Systems, Automation and Control: Extended Papers from the Multiconference on Signals, Systems and Devices 2014 9783110448436, 9783110443769

Table of contents :
Preface of the Series Editor
Preface of the Volume Editor
Contents
Graphical Modeling of Networked Architectures and Real-Time-Requirements for the Analysis of Networked Automation Systems
OBE-based Set-Membership Parameter Estimation of Fractional Models
Improving High Gain Observer Design for Nonlinear Systems Using the Structure of the Linear Part
Fuzzy Network-Based Control for a Class of T–S Fuzzy Systems with Limited Communication
Particle Swarm Optimization-Based Approach for Digital RST Controller Design
Optimal Pattern Generator for Dynamic Walking in humanoid Robotics
Optimal Targeting in Chaos Control, a Discrete Hamiltonian Approach

Citation preview

Nabil Derbel (Ed.) Systems, Automation & Control

Advances in Systems, Signals and Devices

|

Edited by Olfa Kanoun, University of Chemnitz, Germany

Volume 1

Systems, Automation & Control |

Edited by Nabil Derbel

Editor of this Volume Prof. Dr.-Eng. Nabil Derbel University of Sfax Sfax National Engineering School Control & Energy Management Laboratory 1173 BP, 3038 SFAX, Tunisia [email protected]

ISBN 978-3-11-044376-9 e-ISBN (PDF) 978-3-11-044843-6 e-ISBN (EPUB) 978-3-11-044627-2 Set-ISBN 978-3-11-044844-3 ISSN 2365-7493 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2016 Walter de Gruyter GmbH, Berlin/Boston Typesetting: Konvertus, Haarlem Printing and binding: CPI books GmbH, Leck ♾ Printed on acid-free paper Printed in Germany www.degruyter.com

Preface of the Series Editor The Publication and dissemination of scientific results is a big aim of scientists to show the latest developments in their fields and to contribute to the development of science. Nowadays, publications are gaining more and more importance for the progress of science and engineering, because of the higher level of dynamics in these fields promoted by novel technologies in several fields of sciences and especially in the field of electrical engineering and information technology. Today, smart systems involving different technologies provide promising technical solutions which can significantly contribute to improvements of quality, reliability and economic efficiency of technical products. This can be reached by an interlocked synergetic interaction between different technological and scientific fields. For the development of smart systems, generally know-how from different fields is necessary. Moreover this requires the promotion of interdisciplinary thinking. Scientists are demanded to be better informed about up-to-date topics and other disciplines in order to be able to deal with the higher becoming complexity of systems. The series Advances in Systems, Signals and Devices (ASSD) is a fully peer-reviewed international series which addresses relevant fields in electrical engineering related to smart systems having a high level of interaction and innovation potential. Papers are of a high scientific level and are individually accessible and having each a digital object identifier (DOI). The series addresses both academic and application-oriented contributions with the conviction that only with both sides we can significantly improve engineering sciences. A big part of contributions consists of extended selected best papers presented at the International Multi-Conferences on Systems, Signals and Devices (SSD), and ties in with the success of the former journal Transactions on Systems Signals and Devices which reached in total 32 issues organized in 8 volumes. The Multi-Conferences on Systems, Signals and Devices (SSD) is an annual international conference which has been established since 2001 as a forum for scientific exchange between scientists from all over the world in electrical engineering and reached in the meantime an exceptional success. Today, the SSD conference is well-known and recognized in the scientific community for its high scientific level, its extensive internationality and contribution to scientific exchange and networking in the world. The aim of this international series is to promote the international scientific progress in the field of electrical engineering and information technology. It provides an interesting opportunity to report about outstanding results that has been reported during the international SSD conferences and beyond. The series is organized in issues dedicated alternatively to the following main topics: – Sensors, Circuits & Instrumentation Systems (SCI): Sensors and measurement systems, optical sensors, chemical sensors, mechanical sensors, inductive sensors,

VI | Preface of the Series Editor

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–

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capacitive sensors, micro-sensors, thermal sensors, biomedical and environmental sensors, flexible sensors, nano sensors, micro electronic systems, nano systems and nano technology, sensor signal processing, sensor interfaces, modeling, data acquisition, multi sensor data fusion, distributed measurements, device characterization and modeling, custom and semi-custom circuits, analog circuit design, low-voltage, low-power VLSI design, circuit test, packaging and reliability, impedance spectroscopy, wireless sensors, wireless interfaces, wireless sensor networks, energy harvesting, circuits and systems design. Systems, Automation & Control (SAC): System design, system identification, biological and economical models & control, modern control theory, nonlinear observers, control and application of chaos, adaptive/non-adaptive backstepping control techniques, advances in linear control theory, system optimization, multivariable control, large scale and infinite dimension systems, nonlinear control, distributed control, predictive control, geometric control, adaptive control, optimal and stochastic control, robust control, neural control, fuzzy control, intelligent control systems, diagnostics, fault tolerant control, robotics and mechatronics, navigation, robotics and human-machine interaction, hierarchical and man-machine systems. Communication and Signal Processing & Information Technology (CSP): Information technology, communications systems, digital signal processing, image processing, video processing, image and video compression, modulation and signal design, content-based video retrieval, wireless and optical communication, technologies for wireless communication systems, biometry and medical imaging, adaptive and smart antennas, data fusion and pattern recognition, coding compression, communication for e-mobility, microwave active and passive components and circuits, cognitive and software defined radio, vision systems and algorithms. Power Systems & Smart Energies (PSE): Energy systems and energy transmission, renewable energy systems, hybrid renewable energy systems, photovoltaic systems, solar energy, wind energy, energy storage, batteries, thermal energy, combined heat and thermal power generation, electric machine design, electric machines modeling and control, electrical vehicles, technologies for electro mobility, special machines, variable speed drives, variable speed generating systems, automotive electrical systems, monitoring and diagnostics, electromagnetic compatibility, power systems, transformers, power electronics, topologies and control of power electronic converters.

