for the left and the right null space of E. Then, if we define U = [U ⊥ U],

V = [V ⊥ V],

and x = V[x1T x2T ]T ,

system (1) can be rewritten as follows: m (11)

 (12) x1 (t − τi ) + m x2 (t − τi ), i=0 Ai i=0 Ai m m (22) (22) (21) 0 = A0 x2 (t) + i=1 Ai x2 (t − τi ) + i=0 Ai x1 (t − τi ),

E(11) x˙1 (t) =

where E(11) = U ⊥ EV ⊥ , and T

Ai = U ⊥ Ai V ⊥ , (21) Ai = U T Ai V ⊥ , (11)

T

Ai = U ⊥ Ai V (22) Ai = U T Ai V, i = 0, . . . , m. (12)

T

We refer to Ref. [25], from which we take over the following assumption made throughout the chapter. Assumption 1. The matrix U T A0 V is nonsingular. With this assumption, the reformulation of DDAEs as a set of delay differential equations (DDEs) of retarded type coupled with a set of DDEs is well posed and semiexplicit (differentiation index equal to one). The DDEs may induce in the spectrum of DDAEs chains of eigenvalues whose imaginary part tends to infinity while the real part has a finite limit (note that neutral systems can be reformulated as DDAEs); moreover, this limit may be discontinuous with respect to small perturbations of the delay values. Since in these papers we also deal with uncertainties on the delay terms, it is desirable to have continuity of the spectral abscissa w.r.t. delay terms. This is achieved by making the following

191

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CHAPTER 9 Real structured pseudospectra framework

assumption, which implies that the system has retarded dynamics, and that this is preserved in the presence of perturbations. Assumption 2. Matrices U T Ai V = 0 for i = 1, . . . , m. Moreover, for each (i, j) with i = 1, . . . , m, j = 1, . . . , K it must hold that either U T Bi,j = 0 or Ci,j V = 0. Essentially, by introducing this assumption, we eliminate the aforementioned chains of characteristic roots: we refer the reader to Ref. [25] for a detailed description of these assumptions.

3 Computation of the pseudospectral abscissa In this section, we illustrate the algorithm for the computation of the pseudospectral abscissa for Eq. (10), where we summarize the main steps from Ref. [22]. As previously mentioned, the novelty here introduced is the potential interdependency among the uncertainties δAj . We first include the following lemma, which provides a formula for the derivative of an eigenvalue w.r.t. any parameter of the eigenvalue problem; this formula will be widely exploited in this and in the following section, and we refer the reader interested in a proof to Lemma 2.7 in Ref. [26]. Lemma 1. Let F(λ, θ ): C × Cd −→ Cn×n be continuously differentiable, with λ a simple eigenvalue whose corresponding left and right eigenvectors with unit norm are x and y, and θ a set of parameters, then ∂F(λ, θ) x∗ y ∂λ ∂θi , =− ∂F(λ, θ) ∂θi y x∗ ∂λ

i = 1, . . . , d,

(11)

where x∗ is the conjugate transpose vector of x. The core idea that we implement is to find the maximum real part among the points in the pseudospectrum by taking steps in the direction of the (scaled) gradient of the spectral abscissa of the perturbed DEPs in the space of perturbations. In practice, indicating with λ() the rightmost eigenvalue of the DEP (3) perturbed with , we want to solve the following optimization problem: max s.t.

λ()  ∈ R ∗ × Rm , . glob ≤ ε

(12)

The next fundamental theoretical result provides a characterization of the optimal perturbations, namely the perturbations that generate the globally rightmost point in the pseudospectrum, and thus allows us to sensibly reduce the search space of the problem; in particular, it demonstrates that the optimal perturbations affecting the system matrices are always low rank. Before we state the theorem, we define kj as the number of nominal matrices Ai affected by uncertainty δAj .

3 Computation of the pseudospectral abscissa

Theorem 1. Let λRM be the globally rightmost point of the structured ε-pseudospectrum (10) for some optimal ε-bounded perturbation  = (A, τ ) and assume it is a simple eigenvalue. Then,  = (δA 1 , . . . , δA K ), such that (i) There always exists a set of ε-perturbations A for each j = 1, . . . , K rank(δAj ) ≤ ρj∗ := min(2kj , pj , qj ) and for which the rightmost eigenvalue is λRM . (ii) Let x, y be, respectively, the left and the right eigenvectors (with unitary norms) of λRM and such that ⎛ ξ := − x ⎝E +

m 

⎛ ⎝Ai +

i=0

K 

⎞

⎞

Bi,j δAj Ci,j ⎠ (τi + δτi )e−λRM (τi +δτi ) ⎠ y > 0,

(13)

j=1

and let us define X = [R(x) I(x)], and

 Γ0 =  Γi =

1 0

0 1

Y = [R(y) I(y)],

 ,

R(e−(τi +δτi )λRM ) I(e−(τi +δτi )λRM )

−I(e−(τi +δτi )λRM ) R(e−(τi +δτi )λRM )

 ,

for i = 1, . . . , m.

 T T T Now, for each j = 1, . . . , K, matrix m i=0 Bi,j XΓi Y Ci,j can be either zero or nonzero. In the latter case, the optimal perturbation is m

T T T i=0 Bi,j XΓi Y Ci,j  . δAj = −ε   m T  i=0 BTi,j XΓi Y T Ci,j 

(14)

F

(iii) Let x, y be defined as before, then ⎛ ⎛ ⎞ ⎞ K  1 ⎝ ∗⎝ ∂R(λRM ) = R x Ai + Bi,j δAj Ci,j ⎠ λRM e−λRM (τi +δτi ) y⎠ ∂δτi ξ

(15)

j=1

can also be either zero or nonzero. If it is nonzero, then the optimal time-delay perturbation is such that vi |δτi | = ε. Proof. (i) Suppose without loss of generality that the delay terms are free from uncertainties. Moreover, in order to simplify the notation, let us also suppose without loss of generality that each uncertainty δAj perturbs the first kj nominal matrices.

193

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CHAPTER 9 Real structured pseudospectra framework

It is trivial to see that the relation rank(δAj ) ≤ min{pj , qj } always holds. Let us now also assume that 2kj ≤ min{pj , qj }, and thus that ρj∗ = 2kj . For each j = 1, . . . , K, we let rj∗ ≤ ρj∗ be the rank of the space spanned by vectors in ! " Z := R(C1,j y), I(C1,j y), . . . , R(Ckj ,j y), I(Ckj ,j y) ; then we define a matrix Hj such that the first rj∗ columns’ span includes the space spanned by vectors in Z, and the other columns complete the basis of Rqj ×qj , so that Hj is orthogonal. #j H T , j = 1, . . . , K. With this factorization, Thus, we can write δAj = δA j ⎛ 0 = ⎝λRM E −

m  i=0

⎛ = ⎝λRM E −

m  i=0

⎛ = ⎝λRM E −

m  i=0

⎛ ⎝Ai +

K 

⎞

⎞

Bi,j δAj Ci,j ⎠ e−λRM τi ⎠ y

j=1

⎞

Ai e−λRM τi ⎠ y −

m  K 

  #j H T e−λRM τi R(Ci,j y) + jI(Ci,j y) Bi,j δA j

i=0 j=1

⎞ Ai e−λRM τi ⎠ y ⎛

⎞ (1) Hj , R(Ci,j y) + jI(Ci,j y) ⎜ ⎟ .. ⎜ ⎟ ⎜ ⎟ . ⎜ ⎟ m K ∗  ⎜ (rj ) ⎟ −λ τ #j e RM i ⎜Hi , R(Ci,j y) + jI(Ci,j y) ⎟ . Bi,j δA − ⎜ ⎟ ⎜ ⎟ 0 i=0 j=1 ⎜ ⎟ ⎜ ⎟ .. ⎝ ⎠ . 0

From the last expression, it is easy to see that we can get rid of the last qj − rj∗ columns of Hj . Thus, for every j = 1, . . . , K, we define a new matrix Hj by resetting all elements in Hj to zero except for the elements in the first rj∗ #j H T , we obtain matrices with rank r∗ columns. Therefore, defining δAj = δA j j that preserve the eigenvalue λRM  and its right  eigenvector y. We still have to prove that δAj F ≤ δAj F ; this is equivalent to proving that the spectral norm on each row of δAj is larger than the spectral norm of the same row in δAj . Without loss of generality let us investigate the spectral norms of the first row in each matrix. Let     #j = β1 . . . βqj δA (1) #j and the matrices H T and H T defined as follows: be the first row in δA j j T  HjT = pT1 pT2 pT3 . . . pTqj ,

$ %T T T T Hj = p1 . . . prj∗ 0 . . . 0 ,

with pi ∈ Rqj , i = 1, . . . , qj orthonormal vectors.

3 Computation of the pseudospectral abscissa

Thus, 

 δAj (1) =   δAj (1) =

  #j δA HT   (1) j T #j δA H (1) j

=

β1 pT1 + · · · + βqj pTqj ,

=

β1 pT1 + · · · + βr∗ pTr∗ . j

j

Since p1 , . . . , pqj are orthonormal, it follows that            δAj (1)  ≤  δAj (1)  . 2

2

(ii) For each j = 1, . . . , m, let us indicate with (δAj )s,t the coefficient in position (s) (s, t) of matrix δAj , and with Bi,j , Ci,j(t) , respectively, the sth column of Bi,j and the tth row of Ci,j , for s = 1, . . . , pj , t = 1, . . . , qj . From Lemma 1, using some algebra manipulations we obtain that ∂R(λRM ) = ∂δAj

&

' ' & m 1 ∂R(λRM ) (s) =− R x∗ Bi,j Ci,j(t) ye−λRM (τi +δτi ) s=1,...,pj ∂(δAj )s,t s=1,...,pj ξ

1 =− ξ =−

1 ξ

t=1,...,qj

m &  i=0 m 

i=0

t=1,...,qj

'





T Bi,j R xy∗ e−λRM (τi +δτi ) Ci,j(t) s=1,...,pj (s)T

t=1,...,qj T = 0, BTi,j XΓi Y T Ci,j

i=0

where the last step stems from the hypothesis. As the optimal δAj must be ε-bounded, we impose the following constraints:   2 gj := δAj s,t − ε2 ≤ 0, j = 1, . . . , K. s=1,...,pj t=1,...,qj

Since R(λRM ) is a global maximum, from the theory of Lagrange multipliers we have that ∂R(λRM ) − 2μj δAj = 0, ∂δAj

j = 1, . . . , K,

where μj ≥ 0 are the multipliers associated with the gj inequality constraints. RM ) Since we proved that ∂R(λ is nonzero, then μj is positive and gj is an active ∂δAj RM ) constraint; thus δAj is a positive multiple of ∂R(λ and in particular ∂δAj   δAj  = ε for j = 1, . . . , K, from which we derive the thesis. F (iii) The same reasoning used in point (ii) can be applied here; by hypothesis we have that

∂R(λRM )

= 0, ∂δτi

195

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CHAPTER 9 Real structured pseudospectra framework

thus the constraint on the size of δτi must be active. As a consequence, the optimal δτi is such that vi |δτi | = ε. The first statement of the theorem demonstrates that optimal perturbations of system matrices always have low rank; as a consequence, the search space of optimization problem (12) can be reduced, and the new optimization problem that we want to solve reads as follows:

λ()  ∈ S := SF × Sτ ,

max s.t.

where   SF := {A ∈ R∗ : rank(δAj ) ≤ ρj∗ = min(2kj , pj , qj ), δAj F ≤ ε, j = 1, . . . , K},  m  ( ε ε − ,+ . Sτ := vi vi

(16) (17)

i=1

By definition, each point in S defines a perturbation of the nominal DEP. We want to construct a continuous path in S along which the spectral abscissa λ is monotonically increasing. Hence, we consider perturbations (A, τ ) ∈ S as continuously depending on a parameter t (as well as the corresponding rightmost eigenvalue λ(t)), and we use the decomposition

δAj (t) = −εUj (t)Qj (t)Vj (t)T , δτi (t) = vεi qi (t),

t ∈ R+ , j = 1, . . . , K, t ∈ R+ , i = 1, . . . , m,

(18)

where the following properties are satisfied: ⎧ ˙ j (t) = 0, ∀ t ≥ 0 Uj (t)T U ⎪ ⎪ ⎪ ⎨ Vj (t)T V˙ j (t) = 0, ∀ t ≥ 0     ⎪ ∀t≥0 ⎪ ⎪ Qj (t) F ≤ 1, ⎩ |qi (t)| ≤ 1, ∀t≥0 ∗

∗

∗

∗

(19)

with Uj (t) ∈ Rpj ×ρj , Vj (t) ∈ Rqj ×ρj , Qj (t) ∈ Rρj ×ρj for j = 1, . . . , K, and qi (t) ∈ R for i = 1, . . . , m. We refer to Ref. [27] for more information with respect to this kind of parameterization. Indicating with A, B = Trace(AT B) the Frobenius inner product of two matrices, it is easy to prove that the quantities satisfying properties (19) can be characterized, without losing of generality, as the solutions of the following differential equations:

3 Computation of the pseudospectral abscissa

⎧ ˙ j (t) ⎪ U ⎪ ⎪ ⎪ ⎪ ⎪ ⎪V˙ j (t) ⎪ ⎪ ⎪ ⎨ ˙ j (t) Q ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩q˙ i (t)

  = In − Uj (t)Uj (t)T Rj (t),   = In − Vj (t)Vj (t)T Sj (t),   Mj (t) − Mj (t), Qj (t) Qj (t), if Qj (t)F = 1, Mj (t), Qj (t) > 0 = M (t), otherwise, j 0 if |qi (t)| = 1, ri (t)qi (t) > 0 = ri (t), otherwise,

(20)

∗

where we have introduced the arbitrary matrices Rj (t) ∈ Rpj ×ρj , Sj (t) ∈ ∗ ∗ ∗ Rqj ×ρj , Mj (t) ∈ Rρj ×ρj for j = 1, . . . , K and the arbitrary functions ri (t) for i = 1, . . . , m. Now, we want to identify matrices Rj (t), Sj (t), Mj (t) and functions qi (t) such that ˙ R(λ(t)) ≥ 0. This will enable us to individuate, for each t, a direction in the space of perturbations S where the real part of the rightmost eigenvalue λ(t) is monotonically increasing. From Lemma 1, the derivative of an eigenvalue λ w.r.t. the parameter t reads ⎧ ⎛ m K 1 ⎨ ∗ ⎝  dλ(t) = −Bi,j δ A˙ j (t)Ci,j e−λ(τi +δτi (t)) x dt ξ ⎩ i=0 j=1 ⎛ ⎞ ⎞ ⎫ m K ⎬   ⎝Ai + + Bi,j δAj (t)Ci,j ⎠ λδ τ˙i (t)e−λ(τi +δτi (t)) ⎠ y , ⎭ i=0

j=1

with ξ defined as in Theorem 1. Thus, substituting Eq. (20) and considering the easy ˙ j = Mj and q˙ i = ri , we have case where Q R(λ˙ ) =

m  K   ε  ∗ ˙ j Qj VjT Ci,j e−λ(τi +δτi ) y R x Bi,j U ξ i=0 j=1

+

m  K   ε  ∗ ˙ j V T Ci,j e−λ(τi +δτi ) y R x Bi,j Uj Q j ξ i=0 j=1

m  K   ε  ∗ R x Bi,j Uj Qj V˙ jT Ci,j e−λ(τi +δτi ) y ξ i=0 j=1 ⎛ ⎛ ⎞ ⎞ m K   ε + R ⎝x∗ ⎝Ai + Bi,j δAj Ci,j ⎠ λq˙i e−λ(τi +δτi ) y⎠ ξ vi

+

i=0

=

j=1

m  K 1  ε0 X, Bi,j (In − Uj UjT )Rj Qj VjT Ci,j YΓiT ξ i=0 j=1

+

m  K 1  ε0 X, Bi,j Uj Mj VjT Ci,j YΓiT ξ i=0 j=1

197

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CHAPTER 9 Real structured pseudospectra framework

m  K 1  ε0 X, Bi,j Uj Qj SjT (In − Vj VjT )Ci,j YΓiT ξ i=0 j=1 ⎛ ⎛ ⎞ ⎞ m K   ε + R ⎝x∗ ⎝Ai + Bi,j δAj Ci,j ⎠ λe−λ(τi +δτi ) y⎠ q˙i ξ vi i=0 j=1   

+

:= βi

=

K  j=1

+

3 2 m ε  T T T T T Bi,j XΓi Y Ci,j Vj Qj , (In − Uj Uj )Rj ξ

K  j=1

+

K  j=1

i=0

3 2 m ε  T T T T Uj Bi,j XΓi Y Ci,j Vj , Mj ξ i=0

3 m m   ε ε T T T Si , (In − Vj Vj ) Ci,j YΓi X Bi,j Uj Qj + βi q˙ i , ξ ξ vi 2

i=0

i=0

where parameter t has been neglected to simplify the notation. Therefore, since (In − Uj UjT ), (In − Vj VjT ) are positive semidefinite, the steepest ascent is guaranteed by the following choices: Rj =

m 

T V QT , BTi,j XΓi Y T Ci,j j j

i=0

Mj = UjT

m 

Sj =

m  i=0

TV, BTi,j XΓi Y T Ci,j j

Ci,j YΓiT X T Bi,j Uj Qj , (21)

r i = βi .

i=0

Note that the other cases for differential equations (20) give the same expressions. As a consequence of the definition of differential equations (20) with choices (21), at each fixed t the solutions Uj (t), Vj (t), Qj (t), qi (t) define a new perturbed DEP such that R (λ(t + η)) > R (λ(t)) for η > 0 sufficiently small, where λ(t) is the rightmost eigenvalue of the problem. In our algorithm, we move along the trajectory of these solutions using a discretization of the differential equation (20). At each iteration k, we first perform a forward Euler method with adaptive stepsize together with a projection of the new quantities onto the space of perturbations S, in order to satisfy the desired properties listed in Eq. (19). Statement (ii) of Theorem 1 is used for the initialization of the algorithm, which can be interpreted as the first step of the fixed-point iteration suggested in Eq. (14). Delay perturbations are simply initialized following the derivative computed in Eq. (15). We refer the reader interested in a detailed illustration of the method to Algorithm 1 in Ref. [14]. The algorithm converges when the norm of the gradient of the spectral abscissa is under a given tolerance: clearly this condition is also satisfied on local maxima, namely locally but not globally rightmost points of the pseudospectrum. In order to avoid a convergence to such a point, restarting the method from different eigenvalues located on the right part of the original spectrum has proved to be effective. The DEPs

4 Smoothness properties and optimization of the pseudospectral abscissa

in the algorithm are solved using the algorithm and the software described in Ref. [25].

4 Smoothness properties and optimization of the pseudospectral abscissa In the previous section, we presented an exact method for assessing robust stability when the system is subject to perturbations as in Eq. (2). In this section, we make the leap to the design of robustly stabilizing controllers. The robust stabilization approach is based on minimizing the pseudospectral abscissa as a function of controller parameters contained in vector p ∈ Rnp . Considering again Example 1, the vector of controller parameters is p = K T if only gain is optimized, and p = [K T ; τ ] if both gain and delay are used as optimization variables. The latter case will be considered again in the experiment described in Example 3 of the next section. From now on, we take again into account the dependence of the perturbed DEP (3) and of the pseudospectral abscissa on the set of parameters p, as follows: αε (p) :

R np p

−→ −→

R αε (M(λ, p)).

It is important to highlight that the maximum size of the uncertainties is not depending on the set of parameters p, although these parameters may admit an uncertainty too. Let us first briefly summarize the pseudospectral abscissa smoothness properties: as previously mentioned, Assumptions 1 and 2 guarantee that the spectral abscissa is continuous with respect to matrix coefficients and delay terms, and also w.r.t. their potential perturbations. Then, following from the properties of the maximum function, the pseudospectral abscissa is also continuous but not everywhere differentiable; typically it is differentiable almost everywhere, whereas points of nonsmoothness are in most cases generated by the presence of the maximum function and are characterized by switching of the component of pseudospectrum which contains the globally rightmost eigenvalue. For this reason, the function is often nondifferentiable in its local minima. Furthermore, the pseudospectral abscissa is in general nonLipschitz and nonconvex. We would like to minimize αε w.r.t. to the set of controller or design parameters: to this purpose, we use the HANSO algorithm [21], which has been proved to efficiently converge to local minima of nonsmooth, nonconvex functions: example of its applications in this field can be found in Michiels [25] and Gumussoy and Michiels [28], where, respectively, the spectral abscissa and the robust H∞ norm of a system of DDAEs are optimized. The application of the HANSO software only requires the availability of a routine to evaluate the objective function and its gradient, whenever the objective function is differentiable.

199

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CHAPTER 9 Real structured pseudospectra framework

Next theorem provides the explicit expression of the derivative of αε w.r.t. to the design or controller parameters p, whenever this derivative exists. Theorem 2. Let (λRM (p), x(p), y(p)) be, respectively, the globally rightmost point of Λε and its left and rightnormalized eigenvectors; for each set   of parameter p ∈  1 (p), . . . , δA K (p) and τ (p) = δτ1 (p), . . . , δ Rnp let us define A(p) = δA τm (p) as the optimal matrix perturbations and optimal delay perturbations functions such that λRM (p) is simple; then denoting ⎛ ζ := − x∗ ⎝E +

m 

⎛ ⎝Ai +

i=0

K 

⎞

⎞

Bi,j δAj Ci,j ⎠ (τi + δτi )e−λRM (τi +δτi ) ⎠ y

j=1

we can express the derivative of the pseudospectral abscissa w.r.t. each parameter pj as follows: ⎡ ⎛ m ∂R(λRM ) 1 ⎣ ∗ ⎝  ∂Ai −λRM (τi +δτi ) ∂αε = = R x − e ∂pj ∂pj ζ ∂pj i=0 ⎛ ⎞ ⎞ ⎤ m K   ∂τ i −λ (τ + δτ ) ⎝Ai + ⎠ y⎦ , + Bi,j δAj Ci,j ⎠ e RM i i λRM ∂pj i=0

j=1

where we omit the dependence on p to simplify the notation and again assumed τ0 = 0. Observe that the values of the optimal perturbations δAi , δτi does depend on the set of design or controller parameters p: indeed, although the maximum size of the optimal perturbations is prescribed, the optimal perturbations might assume different values in the parameter space. However, due to the optimality conditions, their derivative w.r.t. p does not affect the derivative of the pseudospectral abscissa. The main scope of our work is to tune parameters in order to have a negative pseudospectral abscissa. For this reason, convergence to a local but not global minimum of the function does not represent an issue; even more, from a practical point of view, any negative value for the pseudospectral abscissa ensures a robust stability of our system, regardless of the convergence of the algorithm to a local minimum. HANSO algorithm is intended for unconstrained optimization; however, we might need to force our controller to stay in some region of the parameter space, for example, we constrain the delay terms to have positive values. For this reason, we also include some penalties in our cost function, as we will explain case by case in the following section. In general, the penalties we introduce are continuous but nondifferentiable functions. It is worth to remark that HANSO method can cope with nondifferentiability; moreover, the pseudospectral abscissa function is likely to be nondifferentiable in its minima, so we do not add any nonsmoothness to the cost function.

5 Numerical experiments

5 Numerical experiments Here we report three different numerical experiments: the first two are mathematical examples, respectively, from Fridman and Shaked [1] and Vanbiervliet et al. [4] and prove the effectiveness of the method and the need of the barrier function to constrain delays to be positive. The third example is an application to the cutting machine already introduced in Example 2. Example 3. We consider the system introduced in Example 1, where 4 5 4 5 A=

−0.08 0.2 −0.06

−0.03 −0.04 0.2

0.2 −0.005 −0.07

,

B=

−0.1 −0.2 0.1

,

C = I3 .

For a nominal delay τ = 1, this system is unstable, therefore, we introduce a static feedback u(t) = Ky(t) and we consider τ itself as an optimization variable: thus, the vector of controller parameters is p = [K T ; τ ]. As in Example 1, we include uncertainties δA, δB on the matrix coefficients, while C remains unperturbed; we also assume an uncertainty δτ on the delay τ satisfying |δτ | ≤ 0.5. Thus, we set δA1 = δA, δA2 = δB, ⎡ ⎡ ⎤T ⎤ ⎤ I I I ε = 0.05, B0,1 = AF ⎣ 0 ⎦ , B1,2 = BF ⎣ 0 ⎦ , C0,1 = ⎣ 0 ⎦ , 0 0 0 ⎡ ⎤T 0 ε . C1,2 = ⎣ 0 ⎦ , v1 = 0.5 I ⎡

With these choices, we allow a maximal relative error of 5% on matrices A, B. Note that a barrier function is needed to prevent the delay τ from assuming negative values: since the uncertainty δτ such that |δτ | ≤ 0.5 is also present, we constrain nominal value τ to be larger than 0.5 adding a continuous penalty to our cost function as follows: αε :

R4

−→

(K, τ )

−→

R ⎧ ⎪ if τ ≥ 0.5 ⎨αε (M(λ, K, τ )), αε (M(λ, K, 0.5)) + 10(0.5 − τ ), if τ < 0.5. ⎪ ⎩

(22)

In Table 1, we report the values of the spectral abscissa α and of the pseudospectral abscissa αε in three different cases: in the first case the system is uncontrolled (p0 = [0; 0; 0; 1]); in the second case the controller p∗ = [1.4784; 2.6415; 2.3611; 1.1351] is a minimizer for the spectral abscissa function α (of course w.r.t. the same controller variables p); in the third case the optimal controller p∗ε = [2.0671; 3.5925; 3.7166; 0.8855] is a minimizer for the pseudospectral abscissa αε in Eq. (22). It is worth remarking that αε (p∗ε ) < 0: this means that our optimization process provided an optimal controller that guarantees stability

201

CHAPTER 9 Real structured pseudospectra framework

Table 1 The spectral abscissa α and the pseudospectral abscissa αε in the uncontrolled system (p = [0 0 0 1]), and in the systems with controllers p∗ and pε∗ . Uncontrolled p∗ pε∗

α +1.0806e−01 −5.2374e−01 −2.5966e−01

αε +1.2659e−01 −1.4720e−01 −1.7420e−01

0.8 0.6 0.4 0.2

( )

202

0 –0.2 –0.4 –0.6 –0.8 –0.7

–0.6

–0.5

–0.4

–0.3

–0.2

–0.1

0

0.1

0.2

( ) FIG. 1 The rightmost eigenvalues for K = 0, for K = Kε∗ and the worst-case scenario for K = Kε∗ .

against system’s uncertainties. We also observe that αε (p∗ε ) < αε (p∗ ) and that α(p∗ε ) > α(p∗ ): this demonstrates the considerably different behaviors of functions α and αε and justifies our optimization approach for the robust stabilization. Finally, in Fig. 1 we compare the rightmost eigenvalues in the uncontrolled system, in the system controlled with p = p∗ε and in the worst-case scenario for p = p∗ε , where the real part of the globally rightmost eigenvalue is the pseudospectral abscissa. Example 4. In this example, we consider system x˙ (t) = A0 x(t) + A1 x(t − τ1 ) + A2 x(t − τ2 ),

5 Numerical experiments

with 4 A0 =

−1 −3 −2

13.5 −1 −1

−1 −2 −4

4

5 ,

A1 =

−5.9 2 2

0 0 0

0 0 0

4

5 ,

A2 =

0 0 0

7.1 −1 0

−70.3 5 6

5 ,

whose stability w.r.t. the delay terms τ1 and τ2 is examined in Ref. [1]. In this example, time delays are considered unperturbed, and we assume an uncertainty δA1 on matrix A0 and an uncertainty δA2 affecting the nonzero part of matrix A1 . Thus, we define   B0,1 = A0 F I3 , C0,1 = I3 , B1,2 = A1 F I3 , C1,2 = 1 0 0 , where I3 is the identity matrix with dimension 3, and we set ε = 0.005. We now consider the minimization of the pseudospectral abscissa, where the delays τ1 and τ2 are assumed as controller parameters. Again, we include some continuous penalties in our cost function in order to prevent the delay terms from being negative αε :

R2

(τ1 , τ2 )

−→ −→

R ⎧ αε (M(λ, τ1 , τ2 )), ⎪ ⎪ ⎪ ⎨α (M(λ, 0, τ )) − 100τ , ε 2 1 ⎪ α ε (M(λ, τ1 , 0)) − 100τ2 , ⎪ ⎪ ⎩ αε (M(λ, 0, 0)) − 100τ1 − 100τ2 ,

if τ1 if τ1 if τ1 if τ1

≥ 0, < 0, ≥ 0, < 0,

τ2 τ2 τ2 τ2

≥ 0, ≥ 0, < 0, < 0.

(23)

Optimizing function αε as defined in Eq. (23), a minimizer is found in p∗ε = [1.8955e − 02; 1.5599e − 02], whereas the minimization of the spectral abscissa α converged to a point p∗ = [3.5490e − 05; 1.3920e − 02]. The controllers are close to the negative half-planes for both variables τ1 , τ2 , however the configuration (τ1 , τ2 ) = (0, 0) is not a minimizer. In Fig. 2 we compare the spectral abscissa and the worst-case scenarios for both controllers p∗ and p∗ε : the observations made for the previous example still hold, as α(p∗ ) < α(p∗ε ) and αε (p∗ε ) < αε (p∗ ). Example 5. We conclude this section analyzing the concrete example of a cutting machine performing a turning operation. We thus examine again the system introduced in Example 2, but this time we only consider an uncertainty δk = δA1 on the cutting coefficient k and an uncertainty δτ on the delay term; in particular, we impose |δk| ≤ 1.5 × 106 and |δτ | ≤ 0.0015. Therefore, we assume     0 , C1,1 = C2,1 = 1.5 × 107 1 0 . B1,1 = −B2,1 = 1 m

and we set ε = 0.1, v = 0.015−1 , with v the weight associated with perturbation δτ . In Fig. 3, the blue dashed line is the level set equal to 0 of the spectral abscissa, therefore the region below is the stability region in the k-τ space for system (6); the red continuous line is the level set equal to 0 of the pseudospectral abscissa αε . The rectangular blocks represent all the perturbations (δk, δτ ) within a distance ε

203

204

CHAPTER 9 Real structured pseudospectra framework

4 3 2 1 0 –1 –2 –3 –4 –2.4

–2.2

–2

–1.8

–1.6

–1.4

–1.2

–1

–0.8

FIG. 2 The rightmost eigenvalues of the spectra obtained for the controllers p ∗ and pε∗ , and the corresponding worst-case scenarios.

from the point marked with a red asterisk. By definition, for each point whose αε = 0 we can build a  = (δk, δτ ) with glob = ε such that the perturbed system is unstable. Finally, in Fig. 3, we have represented the iterations of the minimization process of the following cost function, where penalties are added to constrain the cutting force coefficient k ≥ 2 × 106 := kmin and the delay term τ ≥ 0.005 := τmin αε : R2 −→ R ⎧ ⎪αε (M(λ, k, τ )), ⎪ ⎪ ⎨α (M(λ, k, τ )) − 1000(τ − τ ), ε min min (k, τ ) −→ ⎪ αε (M(λ, kmin , τ )) − 1000(kmin − k), ⎪ ⎪ ⎩ αε (M(λ, kmin , τmin )) − 1000(kmin − k) − 1000(τmin − τ ),

if τ1 ≥ 0, τ2 ≥ 0, if k ≥ kmin , τ < τmin , if k < kmin , τ ≥ τmin , if k < kmin , τ < τmin .

(24)

Note from the graphical interpretation that the only minima of the cost function defined in Eq. (24) are generated by the introduction of the penalty on parameter k: since we are mostly interested in a robustly stable configuration, in this case we stop the algorithm as soon as a negative value for αε is found.

6 Concluding remarks

3

×107

2.5

2

1.5

1

0.5

0 0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

FIG. 3 Red and blue lines represent, respectively, the level sets for αε = 0 and for α = 0. The iterations of the optimization of the pseudospectral abscissa are indicated by the green points.

6 Concluding remarks In this chapter, we illustrated a method to design a robust controller for a class of linear DDAEs affected by uncertainty on the system matrices and on the delay values. The method consists of the minimization of the pseudospectral abscissa w.r.t. some controller or design parameters: the optimization variables can either be matrix coefficients or delay values. The advantage of this method is evident: first, regarding the pseudospectral abscissa computation, a very broad class of

205

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CHAPTER 9 Real structured pseudospectra framework

perturbations is allowed on the system of DDAEs and the nonlinear structure of the associated DEP is taken into account; as a consequence, the pseudospectral abscissa minimization allows to synthesize controllers for real-life applications, as it guarantees an asymptotically stable behavior of the solution despite the uncertainties affecting the system. Applications of this method are envisaged in the design of static or dynamic fixedorder controllers that robustly stabilize uncertain system of DDEs. Moreover, since low-rank properties of optimal matrices perturbations are fully exploited and the algorithm only requires the computation of selected rightmost eigenvalues (for which fast iterative methods can be used), large-scale problems, which stem for instance from the discretization of PDEs with delay, can be dealt with as well.

Acknowledgments This work was supported by the project C14/17/072 of the KU Leuven Research Council, by the project G0A5317N of the Research Foundation-Flanders (FWO—Vlaanderen), and by the project UCoCoS, funded by the European Union Horizon 2020 research and innovation program under the Marie Sklodowska-Curie Grant Agreement No. 675080.

References [1] E. Fridman, U. Shaked, H∞ -control of linear state-delay descriptor systems: an LMI approach, Linear Algebra Appl. 351–352 (2002) 271–302. [2] P. Pepe, I. Karafyllis, Z.P. Jiang, On the Liapunov-Krasovskii methodology for the ISS of systems described by coupled delay differential and difference equations, Automatica 44 (9) (2008) 2266–2273. [3] A. Seuret, K.H. Johansson, Stabilization of time-delay systems through linear differential equations using a descriptor representation, in: Proceedings of the ECC, Budapest, Hungary, 2009, pp. 4727–4732. [4] J. Vanbiervliet, K. Verheyden, W. Michiels, S. Vandewalle, A nonsmooth optimization approach for the stabilization of linear time-delay systems, ESAIM Control Optim. Calc. Var. 14 (3) (2008) 478–493. [5] W. Michiels, K. Engelborghs, P. Vansevenant, D. Roose, The continuous pole placement method for delay equations, Automatica 38 (5) (2002) 747–761. [6] M. Krstic, Delay Compensation for Nonlinear, Adaptive and PDE Systems, Birkhauser, 2007. [7] Z.J. Palmor, Time-delay compensation—Smith predictor and its modifications, in: S. Levine (Ed.), The Control Handbook, CRC and IEEE Press, New York, 1996 . [8] W. Michiels, S.I. Niculescu, On the delay sensitivity of Smith predictors, Int. J. Syst. Sci. 34 (2003) 543–552. [9] S.I. Niculescu, Delay Effects on Stability. A Robust Control Approach, Lecture Notes in Control and Information Sciences, vol. 269, Springer-Verlag, 2001. [10] K. Gu, V.L. Kharitonov, J. Chen, Stability of Time-Delay Systems, Birkhauser, 2003.

References

[11] W. Michiels, S.I. Niculescu, Stability and Stabilization of Time-Delay Systems. An Eigenvalue Based Approach, SIAM, 2007. [12] N. Guglielmi, C. Lubich, Low-rank dynamics for computing extremal points of real pseudospectra, SIAM J. Matrix Anal. Appl. 34 (2013) 40–66. [13] W. Michiels, N. Guglielmi, An iterative method for computing the pseudospectral abscissa for a class of nonlinear eigenvalue problems, SIAM J. Sci. Comput. 34 (4) (2012) A2366–A2393. [14] F. Borgioli, W. Michiels, Computing distance to instability for delay systems with uncertainties in the system matrices and in the delay terms, in: Proceedings of the ECC, Limassol, Cyprus, 2018 (accepted). [15] G. Hu, E.J. Davison, Real stability radii of linear time-invariant time-delay systems, Syst. Control Lett. 50 (2003) 209–219. [16] W. Michiels, K. Green, T. Wagenknecht, S.I. Niculescu, Pseudospectra and stability radii for analytic matrix functions with applications to time-delay systems, Linear Algebra Appl. 418 (1) (2006) 315–335. [17] E. Fridman, Stability of linear descriptor systems with delay: a Lyapunov-based approach, J. Math. Anal. Appl. 273 (2002) 24–44. [18] E. Fridman, U. Shaked, An improved stabilization method for linear time-delay systems, IEEE Trans. Autom. Control 47 (2002) 1931–37. [19] E. Fridman, Tutorial on Lyapunov-based methods for time-delay systems, Eur. J. Control 20 (6) (2014) 271–283. [20] A. Lewis, M.L. Overton, Nonsmooth optimization via BFGS, 2009, Available from: https://cs.nyu.edu/overton/papers/pdffiles/bfgs_inexactLS.pdf. [21] M. Overton, HANSO: a hybrid algorithm for nonsmooth optimization, 2009, Available from: https://cs.nyu.edu/overton/software/hanso/. [22] F. Borgioli, W. Michiels, Robust stabilisation of linear time-delay systems with uncertainties in the system matrices and in the delay terms, in: Proceedings of the 14th IFAC Workshop on Time-Delay Systems, Budapest, Hungary, 2018 (accepted). [23] T. Insperger, G. Stépán, Stability of the milling process, Period. Polytech. Mech. Eng. 44 (1) (2000) 47–57. [24] S. Jayaram, S.G. Kapoor, R.E. DeVor, Analytical stability analysis of variable spindle speed machines, J. Manuf. Eng. 122 (2000) 391–397. [25] W. Michiels, Spectrum based stability analysis and stabilization of systems described by delay differential algebraic equations, IET Control Theory Appl. 5 (16) (2011) 1829–1842. [26] K. Schreiber, Nonlinear Eigenvalue Problems: Newton-Type Methods and Nonlinear Rayleigh Functionals (Ph.D. thesis), TU Berlin, 2008. [27] O. Koch, C. Lubich, Dynamical low rank approximation, SIAM J. Matrix Anal. Appl. 29 (2) (2007) 434–454. [28] S. Gumussoy, W. Michiels, Fixed-order strong H-infinity control of interconnected systems with time-delays, IFAC Proc. Vol. 44 (2011) 12544–12549, 18th IFAC World Congress.

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CHAPTER

The Smith predictor, the modified Smith predictor, and the finite spectrum assignment: A comparative study

10

Tamas G. Molnara , David Hajdub , Tamas Inspergerb a Department

of Applied Mechanics, Budapest University of Technology and Economics, Budapest, Hungary b Department of Applied Mechanics and MTA-BME Lendület Human Balancing Research Group, Budapest University of Technology and Economics, Budapest, Hungary

Chapter outline 1 Introduction....................................................................................... 209 2 Description of predictor feedback controllers .............................................. 210 2.1 Control problem without predictor................................................. 211 2.2 The Smith predictor ................................................................. 212 2.3 The modified Smith predictor ...................................................... 214 2.4 Finite spectrum assignment........................................................ 216 3 Comparison of the predictors .................................................................. 218 3.1 Effect of initial conditions .......................................................... 218 3.2 Equivalence of the MSP and the FSA ............................................ 220 3.3 Implementation issues .............................................................. 220 4 Application of observers........................................................................ 221 5 Summary and conclusions ..................................................................... 222 Acknowledgments .................................................................................. 225 References........................................................................................... 225

1 Introduction Feedback loops in control systems are always associated with time delays due to the finite speed of sensing, signal processing, computation of the control input, and actuation. Feedback delay is usually considered to be a source of unstable behavior, which should be eliminated from the control loop. An effective way to compensate Stability, Control and Application of Time-Delay Systems. https://doi.org/10.1016/B978-0-12-814928-7.00010-X © 2019 Elsevier Inc. All rights reserved.

209

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CHAPTER 10 Smith predictor and the finite spectrum assignment

the destabilizing effect of feedback delays is the application of predictor feedback controllers such as the Smith predictor [1] and its modifications [2–4], the prediction based on optimal control [5], the finite spectrum assignment [6–9], the reduction approach [10], or the predictive pole placement control [11]. An in-depth discussion on time delay compensation is given in Ref. [12]. This chapter gives a tutorial overview of the well-known Smith predictor (SP), the modified Smith predictor (MSP), and the finite spectrum assignment (FSA) technique. Throughout the chapter, the works of Zhong [13] and Michiels and Niculescu [8] are followed and extended by time-domain equations. The core idea of these predictor feedback controllers is to estimate (predict) the future state (or output) of the plant. Using the predicted state (or output) in the feedback loop instead of the actual one, the effect of time delay can be compensated: an accurate prediction is able to completely eliminate the delay from the feedback loop. However, prediction requires an internal model of the system that allows the calculation of the predicted state (or output). Inaccuracies of this model and imperfections in the implementation of the predictor affect the control performance significantly. Hereinafter, the basic approach and the implementation of the SP, the MSP, and the FSA are overviewed. In the literature, the difference between the SP and the FSA is often attributed to the observer-predictor or predictor-observer representations [13–15]. Here, we argue that the difference rather lies in the method of prediction. The SP employs a prediction over the time interval [0, t + τ ] with τ being the feedback delay, while the FSA predicts only over the delay period [t, t + τ ]. In Section 2, the control laws of the SP, the MSP, and the FSA techniques are summarized. The governing equations of the closed control loop are shown both in frequency and time domain, and block diagrams are given. Based on the equations, the relationship between these control strategies is established and the most important differences are pointed out in Section 3. It is highlighted that initial conditions (and disturbances) affect the performance of the SP and the MSP, but do not affect the closed-loop behavior for the FSA technique. Then, it is shown that the governing equations of the MSP and the FSA are practically equivalent, and the difference between them lies in the formulation and the realization of their control laws. Issues related to practical realization are considered in terms of sensitivity to parameter mismatches and initial conditions and implementation of the control laws. In Section 4, the application of observers is discussed, which is a necessary step when the state of the system is not fully available for the predictor. The governing equations of observer-predictor and predictor-observer representations are given and their connection is established. Finally, a summary is given in Section 5.

2 Description of predictor feedback controllers This chapter is devoted to the analysis of single input-single output systems with discrete input delay. For extensions to multiple inputs, multiple, distributed, and varying delays, see Refs. [12,16].

2 Description of predictor feedback controllers

(A)

(B)

FIG. 1 Block diagram of control loops without predictors (A); block diagram of the Smith predictor (B).

2.1 Control problem without predictor Consider the control problem illustrated in the block diagram of Fig. 1A. The control input u is used in order to adjust the output y of the plant to the reference signal r in the presence of a disturbance d. Unless stated otherwise, we neglect the effect of the disturbance d, although comments will be made on disturbance response later in this section. We assume that the input u is subjected to a single point delay τ as indicated by the term e−sτ in the block diagram. The transfer function of the corresponding delay-free plant is indicated by P(s), while the transfer function of the controller is denoted by C(s). Taking the Laplace transforms with zero initial conditions and d = 0, we obtain Y(s) = P(s)e−sτ U(s),

(1)

U(s) = C(s)(R(s) − Y(s)),

(2)

where Y(s), U(s), and R(s) denote the Laplace transform of y(t), u(t), and r(t), respectively. For notational convenience, capital letters are used to indicate frequencydomain quantities. The transfer function of the closed control loop from the reference signal r to the output y becomes T(s) =

C(s)P(s)e−sτ . 1 + C(s)P(s)e−sτ

(3)

It can be seen that the denominator of the transfer function involves the delay, which affects the stability of the closed control loop. Since time delays are typically a source of unstable behavior, they should be eliminated from the feedback loop. In this chapter, this problem is addressed by predictor feedback controllers such as the SP, the MSP, and the FSA.

211

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CHAPTER 10 Smith predictor and the finite spectrum assignment

The delay-free plant P(s) can also be described by the state-space representation  P(s) =

A C

B 0



= C(sI − A)−1 B,

(4)

where A ∈ Rn×n , B ∈ Rn×1 , and C ∈ R1×n are the system, input, and output matrices, respectively, n is the number of state variables, and I is the identity matrix. Note that this model could easily be extended to a multi input-multi output case, but for simplicity we restrict ourselves to the model presented earlier. Throughout this chapter, we assume that the pair (A, B) is controllable, while the pair (C, A) is observable. The state-space representation of the plant P(s)e−sτ in time domain is given by x˙ (t) = Ax(t) + Bu(t − τ ), y(t) = Cx(t),

(5)

where x(t) ∈ Rn is the vector of state variables. In order to establish the relationship between the different predictor feedback controllers in time domain, we consider full-state feedback and proportional (static) output feedback. Note that full-state feedback requires the state of the system to be available for control. If this is not the case, either output feedback can be used or observers can be applied (which is addressed later in Section 4). The control law of delayed state feedback controllers reads u(t) = Kx(t),

(6)

where K ∈ R1×n is a feedback matrix. For the special case K = kC, delayed state feedback reduces to proportional delayed output feedback u(t) = ky(t).

(7)

In these time-domain equations, the disturbance d and the reference signal r are assumed to be zero in order to simplify the analysis. Of course, nonzero reference signal and nonzero disturbance could also be taken into account. Furthermore, the control gain k could also be replaced by any controller given by the transfer function C(s). Note that the SP and the MSP are typically introduced as output predictors with output feedback, while the FSA is usually formulated for state predictors and state feedback. Extensions to state and output feedback, respectively, can easily be done as shown in following sections.

2.2 The Smith predictor The SP is intended to compensate the destabilizing effect of the delay τ in order to achieve the delayed response of the delay-free system [13]. The approach utilizes −sτ˜ of the plant P(s)e−sτ to predict the future behavior of ˜ an internal model P(s)e the system. Note that the dynamics of the plant is never known accurately, there ˜ are always mismatches between the actual plant P(s) and its model P(s), as well as

2 Description of predictor feedback controllers

between the actual delay τ and its estimation τ˜ . Estimation quantities are indicated by tilde throughout the chapter. Therefore, only an estimation y˜ of the output y can be obtained via the internal model: −sτ˜ U(s). ˜ ˜ Y(s) = P(s)e

(8)

sτ˜ , which is no longer ˜ The future output can also be estimated (predicted) by Y(s)e subjected to the input delay. This can be used to correct the output to its predicted value in the control law:

   sτ˜ ˜ ˜ , U(s) = C(s) R(s) − Y(s) − Y(s) + Y(s)e

(9)

cf. Eq. (2). This control law can be realized by the term Z(s) as −sτ˜ , ˜ − P(s)e ˜ Z(s) =P(s)

U(s) =C(s) (R(s) − (Y(s) + Z(s)U(s))) ,

(10)

that is illustrated by the block diagram in Fig. 1B. Finally, the controller with the SP can be described by CSP (s) =

C(s) C(s) = , −sτ˜ ˜ ˜ 1 + C(s)Z(s) 1 + C(s)P(s) − C(s)P(s)e

(11)

by which the closed-loop transfer function becomes TSP (s) =

CSP (s)P(s)e−sτ C(s)P(s)e−sτ = . −sτ −sτ˜ + C(s)P(s)e−sτ ˜ − C(s)P(s)e ˜ 1 + CSP (s)P(s)e 1 + C(s)P(s)

(12)

For the ideal case with perfectly accurate estimation of the plant dynamics and the ˜ delay (P(s) = P(s), τ˜ = τ ), the transfer function simplifies to id (s) = TSP

C(s)P(s)e−sτ , 1 + C(s)P(s)

(13)

cf. Eq. (3). This means that the SP is able to remove the effect of the time delay on the poles of the closed control loop, and achieves the delayed response of the delayfree plant P(s) subjected to controller C(s). In reality, however, a perfect internal ˜ model (P(s) = P(s), τ˜ = τ ) is never achievable, thus the delay cannot be completely eliminated from the control loop. Control performance depends on the mismatches between the internal model and the actual system [17]. Considering disturbance response, the transfer function from the disturbance d to the output y reads   −sτ˜ C(s)P(s)e−sτ ˜ − C(s)P(s)e ˜ 1 + C(s)P(s) P(s)e−sτ WSP (s) = = , −sτ˜ + C(s)P(s)e−sτ ˜ − C(s)P(s)e ˜ 1 + CSP (s)P(s)e−sτ 1 + C(s)P(s)

(14)

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CHAPTER 10 Smith predictor and the finite spectrum assignment

˜ which, in the ideal case P(s) = P(s), τ˜ = τ , becomes  id (s) = WSP

 1 + C(s)P(s) − C(s)P(s)e−sτ C(s)P(s)e−sτ . 1 + C(s)P(s)

(15)

This implies that, in the ideal case, the poles of the disturbance response involve those of the plant P(s). Usually, this argument is used to explain the incapability of the SP ˜ to stabilize unstable plants. Note, however, when P(s) = P(s) and τ˜ = τ , both the ˜ actual plant P(s) and the model P(s) affect the poles (and thus the stability) of the closed control loop, which opens the possibility of stabilization for some extreme model parameter mismatches (for further details, see Ref. [17]). ˜ The delay-free internal model P(s) can be represented in state-space form by  ˜ P(s) =

˜ A ˜ C

˜ B 0

˜ ˜ −1 B, ˜ = C(sI − A)

(16)

˜ B, ˜ and C ˜ are the estimations (nominal values) of matrices A, B, and C. where A, −sτ˜ is represented in time domain as ˜ Accordingly, the internal model P(s)e ˜ x(t) + Bu(t ˜ x˙˜ (t) = A˜ − τ˜ ),

(17)

˜ x(t), y˜ (t) = C˜

where x˜ is the estimation of the state x given by Eq. (5). In the case of proportional output feedback, the control law of the SP reads   u(t) = k y(t) − y˜ (t) + y˜ (t + τ˜ ) ,

while for a state feedback controller it becomes 



u(t) = K x(t) − x˜ (t) + x˜ (t + τ˜ ) .

(18)

(19)

This time-domain representation of the SP was given in Ref. [12] (see Eq. (2.45) ˜ = C, proportional output feedback is achieved by the in Ref. [12]). Note that if C choice K = kC. The block diagram of the SP with state-space representation and state feedback is illustrated in Fig. 2A.

2.3 The modified Smith predictor ˜ The fact that the internal model P(s) used by the predictor affects the stability of the closed control loop gives motivation for an improvement of this model. Here, ˆ a specific modified model P(s) is used, which is no longer simply an estimation of ˆ the plant P(s). The MSP [13] employs a modified model P(s), which is based on state-space representation  ˆ P(s) =

˜ A

˜ B

˜ −A˜ τ˜ Ce

0



˜

˜ −Aτ˜ (sI − A) ˜ −1 B. ˜ = Ce

(20)

2 Description of predictor feedback controllers

(A)

(B)

(C) FIG. 2 Block diagram of the Smith predictor (A) and the modified Smith predictor (B); and the finite spectrum assignment (C) in the case of state feedback.

The corresponding modified quantities are indicated by hat throughout the chapter. This way, the predictor and the control law become −sτ˜ , ˆ ˆ − P(s)e ˜ Z(s) =P(s)    ˆ U(s) =C(s) R(s) − Y(s) + Z(s)U(s) .

(21)

The controller involving the MSP can be described by CMSP (s) =

C(s) ˆ 1 + C(s)Z(s)

,

(22)

while the transfer function from the reference signal to the output becomes TMSP (s) =

CMSP (s)P(s)e−sτ , 1 + CMSP (s)P(s)e−sτ

(23)

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CHAPTER 10 Smith predictor and the finite spectrum assignment

ˆ ˜ ˆ ˆ cf. Eqs. (11), (12). Note that since typically P(0) = P(0), one may use Z(s) − Z(0) ˆ instead of Z(s) in the second row of Eq. (21) in order to guarantee zero static error for the output [13]. −sτ˜ with delay is obtained by ˆ In time domain, the modified model P(s)e ˜ −A˜ τ˜ x˜ (t), yˆ (t) = Ce

˜ −A˜ τ˜ where the estimated state x˜ is governed by Eq. (17). Note that since Ae

(24)

=

˜ ˜ e−Aτ˜ A,

one may introduce the modified state ˜

xˆ (t) = e−Aτ˜ x˜ (t)

(25)

and rewrite Eqs. (17), (24) in the form ˜ x(t) + e−A˜ τ˜ Bu(t ˜ x˙ˆ (t) = Aˆ − τ˜ ), ˜ x(t). yˆ (t) = Cˆ

Accordingly, the control law for proportional output feedback reads   u(t) = kˆ y(t) − y˜ (t) + yˆ (t + τ˜ ) ,

(26)

(27)

where the gain kˆ could be replaced by other controllers given by the transfer function ˆ is C(s). The corresponding state feedback controller with feedback matrix K   ˆ x(t) − x˜ (t) + xˆ (t + τ˜ ) , u(t) = K

(28)

ˆ = kC ˜ = C. ˆ if C which reduces to the case of proportional output feedback with K Note that this control law can also be expressed by the estimated state x˜ instead of the modified state xˆ as  ˜    Aτ˜ u(t) = K e

x(t) − x˜ (t) + x˜ (t + τ˜ ) ,

(29)

˜

ˆ = KeAτ˜ . This shows that the difference between the SP and the MSP where K ˜ techniques is the term eAτ˜ , which plays an important role considering the effect of initial conditions in Section 3.1. The block diagram corresponding to the MSP is the ˆ ˜ one in Fig. 1B with P(s) instead of P(s). A more detailed block diagram with the state-space representation is illustrated in Fig. 2B.