It is a big honor to work together with four international editorial boards consisting of internationally renowned scientists in these four scientific fields contributing to realize the high quality series ASSD. Olfa Kanoun Editor-in-Chief

Preface of the Volume Editor The first volume of the Series “Advances in Systems, Signals and Devices” (ASSD), is dedicated to fields related to “Systems, Automation and Control” (SAC). The scope of this issue encompasses all aspects of the research, development and applications of the science and technology in these fields. Topics of this issue concerns: system design, system identification, biological and economical models & control, modern control theory, nonlinear observers, control and application of chaos, adaptive/non-adaptive backstepping control techniques, advances in linear control theory, system optimization, multivariable control, large scale and infinite dimension systems, nonlinear control, distributed control, predictive control, geometric control, adaptive control, optimal and stochastic control, robust control, neural control, fuzzy control, intelligent control systems, diagnostics, fault tolerant control, robotics and mechatronics, navigation, robotics and human-machine interaction, hierarchical and man-machine systems, etc. Authors have been encouraged to submit novel contributions which include results of research or experimental work discussing new developments in the field of systems, automation and control. The work can be also addressed for editing special issues for novel developments in specific fields. Guest editors are encouraged to make proposals to the editor in chief of the corresponding main field. The aims of this volume, in its own way, to promote an international scientific progress in the fields of systems, automation and control. It provides at the same time an opportunity to be informed about interesting results that have been reported during the international SSD conferences. Nabil Derbel Associate Editor-in-Chief

Advances in Systems, Signals and Devices Series Editor: Prof. Dr.-Ing. Olfa Kanoun Technische Universität Chemnitz, Germany. [email protected]

Editors in Chief: Systems, Analysis & Automatic Control Prof. Dr.-Eng. Nabil Derbel ENIS, University of Sfax, Tunisia [email protected]

Power Electrical Systems Prof. Dr.-Ing. Faouzi Derbel Leipzig Univ. of Applied Sciences, Germany [email protected]

Communication & Signal Processing Prof. Dr.-Ing. Faouzi Derbel Leipzig Univ. of Applied Sciences, Germany [email protected]

Sensors, Circuits & Instrumentation Prof. Dr.-Ing. Olfa Kanoun Technische Universität Chemnitz, Germany [email protected]

Editorial Board Members: Systems, Analysis & Automatic Control Dumitru Baleanu, Çankaya University, Ankara, Turkey Ridha Ben Abdennour, Engineering School of Gabès, Tunisia Naceur Benhadj, Braïek, ESSTT, Tunis, Tunisia Mohamed Benrejeb, Engineering School of Tunis, Tunisia Riccardo Caponetto, Universita’ degli Studi di Catania, Italy Yang Quan Chen, Utah State University, Logan, USA Mohamed Chtourou, Engineering School of Sfax, Tunisia Boutaïeb Dahhou, Univ. Paul Sabatier Toulouse, France Gérard Favier, Université de Nice, France Florin G. Filip, Romanian Academy Bucharest Romania Dorin Isoc, Tech. Univ. of Cluj Napoca, Romania Pierre Melchior, Université de Bordeaux, France Faïçal Mnif, Sultan qabous Univ. Muscat, Oman Ahmet B. Özgüler, Bilkent University, Bilkent, Turkey Manabu Sano, Hiroshima City Univ. Hiroshima, Japan Abdul-Wahid Saif, King Fahd University, Saudi Arabia José A. Tenreiro Machado, Engineering Institute of Porto, Portugal Alexander Pozniak, Instituto Politecniko, National Mexico Herbert Werner, Univ. of Technology, Hamburg, German Ronald R. Yager, Mach. Intelligence Inst. Iona College USA Blas M. Vinagre, Univ. of Extremadura, Badajos, Spain Lotfi Zadeh, Univ. of California, Berkeley, CA, USA

Power Electrical Systems Sylvain Allano, Ecole Normale Sup. de Cachan, France Ibrahim Badran, Philadelphia Univ., Amman, Jordan Ronnie Belmans, University of Leuven, Belgium Frdéric Bouillault, University of Paris XI, France Pascal Brochet, Ecole Centrale de Lille, France Mohamed Elleuch, Tunis Engineering School, Tunisia Mohamed B. A. Kamoun, Sfax Engineering School, Tunisia Mohamed R. Mékidèche, University of Jijel, Algeria Bernard Multon, Ecole Normale Sup. Cachan, France Francesco Parasiliti, University of L’Aquila, Italy Manuel Pérez,Donsión, University of Vigo, Spain Michel Poloujadoff, University of Paris VI, France Francesco Profumo, Politecnico di Torino, Italy Alfred Rufer, Ecole Polytech. Lausanne, Switzerland Junji Tamura, Kitami Institute of Technology, Japan

Communication & Signal Processing Til Aach, Achen University, Germany Kasim Al-Aubidy, Philadelphia Univ., Amman, Jordan Adel Alimi, Engineering School of Sfax, Tunisia Najoua Benamara, Engineering School of Sousse, Tunisia Ridha Bouallegue, Engineering School of Sousse, Tunisia Dominique Dallet, ENSEIRB, Bordeaux, France Mohamed Deriche, King Fahd University, Saudi Arabia Khalifa Djemal, Université d’Evry, Val d’Essonne, France Daniela Dragomirescu, LAAS, CNRS, Toulouse, France Khalil Drira, LAAS, CNRS, Toulouse, France Noureddine Ellouze, Engineering School of Tunis, Tunisia Faouzi Ghorbel, ENSI, Tunis, Tunisia Karl Holger, University of Paderborn, Germany Berthold Lankl, Univ. Bundeswehr, München, Germany George Moschytz, ETH Zürich, Switzerland Radu Popescu-Zeletin, Fraunhofer Inst. Fokus, Berlin, Germany Basel Solimane, ENST, Bretagne, France Philippe Vanheeghe, Ecole Centrale de Lille France

Sensors, Circuits & Instrumentation Ali Boukabache, Univ. Paul, Sabatier, Toulouse, France Georg Brasseur, Graz University of Technology, Austria Serge Demidenko, Monash University, Selangor, Malaysia Gerhard Fischerauer, Universität Bayreuth, Germany Patrick Garda, Univ. Pierre & Marie Curie, Paris, France P. M. B. Silva Girão, Inst. Superior Técnico, Lisboa, Portugal Voicu Groza, University of Ottawa, Ottawa, Canada Volker Hans, University of Essen, Germany Aimé Lay Ekuakille, Università degli Studi di Lecce, Italy Mourad Loulou, Engineering School of Sfax, Tunisia Mohamed Masmoudi, Engineering School of Sfax, Tunisia Subha Mukhopadhyay, Massey University Turitea, New Zealand Fernando Puente León, Technical Univ. of München, Germany Leonard Reindl, Inst. Mikrosystemtec., Freiburg Germany Pavel Ripka, Tech. Univ. Praha, Czech Republic Abdulmotaleb El Saddik, SITE, Univ. Ottawa, Ontario, Canada Gordon Silverman, Manhattan College Riverdale, NY, USA Rached Tourki, Faculty of Sciences, Monastir, Tunisia Bernhard Zagar, Johannes Kepler Univ. of Linz, Austria

Contents Preface of the Series Editor | V Preface of the Volume Editor | VII B. Vogel-Heuser, J. Folmer, G. Frey, L. Liu, H. Hermanns and A. Hartmanns Graphical Modeling of Networked Architectures and Real-Time-Requirements for the Analysis of Networked Automation Systems | 1 R. Huber, K. Röbenack, S. Zipser and S. Wagner Track Estimation: High-Gain Observer vs. Algebraic Derivative Methods | 19 M. Amairi, S. Najar, M. Aoun, Z. Lassoued and M. N. Abdelkrim OBE-based Set-Membership Parameter Estimation of Fractional Models | 37 K. Röbenack Improving High Gain Observer Design for Nonlinear Systems Using the Structure of the Linear Part | 57 M. Kchaou and A. Toumi Fuzzy Network-Based Control for a Class of T–S Fuzzy Systems with Limited Communication | 75 S. Bouallègue, R. Madiouni and J. Haggège Particle Swarm Optimization-Based Approach for Digital RST Controller Design | 95 D. Galdeano, A. Chemori, S. Krut and P. Fraisse Optimal Pattern Generator for Dynamic Walking in humanoid Robotics | 115 Ulrich Vogl Optimal Targeting in Chaos Control, a Discrete Hamiltonian Approach | 141

B. Vogel-Heuser, J. Folmer, G. Frey, L. Liu, H. Hermanns and A. Hartmanns

Graphical Modeling of Networked Architectures and Real-Time-Requirements for the Analysis of Networked Automation Systems

Abstract: Networked Automation Systems (NAS) have gained increasing significance in recent years. The connection of controllers and intelligent devices via networks allows more flexibility and shorter time-to-market in the design of automation systems for single machines as well as for complete manufacturing plants. Since all those systems need to fulfill given real-time requirements the verification of time behavior is an important task in the engineering process. Therefore, methods and tools have to be developed that are able to cope with the prevalent heterogeneous system architectures and the variety of different real-time requirements. This paper outlines a concept for solving this challenging task and as a first step in realizing this vision presents a graphical method for modeling real-time requirements and architectural properties of NAS. Keywords: Networked automation system, real-time requirements, analysis, simulation, model checking.

1 Introduction The term Networked Automation Systems (NAS) in industrial automation refers to an automation system consisting of networked controllers, sensors and actuators. We focus on NAS in customized plant automation. Such NAS can be engineered more generically and modularly following distributed and modular automation principles. The design has to support modularity and reuse of mechatronic components to increase quality while reducing time for start-up and time needed for testing and debugging at

B. Vogel-Heuser, J. Folmer: Institute of Automation and Information Systems, TU München, Germany, e-mails: [email protected], [email protected]. G. Frey, L. Liu: Chair of Automation, Saarland University, Germany, e-mails: [email protected], [email protected]. H. Hermanns, A. Hartmanns: Chair for Dependable Systems and Software, Saarland University, Germany, e-mails: [email protected], [email protected]. De Gruyter Oldenbourg, ASSD – Advances in Systems, Signals and Devices, Volume 1, 2016, pp. 1–17. DOI: 10.1515/9783110448375-002

2 | B. Vogel-Heuser et al.

the customer site. The thereby shortened development time is getting more and more important in a competitive market. Therefore, the potentials to use communication networks increase drastically [1]. Due to non-functional requirements, e.g. changed environmental conditions, different specified control suppliers and different plant layouts based on the available customer assets lead to different plant and network architectures even though the plant functionality is identical to another plant already built. This requires the design and engineering of heterogeneous communication systems by application engineers in machine and plant manufacturing companies for every plant. The communication system manufacturers, however, provide only proprietary engineering tools for their network system, which do not support heterogeneous multivendor communication structures. Because of process requirements, best of bread (sensors and actuators) or costs, NAS are rarely composed of subsystems delivered from only one manufacturer. In industrial automation it is however often necessary to use multiple communication systems and controllers. Depending on climatic conditions (humidity, temperature) and spatial layout (indoor or outdoor area, distance) differentiated network architectures and device technologies may also be necessary. The heterogeneity also results from different real-time requirements of the technical process or the technical systems, i.e. hard real-time requirements for the synchronization of robot axes, or synchronization of press frames along a continuous press [2] and soft real-time requirements for sending new parameters in advance. In addition, most field devices integrate additional services, such as an OPC server or web interface, which use simple non-deterministic communication channels such as TCP/IP. The main requirement to be fulfilled in order to ensure the functionality of the automation system within NAS is to guarantee the timeliness of information and data exchange [3]. However, the design and implementation of NAS is as yet lacking support for modeling the required time behavior and network architecture. Time requirements and properties of the communication components are either given in data sheets or need to be measured during runtime, which is too late for a faithful design process. For this reason simulation and verification technologies for time behavior need to be integrated into the engineering process, i.e. into the design of a NAS. This is especially true for designs using non-deterministic of-the-shelf network technologies like Ethernet. The usage of Ethernet is popular since it allows the easy extension of NAS. However, the impact of possible increased network load due to extensions on the performance of already existing functions in the system [4] has to be evaluated prior to the actual runtime. Different tasks performed by industrial plants’ components result in heterogeneous communication systems. Both slow processes in process industry and rapid processes in manufacturing industry are applied in automation systems. Those processes can be classified into three Quality of Service (QoS) levels [5], cf. Tab. 1. Synchronized motions in general need QoS level 3 [6]. For the exchange of process

Modeling and Analysis of Networked Architectures

| 3

data between controllers and decentralized peripheral devices QoS level 2 is sufficient. Visualization necessitates QoS level 1. Typical communication without time requirements uses simple communication channels like TCP/IP.

Tab. 1. Quality of Service (QoS) levels [5]. QoS levels

1 2 3

Application

Controller-to-Controller, Visualization Controller-todecentralized-periphery Synchronized motion

Latency

Jitter

10–100 ms

./.

1–10 ms

> 1 ms

< 1 ms

2π ν.

3 Principle of the prediction error method for linear fractional systems Consider the generalized structure ARX given by the differential equation: n  i=0

a i Dν ai y(t) =

m 

bj D

νbj

u (t) + ζ (t) ,

(8)

j=0

where y (t) is the system output, u (t) is the input and ζ (t) is an unknown but bounded additive noise. Generally a0 and ν a0 are respectively taken equals to 1 and 0.

40 | M. Amairi et al.

– – – – –

The parameter estimation procedure consists of five steps: Discretization. Linearization. Input/Output Filtering (optional). Pseudo-parameters estimation. Return to the continuous parameters.

3.1 Discretization The discretization of the generalized ARX structure consists in replacing each fractional derivative by its approximation resulting from Grünwald fractional derivative (1). The discretization procedure yields to a non linear equation regarding to the parameters [a1 , a2 , . . . , a n , b0 , b1 , . . . , b m ], so its linearization is necessary to apply any identification recursive algorithm.

3.2 Linearization The linearization procedure consists in replacing the coefficients a i and b j in equation (8) by α i and β j as ai νa h αi = n i ,  a l

l=0

h νal

n 

α i = 1,

(9)

i=0

bj ν

βj =

h bj , n  al h νal

(10)

l=0

where h is the sampling time taken as small as possible to get a good fractional derivative approximation. The linearization yields to a linear form of the generalized ARX solution y(Kh) = −

n 

α i y˜ i (Kh) +

i=0

m 

β j u˜ j (Kh) + ζ (Kh),

(11)

j=0

where y˜ i (Kh) =

K  k=1

 (−1)

k

ν ai k

 y((K − k)h),

(12)

Parameter Estimation of Fractional Models |

u˜ j (Kh) =

K 

 (−1)

k=0

k

ν bj k

41

 u((K − k)h).