2.4 Finite spectrum assignment The FSA concept is originated from time-domain representation. The classical form of FSA is developed to compensate the input delay for the state feedback given by Eq. (6), see Refs. [6,7,9]. Similarly to the SP and the MSP techniques, FSA intends to use a predicted value xp (t + τ˜ ) of the state instead of using the actual one x(t) for feedback. Again, an internal model is used to perform the prediction

2 Description of predictor feedback controllers

˜ p (t) + Bu(t ˜ x˙ p (t) = Ax − τ˜ ),

(30)

cf. Eq. (17). The state x(t) is used as initial condition, and the internal model is solved by the predictor over the estimated delay interval [t, t + τ˜ ]. This leads to the predicted state ˜

xp (t + τ˜ ) = eAτ˜ x(t) +

τ˜ ˜ ˜ eAθ Bu(t − θ)dθ,

(31)

0

which is used for state feedback u(t) = Kxp (t + τ˜ ).

(32)

Consider the case of an ideal internal model (30) and an accurate estimation of ˜ = A, B˜ = B, and τ˜ = τ . Then, Eqs. (31), (32) can be simplified the delay: A as follows. Substitute Bu(t − θ ) from Eq. (5) into Eq. (31) and use the equality x˙ (t + τ − θ ) = −x (t + τ − θ ) (where prime denotes differentiation with respect to θ). Then, integration by parts leads to u(t) = Kx(t + τ ).

(33)

Thus, Eqs. (5), (33) describe a delay-free state feedback, and the closed control loop is associated with an ordinary differential equation. This implies that FSA eliminates the delay from the control loop in case of a perfectly accurate internal model. The spectrum (the set of poles) of the closed-loop system becomes finite, which can be assigned via the control parameters in K. Thus, stability can be achieved for arbitrary (but controllable) pairs of (A, B) and arbitrary delay τ . That is, even an unstable plant can be stabilized by FSA. Elimination of the delay requires, however, that the parameters of the internal model (30) match those of the actual system (5) and that the control law (32) is implemented accurately. The effects of implementation inaccuracies are addressed in Section 3.3, while sensitivity to parameter mismatches is analyzed in Ref. [18]. FSA can also be formulated for output feedback following Ref. [8]. Note that the ˜ p (t + τ˜ ) requires the knowledge of the state x(t) according output yp (t + τ˜ ) = Cx to Eq. (31). If the state is not fully known, the following output can be introduced: ˜

˜ −Aτ˜ xp (t + τ˜ ), yˆ p (t + τ˜ ) = Ce

(34)

˜

cf. Eq. (24) for the MSP. The coefficient e−Aτ˜ allows us to write Eqs. (31), (34) in the form ˜ −A˜ τ˜ yˆ p (t + τ˜ ) = y(t) + Ce

τ˜ ˜ ˜ eAθ Bu(t − θ)dθ,

(35)

0

˜ where the available output y(t) is used instead of the unavailable term Cx(t). Via Eq. (35), the predicted output yˆ p (t + τ˜ ) can be calculated based on the output y(t)

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without knowing the full state x(t). The control law for proportional output feedback becomes ˆ yp (t + τ˜ ). u(t) = kˆ

(36)

For details on other (dynamic) output feedback controllers, see Ref. [8]. Via Laplace transformation, the description of FSA in frequency domain is the following. The integral term in predictors (31), (35) is represented by Zx (s) =

τ˜ 0

  ˜ ˜ ˜ ˜ −1 B, ˜ e−(sI−A)θ Bdθ = I − e−(sI−A)τ˜ (sI − A)

(37)

by which the predicted state and the predicted output become ˜

Xp (s)esτ˜ =eAτ˜ X(s) + Zx (s)U(s),

(38)

˜ −A˜ τ˜ Zx (s)U(s). Yˆ p (s)esτ˜ =Y(s) + Ce

(39)

The controller corresponding to Eqs. (10), (21) is therefore   U(s) = C(s) R(s) − Yˆ p (s)esτ˜ .

(40)

The block diagram associated with a state feedback controller is shown in Fig. 2C. Note that in the case of an unstable plant, Eq. (37) cannot be used directly to realize the control law, see the discussion in Section 3.3.

3 Comparison of the predictors For comparison, a summary of the time-domain governing equations for the SP, the MSP, and the FSA is listed in Table 1 at the end of the chapter. Respectively, the corresponding frequency-domain equations are listed in Table 2. Some key differences between the different predictor concepts are highlighted in the following paragraphs.

3.1 Effect of initial conditions Note that for the SP and the MSP, the actual state x(t) is corrected by subtracting the model state x˜ (t) and adding either the predicted state x˜ (t + τ˜ ) or the modified predicted state xˆ (t + τ˜ ), respectively, as shown by Eqs. (19), (28). In contrast, FSA directly uses the predicted state xp (t + τ˜ ), which is obtained from the internal model with the initial condition xp (t) = x(t). This shows a significant difference between the SP concepts and the FSA approach: the initial conditions of the internal model are taken at different time instants. In what follows, we investigate the effect of initial conditions for the state feedback controllers (19), (29), (32). For the sake of simplicity, we study the following initial conditions for the actual system (5) and model (17), respectively: x(0) = x0 , x(t) ≡ 0 for t < 0 and x˜ (0) = x˜ 0 , x˜ (t) ≡ 0 for t < 0. Note, however, that

3 Comparison of the predictors

the conclusions drawn in this section hold for a general initial condition, too. Since the initial conditions of the actual system (5) are unknown to the controller, they can only be estimated by those of model (17). There are always mismatches between the initial conditions (˜x0 = x0 ), whose effect is shown in the following paragraphs. By solving Eqs. (5), (17), controller (19) for the SP can be given in the form 

t

t ˜ ˜ ˜ u(t) = K eAt x0 + eA(t−θ) B u(θ − τ )d θ − eAt x˜ 0 − eA(t−θ) Bu(θ − τ˜ )dθ 0 0

t+τ˜ ˜ ˜ ˜ eA(t+τ˜ −θ) Bu(θ − τ˜ )dθ . (41) +eA(t+τ˜ ) x˜ 0 + 0

˜ = A, B ˜ = B, τ˜ = τ ), this simplifies to With perfectly matching internal model (A   u(t) = K eAt x0 − eAt x˜ 0 + eA(t+τ ) x˜ 0 + x(t + τ ) − eA(t+τ ) x0    = Kx(t + τ ) + K eAt − eA(t+τ ) x0 − x˜ 0 .

(42)

This shows that mismatches in the initial conditions directly affect the control input. For an unstable plant, the last term in Eq. (42) tends to infinity as t → ∞ if x˜ 0 = x0 . (Note that the same could be shown for output feedback via the substitution K = kC.) This explains the incapability of the SP to stabilize unstable plants even in the case when the internal model is free of parameter mismatches. Recall that a similar conclusion was drawn considering the disturbance response in frequency domain in Section 2.2. In a similar manner, initial conditions can be taken into account for the MSP. By solving Eqs. (5), (17), the state feedback controller (29) becomes 

t ˜ ˜ u(t) = K eAτ˜ eAt x0 + eAτ˜ eA(t−θ) Bu(θ − τ )dθ 0

t ˜ ˜ ˜ ˜ ˜ ˜ eA(t−θ) Bu(θ − τ˜ )dθ + eA(t+τ˜ ) x˜ 0 −eAτ˜ eAt x˜ 0 − eAτ˜ 0

t+τ˜ ˜ A(t+ τ ˜ −θ) ˜ + e Bu(θ − τ˜ )dθ .

(43)

0

˜ = A, B ˜ = B, τ˜ = τ ) leads to A perfect internal model (A   u(t) = K eA(t+τ ) x0 − eA(t+τ ) x˜ 0 + eA(t+τ ) x˜ 0 + x(t + τ ) − eA(t+τ ) x0 = Kx(t+τ ). (44)

For the MSP with perfect internal model, the effect of mismatches between the initial ˜ conditions drops owing to the term eAτ˜ in the controller. This enables the MSP to stabilize unstable plants despite the presence of mismatches between the initial ˜ = A, B ˜ = B, conditions. Note, however, that for an imperfect internal model (A τ˜ = τ ), the effect of initial conditions does not vanish. Therefore, the realization (29) of the MSP may lead to instabilities that are associated with unstable pole-zero

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cancelations. Thus, special care must be taken when implementing the control law— for details, see Section 3.3. For FSA, the state x(t) of the plant is used in Eq. (31) to obtain the predicted state xp (t + τ˜ ). Thus, prediction is done over the delay interval of length τ˜ only and not over the whole time interval [0, t + τ˜ ]. This way, the problem of mismatches between the initial conditions does not show up.

3.2 Equivalence of the MSP and the FSA Note that the same delay-free control law is obtained for the FSA and for the MSP in the ideal case without parameter mismatches, cf. Eqs. (33), (44). In fact, a more general equivalence holds for the control laws of these two techniques. Similarly to Eq. (31), the internal model (17) can be solved over the delay interval τ˜ using the initial value x˜ (t), which yields ˜

x˜ (t + τ˜ ) = eAτ˜ x˜ (t) +

τ˜ ˜ ˜ eAθ Bu(t − θ)dθ.

(45)

0

Expressing the integral term and substituting it into Eq. (31) implies  ˜  xp (t + τ˜ ) = eAτ˜ x(t) − x˜ (t) + x˜ (t + τ˜ ).

(46)

Eqs. (29), (32), (46) show that the governing equations of the closed control loop are in fact the same for the MSP and for the FSA in the case of state feedback. In the case of output feedback, it follows from Eqs. (24), (35), (45) that yˆ p (t + τ˜ ) = y(t) − y˜ (t) + yˆ (t + τ˜ ).

(47)

Thus, the governing equations for the MSP and the FSA are the same also for output feedback, cf. Eqs. (27), (36), (47). This was also pointed out in Ref. [8]. The equivalence of the MSP and the FSA can be verified in frequency domain as well. Substitution of Eq. (37) into Eq. (39) leads to the following relationship between two concepts: ˜ −A˜ τ˜ Zx (s). ˆ Z(s) = Ce

(48)

Then, Eqs. (21), (40) verify the equivalence of the MSP and the FSA.

3.3 Implementation issues Although their equations are equivalent, the MSP and the FSA approaches imply different realizations for the control law. The MSP uses expressions of the estimated output and state in the control laws (27), (28), respectively. Meanwhile, FSA replaces these terms with an integral of past control inputs, see Eqs. (31), (35). The role of this integral is crucial in the implementation of the controller. Realization of the predictor via the right-hand side of Eq. (37) involves unstable ˜ has unstable eigenvalues, hence it is not suitable for pole-zero cancelation if A

4 Application of observers

ˆ stabilizing unstable systems [6,19]. Thus, implementation of the MSP using Z(s) in Eq. (21) is possible for stable plants only, otherwise unstable pole-zero cancelations may occur. For unstable plants, one may use an integral of control inputs as in FSA, see Eqs. (37), (48). For FSA, the integral term is typically realized via approximation by numerical quadrature. According to Refs. [19–22], this approximation may lead to a highfrequency instability phenomenon, which introduces additional conditions for safe implementation. These restrictions can be removed by adding a low-pass filter into the controller [8], or by using a digital controller with a sample-and-hold unit [8,23]. During the implementation of the SP, a similar high-frequency instability phenomenon occurs if the transfer function of the corresponding delay-free closed control loop is proper but not strictly proper. In such cases, additional conditions for safe implementation (also called as practical stability conditions) must be fulfilled [8]. ˜ Furthermore, the computation of the matrix exponential e−Aτ˜ also leads to ˜ numerical issues if A has eigenvalues with large negative real parts. In this case, the so-called unified SP can be applied to overcome this problem [13,24], which uses the SP for the stable subsystem and the MSP for the unstable subsystem of the plant. In connection to controller implementation, the mismatches between the parameters of the internal model and those of the actual system also affect the stability of the closed control loop. Infinitesimal delay mismatches are addressed in Ref. [8], while the effect of finite parameter mismatches is analyzed in Ref. [17] for the SP and in Ref. [18] for the FSA. In addition, there might be additional phenomena (such as noise, nonlinearities, and nonsmoothness) that are not modeled by Eq. (17), but affect the closed-loop dynamics.

4 Application of observers When the state x is not (fully) available for feedback, one may either use the output feedback controllers (18), (27), (36) or a state observer. Here, we consider the case of a Luenberger observer with parameters given by the matrix L ∈ Rn×1 . Based on the order of observation and prediction, we can distinguish observerpredictor and predictor-observer representations [13–15]. Here, we discuss these representations briefly in time domain (more details, frequency-domain description and block diagrams are given in Ref. [13]). In the observer-predictor representation [14], first a state observer is employed that uses the output y   ˜ + LC ˜ xO (t) + Bu(t ˜ x˙ O (t) = A − τ˜ ) − Ly(t).

(49)

Then, the observed state xO is introduced into a state predictor ˜

xp (t + τ˜ ) = eAτ˜ xO (t) +

τ˜ ˜ ˜ eAθ Bu(t − θ)dθ, 0

(50)

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CHAPTER 10 Smith predictor and the finite spectrum assignment

cf. Eq. (31). Using the predicted state xp (t + τ˜ ), control law (32) can be used (where ˜ + BK ˜ and A ˜ + LC ˜ are Hurwitz). Recall that the MSP is usually formulated A for output feedback using an output predictor, while the FSA is more commonly formulated for state feedback using a state predictor. Therefore, the observerpredictor realization with a state predictor is more closely related to FSA [13,15]. Note, however, that the integral term in Eq. (50) could be replaced using Eq. (45), which leads to  ˜ τ˜  A xp (t + τ˜ ) = e

xO (t) − x˜ (t) + x˜ (t + τ˜ ).

(51)

Thus, the control law (29) of the MSP could also be used with the observed state xO (t) instead of the actual x(t). This way, the observer-predictor representation also allows one to realize the extension of the MSP to state feedback. In the predictor-observer representation [13,15], first the output predictor (47) is used, then the predicted output is utilized in a state observer   ˜ + LC ˜ xˆ J (t) + e−A˜ τ˜ Bu(t ˜ x˙ˆ J (t) = A − τ˜ ) − Lˆyp (t).

(52)

This way, the future value of the observed state xˆ J is directly used by the controller ˆ xJ (t + τ˜ ). u(t) = Kˆ

(53)

Since an output predictor is utilized, the predictor-observer realization is said to be more closely related to the control law (28) of the MSP [13,15], where x(t) − x˜ (t) + xˆ (t + τ˜ ) is replaced by xˆ J (t + τ˜ ). Note, however, that the output predictor extension (35) of FSA could also be used in the predictor-observer representation. In this case, control law (32) is applied where xp (t + τ˜ ) is replaced ˜ by xJ (t + τ˜ ) = eAτ˜ xˆ J (t + τ˜ ). The relationship between the observer-predictor and the predictor-observer realizations can be established by ˜

xˆ J (t + τ˜ ) = xO (t) − x˜ (t) + e−Aτ˜ x˜ (t + τ˜ ).

(54)

This can be verified by the differentiation of Eq. (54) with respect to time and the substitution of Eqs. (49), (52). The order of prediction and observation is, therefore, interchangeable in the level of equations, but these are two different realizations of the controller. The observer-predictor realization requires a state predictor, while the predictor-observer requires an output predictor.

5 Summary and conclusions Predictor feedback controllers were discussed by describing the SP, the MSP, and the FSA technique both in frequency and time domain. A detailed comparison was made between these techniques by considering (extensions to) both state and

5 Summary and conclusions

Table 1 Governing equations in time domain for the SP, the MSP, and the FSA with state feedback. Plant Internal model Control law Smith predictor Dependence on initial conditions

x˙ (t) = Ax(t) + Bu(t − τ ) ˜ x(t) + Bu(t ˜ x˙˜ (t) = A˜ − τ˜ )   u(t) = K x(t) − x˜ (t) + x˜ (t + τ˜ ) 

t eA(t−θ ) Bu(θ − τ )dθ u(t) = K eAt x0 +

t0 ˜ ˜ ˜ −eAt x˜ 0 − eA(t−θ ) Bu(θ − τ˜ )dθ 0

t+τ˜ ˜ ˜ ˜ eA(t+τ˜ −θ ) Bu(θ − τ˜ )dθ +eA(t+τ˜ ) x˜ 0 + 0

Perfect internal ˜ = A, model A ˜ = B, τ˜ = τ B Plant Internal model Modified model Modified smith predictor

Control law

Dependence on initial conditions



u(t) = Kx(t + τ ) + K eAt − eA(t+τ )



x0 − x˜ 0



x˙ (t) = Ax(t) + Bu(t − τ ) ˜ x(t) + Bu(t ˜ x˙˜ (t) = A˜ − τ˜ ) ˙xˆ (t) = Aˆ ˜ x(t) + e−A˜ τ˜ Bu(t ˜ − τ˜ )   ˆ x(t) − x˜ (t) + xˆ (t + τ˜ ) u(t) = K    ˜  = K eAτ˜ x(t) − x˜ (t) + x˜ (t + τ˜ ) 

t ˜ ˜ eA(t−θ ) Bu(θ − τ )dθ u(t) = K eAτ˜ eAt x0 + eAτ˜

t0 ˜ ˜ ˜ ˜ ˜ −eAτ˜ eAt x˜ 0 − eAτ˜ eA(t−θ ) Bu(θ − τ˜ )dθ 0

t+τ˜ ˜ ˜ A(t+ τ˜ −θ ) ˜ A(t+ τ ˜ ) x˜ 0 + e Bu(θ − τ˜ )dθ +e 0

Perfect internal ˜ = A, model A ˜ = B, τ˜ = τ B Plant

State prediction

x˙ (t) = Ax(t) + Bu(t − τ ) ˜ p (t) + Bu(t ˜ x˙ p (t) = Ax − τ˜ )

τ˜ ˜ ˜ ˜ τ˜ A xp (t + τ˜ ) = e x(t) + eAθ Bu(t − θ)dθ

Control law

u(t) = Kxp (t + τ˜ )

Perfect internal ˜ = A, model A ˜ = B, τ˜ = τ B

u(t) = Kx(t + τ )

Internal model Finite spectrum assignment

u(t) = Kx(t + τ )

0

output feedback. The governing equations for state feedback in time domain are summarized in Table 1, while those for output feedback in frequency domain are collected in Table 2. It was shown that the governing equations of the MSP and the FSA are practically equivalent, while the approach to formulate and realize these controllers is different.

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Table 2 Governing equations in frequency domain for the SP, the MSP, and the FSA with output feedback.  Plant

Smith predictor Internal model

P(s) =

A

B

C

0

= C(sI − A)−1 B

Y(s) = P(s)e−sτ U(s)  ˜ B˜ A ˜ ˜ −1 B ˜ ˜ P(s) = = C(sI − A) ˜ C 0 −sτ˜ U(s) ˜ ˜ Y(s) = P(s)e

Control law

−sτ˜ ˜ − P(s)e ˜ Z(s) = P(s)

U(s) = C(s) (R(s) − (Y(s) + Z(s)U(s)))  Plant

P(s) =

A

B

C

0

= C(sI − A)−1 B

Y(s) = P(s)e−sτ U(s)  Modified smith predictor

Internal model

Modified model

˜ A ˜ C

˜ P(s) =

B˜

˜ ˜ −1 B ˜ = C(sI − A)

0

−sτ˜ U(s) ˜ ˜ Y(s) = P(s)e  ˜ A B˜ ˜ −1 B ˜ −A˜ τ˜ (sI − A) ˜ ˆ P(s) = = Ce ˜ −A˜ τ˜ 0 Ce −sτ˜ U(s) ˆ ˆ Y(s) = P(s)e

Control law

−sτ˜ ˆ ˆ − P(s)e ˜ Z(s) = P(s)    ˆ U(s) = C(s) R(s) − Y(s) + Z(s)U(s)

 Plant

Finite spectrum assignment

Internal model

P(s) =

A

B

C

0

= C(sI − A)−1 B

Y(s) = P(s)e−sτ U(s)  ˜ B˜ A ˜ ˜ −1 B ˜ ˜ P(s) = = C(sI − A) ˜ C 0

Control law



Zx (s) =

τ˜

˜ ˜ e−(sI−A)θ Bdθ

0

˜ −A˜ τ˜ Zx (s)U(s) Yˆ p (s)esτ˜ = Y(s) + Ce   U(s) = C(s) R(s) − Yˆ p (s)esτ˜

References

The performance of the SP, the MSP, and the FSA depends on the accuracy of the internal model used for prediction. In the case of a perfectly matching internal model, the MSP and the FSA techniques are able to stabilize unstable systems, while this is not possible for the SP due to its disturbance response and the effect of initial conditions. Sensitivity to infinitesimal implementation inaccuracies in the control law can be observed for the MSP and the FSA. In order to avoid unstable pole-zero cancelations, these predictors should be implemented using integrals of past control inputs instead of realizing models given by differential equations. The approximation of these integrals by numerical quadratures affects stability. Thus, either conditions for safe implementation must be met or low-pass filters or digital controllers must be used. Furthermore, an additional term must be taken into account for the MSP to ensure zero static error. Finally, when the state of the plant is not available, observers must be utilized. Based on the order of prediction and observation, observer-predictor and predictorobserver realizations are possible that use state and output predictors, respectively. After realizing the observers and overcoming the difficulties of implementation, predictor feedback controllers become efficient tools for compensating the destabilizing effect of feedback delays.

Acknowledgments The research reported in this chapter was supported by the Higher Education Excellence Program of the Ministry of Human Capacities in the frame of artificial intelligence research area of the Budapest University of Technology and Economics (BME FIKP-MI).

References [1] O.J.M. Smith, Close control of loops with dead time, Chem. Eng. Process. 53 (5) (1957) 217–219. [2] K.J. Astrom, C.C. Hang, B.C. Lim, A new Smith predictor for controlling a process with an integrator and long dead-time, IEEE Trans. Autom. Control 39 (2) (1994) 343–345. [3] Z.J. Palmor, Time-delay compensation: Smith predictor and its modifications, in: W.S. Levine (Ed.), The Control Handbook, CRC and IEEE Press, Boca Raton, FL, 2000, pp. 224–237. [4] W. Michiels, D. Roose, Time delay compensation in unstable plants using delayed state feedback, in: Proceedings of the IEEE Conference on Decision and Control, WeA05-3, Orlando, FL, USA, 2001. [5] D.L. Kleinman, Optimal control of linear systems with time-delay and observation noise, IEEE Trans. Autom. Control 14 (5) (1969) 524–527. [6] A.Z. Manitius, A.W. Olbrot, Finite spectrum assignment problem for systems with delays, IEEE Trans. Autom. Control AC-24 (4) (1979) 541–553.

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[7] Q.G. Wang, T.H. Lee, K.K. Tan, Finite Spectrum Assignment for Time Delay Systems, Springer, London, 1999. [8] W. Michiels, S.I. Niculescu, Stability and Stabilization of Time Delay Systems—An Eigenvalue Based Approach, SIAM Publications, Philadelphia, PA, 2007. [9] M. Jankovic, Forwarding, backstepping, and finite spectrum assignment for time delay systems, Automatica 45 (1) (2009) 2–9. [10] Z. Arstein, Linear systems with delayed controls: a reduction, IEEE Trans. Autom. Control 27 (4) (1982) 869–879. [11] P.J. Gawthrop, E. Ronco, Predictive pole-placement control with linear models, Automatica 38 (3) (2002) 421–432. [12] M. Krstic, Delay Compensation for Nonlinear, Adaptive, and PDE Systems, Birkhäuser, Boston, MA, 2009. [13] Q.C. Zhong, Robust Control of Time-delay Systems, Springer, London, 2006. [14] L. Mirkin, N. Raskin, Every stabilizing dead-time controller has an observer-predictor-based structure, Automatica 39 (10) (2003) 1747–1754. [15] Q.C. Zhong, Bridging finite-spectrum assignment and Smith predictor, Annu. Rev. Control 24 (1) (2003) 125–134. [16] M. Krstic, N. Bekiaris-Liberis, Compensation of infinite-dimensional input dynamics, Annu. Rev. Control 34 (2) (2010) 233–244. [17] D. Hajdu, T. Insperger, Demonstration of the sensitivity of the Smith predictor to parameter uncertainties using stability diagrams, Int. J. Dyn. Control 4 (4) (2016) 384–392. [18] T.G. Molnár, T. Insperger, On the robust stabilizability of unstable systems with feedback delay by finite spectrum assignment, J. Vib. Control 22 (3) (2016) 649–661. [19] S. Mondié, M. Dambrine, O. Santos, Approximation of control laws with distributed delays: a necessary condition for stability, Kybernetika 38 (5) (2002) 541–551. [20] W. Michiels, S. Mondié, D. Roose, Robust stabilization of time-delay systems with distributed delay control laws: necessary and sufficient conditions for a safe implementation, Tech. Rep. TWReport 363, Department of Computer Science, Katholieke Universiteit Leuven, Belgium, 2003. [21] K. Engelborghs, M. Dambrine, D. Roose, Limitations of a class of stabilization methods for delay systems, IEEE Trans. Autom. Control 46 (2) (2001) 336–339. [22] S. Mondié, W. Michiels, Finite spectrum assignment of unstable time-delay systems with a safe implementation, IEEE Trans. Autom. Control 48 (12) (2003) 2207–2212. [23] V. van Assche, M. Dambrine, J.F. Lafay, J.P. Richard, Implementation of a distributed control law for a class of systems with delay, in: Proceedings of the 3rd IFAC Workshop on Time Delay Systems, Santa Fe, NM, 2001, pp. 266–271. [24] Q.C. Zhong, G. Weiss, A unified Smith predictor based on the spectral decomposition of the plant, Int. J. Control 77 (15) (2004) 1362–1371.

CHAPTER

11

Extended dissipative control and filtering for singular time-delay systems with Markovian jumping parameters

Baoyong Zhanga , Guangming Zhuangb a School

of Automation, Nanjing University of Science and Technology, Nanjing, China b School of Mathematical Sciences, Liaocheng University, Liaocheng, China

Chapter outline 1 Introduction....................................................................................... 227 2 Stochastic admissibility ........................................................................ 229 2.1 System model and definitions ..................................................... 229 2.2 Relaxed L-K functional.............................................................. 230 2.3 Stochastic admissibility criteria ................................................... 234 3 Extended dissipativity........................................................................... 236 3.1 Definitions and assumptions ....................................................... 236 3.2 Extended dissipativity criteria...................................................... 237 4 State-feedback control ......................................................................... 239 4.1 Problem formulation................................................................. 239 4.2 Useful lemmas ....................................................................... 240 4.3 Controller synthesis conditions .................................................... 242 5 Output-feedback control ........................................................................ 245 6 Extended dissipative filtering .................................................................. 250 7 Conclusions....................................................................................... 252 Acknowledgments .................................................................................. 253 References........................................................................................... 253 Further reading...................................................................................... 255

1 Introduction Singular systems, which are also called descriptor systems, can be used to describe lots of practical models, such as power systems, electrical networks, economic systems, robotics, and so on [1–3]. More importantly, the singular systems have some special characteristics that do not exist in normal state-space systems [3]. Stability, Control and Application of Time-Delay Systems. https://doi.org/10.1016/B978-0-12-814928-7.00011-1 © 2019 Elsevier Inc. All rights reserved.

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For example, the singular systems are always required to be regular and impulsefree when investigating the controller synthesis problems. Therefore, the concept of admissibility (i.e., the system is regular, impulse-free, and stable) should be considered for singular systems. These special characteristics make the study of singular systems more difficult. On the other hand, because of the singularity of matrix E (see Eq. 1), it is often difficult to derive easy-to-check conditions for analysis and synthesis problems. Due to these reasons, the study of singular systems has received considerable attention over the past decades. Recently, a great deal of research efforts has been devoted to singular timedelay systems with Markovian jumping parameters. The earliest results on this topic were reported in Refs. [4,5], where delay-independent conditions were obtained for solving the problems of stability, stabilization, and guaranteed-cost observer design, respectively. In Refs. [6–9], different approaches were developed to derive delaydependent admissibility conditions for singular time-delay systems with Markovian jumping parameters. The robust normalization problem for uncertain singular Markovian jump systems with time-varying delays was studied in Ref. [10] based on PD state-feedback controllers. The authors in Ref. [11] addressed the admissibility analysis and state-feedback stabilization problems for Itô-type stochastic singular systems with time-varying delays and Markovian jumping parameters. The control and filtering problems based on different performance constraints have been studied when the systems involve external disturbances. For example, the H∞ control and filtering problems were investigated in Refs. [12–16], while the dissipative control and filtering problems were considered in Refs. [17,18]. In Ref. [19], the concept of extended dissipativity was proposed for normal timedelay systems with Markovian jumping parameters. It has been shown in Ref. [19] that the extended dissipativity covers several important performances as special cases, such as the H∞ performance, L2 −L∞ performance, passivity, very-strict passivity, and strict (Q, S, R)-dissipativity. Therefore, based on the extended dissipativity, it is possible to provide a unified framework for designing controllers and filters with different disturbance-attenuation performances. Since it was proposed, the extended dissipativity concept has been applied for a variety of dynamic systems; see, for example, Refs. [20–25]. It is worth noting that the unified filter design framework has been applied in Ref. [25] for singular Markovian jump systems with time-varying delays. On the basis of the work [25], this chapter further investigates the extended dissipative control and filtering problems for continuous-time linear singular systems with constant delays and Markovian jumping parameters. New delay-dependent conditions for the analysis of stochastic admissibility and extended dissipativity are obtained by applying the ideas of the relaxed L-K functional approach proposed in Refs. [26–28]. These conditions are then applied to derive delay-dependent conditions for the existence of memory state-feedback controllers, memory outputfeedback controllers and delayed filters. The main results are expressed in the form of strict linear matrix inequalities (LMIs), which can be easily solved by using the Matlab LMI Control Toolbox.

2 Stochastic admissibility

The significance of the methods proposed in this chapter can be summarized as follows. (1) The employed L-K functional involves mode-dependent matrices in integrals, and it also relaxes the restriction on the positive definiteness of all L-K matrices. The mode dependence and restriction relaxation are efficient for reducing the conservatism of analysis results of time-delay systems, which has been illustrated in Refs. [19,26,28,29], respectively. (2) The extended dissipativity concept enables us to analyze the H∞ performance, L2 − L∞ performance, passivity, verystrict passivity and strict (Q, S, R)-dissipativity in a unified framework. Thus, this concept carries forward the traditional study on robust control theory. (3) Strict LMI approaches are developed for the design of controllers and filters. These approaches generalize the ones in the study of regular systems. Based on these approaches, the main results obtained in this chapter are novel and efficient.

2 Stochastic admissibility 2.1 System model and definitions Consider the following continuous-time linear singular system with constant delay and Markovian jumping parameters: E˙x (t) = A (rt ) x (t) + Aτ (rt ) x (t − τ ) ,

(1)

where x (t) ∈ n is the state vector; τ > 0 represents the constant delay; E ∈ n×n is a real matrix that may be singular (i.e., rank(E) = σ ≤ n); the coefficient matrices A (rt ) ∈ n×n and Aτ (rt ) ∈ n×n are dependent on the mode {rt }, which is a continuous-time Markov process with right continuous trajectories. It is assumed that rt takes values in a finite set {1, 2, . . . , s} with the transition probability given by   Pr rt+h = j|rt = i =



πij h + o (h) , 1 + πii h + o (h) ,

i = j i = j,

(2)

 where h > 0, limh→0 (o (h) /h) = 0, πii = − sj=1,j=i πij , and πij ≥ 0, for j = i. It is noted that πij is the transition rate from mode i at time t to mode j at time t + h. In the rest of this chapter, for each possible rt = i ∈ {1, 2, . . . , s}, a matrix M (rt ) will be denoted by Mi for notational simplicity; for example, A (rt ) = Ai , Aτ (rt ) = Aτ i , and so on. The initial conditions of system (1) are given by rt = r0 ,

x (t) = ϕ (t) ,

∀t ∈ [−2τ , 0]

(3)

where r0 is a positive integer belonging to {1, 2, . . . , s}, and ϕ (t) is a differentiable function defined over the interval [−2τ , 0]. Let x (t, ϕ, r0 ) denote the solution to system (1) under the initial conditions (3). Then, similar to Refs. [3,30], we introduce the following definitions.

229

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CHAPTER 11 Extended dissipative control and filtering

Definition 1 (Chen et al. [30]). System (1) is said to be stochastically stable if there exists a constant δ > 0 such that   ∞   |x (t, ϕ, r0 )|2 dt r0 , ϕ ≤ δ sup |ϕ(θ )|2 , E 0

θ∈[−2τ ,0]

where E {·} is the mathematical expectation operator with respect to the probability measure under consideration, and |x| denotes the Euclidean norm for vector x (i.e., |x|2 = xT x). Definition 2 (Xu and Lam [3]). System (1) is said to be regular if the pair (E, Ai ) is regular; that is, det(λE − Ai ) is not identically zero for every i ∈ {1, 2, . . . , s}. System (1) is said to be impulse-free if the pair (E, Ai ) is impulse-free; that is, deg(det(λE − Ai )) = rank(E) for every i ∈ {1, 2, . . . , s}. System (1) is said to be stochastically admissible if it is regular, impulse-free, and stochastically stable. For singular matrix E with rank(E) = σ < n, there always exist nonsingular matrices M ∈ n×n and N ∈ n×n such that

MEN =

Iσ 0

0 0



.

(4)

Write MAi N as MAi N =

Ai1 Ai3

Ai2 Ai4

,

(5)

where Ai4 ∈ (n−σ )×(n−σ ) . Then, the regularity and impulse-free of the pair (E, Ai ) can be ensured by the nonsingularity of matrix Ai4 . This result is expressed in the following lemma. Lemma 1 (Xu and Lam [3]). The pair (E, Ai ) is regular and impulse-free if and only if Ai4 is nonsingular. According to Definition 2 and Lemma 1, we can immediately obtain the following result, which is useful for deriving our main results. Lemma 2. System (1) is regular and impulse-free if and only if the matrix GMAi NGT is nonsingular, where G = 0 In−σ .

2.2 Relaxed L-K functional The relaxed L-K functional approach proposed in Refs. [26–28] does not require all the Lyapunov matrices to be positive definite, and thus it provides a new insight to improve stability conditions for time-delay systems. It is worth noting that the relaxed L-K functional approach has not been applied to study singular time-delay systems. For this reason, we are going to analyze the stochastic admissibility of system (1) by using the idea of the relaxed L-K functional approach. Specifically, we employ the following mode-dependent functional with real symmetric matrices P (rt ), Q (rt ), Z (rt ), R, and W: V (xt , rt , t) = V1 (t) + V2 (t) + V3 (t) + V4 (t) + V5 (t) ,

(6)

2 Stochastic admissibility

where xt = x (t + θ ), −2τ ≤ θ ≤ 0, and V1 (t) = x (t)T ET P (rt ) Ex (t)  t x (α)T Q (rt ) x (α) dα V2 (t) = t−τ

 0  t

V3 (t) =

−τ t+β  0  t

V4 (t) =

x (α)T Rx (α) dαdβ x˙ (α)T ET Z (rt ) E˙x (α) dαdβ

−τ t+β  0  0 t

V5 (t) =

−τ

t+β

θ

x˙ (α)T ET WE˙x (α) dαdβdθ.

It is worth noting that both the single and double integrals in Eq. (6) involve mode-dependent matrices. Thus, functional (6) is strongly dependent on the system mode. The construction of such a functional is motivated by the works [19,29]. Consequently, we give two properties for functional (6). Lemma 3. Consider the functional defined in Eq. (6). There always exists a constant δ > 0 such that V (x0 , r0 , 0) ≤ δ

|ϕ(θ)|2 .

sup

(7)

θ∈[−2τ , 0]

Proof. Note that the matrices Pi , Qi , Zi , R, and W are symmetric for every i = 1, 2, . . . , s. Choose a constant scalar δ1 that is positive and satisfies 

  δ1 ≥ max λmax ET Pi E , λmax (Qi ) , λmax (Zi ) , λmax (R) , λmax (W) . i∈{1,2,...,s}

Then, it is easily obtained that V (x0 , r0 , 0) ≤ δ1 V˜ (x0 , r0 , 0) ,

(8)

where  0

V˜ (x0 , r0 , 0) = |x (0)|2 + +

 0  0 −τ

β

−τ

|x (α)|2 dα +

|E˙x (α)|2 dαdβ +

 0  0

|x (α)|2 dαdβ

−τ β  0  0 0 −τ

θ

β

|E˙x (α)|2 dαdβdθ.

Moreover, it can be verified that V˜ (x0 , r0 , 0) ≤ |x (0)|2 + +

 0  0 −τ

−τ

 0 −τ

|x (α)|2 dα +

|E˙x (α)|2 dαdβ +

= |x (0)|2 + (1 + τ )

 0 −τ

 0  0

|x (α)|2 dαdβ

−τ −τ  0  0 0 −τ

θ

−τ

|E˙x (α)|2 dαdβdθ

|x (α)|2 dα + (τ + 0.5τ 2 )

 0 −τ

|E˙x (α)|2 dα.

(9)

231

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CHAPTER 11 Extended dissipative control and filtering

For every rt = i ∈ {1, 2, . . . , s}, it follows from Eq. (1) that   |E˙x (α)|2 ≤ δ2 |x (α)|2 + |x (α − τ )|2 ,

where

 δ2 = λmax



ATi ATτ i



(10)



Ai

Aτ i



.

Integrating both sides of Eq. (10) from −τ to 0 yields that 

 0 −τ

|E˙x (α)|2 dα ≤ δ2

0

−τ

 = δ2 = δ2

0

−τ

 0

−2τ

|x (α)|2 dα + |x (α)|2 dα +

 0 −τ

 |x (α − τ )|2 dα

 −τ −2τ

 |x (α)|2 dα

|x (α)|2 dα.

(11)

Recalling the initial conditions in Eq. (3), it is obtained that  0 −2τ

|x (α)|2 dα ≤ 2τ

sup

|ϕ(θ)|2 .

(12)

θ∈[−2τ , 0]

Therefore, the inequality in Eq. (7) can be readily derived based on Eqs. (8), (9), (11), (12). In this case, the constant scalar δ can be chosen as δ = δ1 1 + τ + (1 + 2δ2 )τ 2 + δ2 τ 3 . The proof is completed. It is shown in Lemma 3 that V (x0 , r0 , 0) has an upper bound. Next, we shall show that the functional (6) is semipositive definite (i.e., V (xt , rt , t) ≥ 0). To this end, we need the Jensen’s integral inequality. Lemma 4 (Gu [31] and Shu and Lam [32]). For any constant matrix Z ∈ n×n with Z > 0, scalars b > a, vector function v: [a, b] → n , such that the following integrals are well-defined, then   T    b

(b − a)

b

v (α)T Zv (α) dα ≥

a

b

v (α) dα

Z

a

v (α) dα . a

Lemma 5. The functional (6) is semipositive definite (i.e., V (xt , rt , t) ≥ 0), if Pi > 0, Zi > 0, W > 0 and f1i (τ )  f2i (τ ) 

ET Pi E + 2ET Zi E 

2ET Zi E 2τ Qi + 2ET Zi E

ET Pi E + τ ET WE 

τ ET WE 2 τ R + τ ET WE

≥ 0,

(13)

≥ 0,

where  is used as an ellipsis for terms that are induced by symmetry.

(14)

2 Stochastic admissibility

Proof. It is worth noting that  0 t  t 1 1 T T x (t) E Pi Ex (t) dα + 2 x (t)T ET Pi Ex (t) dα dβ. V1 (t) = 2τ t−τ τ −τ t+β Note also that Zi > 0 and W > 0. Then, it follows from Lemma 4 that

 0 1 − (x (t) − x (t + β))T ET Zi E (x (t) − x (t + β)) dβ β −τ

 0 1 ≥ (x (t) − x (t + β))T ET Zi E (x (t) − x (t + β)) dβ −τ τ  1 t [x (t) − x (α)]T ET Zi E [x (t) − x (α)] dα = τ t−τ

V4 (t) ≥

and

 0  0 1 − (x (t) − x (t + β))T ET WE (x (t) − x (t + β)) dβdθ β −τ θ

 0  0 1 ≥ (x (t) − x (t + β))T ET WE (x (t) − x (t + β)) dβdθ τ −τ θ   1 0 t [x (t) − x (α)]T ET WE [x (t) − x (α)] dαdβ. = τ −τ t+β

V5 (t) ≥

Therefore, it is obtained that

T

 t 1 x(t) x(t) f1i (τ ) dα −x(α) 2τ t−τ −x(α)

T

 0  t 1 x(t) x(t) f2i (τ ) dαdβ, + 2 −x(α) τ −τ t+β −x(α)

V (xt , rt , t) ≥

where f1i (τ ) and f2i (τ ) are defined in Eqs. (13), (14). Therefore, under conditions (13), (14), we have that V (xt , rt , t) ≥ 0. The proof is completed. It is worth noting that, if Qi > 0 and R > 0, then the inequalities in Eqs. (13), (14) are satisfied naturally. However, the feasibility of Eqs. (13), (14) does not require that Qi > 0 and R > 0. This means that Lemma 5 provides relaxed conditions guaranteeing the semipositive definiteness of V (xt , rt , t). It should be also pointed out that the conditions in Eqs. (13), (14) are not strict LMIs, which may be difficult to solve. Regarding this, we give the following lemma that provides strict LMI conditions guaranteeing the semipositive definiteness of V (xt , rt , t). Lemma 6. The functional (6) is semipositive definite (i.e., V (xt , rt , t) ≥ 0), if Pi > 0, Zi > 0, W > 0 and

Pi + 2Zi 

2Zi E 2τ Qi + 2ET Zi E



> 0,

Pi + τ W 

τ WE τ 2 R + τ ET WE

> 0.

(15)

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CHAPTER 11 Extended dissipative control and filtering

Proof. By premultiplying diag{ET , I} and postmultiplying diag{E, I} to the matrices in Eq. (15), we can readily obtain the inequalities in Eqs. (13), (14). Then, the proof can be completed according to Lemma 5.

2.3 Stochastic admissibility criteria Theorem 1. Give any full column rank matrix S ∈ n×(n−σ ) satisfying ET S = 0. System (1) is stochastically admissible if there exist matrices Pi > 0, Zi > 0, W > 0, Qi , R, and Li such that Eqs. (13), (14), as well as the following conditions hold: s  j=1

where

s 

πij Qj − R < 0,

f3i (τ ) 

Γ1i 

⎛ s 

⎞

Γ1i = ET ⎝

πij Zj − W < 0

j=1

Γ2i Γ3i



+



ATi ATτ i





τ Zi + 0.5τ 2 W

(16) 

Ai

Aτ i



< 0,

(17)

πij Pj ⎠ E + ATi (Pi E + SLi ) + (Pi E + SLi )T Ai + Qi + τ R − τ −1 ET Zi E,

j=1

Γ2i = (Pi E + SLi )T Aτ i + τ −1 ET Zi E,

Γ3i = −Qi − τ −1 ET Zi E.

Proof. Consider the functional in Eq. (6). Define the weak infinitesimal generator of the random process {xt , rt } as follows [33]: AV (xt , rt , t) = lim

h→0+

1 [E { V (xt+h , rt+h , t + h)| xt , rt = i} − V (xt , i, t)] . h

Then, we shall compute the weak infinitesimal generator of V (xt , rt , t) along the solutions to system (1). First, it is obtained that ⎛ ⎞ s  AV1 (t) = x (t)T ET ⎝ πij Pj ⎠ Ex (t) + 2˙x (t)T ET Pi Ex (t) . j=1

Note that ET S = 0, which implies ET Pi E = ET (Pi E + SLi ). Then, by recalling the state equation (1), we have ⎛ ⎞ s  AV1 (t) = x (t)T ET ⎝ πij Pj ⎠ Ex (t) + 2 [Ai x (t) + Aτ i x (t − τ )]T (Pi E + SLi )x (t) . (18) j=1

Next, by applying the techniques developed in Ref. [33], we obtain A (V2 (t) + V3 (t)) = x (t)T (Qi + τ R) x (t) − x (t − τ )T Qi x (t − τ ) ⎤ ⎡  t s  x (α)T ⎣ πij Qj − R⎦ x (α) dα. + t−τ

j=1

2 Stochastic admissibility

This, together with the first inequality in Eq. (16), implies that A (V2 (t) + V3 (t)) ≤ x (t)T (Qi + τ R) x (t) − x (t − τ )T Qi x (t − τ ) .

(19)

Similarly, by recalling the second inequality in Eq. (16), we have  t   A (V4 (t) + V5 (t)) = x˙ (t)T ET τ Zi + 0.5τ 2 W E˙x (t) − x˙ (α)T ET Zi E˙x (α) dα t−τ ⎞ ⎛  0  t s  x˙ (α)T ET ⎝ πij Zj − W ⎠ E˙x (α) dαdβ + −τ

t+β

j=1

 t  ≤ x˙ (t)T ET τ Zi + 0.5τ 2 W E˙x (t) − x˙ (α)T ET Zi E˙x (α) dα. 

t−τ

(20)

By applying Lemma 4, it is obtained that −

 t t−τ

x˙ (α)T ET Zi E˙x (α) dα ≤ −τ −1 [x(t) − x(t − τ )]T ET Zi E [x(t) − x(t − τ )] .

(21)

It follows from Eqs. (18)–(21) that AV (xt , i, t) ≤

x(t) x(t − τ )

T

f3i (τ )

x(t) x(t − τ )

,

(22)

where f3i (τ ) is defined in Eq. (17). Obviously, there always exist a sufficient small scalar ε > 0 such that f3i (τ ) ≤ −εI. This, together with Eq. (22), implies that AV (xt , i, t) ≤ −ε |x (t)|2 . Therefore, for any t > 0, it is obtained by applying the Dynkin’s formula that  E {V (xt , rt , t)} − V (x0 , r0 , 0) ≤ −εE

t

 |x (α)|2 dα .

0

Moreover, if the conditions in Eqs. (13), (14) hold, then it follows from Lemmas 3 and 5 that  t  2 |x (α)| dα ≤ ε−1 V (x0 , r0 , 0) ≤ δε−1 sup |ϕ(θ )|2 . E 0

θ∈[−2τ , 0]

Let t approach ∞. Then, by Definition 1, system (1) is stochastically stable. Next, we are going to show that system (1) is regular and impulse-free under conditions (13), (14), (17). For nonsingular matrices M and N satisfying Eq. (4), the following equality holds: ENGT = M −1 (MEN)GT = 0,

(23)

235

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CHAPTER 11 Extended dissipative control and filtering

where G =



0

In−σ



. By using this fact, it is obtained that

! GN T Γ1i NGT = GN T ATi SLi + LiT ST Ai + Qi + τ R NGT .

(24)

On the other hand, the conditions in Eqs. (13), (14) give that 2τ Qi + 2ET Zi E ≥ 0 and τ 2 R + τ ET WE ≥ 0. This, by using Eq. (23), implies that GN T Qi NGT ≥ 0 and GN T RNGT ≥ 0. Then, it follows from Eq. (24) that ! GN T Γ1i NGT ≥ GN T ATi SLi + LiT ST Ai NGT

(25)

Moreover, it follows from Eq. (17) that Γ1i < 0, which further implies GN T Γ1i NGT < 0 since G is full row rank. This, together with Eq. (25), yields that the matrix GN T LiT ST Ai NGT is nonsingular. Note that MEN + GT G = In . Thus,   GN T LiT ST Ai NGT = GN T LiT ST M −1 MEN + GT G MAi NGT .

(26)

Since ET S = 0, which is equivalent to ST E = 0, equality (26) can be rewritten as    GN T LiT ST Ai NGT = GN T LiT ST M −1 GT GMAi NGT . Note that GN T LiT ST Ai NGT is nonsingular and GMAi NGT is square. Then, we have that GMAi NGT is nonsingular. Therefore, by Lemma 2, system (1) is regular and impulse-free. Based on the earlier analysis and according to Definition 2, we can draw the conclusion that system (1) is stochastically admissible. Theorem 1 provides delay-dependent conditions that guarantee the stochastic admissibility of system (1). The conditions in Theorem 1 are efficient for controller and filter synthesis, which will be shown in Sections 4–6. However, these conditions are not easy to be solved due to the existence of the nonstrict LMIs (13), (14). Regarding this, it is necessary for us to give strict LMI conditions for the stochastic admissibility analysis of system (1). This purpose, according to Lemma 6, can be achieved by replacing Eqs. (13), (14) with Eq. (15). The corresponding theorem is presented here. Theorem 2. Give any full column rank matrix S ∈ n×(n−σ ) satisfying ET S = 0. system (1) is stochastically admissible if there exist matrices Pi > 0, Zi > 0, W > 0, Qi , R, and Li such that the strict LMIs in Eqs. (15)–(17) hold.

3 Extended dissipativity 3.1 Definitions and assumptions In this section, we introduce the definition of the extended dissipativity and provide some necessary assumptions. Consider the following singular linear systems with constant delays, Markovian jumping parameters and external disturbances:

3 Extended dissipativity



E˙x (t) z (t)

= =

A (rt ) x (t) + Aτ (rt ) x (t − τ ) + D (rt ) ω (t) , C (rt ) x (t)

(27)

where x (t) ∈ n is the state vector; z (t) ∈ nz is the system output; ω (t) ∈ nω is the external disturbance; A (rt ), Aτ (rt ), C (rt ), and D (rt ) are known mode-dependent matrices; " ∞ and E is the singular matrix. It is assumed that ω (t) is energy bounded; that is, 0 |ω (t)|2 dt < ∞. The following assumptions are adopted throughout this chapter. Assumption 1 (Zhang et al. [34]). For each i ∈ {1, 2, . . . , s}, it is assumed that ker(E) ⊆ ∩si=1 ker(Ci ), where ker(·) denotes the kernel of a matrix. Remark 1. For any nonsingular matrices M and N satisfying Eq. (4), we have that MEN = (MEN)2 . This implies that E = ENME, which can be rewritten as E(I − NME) = 0. Then, under Assumption 1, it is obtained that Ci (I − NME) = 0, which is equivalent to Ci = Ci NME. This fact is helpful to develop strict LMI conditions for extended dissipativity analysis. Assumption 2 (Zhang et al. [19]). The weighting matrices Ψ1 , Ψ2 , Ψ3 , and Ψ4 are given in prior according to design requirements, and they satisfy Ψ1 ≥ 0, Ψ3 > 0, Ψ4 ≥ 0, and ( Ψ1 + Ψ2 ) Ψ4 = 0. Remark 2. It is worth noting that the matrices Ψ1 ≥ 0 and Ψ4 ≥ 0 can be decomposed as Ψ1 = Ψ¯ 1T Ψ¯ 1 ,

Ψ4 = Ψ¯ 4T Ψ¯ 4 .

(28)

This decomposition will be used in the controller and filter synthesis. Definition 3 (Zhang et al. [19]). Give matrices Ψ1 , Ψ2 , Ψ3 , and Ψ4 satisfying Assumption 2. Then, system (27) is said to be extended dissipative if there exists a scalar ρ such that the following inequality holds for any tf ≥ 0 and all energybounded ω (t): E

 t f 0

 # $ J (t) dt − sup E z (t)T Ψ4 z (t) ≥ ρ,

(29)

0≤t≤tf

where J (t) = ω (t)T Ψ3 ω (t) + 2z (t)T Ψ2 ω (t) − z (t)T Ψ1 z (t) . By choosing appropriate values for the weighting matrices, the extended dissipativity defined earlier can reduce to several well-known performances, such as H∞ performance, L2 − L∞ performance, passivity, very-strict passivity, and strict (Q, S, R)-dissipativity. Therefore, the extended dissipativity can be regarded as a general disturbance attenuation performance. The readers are referred to Ref. [19] for more explanations on the extended dissipativity concept.

3.2 Extended dissipativity criteria Theorem 3. Under Assumptions 1 and 2, system (27) is extended dissipative in the sense of Definition 3, if there exist matrices Pi > 0, Zi > 0, W > 0, Qi , R, and Li such that Eq. (16) as well as the following conditions hold:

237

238

CHAPTER 11 Extended dissipative control and filtering

f1i (τ ) ≥

CiT Ψ4 Ci 0

0 0



⎡

Γ1i + CiT Ψ1 Ci f4i (τ )  ⎣  

CiT Ψ4 Ci 0 , (30) 0 0 ⎤ ⎡ T ⎤ ⎡ T ⎤T Ai Ai Γ4i   2 T 0 ⎦ + ⎣ Aτ i ⎦ τ Zi + 0.5τ W ⎣ ATτ i ⎦ < 0, −Ψ3 DTi DTi (31)

f2i (τ ) ≥

,

Γ2i Γ3i 

where f1i (τ ) and f2i (τ ) are defined in Eqs. (13), (14), Γ1i , Γ2i , and Γ3i are the same as those in Theorem 1, and Γ4i = (Pi E + SLi )T Di − CiT Ψ2 . Proof. Consider the functional defined in Eq. (6). Similar to the proof of Lemma 5, it is obtained from Eq. (30) that V (xt , rt , t) ≥ x(t)T CiT Ψ4 Ci x(t) = z(t)T Ψ4 z(t).

(32)

On the other hand, similar to the proof of Theorem 1, it is obtained that ⎡

⎡ ⎤T ⎤ x(t) x(t) AV (xt , i, t) − J (t) ≤ ⎣ x(t − τ ) ⎦ f4i (τ ) ⎣ x(t − τ ) ⎦ , ω(t) ω(t)

(33)

where f4i (τ ) is defined in Eq. (31). Thus, it follows from Eqs. (31), (33) that J (t) ≥ AV (xt , i, t). This implies that E

 t 0

 J (α) dα ≥ E {V (xt , rt , t)} − V (x0 , r0 , 0) .