(13)

3.3 Filtering Due to the linearization step where the input and the output are successively derived, a state variable filter presented as an extension of the Poisson filter to the fractional systems can be used [6]: sη F η ( s ) =   N f , s ν +1 ωc

(14)

where η is the filter order and ω c is his cutoff frequency. The number N f must verify: Nf >

max (ν a i ) . ν

(15)

After the input/output pre-filtering, the equation (11) can be rewritten in a linear regression form y fK = d˜ fK θ˜ + ζ K ,

(16)

 d˜ fK = −˜y0f (Kh), . . . , −˜y fn (Kh), u˜ 0f (Kh), . . . , u˜ fm (Kh) ,

(17)

where

θ˜ = [α0 , . . . , α n , β0 , . . . , β m ] .

(18)

A little modification to the linear regressive form (16) yields to  T y fK = −˜y0f + d˜ fK θ˜ + ζ K ,

(19)

where the pseudo-parameters vector θ˜ and the pseudo-regression vector d˜ K are given by θ˜ = [α1 , α2 , . . . , α n , β0 , β1 , . . . β m ]T ,

 d˜ fK = −˜y1f (Kh) + y˜ 0f (Kh), . . . , −˜y fn (Kh) + y˜ 0f (Kh), u˜ 0f (Kh), . . . , u˜ fm (Kh)

(20) (21)

3.4 Return to continuous parameters Suppose that step of pseudo-parameters estimation is performed, the last step in the parameter estimation procedure (using the linear regression form) is the return to the

42 | M. Amairi et al. ˜ One way to perform this parameters vector θ from the pseudo-parameters values θ. operation is to solve the following linear system Φx = ψ where ⎧ ⎪ ⎨ (α i ν− 1) if i = j h ai Φ i,j = , (22) αi ⎪ ⎩ if i = j νai h (23) ψ i = −α i , x = ( a 1 , a 2 , . . . , a n )T ,

(24)

and the coefficients b j are computed according to bj = βj

n 

ai h

ν b j −ν a i

.

(25)

i=0

4 OBE algorithms for the identification of fractional systems 4.1 Problem statement In this paper the noise is assumed to be unknown-but-bounded, i.e., there exists a known γ, such that for every sample |ζ K | ≤ γ.

(26)

Applying the classic identification algorithms (OLS,RLS,. . .) leads to a punctual parameters estimation and not a set of all acceptable parameters. A parameters vector θ is considered acceptable if and only if it verify   T   ∀K = 1, . . . , N, y fK − d fK θ  ≤ γ, (27) where N is the number of samples used in the identification procedure. The problem considered in this paper is the characterization of the set of all the acceptable parameters denoted Θ    T    (28) Θ = θ ∈ Rp , ∀K = 1, . . . , N  y fK − d fK θ ≤ γ , where p is the number of parameters to be estimated. The characterization of Θ is a set-membership problem and it result to a convex polyhedron where his body is bounded by N faces. Many approaches can be used to englobe the polyhedron with a simple geometric form such as an ellipsoid, an

Parameter Estimation of Fractional Models

|

43

interval or a parallelepiped. All these approaches are used for the development of set-membership methods for the time-domain identification of integer systems, but to best of our knowledge this is not the case of fractional systems except some recent methods using the interval-based global optimization algorithm [1, 2]. In this paper, we develop a set-membership method for the time-domain identification of fractional systems using an Optimal Bounding Ellipsoid (OBE) algorithms to approximate the set of all the acceptable parameters Θ.

4.2 OBE-based parameter estimation of fractional systems 4.2.1 Principle In the following, let the discretization and the linearization steps of the generalized ARX model are performed as described above. We consider the linear regression form (19), the set of all the acceptable parameters Θ K exists in a band Π K !  T 2 f f f p 2 ˜ ˜ ˜ θ) ≤ γ . Π K = θ ∈ R : (y + y˜ − d (29) K

0

K

One way to characterize the set Θ K is the intersection of the last K bands as shown in Fig. 1. This intersection operation yields to a convex polyhedron with a complex geometric form ΛK =

K "

Πi .

(30)

i=1

The OBE approach consists in englobing Λ K with an ellipsoid $ # T −1 ˜ K (θ˜ − c˜ K ) ≤ σ˜ 2K , E˜ K = θ˜ ∈ Rp : (θ˜ − c˜ K ) M

(31)

˜ K is a positive definite matrix. The axis of where c˜ K is the center of the ellipsoid and M ˜ K. the ellipsoid are determined from the eigenvectors of the matrix σ˜ 2K M ˜ The ellipsoid E K contains the intersection of the band Π K and the ellipsoid E˜ K−1 tacking into account the new measures (y fK and u fK ) (Fig. 2) E˜ K ⊃ E˜ K−1 ∩ Π K

(32)

44 | M. Amairi et al.

Π2

Π1

Π3 Λ3

Fig. 1. Characterization of the set of all acceptable parameters by a polyhedron.

Π2

Π1

Π3 Λ3 Π4 E˜ 2

E˜ 3

Fig. 2. Principle of the OBE approach (k=3).

To satisfy the condition (32), the ellipsoid E˜ K must take into account the old information (E˜ K−1 ) and the new observations (Π K ) using the weights (ρ K and δ K ): # T −1 ˜ 2 ˜ K−1 (θ˜ − c˜ K−1 ) + δ K (y K − d TK θ) E˜ K = θ˜ ∈ Rp : ρ K (θ˜ − c˜ K−1 ) M $ ≤ ρ K σ˜ 2K−1 + δ K γ (33) ˜ K and σ˜ K are recursively obtained from their previous values The new values of c˜ K , M weighted by ρ K and δ K ⎡ ⎤  T f f ˜ ˜ ˜ ˜ d M M d δ K K−1 K K−1 ⎥ K ˜ K−1 − ˜K= 1 ⎢ (34) M ⎣M ⎦, ρK ρ K + δ K g˜ K ˜ K d˜ f e˜ K , c˜ K = c˜ K−1 + δ K M K σ˜ 2K = ρ K σ˜ 2K−1 + δ K γ2K −  T e˜ K = y K − d˜ fK c˜ K−1 ,  T ˜ K−1 d˜ f . M g˜ K = d˜ f K

K

ρ K δ K e˜ 2K ρ K + δ K g˜ K

(35) ,

(36) (37) (38)

Parameter Estimation of Fractional Models |

45

Using the general OBE algorithm described by equations (34)–(38), the interval vector containing in guaranteed way the coefficients α i and β j can be determined as



 θ˜ = θ˜ 1 , θ˜ 2 , (39) where ˜ θ˜ 1 = c˜ N − ∆, ˜ θ˜ 2 = c˜ N + ∆, + , ˜N . ∆˜ = diag σ˜ 2N M

(40) (41) (42)

The return to the parameters interval vector [θ] from the pseudo-parameters interval ˜ is performed by solving the interval linear system [Φ] [x] = [ψ] where vector [θ]   ([α]i − 1)  if i = j  h νai , [Φ]i,j =   [α]i  if i = j h νai [ψ]i = −[α]i , -T , [x] = [a1 ], [a2 ], . . . , [a n ] .

(43) (44) (45)

The coefficients [b j ] are determined directly by [b j ] = [β]j

n 

[a i ]h

ν b j −ν a i

.

(46)

i=0

The ellipsoid E N englobing the set of all the acceptable parameters Θ can be partially reconstructed using the interval parameters [θ]. The center of the ellipsoid is the center of [θ]. The diagonal entries of σ2N M N can be determined by   (47) diag σ2N M N = ∆2 , , where ∆ = sup [θ] − c N . The ellipsoid E N is then characterized by his center c N and the diagonal matrix σ2N M N .

4.2.2 OBE algorithms Two families of OBE algorithms exist in literature [8]: OBE group 1: algorithms minimizing the volume of the ellipsoid; OBE group 2: algorithms target the σ2K convergence. Table 1 presents a non exhaustif list of the two OBE families algorithms presented in literature. For the first group, the Fogel and Huang algorithm (FH) [9] and the

46 | M. Amairi et al.

Set Membership Set Approximation (SMSA) [14] are mentioned. For the second group the Dasgupta and Huang algorithm (DH) [7] and the Tan algorithm (TAN) [15] are presented. All these algorithms are obtained from the general OBE algorithm (described by equations (34–38) by an appropriate choice of the weights ρ K and δ K as described in Tab. 1.

Tab. 1. Weights choice in OBE algorithms. ρK

δK

˜K M

1

SMSA

σ˜ 2K−1

1 − λ˜ s K

r˜ K γ˜ 2 λ˜ s K

˜K M σ˜ 2K ˜K M

DH TAN

1 − λ˜ d K 1

λ˜ d K λ˜ t K

˜K M ˜K M

OBE group algorithm Group 1

Group 2

FH

Using the determinant criterion   ˜K , ζ˜K = det σ˜ 2K M

(48)

r˜ K is the value minimizing  1 + r˜ K + R(˜r K ) =

r˜ K e˜ 2K 2 γ + r˜ K g˜ 2K 1 + r˜ K γ−2 g˜ 2K

p ,

(49)

and λ˜ K is the value minimizing R(λ˜ K ) 



1 − λ˜ K ⎝ 

R(λ˜ K ) =

⎞p

⎛ σ˜ 2K 

1 − λ˜ K σ˜ 2K−1

1 + λ˜ K (g˜ K − 1)

⎠ .

(50)

5 Numerical example Consider the BIBO stable fractional system described by the following transfer function:   k b0 (51) G s, θ * =  ν  2ν = 1 + a s ν + a s2ν , 1 2 s s 1 + 2ξ + ω0 ω0

Parameter Estimation of Fractional Models |

47

where θ* = (k, ξ , ω0 )T = (1, −0.5, 2)T is the true parameters vector. The true parameters vector can be also written as θ* = (b0 , a1 , a2 )T = (1, −0.7, 0.5)T . System output y with constant sampling period T s = 0.01s is shown in Fig. 3. The chosen system input is a Pseudo Random Binary Sequence (PRBS) uniformly distributed in [−1, 1] where his power spectral density (PSD) is shown in Fig. 4.

1.5

Output

1

0.5

0

−0.5

Time (s) −1

0

5

10

15

20

25

30

Fig. 3. System output without disturbance.

Periodogram Power Spectral Density Estimate 0 Power/frequency (dB/Hz) −10 −20 −30 −40 −50 −60 −70 Frequency (Hz) −80 0

10

20

30

Fig. 4. Power spectral density (PSD) of the input signal.

40

50

48 | M. Amairi et al.

After the discretization and the linearization steps, a linear regression model of the system is obtained ˜ y K = −˜y0 + d˜ TK θ, where θ˜ = [α1 , β0 ]T ,

 d˜ K = −˜y1 (Kh) + y˜ 0 (Kh), u˜ 0 (Kh) . An unknown-but-bounded noise ζ is added to the regression equation y K = −˜y0 + d˜ TK θ˜ + ζ K , where |ζ K | ≤ γ, ∀K ∈ {1, . . . , 5000}. The noise bound γ is proportional to the Signal-to-Noise Ratio (SNR). An example of a realization of noise with SNR = 20dB (γ = 0.055) is shown in Fig. 5. The disturbed system’s output is presented in Fig. 6.

0.06

Noise

0.04

0.02

0

−0.02

−0.04

−0.06

Time (s) 0

5

10

15

20

25

30

Fig. 5. Bounded noise (RSB = 20dB, γ = 0.055).

The system output is filtered with a state variable filter of cutoff frequency ω c equal to 10 rads−1 and N f equal to 3. Two OBE algorithms (SMSA and DH) presented in Tab. 1 are applied with an ˜ c˜ and σ. ˜ Results and discussions are presented in the appropriate initialization of M, following section.

Parameter Estimation of Fractional Models |

1.5

Output Disturbed output Non disturbed output

1

0.5

0

−0.5

−1

0

5

10

15

20

25

Time (s) 30

Fig. 6. Disturbed and non disturbed system output.

10

10

Volume SMSA algorithm 10

10

10

10

5

0

Time (s)

−5

0

5

10

15

20

25

30

10

Volume DH algorithm 10

10

10

5

0

Time (s)

−5

0

5

10

15

Fig. 7. Evolution of the ellipsoid volume.

20

25

30

49

50 | M. Amairi et al.

6 Results and discussions The results discussed in this section correspond to the case when the system output is altered by a 20dB noise. The same data is used for both algorithms (SMSA and DH) for comparative purpose. All the algorithms are initialized with an ellipsoid of great volume centered at , -, , ˜ N = 109 and c˜ = 0, 0 . Figure 7 shows the origin of the coordinates i.e det σ˜ 2N M the volume layout for each OBE algorithm used in the identification after one data circulation. The algorithm of the first group (SMSA) leads to an ellipsoid of minimal volume whereas the algorithm of the second group (DH) leads to an ellipsoid of a little larger volume. This result is foreseeable because the objective of the second group algorithms is not the systematic ellipsoid volume reduction. The convergence property of OBE algorithms is reported by the evolution of the coefficient σ˜ 2K in Fig. 8. The SMSA algorithm presents a good convergence in spite of its appartenance to the first group. The DH algorithm shows clearly a very good convergence since its goal is the decrease of σ˜ 2 .