(34)

Choose ρ = −V (x0 , r0 , 0). Then, it follows from Eqs. (32), (34) that E

 t 0

 # $ J (α) dα ≥ E z(t)T Ψ4 z(t) + ρ.

(35)

When Ψ4 = 0, the following inequality holds for any tf ≥ 0: E

 t f 0

 J (α) dα ≥ ρ.

(36)

When Ψ4 = 0, according to Assumption 2, we have that Ψ1 = 0 and Ψ2 = 0. In this case, J (t) = ω (t)T Ψ3 ω (t) ≥ 0. Therefore, for any tf ≥ t, the following inequality holds: E

 t f 0

  t  # $ J (α) dα ≥ E J (α) dα ≥ E z(t)T Ψ4 z(t) + ρ.

(37)

0

Based on this fact, condition in Eq. (29) can be readily obtained. Therefore, by Definition 3, system (27) is extended dissipative. The proof is completed. In the following theorem, we provide the strict LMI conditions ensuring the extended dissipativity of system (27).

4 State-feedback control

Theorem 4. Under Assumptions 1 and 2, system (27) is extended dissipative in the sense of Definition 3, if there exist matrices Pi > 0, Zi > 0, W > 0, Qi , R, and Li such that Eqs. (16), (31), as well as the following conditions hold:

Pi + 2Zi − (Ci NM)T Ψ4 Ci NM 

2Zi E 2τ Qi + 2ET Zi E

Pi + τ W − (Ci NM)T Ψ4 Ci NM 

τ WE

>0

(38)

> 0.

(39)



τ 2 R + τ ET WE

Proof. Note that Ci = Ci NME. By premultiplying diag{ET , I} and postmultiplying diag{E, I} to the matrices in Eqs. (38), (39), we can readily obtain the inequalities in Eq. (30). Then, the proof can be completed according to Theorem 3. Remark 3. It is not difficult to find that, if the conditions in Theorem 3 hold, then the conditions in Theorem 1 are also satisfied. This means that the conditions in Theorem 3 not only guarantee the extended dissipativity of system (27), but also ensure the stochastic admissibility of system (27) with zero disturbances.

4 State-feedback control 4.1 Problem formulation Consider system (27) with control input; that is, 

E˙x (t) z (t)

= =

A (rt ) x (t) + Aτ (rt ) x (t − τ ) + B (rt ) u (t) + D (rt ) ω (t) , C (rt ) x (t)

(40)

where u (t) ∈ nu is the control input and B(rt ) is the input coefficient matrix. For this system, we consider a memory state-feedback controller that takes the following form: u(t) = K (rt ) x (t) + Kτ (rt ) x (t − τ ) ,

(41)

where K (rt ) and Kτ (rt ) are the mode-dependent gains to be determined. The delay τ should be known since it is involved in the controller. If the delay τ is not known, we can set Kτ (rt ) = 0, and then the controller reduces to a memoryless one. Let rt = i. Then, combining Eqs. (40), (41) yields the closed-loop system: 

E˙x (t) z (t)

= =

A¯ i x (t) + A¯ τ i x (t − τ ) + Di ω (t) , Ci x (t)

(42)

where A¯ i = Ai + Bi Ki and A¯ τ i = Aτ i + Bi Kτ i . The state-feedback control problem is formulated as to determine the controller gains Ki and Kτ i such that the closed-loop system in Eq. (42) is extended dissipative in the sense of Definition 3. Our objective is to develop strict LMI conditions for the solvability of this problem.

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CHAPTER 11 Extended dissipative control and filtering

4.2 Useful lemmas In this section, we give some lemmas that are useful to establish our main results. Lemma 7. Consider matrices P > 0, E, S, and L with appropriate dimensions. If rank(E) = σ ≤ n, PE + SL is nonsingular, and S is a full column rank matrix satisfying ET S = 0, then there exist matrices P¯ > 0 and L¯ such that ¯ T + S¯ L, ¯ (PE + SL)−1 = PE

(43)

where S¯ ∈ n×(n−r) is a full column rank matrix satisfying ES¯ = 0. Proof. Recalling the nonsingular matrices M and N satisfying Eq. (4), we set M −T PM −1 =



 S PT2 , M −T S = 1 , LN = L1 S2 P3



P1 P2

L2 .

(44)

Then, it can be verified that  −1 P + S L 1 1 N −1 (PE + SL)−1 M T = M −T PM −1 MEN + M −T SLN = 1 P2 + S2 L1

S1 L2 S2 L2

−1 . (45)

Note that ET S = 0. We have that

I 0 = N T ET M T M −T S = σ 0



S1 S = 1 . S2 0

0 0

(46)

This implies S1 = 0. Thus, S2 is nonsingular since S is full column rank. It follows from Eq. (45) that N −1 (PE + SL)−1 M T =



P1 P2 + S2 L1 

0 S2 L2

−1

P−1 1 = −1 −(S2 L2 ) (P2 + S2 L1 )P−1 1

 0 . (S2 L2 )−1

(47)

On the other hand, by noting that N T ET M T M −T S = 0 and MEN = (MEN)T = we have ENM −T S = 0. We choose S¯ = NM −T S. It is obviously that S¯ −1 ¯ is full column rank and ES¯ = 0. Moreover, we choose P¯ 1 = P−1 1 , P2 = −ΦP1 , −1 −1 −1 −1 T −1 P¯ 3 > ΦP1 Φ , and L¯ 2 = S2 L2 S2 , where Φ = S2 L¯ 1 P1 + (S2 L2 ) (P2 + S2 L1 ) and L¯ 1 is any matrix. Then, set

 ¯ ¯T ¯P = N P1 P2 N T , L¯ = L¯ 1 L¯ 2 M −T . ¯ ¯ N T ET M T ,

P2

P3

By using the matrices defined earlier, we have that

¯ ¯ T +S¯ L)M ¯ T = P1 N −1 (PE P¯ 2

P¯ T2 P¯ 3



Iσ 0



0 0 ¯ L1 + 0 S2

P¯ 1 ¯L2 = P¯ 2 + S2 L¯ 1

0 . (48) S2 L¯ 2

4 State-feedback control

By comparing Eqs. (48) with Eq. (47), it is seen that ¯ T + S¯ L)M ¯ T. N −1 (PE + SQ)−1 M T = N −1 (PE This implies the equality (43). Now we are going to prove that P¯ is positive definite. It can be verified that

P¯ 1 P¯ 2





P−1 P¯ T2 1 ¯P3 = −ΦP−1 1





T −P−1 I 1 Φ = −Φ P¯ 3

0 I



P−1 1 0

0 T P¯ 3 − ΦP−1 1 Φ



I 0



−Φ T . I

T Since P > 0, we have that P1 > 0. Note also that P¯ 3 > ΦP−1 1 Φ . Then, we have that

P¯ 1 P¯ T2 > 0.

P¯ 2

P¯ 3

This gives that P¯ > 0. The proof is completed. Remark 4. A similar expression to Lemma 7 has been given in Refs. [35,36], and the Chinese version of the proof has been given in Ref. [36]. In the earlier, we provide the English version of the proof in order to make the result more understandable. Lemma 8. Suppose the relation in Eq. (43) is satisfied. Then, the following equality holds:

T −T

T −1 ¯ ¯ T ) PE ¯ ET PE = PE + S¯ L¯ (EPE + S¯ L¯ . Proof. Note that ET S = 0 and ES¯ = 0. Then, it is obtained that ET PE = (PE + SL)T E,  −T  −1 ¯ T + S¯ L¯ ¯ T + S¯ L) ¯ PE ¯ T + S¯ L¯ = PE E(PE ,  −T  −1 ¯ T + S¯ L¯ ¯ T ) PE ¯ T + S¯ L¯ = PE (EPE .

The proof is completed. Lemma 9. Consider P > 0, L, and S with ET S = 0. For any matrix Z, the following equality holds: (PE + SL)T EZET (PE + SL) = ET (PE + SL)Z(PE + SL)T E. Proof. Both sides of the earlier equality equal to ET PEZET PE. Lemma 10. The inequality YX −1 Y T ≥ Y + Y T − X holds for any matrices X > 0 and Y. Proof. The proof can be established by noting that (X − Y)X −1 (X − Y)T = X − Y − Y T + YX −1 Y T ≥ 0.

241

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CHAPTER 11 Extended dissipative control and filtering

4.3 Controller synthesis conditions Now we are in a position to present conditions for the existence of desired statefeedback controllers in the form of Eq. (41). To make the conditions concise, we introduce the following matrix functions:  √

√ √ πi1 Ui · · · πi,i−1 Ui πi,i+1 Ui   N(Ui )  diag U1 , U2 , . . . , Ui−1 , Ui+1 , . . . , Us ,

M(Ui ) 

···

√ πis Ui ,

where Ui denote the matrix variables, which should be specified according to demand. Theorem 5. Consider the system in Eq. (40). Given any full column rank matrix S¯ satisfying ES¯ = 0, under Assumptions 1 and 2 with decomposition (28), there exists a memory state-feedback controller in the form of Eq. (41) such that closedloop system (42) is extended dissipative, if there exist matrices P¯ i > 0, X¯ i > 0, ¯ i , R, ¯ i < Y¯ i and the ¯ > 0, Q ¯ L¯ i , K¯ i , and K¯ τ i such that EP¯ i ET < X¯ i , Q Y¯ i > 0, Z¯ i > 0, W following LMIs hold:

¯ i + R¯ − U ¯i −U ¯T πii Q i 

¯ T) M(U i ¯ ¯ T) ¯ N(Yi − Ui − U i

¯ −U ¯i −U ¯T πii Z¯ i + W i 

¯ i) M(U ¯ ¯ T) ¯ N(Zi − Ui − U i

⎡

P¯ i + 2Z¯ i ⎣  

2Z¯ i ET ¯ 2τ Qi + 2EZ¯ i ET 

⎡

P¯ i + τ Ξi ⎣   ⎡ Υ1i Υ2i ⎢ ⎢  Υ3i ⎢ ⎢ ⎢   ⎢ ⎢   ⎢ ⎣    

τ

< 0,

(49)

< 0,

(50)



⎤ P¯ i CiT Ψ¯ 4T ⎦ > 0, 0 I

τ Ξi ET  ¯i +U ¯ T − R¯ + τ EΞi ET U i 

2

¯ T CT Ψ2 Di − U i i



(51)

⎤ P¯ i CiT Ψ¯ 4T ⎦ > 0, 0 I

0

¯ T AT + K¯ T BT U i i i i ¯ T AT + K¯ T BT U

¯ T AT + K¯ T BT U i i i i ¯ T AT + K¯ T BT U

−Ψ3   

DTi  ¯i −U ¯T ¯ τ Zi − U i  

DTi 0 ¯ −2τ −2 W 

i

τi

τi i

i

τi

τi i

(52)

Υ4i

⎤

⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ < 0, (53) ⎥ 0 ⎥ ⎥ 0 ⎦ Υ5i

¯ i = P¯ i ET + S¯ L¯ i , Ξi = U ¯i +U ¯ iT − W ¯ and where U ¯ i − τ −1 EZ¯ i ET , ¯ iT ATi + K¯ iT BTi + Ai U ¯ i + Bi K¯ i + Q Υ1i = πii EP¯ i ET + U ¯ i − τ −1 EZ¯ i ET ¯ i + Bi K¯ τ i + τ −1 EZ¯ i ET , Υ3i = −Q Υ2i = Aτ i U $ #   T T T ¯i − U ¯ iT . ¯ N X¯ i − U ¯ C Ψ¯ ¯ T ) , Υ5i = diag −I, −τ −1 R, ¯ T M(U Υ4i = U U i i 1 i i

4 State-feedback control

¯ i +U ¯ i > 0, When the LMI conditions in this theorem are feasible, it is obtained that U ¯ i is nonsingular. In this case, the desired controller gains can be which implies that U obtained as ¯ −1 , Ki = K¯ i U i

¯ −1 . Kτ i = K¯ τ i U i

(54)

Proof. Suppose that the conditions in the theorem are satisfied and the controller gains are constructed as in Eq. (54). By applying Lemma 10, we have that ¯ iT R¯ −1 U ¯ i ≤ R¯ − U ¯ iT ≤ Y¯ i − U ¯i − U ¯ iT and −U ¯ i Y¯ i−1 U ¯i − U ¯ iT . Based on these −U two inequalities, it follows from Eq. (49) that

¯i −U ¯ T R¯ −1 U ¯ T) ¯i πii Q M(U i i < 0. ¯ ¯ −1 ¯ T 

N(−Ui Yi Ui )

By applying the Schur complement equivalence to this inequality, we obtain ⎛ ¯ iU ¯ iT ⎝πii U ¯ −T Q ¯ −1 − R¯ −1 + U i i

 j=i

⎞ ¯ −T Y¯ j U ¯ −1 ⎠ U ¯ i < 0. πij U j j

(55)

¯ j ≤ Y¯ j and πij ≥ 0 when j = i. It follows from Eq. (55) that Note that Q s 

¯ jU ¯ −T Q ¯ −1 − R¯ −1 < 0. πij U j j

(56)

j=1

¯ iU ¯ i−1 and R = R¯ −1 . Then Eq. (56) is the same as the first ¯ i−T Q Define Qi = U inequality in Eq. (16). Similar to the earlier analysis, it follows from Eq. (50) that s 

¯ −1 < 0. ¯ −1 Z¯ j U ¯ −T − W πij U j j

(57)

i=1

¯ i−T and W = W ¯ i−1 Z¯ i U ¯ −1 . Then, Eq. (57) is the same as the second Define Zi = U inequality in Eq. (16). Now, applying the Schur complement equivalence to Eq. (51) yields

P¯ i + 2Z¯ i − P¯ i CiT Ψ4 Ci P¯ i 

2Z¯ i ET ¯ 2τ Qi + 2EZ¯ i ET

> 0,

(58)

# $ ¯ i−T E, U ¯ i−T and postmultiplying where Ψ4 = Ψ¯ 4T Ψ¯ 4 . By premultiplying diag U $ # ¯ i−1 , U ¯ i−1 to the left-hand side of Eq. (58), we obtain diag ET U 



¯ −T E P¯ i + 2Z¯ i − P¯ i CT Ψ4 Ci P¯ i ET U ¯ −1 U i i i 

¯ −T EZ¯ i ET U ¯ −1 2U i i  −T ¯ ¯ −1 ¯ ¯ 2τ Qi + 2EZi ET U Ui i

 ≥ 0. (59)

243

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CHAPTER 11 Extended dissipative control and filtering

¯ i = P¯ i ET + S¯ L¯ i . Thus, according to Lemma 7, for each i ∈ {1, 2, . . . , s}, Note that U there exist Pi > 0 and Li such that ¯ −1 = Pi E + SLi , U i

(60)

where S is a full column rank matrix satisfying ET S = 0. Moreover, by Lemmas 8 and 9, we have that ¯ −T EP¯ i ET U ¯ −1 , E T Pi E = U i i

¯ −T EZ¯ i ET U ¯ −1 . E T Zi E = U i i

(61)

Note that ES¯ = 0. Then, under Assumption 1, we have that Ci S¯ = 0. This implies that   ¯ −1 = Ci . ¯ −1 = Ci P¯ i ET + S¯ L¯ i U Ci P¯ i ET U i i

(62)

¯ iU ¯ i−1 defined earlier, we obtain the first inequality in Eq. (30) ¯ i−T Q Recalling Qi = U from Eqs. (59)–(62). ¯i + U ¯ iT − W ¯ ≤ On the other hand, it follows from Lemma 10 that Ξi = U T −1 ¯ ¯ ¯ Ui W Ui . This, together with Eq. (52), implies that ⎡ ⎢ ⎣

¯ iW ¯ −1 U ¯T P¯ i + τ U i

¯ iW ¯ −1 U ¯ T ET τU i

P¯ i CiT Ψ¯ 4T

 

¯ iW ¯ T R¯ −1 U ¯ i + τ EU ¯ −1 U ¯ T ET τ 2U i i 

0 I

⎤

⎥ ⎦ > 0.

(63)

¯i = U ¯ iT ET , it can be verified Similar to the earlier analysis and using the fact that EU that inequality (63) implies the second inequality in Eq. (30). Next, by using Lemma 10, it follows from Eq. (53) that ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

0

¯ T A¯ T U i i ¯ T A¯ T U

¯ T A¯ T U i i ¯ T A¯ T U

−Ψ3

DTi

DTi

Υ1i

Υ2i

¯ T CT Ψ2 Di − U i i



Υ3i





i

τi

i

τi







¯ T Z¯ −1 U ¯i −τ U i i









¯ −2τ −2 W











0

Υ4i

⎤

⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ ⎥ < 0, 0 ⎥ ⎥ ⎥ 0 ⎦ Υ¯5i

(64)

 ¯ ¯ where Ai + Bi Ki , A¯ τ i = Aτ i + Bi Kτ i , and Υ¯ 5i = diag −I, −τ −1 R,  Ai = $ ¯ iT . Then, by applying the Schur complement equivalence to Eq. (64), ¯ i X¯ i−1 U N −U we have ⎡ ⎢ ⎣

Υ¯1i  

Υ2i Υ3i 

⎤ ⎡ T T ⎤ ¯ A¯ ¯ T CT Ψ2 U Di − U i i   i i ⎥ ⎢ ¯ T ¯ T ⎥ −1 ¯ −1 ¯ ¯ −T 2 ¯ −1 0 ⎦ + ⎣ Ui Aτ i ⎦ τ Ui Zi Ui + 0.5τ W −Ψ3 DTi ⎡ T T ⎤T ¯ A¯ U i i ⎢ ¯ T ¯T ⎥ (65) ⎣ Ui Aτ i ⎦ < 0, DTi

5 Output-feedback control

where ⎛



¯ i + τU ¯i +U ¯ iT CiT Ψ1 Ci U ¯ iT R¯ −1 U ¯ iT ⎝ Υ¯1i = Υ1i + U

⎞ ¯ i. ¯ j−T X¯ j U ¯ j−1 ⎠ U πij U

j=i

Recalling that EP¯ i ET ≤ X¯ i and πij ≥ 0 when j = i, we obtain 

¯ j−T EP¯ j ET U ¯ j−1 ≤ πij U

j=i



¯ j−T X¯ j U ¯ j−1 . πij U

j=i

# $ # $ ¯ i−T , U ¯ i−1 , U ¯ i−T , I and postmultiplying diag U ¯ i−1 , I Then, by premultiplying diag U to the left-hand side of Eq. (65), we obtain ⎡ ⎢ ⎣

Γ¯1i + CiT Ψ1 Ci

UiT A¯ τ i + τ −1 ET Zi E

UiT Di − CiT Ψ2

 

−Qi − τ −1 ET Zi E 

0 −Ψ3

⎡

⎤

A¯ Ti

⎡

A¯ Ti

⎤ ⎥ ⎦

⎤T

⎢ ⎥ ⎥ ⎢ 2 ⎢ ¯T ⎥ ¯T ⎥ +⎢ ⎣ Aτ i ⎦ τ Zi + 0.5τ W ⎣ Aτ i ⎦ < 0, DTi DTi

(66)

¯ i−1 = Pi E + SLi and where Ui = U ⎛ ⎞ s  πij Pj ⎠ E + A¯ Ti Ui + UiT A¯ i + Qi + τ R − τ −1 ET Zi E. Γ¯1i = ET ⎝ j=1

It is obvious that Eq. (66) is the same as the condition in Eq. (31) with Ai and Aτ i replaced by A¯ i and A¯ τ i , respectively. Based on the earlier analysis and using Theorem 3, we have that the closed-loop system in Eq. (42) is extended dissipative. The proof is completed here.

5 Output-feedback control This section aims to study the problem of dynamic output-feedback controller design for singular time-delay systems with Markovian jump systems. To this end, we consider system (40) with a measured output: ⎧ ⎨ E˙x (t) z (t) ⎩ y (t)

= = =

A (rt ) x (t) + Aτ (rt ) x (t − τ ) + B (rt ) u (t) + D (rt ) ω (t) C (rt ) x (t) , Cy (rt ) x (t)

(67)

245

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CHAPTER 11 Extended dissipative control and filtering

where y (t) ∈ ny is the measured output. For system (67), we consider the following dynamic output-feedback controller: 

E˙xc (t) u (t)

= =

Ac (rt ) xc (t) + Acτ (rt ) xc (t − τ ) + Bc (rt ) y (t) , Cc (rt ) xc (t) + Ccτ (rt ) xc (t − τ )

(68)

where xc (t) ∈ n is the controller state, and Ac (rt ), Acτ (rt ), Bc (rt ), Cc (rt ), and Ccτ (rt ) are the mode-dependent gains to be determined. Define an augmented state T  . Then the closed-loop system is obtained as follows: vector x˜ = x (t)T xc (t)T 

E x˙˜ (t) z (t)

= =

Ai x˜ (t) + Aτ i x˜ (t − τ ) + Di ω (t) , Ci x˜ (t)

(69)

where

0 Ai E= , Ai = Bci Cyi E

 Di , Ci = Ci 0 . Di = 0 E 0

Bi Cci Aci



, Aτ i =

Aτ i 0

Bi Ccτ i Acτ i

,

Our objective is to design the dynamic output-feedback controller (68) such that the closed-loop system (69) is extended dissipative. Before presenting the main results, we first give the following lemma. Lemma 11. The inequality ET Ω = Ω T E ≥ 0 has a nonsingular matrix solution Ω, if and only if there exist matrices P > 0 and L such that Ω = PE + SL is nonsingular, where S is any full column rank matrix satisfying ET S = 0. Proof. The sufficiency is obvious since ET (PE + SL) = ET PE ≥ 0. In the following, we prove the necessity. Suppose rank(E) = σ ≤ n. Then, there exist nonsingular matrices M and N satisfying Eq. (4). Set M

−T

ΩN =

Ω1 Ω2

Ω3 Ω4

,

where Ω1 ∈ σ ×σ . Then, it is obtained that N T ET ΩN =

Iσ 0

0 0



Ω1 Ω2

Ω3 Ω4



=

Ω1 0

Ω3 0

.

(70)

It follows from ET Ω = Ω T E ≥ 0 that N T ET ΩN = N T Ω T EN ≥ 0. This, together with Eq. (70), implies that Ω1 ≥ 0 and Ω3 = 0. Thus, M

−T

ΩN =

Ω1 Ω2

0 Ω4

.

5 Output-feedback control

Note that Ω is nonsingular. Thus, Ω1 and Ω4 are nonsingular. This implies that Ω1 > 0. For any full column rank matrix S satisfying ET S = 0, set

S1 −T M S= . S2

By noting Eq. (46), we have that S1 = 0 and S2 is nonsingular. Now, define

 Ω1 (Ω2 − S2 L1 )T P = MT M, L = L1 S2−1 Ω4 N −1 , Ω2 − S2 L1

P3

where L1 ∈ (n−σ )×σ is any real matrix and P3 is any matrix satisfying P3 > (Ω2 − S2 L1 )Ω1−1 (Ω2 − S2 L1 )T . It can be verified that P > 0 and M −T ΩN = M −T (PE + SL)N. This gives that Ω = PE + SL. Theorem 6. Consider the system in Eq. (67). Give any full column rank matrices S and S¯ satisfying ET S = 0 and ES¯ = 0. Then, under Assumptions 1 and 2 with decomposition (28), there exists a memory output-feedback controller in the form of Eq. (68) such that closed-loop system (69) is extended dissipative, if there exist ˜ i , R, ˜ > 0, Q ˜ Li , F, A˜ ci , A˜ cτ i , B˜ ci , C˜ ci , and C˜ cτ i , matrices Xi > 0, Y > 0, Z˜ i > 0, W such that the following LMIs hold: s  j=1

Xi 

⎡ ⎢ ⎢ ⎣ ⎡ ⎢ ⎢ ⎣ ⎡ ˜ Λ1i ⎢ ⎢  ⎢ ⎢ ⎢  ⎢ ⎢ ⎣  

s 

˜ j − R˜ < 0, πij Q

˜ < 0, πij Z˜ j − W

j=1



(NM)T Y

Xi NM

(71)

> 0,

(72)

Xi + 2Z˜ i

2Z˜ i E˜

Y T CiT Ψ¯ 4T



˜ i + 2E˜ T Z˜ i E˜ 2τ Q

0





I

˜ Xi + τ W

˜ E˜ τW

Y T CiT Ψ¯ 4T



˜ E˜ τ 2 R˜ + τ E˜ T W

0





I

Λ2i + τ −1 E˜ T Z˜ i E˜

Λ3i − E˜ T Y T CiT Ψ2

ΛT1i

˜ i − τ −1 E˜ T Z˜ i E˜ −Q

0

ΛT2i



−Ψ3

ΛT3i





Λ˜ 4i







⎤ ⎥ ⎥ > 0, ⎦

(73)

⎤ ⎥ ⎥ > 0, ⎦

⎤ E˜ T Y T CiT Ψ¯ 1T ⎥ ⎥ 0 ⎥ ⎥ ⎥ < 0, 0 ⎥ ⎥ 0 ⎦ −I

(74)

(75)

247

248

CHAPTER 11 Extended dissipative control and filtering

   where E˜ = diag E, ET , Y = NM Λ˜ 1i =

s 

Y , and

˜ i + τ R˜ − τ −1 E˜ T Z˜ i E˜ πij E˜ T Xj E˜ + ΛT1i + Λ1i + Q

j=1

˜ − Λ4i − ΛT Λ˜ 4i = τ Z˜ i + 0.5τ 2 W 4i

A˜ ci (Xi E + SLi )T Ai + B˜ ci Cyi Λ1i = ¯ + Bi C˜ ci Ai Ai (YET + SF)

A˜ cτ i (Xi E + SLi )T Aτ i Λ2i = ¯ + Bi C˜ cτ i Aτ i Aτ i (YET + SF)



I Xi E + SLi (Xi E + SLi )T Di . Λ3i = , Λ4i = T ¯ Di I YE + SF

When the earlier conditions are feasible, it is obtained from Eq. (75) that Λ4i + ¯ and I − (YET + ΛT4i > 0, which implies that the matrices Xi E + SLi , YET + SF ¯SF)(Xi E + SLi ) are nonsingular. Choose matrices H1 , H2 and nonsingular matrix ¯ 1 and V T ET + SH ¯ 2 are nonsingular. Define the nonsingular V such that VET + SH matrix Ui as ! ¯ 2 )−1 I − (YET + SF)(X ¯ Ui = (V T ET + SH i E + SLi ) .

(76)

Then, the desired controller gains are obtained as follows:   ¯ Aci = Ui−T A˜ ci − (Xi E + SLi )T Ai + B˜ ci Cyi (YET + SF) ! ¯ 1 )−1 , −(Xi E + SLi )T Bi C˜ ci (VET + SH

! ¯ − (Xi E + SLi )T Bi C˜ cτ i (VET + SH ¯ 1 )−1 , Acτ i = Ui−T A˜ cτ i − (Xi E + SLi )T Aτ i (YET + SF) Bci = Ui−T B˜ ci ,

¯ 1 )−1 , Ccτ i = C˜ cτ i (VET + SH ¯ 1 )−1 . Cci = C˜ ci (VET + SH

Proof. Suppose the conditions in the theorem are satisfied. Define the following nonsingular matrices:



¯ I EY + F T S¯ T Xi E + SLi I I YET + SF Ω1i = = , Ω2 = , Ω . 3 T ¯T T ¯ Ui

0

0

VE + SH1

0

EV + H2 S

For each i ∈ {1, 2, . . . , s}, define ˜ i Ω −1 , Qi = Ω2−T Q 2

Zi = Ω3−T Z˜ i Ω3−1 ,

˜ −1 , R = Ω2−T RΩ 2

˜ −1 . (77) W = Ω3−T WΩ 3

Then, it follows from Eq. (71) that s  j=1

πij Qj − R < 0,

s  j=1

πij Zj − W < 0.

(78)

5 Output-feedback control

Next, by using Eq. (76), it is not difficult to find that Ω3T Ω1i = Λ4i and EΩ2 = ˜ Then, it can be verified that Ω3 E. Ω2T E T (Ω1i Ω2−1 )Ω2 = Ω2T E T Ω1i = E˜ T Ω3T Ω1i =



E T Xi E E

ET EYET

.

(79)

Note from Eq. (4) that (MEN)2 = MEN, which implies that E = ENME. By observing this fact, it follows from Eqs. (72), (79) that Ω2T E T (Ω1i Ω2−1 )Ω2 = Ω2T E T Ω1i = E˜ T Xi E˜ ≥ 0.

(80)

This further implies that E T (Ω1i Ω2−1 ) = (Ω1i Ω2−1 )T E ≥ 0. Therefore, by applying Lemma 11, there exist matrices Pi > 0 and Li such that Pi E + SLi = Ω1i Ω2−1 ,

(81)

where S is any full column rank matrix satisfying E T S = 0. By using the conditions in Eqs. (80), (81), it can be verified that ˜ Ω2T E T Pi EΩ2 = Ω2T E T (Pi E + SLi ) Ω2 = Ω2T E T Ω1i = E˜ T Xi E.

(82)

Considering Eq. (77), we also have that ˜ Ω2T E T Zi EΩ2 = E˜ T Z˜ i E,

˜ ˜ E. Ω2T E T WEΩ2 = E˜ T W

(83)

On the other hand, recalling the fact that E = ENME, which implies E(I−NME) = 0. Thus, according to Assumption 1, we obtain that Ci (I − NME) = 0; that is, Ci = Ci NME. Therefore C i Ω2 =



Ci YET

Ci



=



Ci NME

Ci YET



˜ = Ci Y E.

(84)

Now, applying the Schur complement equivalence to Eq. (73) yields that

Xi + 2Z˜ i − Y T CiT Ψ4 Ci Y 2Z˜ i E˜ > 0. ˜T ˜ ˜ ˜ 

2τ Qi + 2E Zi E

$ # $ # ˜ I to the left-hand side of Premultiplying diag E˜ T , I and postmultiplying diag E, the earlier inequality gives that

T T  Ω2 E Pi E + 2E T Zi E − CiT Ψ4 Ci Ω2 2Ω2T E T Zi EΩ2 ≥ 0. T T ˜ 

2τ Qi + 2Ω2 E Zi EΩ2

This is equivalent to

E T Pi E + 2E T Zi E − CiT Ψ4 Ci 

2E T Zi E 2τ Qi + 2E T Zi E

≥ 0.

(85)

249

250

CHAPTER 11 Extended dissipative control and filtering

Similarly, it follows from Eq. (74) that

E T Pi E + τ E T WE − CiT Ψ4 Ci 

τ E T WE τ 2 R + τ E T WE

≥ 0.

(86)

By some simple manipulations, it can be verified that ⎧ ⎪ ⎨ Λ1i Λ2i ⎪ ⎩ Λ3i

=

T A Ω = Ω T (P E + SL )T A Ω Ω1i i 2 i i i 2 2

=

T A Ω = Ω T (P E + SL )T A Ω . Ω1i τi 2 i i τi 2 2

=

T D = Ω T (P E + SL )T D Ω1i i i i i 2

(87)

By Lemma 10, we have that  −1 ˜ Λ4i Λ˜ 4i ≥ −ΛT4i τ Z˜ i + 0.5τ 2 W  −1 T Ω τ Z˜ + 0.5τ 2 W ˜ = −Ω1i Ω3T Ω1i i 3   T Ω τ Ω T Z Ω + 0.5τ 2 Ω T WΩ −1 Ω T Ω = −Ω1i 3 3 3 i 3 3 3 1i  −1 T τ Z + 0.5τ 2 W = −Ω1i Ω1i . i

(88)

Then, it follows from Eqs. (75), (77), (82)–(84), (88) that

# $ diag Ω2T , Ω2T , I Γ˜i diag {Ω2 , Ω2 , I} < 0,

(89)

where ⎡ T ⎤T ⎤ ⎡ AT ⎤ Ai Γ˜1i + CiT Ψ1 Ci Γ˜2i Γ˜4i i   ⎢ T ⎥ ⎢ T ⎥ 2 ⎣ ⎦ ˜ = + ⎣ Aτ i ⎦ τ Zi + 0.5τ W ⎣ Aτ i ⎦ ,  Γ3i 0   −Ψ3 DiT DiT ⎛ ⎞ s  = ET ⎝ πij Pj ⎠ E + ATi (Pi E + SLi ) + (Pi E + SLi )T Ai + Qi + τ R − τ −1 E T Zi E, ⎡

Γ˜i

Γ˜1i

j=1

Γ˜2i = (Pi E + SLi )T Aτ i + τ −1 E T Zi E, Γ˜3i = −Qi − τ −1 E T Zi E, Γ˜4i = (Pi E + SLi )T Di − CiT Ψ2 .

It is obvious that Eq. (89) is equivalent to Γ˜i < 0. Considering this, recalling the conditions in Eqs. (78), (85), (86), and according to Theorem 3, we draw the conclusion that the closed-loop system (69) is extended dissipative. The proof is completed here.

6 Extended dissipative filtering This section investigates the design of extended dissipative filters for singular timedelay systems with Markovian jumping parameters. In this case, we shall consider system (67) without control input; that is,

6 Extended dissipative filtering

⎧ ⎨ E˙x (t) z (t) ⎩ y (t)

= = =

A (rt ) x (t) + Aτ (rt ) x (t − τ ) + D (rt ) ω (t) C (rt ) x (t) , Cy (rt ) x (t)

(90)

where y (t) ∈ ny is the measured output. For system (90), we consider the following delayed filter: 

E˙xf (t) zf (t)

= =

Af (rt ) xf (t) + Af τ (rt ) xf (t − τ ) + Bf (rt ) y (t) , C (rt ) xf (t)

(91)

where xf (t) ∈ n is the filter state, and Af (rt ), Af τ (rt ), and Bf (rt ) are the mode-dependent gains to be determined. Define an augmented state vector xˆ =  T and the output error e(t) = z(t) − zf (t). Then the filtering error x (t)T xf (t)T system is obtained as follows: 

E x˙ˆ (t) e (t)

= =

Afi xˆ (t) + Af τ i xˆ (t − τ ) + Dfi ω (t) , Cfi xˆ (t)

(92)

where E= Cfi =



E 0 Ci

0 E





, Afi = −Ci .

Ai Bfi Cyi

0 Afi



, Af τ i =

Aτ i 0



0

, Dfi =

Af τ i

Di 0

,

Our objective is to design filter (91) such that the filtering error system (92) is extended dissipative. Remark 5. In the output equation of filter (91), the coefficient matrix C(rt ) is chosen as same as the one in the output equation of system (90). This is because we want the matrix Cfi in the filtering error system (92) to satisfy Assumption 1; that is, ker(E) ⊆ ker(Cfi ). This assumption is necessary for us to develop strict LMI conditions. Theorem 7. Consider system in Eq. (90). Give any full column rank matrices S and S¯ satisfying ET S = 0 and ES¯ = 0. Then, under Assumptions 1 and 2 with decomposition (28), there exists a delayed filter in the form of Eq. (91) such that the filtering error system (92) is extended dissipative, if there exist matrices Xi > 0, ˜ i , R, ˜ > 0, Q ˜ Li , F, V, A˜ fi , A˜ f τ i , and B˜ fi , such that the conditions in Y > 0, Z˜ i > 0, W Eqs. (71)–(74) as well as the following LMI hold: ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

Θ3i − E˜ T Y T CiT Ψ2

T Θ1i

E˜ T Y T CiT Ψ¯ 1T



Θ2i + τ −1 E˜ T Z˜ i E˜ ˜ i − τ −1 E˜ T Z˜ i E˜ −Q

0

T Θ2i

0





−Ψ3

0







T Θ3i Θ˜ 4i









−I

Θ˜ 1i

0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ < 0, ⎥ ⎥ ⎦

(93)

251

252

CHAPTER 11 Extended dissipative control and filtering

   where E˜ = diag E, ET , Y = NM Θ˜ 1i =

s 

Y − V , and

T +Θ +Q ˜ ˜ i + τ R˜ − τ −1 E˜ T Z˜ i E, πij E˜ T Xj E˜ + Θ1i 1i

j=1

˜ − Θ4i − Θ T , Θ˜ 4i = τ Z˜ i + 0.5τ 2 W 4i

A˜ fi (Xi E + SLi )T Ai + B˜ fi Cyi , Θ1i = ¯ Ai Ai (YET + SF)

A˜ f τ i (Xi E + SLi )T Aτ i Θ2i = , ¯ Aτ i Aτ i (YET + SF)



I Xi E + SLi (Xi E + SLi )T Di . Θ3i = , Θ4i = ¯ Di I YET + SF

¯ 1 and Similar to Theorem 6, we choose matrices H1 and H2 such that VET + SH T T ¯ V E + SH2 are nonsingular, and define Ui as in Eq. (76). Then, the desired filter gains are obtained as follows:   ! ¯ ¯ 1 )−1 , Afi = Ui−T A˜ fi − (Xi E + SLi )T Ai + B˜ fi Cyi (YET + SF) (VET + SH ! ¯ ¯ 1 )−1 , Af τ i = Ui−T A˜ f τ i − (Xi E + SLi )T Aτ i (YET + SF) (VET + SH Bfi = Ui−T B˜ fi .

Proof. The proof can be easily established by following the same method as that in the proof of Theorem 6, and thus it is omitted here.

7 Conclusions In this chapter, the analysis and synthesis problems of continuous-time linear singular systems with constant delays and Markovian jumping parameters were investigated. Strict LMI approaches were developed and relaxed delay-dependent conditions were obtained for the problems of stochastic admissibility analysis, extended dissipativity analysis, state-feedback control, output-feedback control, and delayed filter design. It is worth mentioning that some of the presented lemmas provide elegant properties related to the singular matrix E. These lemmas are efficient for deriving strict LMI conditions. It should be also pointed out that the methods proposed in chapter are very basic, and thus they are applicable for a variety of problems related to singular time-delay systems with Markovian jumping parameters. For example, the proposed methods could be generalized directly for analysis and design of singular systems with time-varying delays, distributed delays, semi-Markovian parameters, and hidden Markovian parameters, respectively. It is also meaningful to apply the proposed methods to study the event-triggered control and filtering problems for networked singular systems with Markovian jumping parameters.

References

Acknowledgments The authors would like to sincerely thank Prof. Dabo Xu for approving us to use his idea in the proof of Lemma 7. This work was supported in part by the Natural Science Fund for Distinguished Young Scholars of Jiangsu Province under Grant No. BK20150034, the National Natural Science Foundation of China under Grant Nos. 61473151 and 61773191, and the Natural Science Foundation of Shandong Province for Outstanding Young Talents in Provincial Universities under Grant No. ZR2016JL025.

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Further reading

[35] D. Xu, Q.L. Zhang, Y.B. Hu, Reduced-order H∞ controller design for uncertain descriptor systems, Acta Autom. Sin. 33 (1) (2007) 44–47. [36] D. Xu, Robust H∞ Control and Model Reduction for Singular Systems (Master thesis), 2006.

Further reading [37] E.-K. Boukas, Control of Singular Systems With Random Abrupt Changes, Springer-Verlag, Berlin, Heidelberg, 2008.

255

CHAPTER

12

Event-triggered reliable control for T-S fuzzy Markov jump systems based on a delay system approach

Lei Sua , Dan Yea,b a College b State

of Information Science and Engineering, Northeastern University, Shenyang, China Key Laboratory of Synthetical Automation of Process Industries, Northeastern University, Shenyang, China

Chapter outline 1 Introduction....................................................................................... 257 2 Problem formulation and preliminaries ...................................................... 258 2.1 The introduction for fuzzy Markov jump model ................................. 259 2.2 Event-triggered mechanism and reliable control problem ..................... 260 3 Stability analysis and event-triggered controller design for FMJSs ..................... 263 3.1 Stability analysis for FMJSs ........................................................ 264 3.2 Event-triggered reliable controller design for FMJSs ........................... 268 4 Numerical example.............................................................................. 271 5 Conclusions....................................................................................... 275 Acknowledgments .................................................................................. 276 References........................................................................................... 276

1 Introduction The past several decades have witnessed an increasing research interest in the fuzzy Markov jump systems (FMJSs), which not only provides a forceful method to solve the stability analysis and control design problem for complex nonlinear systems, but can also tolerate some random abrupt changes encountered in system dynamics [1]. To obtain less conservative results, a mode-dependent Lyapunov-Krasovskii functional (LKF) is more suitable to be chosen than mode-independent LKF in FMJSs. However, it should be pointed out that some coupled terms produced by calculating the derivative of mode-dependent LKF are difficult to eliminate. For instance, in Ref. [2], some coupled terms produced by calculating the derivative of mode-dependent LKF have been removed by adding some restrictive conditions, Stability, Control and Application of Time-Delay Systems. https://doi.org/10.1016/B978-0-12-814928-7.00012-3 © 2019 Elsevier Inc. All rights reserved.

257

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CHAPTER 12 Event-triggered reliable control for Markov jump systems

which may bring some conservatism. Noting that these coupled terms in the restrictive conditions are required to be negative, it can reduce the conservatism if these coupled terms are reserved. This is the main problem to be solved in this chapter. On the other hand, the bandwidth is limited to a networked control system. Therefore, how to save the limited communication resources, becomes a hot topic in the last several years. Fortunately, the event-triggered mechanism emerges at the right moment, which can filter some redundant or unnecessary data by setting a predefined threshold condition [3–6]. In this case, the scarce communication resources in transmitting the data to the controller are greatly saved. Nevertheless, the control effect in the event-triggered mechanism may be weakened when the actuator faults occur, which is ignored in Refs. [7–9]. To solve this shortcoming, reliable control method, which can guarantee high reliability and robustness for industrial system against undesirable faults, has been widely investigated [10–13]. Although the importance of event-triggered mechanism and reliable control method has been realized, few efforts have been paid for the event-triggered reliable control of FMJSs, which motivates the current work. In this chapter, the H∞ event-triggered reliable control issue for T-S FMJSs is investigated. The main contributions are summarized as follows: (1) An event-triggered reliable controller is designed to reduce the utilization of scarce communication resources and tolerate some unexpected actuator failures simultaneously. (2) Due to the introduction of event-triggered mechanism, the asynchronous premise variables are considered to design fuzzy state feedback controller, which are more practical than those synchronous ones. (3) Some novelty integral inequalities are employed to deal with the coupled terms produced by calculating the derivative of mode-dependent LKF, which result in less conservative design conditions. Notation. Throughout this note: Rn and Rm×n stand for the n-dimensional Euclidean space and the set of all m×n real matrices, respectively. U > 0 implies that matrix U is real symmetric and positive definite. The notation QT denotes the transpose of the matrix Q. |·| represents the Euclidean norm of a vector and its induced norm of a matrix. L is the weak infinitesimal operator. In symmetric block matrices or complex matrix expressions, we employ an asterisk (∗) to show a term that is induced by symmetry. E {·} denotes the expectation operator. sym{X} is defined as X + X T .

2 Problem formulation and preliminaries In this chapter, we consider the control problem for FMJSs, where the event-triggered mechanism and the reliability are also considered. Under this case, the considered systems not only can reduce the utilization of limited bandwidth, but can also tolerate some unexpected actuator failures. In what follows, we introduce the fuzzy Markov jump model, the event-triggered mechanism and the reliable controller design in this chapter.

2 Problem formulation and preliminaries

2.1 The introduction for fuzzy Markov jump model Define a probability space (Ω, F, P), where Ω is a sample space, F is the σ -algebra of subsets of the sample space, and P is the probability measure on F. Then let us consider the following FMJS which is described by Plant rule i: IF κ1 (t) is μi1 and κ2 (t) is μi2 and . . . and κs (t) is μis . Then x˙ (t) = Ai (ρ (t)) x (t) + Bi (ρ (t)) u (t) + Ci (ρ (t)) ω (t) ,

(1)

z (t) = Di (ρ (t)) x (t) ,

(2)

where i = 1, . . . , r, r is the number of fuzzy rules, μij , j = 1, . . . , s stand for the fuzzy sets, and κj (t) denote the premise variables; x (t) ∈ n , u (t) ∈ u , and z (t) ∈ m are the system state, control input, and output, respectively; ω (t) ∈ q is the disturbance which belongs to L2 [0, ∞) ; matrices Ai (ρ (t)), Bi (ρ (t)), Ci (ρ (t)), and Di (ρ (t)) are known constant matrices with appropriate dimensions. ρ (t) stands for a homogeneous finite-state Markov jump process with right continuous trajectories and takes discrete values  in a given finite set S = {1, 2, . . . , p} with transition probability matrix (TPM) = {πmn } given by  Pr {ρ (t + ζ ) = n |ρ (t) = m } =

πmn ζ + o (ζ ) , m  = n 1 + πmn ζ + o (ζ ) , m = n,

(3)

where ζ > 0 is the sojourn time, limζ →0 (o (ζ ) /ζ ) = 0 and πmn ≥ 0, for m = n is the transition rate from mode m at time t to mode n at time t + ζ and  πmm = − πmn . n∈S , m=n

Furthermore, the fuzzy basis functions are given by vi (κ (t)) , hi (κ (t)) = r i=1 vi (κ (t))

vi (κ (t)) =

s 

  μij κj (t) ,

(4)

j=1

where  κ (t)  = [κ1 (t) , κ2 (t) , . . . , κs (t)] presents the premise variables vector and μij κj (t) denotes  the grade of membership of κj (t) in μij . Suppose vi (κ (t)) ≥ 0, i = 1, 2, . . .  , r, ri=1 vi (κ (t)) > 0 for all t. Therefore, hi (κ (t)) ≥ 0 for i = 1, 2, . . . , r, and ri=1 hi (κ (t)) = 1 for all t. For simplification, each ρ (t) = m ∈ S, the matrix Ai (ρ (t)) will be represented by Aim and the other symbols are similarly defined. By fuzzy blending, it can be inferred that x˙ (t) = z (t) =

r  i=1 r  i=1

hi (κ (t)) {Aim x (t) + Bim u (t) + Cim ω (t)} ,

(5)

hi (κ (t)) {Dim x (t)} .

(6)

259

260

CHAPTER 12 Event-triggered reliable control for Markov jump systems

2.2 Event-triggered mechanism and reliable control problem 2.2.1 The introduction of event-triggered mechanism To promote the later statement, the following assumptions are necessary for the event-triggered control problem. Assumption 1. The actuator, controller, and the zero-order holder (ZOH) are event-triggered except that the sensor is time-triggered with a constant sampling period T. Assumption 2. There is no data packet dropout phenomenon occurs in the transmission process. Assumption from controller to the ZOH (holding 3. When the data transfers

interval is t ∈ tk T + τtk , tk+1 T + τtk+1 ), there exists a total network-induced delays

τtk ∈ 0, h¯ . Now, the following event-triggered mechanism is introduced with a given predefined threshold which can decide whether the newly sampled data should be delivered to the controller or not: [x (kT) − x (tk T)]T Ψm [x (kT) − x (tk T)] ≤ σm x (kT) Ψm x (kT) ,

(7)

where Ψm > 0 is the weighting matrix to be designed with an appropriate threshold parameter σm ∈ [0, 1), x (kT) is the present sampled state, and x (tk T) is the latest received data. If the sampling data violates the given condition (7), the data will be preserved and transferred to the controller. Utilizing the same technology as in Ref. [14], the event-triggered FMJSs (5), (6) can be turned into a new time-delay system. First, suppose there exists a finite positive integer p such that tk+1 = tk + p + 1. Therefore, the holding interval time of the ZOH constitutes the subaggregate: p   tk T + τtk , tk+1 T + τtk+1 = Φn,k ,



n=0

 where Φn,k = tk T + nT + τik +n , tk T + (n + 1) T + τik +n+1 , n = 0, 1, . . . , p. The delay in a network is supposed as

d (t) = t − tk T − nT, t ∈ Φn,k , where 0 ≤ d (t) ≤ T + h¯ = h. Then, the error between the present sampling instant and the latest transmission instant can be written as ek (sk T) = x (sk T) − x (tk T) ,

(8)

where sk T = tk T + nT stands for the sampling instant from the present transmitted sampling instant tk T to the coming transmitted sampling instant tk+1 T. Then, the event-triggered mechanism can be described as

2 Problem formulation and preliminaries

FIG. 1 The description of event-triggered mechanism in fuzzy Markov jump systems.

eTk (sk T) Ψm ek (sk T) ≤ σm xT (sk T) Ψm x (sk T) .

(9)

In order to give an explicit description about the event-triggered mechanism, a diagram is given in Fig. 1.

2.2.2 The model of reliable control First, let the desired controller take the form as Control rule i: IF κ1 (tk T) is μi1 and κ2 (tk T) is μi2 and . . . and κs (tk T) is μis . Then  u (t) = Ki x (tk T) , t ∈ tk T + τtk , tk+1 T + τtk+1 , where Ki is the controller gain to be designed. When the actuator encounters failures, we adopt the reliable control model in the following: u f (t) = Fm u (t) ,

(10)

where Fm = diag {f1m , f2m , . . . , fπm } stands for the actuator faults function matrix, and 0 ≤ fεlm ≤ fεm ≤ fεum ≤ 1, ε = 1, 2, . . . , π . fεlm and fεum are known constants which are characterized by the acceptable failures of the εth actuator. Next, some matrices should be introduced to depict the fault model F0m = diag (f01m , f02m , . . . , f0πm ) , Hm = diag (h1m , h2m , . . . , hπm ) , Lm = diag (l1m , l2m , . . . , lπm ) ,

(11)

261

262

CHAPTER 12 Event-triggered reliable control for Markov jump systems

where f0εm = (fεlm + fεum ) /2, hεm = (fεum − fεlm ) / (fεum + fεlm ), lεm (fεm − f0εm ) /f0εm , ε = 1, 2, . . . , π, thus one can obtain that Fm = F0m (I + Lm ) ,

|Lm | ≤ Hm ≤ I.

= (12)

Then, the desired controller can be rewritten as u (t) =

r 

hj (κ (tk T)) F0m (I + Lm ) Kj x (tk T) .

(13)

j=1

Thanks to the event-triggered mechanism, there exist delays in the considered systems, therefore, the fuzzy rule of system and controller cannot be synchronized. To overcome this difficulty, the asynchronous premise variables for fuzzy state feedback controller are considered in the following. The membership functions follow some restrictions which are given below 

hj (κ (tk T)) = νj hj (κ (t)) , hj (κ (tk T)) − hj (κ (t)) ≤ ∇j ,

(14)

where νj > 0, ∇j > 0 (j = 1, 2, . . . , r) . Adopting the same technique in Ref. [15], it yields that j

1 = 1 −

∇j ∇j j ≤ νj ≤ 1 + = 2 , hj (κ (t)) hj (κ (t))

hj (κ (t)) > 0,

which implies that



 χ2i min (νi) νi νi max (νi ) νi     = ≤ max . ≤ ≤ ≤ min = j j νj νj νj max νj min νj 2 χ1     j j 2 Defining 2 = max 2 , χ1 = min 1 , and δ =  1 , then we have j

1

j=1,2,...,r

j=1,2,...,r

1 νi ≤ δ. ≤ δ νj Then, substituting Eqs. (8), (13), (14) into Eqs. (5), (6) results in the following closed-loop system: x˙ (t) =

r  r 

 νj hi (κ (t)) hj (κ (t)) Aim x (t) + Bim F0m (I + Lm )Kj x (t − d (t))

i=1 j=1

 −Bim F0m (I + Lm )Kj ek (sk T) + Cim ω (t) , z (t) =

r  r 

νj hi (κ (t)) hj (κ (t)) {Dim x (t)} ,

i=1 j=1

x (t) = φ (t) ,

t ∈ [−h, 0) .

(15) (16)

3 Stability analysis and event-triggered controller design for FMJSs

Remark 1. On the one hand, the reliable control issue for FMJSs has been extensively researched such as in Ref. [16]. However, in this chapter, we consider how to reduce the occupation of scarce communication resources by adopting an event-triggered scheme. On the other hand, the asynchronous premise variables for controller is considered in this chapter, which is more reasonable in practice.

3 Stability analysis and event-triggered controller design for FMJSs Before proceeding further, some necessary definitions and lemmas are given. Definition 1 (Boukas [17]). FMJSs (15), (16) with ω (t) = 0 are stochastically stable if there exists a constant scalar C (x0 , η0 ) > 0 such that the following inequality holds for any initial conditions (x0 , η0 ): E

 ∞ 0

 xT (t) x (t) dt |x0 , η0

≤ C (x0 , η0 ) .