1

~2 σ SMSA algorithm

0.8

0.6

0.4

0.2 Time (s) 0

0

1

5

10

15

20

25

30

~2 σ DH algorithm

0.8

0.6

0.4

0.2 Time (s) 0

0

5

Fig. 8. Evolution of σ˜ 2 .

10

15

20

25

30

Parameter Estimation of Fractional Models |

51

 -T  , From the obtained ellipsoid, an interval of each parameter [θ] = [k], [ξ ], [ω0 ] can be calculated by the projection of the ellipsoid on the parametric axis. It’s clear that this projection will induce a pessimism on the parameters interval vector. Then, to reduce to the maximum this pessimism it’s natural to choose the algorithm that reduces more the ellipsoid volume with a good convergence property. As shown by the evolution of the ellipsoid volume and the coefficient σ˜ 2 , the SMSA algorithm leads to the minimal ellipsoid as the FH algorithm but moreover it presents a better convergence property. Figure 9 shows the evolution of estimated parameters corresponding to the center of the ellipsoid and confirm the good choice of the SMSA algorithm.

2.5 Estimated parameters

SMSA algorithm

ω0

2 1.5 k 1 0.5

ω0 k

0

ξ ξ

Time (s)

−0.5 0

5

10

15

20

25

30

2.5 Estimated parameters

DH algorithm

ω0

2 1.5 k

1 0.5

ω0 k

0

ξ Time (s)

ξ

−0.5 0

5

10

15

Fig. 9. Evolution of the estimated parameters.

20

25

30

52 | M. Amairi et al.

For DH and SMSA algorithm, Tab. 2 shows the final values of the estimates parameters corresponding to the ellipsoid center. It also shows the ellipsoid volume and the σ˜ 2 value. With similar convergence properties, we note that the SMSA algorithm gives an ellipsoid with smaller volume.

Tab. 2. Characteristics of the obtained ellipsoid. Algorithm

Ellipsoid center (k, ξ, ω0 )

Ellipsoid Volume

σ˜ 2

DH SMSA

(1.009, −0.492, 2.015) (0.997, −0.498, 2.016)

0.0624 0.0012

0.0030 0.0030

The ellipsoid center is one of the admissible vector. To characterize the other admissible vectors, the ellipsoid volume must be the smallest possible. To reduce the volume, a well known technique is to recirculate the data several times by taking the initial condition for the next pass, the final terms with the previous pass. Table 3 shows the effect of the recirculation of data on the volume of the ellipsoid using the SMSA algorithm.

Tab. 3. Ellipsoid volume based on the number of recirculation. Number of recirculations 1 5 10 20 50 100

  Ellipsoid Volume ×10−4 12 2.285 1.595 1.325 1.140 1.099

Figure 10 shows the ellipsoid englobing the set of all the acceptable parameters Θ (E N ) obtained by the SMSA algorithm with one data circulation. It’s clear that the ellipsoid contains, in a guaranteed way, the true parameter values θ* = (b0 , a1 , a2 )T = T (1, −0.7, 0.5) . Table 4 shows the results obtained by the SMSA algorithm with 1, 5 and 10 data circulation. The width of the parameter’s intervals decrease due to the recirculation effect on the ellipsoid volume shown in Tab. 4.

Parameter Estimation of Fractional Models |

53

1.1 1.05 b0

1

0.95 0.9 0.85

0.6

−0.6 0.5 a2

−0.7 −0.8

0.4

a1

Fig. 10. Ellipsoid englobing the set of all acceptable parameters with one data recirculation.

Tab. 4. Width of interval parameters based on the recirculation number N r . Nr

[a2 ]

[a1 ]

[b0 ]

1 5 10

[0.3985, 0.6491] [0.4211, 0.5965] [0.4440, 0.5557]

[−0.8755, −0.5765] [−0.8126, −0.6158] [−0.7699, −0.6403]

[0.8311, 1.1001] [0.8444, 1.1000] [0.9069, 1.0843]

7 Conclusion In this paper, a new set-membership identification method for linear fractional systems is proposed. The method formalize the identification problem as an ellipsoidal set estimation problem. Supposed that the differentiation orders are a priori known and the disturbances are unknown but bounded, the method uses the Optimal Bounding Ellipsoid algorithm for estimating only the coefficient of the system. A numerical example is proposed to demonstrate the effectiveness of the proposed algorithm and compare different groups. Results of the ellipsoid identification are discussed showing an very good performance of the proposed method. Future works will look for the pessimism of the proposed approach in order to use the proposed method to identify a real system.

54 | M. Amairi et al.

Acknowledgment: This work was supported by the Ministry of the Higher Education and Scientific Research in Tunisia.

Bibliography [1] [2]

[3] [4] [5] [6] [7]

[8] [9] [10] [11] [12] [13] [14] [15]

M. Amairi. Systèmes d’ordres non entiers et identification par méthodes ensemblistes. PhD Thesis, University of Gabes, Tunisia, 2011. M. Amairi, S. Najar, M. Aoun, and M. N. Abdelkrim. Guaranteed output-error identification of fractional order model. 2nd IEEE Int. Conf. on Advanced Computer Control (ICACC 2010), 2:246–250, 2010. M. Aoun. Systèmes linéaires non entiers et identification par bases orthogonales non entières. PhD Thesis, Université Bordeaux I, France, 2005. J. L. Battaglia, L. Le Lay, J. C. Batsale, A. Oustaloup, and O. Cois. Heat flux estimation through inverted non integer identification models. Int. J. of Thermal Science, 39(3):374–389, 2000. M. Caputo. Linear models of dissipation whose Q is almost frequency independent-II. Geophysical J. Int., 13(5):529–539, 1967. O. Cois. Systèmes linéaires non entiers et identification par modèle non entier: application en thermique. PhD Thesis, Université Bordeaux I, France, 2002. S. Dasgupta and Y. F. Huang. Asymptotically convergent modified recursive least-squares with data-dependent updating and forgetting factor for systems with bounded noise. IEEE Trans. on Information Theory, 33(3):383–392, 1997. T. Q. K. Dinh. Contributions à l’identification ensembliste ellipsoïdale. PhD Thesis, Institut National Polytechnique de Grenoble, France, 2005. E. Fogel and Y. F. Huang. On the value of information in system identification – Bounded noise case. Automatica, 18(2):229–238, 1982. A. K. Grünwald. Uber begrenzte derivationen und deren anwendung. Zeitschrift für Mathematik und Physik, 12:441–480, 1867. R. L. Magin. Fractional calculus in bioengineering. Critical reviews in biomedical engineering, 32(3–4), 2004. D. Matignon. Stability properties for generalized fractional differential systems. ESAIM: proceedings, 5:145–158, 1998. K. Miller and B. Ross. Introduction to the fractional calculus and fractional differential equations. John Wiley & Sons New York, 1993. M. Nayeri, M. S. Liu, and J. R. Deller. An interpretable and converging set-membership algorithm. IEEE Int. Conf. on Acoustics, Speech, and Signal Processing, 4:472–475, 1993. G. Tan, C. Wen, and Y. C. Soh. Identification for systems with bounded noise. IEEE Trans. on Automatic Control, 42(7):996–1001, 1997.

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Biographies Messaoud Amairi was born in Tunisia in 1980. He received his Ph.D. in Electrical Engineering in 2011. He is currently an Associate Professor in automatic control and electrical engineering at the National Engineering School of Gabes (Tunisia) and a member of the Modeling, Analysis and Control of Systems (MACS) laboratory. His current research interests include automatic control, system identification, and fractional differentiation.

Slaheddine Najar was born in Tunisia in 1958. He obtained his Electrical engineering and instrumentation Diploma from the Technical and Scientific Faculty of Sfax (Tunisia 1982) and his Master Degree of Energetic from the Paris VII University, and then the Doctorate in Energetic engineering from the Aix-Marseille III University (1985). In 1992 he obtained a PhD in Physical Sciences from the Polytechnic School of Palaiseau (France). He is Associate Professor at the Electrical Engineering Department (control) of the National Engineering School of Gabes (Tunisia) and a member of the Modeling, Analysis and Control of Systems (MACS) laboratory. Mohamed Aoun was born in Tunisia in 1975. He received his Ph.D. in automatic control in 2005 from the University of Bordeaux, France. He is currently Associate Professor in automatic control, electrical engineering, and computer engineering at ENIG, Tunisia and member of its MACS research laboratory. His research interests include automatic control, system identification, fault diagnosis, and fractional differentiation.

Mohamed Naceur Abdelkrim was born in Tunisia in 1958. He obtained a Diploma in Technical Sciences in 1980, his Master Degree in Control in 1981 from the ENSET school of Tunis (Tunisia ),and his PhD in Control in 1985 and the Doctorate in Sciences Degree (Electrical engineering) in 2003 from the ENIT School of Tunis. Since 2003 he is an Associate Professor at the Electrical Engineering Department (Control) of the National Engineering School of Gabès (Tunisia) and he is manager of the Modeling, Analysis and Control Systems (MACS) laboratory.

K. Röbenack

Improving High Gain Observer Design for Nonlinear Systems Using the Structure of the Linear Part

Abstract: This paper deals with high gain observer design for Lipschitz nonlinear systems. High gain observers are very popular in applications because of their simple implementation with a constant observer gain. In practice, the gain is usually chosen based on eigenvalue placement sufficiently far in the complex left half-plane, where the convergence of the nonlinear observer is verified by simulation. Although this strategy works well in many applications, the exact choice of the observer gain is more complicated. In particular, existing design methods often result in a severe restriction of the maximum allowed Lipschitz constant. In many cases, these bounds on the Lipschitz constant are very conservative. We will show that the maximum admissible Lipschitz constant can be increased significantly if the structure of the system is taken into consideration. Keywords: Nonlinear observer, high gain deisgn, stability radius, Riccati inequality

1 Introduction The task of a state observer is the estimation of the state from measurements of the input and the output. The problem of observer design for linear time-invariant systems has been solved by Luenberger [21, 22] in the sixties. His work has been extended into several directions such as observer design for time-varying and nonlinear systems [18, 23, 24, 35, 38], the simultaneous adaption of unknown parameters [1, 2, 19], or the design of unknown input observers [3, 8, 17, 33] as well as functional observers [7, 11]. Several techniques for nonlinear observer design are subject to restrictive existence conditions [20] or require complicated calculations [34]. In this contribution we consider the observer design for nonlinear systems whose dynamics can be decomposed into a linear and a nonlinear part. Taking this decomposition of the nonlinear system into account, Thau [36] suggested an observer with a constant observer gain. Provided the system’s nonlinearity is Lipschitz continuous, the dynamics of the nonlinear system can under additional conditions be dominated by the dynamics of

K. Röbenack: Technische Universität Dresden, Faculty of Electrical and Computer Engineering, Institute of Control Theory, Dresden, Germany, email: [email protected]. De Gruyter Oldenbourg, ASSD – Advances in Systems, Signals and Devices, Volume 1, 2016, pp. 57–73. DOI: 10.1515/9783110448375-005

58 | K. Röbenack

the linear part through an appropriate choice of the gain matrix such that the error dynamics is globally asymptotically stable. Usually, this approach results in large entries of the observer gain. For this reason, this design strategy became known as high gain observer design. High gain observer design is very popular due to the simple implementation once the constant observer gain is known. However, both the existence and the actual calculation of the gain matrix seem to be more complicated. Earlier existence results provide only very conservative bounds with respect to the maximum allowed Lipschitz constant [28, 40]. The most complete result was provided by Rajamani [29], where sufficient and (in some sense) necessary existence conditions for a stabilizing observer gain are formulated. The above mentioned contributions suggest the existence of a strong relationship between the linear part and the maximum value of the Lipschitz constant of the nonlinearity in order to guarantee the convergence of the observer. However, high gain observer design is not always carried out in the original coordinates. A now very commonly used observer was suggested in [6, 13], where the high gain design is carried out in the observability canonical form. A similar approach has been suggested in [30, 32], where the observer canonical form is used. In both cases, the convergence of the observer can be ensured for any (finite) Lipschitz constant through suitable eigenvalue placement. In high gain observer design, many authors take no structural information on the nonlinearities into account (see [28, 29]). An exception is the work [9], which uses a generalized Lipschitz condition to describe the system’s structure in terms of positive matrices. Unfortunately, this description is not well-suited to model sparse nonlinearities. Alternatively, some papers use a very special structure such as the observability canonical form in [6, 13] or the observer canonical form in [32]. However, the later approaches require an additional, usually nonlinear transformation of the observer gain. In this paper we want to improve existing results on high gain observer design by taking the system’s structure into account. Our approach offers an additional framework for slightly structured systems such as non-triangular systems [10]. Since most physically modeled systems already have a certain structure, we suggest the implementation of the observer in the original coordinates taking the available structural information into account. The paper is structured as follows: In section 2 we will first remind the reader of existing results for unstructured systems. Then we will investigate the stability of structured systems, where the existence of a stabilizing observer gain will mainly be formulated in terms of algebraic Riccati inequalities. The existence condition of a high gain observer is linked to the stability radius in section 3. With these results we take a new look at the high gain observer design based on the observability canonical form in section 4. The advantage of a more detailed consideration of the system’s structure is illustrated on an example system in section 5. We will draw some conclusions in section 6.