(17)

Definition 2 (Chen et al. [18]). FMJSs (15), (16) are stochastically stable and satisfy an H∞ performance γ , if the following two demands are satisfied. Meanwhile: (1) FMJSs (15), (16) are stochastically stable in view of Definition 1 and (2) under zero initial condition, there exists a scalar γ > 0 such that the following condition is satisfied: E

 η 0

   η   ωT (t) ω (t) dt , zT (t) z (t) dt ≥ E γ 2

(18)

0

for any η ≥ 0 and any nonzero ω (t) ∈ L2 [0, ∞). Lemma 1 (Seuret and Gouaisbaut [19]). For a given symmetric positive-definite matrix V, and any differentiable function Γ in [a, b] → n , then the following inequality holds:  b a

Γ˙ T (s) V Γ˙ (s) ds ≥

 1  (Γ (b) − Γ (a))T V (Γ (b) − Γ (a)) b−a ⎡ T  b 2 3 ⎣ Γ (b) + Γ (a) − Γ (s) ds + b−a b−a a    b 2 × V Γ (b) + Γ (a) − Γ (s) ds . b−a a

(19)

263

264

CHAPTER 12 Event-triggered reliable control for Markov jump systems

Lemma 2 (Shen et al. [20]). For any scalars 0 ≤ d (t) ≤ h and matrices Um ∈  X1 X2 and X = ∈ 2n×2n satisfying X3 X4

n×n

 Λm =

diag {Um , 3Um } ∗

X diag {Um , 3Um }

 ≥ 0,

(20)

if there exists a vector function x: [t − h, t] → n such that the integrations in the following are well defined, then −h −

 t

 0  t

t−h

−h t+θ

T Λ Φ ξ (t) , x˙ T (s) Um x˙ (s) ds ≤ −ξ T (t) Φ12 m 12

x˙ T (s) Um x˙ (s) dsdθ ≤ −2 [x (t) − ξ1 (t)]T Um [x (t) − ξ1 (t)] −

h − d (t) [x (t) − ξ2 (t)]T Um [x (t) − ξ2 (t)] , h

where

xT (t) xT (t − d (t)) xT (t − h) ξ1T (t) ξ2T (t) ,  t  t−d(t) 1 1 ξ1 (t) = x (s) ds, ξ2 (t) = x (s) ds, d (t) t−d(t) h − d (t) t−h     I −I 0 0 0 0 I −I 0 0 Φ1 = , Φ2 = , I I 0 −2I 0 0 I I 0 −2I

ξ T (t) =

 Φ12 =

Φ1 Φ2



3.1 Stability analysis for FMJSs In this section, a sufficient condition, which ensures FMJSs (15), (16) are stochastically stable and satisfy an H∞ performance γ , is given first. Based on the condition, the controller gains Ki can be obtained under a set of unknown Fm by solving a convex optimization problem. For simplification, we first denote ei (i = 1, 2, . . . , 8) ∈ 8n×n are elementary matrices, for example, eT3 = 0 0 I 0 0 0 0 0 , then Θ T (t) = 1m =



ξ T (t)



x˙ T (t)

ωT (t)

πmn (Qn + Rn ) − S,

eT (sk T) 3m = h

n∈ S

Π=



e1 + e2



,



2m =



πmn Rn − S,

n∈S

πmn Un − W,

n∈S

−e1 + e2 + e3 + e4

−e3 + e4

−2e2

−2e4





.

 Pm P12 11 Theorem 1. For given scalars h, γ , and σm , if there exist matrices > ∗ P22 S > 0, W > 0, φ1 , φ2 , Kj , such that the 0, Qm > 0, Um > 0, Ψm > 0, Rm > 0,  following matrix inequalities hold for η ∈ δ, 1δ , m ∈ S,

.

3 Stability analysis and event-triggered controller design for FMJSs

Υ1iim + Υ2m (1) < 0,

i = 1, 2, . . . , r,

Υ1iim + Υ2m (2) < 0,

i = 1, 2, . . . , r,

(22)

Υ1ijm + ηΥ1jim + Υ2m (1) < 0,

1 ≤ i < j ≤ r,

(23)

Υ1ijm + ηΥ1jim + Υ2m (2) < 0,

1 ≤ i < j ≤ r,

(24)

 Λm =

where

(21)

k = 1, 2, 3,

km < 0, diag {Um , 3Um } ∗

⎛ Υ1ijm = e1 ⎝Qm + Rm + hS +

 n∈S

⎞

(25)



X diag {Um , 3Um }

> 0,

(26)

  πmn Pn11 + sym P12 + φ1T Aim + 3m

+DTim Dim ⎠ eT1 + sym(e1 φ1T Bim F0m (I + Lm )Kj eT2 − e1 P12 eT3   T T T T T − e1 3m eT4 + e1 Pm 11 − φ1 + Aim φ2 e6 + e1 φ1 Cim e7  −e1 φ1T Bim F0m (I + Lm ) Kj eT8 + e2 σm Ψm eT2 − e3 Rm eT3     h2 W T 2 − sym (φ2 ) eT6 + sym e4 3m e4 + e6 h Um + 2   + sym e6 φ2T Cim eT7 − e6 φ2T Bim F0m (I + Lm ) Kj eT8   − e7 γ 2 I eT7 − e8 (Ψm I) eT8 − Π T Λm Π ,   Υ2m (1) = sym e4 hPT12 eT6 + e4 hP22 eT1 − e4 hP22 eT3 + e4 h1m eT4 ,   Υ2m (2) = sym e5 hPT12 eT6 + e5 hP22 eT1 − e5 hP22 eT3 + e5 h2m eT5 + (e1 − e5 ) h3m (e1 − e5 )T ,

then, FMJSs (15), (16) are stochastically stable and satisfy an H∞ performance γ . Proof. The Lyapunov functional candidate is chosen as V (t) =

3 

Vl (t) ,

(27)

l=1

where  V1 (t) = V2 (t) =

$ t x (t) t−h x (α) dα

 t

V3 (t) = h

T 

Pm 11 ∗

xT (α) Qm x (α) dα +

t−d(t)  0  t

−h t+β

P12 P22  t t−h



$ t x (t) t−h x (α) dα



xT (α) Rm x (α) dα +

x˙ T (α) Um x˙ (α) dαdβ +

 0  0  t −h sita t+β

,  0  t −h t+β

xT (α) Sx (α) dαdβ,

x˙ T (α) W x˙ (α) dαdβdθ,

265

266

CHAPTER 12 Event-triggered reliable control for Markov jump systems

Then, we use the weak infinitesimal operator L which is defined in Ref. [21], and one has LV1 (t) = Θ T (t)

⎧ ⎨ ⎩

e1

 n∈S

  T T T πmn Pn11 eT1 + sym e1 Pm 11 e6 − e1 P12 e3 + e1 P12 e1

  d (t)  sym e4 hPT12 eT6 + e4 hP22 eT1 − e4 hP22 eT3 + h   h − d (t)  T T T T sym e5 hP22 e1 + e5 hP12 e6 − e5 hP22 e3 Θ (t) , + h   LV2 (t) = Θ T (t) e1 (Qm + Rm + hS) eT1 − e3 Rm eT3 Θ (t)  t  t−d(t) + xT (α) 1m x (α) dα + xT (α) 2m x (α) dα,

(29)

 

 W T e6 Θ (t) LV3 (t) = Θ T (t) h2 e6 Um + 2  t  0  t −h x˙ T (α) Um x˙ (α) dα + x˙ T (α) 3m x˙ (α) dαdβ.

(30)

t−d(t)

t−h

t−h

−h t+β

(28)

Noting that 1m < 0 and 2m < 0, we have  d (t) T e4 h1m e4 Θ (t) , h t−d(t)    t−d(t) h − d (t) T T T e5 h2m e5 Θ (t) . x (α) 2m x (α) dα ≤ Θ (t) h t−h 

 t

xT (α) 1m x (α) dα ≤ Θ T (t)

Then, in view of Lemma 1, it leads to −h

 t t−h

x˙ T (α) Um x˙ (α) dα +

 0  t −h t+β

x˙ T (α) 3m x˙ (α) dαdβ

 ≤ −Θ T (t) Π T Λm Π Θ (t) + Θ T (t) 2 [e1 − e4 ]T 3m [e1 − e4 ]  h [e1 − e5 ]T 3m [e1 − e5 ] Θ (t) . + h − d (t)

(31)

Furthermore, for any matrices φ1T and φ2T of appropriate dimensions, the following equation holds: 0=

r  r 

   νj hi (κ (t)) hj (κ (t)) 2 xT (t) φ1T + x˙ T (t) φ2T

i=1 j=1

× −˙x (t) + Aim x (t) + Bim F0m (I + Lm ) Kj x (t − d (t))

 −Bim F0m (I + Lm ) Kj ek (sk T) + Cim ω (t) .

(32)

3 Stability analysis and event-triggered controller design for FMJSs

h−d(t) Then, in view of the fact that d(t) = 1 and combining with Eqs. (28)–(30) h + h and adding the right-hand side of Eq. (32), it can be inferred that

  E LV (t) + zT (t) z (t) − γ 2 ωT (t) ω (t) ⎧    r ⎨r−1  vi d (t) Θ T (t) Υ1ijm + Υ1jim νj hi (κ (t)) hj (κ (t)) ≤E ⎩ h vj i=1 j>i       vi h − d (t) Θ T (t) Υ1ijm + Υ1jim + Υ2m (2) Θ (t) +Υ2m (1) Θ (t) + h vj  r r  

d (t)  T Θ (t) Υ1ijm + Υ2m (1) Θ (t) + νi h2i (κ (t)) h i=1 j=i  

h − d (t) Θ T (t) Υ1ijm + Υ2m (2) Θ (t) . + h

Furthermore, denoting ℘1 =

ν j δ− 1δ

δ− νi

≥ 0, ℘2 =

νi 1 νj − δ δ− 1δ

≥ 0, we have

  E LV (t) + zT (t) z (t) − γ 2 ωT (t) ω (t) ⎧     r ⎨r−1  1 d (t) T Θ (t) Υ1ijm + Υ1jim + Υ2m (1) Θ (t) νj hi (κ (t)) hj (κ (t)) ℘1 ≤E ⎩ h δ i=1 j>i    1 h − d (t) T Θ (t) Υ1ijm + Υ1jim + Υ2m (2) Θ (t) + h δ 

h − d (t) T d (t) T Θ (t) Υ1ijm + δΥ1jim + Υ2m (1) Θ (t) + Θ (t) Υ1ijm + ℘2 h h    r  r

d (t) T Θ (t) [Υ1iim + Υ2m (1)] Θ (t) +δΥ1jim + Υ2m (2) Θ (t) νi h2i (κ (t)) + h i=1 j=i  h − d (t) T Θ (t) [Υ1iim + Υ2m (2)] Θ (t) . (33) + h

According to conditions (21)–(24), it can be achieved that   E LV (t) + zT (t) z (t) − γ 2 ωT (t) ω (t) < 0.

(34)

Thus, under the zero initial condition, for any η, it implies that  η   η  zT (t) z (t) dt ≥ E γ 2 ωT (t) ω (t) dt . E 0

0

In this way, condition (18) is ensured for any nonzero ω (t) ∈ L2 [0, ∞). Besides, when ω (t) = 0, in view of Eqs. (34), there exists a scalar c > 0 such that LV (t) ≤ −cxT (t) x (t) .

267

268

CHAPTER 12 Event-triggered reliable control for Markov jump systems

Then, applying Dynkin’s formula and Gronwall-Bellman lemma, one has 

∞

xT (t) x (t) dt < ∞.

0

Under this circumstances, FMJSs (15), (16) with ω (t) = 0 are stochastically stable. Thus, according to Definition 1, FMJSs (15), (16) are stochastically stable and satisfy an H∞ performance γ . The proof is completed.

3.2 Event-triggered reliable controller design for FMJSs In the previous section, the stability analysis is investigated. However, the controller gain K is coupled with some unknown parameters, which is difficult to overcome. In the following, according to Theorem 1, the controller design for FMJSs (15), (16) can be solved via a simple matrix decoupling method. Theorem 2. For given  scalars h, γ , α1, α2 , and σm , if there exist constant  > 0 ¯ 12 P¯ m P ¯ m > 0, U ¯ m > 0, W ¯ > 0, Ψ¯ m > 0, S¯ > 0, R¯ m > 0, and matrices 11 ¯ > 0, Q ∗ P22     ¯ Z, X= X¯ 2×2 , such that the following matrix inequalities hold for η ∈ δ, 1δ , m ∈ S,   ⎡

Ω1iim + Ω2m (1) ∗

Ω3iim Ω4

Ω1iim + Ω2m (2) ∗

Ω3iim Ω4

 < 0,

i = 1, 2, . . . , r,

(35)

< 0,

i = 1, 2, . . . , r,

(36)



⎤ Ω1ijm + ηΩ1jim + Ω2m (1) Ω3ijm Ωˆ 3ijm ⎣ ∗ −I 0 ⎦ < 0, ∗ ∗ Ω4 ⎤ ⎡ Ω1ijm + ηΩ1jim + Ω2m (2) Ω3ijm Ωˆ 3ijm ⎣ ∗ −I 0 ⎦ < 0, ∗ ∗ Ω4 ¯ km < 0, k = 1, 2, 3,      ¯ ¯m ¯ m , 3U diag U  > 0,  X Λ¯ m = ¯m ¯ m , 3U ∗ diag U

1 ≤ i < j ≤ r,

(37)

1 ≤ i < j ≤ r,

(38)

where ⎛ ¯ m + R¯ m + hS¯ + Ω1ijm = e1 ⎝Q

 n∈S

πmn P¯ n11 + sym(P¯ 12 + α1 Aim Z

  ¯ 3m eT ¯ 3m ) eT + sym e1 α1 Bim F0m Yj eT − e1 P¯ 12 eT − e1  + 1 2 3 4    m T T T T +e1 P¯ 11 − α1 z + α2 Z Aim + e6 e1 α1 Cim e7 − e1 α1 Bim F0m Yj eT8

(39) (40)

3 Stability analysis and event-triggered controller design for FMJSs





  ¯ h2 W ¯ 3m eT +e6 h2 U ¯m + − sym (α2 Z) eT6 + e2 σm Ψ¯ m eT2 − e3 R¯ m eT3 + sym e4  4 2

      + sym e6 α2 Cim eT7 − e6 α2 Bim F0m Yj eT8 − e7 γ 2 I eT7 − e8 Ψ¯ m I eT8 − Π T Λ¯ m Π ,   ¯ 1m eT , Ω2m (1) = sym e4 hP¯ T12 eT6 + e4 hP¯ 22 eT1 − e4 hP¯ 22 eT3 + e4 h 4   T T T T Ω2m (2) = sym e5 hP¯ e + e5 hP¯ 22 e − e5 hP¯ 22 e 12 6

1

3

¯ 2m eT + (e1 − e5 ) h ¯ 3m (e1 − e5 )T , + e5 h 5 Ω3ijm =

⎡

0

Dim

0

0

√

0

0

0 0

δDjm T BT Ωˆ 3ijm = ⎣ α1 F0m im 0

0

0 0 0

Hm Yj

0

0 0 0

0 0 0

T

,

⎡

−I Ω4 = ⎣ ∗ ∗

0 T BT α2 F0m im 0

⎤ 0 0 ⎦, −I ⎤T

0 −I ∗

0 0 −Hm Yj

0 0 ⎦ , 0

then, FMJSs (15), (16) are stochastically stable and satisfies an H∞ performance γ . Besides, the desired controller gains can be achieved as: Kj = Yj Z −1 .

(41)

Proof. Define three diagonal matrices q1 = diag q3 = diag



Z

Z

Z

Z

q2

I

Z

I



,

q2 = diag



Z

Z

Z

Z

Z

Z



,

and    m  T ¯ ¯ ¯ ¯ ¯ diag P¯ m 11 , Σm , Qm , Rm , Um , W, = q2 diag P11 , Σm , Qm , Rm , Um , W q2 , P¯ 12 = Z T P12 Z, P¯ 22 = Z T P22 Z, S¯ = Z T SZ, φ1 = α1 Z −1 , φ2 = α2 Z −1 .

First, it can be obtained that Λ¯ m = qT1 Λm q1 ,

¯ km = Z T km Z, 

k = 1, 2, 3,

then, conditions (39), (40) can guarantee that Eqs. (25), (26) are satisfied simultaneously. Moreover, it is easy to see that applying the Schur complement to LV (t) + zT (t) z (t) − γ 2 ωT (t) ω (t) and premultiplying and postmultiplying Eqs. (21)–(24) by qT3 and their transpose, respectively, we have T Sˆ T < Γ +  Sˆ Sˆ T +  −1 Tˆ H T H Tˆ T , Γ1 + Sˆim Lm TˆjT + Tˆj Lm im im j m m j 1 im

(42)

T Sˇ T < Γ +  Sˇ Sˇ T +  −1 Tˇ H T H Tˇ T , Γ2 + Sˇim Lm TˇjT + Tˇj Lm im im j m m j 2 im

(43)

269

270

CHAPTER 12 Event-triggered reliable control for Markov jump systems

where  Γ1 =

Ω˜ im −I

Ω1iim + Ω2m (l) ∗

 ,

 Ωˇ im , l = 1, 2 diag {I, I}

T Ω˜ im = Dim 0 0 0 0 0 0 0 , T  0 0 0 0 0 0 0 D Ωˇ im = √ jm δDim 0 0 0 0 0 0 0

T Tˆj = 0 Yj 0 0 0 0 0 −Yj 0 ,

T = α F T BT T T Sˆim 1 0m im 0 0 0 0 α2 F0m Bim 0 0 0 ,

TˇjT = 0 Yj 0 0 0 0 0 −Yj 0 0 ,

T = α F T BT T T Sˇim 1 0m im 0 0 0 0 α2 F0m Bim 0 0 0 0 . 

Γ2 =

Ω1ijm + ηΩ1jim + Ω2m (l) ∗

Applying the Schur complement to Eqs. (42), (43) gives rise to Eqs. (35), (38). This completes the proof. Remark 2. It should be pointed out that some coupled terms such as km (k = 1, 2, 3) will be produced when calculating the derivative of mode-dependent Lyapunov functional. In Ref. [2], these coupled terms have been removed by adding some restrained conditions km < 0 (k = 1, 2, 3). Noting that these coupled terms are negative in Ref. [2], which could reduce the conservatism if these coupled terms are retained. In this chapter, these coupled terms are reserved by some novel integral inequalities. Moreover, we preserve the item x˙ (t) by adopting Eq. (32), which results in the use of the inequality −XS−1 X T ≤ S − X T − X is avoided. Therefore, a corresponding criterion is given in the following corollary on condition that these coupled terms are removed. Corollary1. For given  scalars h, γ , α1, α2 , and σm , if there exist constant  > 0, ¯ 12 P¯ m P ¯ m > 0, U ¯ m > 0, W ¯ > 0, Ψ¯ m > 0, S¯ > 0, R¯ m > 0, and matrices 11 ¯ > 0, Q ∗ P22     ¯ Z, X= X¯ 2×2 , such that the following matrix inequalities hold for η ∈ δ, 1δ , m ∈ S,   ⎡

∃1ijm + ∃2m (1) ∗

∃3ijm ∃4

∃1ijm + ∃2m (2) ∗

∃3ijm ∃4

 < 0,

i = 1, 2, . . . , r,

< 0,

i = 1, 2, . . . , r,



∃1ijm + η∃1jim + ∃2m (1) ⎣ ∗ ∗

∃3ijm −I ∗

⎤ ∃ˆ 3jim 0 ⎦ < 0, ∃4

1 ≤ i < j ≤ r,

4 Numerical example

⎡

⎤ ∃1ijm + η∃1jim + ∃2m (2) ∃3ijm ∃ˆ 3jim ⎣ ∗ −I 0 ⎦ < 0, ∗ ∗ ∃4 ¯ km < 0, k = 1, 2, 3,      ¯m ¯ m , 3U X¯ diag U ¯   > 0, Λm = ¯m ¯ m , 3U ∗ diag U

1 ≤ i < j ≤ r,

(44)

where ⎛ ¯ ∃1ijm = e1 ⎝¯ Qm + R¯ m +hS+ 

 n∈S

⎞



πmn P¯ n11 +sym(P¯ 12 +α1 Aim Z)⎠ eT1 +sym e1 α1 Bim F0m Yj eT2 

T T T T − e1 P¯ 12 eT3 + e1 P¯ m 11 − α1 z + α2 Z Aim e6 + e1 α1 Cim e7





−e1 α1 Bim F0m Yj eT8 +e2 σm Ψ¯ m eT2 − e3 R¯ m eT3 +e6   + sym e6 α2 Cim eT7 − e6 α2 Bim F0m Yj eT8     − e7 γ 2 I eT7 − e8 Ψ¯ m I eT8 − Π T Λ¯ m Π ,   ∃2m (1) = sym e4 hP¯ T12 eT6 + e4 hP¯ 22 eT1 − e4 hP¯ 22 eT3 ,   ∃2m (2) = sym e5 hP¯ T12 eT6 + e5 hP¯ 22 eT1 − e5 hP¯ 22 eT3 , ∃3ijm =

⎡

Dim

0

0

√

δDjm T BT ∃ˆ 3ijm = ⎣ α1 F0m im 0

0

0 0 0

Hm Yj

0

0

0 0 0

0 0 0

0

T

0 0 0

,

2

⎡

−I ∃4 = ⎣ ∗ ∗

0 T BT α2 F0m im 0



¯ h2 W ¯ m+ h2 U − sym (α2 Z)

0 −I ∗

0 0 −Hm Yj

eT6

⎤ 0 0 ⎦, −I ⎤T 0 0 ⎦ , 0

and the other parameters follow the same definitions as those in Theorem 2.

4 Numerical example In this section, an example is utilized to illustrate the reduced conservatism whether these coupled terms km are reserved or not. Then, the control effect with or without fault is investigated and the advantages of event-triggered mechanism are also elaborated. Example 1. Consider FMJSs (15), (16) with two modes and two rules. The following parameters are assumed as  A11 =

−4.0 −1.5

2.0 −3.75



 ,

A12 =

−2.1 1.95

0.45 0

 ,

271

272

CHAPTER 12 Event-triggered reliable control for Markov jump systems

 A21 =  B11 = 

2.25 −0.8 0.21 0.35

,



 

B12 =



−2.625 −0.375

A22 =

,



0.15 0.34



2.0 −3.25

 ,

, 

0.55 B22 = , 0.23     0.25 0.44 0.15 0.56 , C12 = , C11 = 0.23 0.31 0.78 0.72     0.16 0.36 0.18 0.21 , C22 = , C21 = 0.54 0.22 0.33 0.49    0 −0.1 −3.2 , D12 = D21 = D11 = D22 = 0.25 −0.2 −0.4 B21 =

0.53 1.2



1.25 −2.0

,

−0.1 0.3

 .

Then, two possible actuator fault modes are considered in the following: (1) Fault mode 1: The actuator suffers from a loss of effectiveness, F01 = f011 ∈ [0.075, 0.525] , which implies that F01 = 0.3, H1 = 0.75. (2) Fault mode 2: The actuator encounters a loss of effectiveness, F02 = f021 ∈ [0.3, 0.9] , which implies that F02 = 0.6, H2 = 0.5. Furthermore, in order to illustrate the necessity of the retention of km (k = 1, 2, 3) and the employed Eq. (32), as noted in Remark 2, the other parameters are given  as σ1 = 0.5, σ2 = 0.5, α1 = 1, α2 = 1, δ = 1.25, γ = 1.6, and the TPM −0.1 0.1 Π = . By solving the conditions in Theorem 2, we can find the 0.8 −0.8 maximum upper bound of time delay h = 0.5334. Nevertheless, there is no feasible solution that can be found by solving the conditions in Corollary 1, which shows that the presented method in this chapter achieves less conservatism. On the other hand, we aim to research the relation between the number of transmissive data n and the threshold σm , the other parameters are the same as before besides h = 0.25, the sampling period is assumed as T = 0.12 s and the whole simulation time is chosen as 50 s. The relation among the number of transmissive data n and the threshold σm is shown in Table 1. From Table 1, we can obtain that the number of transmissive data n decreases greatly with the increment of the threshold σm . Consequently, in some situations, it is of significance to research the selection issue of these values.

Table 1 The relation between the threshold σm and the number of transmissive data n in Example 1. σm n Percentage

0 416 100

0.1 58 13.94

0.3 26 6.25

0.5 13 3.13

0.7 7 1.68

0.9 6 1.44

4 Numerical example

Then, we consider a specific case with σ1 = 0.5, σ2 = 0.5, α1 = 1, α2 = 1, h = 0.25, γ = 1.6, and ∇j = 0.12 (i.e., δ = 1.25). The other parameters are the same as before. By using the Matlab LMI Toolbox to solve these conditions (35)–(40), the desired controller gains are achieved as follows:



K1 = 0.1772 −0.2019 , K2 = −0.4343 −0.0985 and the event-triggered matrices   0.2742 0.0621 Ψ1 = , 0.0621

0.0884

 Ψ2 =

0.3344 0.0595



0.0595 0.0822

.

Besides, we set Fm = I, then the corresponding controller gains and eventtriggered matrices without fault can also be obtained:



−0.1107 , K2 = −0.1882 −0.0539 ,     0.0846 0.0132 0.0931 0.0125 , Ψ2 = . Ψ1 = 0.0132 0.0310 0.0125 0.0290 K1 =



0.3150

The fuzzy basis functions are supposed as h1 (x1 (t)) =

1



  π ∗

1 + e−7 x1 (t)+ 4

1−

1



  π

1 + e−7 x1 (t)− 4

,

h2 (x1 (t)) = 1 − h1 (x1 (t)) .

In the following simulation, the exogenous disturbance is ω (t) =

√

0.4 1+t2

Here, the sampling period is T = 0.12 s and the initial condition is x0 =

√

0.4 1+t2

0.25 −0.2

T 

. .

The considered actuator fault case is uf (t) = Fm u (t) , where

* F1 =

1 0.525 1

0 ≤ t < 10 10 ≤ t ≤ 30 , t > 30

m = 1, 2, * F2 =

1 0.9 1

0 ≤ t < 10 10 ≤ t ≤ 30 . t > 30

Then, the possible mode evolution, the response curves for states and the release intervals are given in Figs. 2–6, respectively. From Figs. 2 and 6, it can be seen that the performance of system is greatly degenerated when the actuator encounters faults. Fortunately, Fig. 4. demonstrates the good performance of system can be still achieved, once the reliability is taken into account. On the other hand, the scarce communication resources are saved greatly when adopting the event-triggered method in this chapter, which is shown in Figs. 3 and 5. Therefore, it is significant and necessary to address the reliability and event-triggered control issue when some communication resources are limited or some unexpected failures occur in actuators.

273

2.5

3

x1(t)

Mode

(t)

State responses

2 1.5

x2(t)

2 1 0 50

1

t (s)

0.5 0

–0.5

0

10

20

30

40

50

t (s)

FIG. 2 The state responses of closed-loop system and possible mode evolution in normal case. 10

Transmit interval

8 6 4 2 0 0

10

20 t (s)

30

40

FIG. 3 The release instants in normal case. 2

3

x1(t)

Mode

(t)

State responses

1.5 1

2

x2(t)

1 0 0

50

t (s)

0.5 0

–0.5

0

10

20

t (s)

30

40

50

FIG. 4 The state responses of closed-loop system and possible mode evolution in faulty case.

5 Conclusions

14

Transmit interval

12 10 8 6 4 2 0 0

10

20

30

40

50

t (s)

FIG. 5 The release instants in faulty case. 2

3

x1(t)

Mode

(t)

State responses

1.5

2

x2(t)

1 0

1

0

50

t (s) 0.5 0

–0.5 0

10

20

30

40

50

t (s)

FIG. 6 The state responses of closed-loop system and possible mode evolution in faulty case without reliable control.

5 Conclusions In this chapter, the event-triggered reliable controller for FMJSs has been designed such that the limited communication resources are saved and simultaneously can tolerate some unexpected actuator failures. Some sufficient conditions, which assure the resulting closed-loop system is stochastically stable with an H∞ performance index, are established. Some novel integral inequalities are applied to handle with some cross terms produced by the constructed mode-dependent LKF. In this case, less

275

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CHAPTER 12 Event-triggered reliable control for Markov jump systems

conservative results are obtained. An example has been given to show the superiority of the proposed scheme.

Acknowledgments This work is supported in part by National Natural Science Foundation of China (Grant No. 61773097), the Fundamental Research Funds for the Central Universities (Grant Nos. N160402004, N170406013).

References [1] Y. Xu, R.Q. Lu, K.X. Zhou, Z.X. Li, Nonfragile asynchronous control for fuzzy Markov jump systems with packet dropouts, Neurocomputing 175 (2016) 443–449. [2] R. Sakthivel, M. Joby, K. Mathiyalagan, S. Santra, Mixed H and passive control for singular Markovian jump systems with time delays, J. Frankl. Inst. 352 (10) (2015) 4446–4466. [3] C. Peng, Y. Song, X. Xie, M. Zhao, M.R. Fei, Event-triggered output tracking control for wireless networked control systems with communication delays and data dropouts, IET Control Theory Appl. 10 (17) (2016) 2195–2203. [4] S. Weng, D. Yue, Distributed event-triggered cooperative attitude control of multiple rigid bodies with leader-follower architecture, Int. J. Syst. Sci. 47 (3) (2016) 631–643. [5] L.J. Zhang, J.A. Fang, X.F. Li, J.L. Liu, Event-triggered output feedback H control for networked Markovian jump systems with quantizations, IEEE Trans. Cybern. 45 (11) (2015) 2449–2460. [6] X.M. Zhang, Q.L. Han, B.L. Zhang, An overview and deep investigation on sampled– data-based event-triggered control and filtering for networked systems, IEEE Trans. Ind. Inf. (13) (2017) 4–16, https://doi.org/10.1109/TII.2016.2607150. [7] M. Abdelrahim, R. Postoyan, J. Daafouz, D. Nešic, Stabilization of nonlinear systems using event-triggered output feedback controllers, IEEE Trans. Autom. Control 61 (9) (2016) 2682–2687. [8] H. Li, Z. Chen, L. Wu, H. Lam, Event-triggered control for nonlinear systems under unreliable communication links, IEEE Trans. Fuzzy Syst. 25 (2017) 813–824 https://doi.org/10.1109/TFUZZ.2016.2578346. [9] P. Shi, H. Wang, C.C. Lim, Network-based event-triggered control for singular systems with quantizations, IEEE Trans. Ind. Electron. 63 (2) (2016) 1230–1238. [10] H. Hu, B. Jiang, H. Yang, Reliable guaranteed-cost control of delta operator switched systems with actuator faults: mode-dependent average dwell-time approach, IET Control Theory Appl. 10 (1) (2016) 17–23. [11] R. Sakthivel, M. Joby, P. Shi, K. Mathiyalagan, Robust reliable sampled-data control for switched systems with application to flight control, Int. J. Syst. Sci. 47 (15) (2016) 3518–3528. [12] S. Santra, H.R. Karimi, R. Sakthivel, S.M. Anthoni, Dissipative based adaptive reliable sampled-data control of time-varying delay systems, Int. J. Control Autom. Syst. 14 (1) (2016) 39–50.

References

[13] M. Shen, C.C. Lim, P. Shi, Reliable H static output control of linear time-varying delay systems against sensor failures, Int. J. Robust Nonlinear Control 27 (2017) 3109–3123, https://doi.org/10.1002/rnc.3729. [14] H. Wang, P. Shi, C.C. Lim, Q. Xue, Event-triggered control for networked Markovian jump systems, Int. J. Robust Nonlinear Control 25 (17) (2015) 3422–3438. [15] C. Peng, D. Yue, Q.L. Han, Communication and Control for Networked Complex Systems, Springer, Heidelberg, 2015. [16] J. Tao, R. Lu, P. Shi, H. Su, Z.G. Wu, Dissipativity-based reliable control for fuzzy Markov jump systems with actuator faults, IEEE Trans. Cybern. 47 (2017) 2377–2388, https://doi.org/10.1109/TCYB.2016.2584087. [17] E.K. Boukas, Stabilization of stochastic nonlinear hybrid systems, Int. J. Innov. Comput. Inf. Control 1 (2005) 131–141. [18] J. Chen, C. Lin, B. Chen, Q.G. Wang, Mixed H and passive control for singular systems with time delay via static output feedback, Appl. Math. Comput. 293 (2017) 244–253. [19] A. Seuret, F. Gouaisbaut, Jensen’s and Wirtinger’s inequalities for time-delay systems, in: 11th IFAC Workshop on Time-Delay Systems, 2013, pp. 343–348. [20] H. Shen, Z.G. Wu, J.H. Park, Reliable mixed passive and H∞ filtering for semi-Markov jump systems with randomly occurring uncertainties and sensor failures, Int. J. Robust Nonlinear Control 25 (7) (2015) 3231–3251. [21] J. Huang, Y. Shi, Stochastic stability and robust stabilization of semi-Markov jump linear systems, Int. J. Robust Nonlinear Control 23 (18) (2013) 2028–2043.

277

CHAPTER

Stability of a class of linear fractional-delay systems

13

Zaihua Wang, Song Liang State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

Chapter outline 1 Introduction....................................................................................... 279 2 Problem statement and preliminaries ........................................................ 281 2.1 Asymptotic stability of linear systems ............................................ 282 2.2 Mittag-Leffler stability of linear systems ......................................... 282 3 Effective algorithms for stability test ......................................................... 283 3.1 Evaluation of the test integral...................................................... 284 3.2 Nyquist frequency plot .............................................................. 285 3.3 Calculation of the rightmost characteristic roots ................................ 288 4 Stability of a class of systems with delay-dependent coefficients ...................... 289 5 Concluding remarks ............................................................................. 291 Acknowledgment.................................................................................... 292 References........................................................................................... 292

1 Introduction After a long silence of more than 300 years, fractional calculus and its applications have been a hot topic in science and engineering since the second half of the last century. In mechanics, for example, fractional derivative has been often used to model viscoelastic materials with memory. Because viscoelastic materials behave between elasticity and viscosity, it is reasonable to assume that the stress is proportional to a fractional derivative of the strain with the fractional-order between 0 and 1. This is the Scott-Blair’s law for viscoelasticity, which covers two extremes: Hooke’s law with order 0 for elasticity and Newton’s law with order 1 for viscosity. This law works also in modeling protein adsorption kinetics as well as in modeling learning and forgetting in psychology [1]. Fractional derivative models are substantially more accurate and appropriate for modeling some real materials, such as aircraft tire [2] and aging of polymers [3]. More applications of fractional calculus in solid mechanics can be referred to a review article [4]. Control theory is another discipline Stability, Control and Application of Time-Delay Systems. https://doi.org/10.1016/B978-0-12-814928-7.00013-5 © 2019 Elsevier Inc. All rights reserved.

279

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CHAPTER 13 Stability of a class of linear fractional-delay systems

that impulses the development of fractional calculus. For example, the PIλ Dμ control proposed by Podlubny [5] extends greatly the most popular and well-developed PID control, by introducing two parameters λ and μ ranging from −∞ to +∞. When the control parameters are properly chosen, the PIλ Dμ control has better control effect. The CRONE control proposed by Oustaloup is a robust control based on fractional differentiation for linear systems and it has several industrial applications proving its efficiency [6]. Some fundamentals and applications of fractional systems and controls are refereed to see the monograph [7]. Fractional-delay systems stand for dynamical systems with fractional derivatives and time delays. Stability of a fractional-delay system is usually referred to Lyapunov’s asymptotic stability or Mittag-Leffler stability or finite time stability [7–13]. For asymptotic stability, the solution is estimated on the basis of the classical exponential functions and a nonzero integral over the right half infinite interval, while in the study of Mittag-Leffler stability, the solution is estimated by using the Mittag-Leffler functions. Mittag-Leffler stability and generalized Mittag-Leffler stability imply asymptotic stability [10,11]. Finite time stability requires that the solution tends to a steady solution within a finite time. The paper [8] presents a survey on the stability of fractional systems by using frequency domain methods, while the paper [12] collects some results about stability of fractional systems based on time domain methods and frequency domain methods. Frequency domain methods determine the local stability in a neighborhood of an equilibrium, while time domain methods such as the Lyapunov’s direct method and its widely used variant-LMI (linear matrix inequality) method can be used to study not only local stability but also nonlocal stability [11,13]. When the local stability is addressed as in many practical applications, frequency domain methods usually give less conservative results than the Lyapunov’s direct method. Similar to the stability of an ordinary differential equation, the local stability of an equilibrium of a nonlinear fractional system can be determined by that of the linearized system [14], via analyzing the root location of the characteristic equation, if the linearized system has no characteristic roots with zero real part. In the presence of time delay(s), there are roughly two main routines that have been widely applied for integer-order time-delay systems to deal with the exponential functions in the characteristic equations caused by the delays. The first routine is to convert the exponential functions into rational functions via the Reksius transform [15,16]. This technique can be useful for integer-order time-delay systems, for which the Rekasius transform converts the characteristic functions into polynomials, a form that is ready for the application of the Routh-Hurwitz stability criterion. In the applications of this transform, incorrect results might be obtained if the degenerate cases were not considered, as pointed out in Ref. [17]. The second routine is to keep the exponential functions unchanged, with a combination of some methods such as D-subdivision method, or the method of stability switch, or the argument principle. In this way, a number of results have been obtained, say, Refs. [18–24], and they have been shown effective in applications. The methods and algorithms based on the argument principle show some advantages over the other available methods and algorithms, such as simple representation and easy implementation.

2 Problem statement and preliminaries

In this chapter, the local linear stability in Lyapunov’s sense will be addressed. Three argument principle-based results and a closed-form formula for the analysis of stability switch will be introduced and demonstrated with case studies.

2 Problem statement and preliminaries For clarity in the presentation and for simplicity in mathematics, this chapter focuses on the following class of autonomous fractional-delay systems of retarded type: C Dα x(t) = f (x(t), x(t − τ ), x(t − τ ), . . . , x(t − τ )), m 1 2 0 t

(1)

where τ1 , τ2 , . . . , τm ≥ 0 are the delays, α = [α1 , α2 , . . . , αn ]T ∈ Rn+ with positive rational orders αi , x(·) = [x1 (·), x2 (·), . . . , xn (·)]T ∈ Rn is the state vector, f (·) = α C α1 C α2 C αn C αi [f1 (·), f2 (·), . . . , fn (·)]T ∈ Rn , C 0 Dt = diag(0 Dt , 0 Dt , . . . , 0 Dt ), and 0 Dt xi (t) is the Caputo’s fractional derivative defined by  t 1 x([α]) (ξ ) C Dα x(t) = dξ , 0 t Γ ([α] − α) 0 (t − ξ )α−[α]+1

(2)

in which [α] is the integer satisfying [α] − 1 ≤ α < [α], x([α]) (t) is the [α]-th order derivative of x(t), and Γ (z) is the Gamma function satisfying Γ (z + 1) = zΓ (z) for C α λt α λt α λt α λt z > 0. For noninteger α, one has C 0 Dt e  = λ e , but −∞ Dt e = λ e . An equilibrium x∗ of Eq. (1) is defined as a solution of f (x∗ , x∗ , . . . , x∗ ) = 0. Assume that f (0, 0, . . . , 0) = 0 without loss of generality, and assume that x∗ = 0 is a hyperbolic equilibrium (no characteristic roots with zero real part) [14], then the local stability of x∗ = 0 of Eq. (1) can be determined by that of the linearized system C Dα x(t) = Ax(t) + 0 t

m 

Bi x(t − τi ),

(3)

i=1 α C α where A, Bi ∈ Rn×n (i = 1, 2, . . . , m). Denote C 0 Dt x(t) simply by 0 Dt x(t) if αi ’s equal to α. Eq. (3) may come from practical applications. In vibration control, for example, a generalization of the classical mass-damping-spring system under harmonic excitation and control can be written in the following form: α m¨x(t) + c C 0 Dt x(t) + k x(t) = f0 sin(ωt) + u (0 < α < 1),

where the damping is described by fractional derivative if viscoelastic material is used, u is the control used to reduce the excited vibration. When α = 1/2, u = −kp x(t − τ1 ) − kd x˙ (t − τ2 ), by introducing a state vector y(t) = [x(t), x1 (t), x2 (t), x3 (t)]T and a excitation vector f (t) = [0, 0, 0, f0 sin(ωt)]T , and ⎡

0 1 ⎢ 0 0 A=⎢ ⎣ 0 0 − mk − mc

0 1 0 0

⎤ 0 0⎥ ⎥, 1⎦ 0

⎡

0 ⎢ 0 B1 = ⎢ ⎣ 0 k

⎤ 0 0 0 0 0 0⎥ ⎥ 0 0 0 ⎦,

− mp 0 0 0

⎡

0 ⎢0 B2 = ⎢ ⎣0 0

0 0 0 0 0 0 0 − kmd

⎤ 0 0⎥ ⎥, 0⎦ 0

281

282

CHAPTER 13 Stability of a class of linear fractional-delay systems

the equation of the controlled vibration system becomes C D1/2 y(t) = Ay(t) + B y(t − τ ) + B y(t − τ ) + f (t). 1 1 2 2 0 t

If the linear elastic force k x(t) is replaced with a nonlinear function, say, k1 x(t) + k3 x3 (t) as used in many applications, then a nonlinear fractional-delay system of form (1) with excitation is obtained.

2.1 Asymptotic stability of linear systems The solution of Eq. (3) under given initial conditions can be represented as a contour integral by using Laplace transform and inverse Laplace transform. Moreover, by using the residue theorem for contour integral, it is found that the solution equals to the summation of two terms [4]. The first term is the summation of residues at the roots of the characteristic equations p(λ) = 0 with ⎛ p(λ) = det ⎝diag(λα1 , λα2 , . . . , λαm ) − A −

m 

⎞ Bi e−λτi ⎠ .

(4)

i=1

Such a summation resulted from the residue calculation is called eigenfunction expansion of Eq. (3). If there is at least one noninteger order, p(λ) is multivalued, and when a branch cut is made along the negative real axis of the complex plane, it becomes a single-valued function that is necessary in the calculation of the contour integral. The second term is a nonzero infinite integral taken along the two sides of the branch cut along the negative real semiaxis of the complex plane, which cannot be counteracted when there is a fractional power in p(λ). This integral tends to 0 as the time t goes to +∞, and it disappears when the system is degenerated to an integer-order one. Therefore, the zero solution of Eq. (3) is asymptotically stable if all the roots of p(λ) = 0 have negative real parts.

2.2 Mittag-Leffler stability of linear systems It is referred to Refs. [10,11] for a general discussion on Mittag-Leffler stability of fractional systems. When linear stability via root location is focused, note that the λt solution of x˙ (t) = ax(t) with x(0) = x0 is x(t) m= e x0 , thus, for the retarded timedelay system described by x˙ (t) = Ax(t) + i=1 Bi x(t − τi ), one always look for solution of the form eλt c, so that  to find an eigenfunction of the solution,  expansion −λτi = 0. The zero where λ is called a root of det diag(λ, λ, . . . , λ) − A − m B e i i=1 solution is asymptotically stable if characteristic roots have negative real parts only. Although the stability of the zero solution of Eq. (3) is guaranteed if all the characteristic roots have negative real parts, it is NOT possible to look for solution of the form eλt c for fractional-delay equation (3) due to the presence of the infinite α integral. For a fractional differential equation C 0 Dt x(t) = a x(t) with x(0) = x0 , the α solution is x(t) = Eα (a t )x0 , where Eα (x) is the Mittag-Leffler function, defined by

3 Effective algorithms for stability test

Eα (x) =

∞  n=0

xn . Γ (nα + 1)

Furthermore, x(t) → 0, (t → +∞), if |arg(a)| > α π2 . Thus, the solution has an expansion of Mittag-Leffler functions. In this way, estimation of the solution as well as the stability of the solution can be studied on the basis of Mittag-Leffler functions. When a delay is involved, analysis on the characteristic function resulted from the above-mentioned asymptotic stability is easier understood and analyzed than that of the characteristic function obtained on the basis of Mittag-Leffler functions [22].

3 Effective algorithms for stability test Hereafter, the stability means asymptotic stability, and α1 = α2 = · · · = αn = α is assumed for simplicity. Let μ = λα , then p(λ) can be represented as p(λ) = μn +

n 

βi (e−λτ1 , . . . , e−λτm )μn−i = λnα +

i=1

n 

βi (e−λτ1 , . . . , e−λτm )λ(n−i)α , (5)

i=1

where each βi (z1 , . . . , zm ) is a polynomial with respect to z1 = e−λτ1 , . . . , zm = e−λτm . The complex variable function p(λ) is an analytic function in C except for a possible singularity at λ = 0. Assume that p(λ) has no roots on the imaginary axis. Let C be the contour in the complex plane of λ shown in Fig. 1, consisted of a directed arc {C1 : λ = Reiθ , θ ∈ (−π/2, π/2)} and a directed line {C2 : λ = iω, ω ∈ (−R, R)}, and let C arg p(λ) denote the change in the argument of p(λ) over the contour C, then the Argument Principle gives the finite number N of roots of p(λ) in (λ) > 0 as

iR C1 C

2

0

R

C

1

−i R

FIG. 1 The contour C.

283

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CHAPTER 13 Stability of a class of linear fractional-delay systems

N =

lim

R→+∞

C arg(p(λ)) . 2π

(6)

The trivial solution x = 0 of Eq. (1) with characteristic function given in Eq. (5) is asymptotically stable if N = 0. The stability criteria developed on the basis of Eq. (6) are shown effective in stability test. Among these criteria, the integral test and the graphical test are most convenient in applications.

3.1 Evaluation of the test integral By direct calculation, the integer N defined in Eq. (6) can be represented as N =

ω→+∞  1 nα − arg(p(iω)) . 2 π ω→0

(7)

The asymptotic stability is guaranteed if N = 0, namely +∞  nα 1 arg(p(iω)) . = π 2 0

(8)

This is a generalization of the Mikhailov’s criterion from retarded time-delay systems [25] to retarded fractional-delay systems [22]. Moreover, let R1 (ω) and S1 (ω) be the real and imaginary parts of p(iω), that is, p(iω) = R1 (ω) + iS1 (ω),

(9)

then the chain rule gives p (iω) · i = R 1 (ω) + iS1 (ω), thus d dω

argp(iω) =

S1 (ω)R1 (ω) − S1 (ω)R 1 (ω) R21 (ω) + S12 (ω)

 p ( iω) . = p(iω) 

So the Mikhailov-type criterion (8) can also be stated as follows [22]: assume that p(iω) = 0 for all real ω, then N = 0 holds if  

 nα 1 +∞ p (iω) dω = . π 0 p(iω) 2

(10)

Unlike its equivalent form (6) that is hardly tested, this condition can be easily checked for a given system, for which lots of well-developed algorithms for numerical integration can be used. Moreover, one has the following theorem. Theorem 1. Assume that p(iω) = 0 for all real ω, then N = 0 holds if there is a sufficient large Ω > 0 such that    

  nα 1 Ω p (iω)   1 − dω < .   2  2 π 0 p(iω)

(11)

Theorem 1 was proved in Ref. [22] for Eq. (3), and an estimation for the upper limit Ω (delay-dependent or delay-independent) can be easily obtained [24]. It implies that the asymptotic stability can be judged by using a rough estimation of the testing integral, without any restrictions on the delays.

3 Effective algorithms for stability test

π

π

Actually, let R2 (ω) = (p(iω)e−nα 2 i ) and S2 (ω) = (p(iω)e−nα 2 i ), namely π

R2 (ω) + iS2 (ω) = p(iω)e−nα 2 i ,

(12)

then R2 (ω) = ωnα + b1 ω(n−1)α + · · · ,

S2 (ω) = c1 ω(n−1)α + · · ·

where b1 = b1 (ω, τ1 , . . . , τm ), c1 = c1 (ω, τ1 , . . . , τm ), and the other coefficients are π all bounded functions. Here the factor e−nα 2 i is introduced to rotate the plot of p(iω) in the complex plane around the origin by the angle nαπ/2 clockwise, so that the plot of R2 (ω) + iS2 (ω) approaches to the positive real axis [23]. Moreover, direct calculation from Eq. (12) gives 

 d S2 (ω) p (iω) arctan = . dω R2 (ω) p( iω) Hence, the maximal root of R2 (ω) plays an important role in the stability test. It is easy to find the minimal values of the coefficients bi in R2 (ω) by replacing cos(ωτj ) and sin(ωτj ) with ±1, a delay-independent function R∗ (ω) can be obtained, satisfying R2 (ω) ≥ R∗ (ω),

∀τ1 , τ2 , . . . , τm ≥ 0).

Theorem 2. Assume that p(iω) = 0 for all real ω, and let Ω∗ be the maximal positive root of R∗ (ω), then for all Ω > Ω∗ one has    

  nα  1 1 Ω p ( iω)   − dω < .   2  2 π 0 p(iω)

(13)

Therefore, the key step of the stability test is to choose an arbitrary positive number Ω > Ω∗ and to calculate   

 1 Ω nα p ( iω) − dω . N = round 2 π 0 p(iω) 

(14)

If R(ω) = 0 has no positive root, then N is the closest integer neighboring nα/2. When N = 0, the trivial solution of the system is asymptotically stable, otherwise, the solution is unstable [24].

3.2 Nyquist frequency plot The Mikhailov-type criterion (8) also leads to a graphical test method. In fact, for a given constant c > 0, let Π (λ) =

p(λ) , (λ + c)nα

(15)

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CHAPTER 13 Stability of a class of linear fractional-delay systems

where p(λ) is the characteristic function defined by Eq. (4), and c > 0 is a fixed number. The argument rule implies that argΠ (iω) = argp(iω) − arg((iω + c)nα ), then it holds +∞ nαπ nαπ argΠ (iω)|+∞ = arg(f (iω))|+∞ − arg ((iω + c)n )0 = − = 0. 0 0 2 2 Moreover, let R3 (ω) and S3 (ω) be the real and imaginary parts of Π (iω), that is, Π (iω) = R3 (ω) + iS3 (ω),

(16)

then the following theorem holds [21,24]. Theorem 3. Assume that p(iω) = 0 for all real ω, then N = 0 if there is a constant c > 0 such that the Nyquist frequency plot defined by the curve of {(R3 (ω), S3 (ω)): − ∞ < ω < +∞} does not encircle the origin of the complex plane. The curve always ends at (1, 0) on the real axis as ω → ±∞. For a given retarded fractional-delay system, the stability test can be very convenient and effective, by using the earlier three theorems, in particularly by using Theorems 2 and 3. Example 1. Consider a retarded fractional-delay system described by C D3/2 x(t) − 3 x˙ (t) − 3 x˙ (t − τ ) + 4 C D1/2 x(t) + 8x(t) = 0, 0 t 0 t 2 2

(17)

whose characteristic function is 3 3 p(λ) = λ3/2 − λ − λ e−λτ + 4λ1/2 + 8. 2 2 When τ = 0, p(λ) = λ3/2 − 3λ + 4λ1/2 + 8. With μ = λ1/2 , p(μ2 ) = μ3 − 3μ2 + 4μ + 8, which has three roots: μ = −1, 1.2500 ± 2.2220i. The zero solution is asymptotic stable because the three roots satisfy |arg(μ)| > 12 π2 = π4 . π When τ > 0, by using (iω)α = eiα 2 ωα , direct calculation gives √ √ 2 3/2 3 ω R1 (ω) = − − cos(ωτ ) + 2 2ω1/2 + 8, 2 2 √ √ 3 2 3/2 3 S1 (ω) = ω − ω + cos(ωτ ) + 2 2ω1/2 . 2 2 2

When τ = 0.1, Ω = 5, 10, 50, 100, 1000, one has 

  p (iω) 1 Ω 3 dω = −0.4412, −0.1416, 0.0221, 0.0215, 0.0097. − 4 π 0 p(iω) All are within (−1/2, 1/2). Thus, the zero solution of the fractional-delay system (17) is asymptotic stable, according to Theorem 1.

3 Effective algorithms for stability test

In order to use Theorem 2, the functions R2 (ω) and S2 (ω) are found to be √ √ 3 2 (cos(ωτ ) + sin(ωτ ) − ω) − 4 2, R2 (ω) = ω3/2 + 4 √ √ √ 3 2 S2 (ω) = (cos(ωτ ) − sin(ωτ ) + ω) − 4 ω − 4 2, 4

and the function R∗ (ω) that is independent of the delay τ is R∗ (ω) = ω

3/2

√ √ √ √ 3 2 3 2 11 2 3/2 + (−1 − 1 − ω) − 4 2 = ω − ω− . 4 4 2

The unique positive root of R∗ (ω) is Ω∗ = 5.7850. According to Theorem 2, the zero solution of the fractional-delay system is asymptotic stable because for an arbitrary chosen Ω = 5.8 that is a little bit larger than Ω∗ , one has 1 3 1 − < − 2 4 π

 0

Ω



 p (iω) 1 dω = −0.3759 < . p(iω) 2

This means that N = 0. Compared with the infinite integral over [0, +∞), the definite integral over [0, 5.8] can be calculated with much less computational cost. The asymptotical stability can be validated by using Theorem 3 with c = 1, because as shown in Fig. 2A, the Nyquist frequency plot {(R3 (ω), S3 (ω)): − ∞ < ω < +∞} of Π (iω) does not encircle the origin of the complex plane.

(A)

(B)

FIG. 2 Stability test of the fractional-delay system (17). (A) The Nyquist plot of Π(iω) for τ = 0.1. (B) The plot of N (τ ) for τ ∈ [0, 1].

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CHAPTER 13 Stability of a class of linear fractional-delay systems

The stability can also be identified by using Theorem 2 even if τ is not given but can be chosen from an interval, say τ ∈ [0, h]. Divide [0, h] with the nodes τi = i Mh for sufficient large integer M, then, the plot of {(τi , N (τi )): i = 0, 1, . . . , M} shows the stability of Eq. (17), depending on whether N (τi ) = 0 or not, where 

  3 p ( iω)  1 Ω dω. N (τi ) = − 4 π 0 p(iω) τ =τi As shown in Fig. 2B, the zero solution is asymptotic stable for all τ ∈ [0, 1].

3.3 Calculation of the rightmost characteristic roots A more direct method for the stability test is to calculate the rightmost characteristic root(s). Let the abscissa σ be defined by σ = max{ (λ): p(λ) = 0}. The zero solution of Eq. (3) is asymptotically stable (unstable) if and only if σ < 0 (σ > 0). For the stable case with a negative σ , the larger the absolute of σ is, the faster the solution decays to zero. Obviously, p(λ) = 0 has no roots on (λ) = σ if and only if p(σ +λ) = 0 has no roots on the imaginary axis. According to Theorem 2, assume that there is a a ∈ R such that for sufficiently large Ω > 0 one has 

    p (a + iω) 1 Ω nα N (a) = round dω > 0 (or < 0). − 2 π 0 p(a + iω) In the case of N (a) > 0 (or N (a) < 0), the abscissa of p(a+λ) = 0 must be positive (or negative), namely a < σ (or a > σ ). Thus, two bounds a1 , a2 ∈ R can be found for an estimation a1 < σ < a2 . In this way, the abscissa σ of p(λ) can be estimated to be σ0 = (a1 + a2 )/2 when a2 − a1 is small enough. The smaller the difference a2 − a1 is, the better the estimation of σ is. For a given fractional-delay system with characteristic function p(λ), the value of σ can be estimated graphically as follows. Step 1. For the stable case, find a a2 < 0 such that σ ∈ (a2 , 0), and for the unstable case, find a a1 > 0 such that σ ∈ (0, a1 ). π Step 2. Calculate the real part R2 (ω) of p(iω)e−nα 2 i , and find an uniform upper limit Ω within σ ∈ [a2 , 0] for the stable case or σ ∈ [0, a1 ] for the unstable case, according to Theorem 2. Step 3. Grid σ ∈ [a2 , 0] or σ ∈ [0, a1 ] with σi (i = 1, 2 . . . , l) and calculate   

 1 Ω nα p (σi + iω) − dω . N (σi ) = round 2 π 0 p(σi + iω) 

(18)

Step 4. Plot {(σi , N (σi )): i = 1, 2 . . . , l}. If N (σi ) jumps from 0 to a nonzero integer at some σi , then σi can be regarded as an estimation of σ . The smaller the griding stepsize is, a more accurate estimation of σ one has.

4 Stability of a class of systems with delay-dependent coefficients

Step 5. With a sufficient accurate estimation σˆ of σ , an estimation ωˆ of ω can be obtained numerically from f (σˆ + iω) ≈ 0. Let σˆ + iωˆ be the initial guess, and let ε > 0 be a given tolerance, then the Newton-Raphson iteration scheme p(λi ) , λi+1 = λi −

p (λi )

i = 0, 1, 2, . . .

(19)

gives the rightmost characteristic root(s) λk+1 if |λk+1 − λk | < ε. For example, for system (17) with τ = 0.1 and Ω = 10, one has  

 1 Ω 3 p (−4 + iω) − dω = 1.8624, 4 π 0 p(−4 + iω)  

 1 Ω 3 p (−3.5 + iω) − dω = −0.0720, 4 π 0 p(−3.5 + iω)

which means that −4 < σ < −3.5. If σˆ = −3.75, then ωˆ = 5.3157 from p(−3.75 + iω) ≈ 0. With ε = 10−5 , λ0 = −3.75 + 5.3157i, the Newton-Raphson method gives the rightmost characteristic roots λ5 = −3.7282 ± 5.3226i after five iterations. The Nyquist plot of Π (−3.7282 + iω) can be used to validate the computational result.

4 Stability of a class of systems with delay-dependent coefficients For simplicity, in this section, let us study the asymptotic stability of fractional-delay systems with characteristic functions described by p(λ) = ξ(λ1/n , τ ) + η(λ1/n , τ )e−λτ ,

(20)

where ξ(z, τ ) and η(z, τ ) are real polynomials with respect to z, deg ξ > deg η, and n ∈ N. As τ varies from 0 to +∞ or to a given positive number, the stability may change from stable to unstable or vice versa. This is the phenomenon of stability switch. Because the stability is guaranteed if (λ) < 0 for all characteristic roots, the fractional-delay system loses its stability only if τ passes through a critical delay value for which p(iω) = 0, a case between stable and unstable. Thus, in the applications of the method of stability switch, the first step is to find out the critical delay values from p(iω) = 0. For example, in the Maple platform, one can use the command plots[implicitplot] to find an estimation of critical values (ω, τ ) an initial guess from (p(iω)) = 0 and (p(iω)) = 0, and then to refine the critical values by using the Newton-Raphson iteration method. The system has the same stability in each delay interval between two adjacent critical delay values. A computational algorithm based on the Orlando formula was also be proposed in Ref. [26] for calculating the critical values. The second step is, once a critical point (τj , λ) = (τj , iωj ) obtained from p(iω) = 0 is in hand, to determines the crossing

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CHAPTER 13 Stability of a class of linear fractional-delay systems

direction (also called “root tendency” in the literature) of the characteristic root with respect to the change of τ , by calculating  S=

 dλ . dτ (τj ,λ)=(τj ,iωj )

(21)

The system increases (or decreases) a pair of conjugate characteristic roots with positive real parts if S > 0 (or S < 0), as τ passes τj from the left to the right. Of course, the value of S can be calculated numerically in a straightforward way without any difficulty. A useful generalization of the results in Refs. [27,28] is that the crossing direction can be determined by using a simple analytical formula in terms of a Jacobian determinant of two auxiliary functions F(ω, τ ) and G(ω, τ ) derived directly from the characteristic function p(λ) [29]. In this way, there is no need to check the stability at each point of the given delay interval. For this purpose, let ξ((iω)1/n , τ ) = ξR (ω1/n , τ ) + iξI (ω1/n , τ ),

η((iω)1/n , τ ) = ηR (ω1/n , τ ) + iηI (ω1/n , τ ),

where the subscripts R and I stand for the real and imaginary parts of a complex number, respectively. From p(i ω) = 0, namely, ξ((iω)1/n , τ ) + η((iω)1/n , τ )e−iωτ = 0,

(22)

one has a relationship between the amplitudes of the two complex numbers: def

F(ω, τ ) = |ξ |2 − |η|2 = 0, where F(ω, τ ) = (ξR (ω1/n , τ ))2 + (ξI (ω1/n , τ ))2 − ((ηR (ω1/n , τ ))2 + (ηI (ω1/n , τ ))2 ).

(23)

In addition, a relationship between the phase angles of the complex numbers in Eq. (22) can be obtained: there is an integer k such that 2kπ + G(ω, τ ) = 0, where G(ω, τ ) = ωτ + arctan

ηI (ω1/n , τ ) ξI (ω1/n , τ ) − arctan . 1/n ξR (ω , τ ) ηR (ω1/n , τ )

(24)

Theorem 4. Let p(λ) be the characteristic function in Eq. (20), let J be the Jacobian determinant of F(ω, τ ), G(ω, τ ) with respect to ω and τ , defined by    ∂F ∂G ∂G ∂F − =  J(ω, τ ) = ∂ω ∂τ ∂ω ∂τ 

∂F ∂ω ∂G ∂ω

∂F ∂τ ∂G ∂τ

   ,  

(25)

then at the critical point (ω∗ , τ∗ ) for which p(iω∗ ) = 0, one has sgn(S) = sgn(J(ω∗ , τ∗ )).

(26)

For a special case when the coefficients of p(λ) are independent of the delay, one ∂G ∂F = 0, = ω, thus, Theorem 4 gives has F(ω, τ ) = F(ω), and ∂τ ∂τ sgn(S) = sgn(ω∗ F (ω∗ )) = sgn(F (ω∗ )),

5 Concluding remarks

which is independent of the critical delay values. In this case, at every critical point obtained from the same critical frequency ω∗ , the crossing direction is the same, no need to check the signs one by one for the critical delay values. Example 2. Consider a vibration system with a delayed PDα feedback α x¨ (t) + 2ξ x˙ (t) + x(t) = ux(t − τ ) + v C 0 D x(t − τ ),

(27)

where α = 1/2. The characteristic function reads p(λ) = λ2 + 2ξ λ + 1 − (u + vλα )e−λτ .

(28)

For a given number γ > 0, let q(λ) = p(−γ + λ), then the fractional-delay system (27) is γ -stable, namely (λ) < −γ for all roots of p(λ), if and only if (λ) < 0 for all roots of q(λ). In this case, one has q(λ) = ζ (λ) + η(λ, τ )e−λτ , where ζ (λ) = λ2 + 2(ξ − γ )λ + γ 2 − 2ξ γ + 1 and η(λ, τ ) = −(u + v(λ − γ )α )eγ τ . The coefficients in η(λ, τ ) are delay-dependent. Let 1 = γ 2 − ω2 − 2ξ γ + 1, 2 = 2ω(ξ − γ ),       v v eγ τ γ τ 2 2 2 ω + γ − 2γ e , 4 = − 2 ω2 + γ 2 + 2γ , 3 = − u + 2 2

then the two corresponding auxiliary functions read F(ω, τ ) = (21 + 22 ) − (23 + 24 ),

G(ω, τ ) = ωτ + arctan

2  − arctan 4 . 1 3

For a special case when γ = ξ = 0.1, u = −0.1, v = −0.05 and τ ∈ [0, 4], two critical points (ω, τ ) are found to be (ω1 , τ1 ) = (1.0648, 0.2595) and (ω2 , τ2 ) = (0.8890, 3.8284). Thus, J(ω1 , τ1 ) = 0.6462 > 0 and J(ω2 , τ2 ) = −0.6137 < 0. At τ = 0, p(λ) has two roots −0.1181 ± 1.0606i, satisfying (λ) = −0.1181 < −γ . It means that Eq. (27) is γ -stable when τ = 0. Hence, Eq. (27) is γ -stable for τ ∈ [0, 0.2595) ∪ (3.8285, 4], and it is not γ -stable when τ ∈ (0.2595, 3.8285). The system undergoes two times of γ -stability switches.

5 Concluding remarks Under the framework of local asymptotic stability of an equilibrium, the basic ideas used in the stability analysis for fractional-delay systems are similar to these used for time-delay systems of integer orders. This chapter introduces three effective algorithms for the stability test of a class of linear fractional-delay systems, and one analytical formula in closed form for the analysis of stability switches of a class of linear fractional-delay systems with delay-dependent coefficients. The results are generalizations of the ones for integer-order time-delay systems. One of the main advantages of the algorithms for stability test is free of constraints on the delays.

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The algorithms are presented for linear retarded fractional-delay systems only, but they work actually also for linear neutral fractional-delay systems.

Acknowledgment This work was supported by the NSF of China under Grants 10825207, 11372354.

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14

Sliding mode control for Markovian jumping systems with time delays

Hongyan Yang, Shen Yin Harbin Institute of Technology, Harbin, China

Chapter outline 1 Problem formulation............................................................................. 1.1 System description .................................................................. 1.2 Design goal ........................................................................... 2 Main results ...................................................................................... 2.1 Augmented system and SMO formulation........................................ 2.2 Derivation of the error dynamics................................................... 2.3 Stability analysis of the overall closed-loop plant ............................... 2.4 Reachability of the sliding mode surface ........................................ 2.5 The complete system synthesis algorithm........................................ 3 Simulation study ................................................................................. 3.1 Simulation setup..................................................................... 3.2 Results illustration and discussion ................................................ 4 Conclusions and future work .................................................................. References...........................................................................................

296 296 297 297 298 300 301 303 305 306 307 307 309 313

Markov jump systems (MJSs) have drawn increasing attention during the past decades since such classes of hybrid systems provide a powerful framework for modeling a variety of realistic systems including communication networks, power systems, aerospace industry, and so on. Fruitful technologies such as adaptive control, sliding mode control (SMC), fault tolerant control, and H∞ control have been employed to research on MJSs. It is worth mentioning that the SMC technology has been recognized as an excellent method because of its strong robustness. On the other hand, it should be noted that time delay is a source of instability which frequently occurs in various practical systems such as biological, mechanical, and economical processes. Many researchers have paid attention on dynamic systems with time delay and various results have been reported [1]. A lot of excellent approaches have been utilized to cope with time-delay phenomenon in the existing works such as Lyapunov-Krasovskii function approach, the Razumikhin Stability, Control and Application of Time-Delay Systems. https://doi.org/10.1016/B978-0-12-814928-7.00014-7 © 2019 Elsevier Inc. All rights reserved.

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lemma-based approach, and observer-based approach. For MJSs with time delay, the observer-based approach especially sliding mode observer (SMO)-based approach has attracted much attention of researchers [2]. This chapter focuses on the SMO-based robust control problem for a class of Itô stochastic MJSs with output disturbances and time delays. First, we will construct a proportional-derivative SMO and synthesize an SMO-based controller. Then, for the purpose of guaranteeing the stochastic stability for the closed-loop plant, we will propose Theorem 1. Furthermore, Theorem 2 is derived to ensure the reachability of the sliding mode surface. During these procedures, the main contributions of this chapter lie in the following twofolds: first, we obtain the controller gain and the observer coefficients by solving only one linear matrix inequality (LMI) problem which reduces the computational load and the design complexity than those methods with two LMIs. Second, by employing the proposed descriptor SMO, we can identify the disturbances directly. For the plants considered in this chapter, the following techniques are employed: • LMI technology • SMO technology The rest of this chapter is structured as below. In the following section, the system model is established and necessary assumptions are provided, and the design goal is clarified as well. In Section 2, the technical core is presented together with rigorous theoretical derivations and an applicable algorithm. In Section 3, the feasibility and stochastic stability are testified with an illustrative numerical example. Finally, the contributions are summarized in Section 4.

1 Problem formulation 1.1 System description During the past decades, Itô stochastic plants have played an important role in many industry and science fields, and thus much attention have been paid on such kind of systems. At this step, consider the following MJS with time delay, output disturbances, and Itô stochastic process defined in the fixed probability space (Ω, F, P): ⎧ dx(t) = [A(rt )x(t) + B(rt )u(t) + Ah (rt )x(t − h)]dt ⎪ ⎪ ⎨ + [Bw (rt )x(t) + Bh (rt )x(t − h)]dw(t) ⎪ x(t) = φ(t), t ∈ [−h, 0], r(0) = r0 ⎪ ⎩ ys (t) = C(rt )x(t) + Dd (rt )d(t).

(1)

In the earlier plant, A(rt ), B(rt ), C(rt ), and Dd (rt ) refer to the system matrices in appropriate dimension and will be represented by Ai , Bi , Ci , and Ddi , where rt = i, i ∈ S  {1, 2, . . . , N} for simplicity purpose. x(t) ∈ Rn , ys (t) ∈ Rp , u(t) ∈ Rm ,

2 Main results

and d(t) ∈ Rd denote the state variable, the measurement output, the input, and the bounded disturbances, respectively. {rt , t ≥ 0} represents a Markov process. The transition probability is defined as follows:  Pij = Pr{rt+ = j|rt = i} =

1 + πii  + o() πij  + o()

if i = j if i  = j

(2)

with  the generator matrix Π  [πij ], where i, j ∈ S, πij > 0, i = j, and πii = − j=i πij < 0;  > 0 and lim→0 o()/ = 0. Before the design procedure, we provide some definitions and assumptions at first: Assumption 1. The output disturbances d(t) ≤ rd with known constant rd > 0. Assumption 2. Bi and Ddi are full column rank matrices, and matrix Ci is in full row rank. Assumption 3. (Ai , Ci ) is observable. Definition 1 (Liu and Shi [3]). For arbitrary u(t) ≡ 0, r0 ∈ S, and x0 ∈ Rn , if the following condition holds: E

 ∞ 0

x(t)2 |x0 , r0 < ∞,

(3)

then plant (1) is said to be stochastic stable. Definition 2 (Skorokhod [4]). Let C2 (Rn ×S; R+ ) denote the family of V(x(t), i) on Rn × S which are twice continuously differentiable in x(t). For V ∈ C2 (Rn × S; R+ ), define an infinitesimal operator as follows: LV(x(t), i) = lim

1

→0+ 

[E{V(x(t + ), rt+ )|x(t), rt = i} − V(x(t), i)],

where functions V(x(t), i) are nonnegative.

1.2 Design goal In this chapter, to achieve effective robust control for time-delay MJSs with output disturbances, we aim to construct an observer-based controller. As shown in Fig. 1, we propose an architecture to illustrate the whole control plant to be designed. Three blocks (the SMO block in dotted box, the plant model block in dashdotted box, and the controller block in solid box) have different functionalities.

2 Main results Section 2.1 deals with the augmented system construction issues, corresponding to the dashdotted box in Fig. 1. The superiority of observer and controller design based on the augmented system is that the output disturbance signals are treated as state variables in the augmented system. In this case, the output disturbances can

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CHAPTER 14 Markovian jumping systems with time delays

FIG. 1 The architecture of the control system.

be estimated as long as the augmented state variables can be precisely estimated. This equivalent transformation is essential to the subsequent study. In Section 2.2, the error plant is constructed. Sections 2.3 and 2.4 provide Theorem 1 to ensure the performance of the observer and Theorem 2 to show the reachability and convergence of the sliding mode surface in finite time, respectively, corresponding to the dotted box in Fig. 1. Theorem 1 presents sufficient conditions for the sliding mode function to converge to zero, and subsequently, for the residual signals in the error system to converge to zeros. This part is significant for estimating the augmented state variables, or in other words, to reconstruct the state variables and the output disturbances. These variables are the basis of the observer-based controller. Section 2.3 proves the Itô stochastic stability via Theorem 1, and in the meantime provides a solution for the observer-based controller that guarantees the robust control capability. This section corresponds to the solid box and the entire closed-loop system. It is worth noting that the disturbance vector d(t) is reconstructed by employing our descriptor SMO. In Section 2.5, the design approaches are summarized and a complete robust control system synthesis algorithm is provided.

2.1 Augmented system and SMO formulation First, some augmented matrices and variables are defined for plant (1) in this section.  x(t) , x¯ (t)  Ddi d(t)

B¯i 



Bi

0p×m

 ,

(4)

2 Main results

E¯ 



 0n×p , 0p×p

In 0p×n



0 N¯ i  n×d , Ddi

C¯ i  Ci Ip , 

0n×p Ai , A¯i  0p×n −Ip 

Bwi 0n×p , B¯ wi  0p×n 0p×p



Ahi A¯ hi  0p×n

Bhi B¯ hi  0p×n

(5) (6)

 0n×p , 0p×p  0n×p . 0p×p

(7) (8)

Then, we can construct the following augmented plant: ⎧ ¯ ¯ ¯ ¯ ¯ ⎪ ⎨ Ed¯x(t)= [Ai x¯ (t) + Bi u(t) + Ahi x¯ (t − h) + Ni d(t)]dt +[B¯ wi x¯ (t) + B¯ hi x¯ (t − h)]dw ⎪ ⎩ ys (t) = C¯ i x¯ (t).

(9)

It is observed that plant (9) is singular, thus we hold

 E¯ rank ¯ = n + p = n¯ . Ci

(10)

By defining the following matrices and referring to Lemma 1 in Ref. [3], L¯ Di = 0p×n

WiT

T

,

(11)

S¯ i  (E¯ + L¯ Di C¯ i ),

(12)

where W = diag{l1 , l2 , . . . , lp } and li > 0 for i = 1, 2, . . . , p, we derive the following S¯ i and S¯ i−1 . S¯ i =

 0n×p , Wi



In Wi C i

S¯ i−1 =



In −Ci

 0n×p . Wi−1

(13)

By now, we can propose the following descriptor SMO: ⎧ ¯ ¯ ¯ ¯ ¯ ˆ ¯ ¯ ˆ ⎪ ⎨ Si d¯z(t) = [(Ai − Lpi Ci )x¯ (t) + (Ahi − Lhi Ci )x¯ (t − h) + B¯ i u(t) + L¯ pi ys (t) + L¯ hi ys (t − h) + L¯ si us (t)]dt ⎪ ⎩ xˆ¯ = z¯(t) + S¯ −1 L¯ Di ys (t),

(14)

i

where the observer gains L¯ Di ∈ Rn¯ ×p , L¯ si ∈ Rn¯ ×(a+d) , and L¯ pi , L¯ hi ∈ Rn¯ ×p will be designed in Section 2.3. In this observer, the estimation of x¯ (t) is xˆ¯  [ˆxT (t), dˆ T (t)]T and z¯(t)  [zTx (t), zTd (t)]T represents the intermediate variable. It can be seen that by selecting a suitable L¯ Di , we hold S¯ i  (E¯ + L¯ Di C¯ i ) is ˆ nonsingular. Besides, the real estimation of d(t) can be derived as (DT Ddi )−1 DT d(t) di

di

since (DTdi Ddi )−1 exists because of Ddi in full column rank. For analysis convenience, we adopt two lemmas as follows: ˜ C) ˜ with A˜ ∈ Rn×n , C˜ ∈ Rp×n , we hold Lemma 1 (Chen [5]). Consider a pair (A, the following two equivalent statements:

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(1) A˜ is stable. ˜ C) ˜ is observable, then A˜ T P˜ + P˜ A˜ = −C˜ T C˜ has a unique solution. (2) If (A, Lemma 2 (Liu and Shi [3]). For each i ∈ S, under Assumption 3, there exists a gain matrix L¯ pi such that S¯ i−1 (A¯ i − L¯ pi C¯ i ) is Hurwitz.

2.2 Derivation of the error dynamics We aim to construct the error plant in this section. First, by considering Eq. (14), we have S¯ i dxˆ¯ (t) = [(A¯ i − L¯ pi C¯ i )xˆ¯ (t) + (A¯ hi − L¯ hi C¯ i )xˆ¯ (t − h) + B¯ i u(t) + L¯ pi ys (t) + L¯ hi ys (t − h) + L¯ si us (t)]dt + L¯ Di dys (t).

(15)

At the same time, we add L¯ Di C¯ i d¯x(t) to both sides of Eq. (9) to obtain the following equation. S¯ i d¯x(t) = [A¯ i x¯ (t) + B¯ i u(t) + A¯ hi x¯ (t − h) + N¯ i f¯ (t)]dt + [B¯ wi x¯ (t) + B¯ hi x¯ (t − h)]dw + L¯ Di C¯ i x˙¯ (t).

(16)

Then, by defining eTx (t)  xˆ (t) − x(t),

(17)

e¯ (t)  xˆ¯ (t) − x¯ (t) = [eTx (t), eTd (t)]T ,

(18)

and subtracting Eq. (16) from Eq. (15), we hold the error plant as follows: S¯ i d¯e(t) = [(A¯ i − L¯ pi C¯ i )¯e(t) + (A¯ hi − L¯ hi C¯ i )¯e(t − h) + L¯ si us (t) − N¯ i f¯ (t)]dt − [B¯ wi x¯ (t) + B¯ hi x¯ (t − h)]dw.

(19)

Furthermore, to construct us (t), we firstly define the sliding mode surface: s(t, i) = N¯ iT S¯ i−T P¯ i e¯ (t)

(20)

for each i ∈ S with P¯ i satisfying N¯ iT S¯ i−T P¯ i = Hi C¯ i ,

P¯ i > 0,

(21)

where Hi ∈ Rp will be determined later. Till now, us (t) can be constructed: us (t) = −(rd + )sgn(s(t, i)) − 0.5

N  j=1

πij (N¯ iT S¯ i−T P¯ j S¯ i N¯ i )−1 s(t, i)sgn(δi ),

(22)

2 Main results

 ¯ T ¯ −T ¯ ¯ ¯ −1 where δi = s(t, i)T N j=1 πij (Ni Si Pj Si Ni ) s(t, i) rd ,  > 0 will be designed later and rd has been defined in Assumption 1. We formulate the overall closed-loop plant in the following form: ⎧ dx(t) = [Ai x(t) + Bi u(t) + Ahi x(t − h)]dt ⎪ ⎪ ⎪ ⎪ ⎨ + [Bwi x(t) + Bhi x(t − h)]dw(t)

⎪ d¯e(t) = S¯ i−1 [(A¯ i − L¯ pi C¯ i )¯e(t) + (A¯ hi − L¯ hi C¯ i )¯e(t − h) ⎪ ⎪ ⎪ ⎩ + L¯ si us (t) − N¯ i d(t)]dt − S¯ i−1 [B¯ wi x¯ (t) + B¯ hi x¯ (t − h)]dw(t).

(23)

2.3 Stability analysis of the overall closed-loop plant The observer-based input signal is u(t) = Ki xˆ (t) = K¯ i xˆ¯ (t),

(24)

with the gain matrix Ki designed such that Ai + Bi Ki is Hurwitz and K¯ i = [Ki , 0m×d ]. By adopting Eq. (24) to the original plant, we derive the following equation: dx(t) = [(Ai + Bi Ki )x(t) + Bi K¯ i e¯ (t) + Ahi x(t − h)]dt + [Bwi x(t) + Bhi x(t − h)]dw.

(25)

Thus the following closed-loop plant can be obtained: ⎧ dx(t) = [(Ai + Bi Ki )x(t) + Bi K¯ i e¯ (t) + Ahi x(t − h)]dt ⎪ ⎪ ⎪ ⎪ ⎨ + [Bwi x(t) + Bhi x(t − h)]dw(t) ⎪ d¯e(t) = S¯ i−1 [(A¯ i − L¯ pi C¯ i )¯e(t) + (A¯ hi − L¯ hi C¯ i )¯e(t − h) ⎪ ⎪ ⎪ ⎩ + L¯ si us (t) − N¯ i d(t)]dt − S¯ i−1 [B¯ wi x¯ (t) + B¯ hi x¯ (t − h)]dw(t),

(26)

where Fi = [Bi Ki , 0n×d ]. Till now, Theorem 1 of this chapter related to the stability condition of plant (26) is proposed. Theorem 1. By applying us (t) Eq. (22) to Eq. (19), plant (26) is stochastic stable, if there exist matrices Ki ∈ Rm×n and positive-definite matrices P¯ i , Q2 ∈ Rn¯ ׯn , R¯ i , Q1 ∈ Rn×n , Hi ∈ R(d)×p , and Y¯ i , Y¯ hi ∈ Rn¯ ×p for each i ∈ S, such that Eqs. (27), (28) hold: N¯ iT S¯ i−T P¯ i = Hi C¯ i , ⎡ Γ11i ⎢ ∗ ⎢ Ψi = ⎣ ∗ ∗

where

T¯ i = B¯ wi = BTwi 0n×p ,

T¯ hi = B¯ hi = BThi 0n×p ,

Γ12i Γ22i ∗ ∗

Γ13i 0 Γ33i ∗

⎤

0 0 ⎥ ⎥ < 0, Γ34i ⎦ −Q2

(27)

(28)

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CHAPTER 14 Markovian jumping systems with time delays

Γ11i = R¯ i (Ai + Bi Ki ) + (Ai + Bi Ki )T R¯ i +

N 

πij R¯ j + BTwi Ri Bwi + T¯ iT S¯ i−T P¯ i S¯ i−1 T¯ i ,

j=1

Γ12i = R¯ i Ahi + BTwi R¯ i Bhi + T¯ iT S¯ i−T P¯ i S¯ i−1 T¯ hi , T S¯ −T P ¯ i S¯ −1 T¯ hi , Γ22i = −Q1 + BThi R¯ i Bhi + T¯ hi i i

Γ13i = Ri Bi Ki 0n×d ,

Γ33i = P¯ i S¯ i−1 A¯ i − Y¯ i C¯ i + A¯ Ti S¯ i−T P¯ i − C¯ iT Y¯ iT +

N 

πij P¯ j + Q2 ,

j=1

Γ34i = P¯ i S¯ i−1 A¯ i − Y¯ hi C¯ i .

(29)

Besides, the gain matrices L¯ si and L¯ pi can be derived as follows: ¯ L¯ hi = S¯ i P¯ −1 i Yhi ,

(30)

¯ L¯ pi = S¯ i P¯ −1 i Yi ,

(31)

¯ −T T ¯ L¯ si = S¯ i P¯ −1 i Ci Hi = Ni .

(32)

Proof. Recalling the error plant (26), we consider the following Lyapunov function: V(t) = Ve (t) + Vx (t),

(33)

h with Ve (t) = e¯ T (t)P¯ i e¯ (t) + t−h e¯ T (t)Q2 e¯ (t) and Vx (t) = xT (t)R¯ i x(t) + h T t−h x (t)Q1 x(t). Then, the infinitesimal operators can be derived along Eq. (26) by Definition 2, ⎡ LVe (t) = e¯ T (t) ⎣P¯ i S¯ i−1 (A¯ i − L¯ pi C¯ i ) + (A¯ i − L¯ pi C¯ i )T S¯ i−T P¯ i +

N 

⎤ πij P¯ j + Q2 ⎦ e¯ (t)

j=1

+ 2¯eT (t)P¯ i S¯ i−1 [(A¯ i − L¯ hi C¯ i )¯e(t − h) + L¯ si us (t) − N¯ i f¯ (t)] + [B¯ wi x¯ (t) + B¯ hi x¯ (t − h)]T S¯ i−T P¯ i S¯ i−1 [B¯ wi x¯ (t) + B¯ hi x¯ (t − h)] − e¯ T (t − h)Q2 e¯ (t − h),

(34)

LVx (t) = xT (t)[R¯ i (Ai + Bi Ki ) + (Ai + Bi Ki )T R¯ i + Q1 ]x(t) + 2xT (t)R¯ i Ahi x(t − h) + 2xT (t)R¯ i B¯ i K¯ i e¯ (t) + [Bwi x(t) + Bhi x(t − h)]T R¯ i [Bwi x(t) + Bhi x(t − h)] ⎛ ⎞ N  πij R¯ j ⎠ x(t) − xT (t − h)Q1 x(t − h). (35) + xT (t) ⎝ j=1

2 Main results

By considering N¯ iT S¯ i−T P¯ i = Hi C¯ i and L¯ si = (P¯ i S¯ i−1 )−1 C¯ iT HiT , we have 2¯eT (t)P¯ i S¯ i−1 [L¯ si us (t) − N¯ i d(t)] = 2¯eT (t)C¯ iT HiT − 2¯eT (t)P¯ i S¯ i−1 N¯ i d(t) ≤ −2sT (t, i)us (t) + 2¯eT (t)C¯ iT HiT d(t) ≤ −2sT (t, i)(rd + )sgn(s(t, i)) − sT (t, i)

N 

πij (N¯ i S¯ i−T P¯ j S¯ i N¯ i )−1 s(t, i)

j=1

+ 2¯eT (t)C¯ iT HiT d(t)

⎛ ⎞ N  −T −1 πij (N¯ i S¯ i P¯ j S¯ i N¯ i ) ⎠ ≤ −2(rd + )s(t, i) + 2rd s(t, i) − s(t, i) ⎝ j=1

≤ −2s(t, i).

(36)

From Eq. (30), we obtain Y¯ i = P¯ i S¯ i−1 L¯ pi ,

L¯ si = N¯ i ,

Y¯ hi = P¯ i S¯ i−1 L¯ hi .

(37)

By substituting Eqs. (36), (37) into Eq. (33), we have LV(t) = LVx (t) + LVe (t) ≤ z(t)T Ψi z(t),

(38)



T where z(t) = xT (t) xT (t − h) e¯ T (t) e¯ T (t − h) . It can be observed that Ψi < 0 from Eq. (28), thus LV(t) < 0 can be reached. Therefore, the stochastic stability of plant (26) is concluded. The proof is completed. Besides, it should be noticed that condition (27) is not an LMI problem. We overcome this difficulty by converting Eq. (27) into a minimization problem as follows:

−βi In¯ ∗

 (N¯ iT S¯ i−T P¯ i − Hi C¯ i )T < 0. −Id

(39)

That is to say, finding the minimum βi in the earlier constraint is equivalent to solving Eq. (27). The details can be referred to Liu and Shi [3]. Remark 1. Theorem 1 provides a sufficient condition to stabilize the closed-loop plant (23) in the form of an LMI problem. The residual signal e¯ is suppressed to ensure the estimated xˆ¯ to converge to the true values x¯ . In this sense, the estimation precision of the output disturbances is guaranteed. Moreover, the observer gain matrices and the controller gain can be solved simultaneously.

2.4 Reachability of the sliding mode surface In this section, we will present the second theorem of this chapter to illustrate the validity of the proposed observer. The convergence and reachability of the sliding mode surface in finite time will be proved.

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Theorem 2. For each i ∈ S, if there exist matrices P¯ i , R¯ i which are positive definite and parameter matrix Hi such that formulas (27), (28) hold, then the sliding mode input us (t) guarantees the sliding motion to be driven to the sliding mode surfaces s(t, i) = 0 in finite time. Proof. Inspired by Kolmanovskii and Myshkis [6], we solve e¯ (t) as follows: e¯ (t) =

 t 0

S¯ i−1 [(A¯ i − L¯ pi C¯ i )¯e(τ ) + (A¯ hi − L¯ hi C¯ i )

× e¯ (τ − h) + L¯ si us (τ ) − N¯ i d(τ )]dτ  t S¯ i−1 [B¯ wi x¯ (τ ) + B¯ hi x¯ (τ − h)]dw(τ ). −

(40)

0

Taking Eq. (40) and s(t, i) into consideration, we have s(t, i) = N¯ iT S¯ i−T P¯ i S¯ i−1

 t 0

[(A¯ i − L¯ pi C¯ i )¯e(τ )

+ (A¯ hi − L¯ hi C¯ i )¯e(τ − h) + L¯ si us (τ )  t x¯ (τ )dw(τ ) − N¯ i d(τ )]dτ − Hi C¯ i S¯ i−1 B¯ wi 0  t x¯ (τ − h)dw(τ ). − Hi C¯ i S¯ i−1 B¯ hi

(41)

0

It is noticed that C¯ i S¯ i−1 B¯ wi = 0,

(42)

C¯ i S¯ i−1 B¯ hi = 0.

(43)

Considering Eqs. (42), (43) together, Eq. (41) can be changed as s(t, i) = N¯ iT S¯ i−T P¯ i S¯ i−1

 t 0

[(A¯ i − L¯ pi C¯ i )¯e(τ )

+ (A¯ hi − L¯ hi C¯ i )¯e(τ − h) + L¯ si us (τ ) − N¯ i d(τ )]dτ .

(44)

Then, we have s˙(t, i) = N¯ iT S¯ i−T P¯ i S¯ i−1 [(A¯ i − L¯ pi C¯ i )¯e(t) + (A¯ hi − L¯ hi C¯ i )¯e(t − h) + L¯ si us (t) − N¯ i d(t)].

(45)

Now, we select the Lyapunov function Vs (t) as follows: Vs (t) = 0.5sT (t, i)(N¯ iT S¯ i−T P¯ i S¯ i−1 N¯ i )−1 s(t, i).

(46)

For simplicity purpose, we define Gi = N¯ iT S¯ i−T . By reconsidering Eq. (45), we have LVs (t, i) = sT (t, i)(Gi P¯ i GTi )−1 Gi P¯ i S¯ i−1 × [(A¯ i − L¯ pi C¯ i )¯e(t) + (A¯ hi − L¯ hi C¯ i )¯e(t − h) + L¯ si us (t) − N¯ i d(t) + 0.5sT (t, i)

N  j=1

πij (N¯ iT S¯ i−T P¯ i S¯ i−1 N¯ i )−1 s(t, i)].

(47)

2 Main results

Recall that N¯ i S¯ i−T Pi = Hi C¯ i , the following can be derived: sT (t, i)(Gi P¯ i GTi )−1 Gi P¯ i S¯ i−1 × [L¯ si us (t) − N¯ i d(t)] = sT (t, i)(Gi P¯ i GTi )−1 Gi P¯ i S¯ i−1 N¯ i [us (t) − d(t)] = sT (t, i)(Gi P¯ i GTi )−1 Gi P¯ i S¯ i−1 N¯ i us (t) − sT (t, i)(Gi P¯ i GTi )−1 Gi P¯ i S¯ i−1 N¯ i f¯ (t) = sT (t, i)(Gi P¯ i GTi )−1 Gi P¯ i GTi us (t) − sT (t, i)(Gi P¯ i GTi )−1 Gi P¯ i GTi d(t) = sT (t, i)(us (t) − d(t)) − 0.5sT (t, i)

N 

¯ −1 s(t, i) πij (N¯ iT S¯ i−T P¯ i S¯ i−1 N) i

j=1

< −s(t, i) − 0.5sT (t, i)

N 

¯ −1 s(t, i). πij (N¯ iT S¯ i−T P¯ i S¯ i−1 N) i

(48)

j=1

Substituting Eq. (48) into Eq. (47), we have LVs (t, i) < −s(t, i) + s(t, i)(Gi P¯ i GTi )−1 Gi P¯ i S¯ i−1 × (A¯ i − L¯ pi C¯ i )¯e(t) + s(t, i)(Gi P¯ i GTi )−1 × Gi P¯ i S¯ i−1 (A¯ i − L¯ pi C¯ i )¯e(t − h).

By defining δi = (Gi P¯ i GTi )−1 Gi P¯ i S¯ i−1 (A¯ i − L¯ pi C¯ i ),

(49)

ρi = (Gi P¯ i GTi )−1 Gi P¯ i S¯ i−1 (A¯ hi − L¯ hi C¯ i ),

(50)

and recalling Eq. (49), one gets LVs (t, i) < −s(t, i)( − δi ¯e(t) − ρi ¯e(t − h)).

(51)

Then, for each i ∈ S, we define two domains as follows: Ωi (δi , ρi )  { − δi ¯e(t) − ρi ¯e(t − h) > 0}, Ω

s 

Ωi (δi , ρi ).

(52) (53)

i=1

From Eqs. (51), (53), we know that LVs (t, i) < 0 is in the domain Ω. Then, in Theorem 1, the stochastic stabilization of Eq. (26) has already been proved. At this time, the purpose that the trajectories of e¯ (t) enter Ω in finite time and remains there is further completed. Until now, we complete the proof.

2.5 The complete system synthesis algorithm The aforementioned two theorems provide solid theoretical derivation and rigorous analysis for SMO design and observer-based controller design for plant (1). Now we provide a complete controller design procedure and the robust control plant synthesis algorithm in the following part. Step 1 demonstrates the plant model block

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CHAPTER 14 Markovian jumping systems with time delays

(see dashdotted box); Steps {2, 4, 5} show the SMO block (see dotted box) and Step 3 focuses on the controller block (see solid box) in Fig. 1.

Algorithm 1 Observer-based robust control system synthesis Step 1. Construct the augmented plant as shown in Eq. (9). ¯ A¯ i , A¯ hi , B¯ i , C¯ i , B¯ wi , B¯ hi , and N¯ i According to Eqs. (4)–(8), calculate the coefficient matrices E, of the augmented plant, respectively. Step 2. Choose suitable gain matrix L¯ Di to ensure that S¯ i is nonsingular according to Eq. (12). Step 3. Controller design. Search the gain matrix K¯ i by solving an equivalent LMI problem shown in Theorem 1. Step 4. Establish the SMO as shown in Eq. (14). (i) Solve for observer-based coefficient matrices P¯ i , Hi , Y¯ i , and Y¯ hi via the equivalent LMI problem in Eqs. (28), (39). (ii) Calculate the SMO coefficient matrices according to Eqs. (31), (32), that is, the solutions are L¯ si , L¯ pi , and L¯ hi . (iii) Construct the sliding mode input us (t) and the sliding mode surface function s(t) according to Eqs. (20), (22). Step 5. Inspect the error signals e¯ (t). If e¯ (t) converges to zeros, the robust control system design is completed. Otherwise, back to Step 4 and tune the parameter in Eq. (22) to suppress the phenomenon of chattering.

Remark 2. As elaborated in the previous section, Theorem 1 presents an equivalent transformation from the closed-loop plant to an LMI problem where the controller gain K¯ i and the observer-related parameters can be simultaneously calculated. Likewise, from the robust control system design algorithm’s viewpoint, Steps 3 and 4(i) solve one LMI problem for two purposes at the same time. Compared with the similar theorems in some existing works, the integrated scheme from the newly derived Theorem 1 in this work reduces two LMI problems to one, which could simplify the design procedure and reduce the computational load. Remark 3. The reason why the controller design is carried out before observer design lies in the following. As a model-based robust control approach, the closedloop form (26) reveals its stability under the condition that (Ai + Bi Ki ) is Hurwitz. This indicates that Ki can be tuned independently from the observer although the controller is an observer-based one. The online configuration will not alter because the faulty conditions and perturbations have already been considered and dealt with by means of the augmentation of state variables.

3 Simulation study The results proposed in this chapter are testified by an example in this section. The validity of the derivations, the algorithm employed and intermediate results will be presented. Finally, the performance of the robust control system will be further discussed.

3 Simulation study

3.1 Simulation setup Recall MJS (1). ⎧ ⎨ dx(t) = [Ai x(t) + Bi u(t) + Ahi x(t − h)]dt + [Bwi x(t) + Bhi x(t − h)]dw(t) ⎩ ys (t) = Ci x(t) + Ddi d(t).

(54)

We select the plant parameters in the following form:



 −0.1 −5 0 , A2 = , −2 −0.4 −4



 0.4 0 3 0 = , Ah2 = , 0 0.1 0 2



= −1 0 , B2 = −1 0 ,



 −0.1 −0.1 −0.5 0 = , Bw2 = , 0 −0.2 −0.4 −0.4



 0.4 0 3 0 = , Bh2 = , 0 0.1 0 2



 0.1 −0.2 0.5 0.2 = , C2 = , 0.1 0.5 0.1 −0.1

T = Dd2 = 0.1 0.2 ,

 −0.6 0.6 = . 0.8 −0.8

A1 = Ah1 B1 Bw1 Bh1 C1 Dd1 Π

−1 0

From the aforementioned demonstration, we can obtain the mode transition and the mode switching which has been shown in Fig. 2.

3.2 Results illustration and discussion First, by choosing L¯ Di as follows: ⎡

1 ⎢0 L¯ D1 = L¯ D2 = ⎢ ⎣1 0

⎤T 0 1⎥ ⎥ , 0⎦ 1

(55)

we can obtain the corresponding matrices S¯ 1 and S¯ 2 ⎡

1.1000 ⎢ ¯S1 = ⎢0.1000 ⎣0.1000 0.1000

−0.2000 1.5000 −0.2000 0.5000

1.0000 0 1.0000 0

⎤ 0 1.0000⎥ ⎥, 0 ⎦ 1.0000

(56)

307

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CHAPTER 14 Markovian jumping systems with time delays

State jumps 3 r(t) 2.5

2

1.5

1

0.5

0

0

1

2

3

4

5 Time (s)

6

7

8

9

10

FIG. 2 Mode switch probability.

⎡

1.5000 ⎢0.1000 ⎢ S¯ 2 = ⎣ 0.5000 0.1000

0.2000 0.9000 0.2000 −0.1000

1.0000 0 1.0000 0

⎤ 0 1.0000⎥ ⎥, 0 ⎦ 1.0000

(57)

which are nonsingular. This satisfied the necessary condition in Step 2 of Algorithm 1. Furthermore, by the Matlab LMI Toolbox, we solve Ki as presented in Steps 3 and 4(i) K1 = −0.5000

0.1000 ,

K2 = 3.5153

0.5824 .

(58)

The eigenvalues of (A1 + B1 K1 ) and (A2 + B2 K2 ) are −0.5, −2 and −4.5153, −2. According to Steps 4(ii) and 4(iii), the parameters L¯ s1 , L¯ s2 , L¯ p1 , L¯ p2 , L¯ h1 ,L¯ h2 , and us (t) can be obtained. ⎡

0.5005 ⎢ ¯Lp1 = ⎢−0.5733 ⎣ 0.7366 0.1152

⎤ −0.5059 −0.6815⎥ ⎥, 0.4676 ⎦ 1.4317

⎡

56.3452 ⎢ ¯Lp2 = ⎢−25.8981 ⎣ 1.6099 −1.2210

⎤ 7.5808 −5.9611⎥ ⎥, −0.3155⎦ 1.0885

4 Conclusions and future work

⎡

⎤ 0.0659 −0.0672 ⎢ 0.3145 −0.2561⎥ ⎥, L¯ h1 = ⎢ ⎣−0.5584 0.0078 ⎦ −0.1094 −0.3727 ⎡ ⎤T −0.0000 ⎢−0.0000⎥ ⎥ L¯ s1 = L¯ s2 = ⎢ ⎣ 0.1000 ⎦ , 0.2000

⎡

−3.6133 ⎢ 0.9957 L¯ h2 = ⎢ ⎣−0.3293 0.0469

us (t) = −0.5001 × sgn(s(t)).

⎤ −2.2096 1.3183 ⎥ ⎥, −0.0068⎦ −0.3250

(59)

At present, we have completed the design procedure of the robust control system. Then, we can carry out the simulation by selecting the initial condition as x(0) = [1, −1]T and the time delay as h = 0.5. The disturbance signals are given as d(t) = 0.2 cos(10t) + 0.1 sin(10t) + 0.2.

(60)

Fig. 5 demonstrates the trajectories of s(t) which have achieved satisfying convergence. Meanwhile, the classical chattering phenomenon which can be suppressed by tuning the coefficient of us in the SMC has been shown. Fig. 6 presents the estimated and the true output disturbance. Its observed that the effectiveness of the proposed SMO has been verified. Finally, we present the estimation error in Fig. 7 and the state variables in Figs. 3, 4, and 8. It can be seen that the chattering phenomenon still exists in Fig. 7; however, the stability of the closed-loop plant is not affected. In addition, the stochastic stability of the developed approach in this chapter has been demonstrated by carrying out similar simulations for many times.

4 Conclusions and future work In this chapter, we have proposed the robust control problem for MJS with disturbances and time delay. First, by augmented algorithm, the original plant has been augmented into a novel plant where the disturbance vectors have been treated as state variables. Then, we proposed the developed observer which inherits the advantages of both the SMO and the PD observer. Furthermore, we provided Theorem 1 to show a sufficient condition to stabilize the closed-loop plant and Theorem 2 to guarantee the convergence and reachability of the sliding mode surface. Finally, we tested the proposed approach with an example. The main work mentioned earlier has certain merits for the modern industry. However, the proposed results are established under the condition that the upper bounds of the fault vectors are known and the plants considered are linear. Further investigation on more general and realistic forms that are employed to robotic arm BU-RSL-10 or mobile manipulators are expected in the future.

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CHAPTER 14 Markovian jumping systems with time delays

True VS estimated values of x1

1.2

x1(t)

1

ˆx1 (t) 0.8

0.6

0.4

0.2

0

−0.2

0

1

2

3

4

5 Time (s)

6

7

8

9

10

FIG. 3 Trajectories of the state variables x1 . True VS estimated values of x2

0 −0.1 −0.2 −0.3 −0.4 −0.5 −0.6 −0.7

x2(t) −0.8

ˆx2 (t)

−0.9 −1

0

1

2

3

FIG. 4 Trajectories of the state variables x2 .

4

5 Time (s)

6

7

8

9

10

4 Conclusions and future work

Sliding mode function values 1−1 1−2 0.015 s1(t) s2(t)

0.01

0.005

0

−0.005

−0.01

−0.015

0

1

2

3

4

5 Time (s)

6

7

8

9

10

8

9

10

FIG. 5 Trajectories of s(t) when us (t) = −0.5001sgn(s(t)). True VS estimated values of d(t)

0.25 d(t) 0.2

ˆ d(t)

0.15 0.1 0.05 0 −0.05 −0.1 −0.15 −0.2 −0.25

0

1

2

3

4

5 Time (s)

FIG. 6 Output disturbance estimation performance.

6

7

311

312

CHAPTER 14 Markovian jumping systems with time delays

–3

10

ALL error terms

x 10

¯e1 (t) 8

¯e2 (t)

6 4 2 0 −2 −4 −6 −8

0

1

2

3

4

5 Time (s)

6

7

8

10

9

FIG. 7 Residual signals of the state variables. 1.5 x1(t) x2(t) 1

0.5

0

−0.5

−1

0

1

2

3

FIG. 8 Trajectories of the state variables.

4

5 Time (s)

6

7

8

9

10

References

References [1] H. Yang, Y. Jiang, S. Yin, Fault-tolerant control of time-delay Markov jump systems with Itô stochastic process and output disturbance based on sliding mode observer, IEEE Trans. Ind. Inf. (2018), https://doi.org/10.1109/TCYB.2016.2574754. [2] S. Yin, H. Yang, O. Kaynak, Sliding mode observer-based FTC for Markovian jump systems with actuator and sensor faults, IEEE Trans. Autom. Control 62 (7) (2017) 3551–3558. [3] M. Liu, P. Shi, Sensor fault estimation and tolerant control for Ito stochastic systems with a descriptor sliding mode approach, Automatica 49 (2013) 1242–1250. [4] A.V. Skorokhod, Asymptotic Methods in the Theory of Stochastic Differential Equations, American Mathematical Society, 1989. [5] C. Chen, Linear System Theory and Design, Oxford University Press, New York, 1999. [6] V.B. Kolmanovskii, A. Myshkis, Applied Theory of Functional Differential Equations, Kluwer, Dordrecht, The Netherlands, 1992.

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Networked predictive control for linear systems with communication delay in the feedback channel

15

Jian Suna,b , Jie Chena,b a School

b Key

of Automation, Beijing Institute of Technology, Beijing, China Laboratory of Intelligent Control and Decision of Complex Systems, Beijing Institute of Technology, Beijing, China

Chapter outline 1 Introduction....................................................................................... 315 2 Preliminaries ..................................................................................... 317 3 Networked predictive control.................................................................. 319 3.1 NPC: State feedback case .......................................................... 319 3.2 NPC: Observer-based output feedback case ..................................... 319 4 Stability analysis ................................................................................ 320 4.1 NPC: State feedback case .......................................................... 320 4.2 NPC: Observer-based output feedback case ..................................... 322 5 A numerical example............................................................................ 325 5.1 State feedback case ................................................................. 326 5.2 Observer-based output feedback case ............................................ 328 6 Conclusions....................................................................................... 329 References........................................................................................... 329

1 Introduction In the past few decades, networked control systems (NCSs) have received much attention and having being a hot research topic. Compared with the traditional control system which has a point-to-point structure, NCSs have many advantages such as ease of maintenance, efficient resource sharing, and energy saving [1,2]. However, NCSs have several drawbacks coming from a real-time network’s introduction, such as network-induced delay, data disorder, data dropout, and quantized error, which bring great challenges to conventional control theories built on ideal assumptions. Stability, Control and Application of Time-Delay Systems. https://doi.org/10.1016/B978-0-12-814928-7.00015-9 © 2019 Elsevier Inc. All rights reserved.

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CHAPTER 15 Networked predictive control for linear systems

Numerous control methodologies have been developed to solve these problems. For example, it is quite natural to model NCSs as time-delay systems and rich results for time-delay systems can be used to stability analysis and controller design for NCSs [3–6]. A switched system approach was adopted to solve the problems of the output feedback controller design, an exponential stabilization and disturbance attenuation of NCSs based on the switched Lyapunov function approach as well as the average dwell-time approach [7–9]. When the network-induced delay is assumed in following a Markov chain, the Markovian jump system approach was applied to NCSs [10–12]. In addition to the earlier methods, adaptive control methods, fuzzy control methods, and neural network control methods were also utilized to realize the distributed coordination the fault detection, and the scheduling of NCSs [13–15]. For some other methodologies, please refer to some good survey papers [16–19] and references therein. Generally speaking, the existing methods can be classified into two categories. One is to design a controller that is sufficiently robust to network constraints and the other is to design a controller to compensate for them actively. In this chapter, the second class of methods are concerned. Recently, a new model-based method called networked predictive control (NPC) has been proposed to actively compensate for network-induced delay and data dropout [20–31]. This method has a good ability to compensate for the network constraints and can yield a very good control performance that has been confirmed by a great deal of simulations and experiments. It is very interesting that NPC can obtain a control performance similar to local control (i.e., there is no network in the system). However, the problem of stability analysis of an NPC system is a challenging topic which has not been completely solved. In existing results, only some sufficient stability conditions have been obtained using the well-known common Lyapunov function approach [20–27] or the switched Lyapunov function approach [28]. To the best of authors’ knowledge, there is no necessary and sufficient stability condition for an NPC system when taking both the network-induced delay and data dropout into consideration. In addition, in the results mentioned earlier, the network-induced delay is often assumed to be measured accurately, which is not a easy task in practice. Due to the network protocols and time synchronization scheme and some other reasons, the measurement of the network-induced delay often has some errors. When there is a mismatch in the network-induced delay, what is the performance of NPC method and how to analyze the stability of the closed-loop system are two questions to be answered. In this chapter, a state feedback networked predictive controller and an observerbased output feedback networked predictive controller are both designed to compensate for the network-induced delay actively. For the state feedback case, a necessary and sufficient stability condition for the closed-loop system is derived when there is no delay mismatch and a sufficient stability condition for the closedloop system is derived when there is a delay mismatch. For the observer-based output feedback case, a new formulation of the NPC method is presented. A necessary and sufficient stability condition is obtained when there is no delay mismatch and a sufficient stability condition for the closed-loop system is derived when there is

2 Preliminaries

a delay mismatch. Finally, some simulations are given to illustrate the efficacy and effectiveness of the proposed methods.

2 Preliminaries Consider the following discrete-time plant x(k + 1) = Ax(k) + Bu(k) y(k) = Cx(k)

(1) (2)

where x(k) ∈ Rn , u(k) ∈ Rm , and y(k) ∈ Rp are the state vector, the control input vector, and the output vector, respectively. A, B, and C are constant matrices with appropriate dimensions. For the sake of simplicity, but without loss of generality, we only consider an NCS with random delay and data dropout in the feedback channel. The following assumptions are made [22,25]. Assumption 1. Network-induced delay d(k) is bounded by τ1 h ≤ d(k) ≤ τ2 h with h being the sampling interval. The number of successive data packet dropouts is not larger than N with τ2 ≥ τ1 ≥ 0, and N ≥ 0 being integers. Assumption 2. All signals in the system are transmitted with time-stamps and all components in the system are synchronized. Let {tk = kh|k = 1, 2, . . .} be the sampling instants. The sensor samples the output of the plant at tk and sends the output of the plant y(tk ) together with its time-stamp to the controller through a network. During the transmission, a so-called “packet disorder” will happen, which means that data packets sent earlier (later) arrive later (earlier). In addition, no data packet or more than one data packet will reach the controller node in a sampling interval. To deal with such a problem, a logic zeroorder-hold (ZOH) [32,33] is introduced at the controller node. The mechanism of the logic ZOH can be described as follows. Logic ZOH. Step 1: Let k = 0, 0 = 0, and t0 = 0. Step 2: At the sampling instant tk , ZOH updates its output by y(t) = y(k h) for tk ≤ t ≤ tk+1 ; Step 3: If there are some data packets reaching the ZOH during tk ≤ t ≤ tk+1 , compare their time-stamps and denote the largest one by ϑ. If ϑ > k , then the ZOH stores y(ϑh) and lets k+1 = ϑ; Step 4: Let k = k + 1 and go to Step 2. A data packet successfully transmitted from the sensor to the ZOH and subsequently stored by the ZOH is called an effective data packet. The time-stamp sequence of the effective data packets is denoted by {Ti h|i = 1, 2, . . .}. The time of an effective data packet is used to update the ZOH is called an updating instant. The updating instant sequence is denoted by {Si h|i = 1, 2, . . .}. It is easy to see that the controller use y(Ti h) to develop the control law at the updating instant Si h. To make the earlier definitions clear, an illustrative example is shown in Fig. 1. It is easy to see that y(h) reach the ZOH on a instant between 2h and 3h and there are no other measurements arriving at the ZOH. Therefore, we can obtain S1 = 3

317

318

CHAPTER 15 Networked predictive control for linear systems

1

3

2

2

4

3

5

1

6

2

7

4

8

3

9

4

FIG. 1 An example to show {Ti } and {Si } of the logic ZOH.

and T1 = 1. During 2h and 4h, there are two measurements y(2h) and y(3h) reaching the ZOH, and “packet disorder” happens. Since y(3h) is newer than y(2h), y(2h) is discarded. Therefore, we have S2 = 5 and T2 = 3. y(5h) is dropped out. During 7h and 8h, only y(7h) reaches the ZOH, and hence T2 = 7 and S3 = 8. Between 8h and 9h, there are three measurements arriving at the ZOH. Among them, y(8h) is the most recent one and the other two are discarded. Therefore, T4 = 8 and S4 = 9. In the following sections, y(kh), k = 1, 2, . . . is denoted by y(k) in the absence of ambiguity. Define τ (k) = k − Ti , k ∈ {Si , Si + 1, . . . , Si+1 − 1}. τ (k) is called actual delay. Clearly, x(k − τ (k)) is the most recent measurement signal available at the sampling time k. For Si , Ti , and τ (k), the following lemma can be obtained which plays an important role in the derivation of the main results. Lemma 1. For Si and Ti , denote θi = Si+1 − Si , ηi = Ti+1 − Ti , the following inequalities hold 1 ≤ θi ≤ τ2 − τ1 + N + 1

(3)

1 ≤ ηi ≤ τ2 − τ1 + N + 1 τ1 ≤ τ (k) ≤ τ2 + N

(4) (5)

where τ2 , τ1 , and N are as defined in Assumption 1. Proof. Clearly, the minimum of θi and ηi is 1 and the minimum of τ (k) is τ1 . Since the maximum of successive data packet dropouts is N, Ti+1 − Ti reaches its maximum only if (i) y(Ti + 1), y(Ti + 2), . . ., y(Ti + N) are all dropped out; (ii) y(Ti + N + 1) suffers the maximum delay; (iii) y(Ti + N + 1), y(Ti + N + 2), . . ., y(Ti + N + τ2 − τ1 ) are not effective data packets and y(Ti + N + τ2 − τ1 + 1) is stored by the ZOH as an effective data packet. In such cases, Ti+1 − Ti = τ2 − τ1 + N + 1. Similarly, we can prove the maximum of Si+1 − Si is τ2 − τ1 + N + 1 and the maximum of τ (k) is τ2 + N.

3 Networked predictive control

3 Networked predictive control In this section, two networked predictive controllers are introduced.

3.1 NPC: State feedback case It is assumed that all the states of the system are available. From discussions in the previous section, the most recent measurement signal available at the sampling time k is x(k − τ (k)). The following NPC method can be used to compensate for the network-induced delay and data dropout actively [34]. Step 1. Based on the most recent measurement signal x(k − τ (k)), the current state of the system x(k) can be obtained iteratively x(k) = Ax(k − 1) + Bu(k − 1) = A2 x(k − 2) + ABu(k − 2) + Bu(k − 1) = A3 x(k − 3) + A2 Bu(k − 3) + ABu(k − 2) + Bu(k − 1) .. . = Aτ (k) x(k − τ (k)) +

τ (k)

Ai−1 Bu(k − i)

(6)

i=1

Step 2. The following state feedback control law is obtained u(k) = Kx(k) = KAτ (k) x(k − τ (k)) + K

τ (k)

Ai−1 Bu(k − i)

(7)

i=1

where K is the controller gain matrix.

3.2 NPC: Observer-based output feedback case When some states of the system are not available, the earlier state feedback NPC cannot be used. Clearly, the most recent measurement signal available at the sampling time k is y(k − τ (k)). The following observer-based NPC method can be used to compensate for the network-induced delay and data dropout. Step 1: Based on the received measurement signal y(k − τ (k)), the current state of the system can be predicted by the following iteration: xˆ (k − τ (k) + 1) = Aˆx(k − τ (k)) + Bu(k − τ (k)h)   + L y(k − τ (k)) − Cˆx(k − τ (k)) xˆ (k − τ (k) + 2) = Aˆx(k − τ (k) + 1) + Bu(k − τ (k) + 1)

319

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CHAPTER 15 Networked predictive control for linear systems

xˆ (k − τ (k) + 3) = Aˆx(k − τ (k) + 2) + Bu(k − τ (k) + 2) .. . xˆ (k) = Aˆx(k − 1) + Bu(k − 1) = Aτ (k) xˆ (k − τ (k)) +

τ (k)

Ai−1 Bu(k − i)

i=1   τ (k)−1 L y(k − τ (k)) − Cˆx(k − τ (k)) +A

(8)

Step 2: The following observer-based output feedback control law is applied to the plant u(k) = K xˆ (k)

(9)

where L is the observer gain matrix, and K is the controller gain matrix. K and L can be determined by some standard methods such as the pole assignment method.

4 Stability analysis In this section, some stability conditions for NPC systems are presented.

4.1 NPC: State feedback case In this section, we assumed that actual delay τ (k) is measured as τ˜ (k) with τ (k) − τ˜ (k) not being always 0. It means that there is a mismatch between τ (k) and its measurement τ˜ (k)(k). Furthermore, it is assumed that τ˜ (k) satisfies d1 ≤ τ˜ (k) ≤ d2 where d1 ≥ 0 and d2 ≥ 0 are both integers. When the actual delay τ (k) is measured as τ˜ (k), Eqs. (6), (7) become xˆ (k) = Aτ˜ (k) x(k − τ (k)) +

τ ˜ (k)

Ai−1 Bu(k − i)

(10)

i=1

u(k) = KAτ˜ (k) x(k − τ (k)) + K

τ ˜ (k)

Ai−1 Bu(k − i)

(11)

i=1

Substitute Eq. (11) into Eq. (1) and one can obtain x(k + 1) = Ax(k) + BKAτ˜ (k) x(k − τ (k)) + BK

τ ˜ (k)

Ai−1 Bu(k − i)

(12)

i=1

Furthermore, τ ˜ (k) i=1

Ai−1 Bu(k − i) = x(k) − Aτ˜ (k) x(k − τ˜ (k))

(13)

4 Stability analysis

Substituting Eq. (13) into Eq. (12), we can obtain the closed-loop system described by x(k + 1) = (A + BK)x(k) + BKAτ˜ (k) x(k − τ (k)) − BKAτ˜ (k) x(k − τ˜ (k))

(14)

It is clear that system (14) is a system with two time delays and it involves the difference between x(k−τ (k)) and x(k−τ˜ (k)). Clearly, we can see that if τ (k) = τ˜ (k), Eq. (14) turns into x(k + 1) = (A + BK)x(k), which means that the network-induced delay is compensated completely. Therefore, if there is no delay mismatch, we have the following theorem. Theorem 1. When there is no delay mismatch, the NPC systems are asymptotically stable for any delay and data dropout satisfying Assumption 1 if and only if eigenvalues of A + BK are within the unit circle. When a delay mismatch exists, we denote M = max{τ2 + N,d2 } and define a new variable ξ T (k) = xT (k) xT (k − 1) xT (k − 2) · · · xT (k − M) , and Eq. (14) can be reformulated as ξ(k + 1) = Ξσ (k), (k) ξ(k)

(15)

where {σ (k), (k)} ∈ I1 × I2 , I1 = {1, 2, . . . , τ2 + N}, I2 = {1, 2, . . . , d2 } and ⎡

⎤

i−1

⎢ A + BK ⎢ ⎢ Ξi,j = ⎢ ⎢ ⎣ ⎡

  0 ··· 0

BKAj

0

···

0

Mn×Mn

⎤

j−1

⎢ 0 ⎢ ⎢ +⎢ ⎢ ⎣

  0 ··· 0

−BKAj

0

···

I

Mn×Mn

0 0 .. . 0

0 0 .. . 0

⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎦

Using the switched Lyapunov function approach [35], a sufficient stability condition for system (15) is derived in the following theorem. Theorem 2. For the given controller gain matrix K, the NPC system is asymptotically stable for any delay and data dropout satisfying Assumption 1 if there exist matrices Pi,j > 0 and any matrices Gi,j such that 

−Pi,j Gi,j Ξi,j

∗ Pl,k − Gi,j − GTi,j

 0, it is clear that x(Si + γ ) = (A + BK)γ x(Si ) −

γ 

(A + BK)j−1 BKAdi +γ −j−1 (A − LC)e(Ti |Ti−1 )

(25)

j=1

Therefore, we can obtain x(Si+1 ) = (A + BK)θi x(Si ) −

θi 

(A + BK)j−1 BKAdi +θi −j−1 (A − LC)e(Ti |Ti−1 )

(26)

j=1

 Define a new vector ξi =

x(Si ) e(Ti |Ti−1 )

 , the closed-loop system can be

described as ξi+1 = Ξi ξi

where

 Ξi =

(A + BK)θi 0

Ξi12 η −1 i A (A − LC)

(27)



i (A + BK)j−1 BKAdi +θi −j−1 (A − LC). with Ξi12 = − θj=1 It is clear that Ξi switches according to θi , ηi , and di . Therefore, the closed-loop system (29) is a switched system. According to the switched system theory [36], a switched linear system with a block upper-triangular structure is asymptotically stable if and only if each of its block diagonal subsystems is asymptotically stable. Therefore, the closed-loop system (29) is asymptotically stable if and only if eigenvalues of (A + BK)θi and Aηi −1 (A − LC) are within the unit circle. Clearly, eigenvalues of (A + BK)θi are within the unit circle is equivalent to eigenvalues of A + BK are within the unit circle. Therefore, we have the following theorem.

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CHAPTER 15 Networked predictive control for linear systems

Theorem 3. The NPC system is asymptotically stable for any delay and data dropout satisfying Assumption 1 if and only if eigenvalues of A + BK and Aκ−1 (A − LC), for all 1 ≤ κ ≤ τ2 − τ1 + N + 1 with κ being integers, are within the unit circle. Remark 1. From Theorem 3, it can be seen that the state feedback controller and the state observer can be designed independently by guaranteeing eigenvalues of A + BK and Aκ−1 (A − LC), for all 1 ≤ κ ≤ τ2 − τ1 + N + 1, are within the unit circle. This property is in accordance with the separation principle.

4.2.2 With delay mismatch When the network-induced delay τ (k) is measured as τ˜ (k), Eq. (8) turns into xˆ (k) = Aτ˜ (k) xˆ (k − τ˜ (k)) +

τ ˜ (k)

Ai−1 Bu(k − i)

i=1   τ ˜ (k)−1 L y(k − τ (k)) − Cˆx(k − τ˜ (k)) +A

(28)

It is easy to see x(k) = Aτ˜ (k) x(k − τ˜ (k)) +

τ ˜ (k)

Ai−1 Bu(k − i)

(29)

i=1

Define e(k) = x(k) − xˆ (k), from Eqs. (28), (29), one can obtain   e(k) = Aτ˜ (k) ex(k − τ˜ (k)) − Aτ˜ (k)−1 LC x(k − τ (k)) − xˆ (k − τ˜ (k)) = Aτ˜ (k)−1 (A − LC)e(k − τ˜ (k)) − Aτ˜ (k)−1 LC [x(k − τ (k)) − x(k − τ˜ (k))]

(30)

x(k + 1) = (A + BK)x(k) − BKe(k)

(31)

and     Define ζ T (k) = eT (k) eT (k − 1) · · · eT (k − d2 ) and ψ T (k) = ξ T (k) ζ T (k) , one can obtain ψ(k + 1) = Φσ (k), (k) ψ(k)

where {σ (k), (k)} ∈ I1 × I2 , I1 = {1, 2, . . . , τ2 + N}, I2 = {1, 2, . . . , d2 } and  Φi,j = ⎡

Υ1 Υ3

Υ2 Υ4

⎢ A + BK ⎢ ⎢ Υ1 = ⎢ ⎢ ⎣

 ⎤

M−1

  0 ··· 0

I

Mn×Mn

0 0 .. . 0

⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(32)

5 A numerical example

⎡

 0 0 .. . 0

⎢ −BK ⎢ ⎢ Υ2 = ⎢ 0 ⎢ . ⎣ .. 0 ⎡ j−1   ⎢ 0 ··· 0 ⎢ Υ3 = ⎢ ⎣ ⎡

d

2 ··· ··· .. . ···

⎤

 0 0 .. . 0

⎥ ⎥ ⎥ ⎥ ⎥ ⎦

0

⎡

⎤

Aj−1 LC

0

···

d2 n×d2 n

0 .. . 0

Aj−1 (A − LC)

I

d2 n×d2 n

0

···

Ai−1 LC

0

d2 n×d2 n

⎤

j−1

  ⎢ 0 ··· 0 ⎢ Υ4 = ⎢ ⎣

⎤

i−1

  ⎥ ⎢ 0 ··· 0 ⎥ ⎢ ⎥+⎢ ⎦ ⎣

0 .. . 0

0

···

0 .. . 0

⎥ ⎥ ⎥ ⎦

⎥ ⎥ ⎥ ⎦

Using the switched Lyapunov function approach [35], a sufficient stability condition for system (32) is derived in the following theorem. Theorem 4. For given controller gain matrix K, the NPC systems are asymptotically stable for any delay and data dropout satisfying Assumption 1 if there exist matrices Pi,j > 0 and any matrices Gi,j such that 

−Pi,j Gi,j Φi,j

∗ Pl,k − Gi,j − GTi,j

 0 (τr > τp ) .................................................................... 339 3.3 θ < 0 (τr < τp ) ..................................................................... 342 4 Comparison against computer simulations .................................................. 342 4.1 θ > 0 (τr > τp ) ..................................................................... 343 4.2 θ < 0 (τr < τp ) ..................................................................... 343 5 Discussion ........................................................................................ 344 Appendix ............................................................................................. 345 Ruin probability ........................................................................... 345 Expected duration of the betting........................................................ 346 Acknowledgments .................................................................................. 346 References........................................................................................... 347

1 Introduction The origin of the concept of probability is closely connected to games or gamblings using dice (astragali), whose outcome depends on chance. On the other hand, the concept of delays is more common in ordinary livings. Although both concepts are rather intricate when treated mathematically, we will attempt here combining them through a simple mathematical model. The gambler’s ruin is a classic problem in probability theories [1]. A gambler who has an initial asset takes on betting under certain probabilities of win and loss, until he is broken or reaches to a specific asset level. One of the simplest models can be described as a restricted random walk with two absorbing boundaries. The position Stability, Control and Application of Time-Delay Systems. https://doi.org/10.1016/B978-0-12-814928-7.00016-0 © 2019 Elsevier Inc. All rights reserved.

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of the walker indicates his asset and the walker can probabilistically gain or lose a unit of his asset at each time step, and moves accordingly until he reaches either boundary. This model has been studied extensively and various extensions are made. Inclusion of nonmoving time steps, and extensions to n-players gambler’s ruin are such examples [2–5]. Also, an interesting application of gambler’s ruin is found for a development of alternative quantum mechanics. In this proposed theory, a superposed quantum state collapses to one of the eigenstates through a competition of amplitudes based on gambler’s ruin [6–8]. We will propose and investigate yet another extension of this problem in this chapter, which we term as “delayed gambler’s ruin” (DRG). The main feature of this model is that it includes delays in the gain or loss of a unit in the gambler’s asset. This reflects that, in reality, payments and/or incomes often do not take place immediately at the time of corresponding events, such as a purchase with a credit card. Our proposed model, thus, moves according to the past results of gambling. Delays in dynamics have also been studied over the past 50 years, often with feedback control systems [9,10]. It has been shown that even a simple first-order ordinary differential equation can show rather complex behaviors with delays. They are called “delay differential equations.” Stochastic elements are further introduced onto both dynamical models by “stochastic delay differential equations” [11,12] or probabilistic models by “delayed random walks” [13,14]. The DRG we propose here can be thought of one example of these models incorporating stochasticity and delay. The outline of this chapter is as follows. First, we review simple random walks on which the gambler’s ruin is based. Basic properties of simple random walks, as well as related concepts, are described. Then, the standard gambler’s ruin is explained. Finally, our formulation of the DRG is introduced. We investigate the model with a particular focus on how delays affect the probability of ruin. It will be shown both by analysis and by computer simulations that an approximation using averaging can describe the behavior of the model for small delays.

2 Simple random walks We, here, describe simple random walks. Assume that we are tossing a coin which has a property of resulting in “head” with probability p and “tail” with q = 1−p. Starting with S0 = 0 point, we give +1 point for the case of head and −1 point for tail. Thus, after repeating coin tossing n times, the point is Sn , which is an accumulation of points ±1. This Sn is defined as simple random walks. In slightly more mathematically toned descriptions, we consider independent stochastic variables Xi (i = 1, 2, 3, . . .) each of which takes two values ±1 with the following probabilities: Xi = +1 with p Xi = −1 with q = 1 − p

2 Simple random walks

p

q

–10

–5

+5

0

+10

FIG. 1 Schematic representation of a simple random walk.

We define a simple random walk Sn as a sum of Xi : Sn = X1 + X2 + · · · + Xn The initial condition S0 needs to be specified. For now, we assume S0 = 0. The name “random walks” comes from a representation of this definition by a drunkard stepping right (+1) and left (−1) with uncertainty. Starting from the origin, his position after n steps is given as Sn (Fig. 1). We now list basic properties of simple random walks. (a) The average and variance for each step are given as follows: E[Xi ] = p − q V[Xi ] = 4pq

(b) By statistical independence of Xi , the average and variance of the random walk at n steps are given as follows: E[Sn ] = n(p − q) V[Sn ] = 4npq

(c) We consider the probability distribution P(Sn = s) for the random walker to be at the position s after n steps. First, by defining j+ : total number of + 1 steps j− : total number of − 1 steps

Then we have n = j+ + j− ,

s = j+ − j −

so that n+s 2 n−s j− = 2

j+ =

335

336

CHAPTER 16 Delayed gambler’s ruin

With these, the probability distribution is given as follows: P(Sn = s) =

  n

j+ j− j+ p q =

 n   n+s   n−s  (−n ≤ s ≤ n, and n + s is even) n+s p 2 q 2 2

0

(other cases)

When the probabilities of taking a step to the right and left are equal p = q = 12 , it is called simple symmetric random walks. In terms of coin tossing, we are now using a fair coin which gives head and tail with equal probability. In the case of simple symmetric random walks, its properties are simpler. (a) The average and variance for each step: E[Xi ] = 0 V[Xi ] = 1

(b) The average and variance at n steps: E[Sn ] = 0 V[Sn ] = n

(c) The probability distribution P(Sn = s): P(Sn = s) =

  n  1 n j+

0

2

=

 n  1 n n+s 2

2

(−n ≤ s ≤ n, and n + s is even) (other cases)

2.1 Gambler’s ruin The simple random walks, which we discussed in the previous section, did not have boundaries. With the number of steps n increasing, there are more chances that the random walker can be found further away from the origin. We now consider a case where the range of positions random walkers can move around is limited. This is done by placing boundaries (walls). From here, we assume that there are two absorbing boundaries at the origin s = 0 and at s = A(> 0). A random walker which starts at 0 < S0 < A can move in the interval until it reaches to either of the two boundaries. At that point, the walker is absorbed and the walk is over. Gambler’s ruin is an analogy of this restricted random walks with two absorbing barriers to a gambling situation. A gambler attends a gamble with the initial asset of S0 = s points. At each bet, he either wins one point or loses one point with a given probability. He ends his betting either when his assets become zero (“broken”) or reach his intended level A.

2 Simple random walks

We now define some notations to analyze this problem. p: The probability of the gambler’s winning a point. (We also set q = 1 − p as the probability of losing.) • Ut : Gambler’s asset after t betting. • PA (s): Probability that a gambler with the initial asset of s points to become broken. • Xt = ±1: The result of betting at t. •

By mapping this problem to restricted simple random walks with each step as Xt and with two absorbing boundaries at 0, A as discussed earlier, following results have been obtained. (Details are given in “Appendix” section.)

PA (s) =

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩



A−s A  A  s q − qp p  A q −1 p

p = q = 12



(p  = q)

We note that as the initial asset s becomes smaller, PA (s) becomes larger. This is natural as the less asset one has at the beginning, more likely that he will become broken (Fig. 2).

1.0 (i) (h) (g)

0.8 (f)

PA(s)

0.6 (e)

0.4 (d) (c)

0.2 (b) (a)

0

20

40

s

60

80

100

FIG. 2 Ruin probabilities PA (s). We set A = 100 and (a) p = 0.6, q = 0.4, (b) p = 0.55, q = 0.45, (c) p = 0.52, q = 0.48, (d) p = 0.51, q = 0.49, (e) p = 0.5, q = 0.5, (f) p = 0.49, q = 0.51, (g) p = 0.48, q = 0.52, (h) p = 0.45, q = 0.55, (i) p = 0.4, q = 0.6.

337

338

CHAPTER 16 Delayed gambler’s ruin

3 Delayed gambler’s ruin We will focus on how this probability of ruin is affected by the inclusion of delay. Two delays are introduced into the earlier gambler’s ruin model: • τp : delay in payment of a point, • τr : delay in receipt of a point. At each bet, the paying or receiving of the point is deferred with the earlier delays. τ ,τ We also denote the probability of ruin in the DRG as PAr p (s), which we will focus on the following analysis. We also denote the stopping time, the time duration of the gambler’s betting (i.e., the time between the beginning to the end of his betting, either by broken or by reaching A), as Tτr ,τp . It turns out the important parameter in analyzing this model is the difference between these two delays: • θ = τr − τp : the difference of delay in payment and receipt. With the earlier setup, we now start our analysis by considering different cases of θ . In the following, we assume that the initial asset is further away from the boundary than the difference between delays. A − s >| θ |

3.1 θ = 0

and

s >| θ |

(τr = τp )

We first consider the case of θ = 0, which means two delays are the same, τr = τp . In this case, the gambler’s asset Ut just after the tth betting is given as follows: ⎧ (1 ≤ t ≤ τr ) ⎪ ⎨ s t−τ r Ut = Xk (τr < t) ⎪ ⎩ s+ k=1

(We can replace τr by τp .) Naturally, the gambler’s betting does not end at least before t = τr . Hence, the stopping time satisfies Tτr ,τp > τr , and the probability for him to be broken is given as ⎛ ⎞ Tτr ,τp −τr τr ,τp Xk = 0⎠ PA (s) = P(UTτr ,τp = 0) = P ⎝s + k=1

Let T0,0 be the stopping time when there are no delays both in receipt and payment. Then, we can see from the definition of the model that T0,0 = Tτr ,τp − τr This means that we can reduce the problem for the case of θ = 0 to the original gambler’s ruin, leading to the following ruin probability for τr = τp :

3 Delayed gambler’s ruin

τr ,τp

PA

(s) = P(UTτr ,τp = 0) ⎞ ⎛ Tτr ,τp −τr Xk = 0⎠ = P ⎝s + k=1

⎛

T0,0



= P ⎝s +

=

3.2 θ > 0

Xk = 0⎠

k=1

⎧ A−s ⎪ ⎪ ⎪ ⎨ A ⎪ ⎪ ⎪ ⎩

⎞



p = q = 12

 A  s q − qp p  A q −1 p

 (1)

(p  = q)

(τr > τp )

Let us now consider the case when the delay in receipt is longer than that of payment, τr > τp (θ > 0). In this case, the asset Ut of the gambler at t is given as follows:

Ut =

⎧ s ⎪ ⎪ t−τp ⎪ ⎪ ⎪ ⎪ ⎨s − δ−1,Xk

(0 ≤ t ≤ τp ) (τp < t ≤ τr )

k=1 ⎛ ⎞ ⎛ ⎪ t−τ ⎪ r ⎪ ⎪ ⎪ Xk ⎠ − ⎝ ⎪s + ⎝

⎩

t−τp



k=t−τr +1

k=1

⎞ δ−1,Xk ⎠

(τr < t)

Here, δi,j is a Kronecker delta,  δi,j =

1 0

(i = j) (i  = j)

Since we have set the initial condition as s >| θ |, the stopping time Tτr ,τp satisfies Tτr ,τp > τr . Hence, the probability of the ruin is formally written down as τr ,τp

PA

(x) = P(UTτr ,τp = 0) ⎛ ⎞ ⎛ ⎛T τr ,τp −τr ⎜ ⎜ Xk ⎠ − ⎝ = P ⎝s + ⎝ k=1

Tτr ,τp −τp



k=Tτr ,τp −τr +1

⎞

⎞

⎟ ⎟ δ−1,Xk ⎠ = 0⎠

By the condition of the ruin takes place for the first time at time Tτr ,τp , the results of the bet at earlier times are restricted. We will show this in the following. First, we assume that XTτr ,τp −τp = 1, then by the condition of the ruin at Tτr ,τp for the first time leads to

339

340

CHAPTER 16 Delayed gambler’s ruin

⎛

⎞

Tτr ,τp −τr

0 = UTτr ,τp = s + ⎝



⎛

⎞

Tτr ,τp −τp



Xk ⎠ − ⎝

δ−1,Xk ⎠

k=Tτr ,τp −τr +1

k=1

Hence, the same condition requires that XTτr ,τp −τr = −1

and ⎛T s+⎝

τr ,τp −τr



⎛

⎞

⎜ Xk ⎠ − ⎝

(2) ⎞

Tτr ,τp −τp −1



k=Tτr ,τp −τr +1

k=1

⎟ δ−1,Xk ⎠ = 0

(3)

are simultaneously satisfied. However, these assumptions and restriction lead to 0 = UTτr ,τp ⎛T =s+⎝

τr ,τp −τr



⎛

⎞

⎜ Xk ⎠ − ⎝

Tτr ,τp −τp



k=Tτr ,τp −τr +1

k=1

⎞ ⎟ δ−1,Xk ⎠

(by XTτr ,τp −τp = 1) ⎞ ⎞ ⎛ T ⎛T τr ,τp −τr τr ,τp −τp −1 ⎟ ⎜ Xk ⎠ − ⎝ δ−1,Xk ⎠ =s+⎝ k=Tτr ,τp −τr +1

k=1

⎛T

τr ,τp −τr −1

=s+⎝



⎞

⎛

⎜ Xk ⎠ + XTτr ,τp −τr − ⎝

Tτr ,τp −τp −1



k=Tτr ,τp −τr +1

k=1

⎞ ⎟ δ−1,Xk ⎠

(by XTτr ,τp −τr = −1) ⎞ ⎞ ⎛T ⎛T τr ,τp −τr −1 τr ,τp −τp −1 ⎟ ⎜ Xk ⎠ − ⎝ δ−1,Xk ⎠ =s+⎝ k=1

k=Tτr ,τp −τr

= UTτr ,τp −1

This means that the ruin takes place at Tτr ,τp − 1, one time step earlier, which contradicts that the gambler’s ruin occurs at Tτr ,τp for the first time. Thus, the assumption cannot be accepted and XTτr ,τp −τp = 1, and must be XTτr ,τp −τp = −1. Even with XTτr ,τp −τp = −1, it is apparent that XTτr ,τp −τr = −1 is also required. In other words, for the ruin to takes place at Tτr ,τp for the first time, the amount of points the gambler received at that time must be negative. Incorporating these factors, we can further reduce the expression for the ruin probability for θ > 0 to the following

3 Delayed gambler’s ruin

τr ,τp

PA

(s) = P(UTτr ,τp = 0) ⎛ ⎞ ⎛ ⎛T τr ,τp −τr ⎜ ⎜ Xk ⎠ − ⎝ = P ⎝s + ⎝

⎞

Tτr ,τp −τp



k=Tτr ,τp −τr +1

k=1

⎞

⎟ ⎟ δ−1,Xk ⎠ = 0⎠

(by XTτr ,τp −τp = −1) ⎛ ⎞ ⎞ ⎛T ⎞ ⎛ T τr ,τp −τr τr ,τp −τp −1 ⎜ ⎟ ⎟ ⎜ = P ⎝s + ⎝ Xk ⎠ − ⎝ δ−1,Xk ⎠ − 1 = 0⎠ k=Tτr ,τp −τr +1

k=1

⎛

⎛T

−τr

τr ,τp ⎜ = P ⎝(s − 1) + ⎝

⎞

⎛

⎜ Xk ⎠ − ⎝

⎞

Tτr ,τp −τp −1



k=Tτr ,τp −τr +1

k=1

⎞

⎟ ⎟ δ−1,Xk ⎠ = 0⎠

At this point, the initial amount of the asset is shifted by one point, but further simplification is hindered by the term containing the Kronecker’s delta. Even though the exact evaluation is not simple, we now employ an approximate assumption to replace this term by its expectation value. Namely, we assume ⎡

Tτr ,τp −τp −1





δ−1,Xk ≈ E ⎣

k=Tτr ,τp −τr +1

⎤

Tτr ,τp −τp −1

δ−1,Xk ⎦ = q(θ − 1)

k=Tτr ,τp −τr +1

With this assumption, the probability of ruin is approximated as ⎛

⎛

⎞

⎞

Tτr ,τp −τr τr ,τp PA (s) ≈ P ⎝(s − 1) + ⎝ Xk ⎠ − q(θ − 1) = 0⎠ k=1

⎛

⎛T

= P ⎝{s − 1 − q(θ − 1)} + ⎝

τr ,τp −τr



⎞

⎞

Xk ⎠ = 0⎠

k=1

This equation is in the same form as Eq. (1) by identifying the initial asset is decreased by 1 + q(θ − 1). Thus, this approximation leads to the ruin probability as

τr ,τp

PA

(s) ≈

⎧ A−{s−(1+q(θ−1))} ⎪ ⎪ ⎪ A ⎨



⎪ ⎪ ⎪ ⎩

(p  = q)

 A  (s−(1+q(θ−1))) q − qp p  A q −1 p

p = q = 12

 (4)

This is a natural result considering that the receipt of points is more delayed than the payments. Hence, the gambler’s asset tends to be lower at any time points, leading to a higher probability of ruin compared to the case of no delays or the same delays. This approximation accounts for these effects by shifting the initial assets to

341

342

CHAPTER 16 Delayed gambler’s ruin

lower points. We will compare this approximation with computer simulations in the following section.

3.3 θ < 0

(τr < τp )

We now consider the opposite case with τr < τp . By essentially the same arguments, we arrive at the ruin probability for this case as follows: τr ,τp

PA

(s) = P(UTτr ,τp = 0) ⎞ ⎞ ⎛ ⎞ ⎛ T ⎛T τr ,τp −τp τr ,τp −τr −1 ⎟ ⎟ ⎜ ⎜ Xk ⎠ + ⎝ δ1,Xk ⎠ = 0⎠ = P ⎝s + ⎝ k=1

k=Tτr ,τp −τp +1

We again use the average of the term containing the Kronecker’s delta. ⎤ ⎡ Tτr ,τp −τr −1 δ1,Xk ⎦ = p(−θ − 1) E⎣ k=Tτr ,τp −τp +1

This leads to the following approximation for the ruin probability: ⎛

⎛

⎞

⎞

Tτr ,τp −τp τr ,τp PA (s) ≈ P ⎝{s + p(−θ − 1)} + ⎝ Xk ⎠ = 0⎠ k=1

(5)

Again, this equation is in the same form as Eq. (1) by identifying the initial asset is increased by p(−θ − 1). Thus, this approximation leads to the ruin probability as τr ,τp

PA

(s) ≈

⎧ A−{s+p(−θ−1)} ⎪ ⎪ ⎪ A ⎨



⎪ ⎪ ⎪ ⎩

(p  = q)

 A  (s+p(−θ−1)) q − qp p  A q −1 p

p = q = 12

 (6)

As in the case for θ > 0, we can see that this is an effective approximation to account for the decrease of the ruin probability using an increase of the initial asset points.

4 Comparison against computer simulations τ ,τ

In this section, we will compare our approximate results for PAr p (x) with computer simulations. We will fix the following parameters: • A = 100 • p = 9/19 (q = 11/19) Also, we take 10,000 trials to obtain average values from computer simulations.

4 Comparison against computer simulations

4.1 θ > 0

(τr > τp )

For the simplicity, we set τp = 0 and vary τr , and initial asset points x. The results are given in the following five tables. Columns A and B are, respectively, the estimations from computer simulations, and from our approximation Eq. (4). Although data are limited due to constraints on computational times, for the ranges of τr (= θ ), the discrepancy is less than 2-point percentiles. We also present a graph of comparison for the case of s = 90 as an example (Figs. 3 and 4).

4.2 θ < 0

(τr < τp )

Again, for the simplicity, we set τr = 0 and vary τp and initial asset points x. The results are given in the following five tables. Columns A and B are, respectively, the estimations from computer simulations, and from our approximation Eq. (6) (Figs. 5 and 6).

FIG. 3 The case θ > 0 with various s and τr : (A) s = 95, (B) s = 90, (C) s = 85, (D) s = 70, (E) s = 50.

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CHAPTER 16 Delayed gambler’s ruin

0.82 0.80 0.78 PA(x)

344

0.76 0.74 0.72 0.70

0

2

4

tr

6

8

10

FIG. 4 A graph of comparison for the case θ > 0 with s = 90 and various τr .

FIG. 5 The case θ < 0 with various s and τp : (A) s = 90, (B) s = 85, (C) s = 70.

5 Discussion We have presented an extension of gambler’s ruin to include delays in gaining and losing a unit of gambler’s assets. Although exact analysis of the ruin probability is

Appendix

0.65

PA(x)

0.60

0.55

0.50

0

2

4

tp

6

8

FIG. 6 A graph of comparison for the case θ < 0 with s = 90 and various τp .

difficult when there is a difference between delays associated with gain and loss, we proposed an approximation scheme. The scheme essentially finds shifts in the initial assets to account for the effects of the delays and reduces the problem to a normal gambler’s ruin with a shifted initial assets. These effective shifts increases (decreases) initial assets when the gaining (losing) delay is shorter leading to changes in the ruin probability. Although the comparison against computer simulations are preliminary, our approximation can function well for small delay differences, particularly for the case that the initial asset is closer to the midpoints between two boundaries. Further analysis is left for the future. Although the model presented here is simple, we can consider extensions and developments to different directions. For example, multiplayer transactions with delays and uncertainties can be viewed as an extension of this model. It may connect to socioeconomic phenomena relating to corporate transactions and bankruptcies.

Appendix In this appendix, we briefly describe some properties of the gambler’s ruin.

Ruin probability We will start with a derivation of the ruin probability. As in the main text, we set the following with the boundaries at 0 and A:

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CHAPTER 16 Delayed gambler’s ruin

p: The probability of the gambler’s winning a point. (We also set q = 1 − p as the probability of losing.) • PA (s): Probability that a gambler with the initial asset of s points to become broken.

•

Consider the cases after the first betting. The gambler wins with the probability p and his asset increases s to s + 1. The ruin probability is now PA (s + 1). We can similarly deduce the ruin probability becomes PA (s − 1) if he loses at the first bet with the probability q. As there are only these two independent outcomes at the first bet, we see the ruin probability PA (s) is given by the following: PA (s) = pPA (s + 1) + qPA (s − 1),

1≤s≤A−1

(7)

where we have adopted the conventions that PA (0) = 1, PA (A) = 0. We can solve this difference equation through a standard method [1], leading to the expression given in the main text.

Expected duration of the betting We can also compute average number of betting before the gambler reaches either boundary 0 or A. Let us denote the expected duration of the betting starting from the initial asset of s as DA (s). In a similar argument as for the ruin probability, we derive the difference equation by noting that after the first betting, the expected duration increases by 1. This leads to DA (s) = p(1 + DA (s + 1)) + q(1 + DA (s − 1)),

1≤s≤A−1

(8)

where we have adopted the conventions that DA (0) = 0, DA (A) = 0. Again, this difference equation can be solved by a standard method [1]. The results are given as ⎧   ⎪ p = q = 12 ⎪ ⎨s(A − s)  s 1− qp DA (s) = s A ⎪ − (p  = q) ⎪ ⎩ q−p q−p 1− q A p

We note that when p = q = 1/2, the expression becomes simple as for the case of the ruin probability. On the other hand, even for small s, the expected duration can be rather large with increasing A.

Acknowledgments This work was in part supported by research funds from the Ohagi Hospital (Hashimoto, Wakayama, Japan) and the NT Engineering Corporation (Takahama, Aichi, Japan).

References

References [1] N. Bailey, Elements of Stochastic Process With Applications to the Natural Sciences, Wiley, 1964. [2] A. Gut, Probability: A Graduate Course, Springer, New York, 2005. [3] A. Gut, The gambler’s ruin problem with delays, Stat. Prob. Lett. 83 (2013) 2549–2552. [4] A.L. Rocha, F. Stern, The gambler’s ruin problem with n players and asymmetric play, Stat. Prob. Lett. 44 (1999) 87–95. [5] A.L. Rocha, F. Stern, The asymmetric n-player gambler’s ruin problem with equal initial fortunes, Adv. Appl. Math. 33 (2004) 512–530. [6] P.M. Pearle, Reduction of the state vector by a nonlinear Schrodinger equation, Phys. Rev. D 13 (1976) 857–868. [7] P.M. Pearle, Might god toss coins? Found. Phys. 12 (1982) 249–263. [8] P.M. Pearle, Combining stochastic dynamical state-vector reduction with spontaneous localization, Phys. Rev. A 39 (1989) 2277–2289. [9] L. Glass, M.C. Mackey, From Clocks to Chaos: The Rhythms of Life, Princeton University Press, Princeton, NJ, 1988. [10] G. Stepan, Retarded Dynamical Systems: Stability and Characteristic Functions, Wiley, New York, 1989. [11] U. Kuchler, B. Mensch, Langevins stochastic differential equation extended by a time-delayed term, Stochastics Stochastic Rep. 40 (1992) 23–42. [12] T.D. Frank, P.J. Beek, Stationary solutions of linear stochastic delay differential equations: applications to biological systems, Phys. Rev. E 64 (2001) 021917. [13] T. Ohira, J. Milton, Delayed random walks, Phys. Rev. E 52 (1995) 3277–3280. [14] T. Ohira, T. Yamane, Delayed stochastic systems, Phys. Rev. E 61 (2000) 1247–1257.

347

CHAPTER

Event-triggered and self-triggered control for networked feedback systems with delay and packet dropout

17 Dan Ma, Zhuoyu Li

College of Information Science and Engineering, State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang, China

Chapter outline 1 Introduction....................................................................................... 349 2 Problem formulation............................................................................. 351 3 Event-triggered control ......................................................................... 354 3.1 Event generator and switching controller synthesis: The short network-induced delay case ........................................................ 354 3.2 Event generator and switching controller synthesis: Combined the short network-induced delay and the packet dropout case ........................... 357 4 Self-triggered control ........................................................................... 359 5 An example ....................................................................................... 361 6 Conclusions....................................................................................... 364 Acknowledgments .................................................................................. 364 References........................................................................................... 364

1 Introduction Networked control systems (NCSs), in which actuators, sensors, and controllers exchange information through the shared band-limited digital communication network, have received considerable attention in the last two decades [1–6]. Although NCSs allow for reduced wiring as well as for lower installation cost, the networkinduced delay and the packet dropout often occur inevitably. The reason is that the network bandwidth is limited and shared by multiple real-time tasks. In general, the network-induced delay is typically time varying, and in the extreme results in packet dropouts, even leads to the instability of the system. A challenging problem Stability, Control and Application of Time-Delay Systems. https://doi.org/10.1016/B978-0-12-814928-7.00017-2 © 2019 Elsevier Inc. All rights reserved.

349

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CHAPTER 17 Event-triggered and self-triggered control for networked feedback systems with delay and packet dropout

herein is to cope with the negative impact of network-induced delays and packet dropouts. Meanwhile, for networked feedback control systems, how to guarantee the performance of the system by less traffic requirement and better resource utilization is of great importance. Event-triggered control is currently attracting more and more attention due to its abilities to significantly reduce the communication and computation resources in embedded control systems and distributed systems [6–15]. Unlike the traditional time-triggered control, in which the controller updates control law periodically, the event-triggered control is aperiodic. The key point is that the transmission instants in a networked feedback control loop are generated by the triggered events. Generally speaking, an event generator provides the even-triggered conditions, under which the ability, such as convergence and stability, of the systems can still be achieved. The controller updates as long as the event is violated. The aperiodic event-triggered control can be usually realized by the hardware and the software. The hardware and software realization are typically called event-triggering and self-triggering, respectively. Different from the event-triggered control, which needs to verify a specific condition continuously and determine whether the control task is triggered, the self-triggered control provides that the next triggered-time based on the receiving data of the systems. The related work can be found in Refs. [16–18]. In this chapter, we focus on the stabilization of networked feedback control systems with short network-induced delays and packet dropouts under the event- and self-triggered state feedback control, respectively. Distinct to the time-delay system approach adopted in Ref. [11], we model the NCS by using a switched system with one-step delay, whose distinctive advantage enables us to bypass the difficult construction of complex Lyapunov-Krasovskii functionals that are otherwise necessary. On the other hand, the switched system model can also reduce the conservatism of the robust control, which customarily uses the bound of delays instead of the timevarying characteristics of the network-induced delay [19]. Intuitively, the switched system model indicates the time-varying characteristics of the network-induced delay exactly by switchings of the subsystems. Furthermore, different from the most of the existing literature, we adopt the periodic event-triggered mechanism [10], under which the sampled state needs to be compared in the event generator. This kind of triggering mechanism both saves the computation and communication resources and ensures the strictly positive lower bound of the interevent times as well, which avoids the Zeno behavior. Finally, a self-triggered condition is provided as a more flexible way to adjust the triggered interval, which is related to the variation rate of the network-induced delay. Moreover, the triggered interval is monotonically increasing with the average dwell time, implying that with a slower rate in delay variation, the longer is the triggered interval. This chapter is organized as follows. Section 1 ends with the basic notations to be used in this chapter. Section 2 formulates the problem and presents a discretetime switched system model with limited subsystems to describe the NCS with short network-induced delays. Section 3 provides a codesign condition on the existence of the event-triggered mechanism and the switching controller to ensure the stabilization

2 Problem formulation

of the NCS. In the joint presence of short network-induced delays and packet dropouts, the NCS is also shown to be stabilized under a proper switching controller and an event-triggered mechanism. Section 4 extends the codesign method from the event-triggered control to the self-triggered control. Section 5 shows the feasibility and efficiency of the proposed method by using an example. At the end of this chapter, conclusion and remarks are given. Notation. R and N denote the sets of real and natural numbers. I is the identity matrix of any dimension. P > 0 is a symmetric positive definite matrix P.  ·  stands for the Euclidian norm of vectors and its induced-spectral norm of matrices.

2 Problem formulation Consider an NCS depicted in Fig. 1. The plant is a continuous-time linear system in the state-space form x˙ (t) = Ap x(t) + Bp u(t),

(1)

where x(t) ∈ Rn is the system state, u(t) ∈ Rm is the control input, and Ap and Bp are the constant matrices with appropriate dimensions. The plant is controlled by a discrete-time controller. The information exchanges among sensors, controllers, and actuators through the shared communication network. In order to reduce the usage of computation and communication resources, an event generator is adopted to determine whether the current sampled-state measurement is transmitted to the controller via the networks or not. The gap between current state and past sampled state is as a measure novelty in feedback. The controller receives the sampled state through the networks as long as the gap exceeds a predesigned threshold.

Plant

Actuator

x(t )

Sensor

x( k )

Event generator

ZOH

x (bk ) Network

x(rk )

Discrete-time controller

FIG. 1 An NCS with an event generator.

351

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CHAPTER 17 Event-triggered and self-triggered control for networked feedback systems with delay and packet dropout

Without loss of generality, the sensor is time driven with a constant sampling period T. The sampling instants belong to Ω1 = {0, T, 2T, . . . , kT, . . .}, k ∈ N. The event generator is used to determine whether the sampled state is transmitted to the controller via the network. Denote the transmission instants of the event generator by Ω2 = {0, b1 T, b2 T, . . . , bk T, . . .} ⊆ Ω1 , bk ∈ N. The controller is event driven. Once the controller receives the latest sampled state, the controller updates immediately. The sampling instants received by the controller are denoted by Ω3 = {0, r1 T, r2 T, . . . , rk T, . . .} ⊆ Ω2 , rk ∈ N. The actuator is time driven, in which a buffer contains the most recent data sent by the controller. In order to describe the distribution characteristics of network-induced delays, the actuator reads the buffer periodically at a higher frequency than the sampling frequency. Moreover, we denote the reading buffer period of the actuator is T0 , so thus T0 = T/N. The short network-induced delay, that is, τk ≤ T and the bounded maximum allowable number of successive packet dropouts dMANSPD will be considered in this chapter. In particular, if all the sampled states are received by the controller, then we say Ω3 = Ω2 . Otherwise, Ω3 ⊂ Ω2 means the packet dropouts occur. Next, we give a timing diagram of the NCS in Fig. 2 in any sampling period [kT, (k + 1)T), k ∈ N. Since the network-induced delay τk is less than one sampling period T, there will exist u(k −1) and u(k) to control the plant in one sampling period [kT, (k + 1)T). As depicted in Fig. 2, the feedback control based on the sampled-state measurement x(k) arrived at the actuator at the instant t1 , which belongs to [kT + mT0 , kT + (m + 1)T0 ), m ∈ Z0 = {0, 1, . . . , N}. The actuator reads the buffer at time kT +(m+1)T0 and controller updates the control law by using the new arrival. Let us denote the control interval of u(k) and u(k − 1) by n0 (k)T0 and n1 (k)T0 , respectively. In the sequel, we know that 

n0 (k), n1 (k) ∈ Z0 , n0 (k) + n1 (k) = N, n0 (k)T0 + n1 (k)T0 = NT0 = T,

where n1 (k)T0 indicates the size of the network-induced delay interval in one sampling period.

u (k )

... kT kT + T0

t1

(k + 1)T

u (k ) u (k – 1)

FIG. 2 A timing diagram of an NCS with short network-induced delays.

2 Problem formulation

By using a mapping: R2 → M = {0, 1, . . . , N}: [n0 (k) n1 (k)] → σ (k) to describe the relationship between the time-varying characteristics of the networkinduced delay and the switchings, where σ (k) is a nonnegative integer, we give the following indication: [n1 (k) n0 (k)] = [0 N] → σ (k) = 0, [n1 (k) n0 (k)] = [1 N − 1] → σ (k) = 1, .. . [n1 (k) n0 (k)] = [N 0] → σ (k) = N.

In light of the existing modeling method in Ref. [20] and combining with the earlier analysis of the network-induced delay, system (1) can be rewritten as a switched system Sσ (k) : x(k + 1) = Ax(k) + B0σ (k) u(k) + B1σ (k) u(k − 1),

(2)

where A = eAp T , B0σ (k) =

n0 (k)−1

A0 = eAp T0 , Ai0 B0 ,

B0 =

B1σ (k) =

i=0

 T0

0 n1 (k)−1

eAP τ Bp dτ , i+n0 (k)

A0

B0 .

i=0

It can be shown that system (2) is a switched system with N + 1 subsystems. For σ (k) = i, Si denotes the ith subsystem. We use Ω4 = {0, k1 T, . . . , kj T, . . .} ⊆ Ω1 , kj ∈ N to denote the set of switching instants. In what follows, we adopt multiple discrete-time event-triggered state feedback controller switchings to stabilize the NCS, which synchronously switch with the subsystems. This typically leads to better performance achieved by a single controller. Let us consider the following switching event-triggered feedback controller: u(k) = Kσ (k) x(rk ),

k ∈ [rk , rk+1 ),

(3)

where Kσ (k) is the switching state feedback controller gains to be designed later. The state-based event-triggered condition is given by rk+1 = min{k > rk | e(k) ≥ δ1 x(rk )},

(4)

where e(k) = x(k) − x(rk ), rk is the event-triggered instant, 0 < δ1 < 1 is the event threshold to be determined. Remark 1. Eq. (4) is a periodic event-triggered condition. The triggering instants are always integer times of the sampling period T, thus the Zeno behavior can be avoided. In the extreme case, δ1 = 0, rk+1 = rk +1, which leads to the time-triggered generator.

353

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CHAPTER 17 Event-triggered and self-triggered control for networked feedback systems with delay and packet dropout

Combining with Eqs. (2), (3), we have the closed-loop event-triggered switched system with one-step input delay 

x(k + 1) = Ax(k) + B0σ (k) u(k) + B1σ (k) u(k − 1), u(k) = Kσ (k) x(rk ), k ∈ [rk , rk+1 ).

(5)

Our goal is to design an event-triggered or a self-triggered switching controller to stabilize the NCS (5) simultaneously.

3 Event-triggered control In this section, we first provide a sufficient condition on the existence of the event generator and the switching controller to simultaneously stabilize the NCS (5) with short network-induced delays. Then, the stabilization of networked feedback control systems in the joint presence of short network-induced delays and packet dropouts is extended.

3.1 Event generator and switching controller synthesis: The short network-induced delay case Theorem 1. For some given positive constants λ < 1, μ ≥ 1, ξ > 0, if there exist ˆ i > 0, and Ri with appropriate dimensions, such 0 < δ1 < 1, and matrices Pˆ i > 0, Q that the following inequalities ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

ˆ T + RT B T PA i 0i RTi BT1i

−Pˆ i

−λ2 Pˆ i

0

0

∗

ˆi −2λ2 Pˆ i + λ2 Q

0

∗

∗

−2ξ Pˆ i + ξ 2 I

∗

∗

∗

−RTi BT0i −Pˆ i

∗

∗

∗

∗

−δ1−2 I

∗

∗

∗

∗

∗ μPˆ i ≥ Pˆ j ,

ˆi ≥ Q ˆ j, μQ

i, j ∈ M,

0 Pˆ i 0

Pˆ i

⎤

⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ < 0, 0 ⎥ ⎥ ⎥ 0 ⎥ ⎦ ˆ −Qi

(6)

(7)

hold, then under the event-triggered mechanism (4) and the discrete-time switched state feedback controller (3) with any switching signal satisfying the average dwell time τa > τa∗ :=

ln μ , 2 ln λ−1

system (5) is exponentially stabilized with the decay rate 1

α = λμ 2τa .

(8)

3 Event-triggered control

Moreover, the feedback control gains Ki = Ri Pˆ −1 i ,

i ∈ M.

(9)

Proof. Choose the multiple Lyapunov candidates for the NCS (5) as follows: Vi (k) = xT (k)Pi x(k) + xT (k − 1)Qi x(k − 1),

(10)

where Pi > 0 and Qi > 0, i ∈ M are symmetric positive definite matrices. For any sampling period [kT, (k + 1)T), in order to guarantee the exponential stabilization of system (5), the difference of Vi (k) along the ith subsystem of system (5) needs to be Vi (k + 1) − λ2 Vi (k) = xT (k + 1)Pi x(k + 1) + xT (k)(Qi − λ2 Pi )x(k) − λ2 xT (k − 1)Qi x(k − 1), = [Ax(k) + B0i Ki x(rk ) + B1i Ki x(k − 1)]T Pi [Ax(k) + B0i Ki x(rk ) + B1i Ki x(k − 1)] + xT (k)(Qi − λ2 Pi )x(k) − λ2 xT (k − 1)Qi x(k − 1).

Combining with the event-triggered condition (4), we get Vi (k + 1) − λ2 Vi (k) ≤ [Ax(k) + B0i Ki x(rk ) + B1i Ki x(k − 1)]T Pi [Ax(k) + B0i Ki x(rk ) + B1i Ki x(k − 1)] + xT (k)(Qi − λ2 Pi )x(k) − λ2 xT (k − 1)Qi x(k − 1) − eT (k)e(k) + δ12 xT (rk )x(rk ), = [Ax(k) + B0i Ki (x(k) − e(k)) + B1i Ki x(k − 1)]T Pi [Ax(k) + B0i Ki (x(k) − e(k)) + B1i Ki x(k − 1)] + xT (k)(Qi − λ2 Pi )x(k) − λ2 xT (k − 1)Qi x(k − 1) − eT (k)e(k) + δ12 [x(k) − e(k)]T [x(k) − e(k)], ⎡ ⎤T ⎡Σ (A + B K )T P B K −(A + B K )T P B K − δ 2 I ⎤ ⎡ ⎤ i 1i i i 0i i 1 0i i 0i i x(k) x(k) 1 ⎢ ⎥ = ⎣x(k − 1)⎦ ⎣ ∗ Σ2 −KiT BT1i Pi B0i Ki ⎦ ⎣x(k − 1)⎦ , e(k) e(k) ∗ ∗ Σ 3

where Σ1 = (A + B0i Ki )T Pi (A + B0i Ki ) − λ2 Pi + Qi + δ12 I, Σ2 = −λ2 Qi + KiT BT1i Pi B1i Ki , Σ3 = (δ12 − 1)I + KiT BT0i Pi B0i Ki .

Using the Schur complement three times, Eq. (6) is equivalent to ⎡

1

⎢ ⎣ ∗ ∗

Pˆ i (A + B0i Ki )T Pi B1i Ri

−Pˆ i (A + B0i Ki )T Pi B0i Ri − δ12 Pˆ i Pˆ i

2

−RTi BT1i Pi B0i Ri

∗

3

⎤ ⎥ ⎦ < 0,

(11)

355

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CHAPTER 17 Event-triggered and self-triggered control for networked feedback systems with delay and packet dropout

where

1 = Pˆ i (A + B0i Ki )T Pi (A + B0i Ki )Pˆ i − λ2 Pˆ i + Pˆ i Qi Pˆ i + δ12 Pˆ i Pˆ i , ˆ i + Ri BT Pi B1i Ri ,

2 = −2λ2 Pˆ i + λ2 Q 1i

3 = −2ξ Pˆ i + ξ 2 I + δ12 Pˆ i Pˆ i + Ri BT0i Pi B0i Ri .

ˆ i − Pˆ i )Qi (Q ˆ i − Pˆ i ) > 0, which On the other hand, noticing that Qi > 0, we have (Q ˆ yields to −Pˆ i Qi Pˆ i < −2Pˆ i + Qi . Similarly, −Pˆ i Pˆ i < −2ξ Pˆ i + ξ 2 I holds for any arbitrary scalar ξ . Using the earlier two inequalities, multiplying by the diagonal matrix diag{Pi , Pi , Pi } and its transpose on both sides of Eq. (11), and setting Ri = Ki Pˆ i , we get Vi (k + 1) − λ2 Vi (k) < 0,

i ∈ M.

(12)

This indicates that each subsystem of Eq. (5) is exponentially stable. Hence, for any switching interval [kj , kj+1 ), we have Vσ (kj ) (k) < λ2(k−kj ) Vσ (kj ) (kj ),

j = 0, 1, . . . .

(13)

j = 0, 1, . . . .

(14)

Due to Vσ (k) (k) = Vσ (kj ) (k), we get Vσ (k) (k) < λ2(k−kj ) Vσ (kj ) (kj ),

Since the system state is continuous at any switching instant, from inequality (7), we know that Vσ (kj ) (kj ) ≤ μVσ (k− ) (kj− ), j

j = 0, 1, . . . .

(15)

According to Eqs. (14), (15), we can obtain Vσ (k) (k) = Vσ (kj ) (k) ≤ λ2(k−kj ) Vσ (kj ) (kj ) ≤ μλ2(k−kj ) Vσ (k− ) (kj− ) j

≤ μ2 λ2(k−kj−1 ) Vσ (kj−1 ) (kj−1 ) .. .

(16)

≤ μNσ (k0 ,k) λ2(k−kj ) λ2(kj −kj−1 ) · · · λ2(k1 −k0 ) Vσ (k0 ) (k0 ) k−k0

≤ μ τa λ2(k−k0 ) Vσ (k0 ) (k0 ) = α 2(k−k0 ) Vσ (k0 ) (k0 ).

Denoting ε1 = min{λmin (Pi )}, ε2 = max{λmax (Pi )}, and ε3 = min{λmin (Qi )}, i∈M

i∈M

i∈M

Eq. (16) yields to ε1 x(k)2 ≤ ε1 x(k)2 + ε3 x(k − 1)2 ≤ Vσ (k) (k) < ε2 α 2(k−k0 ) x(k0 )2 .

(17)

3 Event-triggered control

Thus, x(k) ≤

ε2 (k−k0 ) x(k0 ) . α ε1

(18)

This completes the proof. Remark 2. Theorem 1 indicates that the exponential decay rate α is monotonically increasing with the parameter λ, and decreasing with the average dwell time τa . According to the modeling process of the NCS (5), the variation rate of the networkinduced delays can be reflected by the average dwell time τa . The smaller the variation rate of network-induced delay is, the larger the average dwell time τa is.

3.2 Event generator and switching controller synthesis: Combined the short network-induced delay and the packet dropout case Once the packet dropouts occur in the network transmission, the event-triggered mechanism (4) cannot be directly used to determine whether the sampled state is transmitted or not. The packet dropouts need to be incorporated to the eventtriggered mechanism. Toward this end, a novel event-triggered condition is provided as follows: bk+1 = min{k > bk | ˜e(k) ≥ δ2 x(bk )},

(19)

where e˜ (k) = x(k) − x(bk ), bk is the transmission instants of the event generator. The following theorem further gives a sufficient condition on the existence of a switching controller and an event generator to stabilize the NCS with short networkinduced delays and packet dropouts. Theorem 2. For some given positive constants λ < 1, μ ≥ 1, ξ > 0, if there exist 1

0 ≤ δ2 ≤ ((1 + δ1 )(1 + ε)−dMANSPD ) dMANSPD +1 − 1,

(20)

   

 0i Ki   B1i Ki (1+δ1 )  + where 0 < δ1 < 1, ε = max A − I +  B1−δ    , and positive 1−δ 1 1 ∀i∈M

ˆ i > 0 and matrix Ri with appropriate dimensions such definite matrices Pˆ i > 0, Q that Eqs. (6), (7) hold, then under the event-triggered mechanism (19), the controller (3) and any switching signal with the average dwell time satisfying Eq. (8), system (5) with the maximum allowable number of successive packet dropouts dMANSPD is exponentially stabilized. Furthermore, the feedback control gains are given by 1 Eq. (9), and the exponential decay rate is α = λμ 2τa . Proof. Consider a successful transmission interval [rk , rk+1 ), in which there exists dk packet dropouts, that is, rk = b0 < b1 < b2 < · · · < bdk < bdk +1 = rk+1 . For all l = 1, . . . , dk , during the triggering interval, we have   x(bl+1 − 1) − x(bl ) ≤ δ2 x(bl ) ,

(21)

357

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CHAPTER 17 Event-triggered and self-triggered control for networked feedback systems with delay and packet dropout

which yields to     x(bl+1 − 1) ≤ x(bl+1 − 1) − x(bl ) + x(bl ) ≤ (1 + δ2 ) x(bl ) .

(22)

Since     x(rk ) = x(bl+1 − 1) − x(bl+1 − 1) + x(rk ) ≤ x(bl+1 − 1) + δ1 x(rk ) ,

(23)

then we have x(rk ) ≤

 1  x(bl+1 − 1) . 1 − δ1

(24)

Consequently, we know that     x(bl+1 − 2) ≤ x(bl+1 − 2) − x(rk ) + x(rk ) ≤ (1 + δ1 ) x(rk )  1 + δ1  x(bl+1 − 1) . ≤ 1 − δ1

(25)

Consider system (5) at the instant k = bl+1 , then we have x(bl+1 ) = Ax(bl+1 − 1) + B0i Ki x(rk ) + B1i Ki x(bl+1 − 2),

i ∈ M.

Next, in view of the packet dropouts in the network transmission, for the interval [b1 , rk+1 ), combining with the earlier procedure, we know that    x(bl+1 ) − x(bl+1 − 1) = Ax(bl+1 − 1) + B0i Ki x(rk )

 +B1i Ki x(bl+1 − 2) − x(bl+1 − 1)   ≤ (A − I)x(bl+1 − 1) + B0i Ki x(rk )   + B1i Ki x(bl+1 − 2)   ≤ ε x(bl+1 − 1) .

(26)

In light of Eq. (22), it follows that   x(bl+1 ) − x(bl+1 − 1) ≤ ε(1 + δ2 ) x(bl ) ,

(27)

which further leads to       x(bl+1 ) ≤ x(bl+1 ) − x(bl+1 − 1) + x(bl+1 − 1) − x(bl ) + x(bl ) ≤ [(1 + ε)(1 + δ2 )] x(bl ) ≤ [(1 + ε)(1 + δ2 )]l+1 x(rk ) .

(28)

4 Self-triggered control

For any k ∈ [bdk , bdk +1 ), combining with Eqs. (21), (26), (28), we can get    k −1     d x(k) − x(rk ) ≤ x(k) − x(bdk ) +  (x(bl+1 ) − x(bl+1 − 1))    l=0    dk −1    + (x(bl+1 − 1) − x(bl ))    l=0  ≤

dk 

δ2 x(bl ) +

d k −1

l=0

ε(1 + δ2 ) x(bl )

l=0

⎛ ⎞ dk d k −1  l l δ2 ((1 + ε)(1 + δ2 )) + ε(1 + δ2 )((1 + ε)(1 + δ2 )) ⎠ x(rk ) ≤⎝ l=0

l=0

= ((1 + δ2 )dk +1 (1 + ε)dk − 1) x(rk ) . (29)

According to Eqs. (20), (29), it leads to x(k) − x(rk ) ≤ δ1 x(k) ,

(30)

where δ1 = (1 + δ2 )dk +1 (1 + ε)dk − 1. The earlier analysis indicates the novel event-triggered condition (19) can guarantee (4). Combining with the existence conditions on the switching controller in Theorem 1, the proof is completed. Remark 3. From Eq. (20), we can see that the proposed event-triggered threshold depends on the maximum allowable number of successive packet dropouts dMSNSPD . Since dMSNSPD ≥ 0, this implies that δ2 ≤ δ1 . Thus we conclude that the packet dropouts increase the number of the event-triggered instants in order to guarantee the stabilization of the NCS with delays and packet dropouts. In the extreme that, if dMSNSPD = 0 and δ2 = δ1 , that is, no packet dropout occurs in the network transmission, Theorem 2 reduces to Theorem 1.

4 Self-triggered control In this section, we study the stabilization problem of the NCS subject to short network-induced delays and packet dropouts under a self-triggered mechanism. The following theorem gives a sufficient condition on the stabilization of the NCS with short network-induced delays and packet dropouts under the self-triggered control and the switching controller simultaneously. Theorem 3. For some given positive constants λ < 1, μ ≥ 1, ξ > 0, if there exist ˆ i > 0 and matrix Ri with appropriate dimensions positive definite matrices Pˆ i > 0, Q such that Eqs. (6), (7), (20) hold, then under the switching controller (3) with the average dwell-time satisfying Eq. (8), and the self-triggered mechanism

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CHAPTER 17 Event-triggered and self-triggered control for networked feedback systems with delay and packet dropout

 bk+1 = bk + max 1,

  δ (1 − A) x(bk ) , lnA 1 − 2 Θ



(31)

where Θ = ψ1

bk −1 ε2 λμ 2τa x(k0 ) + ψ2 x(bk ) , ε1

ψ1 = max {B1i Ki } , ψ2 = max {B0i Ki − I} , ∀i∈M

∀i∈M

system (5) with the maximum allowable number of successive packet dropouts dMANSPD is exponentially stabilized. Furthermore, the exponential decay rate is 1 α = λμ 2τa , and the feedback control gains are given by Eq. (9). Proof. For any interval [bk , bk+1 ), combining with the dynamics of the NCS (5), we compute the state error e˜ (k + 1) = x(k + 1) − x(bk ) = Ax(k) + B0i Ki x(bk ) + B1i Ki x(k − 1) − x(bk ) = A˜e(k) + B1i Ki x(k − 1) + (B0i Ki + A − I)x(bk ),

(32) i ∈ M.

This leads to ˜e(k + 1) ≤ A ˜e(k) + B1i Ki  x(k − 1) + B0i Ki − I x(bk ) ≤ A ˜e(k) + ψ1 x(k − 1) + ψ2 x(bk ) .

(33)

Based on Theorem 2, we know that the system state satisfies x(k − 1) ≤

ε2 (k−1−k0 ) x(k0 ) ≤ α ε1



ε2 (bk −1) x(k0 ) . α ε1

(34)

x(k0 ) + ψ2 x(bk ) .

(35)

Combining with Eqs. (33), (34), we have ˜e(k + 1) ≤ A ˜e(k) + ψ1



ε2 ε1 λμ

bk −1 2τa

Thus, for k ∈ [bk , bk+1 ), ˜e(k) ≤ Ak−bk ˜e(bk ) + (1 − A)−1 (1 − Ak−bk )Θ = (1 − A)−1 (1 − Ak−bk )Θ.

If the interevent interval satisfying Eq. (31) holds, we have (1 − A)−1 (1 − Ak−bk )Θ ≤ δ2 x(bk ) ,

k ∈ [bk , bk+1 ).

Define (1 − A)−1 (1 − ATk )Θ = δ2 x(bk ), which implies that   δ2 (1 − A) x(bk ) Tk = bk+1 − bk = lnA 1 − . Θ

(36)

5 An example

This indicates that the self-triggered condition (31) can guarantee the eventtriggered condition (19), which completes the proof. Remark 4. The interevent interval Tk is related to the average dwell time τa , which implies that the self-triggered mechanism depends on the variation rate of the network-induced delay. The relationship can be summarized as the following two aspects: (i) Θ is decreasing with the increasing of the average dwell time τa . If A < 1, then the interevent interval Tk is monotonically increasing, that is,   δ2 (1 − A) x(bk h) ↓→ Tk ↑ . τa ↑→ Θ ↓→ 1 − Θ (ii) Θ is decreasing with the increasing of the average dwell time τa . If A ≥ 1, the interevent interval Tk is also monotonically increasing, that is,   δ2 (1 − A) x(bk h) ↑→ Tk ↑ . τa ↑→ Θ ↓→ 1 − Θ Therefore, the intertrigger interval Tk is monotonically increasing with the average dwell time τa , which implies that with a slower rate in network-induced delay variation, the longer is the triggered interval.

5 An example In this section, we give a numerical example to illustrate the effectiveness of the −8 1 , Bp = main results. Consider the plant (1) with the parameters Ap = 0.1 0.01   −0.2 0 . Set the sensor sampling period T = 0.2 s, and the actuator sampling 0 0.1 period T0 = 0.1 s, which implies that N = 2. Hence, the NCS (5) has three subsystems. Suppose that the number of successive packet dropouts in the network transmission is bounded by dMANSPD = 2. Next, according to the modeling process of the NCS (5), we can obtain 

 0.2019 1.2214 , 1.0202 1.0020    0 0 , B01 = B10 = 0 0    0 0 , B12 = B02 = 0 0 A=



 −0.0422 0.0263 , −0.0541 0.0306   0.0138 0.0105 −0.0284 , B11 = −0.0201 0.0100 −0.0340  −0.0422 0.0263 . −0.0541 0.0306 B00 =

0.0158 0.0206

 ,

361

CHAPTER 17 Event-triggered and self-triggered control for networked feedback systems with delay and packet dropout

Choosing μ = 1.04, λ = 0.9539, ξ = 4 and solving Eqs. (6), (7) by Theorem 2, we get δ1 = 0.0023, δ2 = 0.00017;    5420.50 −3442.71 ˆ ˆP0 = , Q0 = −3442.71 2191.19   1,088,218.66 −691,480.11 ; R0 = 1,857,449.10 −1,180,425.81    5428.46 −3444.67 ˆ ˆP1 = , Q1 = −3444.67 2190.40   216,135.40 −123,120.83 ; R1 = 367,021.72 −209,072.76    5347.72 −3399.09 ˆ ˆP2 = , Q2 = −3399.09 2165.14   −14,626.41 8125.68 . R2 = −25,047.14 13,914.91

10,506.45 −6681.07

−6681.07 4256.42

10,560.95 −6720.51

−6720.51 4284.49

10,644.80 −6765.70

−6765.70 4308.14

 ,

 ,

 ,

Furthermore, we can have the feedback control gain 



156.62 −69.48 K0 = , 245.63 −152.78





1991.96 3076.38 K1 = , 3382.56 5224.03

and the average dwell time τa > τa∗ = 1 2τa

ln μ 2 ln λ−1





−163.53 −252.98 K2 = , −280.04 −433.22

= 0.4155. Taking τa = 0.5, we have

α = λμ = 0.9921. Fig. 3 gives a switching signal σ (k) with the average dwell time τa = 0.5 >τa∗ . Fig. 4 shows the trajectories of the system state converges to zero. Figs. 5 and 6 give the event-triggered instants and the time-triggered instants, respectively.

2

1.5

Switching signal

362

1

0.5

0

0

1

FIG. 3 Switching signal.

2

3

4

5 Time (s)

6

7

8

9

10

5 An example

1.5

System state 1 System state 2

x (k)

1

0.5

0

–0.5 0

1

2

3

4

5 Time (s)

FIG. 4 System state.

FIG. 5 Event-triggered instants.

FIG. 6 Time-triggered instants.

6

7

8

9

10

363

364

CHAPTER 17 Event-triggered and self-triggered control for networked feedback systems with delay and packet dropout

The simulation results indicate that the event-triggered mechanism reduces the computation of the controller and saves the communication resources with respect to the traditional periodic sampling control.

6 Conclusions A codesign method of the event-triggered generator and the switching feedback controller for NCSs with short network-induced delays and packet dropouts is investigated in this chapter. Both the event threshold and the switching frequency depend on the variation rate of the network-induced delays and the maximal number of successive packet dropouts to stabilize the networked feedback system. Moreover, a self-triggered condition as the flexible software realization is also developed. Finally, a numerical example shows that the updating frequency of the control law can be reduced and simultaneously maintain the system performance.

Acknowledgments The authors gratefully acknowledge the support of the Natural Science Foundation of China under Grants 61603079 and 61773098.

References [1] G.C. Walsh, H. Ye, L.G. Bushnell, Stability analysis of networked control systems, IEEE Trans. Control Syst. Technol. 10 (3) (2002) 438–446. [2] P. Antsaklis, J. Baillieul, Special issue on technology of networked control systems, Proc. IEEE 95 (1) (2007) 5–8. [3] H. Zhang, Z. Zhang, Z. Wang, New results on stability and stabilization of networked control systems with short time-varying delay, IEEE Trans. Cybern. 46 (12) (2015) 2772–2781. [4] C. Tan, L. Li, H. Zhang, Stabilization of networked control systems with both network-induced delay and packet dropout, Automatica 59 (2015) 194–199. [5] A. Selivanov, E. Fridman, Observer-based input-to-state stabilization of networked control systems with large uncertain delays, Automatica 74 (2016) 63–70. [6] D. Liu, G.H. Yang, Robust event-triggered control for networked control systems, Inf. Sci. 459 (2018) 186–197, https://doi.org/10.1016/j.ins.2018.02.057. [7] K.E. Årzén, A simple event-based PID controller, in: Proceedings of IFAC World Congress, 1999, pp. 8687–8692. [8] P. Tabuada, Event-triggered real-time scheduling of stabilizing control tasks, IEEE Trans. Autom. Control 52 (9) (2007) 1680–1685. [9] M.C.F. Donkers, W.P.M.H. Heemels, Output-based event-triggered control with guaranteed L∞ -gain and improved and decentralized event-triggering, IEEE Trans. Autom. Control 57 (6) (2012) 1362–1376.

References

[10] D.P. Borgers, V.S. Dolk, W.P.M.H. Heemels, Dynamic periodic event-triggered control for linear systems, IEEE Trans. Autom. Control 58 (4) (2013) 847–861. [11] D. Yue, E. Tian, Q.L. Han, A delay system method to design of event-triggered control of networked control systems, in: IEEE Conference on Decision and Control and European Control Conference, December, 2011, pp. 1668–1673. [12] X. Wang, M.D. Lemmon, Event-triggering in distributed networked control systems, IEEE Trans. Autom. Control 56 (3) (2011) 586–601. [13] X.M. Zhang, Q.L. Han, Event-triggered dynamic output feedback control for networked control systems, IET Control Theory Appl. 8 (4) (2014) 226–234. [14] M. Cao, F. Xiao, L. Wang, Event-based second-order consensus control for multi-agent systems via synchronous periodic event detection, IEEE Trans. Autom. Control 60 (9) (2015) 2452–2457. [15] W. Hu, L. Liu, G. Feng, Output consensus of heterogeneous linear multi-agent systems by distributed event-triggered/self-triggered strategy, IEEE Trans. Cybern. 47 (8) (2017) 1914–1924. [16] X. Wang, M.D. Lemmon, Self-triggered feedback control systems with finite-gain stability, IEEE Trans. Autom. Control 54 (3) (2009) 452–467. [17] Y. Tang, H. Gao, J. Kurths, Robust H∞ self-triggered control of networked systems under packet dropouts, IEEE Trans. Cybern. 46 (12) (2015) 3294–3305. [18] C. Peng, Q.L. Han, On designing a novel self-triggered sampling scheme for networked control systems with data losses and communication delays, IEEE Trans. Ind. Electron. 63 (2) (2016) 1239–1248. [19] C. Peng, T.C. Yang, Event-triggered communication and H∞ control co-design for networked control systems, Automatica 49 (5) (2013) 1326–1332. [20] W.A. Zhang, L. Yu, New approach to stabilisation of networked control systems with time-varying delays, IET Control Theory Appl. 2 (12) (2008) 1094–1104.

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CHAPTER

18

Observer-based stabilization of nonlinear discrete-time systems using sliding window of delayed measurements

Noussaiba Gasmia , Mohamed Boutayeba , Assem Thabetb , Mohamed Aounb a CRAN

b MACS

UMR CNRS 7039, University of Lorraine, Cosnes et Romain, France Laboratory, National Engineering School of Gabes (ENIG), University of Gabes, Gabes, Tunisia

Chapter outline 1 2 3 4

Introduction....................................................................................... Problem statement............................................................................... Convergence analysis........................................................................... Converting BMI into LMI ........................................................................ 4.1 Particular solution: The case of two measurements ............................ 4.2 Particular solution: The case of M measurements .............................. 5 Discussion on the enhancement .............................................................. 5.1 Classical approach................................................................... 5.2 Sliding window approach vs. classical approach ................................ 6 Illustrative examples ............................................................................ 6.1 Example 1 ............................................................................ 6.2 Example 2 ............................................................................ 7 Conclusion ........................................................................................ 8 Appendix .......................................................................................... 8.1 Differential mean value theorem .................................................. 8.2 Schur’s lemma ....................................................................... References...........................................................................................

367 369 374 375 376 376 377 377 378 379 380 383 383 384 384 384 384

1 Introduction Most of the existing works on control design for linear and nonlinear systems assume that all the components of the state vector are available [1–3]. On the other hand, the Stability, Control and Application of Time-Delay Systems. https://doi.org/10.1016/B978-0-12-814928-7.00018-4 © 2019 Elsevier Inc. All rights reserved.

367

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CHAPTER 18 Stabilization of nonlinear discrete-time systems

size of the output vector is smaller than the size of the state vector for several reasons (technical implementation, cost, etc.). Therefore, at a given time t, the state cannot be deduced algebraically from the output measurements. In control theory, a state observer is a dynamical system that mirrors the behavior of a physical system. From the output measurements, he/she provides an estimate of the states of the system and gives an approximation as close as possible to reality. For this reasons, observer design for linear and nonlinear systems attracts the attention of many researchers [4–7]. The state observer theory was first introduced by Kalman and Bucy in Ref. [8] for a linear system in a stochastic environment. Then, Luenberger presented a general theory of observers for deterministic linear systems [9]. After that, many works on linear and nonlinear estimation were developed using the famous Luenberger observer [10–12]. The existing works on observer design for Lipschitz systems deal with the continuous case [13–15], and also the discrete case [4,11]. However, there is no result that deals with the design of sliding window filter for this class of nonlinear systems despite the superiority and the good performances proved with the extended Kalman filter in Refs. [16,17]. In the literature, there are many researchers that focus on the observer-based controller design for nonlinear systems. For instance, in Ref. [1], linear matrix inequality (LMI) condition for observer-based stabilization of Lipschitzian systems is given. The authors proposed to compute the controller and observer gains in two steps. On the other hand, in Ref. [18], the proposed design methodology uses a diagonal Lyapunov matrix and allows to compute the observer and controller gains simultaneously via a unique LMI. Then, a recent publication of Kheloufi et al. [19] presents a useful design procedure to synthesize a decentralized observer-based stabilization for nonlinear interconnected systems using a symmetric Lyapunov function. To linearize the obtained constraint, the authors proposed to use a slack variable. In this chapter, the problem of designing H∞ sliding window observer-based controller for nonlinear discrete-time systems in the presence of disturbances is addressed. The considered nonlinearity is assumed to be Lipschitz. The main contribution lies in the use of a sliding window of delayed measurements in the observer structure and a sliding window of delayed states in the control low. Thanks to the introduction of a relevant Lyapunov function with a slack variable technique inspired from Ref. [20], the observer and the controller gains can be computed simultaneously through a less restrictive constraint. In the following, we summarize the improvements with respect to existing results: • First, this chapter investigates the problem of nonlinear control in the presence of disturbances with bounded energy. • One of the practical points guaranteed by the constraint presented in this contribution is the possibility of computing the controller and the observer gains simultaneously through the same LMI, which is not the case in many existing works [1,21]. • Sliding windows of delayed measurements and delayed states are added to the Luenberger observer and the classical feedback control law, respectively, in

2 Problem statement

•

order to ensure better performances and to provide a more general and less restrictive LMI condition. As we know, the problem of using previous measurements with an observer-based controller has not been tackled before. In order to avoid bilinearities caused by the Lyapunov function, a slack variable is introduced to linearize the obtained constraint [19,20].

The remainder of this chapter is organized as follows. In the following section, the problem formulation is introduced. Then, the synthesis procedure of the sliding window H∞ observer-based controller is detailed in Section 3. Section 4 presents an interesting comparison with the classical approach. Finally, two numerical examples are considered to illustrate the effectiveness and the superiority of the proposed design methodology. Notation. The following notation will be used throughout this chapter: ⎛ ⎞T ith  ⎜ n 1 , 0, . . . , 0 ⎟ • en (i) = ⎝ 0, . . . , 0,   ⎠ ∈ R , n  1, is a vector of the canonical n-components basis of Rn . • ST is the transposed matrix of S. • S is a square matrix. The notation S > 0 (S < 0) means that S is positive definite (negative √ definite). • S = ST S is the Euclidean vector norm. • In represents the identity matrix of dimension n. •  S l2 represents the l2 norm of the vector S ∈ Rn and is defined as ∞ 2  S l2 = k=0  Sk  . • In a matrix, the notation () is used for the blocks induced by symmetry.

2 Problem statement This section is interested in providing an efficient new H∞ observer-based controller. The idea consists in adding a sliding window of delayed measurements to the Luenberger observer and to use a new reformulation of the feedback controller including delayed states. The interest of using delayed states and measurements on the feedback controller and the observer’s structure increases the degree of freedom of the constraint to be verified. Consider the class of nonlinear discrete-time systems described by 

x(k + 1) = Ax(k) + Bu(k) + f (x(k)) + W1 ω(k) y(k) = Cx(k) + W2 ω(k)

(1)

where x(k) ∈ Rn , u(k) ∈ Rm , y(k) ∈ Rp , and ω(k) ∈ Rr are the state, the input, the output, and the disturbance vectors, respectively. A ∈ Rn×n , B ∈ Rn×m , C ∈ Rp×n ,

369

370

CHAPTER 18 Stabilization of nonlinear discrete-time systems

W1 ∈ Rn×r , and W2 ∈ Rp×r are constant matrices of adequate dimensions. f : Rn → Rn is the nonlinear function which is assumed to be globally Lipschitz. The pairs (A, B) and (A, C) are assumed to be stabilizable and detectable, respectively. Now, let us consider the following sliding window observer: ⎛

⎞ y(k) − Cˆx(k) ⎜ ⎟ y(k − 1) − Cˆx(k − 1) ⎜ ⎟ xˆ (k + 1) = Aˆx(k) + Bu(k) + f (ˆx(k)) + L ⎜ ⎟ .. ⎝ ⎠ . y(k − N + 1) − Cˆx(k − N + 1)

(2)

coupled with a state estimated feedback controller ⎛

⎞ xˆ (k) N ⎜ xˆ (k − 1) ⎟  ⎜ ⎟ u(k) = Ki xˆ (k − i + 1) = K ⎜ ⎟ .. ⎝ ⎠ . i=1 xˆ (k − N + 1)

(3)

    where xˆ (k), N, L = L1 L2 · · · LN and K = K1 K2 · · · KN represent, respectively, the state estimate, the number of measurements, the observer, and the controller gain matrices. Remark 1. The idea behind using a sliding window approach in the synthesis of the H∞ observer-based controller is to improve the disturbance rejection by using a fixed number of delayed states (x(k), x(k − 1), . . . , x(k − N + 1)) and measurements (y(k), y(k − 1), . . . , y(k − N + 1)). This will increase the accuracy and the robustness of estimation. This new formulation is a significant contribution, contrary to conventional approaches considering only the last available information. The state Eq. (1) can be rewritten in a new form containing the delayed measurements. So, the new state vector is described by X (k + 1) = AX (k) + FBu(k) + Ff (x(k)) + W1  (k)

where ⎛

⎞ x(k) ⎜ x(k − 1) ⎟ ⎜ ⎟ X (k) = ⎜ ⎟ .. ⎝ ⎠ . x(k − N + 1) ⎛ ⎞ ω(k) ⎜ ω(k − 1) ⎟ ⎜ ⎟  (k) = ⎜ ⎟ .. ⎝ ⎠ . ω(k − N + 1)

(4)

2 Problem statement

⎛

A ⎜In ⎜ ⎜ A=⎜ ⎜0 ⎜. ⎝ .. ⎛

0 W1 0 .. . 0

⎜ ⎜ W1 = ⎜ ⎝

0 0 .. . .. . ···

··· ··· .. . .. . 0

··· ···

···

0

..

⎞ 0 0⎟ ⎟ .. ⎟ .⎟ ⎟ .. ⎟ .⎠

. In 0 ⎞ ··· 0 · · · 0⎟ ⎟ .⎟ .. . .. ⎠ 0 0

0 0 .. . ···

and  F = In

0

T

Then, we obtain a new reformulation of the sliding window observer (2) Xˆ (k + 1) = AXˆ (k) + FBu(k) + Ff (ˆx(k)) + F LCε(k) + F LW2  (k)

(5)

where ε(k) = X (k) − Xˆ (k) is the estimation error ⎛ N times ⎞    C = block-diag ⎝C, . . . , C⎠

and

⎛

N times

⎞

   W2 = block-diag ⎝W2 , . . . , W2 ⎠

Then the dynamic of the estimation error ε(k + 1) is given by ε(k + 1) = X (k + 1) − Xˆ (k + 1)

(6)

Using Eqs. (3)–(6), the state estimate and the dynamic of the estimation error can be rewritten as follows: Xˆ (k + 1) = (A + F BK)Xˆ (k) + Ff (ˆx(k)) + F LCε(k) + F LW2  (k)   ε(k + 1) = (A − F LC)ε(k) + F f x(k), xˆ (k) + (W1 − F LW2 ) (k)

with x(k) = F T X (k) xˆ (k) = F T Xˆ (k)

(7) (8)

371

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CHAPTER 18 Stabilization of nonlinear discrete-time systems

and   f x(k), xˆ (k) = f (x(k)) − f (ˆx(k)) As stated previously, f (.) is globally Lipschitz. We assume that f (0) = 0, then by applying Lemma 1 (see Section 8) to this nonlinear function, we obtain f (ˆx(k)) = Σ1 (Θ)ˆx(k) = Σ1 (Θ)F T Xˆ (k)

(9)

and 



f x(k), xˆ (k) = Σ1 (Θ)(x(k) − xˆ (k)) = Σ1 (Θ)F T ε(k)

(10)

with Σ1 (Θ) =

n,n 

ϕij Hij

i,j=1

Hij = en (i)eTn (j)

The parameter Θ belongs to the bounded convex set Hn , for which the set of vertices is defined by    VHn = ϕij ∈ Rn×n , ϕij ∈ f ij , f ij The observer gain matrix L and the control gain matrix K are unknown matrices to be determined such that xˆ k converges asymptotically to xk . Hence, we can define an augmented system described by the following structure: ¯ x(k) + Σ x¯ (k) + W ¯ (k) x¯ (k + 1) = A¯

(11)

with   Xˆ (k) ε(k)   A + F BK F LC A¯ = 0 A − F LC   T 0 FΣ1 (Θ)F Σ= 0 FΣ1 (Θ)F T

x¯ (k) =

and ¯ = W



F LW2 W1 − F LW2



The synthesis of the sliding window H∞ observer-based controller corresponding to the augmented system (11) return to search the gain matrices L and K that guarantee

2 Problem statement

the convergence of the vector x¯ asymptotically toward zero; that is, we must find the parameters L and K such that  H(x(k) − xˆ (k)) l2 ≤ μ  ω(k) l2

for

x(0) − xˆ (0) = 0

(12)

H is a known matrix and μ > 0 is the disturbance attenuation level that will be minimized. The resolution of this problem returns to search a Lyapunov function V(k) so that ¯ F¯ T x¯ (k) − Λ = V(k) + x¯ T (k)FH

with

 H=

and

0 HT









0 0

H =

0

 F F¯ = 0

μ2 T  (k) (k) < 0 N

0 F

0

(13)



HT H



Then, let us consider the following candidate Lyapunov function for system (1): V(k) = x¯ T (k)P¯x(k)

(14)

where P = PT > 0 is the Lyapunov matrix. Usually, to solve this kind of problem, most existing works on observer-based controller design for linear and nonlinear systems, as Refs. [18–22], use a particular form of the matrix P = diag(P1 , P2 ). This specific form allows to overcome the problem of bilinearities and to simplify the calculation. The resulting LMI remains restrictive due to the use of this particular form. In this contribution, a symmetric Lyapunov matrix P is used to get a more relaxed LMI. For that, consider the following form of the Lyapunov matrix:  P=

P11 PT12

P12 P22

 (15)

Define V(k) = V(k + 1) − V(k). Then, along the solution of the augmented system (11), we have 

x¯ k Λ= ωk

T



x¯ k Ξ ωk

 (16)

where ⎛ Ξ =⎝

¯ F¯ T (A¯ + Σ)T P(A¯ + Σ) − P + FH ()

⎞ ¯ (A¯ + Σ)T PW ⎠ μ2 ¯ − ¯ T PW Ir×N W N

(17)

373

374

CHAPTER 18 Stabilization of nonlinear discrete-time systems

Note that Λ < 0 is satisfied if Ξ < 0 which is equivalent to ⎛

¯ F¯ T −P + FH

0

⎝

⎞

⎠+ μ2 − Ir×N N

0

   (A¯ + Σ)T P A¯ + Σ T ¯ W

 ¯ 0 means that A is positive definite. (∗) denotes the transpose of the off-diagonal parts of a matrix.

2 Problem statement We consider a bilinear system affected by multiple time delays, where the dynamics is described by the following state space model: ⎛ x˙ (t) = ⎝A0 +

m  i=1

⎞

m  Ai ui (t)⎠ x(t) + Ad0 x(t − τ0 ) + Adi ui (t)x(t − τi ) + Bu(t) + Gθ + Dw(t) i=1

(1a) θ˙ = 0

(1b)

y(t) = Cx(t)

(1c)

z(t) = Lx(t)

(1d)

407

408

CHAPTER 20 Functional adaptive H∞ observer design

where x ∈ Rn , u ∈ Rm , y ∈ Rp , and w ∈ Rs are the state vector, the input, the output, and the disturbance vector, respectively. θ ∈ Rq is the unknown parameters vector. z ∈ Rr is a vector of the states to be estimated. τi , i = 0, . . . , m are known constant delays. The matrices Ai , Adi i = 0, . . . , m, B, L, C, G, and D are known with constant values and with appropriate dimensions. Assumption 1. The input u(t) is continuous and bounded, such that u(t) ∈ U ⊂ Rm , where U = {u: t → Rm /∀t ∈ R+ , ui,min ≤ ui (t) ≤ ui,max , μi,min ≤ u˙ i (t) ≤ μi,max

for i = 1, . . . , m}

(2)

Since Assumption 1 holds, we have put the inputs ui (t), for i = 1, . . . , m, and their derivatives in the same vector δ as follows: ⎡

T δ2m

⎤

⎡

⎤ u1 ⎢ ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢um ⎥ δ=⎢ ⎥=⎢ ⎥ ⎢δm+1 ⎥ ⎢ u˙ 1 ⎥ ⎢ . ⎥ ⎢ . ⎥ ⎢ . ⎥ ⎢ . ⎥ ⎣ . ⎦ ⎣ . ⎦ δ1 .. . δm

(3)

u˙ m

one can see that the vector δ belongs to a convex polytope, described by P = [u1,min , u1,max ] × · · · × [um,min , um,max ] × [μ1,min , μ1,max ] × · · · × [μm,min , μm,max ] (4)

Let us note Φ the set of vertices of the convex polytope P, where its cardinality is equal to 22m , and described by Φ = {σ = [φ1 , . . . , φ2m ]T ∈ R2m /∀i ∈ [0, m], φi ∈ {ui,min , ui,max } η(t) ˙ = N0 η(t) +

m 

and ∀i ∈ [m + 1, 2m], φi ∈ {μi,min , μi,max }}

Ni ui η(t) + Nd0 η(t − τ0 ) +

i=1 m 

+ M0 y(t) +

m 

(5)

Ndi ui η(t − τi ) + Fu(t)

i=1

Mi ui y(t) + Md0 y(t − τ0 ) +

i=1

m 

Mdi ui y(t − τi ) + Q0 θˆ

(6a)

i=1

θ˙ˆ (t) = Qθ η(t) + Mθ y(t)

(6b)

zˆ(t) = η(t) + Ey(t)

(6c)

where η ∈ Rr is the observer state vector. zˆ ∈ Rr and θˆ ∈ Rq are, respectively, the estimates of z and θ . The matrices Ni ∈ Rr×r , Mi ∈ Rr×p , Ndi ∈ Rr×r , and Mdi ∈ Rr×p , for i = 0, . . . , m, F ∈ Rr×m , Q0 ∈ Rr×q , Mθ ∈ Rq×p , Qθ ∈ Rq×r , and E ∈ Rr×p are unknown matrices, which should be determined such that the estimation of the states functional zˆ converge to its real values z. As mentioned in the

2 Problem statement

introductory section, the problem of the design of observer can be formulated as the problem of stability analysis of the estimation error. The aim from the observer design is to provide a simultaneous state and unknown parameters estimation which converge asymptotically to their real values (the system states and the system unknown parameters, respectively), that is, the estimation errors of both the states and the unknown parameters vectors converge asymptotically to zero. Namely, zero is an asymptotically stable equilibrium for the estimation errors. For that reason, the stability of the estimation errors will be studied in the sequel. So, we note ered (t) the estimation error vector described by  ez (t) ered (t) = eθ (t)

(7)

where ez (t) = z(t) − zˆ(t) is the estimation error of the functional z(t), given by ez (t) = z(t) − zˆ(t) = Lx(t) − η(t) − Ey(t) = (L − EC)x(t) − η(t) = Ψ x(t) − η(t)

(8)

and eθ (t) is the estimation error of the unknown parameter vector θ given by eθ (t) = θ − θˆ (t) is the estimation error of the unknown parameter vector θ. The convergence of the adaptive observer (6) is guaranteed via the following theorem. Theorem 1. System (6) is an adaptive observer for the delayed considered system described by Eq. (1), if the estimation error system given by the following differential equation: e˙ red (t) = N(u)ered (t) + Nd0 ered (t − τ0 ) +

m 

Ndi ui (t)ered (t − τi ) + Dw(t)

(9)

i=1

where

N(u) =

N0 +

m

i=1 Ni ui (t) Qθ

ΨG 0



N d0 =

N d0 0



0 0

N di =

N di 0

0 0



D=



ΨD 0

is stable, and the following equations hold for i = 0, . . . , m: Ψ G = Q0

(10a)

ΨB = F

(10b)

Qθ Ψ + Mθ C = 0

(10c)

Ψ Ai − Ni Ψ − Mi C = 0

(10d)

Ψ Adi − Ndi Ψ − Mdi C = 0

(10e)

Proof. We assumethat the gain Q0 satisfy Eq. (10a). A computation of the dynamics of the states estimation error leads to the following differential equations:

409

410

CHAPTER 20 Functional adaptive H∞ observer design

e˙ z (t) = N0 ez (t) +

m 

Ni ui (t)ez (t) + Nd0 ez (t − τ0 ) +

i=1

m 

Ndi ui (t)ez (t − τi ) + (Ψ B − F)u(t)

i=1

+ (Ψ A0 − N0 Ψ − M0 C)x(t) +

m 

(Ψ Ai − Ni Ψ − Mi C)ui (t)x(t) + Ψ Geθ (t)

i=1

+ (Ψ Ad0 − Nd0 Ψ − Md0 C)x(t − τ0 ) +

m  (Ψ Adi − Ndi Ψ − Mdi C)ui (t)x(t − τi ) i=1

(11)

and the dynamics of the unknown parameters estimation error is as follows: e˙ θ (t) = Qθ ez (t) − (Qθ Ψ + Mθ C)x(t)

(12)

One can see that since Eqs. (10b)–(10e) are satisfied, the estimation error system dynamics, given by Eq. (9), is obtained. Those equations ensure the unbiasedness of the estimation error. For that reason, the dynamics of the estimation error must be independent of the state x(t), x(t − τi ), for i = 0, . . . , m, and the input u(t). So, the terms depending on those vectors are canceled Eqs. (11), (12). Based on Theorem 1, the observer design is made into two steps. The first step (see Section 3) consists in the resolution of Eqs. (10c)–(10e) which allows to deduce the existence condition of the proposed adaptive observer and the parameterization of the observer gains Ni , Mi , Ndi , Mdi (for i = 0, . . . , m), E and Qθ with a unique gain. Thus, from this parameterization, the complexity of the problem analysis is reduced. One can see that the resolution of Eqs. (10c)–(10e) allows to obtain the observer gains Ni , Mi , Ndi , Mdi (for i = 0, . . . , m), E and Qθ , and then to deduce the gains F, Q0 , and Mθ . In a second step (see Section 4), using the parameterization of the observer gains made in the first step, the stability conditions of the augmented estimation error system described by Eq. (9) are given in terms of the solvability of LMIs on the vertices of the convex polytope P, using a Lyapunov-Krasovskii approach based on a descriptor transformation which allows to compute the observer gains. Before starting the observer design, we give some useful relations used in this chapter. For x ∈ Rn and y ∈ Rn , the following well-known inequalities hold ∀ > 0: 1 xyT + yxT ≤ xxT + yyT

(13)

3 Existence conditions and gain parameterization In this respect, Eqs. (10c)–(10e) can be rewritten as follows: 0 = Qθ L + K0θ C

(14a)

LAi = J z CAi + Niz L + Kiz C,

for i = 1, . . . , m

(14b)

LAdi = J z CAdi + Ndz L + Kdz C, i i

for i = 1, . . . , m

(14c)

3 Existence conditions and gain parameterization

where K i = Mi − N i E

(15a)

Kdi = Mdi − Ndi E Kθ = Mθ − Qθ E

(15b) (15c)

Therefore, Eqs. (14a)–(14c) can be rewritten in a compact form assimilated to a linear equation as follows: Bred = Xred Ared

(16)

where Bred and Ared are known matrices described as follows:

 LA0 LA1 . . . LAm LAd0 . . . LAdm 0 ... ... 0 0 ... 0 ⎤ ⎡ L 0 ... 0 0 ... 0 ⎢ 0 L 0 0 0 ... 0 ⎥ ⎥ ⎢ ⎥ ⎢ . . ⎢ 0 . 0 0 ... 0 ⎥ 0 ⎥ ⎢ ⎢ 0 0 0 L 0 ... 0 ⎥ ⎥ ⎢ ⎥ ⎢ 0 ... 0 L 0 0 ⎥ ⎢ 0 ⎥ ⎢ .. ⎥ ⎢ ⎥ ⎢ 0 . 0 0 . . . 0 0 ⎥ ⎢ ⎥ ⎢ 0 0 . . . 0 0 0 L ⎥ ⎢ ⎢ 0 ... 0 0 ... 0 ⎥ Ared = ⎢ C ⎥ ⎢ 0 C 0 0 0 ... 0 ⎥ ⎥ ⎢ ⎥ ⎢ .. ⎥ ⎢ . 0 0 ... 0 ⎥ 0 ⎢ 0 ⎥ ⎢ 0 0 C 0 ... 0 ⎥ ⎢ 0 ⎥ ⎢ ⎢ 0 0 ... 0 C 0 0 ⎥ ⎥ ⎢ ⎥ ⎢ .. ⎥ ⎢ 0 . 0 0 . . . 0 0 ⎥ ⎢ ⎣ 0 0 ... 0 0 0 C ⎦ CA0 CA1 . . . CAm CAd0 . . . CAdm

Bred =

(17a)

(17b)

and Xred is an unknown matrix containing the observer gains to be determined and described as follows:

Xred =

N 0 N 1 . . . Nm N d0 . . . Ndm K 0 K 1 . . . Km K d0 . . . Kdm E 0 . . . 0 Kθ 0 . . . 0 0 ... 0 0 Qθ 0 . . . 0



(18)

In fact, Eq. (16) admits a solution if and only if the following rank condition is satisfied: 

rank(Ared ) = rank

Bred Ared



(19)

and the general solution has the following form: †

†

Xred = Bred Ared + Zred (Ik − Ared Ared )

(20)

411

412

CHAPTER 20 Functional adaptive H∞ observer design

†

where k = 2(m + 1)(n + p) + p, Zred ∈ R(r+q)×k is an arbitrary matrix, and Ared is any generalized inverse of Ared , which fulfils the following relation [15]: †

Ared = Ared Ared Ared

If condition (19) is satisfied and using Eqs. (18), (20), the observer’s gains can be written in the following form:

N0 Qθ =

N1 0

... ...

Nm 0

N d0 0

... ...

Ndm 0

K0 Kθ

K1 0

... ...

Km 0

K d0 0

... ...

Kdm 0

 E 0 

N0,1 N1,1 . . . Nm,1 Nd0 ,1 . . . Ndm ,1 K0,1 K1,1 . . . Km,1 Kd0 ,1 . . . Kdm ,1 E1 0 0 ... 0 0 ... 0 0 0 ... 0 0 ... 0 0   + Zred,1 Zred,2 

N . . . Nm,2 Nd0 ,2 . . . Ndm ,2 K0,2 K1,2 . . . Km,2 Kd0 ,2 . . . Kdm ,2 E2 N × 0,2 1,2 Qθ,2 0 . . . 0 0 . . . 0 Kθ,2 0 . . . 0 0 ... 0 0 (21)

where Zred,1 ∈ Rr×kred and Zred,2 ∈ Rq×kred . With the observer gains expressions given in Eq. (21), system (9) is rewritten as follows: e˙ red (t) = N(u)ered (t) + Nd0 ered (t − τ0 ) +

m 

Ndi ui (t)ered (t − τi ) + Dw(t)

(22)

i=1

where   (N0,1 + Zred,1 N0,2 ) + m i=1 (Ni,1 + Zred,1 Ni,2 )ui (t) (L − E1 C − Zred,1 E2 C)G 0 Zred,2 Qθ,2  

(Nd0 ,1 + Zred,1 Nd0 ,2 ) 0 (Ndi ,1 + Zred,1 Ndi ,2 ) 0 N d0 = N di = 0 0 0 0 

(L − E1 C − Zred,1 E2 C)D D= 0

N(u) =

According to Theorem 1, since Eqs. (10a)–(10e) and condition (19) are satisfied, it remains to give the stability conditions of the estimation error where the dynamics is described by Eq. (22).

4 Stability conditions for gain computation Based on the observer gains parameterization given by Eq. (21), we assume that Assumption 1 and condition (19) are satisfied. Now, it remains to ensure the stability of the estimation error ered (t). Using a Lyapunov-Krasovskii approach for the stability analysis of time-delay systems, the stability conditions of the estimation error are given in the following theorem.

4 Stability conditions for gain computation

Theorem 2. Assume that Assumptions 1 holds and condition (19) is satisfied. The estimation error, which dynamics is described by Eq. (22) is quadratically stable for σ j ∈ Φ, j = 1, . . . , 22m , that is, system (6) is a reduced order adaptive observer for system (1), if there exist matrices P(σ j ) ∈ R(r+q)×(r+q) where

•



P(σ j ) = PT (σ j ) =

P01 PT02

 m  j Pi1 + σi PTi2 P03 i=1 P02



Pi2 Pi3

 >0

(23)

Pi1 ∈ Rr×r , Pi2 ∈ Rr×q and Pi3 ∈ Rq×q , for i = 0, . . . , m, M ∈ R(r+p)×(r+p) , such that

•

M=

S2 M22 M22

M11 S1 M11



(24)

where M11 ∈ Rr×r and M22 ∈ Rq×q are nonsingular matrices. S1 ∈ Rq×r and S2 ∈ Rr×q are some tuning matrices. Y11 ∈ Rr×p and Y22 ∈ Rq×p scalars > 0 and γ > 0,

• •

such that the following LMIs ⎡ j j j a(1,1) a(1,2) a(1,3) ⎢ j j ⎢ ∗ a(2,2) a(2,3) ⎢ ⎢ ⎢ ∗ a(3,3) ⎢ ∗ ⎢ ⎢ ∗ ∗ ∗ ⎢ ⎢ ⎢ ∗ ∗ ∗ ⎢ ⎢ ∗ ∗ ⎢ ∗ ⎢ ⎢ ⎢ ∗ ∗ ∗ ⎢ ⎢ ∗ ∗ ∗ ⎢ ⎢ ⎢ .. .. .. ⎢ . . . ⎢ ⎢ ⎢ ∗ ∗ ∗ ⎢ ⎢ ∗ ∗ ∗ ⎣ ∗

∗

∗

j

a(1,4)

a0

j

0

a1

a(2,4) S1 a0

0

S1 a1

a(3,4) a0 a(4,4) S1 a0

0 0

j

j

0

...

0

. . . S1 am

a1 j S1 a1

0 0

... ...

... ...

... ...

j

j

−1 I

r

0

∗

−1 I

q

... 0

∗ ∗

∗ ∗

∗ ∗

−1 I

r

.. .

.. .

∗ ∗

∗ ∗

... ...

... ...

... ...

∗ ...

∗

∗

∗

∗

∗

...

0

∗

−1 I

q

... 0

..

..

..

.

.

0

a(1,5)

⎤

⎥ S1 a(1,5) ⎥ ⎥ ⎥ ⎥ j am 0 a(1,5) ⎥ ⎥ j S1 am 0 S1 a(1,5) ⎥ ⎥ ⎥ ... 0 0 ⎥ ⎥ ⎥ ... 0 0 ⎥ ⎥ 0 ∈ R(r+q)×(r+q) , M ∈ R(r+q)×(r+q) and κ > 0. The derivative of the Lyapunov function V(ered ) along the trajectory of system (27) is given by T   

P(u) M Ir+q 0 e˙ red (t) ered (t) ˙ V(ered ) = εred (t) 0 0 ε˙ red (t) 0 M T  T

  ˙ 0 ered (t) e˙ red (t) Ir+q 0 P(u) ered (t) P(u) + + ε˙ red (t) 0 0 M T M T εred (t) εred (t) 0  m t−τ i 1 T (s)ε (s)ds − εred red κ t i=0 ⎡ ⎤T ⎡ ⎤ ⎤⎡ a(1,2) MD a(1,1) ered (t) ered (t) T = ⎣εred (t)⎦ ⎣ ∗ MD⎦ ⎣εred (t)⎦ −M − M w(t) w(t) ∗ ∗ 0 m m  t−τi   1 Si (t) − e˙ Tred (s)˙ered (s)ds + κ t i=0

  0 ered (t) 0 εred (t)

(29)

i=0

where ˙ a(1,1) = P(u) + MN(u) + NT (u)M T +

m 

MNdi ui (t) +

i=0

a(1,2) = P(u) − M + NT (u)M T +

m  i=0

NTdi ui (t)M T

m 

NTdi ui (t)M T

(30a)

i=0

(30b)

415

416

CHAPTER 20 Functional adaptive H∞ observer design

T   t−τi  P(u) M 0 ered (t) e˙ red (s)ds εred (t) 0 M Ndi ui (t) t T  

 t−τi P(u) 0 ered (t) 0 e˙ Tred (s)ds + T T εred (t) Ndi ui (t) M M t

Si (t) =

(30c)

According to the Lyapunov-Krasovskii approach for the stability of time-delay systems, the derivative of the Lyapunov function candidate has to be negative (see Ref. [16]). For this reason, an upper bound to the derivative of V(ered ) will be given ˙ red ), the term Si (t), for i = 0, . . . , m, can in the sequel. From the expression of V(e have an upper bound. Thus, an upper bound is given using inequality (13) as follows: T  T  P(u) 0 P(u) M 0 ered (t) εred (t) 0 M Ndi ui (t) Ndi ui (t) MT  t−τi 1 T (s)ε (s)ds + εred red

t

Si (t) ≤

0 MT

 ered (t) εred (t)



(31)

Hence, the derivative of the Lyapunov function V(ered ) has an upper bound as follows: ⎡

⎤T ⎡ ered (t) a(1,1) ˙ red ) ≤ ⎣εred (t)⎦ ⎣ ∗ V(e w(t) ∗

a(1,2) a(2,2) ∗

⎤⎡ ⎤ MD ered (t) MD⎦ ⎣εred (t)⎦ 0 w(t)

(32)

where ˙ a(1,1) = P(u) + MN(u) + NT (u)M T +

m 

MNdi ui (t) +

i=0

+

m 

m 

NTdi ui (t)M T

i=0

MNdi NTdi M T u2i (t)

(33a)

i=0

a(1,2) = P(u) − M + NT (u)M T +

m 

NTdi ui (t)M T +

i=0

a(2,2) = −M − M T +

m 

m 

MNdi NTdi M T u2i (t)

(33b)

i=0

MNdi NTdi M T u2i (t)

(33c)

i=0

The H∞ approach ensures that the mapping from w(t) to ered (t) has an L2 gain attenuation less or equal to the scalar γ , with γ > 0 called the level of disturbance attenuation and satisfies the following inequality under the zero initial conditions: Jered w =

 ∞ 0

(eTred ered − γ 2 wT w)dt < 0

(34)

4 Stability conditions for gain computation

Since the initial conditions are null, V(ered (0)) = 0 and V(ered (∞)) ≥ 0, the criterion Jered w described by Eq. (34) can be upper bounded as follows: Jered w ≤ ≤

 ∞ 0

 ∞ 0

(eTred ered − γ 2 wT w)dt + V(ered (∞)) − V(ered (0)) ˙ red ))dt < 0 (eTred ered − γ 2 wT w + V(e

(35)

Then, using inequality (32), inequality (35) becomes ⎡ ⎤ ⎡  ∞ ered (t) T a(1,1) + Ir ⎣εred (t)⎦ ⎣ ∗ Jered w ≤ 0 w(t) ∗

a(1,2) a(2,2) ∗

⎤⎡ ⎤ MD ered (t) MD ⎦ ⎣εred (t)⎦ dt < 0 w(t) −γ 2 Is

(36)

˙ However, to ensure the stability of the estimation error ered (t), V(e) must be ˙ red ) < 0, which is equivalent to the following inequality negative definite, that is, V(e ⎡

a(1,1) + Ir ⎣ ∗ ∗

a(1,2) a(2,2) ∗

⎤ MD MD ⎦ < 0 −γ 2 Is

(37)

where a(1,1) , a(1,2) , and a(2,2) are given, respectively, by Eqs. (33a)–(33c). NeverT T 2 theless, we note the presence of the nonlinear term m i=0 MNdi Ndi M ui (t) in the blocks a(1,1) , a(1,2) , and a(2,2) . To overcome this problem, Schur’s lemma is applied to inequality (37), which leads to the following inequality: ⎡

a(1,1) ⎢ ∗ ⎢ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎣ ∗

a(1,2) a(2,2)

MNd0 MNd0

MNd1 MNd1

... ...

MNdm MNdm

∗ ∗

−1 I

r+q

∗

−1 I

r+q

0

0 0

∗ ∗

... 0 .. . ∗

∗ ∗

∗ ∗

−1 I

r+q

∗

∗

∗

∗

∗

⎤ MD MD ⎥ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 (X < 0) means that X is positive definite√ (negative definite). X = X T X is the Euclidean vector norm. In a matrix, the notation () is used for the blocks induced by symmetry.

1.2 Preliminaries Lemma 1 (Differential mean value theorem). Let φ: Rn → Rq . Let x1 ∈ Rn , x2 ∈ Rn , suppose that φ is differentiable on Co(x1 , x2 ). Then, there are constant vectors z1 , . . . , zq ∈ Co(x1 , x2 ), zi = x1 , zi = x2 , i = 1, . . . , q such that ⎛

⎞

q,n

∂φi φ(x1 ) − φ(x2 ) = ⎝ eq (i)eTn (j) (zj )⎠ (x1 − x2 ) ∂xj i,j=1

(1)

where Co(x1 , x2 ) = {λx1 + (1 − λ)x2 , 0 ≤ λ ≤ 1} is the convex domain of x1 , x2 . Proof. The proof of this theorem is given in Ref. [21] Lemma 2 (The Schur complement lemma [18]). Given constant matrices M, N, and Q of appropriate dimensions where M and Q are symmetric, then Q > 0 and M + N T Q−1 N < 0 if and only if

M N

 NT 0, the H∞ sliding window observer design problem corresponding to system (5) and observer (6) is solvable if there exist matrices Qij ∈ Rn×n and P = PT > 0 ∈ Rn×m of appropriate dimensions such that the following BMI is feasible: min ⎛⎛ ⎜⎜ ⎜⎝ ⎜ ⎝

μ subject to −P + F F T () ()

0 μ2

− N Ir×N () ()

(A − F LC)T P

⎞

⎟ (W1 − F LW2 )T P⎠ −P



Ξ1

···

⎞ ⎟ Ξn ⎟ ⎟ 0 is the disturbance attenuation level to be minimized. Then, it is sufficient to find a Lyapunov function V(k) so that Γ = V(k) + (x(k) − xˆ (k))T (x(k) − xˆ (k)) −

μ2 T  (k) (k) < 0 N

(14)

3 Robust sliding window observer synthesis

Or x(k) = F T X (k) and x(k) − xˆ (k) = F T ε(k), then, inequality (14) is equivalent to Γ = V(k) + X T (k)F F T X (k) −

μ2 T  (k) (k) < 0 N

(15)

Then, let us consider the following candidate Lyapunov function: V(k) = X T (k)PX (k)

(16)

where P = PT > 0 is the Lyapunov matrix. Define V(k) = V(k + 1) − V(k). Then, we obtain

T  X (k) X (k) Π  (k)  (k)

Γ =

(17)

with Π=

(A − F LC + F Σ F T )T P(A − F LC + F Σ F T ) − P + F F T

(A − F LC + F Σ F T )T P(W1 − F LW2 )



2 (W1 − F LW2 )T P(W1 − F LW2 ) − μN Ir×N

()

(18)

Note that Γ < 0 is satisfied if Π < 0 which is equivalent to 



−P + F F T

0

0

2 − μN Ir×N

+

 (A − F LC + FΣF T )T PP−1 P T (W1 − F LW2 )   × (A − F LC + FΣF T ) (W1 − F LW2 ) < 0

(19)

Using Schur’s complement, inequality (19) is equivalent to ⎛

−P + F F T

0

() ()

− μN Ir×N ()

⎜ ⎝

(A − F LC + FΣF T )T P

2

(W1 − F LW2 )T P −P

⎞ ⎟ ⎠ 0, m − 1 < α˜ ≤ m, m ∈ N and for α˜ ≤ 0, m = 0.

443

444

CHAPTER 22 Identification of fractional Hammerstein system with delay

Definition. The Caputo’s operator of order α˜ of a function g is defined as C Dα˜ g(t) = a t

 t 1 g(m) (τ ) dτ ˜ Γ (m − a) a (t − τ )α−m+1

(2)

for α˜ > 0, m − 1 < α˜ ≤ m, m ∈ N for α˜ ≤ 0, m = 0. Definition. The GL operator of order α˜ of a function g is defined as t

GL Dα˜ g(t) = lim a t h→0

h 1 

hα˜ j=0

(−1)j

  α˜ j

g(t − jh)

where h is the time increment, j is the number of samples, and

(3)

  α˜ is the Newton j

binomial term given by the following relation:    1 α˜ = α( ˜ α−1)...( ˜ α−j+1) ˜ j j!

for j = 0 for j > 0

(4)

These definitions show that fractional differintegral operators are global operators having a memory of all the past, which makes them suitable for modeling hereditary effects in most materials and processes. However, for numerical computation of the fractional derivative, the drawback of the infinite memory which requires an important number of terms is circumvented by considering a limited length memory denoted L. A good approximation can be obtained using the discrete GL difference operator α˜ with the assumption of a sampling interval h = 1, and initial time equal to zero. α˜ g(k) =

L  (−1)j j=0

  α˜ g(k − j) j

(5)

where Eq. (5) can be written under the form α g(k) = 

L 

β(j)g(k − j)

with β(j) = (−1)j

j=0

β(0) = 1 β(j) = β(j − 1) (j−1)(α−1) j

for j = 1, . . . , L

  α j

(6)

(7)

In this study, the discrete GL operator, which is the most adequate to simulate discrete fractional systems is used for numerical calculations.

2 Fractional calculus

2.2 Fractional-order state space model Let us consider the discrete fractional-order state space model, based on GL difference operator given in Ref. [29], described by the following equations: α x(k + 1) = Ax(k) + Bu(k)

(8)

y(k) = Cx(k) + Du(k)

where x(k) ∈ Rn represents the state vector; u(k) and y(k) ∈ R represent, respectively, the input and the output of the system; A ∈ Rn×n , B ∈ Rn×1 , C ∈ R1×n , and D ∈ R1 are the system matrices; α is the fractional orders vector; and α x is the fractional state variables vector as follows:  α = α1

α2

 αn ,

···

 α x = α1 x1

α2 x2

···

αn xn

T

The fractional orders are completely different and the system is called a fractional noncommensurate system or a generalized fractional system; in the special case where the state variables are differentiated to the same order α, ˜ the system is called a commensurate-order system with α˜ = α1 = α2 = · · · = αn

and

 α˜ x(k + 1) = α˜ x1 (k + 1)

...

T

xn (k + 1)

The simulation of the fractional state space model (8) is performed using the GL operator α x(k + 1) = Ax(k) + Bu(k) x(k + 1) = α x(k + 1) −

k+1 

β(j)x(k + 1 − j)

(9)

j=1

y(k) = Cx(k) + Du(k)

with β(j) the matrix of the elements βi (j) as follows: β(j) = diagonal[βi (j)]

for i = 1, 2, . . . , n

with βi (j) = (−1)j

  αi j

(10)

Using a limited memory L, Eq. (9) can be rewritten as α x(k + 1) = Ax(k) + Bu(k) x(k + 1) = α x(k + 1) − [β(1)x(k) + β(2)x(k − 1) + · · · + β(L)x(k − L)]

(11)

y(k) = Cx(k) + Du(k)

The fractional system in Eq. (11) is a system with multiple state delays; it will be used to describe the linear part of the Hammerstein system.

445

446

CHAPTER 22 Identification of fractional Hammerstein system with delay

3 Problem definition Consider a fractional Hammerstein system shown in Fig. 1, which consists of a static nonlinear block (NL) followed by a fractional linear block (FL); the linear part is a fractional controllable fractional state space model equation (12) α x(k + 1) = A0 x(k) + B0 u˜ (k)

(12)

y˜ (k) = C0 x(k) + D0 u˜ (k)

where u(k) and y(k) are, respectively, the system input and the system output; y˜ (k) is the output of the linear part; and u˜ (k) is the nonlinear part output which is an intermediate variable. In this work, the nonlinear block is represented by a polynomial of order r with unknown coefficients pi (i = 1, 2, . . . , r). u˜ (k) = f (u(k)) =

r 

pi ui (k)

(13)

i=1

The system overall output is y(k) = y˜ (k) + v(k), where v(k) is the noise. Replacing u˜ (k) in Eq. (12) yields to the fractional Hammerstein model equation r 

α x(k + 1) = A0 x(k) + B0 y(k) = C0 x(k) + D0

pi ui (k)

i=1 r 

(14) pi ui (k) + v(k)

i=1

In order to normalize the Hammerstein model, the nonlinear part first coefficient p1 is set equal to 1 (p1 = 1). The Hammerstein model equation (14) is a state space model containing nonlinear terms, thus it is suitable to use the polynomial nonlinear state space (PNLSS) model which is the generalization of the state space model to nonlinear case; in our study, we extend it to the fractional case given by equations α  x(k + 1) = Ax(k) + Bu(k) + Eη(k) y(k) = Cx(k) + Du(k) + Fζ (k)

(15)

where the matrices A, B, C, and D describe the state space model linear part and the nonlinear part is represented by the matrices E ∈ R1×nη and F ∈ R1×nζ ; ζ (k) and η(k) are vectors containing the monomials expansion of u(k) and x(k), of degree 2 up to r. v(k) u(k)

FIG. 1 Hammerstein system.

NL

u˜(k)

FL

y˜(k)

+

y(k)

4 Identification method

Let us derive the relationship between the fractional Hammerstein system and the FPNLSS representation α x(k + 1) = A0 x(k) + B0 u(k) + B0

r 

pi ui (k)

i=2 r 

y(k) = C0 x(k) + D0 u(k) + D0

(16) pi ui (k) + v(k)

i=2

where A = A0

B = B0

C = C0 D = D0 E = [p2 B0 · · · pr B0 ] F = [p2 D0 · · · pr D0 ]

(17) (18)

The vectors ζ (k) and η(k) containing the powers of u(k) are equals under the form  ζ (k) = η(k) = u2 (k)

u3 (k)

···

ur (k)

T

(19)

The identification of the fractional Hammerstein system described by the FPNLSS (Eq. 15) is presented in the following section.

4 Identification method Let us consider the fractional Hammerstein system described by the FPNLSS model α  x(k + 1) = Ax(k) + Bu(k) + Eη(k) y(k) = Cx(k) + Du(k) + Fζ (k)

(20)

The fractional linear part is assumed to be completely observable and controllable with matrices ⎡

0 ⎢ .. ⎢ A0 = ⎢ . ⎣0 a1

1

0

0 a2

... ...

⎡ ⎤ ⎤ 0 0 ⎢ .. ⎥ .. ⎥ ⎢ ⎥ .⎥ ⎥ , B0 = ⎢ . ⎥ , C0 = [c1 , c2 , . . . , cn−1 , cn ], D0 = [d] ⎣0⎦ ⎦ 0 1 an−1 an 1 (21)  ...

 and α = α1 α2 · · · αn being the fractional orders vector. The matrices linking the fractional Hammerstein model to the FPNLSS representation is given as follows: A = A 0 B = B 0 C = C0 ⎡ 0 0 0 ... ⎢ .. ⎢ E=⎢ . ⎣0 0 ... 0 p2 p3 . . . pr−1

D = D0 ⎤ 0 .. ⎥ .⎥ ⎥ , F = [p2 D0 0⎦ pr

···

pr D0 ]

(22)

447

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CHAPTER 22 Identification of fractional Hammerstein system with delay

The objective of this work is the matrices A, C, D, E, F coefficients estimation and the fractional orders vector α. Hence, the parameter vector θ to be estimated is  θ= a

c

d

p

  α = θ˜

 α ∈ R nθ

with  θ˜ = a

c

 p ,

d

p = [1, p2 , . . . , pr ],

a = [a1 , . . . , an ],

c = [c1 , . . . , cn ],

α = [α1 , . . . , αn ]

(23)

The identification procedure is based on a nonlinear optimization approach using L-M algorithm, which is a blend of two well-known optimization methods: the gradient method and the Gauss-Newton method. It is based on the calculation of the gradient and the Hessian by developing the parametric sensitivity functions [30]. The mean quadratic prediction error of the output evaluates cost function J(θ ), it is given by J(θ) =

K K 1  2 1  ε (k) = (y(k) − yˆ (k))2 K K k=1

(24)

k=1

where K is the samples number, ε(k) is the prediction error, and yˆ (k) is the estimated output. The updating equations of the algorithm are as follows: ⎧ ⎪ θ (i+1) = θ (i) − {[J  + λI]−1 J  }θ=θ ˆ (i) ⎪ ⎪   ⎪ ⎪ K K ∂ yˆ (k) ⎪ 2  ⎪ = −2 Jθ = − K k=1 ε(k) ∂θ ⎪ k=1 ε(k)σyˆ (k)/θ the Gradient K ⎪ ⎨   T   ∂ y ˆ (k) ∂ y ˆ (k) K 2 2 T Jθ = K k=1 ∂θ = K K k=1 σyˆ (k)/θ σyˆ (k)/θ the Hessian ⎪ ∂θ ⎪ ⎪ ⎪ ˆ (k) ⎪ ⎪ σyˆ (k)/θ = ∂ y∂θ the output sensitivity function ⎪ ⎪ ⎪ ⎩ λ : a tuning parameter for the convergence

(25)

The gradient J  and the Hessian J  are calculated based on the sensitivity functions computation σyˆ (k)/θ . For the vector θ˜ , they are obtained by differentiating the FPNLSS with respect to each parameters of θ˜ 

α x(k + 1) = Ax(k) + Bu(k) + Eη(k)

y(k) = Cx(k) + Du(k) + Fζ (k)   ⎧ ∂x(k+1) α ⎪  = ∂A˜ x(k) + A ∂x(k) + ∂B˜ u(k) + B ∂u(k) + E ∂η(k) + ∂E˜ η(k) ⎪ ⎪ ˜ ∂ θ ∂ θi ∂ θ˜i ∂ θi ∂ θ˜i ∂ θ˜i ∂ θi ⎨ i ∂ yˆ (k) = ∂C˜ x(k) + C ∂x(k) + ∂D˜ u(k) + D ∂u(k) + F ∂η(k) + ∂F˜ η(k) ∂ θ˜i ∂ θi ∂ θ˜i ∂ θi ∂ θ˜i ∂ θ˜i ∂ θi i = 1, . . . , nθ˜

⎪ ⎪ ⎪ ⎩

where

∂B ∂ θ˜i

= 0,

∂u(k) ∂ θ˜i

= 0, and

∂η(k) ∂ θ˜i

= 0.

(26)

(27)

5 Simulation examples

Eq. (27) is reduced to  ⎧   ∂A 0 x(k) + ∂E η(k) α [σ ⎪ ⎪  ] = Aσ + ⎪ x(k+1)/θ˜i x(k)/θ˜i ⎪ ∂ θ˜i ∂ θ˜i u(k) ⎨  x(k)  ⎪ σyˆ (k)/θ˜ = Cσx(k)/θ˜ + ∂C˜ ∂D˜ + ∂F˜ η(k) ⎪ ⎪ i i ∂ θi ∂ θi ∂ θi u(k) ⎪ ⎩ i = 1, . . . , nθ˜

(28)

Note that σx(k)/θ˜i = ∂x(k) and σyˆ (k)/θ˜i = ∂ yˆ (k) are, respectively, the state sensitivity ∂ θ˜i ∂ θ˜i function and the output sensitivity function. The overall sensitivity functions model can be written under the FPNLSS form 

α [σx(k+1)/θ˜ ] = As σx(k)/θ˜ + Bs us (k) + Es ηs (k) σyˆ (k)/θ˜ = Cs σx(k)/θ˜ + Ds us (k) + Fs ηs (k)

(29)

As = Diagonal block [A], Cs = Diagonal block[C],     Bs = ∂A˜ 0 , Ds = ∂C˜ ∂D˜ , ∂θ ∂θ ∂θ     ∂E ∂F , Fs = , Es = ∂ θ˜ ∂ θ˜  T us (k) = x(k) u(k) , ηs (k) = [u2 (k) · · · ur (k)]T .

(30)

where

The sensitivity functions with respect to the fractional orders α are calculated using the output Taylor series of order 1 with respect to each αi (i = 1, 2, . . . , n) [16] 

yˆ (k) yˆ (k, αi + δαi ) − yˆ (k, αi ) ≈ δαi ∂∂α = δαi σyˆ (k)/αi i i = 1, . . . , n

(31)

The overall sensitivity functions vector σyˆ (k)/θ of the model is expressed as  σyˆ (k)/θ = σyˆ (k)/θ˜

σyˆ (k)/α

T

Using the parametric sensitivity functions, the gradient Jθ and the Hessian Jθ are expressed as follows: 

 ε(k)(σyˆ (k)/θ ) Jθ = − K2 K  k=1 T Jθ = K2 K (σ k=1 yˆ (k)/θ )(σyˆ (k)/θ )

(32)

5 Simulation examples Two simulation examples are considered, the first of one is a commensurate-order fractional Hammerstein system and a noncommensurate-order system in the second

449

450

CHAPTER 22 Identification of fractional Hammerstein system with delay

example. The input u(k) is a persistent excitation sequence of zero mean and unit variance, the disturbance v(k) is a white noise sequence of zero mean and the data length is K = 500. The first step is to obtain a good structure for each example. It is performed by the analysis the criteria evolution for different structures; the best structure corresponding to the smallest criterion. The identification is carried out in the absence of noise and in the presence of noise using the Monte Carlo simulations for different signal-to-noise ratios SNR = 34 dB and SNR = 25 dB.

Example 1: Fractional commensurate case The fractional commensurate Hammerstein system has its linear part and the nonlinear one as follows: u˜ (k) = f (u(k)) = u(k) + 0.75u2 (k) + 0.35u3 (k) α˜  x(k + 1) = A0 x(k) + B0 u˜ (k) y˜ (k) = C0 x(k) + D0 u(k)

(33) (34)

with ⎡

0 A0 = ⎣ 0 0.40

1 0 −0.10

⎤ 0 1 ⎦, −0.60

⎡ ⎤ 0 B0 = ⎣0⎦ , 1

(35)

C0 = [−0.20, −0.80, −0.70], D0 = [0.10].

The fractional order α˜ = 0.3.  θ = 0.40

−0.10

−0.60

−0.20

−0.80

−0.70

0.10

0.75

0.35

0.30



(36)

Before estimating the system parameters, the best structure is chosen based on several tests for different orders n and r. The evolution of the criteria J is shown in Fig. 2 and the obtained values of J are listed in Table 1. The results show that the orders n = 3 and r = 3 correspond to the best structure with J ≈ 1e − 31. Based on the best structure, the presented method is applied for the estimation of the parameter vector θ . Fig. 3 plots the simulation results for the noise-free case, the error is null and the estimated output overlaps with the data. In the presence of noisy measurements, a Monte Carlo simulation is performed for 50 sets of computer realizations for SNR = 34 dB and SNR = 25 dB. The results are summarized in Table 2, where the estimated parameters mean value is recorded with a satisfactory criterion accuracy (J ≈ 10−4 for SNR = 34 dB) and (J ≈ 10−1 for SNR = 25 dB). Figs. 4 and 5 show, respectively, the simulated versus the estimated outputs. It can be concluded that the obtained models show a perfect adequacy with the data.

5 Simulation examples

6

n = 3, r = 2 n = 2, r = 3 n = 3, r = 3

5

Criterion J

4

3

2

1

0

0

2

4

6

8 10 12 14 Number of iterations

16

18

20

22

FIG. 2 Evolution of the criteria versus the number of iterations for the commensurate example.

Table 1 Structure test results of the commensurate example Structure

n=2 r=3

n=3 r=2

n=3 r=3

J

0.670

0.433

7.4e − 31

The statistical performance of the estimator is analyzed on a Monte Carlo simulation, for different amount of noise and a good efficiency of the optimization method is obtained.

Example 2: Fractional noncommensurate example The Hammerstein linear part is a noncommensurate fractional state space model of order n = 2, with the fractional orders vector α = [0.4 0.6]. The linear part matrices are given here: 

0 A0 = −0.37

 1 , −0.58

C0 = [−0.10, −0.20],

  0 B0 = 1

D0 = [0.10]

The nonlinearity is described by the following polynomial of order r = 3: u˜ (k) = f (u(k)) = u(k) + 0.5u2 (k) + 0.25u3 (k)

(37)

451

CHAPTER 22 Identification of fractional Hammerstein system with delay

10

Simulated Estimated

5

Output

0 –5

Zoom –10 –15 0

50

100

150

200

250

300

350

400

450

500

110

120

130

140

150

160

170

180

190

200

200

250

k

5 0 –5 –10 100

(A) 4

× 10–1

3 2

Prediction error

452

1 0 –1 –2 –3 –4 –5 0

50

100

150

300

350

400

450

500

k

(B) FIG. 3

Identification results for the commensurate example for the noise-free case. (A) Simulated and estimated outputs; (B) prediction error.

Table 2 Monte Carlo simulation results of the commensurate example a2

a3

c1

c2

c3

a1

0.400 −0.100 −0.600 −0.200 −0.800 −0.700 0.100 0.749 0.350 0.320 1.5e − 4

25 dB

0.397 −0.105 −0.586 −0.206 −0.792 −0.709 0.098 0.736 0.344 0.314 0.103

Exact values 0.400 −0.100 −0.600 −0.200 −0.800 −0.70

d

p2

p3

α˜

SNR 34 dB

J

0.100 0.750 0.350 0.300 −

5 Simulation examples

20

Output

0 Zoom

–20

Simulated Estimated

–40 0

100

200

5

300

k

400

500

0 –5 –10 –15 100

110

120

130

140

150

160

250

300

170

180

190

200

(A) 0.03 0.02 0.01 0 –0.01 –0.02 –0.03 05

(B)

0

100

150

200

350

400

450

500

k

FIG. 4 Identification results for the commensurate example for SNR = 34 dB. (A) Simulated and estimated outputs; (B) prediction error.

453

CHAPTER 22 Identification of fractional Hammerstein system with delay

10 Simulated Estimated 0

Output

454

Zoom

–10

k

–20 0

50

100

150

200

250

300

350

400

450

500

110

120

130

140

150

160

170

180

190

200

300

350

5

0

–5

–10 100

(A) 1 0.8 0.6 0.4 0.2 0 –0.2 –0.4 –0.6 –0.8 –1 0

(B)

50

100

150

200

250

400

450

500

k

FIG. 5 Identification results for the commensurate example for SNR = 25 dB. (A) Simulated and estimated outputs; (B) prediction error.

5 Simulation examples

Table 3 Structure test results of the noncommensurate example Structure

n=2 r=2

n=2 r=3

n=3 r=3

J

0.132

1.2e − 33

1.1e − 07

1.4 n = 3, r = 2 1.2

n = 2, r = 2 n = 2, r = 3

Criterion J

1 0.8 0.6 0.4 0.2 0

0

2

4

6

8

10 12 14 Number of iterations

16

18

20

22

FIG. 6 Evolution of the criteria versus the number of iterations for the noncommensurate example.

The parameters vector to be estimated is as follows:  θ = −0.37

−0.58

−0.10

−0.20

0.10

0.50

0.25

0.40

0.60



(38)

The first step is the examination of the best structure. The obtained values of J for differents n and r ((n = 2, r = 2), (n = 2, r = 3), and (n = 3, r = 3)) are tabulated in Table 3 and the evolution of the different criteria is depicted in Fig. 6. The obtained results show that the best structure is recorded for the exact orders (n = 2, r = 3). The obtained results for the noise-free data and in the presence of noise, that is, SNR = 34 dB and SNR = 25 dB are shown, respectively, in Figs. 7–9. The values of the criterion J and the mean of the parameters using Monte Carlo simulations for 50 runs of noisy data are tabulated in Table 4. On the basis of the presented results, the following conclusions can be drawn: •

In the absence of noise, from the plot illustrated in Fig. 7, we can conclude that the prediction error is null  ≈ 10−16 ) and the estimated output overlap with the simulated one.

455

CHAPTER 22 Identification of fractional Hammerstein system with delay

2

0

Output

456

Zoom

–2

Simulated Estimated

–4 0

50

100

150

200

250

300

k

350

400

450

500

1 0 –1 –2 –3 100

110

120

130

140

150

160

170

180

190

200

(A) 1.5

× 10

–16

1

0.5

0

–0.5

–1

–1.5 0

(B)

50

100

150

200

250

300

350

400

450

500

k

FIG. 7 Noise-free identification results of the noncommensurate example. (A) Simulated and estimated outputs; (B) prediction error.

•

Figs. 8 and 9 depict the obtained results for noisy data. It shows that the prediction errors are very low and the estimated output and the simulated one overlap. • The mean values of the parameters and the errors are given in Table 4. It show that all the parameters are recovered in each case and the errors are very low. On the basis of these numerical simulations, we can conclude that the developed method efficiency is confirmed in the presence of noise and for noise-free data.

5 Simulation examples

2 Simulated Estimated

Output

1 0

Zoom

–1 –2 0

50

100

150

200

250

300

350

400

450

500

110

120

130

140

150

160

170

180

190

200

200

250

300

1

k

0 –1

(A)

–2 100 10

5

–3

0

–5

–10

(B)

0

50

100

150

350

400

450

500

k

FIG. 8 Identification results for the noncommensurate example for SNR = 34 dB. (A) Simulated and estimated outputs; (B) prediction error.

457

CHAPTER 22 Identification of fractional Hammerstein system with delay

2 1

Output

458

0

Zoom

–1

Simulated –2

Estimated

–3 0

50

100

150

200

250

300

350

400

450

500

110

120

130

140

150

160

170

180

190

200

450

500

2

k

1 0 –1 –2 100

(A) 0.2 0.15 0.1 0.05 0 –0.05 –0.1 –0.15 0

(B)

50

100

150

200

250

300

350

400

k

FIG. 9 Identification results for noncommensurate example for SNR = 25 dB. (A) Simulated and estimated outputs; (B) prediction error.

References

Table 4 Monte Carlo simulation results of the noncommensurate example SNR

a1

34 dB

−0.370 −0.580 −0.100 −0.200 0.100 0.500 0.250 0.435 0.538 2.8e − 6

a2

c1

c2

d

p2

p3

25 dB

−0.370 −0.579 −0.099 −0.199 0.100 0.502 0.252 0.419 0.447 0.004

α1

α2

J

Exact values −0.370 −0.580 −0.100 −0.200 0.100 0.500 0.250 0.400 0.600 −

6 Conclusion This chapter presents a novel identification method for the class of fractional nonlinear block-structured systems with time-delay. The fractional Hammerstein system is considered. The flexible polynomial nonlinear state space representation is used to describe the Hammerstein model. This allows a better parameterization of the model and reduces greatly the computational load. The robust L-M algorithm is developed for the parameters estimation of the Hammerstein model as well as its fractional orders. The proposed approach is based on the on parametric sensitivity functions computation, which are implemented as an FPNLSS model. Various simulations show the good performance of the proposed algorithm for the commensurate and the noncommensurate systems. In future work, it is worthwhile to investigate other block-oriented structures identification such as Wiener, Hammerstein-Wiener, etc.

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461

Index Note: Page numbers followed by f indicate figures, t indicate tables, and b indicate boxes.

A Absolute head-to-tail string stability, 72–74 Absolute internal stability, 72–74 Adaptive cruise control (ACC), 54, 163–164 Argument principle, 280, 283–284

B Banach space, 90 Bilinear matrix inequality (BMI), 374–375, 424 Bilinear time-delay system, functional H∞ adaptive observer constant delays, 389 control loop devices, 406 convergence analysis, 388–389 existence conditions, 392–393 stability conditions, 393–397 convex polytope, 389–390 delay-dependent stability problems, 388 delay independent analysis methods, 388 differential Riccati matrix, 387–388 estimation errors, 409–410 existence conditions and gain parameterization, 410–412 fault-tolerant control, 405 high-gain observer, 387–388 linear estimation error, 387–388 LMIs, 407, 410 Lyapunov-Krasovskii approach, 407, 410 simulation and results, 397–399, 418–421 stability conditions, gain computation, 412–418 state space model, 389, 407–408 unknown parameters vector, 406 Bilinear transformation, 2, 9–10, 19 BMI. See Bilinear matrix inequality (BMI)

C CACC. See Cooperative adaptive cruise control (CACC) Cartesian impedance control, 25–26 CCC. See Connected cruise control (CCC) Characteristic quasipolynomial approximation, 2, 8, 12–13, 16–19 Cluster treatment of characteristic roots (CTCR) paradigm, 2–3, 8–9, 13–14, 15f, 16, 16f, 17f, 18, 20, 124, 133–134

Commensurate-order system, 445 Connected cruise control (CCC) adaptive control scheme, 54–55 beyond-line-of-sight identification, 54–55 connected vehicle system, 71–74 control framework, 55–58 frequency-domain stability analysis, 75 head-to-tail transfer function, 60–66 numerical simulations, 66 stability conditions, 63–64 stability diagrams, 64–66 head-to-tail string stability, 59–60, 71–74, 73f, 75 high level control, 54–55 internal stability and instability, 58, 59f motif-based approach, 54–55, 66 robust string stability, 178–180 stochastic delays, 54–55 string stability and instability, 58–59, 59f time-domain stability analysis, 68–70, 75 uniform flow equilibrium, 58 Connected vehicle system, 71–74, 169–173 Continuous-time approximation, 8–9, 14, 15f, 16, 16f, 17f Convergence analysis, 388–389 existence conditions, 392–393 stability conditions, 393–397 Cooperative adaptive cruise control (CACC), 54, 164

D Delay-dependent stability (DDS), 2 advantages, 20 algorithm steps, 5–8, 6b continuous-time approximation, 8–9, 19 discrete-time approximation, 9–10, 19 framework algorithm, 5–6, 6b neutral quasipolynomial approximation, 10–11 associated characteristic exponential polynomial approximation, 11–12, 16–17 characteristic quasipolynomial approximation, 12–13, 16–19 retarded system, 13–16 Delay differential algebraic equations (DDAEs), 187–191, 199 Delay differential equations (DDEs), 191–192

463

464

Index

Delayed gambler’s ruin (DRG) betting, expected duration of, 346 computer simulations, 342–343 delay differential equations, 334 Kronecker delta, 339, 341–342 in payment, 338 probability, concept of, 333 quantum mechanics, 333–334 in receipt, 338 ruin probability, 345–346 simple random walks, 334–337 stochastic delay differential equations, 334 stopping time, 338 Delay eigenvalue problems (DEPs), 186–188, 190–192, 198–199 Delay scheduling, 124 Differential algebraic equations (DAEs), 144–146 Differential mean value theorem, 425 Directed spanning tree, 102 Discrete-time approximation, 9–10, 14, 15f, 16, 16f, 17f Discrete-time systems BMI into LMI N measurements, 431 nonconvex matrix inequalities, 430 two measurements, 430 differential mean value theorem, 425 enhancement classical approach, 432 computational complexity, 433 H∞ sliding window observer, 431 LMI feasibility, 433 sliding window, delayed measurements, 431 Kalman filter, 423–424 Lipschitz systems, 423–424, 432 nonlinear discrete-time systems, 425–426 Schur complement lemma, 425 simulation results, 434–437 sliding window observer synthesis, 428–430 state observer theory, 423–424 Young’s relation, 425 Distributed operational space control, 45–46 DRG. See Delayed gambler’s ruin (DRG) D-subdivision method, 280 Dynkin’s formula, 233–234, 264–268

E Euclidean norm, 258 Euler discretization, 434–435 Event generator and switching controller synthesis, NCSs, 354–357 Event-triggered mechanism, 258, 260–261, 261f

Extended dissipativity, singular time-delay systems criteria, 237–239 definitions and assumptions, 236–237 filters, 250–252

F Finite spectrum assignment (FSA), 209–210 closed control loop system, 217 delay-free state feedback, 217 effect of initial conditions, 218, 220 frequency domain, 218, 224t implementation issues, 220–221 internal model, 216–217 and MSP, equivalence of, 220 observer-predictor realization, 221–222 output feedback, 217–218 predictor-observer realization, 222 vs. Smith predictor, 210 state feedback, 215f, 216–218 time-domain, 216–218, 223t Finite time stability, 280 Floquet theory, 128–129 FMJSs. See Fuzzy Markov jump systems (FMJSs) Forward Euler method, 198 Fractional derivative models argument principle, 280, 283–284 autonomous fractional-delay systems, 281 Caputo’s fractional derivative, 281 characteristic equation, 280 CRONE control, 279–280 delay-dependent coefficients, class of systems with, 289–291 finite time stability, 280 frequency domain methods, 280 Gamma function, 281 hyperbolic equilibrium, 281–282 integer-order time-delay systems, 280 linear elastic force, 281–282 linear systems asymptotic stability of, 282 Mittag-Leffler stability, 282–283 Lyapunov’s asymptotic stability, 280 mass-damping-spring system, 281–282 PID control, 279–280 stability test, algorithms for abscissa, calculation of, 288–289 Nyquist frequency plot, 285–288 test integral, evaluation of, 284–285 time domain methods, 280 viscoelastic materials, 279–280 Fractional polynomial nonlinear state space (FPNLSS) model, 442–443, 447–449

Index

Frequency-domain approach, 100 Frequency-domain stability analysis, 75 head-to-tail transfer function, 60–66 numerical simulations, 66 stability conditions, 63–64 stability diagrams, 64–66 FSA. See Finite spectrum assignment (FSA) Fuzzy Markov jump systems (FMJSs) event-triggered mechanism, 258, 260–261, 261f event-triggered reliable controller design, H∞ performance, 263–264, 268–271 mode-dependent LKF, 257–258 numerical example, 263–264, 271–273 problem formulation and preliminaries probability space, 259 TPM, 259 reliable control model, 261–263 stability analysis, 264–268

Gain scale (GS), 32–35, 34f, 41–42, 42f Gamma function, 443–444 Gauss-Newton method, 443, 448 Graphical test, 283–284 Graph theory, 102 Gronwall-Bellman lemma, 264–268 Grünwald-Letnikov (GL), 443–444

controlled system, pitch-flap stability boundaries control law, 124, 129–131 control without delay, hovering flight, 131–133 delayed control, hovering flight, 133–137 horizontal flight, 137–139 delay scheduling, 124 dynamic model, 125–126, 126t flapping motion, 124 higher harmonic control, 123 hinge arrangement and degrees of freedom, 124, 125f individual blade control, 123 lead-lag motion, 124 pitch-flap flutter, 124–125 pitching/feathering motion, 124 Stammer’s model, 124–125 uncontrolled system, pitch-flap stability boundaries horizontal flight, 128–129, 130f hovering flight, 127–128, 128f Higher harmonic control (HHC), 123 High-level controllers (HLCs), 23–24, 24f H∞ method, 79–80, 85–90, 94 Hooke’s law, 279–280 Hopf bifurcation (HB), 145–147 Hybrid algorithm for nonsmooth optimization (HANSO) method, 187, 199–200

H

I

Hammerstein system classical integer case models, 442 FPNLSS, 447–449 fractional calculus fractional-order operator, 443–444 fractional-order state space model, 445 fractional nonlinear time-delay systems, 442 fractional state space model equation, 446 identification method, 447–449 Levenberg-Marquardt algorithm, 442 partial differential equations, 441 PNLSS model, 446 simulation examples fractional commensurate case, 450–451, 452f, 452t, 453f, 454f noncommensurate fractional state space model, 451–456, 455f, 455t, 456f, 457f, 458f, 459t Head-to-tail string stability, 59–60, 63–66, 70–74, 73f, 75, 170–172 Head-to-tail transfer function (HTTF), 60–66, 174, 180 connected vehicle system, 171–173, 173f Helicopter rotors

Impedance analysis, SEAs impedance transfer function, 35–36 load inertia, effect of, 38, 40f time delays and filtering, effects of, 36–38, 37f varying natural frequencies, 38, 39f Impedance transfer function, SEAs, 35–36 Individual blade control (IBC), 123 Input/output (I/O) models, 442–443 Integral quadratic constraints (IQCs) advantages, 100–101 definitions, 102–103 time-varying input delays, MASs LFT system, 104–105 LTI form, 105 notations and graph theory, 101–102 numerical example, 115–117 output-feedback distributed protocol, 109–114 robust control theory, 100–103 state-feedback distributed protocol, 105–109, 120–121 Integral test, 283–284 Interior-point optimization method, 379 Itô stochastic process, 296–297

G

465

466

Index

J

M

Jacobian determinant, 289–290 Jensen’s inequality, 86–90, 232–233

Markov jump systems (MJSs), 228, 245–246, 252, 295. See also Singular time-delay systems FMJSs (see Fuzzy Markov jump systems (FMJSs)) SMO-based approach, 295–296 Mikhailov’s criterion, 284–286 Mittag-Leffler stability, 280, 282–283 Modified Smith predictor (MSP), 209–210 control law, 214–216 effect of initial conditions, 218–220 frequency-domain equations, 218, 224t and FSA, equivalence of, 220 implementation issues, 220–221 observer-predictor realization, 221–222 predictor-observer realization, 222 state-space representation, 214–216, 215f time domain, 216, 218, 223t transfer function, 214–216 Monodromy matrix, 128–129, 139 Monte Carlo simulations, 450–451, 452t, 455, 459t MSP. See Modified Smith predictor (MSP) Multiagent systems (MASs), consensus control dynamic IQCs (see Integral quadratic constraints (IQCs)) external disturbance, 103–104 first- and second-order MASs, 99–100 leader-following consensus problem, 103–104 multiagent coordination tasks, 99–100 nonuniform control input delays, 103–104 physical constraints and system limitations, 99–100 time-delay MASs, 100

K Kronecker delta, 339, 341–342 Kronecker multiplication method, 2

L Lagrange multipliers, 195 Laplace transform, 61, 166, 171, 211, 218, 282 Laplacian matrix, 115–117 Leader-following consensus problem, MASs, 103–104 output-feedback distributed protocol, 109–117, 119f state-feedback distributed protocol, 105–109, 115–117, 120–121 Levenberg-Marquardt (L-M) algorithm, 442, 448 LFT. See Linear fractional transformation (LFT) L’Hospital rule, 167–168 Linear fractional transformation (LFT), 104–109, 112–114 Linear matrix equalities (LMEs), 79–80 Linear matrix inequalities (LMIs), 2, 70, 79–80, 85–93, 185–186, 228, 233–234, 236, 239, 247, 280, 296, 303, 367–368, 387–388, 407, 426, 431, 433 Linear time-invariant (LTI), 105, 124, 127 Linear time-invariant time-delay systems (LTI-TDSs), 1–2, 133–134, 139 definition, 3–4 exponential stability and spectral properties, 4–5 numerical gridding DDS algorithm (see Delay-dependent stability (DDS)) Linear time-periodic (LTP) system, 124, 127–129, 137–139 LMIs. See Linear matrix inequalities (LMIs) Load inertia, 38, 40f Low-level controllers (LLCs), 23–24, 24f Lyapunov function, 100, 117, 120–121, 269, 302–305, 315–316, 321, 324–325, 428–429, 432 Lyapunov-Krasovskii approach, 2, 392, 407, 410, 412–418 Lyapunov-Krasovskii functionals (LKFs), 75, 80, 85–90, 92–93, 185–186, 257–258 Lyapunov matrices, 230–232 Lyapunov stability theory, 145

N Networked control systems (NCSs), 315–316 closed-loop event-triggered switched system, 354 codesign method, 350–351 continuous-time linear system, 351 discrete-time controller, 351 event generator and switching controller synthesis packet dropout and short network-induced delay, 357–359 short network-induced delay, 354–357 network-induced delays, 352, 352f numerical example, 361–364 sampling instants, 352 self-triggered control, 359–361 switched system, 353 time-triggered generator, 353

Index

Networked predictive control (NPC) effective data packet, 317 Markovian jump system approach, 315–316 network-induced delay, 315–317, 326, 326f observer-based output feedback, 319–320 numerical example, 328–329 stability analysis, 322–325 packet disorder, 317–318 stability conditions, 316 state feedback, 319 numerical example, 326–328 stability analysis, 320–321 time-stamps, 317 ZOH, 317–318 Network-induced delay, 315–317, 326, 326f Neutral quasipolynomial approximation, 10–11 associated characteristic exponential polynomial approximation, 11–12, 16–17 characteristic quasipolynomial approximation, 12–13, 16–19 Newton-Leibniz formula, 69–70 Newton-Raphson iteration method, 289–290 Newton’s law, 279–280 Newton’s zero point extrapolation principle, 2, 7–8, 16, 19–20 Nonlinear discrete-time systems augmented system, 372–374 BMI into LMI controller gain, 375 M measurements, 376–377 two measurements, 376 convergence analysis, 374–375 enhancement classical approach, 377–378 computational complexity, 379 LMI feasibility, 378–379 H∞ observer-based controller, 369–373 Luenberger observer, 369 Lyapunov function, 373 nonlinear discrete-time systems, 369–370 nonlinear function, 370–372 numerical examples Euler discretization, 379 Lipschitzian nonlinear function, 381, 383 one-link flexible joint robot, 380–383 state-space model, 383 Schur’s lemma, 373–374 sliding window observer, 370–372 state observer theory, 367–368 Nonlinear time-delay systems, UIO. See Unknown input observer (UIO) NPC. See Networked predictive control (NPC)

O Operational space controller (OSC), 45–46 Orlando formula, 289–290 Output-feedback control, 245–250 Output-feedback distributed protocol, 109–117, 119f

P Pencil method, 2 Period doubling bifurcation, 129 Period resolution, 139 Phase measurement units (PMUs), 143–144 Pitch divergence controlled system, pitch-flap stability boundaries control without delay, hovering flight, 131–132, 132f delayed control, hovering flight, 135 horizontal flight, 137, 138f, 139, 140f uncontrolled system, pitch-flap stability boundaries horizontal flight, 128–129, 130f hovering flight, 127–128, 128f Pitch-flap flutter, 124–125 controlled system, pitch-flap stability boundaries control without delay, hovering flight, 131–132, 132f delayed control, hovering flight, 136–137 horizontal flight, 137, 138f, 139, 140f uncontrolled system, pitch-flap stability boundaries horizontal flight, 128–129, 130f hovering flight, 127–128, 128f Plant stability, 166–167, 167f Polynomial nonlinear state space (PNLSS) model, 443, 446 Polytopic approach, 392, 417–418 Power system and time delays, 144 controller designs, influences on, 144 DAEs, 144–145 eigenvalue trajectories, time-delay impact on critical eigenvalue loci, 151, 153, 153f error index, 152–153 intersection of eigenvalue loci, 153–154, 154f single-machine-infinite-bus system, 149–150, 149f, 151, 151t system eigenvalue loci, 151–152, 151f, 152f PMUs and WAMS, 143–144 small signal stability region boundary of, 145–147 DAE model, 145–146 definition, 145–147

467

468

Index

Power system and time delays (Continued) optimization-based boundary tracing algorithm, 147–149 single-machine-infinite-bus system, 154–156 WSCC 3-generator-9-bus system, 156–158 “Predictor-corrector” framework, 148, 148f Predictor feedback controllers control problem without predictors, 211–212, 211f finite spectrum assignment, 209–210 closed control loop system, 217 delay-free state feedback, 217 effect of initial conditions, 218, 220 frequency domain, 218, 224t implementation issues, 220–221 internal model, 216–217 and MSP, equivalence of, 220 observer-predictor realization, 221–222 output feedback, 217–218 predictor-observer realization, 222 vs. Smith predictor, 210 state feedback, 215f, 216–218 time-domain, 216–218, 223t modified Smith predictor, 209–210 control law, 214–216 effect of initial conditions, 218–220 frequency-domain equations, 218, 224t and FSA, equivalence of, 220 implementation issues, 220–221 observer-predictor realization, 221–222 predictor-observer realization, 222 state-space representation, 214–216, 215f time domain, 216, 218, 223t transfer function, 214–216 Smith predictor, 209–210 closed-loop transfer function, 213 control law, 211f, 212–214 delay-free internal model, 214 disturbance response, 213–214 effect of initial conditions, 218–219 frequency-domain equations, 218, 224t vs. FSA, 210 implementation issues, 221 state-space representation, 214, 215f time domain equations, 218, 223t Proportional integral derivative (PID) controllers, 26 Pseudospectral abscissa, 186–187 advantage, 205–206 computation delay perturbations, 198 differential equations, 196–198 forward Euler method, 198

optimal perturbations, 192–196 perturbed DEPs, 192, 196, 198–199 numerical experiments, 201–204 robust stability, uncertain system DDAEs, 187–191 DDEs, 191–192 ˙ ÉZ-pseudospectral abscissa, 191 perturbed DEP, 188, 190–191 real-valued uncertainties, 188 smoothness properties and optimization, 199–200 Puiseux series expansion technique, 2

Q Quadratic extrapolation method, 2, 10 Quasi-Polynomial mapping Rootfinder (QPmR) software, 3, 8, 16

R Rekasius’s substitution, 174–177 Reksius transform, 280 Reliable control model, 261–263, 268–271 Riemann-Liouville (RL) operator, 443 Rightmost characteristic root(s), 288–289 Robotic control systems, 24–25 centralized control with HLCs, 23–24, 24f decentralized control with LLCs, 23–24, 24f distributed control, SEAs (see Series elastic actuators (SEAs)) Robust control theory, 100–103 Robust stability robust string stability, V2X communication connected cruise control design, 178–180 predecessor-follower system, uncertainties in, 174–178 uncertain system, pseudospectral abscissa DDAEs, 187–191 DDEs, 191–192 ˙ ÉZ-pseudospectral abscissa, 191 perturbed DEP, 188, 190–191 real-valued uncertainties, 188 Root tendency (RT), 134–135 Routh-Hurwitz stability criterion, 280

S Saddle-node bifurcation (SNB), 145–147 Schur complement, 242–245, 249–250, 269, 355–356, 429–430 Scott-Blair’s law, 279–280 SEAs. See Series elastic actuators (SEAs) Second-order finite-dimensional model, 9–10 Self-triggered control, 359–361

Index

Semidiscretization method, 139, 140f, 141 Series elastic actuators (SEAs) advantages, 25–26 controller design evaluation, 39–42 critically damped controller gain design criterion, 26, 31–32, 34f distributed operational space control, 45–46 gain controller design procedure, 34b gain scale, 32–35, 34f impedance analysis impedance transfer function, 35–36 load inertia, effect of, 38, 40f time delays and filtering, effects of, 36–38, 37f varying natural frequencies, 38, 39f modeling of, 27–30 passivity-based controller gains, 26–27 PID controllers, 26 step response implementation, 42–45 torque and impedance control, trade-off between, 32–35 Valkyrie robot, 27, 27f, 32–33 Signal-to-noise ratios (SNR), 450, 455 Single-machine-infinite-bus system, 149–150, 149f, 151, 151t, 154–156 Singularity-induced bifurcation (SIB), 145–147, 154 Singular time-delay systems descriptor systems, 227–228 extended dissipativity criteria, 237–239 definitions and assumptions, 236–237 filters, 250–252 Markovian jumping parameters, 228, 252 output-feedback control, 245–250 state-feedback control controller synthesis conditions, 242–245 problem formulation, 239 useful lemmas, 240–241 stochastic admissibility criteria, 234–236 relaxed L-K functional approach, 230–234 system model and definitions, 229–230 Sliding mode control (SMC), MJSs augmented system and SMO formulation, 298–300 design goal, 297 error dynamics, 300–301 overall closed-loop plant, stability analysis of, 301–303 simulations, 306–309 sliding mode surface, reachability of, 303–305 synthesis algorithm, 305–306 system description, 296–297

Sliding mode observer (SMO)-based approach, 295–296 Small signal stability region (SSSR) definition, 144 power system and time delays boundary, 145–147 critical eigenvalue loci, 151, 153, 153f DAE model, 145–146 definition, 145–147 error index, 152–153 intersection of eigenvalue loci, 153–154, 154f optimization-based boundary tracing algorithm, 147–149 single-machine-infinite-bus system, 149–150, 149f, 151, 151t, 154–156 system eigenvalue loci, 151–152, 151f, 152f WSCC 3-generator-9-bus system, 156–158 Smith predictor (SP), 209–210 closed-loop transfer function, 213 control law, 211f, 212–214 delay-free internal model, 214 disturbance response, 213–214 effect of initial conditions, 218–219 frequency-domain equations, 218, 224t vs. FSA, 210 implementation issues, 221 state-space representation, 214, 215f time domain equations, 218, 223t Sobolev norm, 80, 90 Sobolev space, 80, 90 SSSR. See Small signal stability region (SSSR) Stammer’s model, 124–125 State-feedback control, singular time-delay systems controller synthesis conditions, 242–245 problem formulation, 239 useful lemmas, 240–241 State-feedback distributed protocol, 105–109, 115–117, 120–121 Stochastic admissibility, singular time-delay systems criteria, 234–236 relaxed L-K functional approach, 230–234 system model and definitions, 229–230 String instability, 53, 58–59, 59f String stability, 167–168, 167f Structured singular values, 177, 180 Switched system theory, 323–324 Sylvester’s equations, 421

T Taylor series expansion, 2, 8–9, 19 Time-domain stability analysis, 68–70, 75

469

470

Index

Tip speed ratio, 126 Transition probability, 229, 296–297 Transition probability matrix (TPM), 259 Tustin transformation. See Bilinear transformation

U Uncertain delay systems, pseudospectral approach. See Pseudospectral abscissa Unknown input observer (UIO), 80 H∞ method, 79–80, 85–90, 94 state x and unknown input u, 81–82 W1,2 method disturbance attenuation level, optimal value of, 96 robustness criterion, 90–93 Sobolev norm, 80, 90 Sobolev space, 80, 90 Upper linear fractional transformation (upper LFT), 176

V Valkyrie robot, SEAs, 27, 27f, 32–33 Vehicle-to-everything (V2X) communication ACC, 164 CACC, 164 modeling and control design with delays car-following models, 164–168, 165f connected vehicle system, 169–173 plant and string stability, 166–168, 167f range policy function, 165–166, 165f transfer function, 165f, 166 robust string stability connected cruise control design, 178–180 predecessor-follower system, uncertainties in, 174–178

Vehicle-to-vehicle (V2V) communication, 54 CACC, 54 time-delayed CCC adaptive control scheme, 54–55 beyond-line-of-sight identification, 54–55 connected vehicle system, 71–74 control framework, 55–58 frequency-domain stability analysis (see Frequency-domain stability analysis) head-to-tail string stability, 59–60, 71–74, 73f, 75 high level control, 54–55 internal stability and instability, 58, 59f motif-based approach, 54–55, 66 stochastic delays, 54–55 string stability and instability, 58–59, 59f time-domain stability analysis, 68–70, 75 uniform flow equilibrium, 58

W Wide area measurement system (WAMS), 143–144 W1,2 method disturbance attenuation level, optimal value of, 96 robustness criterion, 90–93 Sobolev norm, 80, 90 Sobolev space, 80, 90 WSCC 3-generator-9-bus system, 156–158

Y Young’s inequality, 80, 86–90

Z Zero-order hold (ZOH), 260, 317