Improving High Gain Observer Design for Nonlinear Systems

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2 High gain observer design 2.1 Existing results on unstructured systems We consider the observer design for nonlinear systems of the form x˙ = Ax + Φ(x, u)

(1a)

y = Cx

(1b)

with appropriate matrices A, C and a nonlinear map Φ. The task of an observer is the estimation of the n-dimensional state vector x from measurements of the input u and the output y. The nonlinear part consists of the map Φ, which is assumed to be Lipschitz continuous uniformly in u with a Lipschitz constant γ > 0: ∀x, xˆ ∈ Rn :

Φ(x, u) − Φ(ˆx , u) ≤ γ x − xˆ .

(2)

Here,  ·  denotes the euclidean norm. The observer has the form xˆ˙ = A xˆ + Φ(ˆx , u) + L(y − C xˆ )

(3)

with a constant observer gain L. We want to calculate the gain matrix L such that the error dynamics x˜˙ = (A − LC)˜x + Φ(x, u) − Φ(ˆx , u)

(4)

is globally asymptotically stable, where x˜ = x − xˆ denotes the observation error. In this context we would like to recall the important results from Rajamani [29]. The sufficient existence conditions are given in the following theorem [29, Th. 2]: Theorem Consider system (1) with (A, C) observable and Φ bounded by (2). The error dynamics (4) of the observer (3) is asymptotically stable if L can be chosen such that A − LC is stable and min σmin (A − LC − jωI) > γ, ω≥0

(5)

where σmin denotes the smallest singular value. Remark In the proof of this theorem, the observability of (A, C) is only used to ensure that L can be chosen such that A − LC is stable. To achieve that, observability is sufficient but not necessary. More precisely, A − LC can be stabilized through an appropriate choice of L if and only if (A, C) is detectable, that is   sI − A rank =n C

60 | K. Röbenack for all s ∈ C with (s) ≥ 0, see [14, 37]. Roughly speaking, only the unstable modes have to be observable. Remark In [29], the proof is splitted into several parts. In the first two parts it is shown that under condition (5) there exists a symmetric positive definite matrix P solving the algebraic Riccati equation (A − LC)T P + P(A − LC) + γ2 P2 + (1 + ε)I = 0

(6)

for sufficiently small ε > 0. In the last part of the proof, this matrix P is used in the Lyapunov function V(˜x) = x˜ T P x˜ in order to show the asymptotic stability of the error dynamics (4). Note that the solvability of the Riccati equation (6) for sufficiently small ε > 0 is equivalent to the solvability of the Riccati inequality (A − LC)T P + P(A − LC) + γ2 P2 + I < 0,

(7)

where the order relation is understood in the sense of definiteness. In contrast to previous work [36, 40, 28], the bound (5) is strict, i.e., condition (5) is not only sufficient, but also in some sense necessary [29, Th. 3]: Theorem If the observer gain L is chosen such that min σmin (A − LC − jωI) ≤ γ, ω≥0

then there exists at least one matrix E ∈ Cn×n such that the function Φ(x, u) = Ex has the Lipschitz constant γ and the error dynamics (4) is unstable. Remark Condition (5) is also related to results from H∞ -theory [29, p. 399]. Using elementary results on the singular value decomposition, we have −1  min σmin (A − LC − jωI) = sup σmax (A − LC − jωI)−1 ω≥0

(8a)

ω≥0

. .−1 . . = .(sI − (A − LC))−1 . . ∞

(8b)

Therefore, condition (5) is equivalent to . .−1 . . γ < .(sI − (A − LC))−1 . . ∞

(9)

Remark In [29] as well as in this contribution, full order observers with a constant observer gain are considered, i.e., the dynamic order of the observer coincides with the dimension n of the given plant (1). In [41], the results from [29] are generalized for reduced order observers. On the other hand, the above mentioned relation to H∞ -theory can be used to design observers with a dynamic gain, i.e., the observer gain is a transfer function. This approach results in observers of higher dimension than the system, but with less restrictive bounds on the Lipschitz constant [26, 27].

Improving High Gain Observer Design for Nonlinear Systems

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2.2 Structured nonlinear systems In most applications, the nonlinearity Φ occurring in (1) will neither depend on all variables nor influence all differential equations in (1). In the following, these dependencies and influences are described by appropriate matrices B and D such that Φ(x, u) = BF(Dx, u)

(10)

defines a new (in some sense reduced) nonlinear map F. Again, the map F is assumed to be bounded by the Lipschitz constant γ, i.e., ∀z, zˆ ∈ im D :

F(z, u) − F(ˆz , u) ≤ γ z − zˆ ,

(11)

where im D denotes the column space (image, or range) of the matrix D. The structured nonlinear system resulting from (10) reads as x˙ = Ax + BF(Dx, u)

(12a)

y = Cx

(12b)

with the associated observer xˆ˙ = A xˆ + BF(D xˆ , u) + L(y − C xˆ ).

(13)

The observation error x˜ is governed by the error dynamics x˜˙ = (A − LC)˜x + B[F(Dx, u) − F(D xˆ , u)].

(14)

2.3 Convergence of structured systems Now, we will formulate a sufficient existence condition similar as in Theorem 2.1. However, the bound on the Lipschitz constant will be expressed in terms of a new Riccati inequality, which can be seen as a generalization of (7) for the structured case. Theorem Consider system (12) with a detectable pair (A, C) and a nonlinearity F bounded by (11). Moreover, assume that the matrix L can be chosen such that the Riccati inequality (A − LC)T P + P(A − LC) + γ2 PBB T P + D T D < 0

(15)

has a symmetric positive definite solution P. Then the error dynamics (14) of the observer (13) is asymptotically stable.

62 | K. Röbenack

Proof: Let P be a symmetric positive definite solution of (15). Then, the Lyapunov candidate function V(˜x) = x˜ T P x˜ is positive definite and radially unbounded. Its time derivative along (14) reads as V˙ = x˜ T [(A−LC)T P + P(A−LC)]˜x + 2˜x T PB[F(Dx, u)−F(D xˆ , u)].

(16)

Using standard inequalities in connection with the Lipschitz property (11) we obtain 2˜x T PB[F(Dx, u) − F(D xˆ , u)] ≤ |2˜x T PB[F(Dx, u) − F(D xˆ , u)]| ≤ 2x˜ T PBF(Dx, u) − F(D xˆ , u) ≤ 2γx˜ T PBD x˜  ≤ γ2 x˜ T PBB T P x˜ + x˜ T D T D x˜ . Using (15) and (16) we have V˙ ≤ x˜ T [(A − LC)T P + P(A − LC) + γ2 PBB T P + D T D]˜x < 0, for all x ∈ Rn with x  = 0, which proves the asymptotic stability of the error dynamics (14). For γ = 0, the Riccati inequality (15) becomes a Lyapunov inequality. In this case, there exists a symmetric positive definite solution P because the detectability of (A, C) implies the stability of (A − LC) through an appropriate choice of L. For reasons of continuity, the Riccati inequality (15) has also a symmetric positive definite solution P for sufficiently small γ > 0. Now, we define the bound of γ up to which Theorem 2.3 guarantees the convergence of the observer: γmax = sup{γ > 0 : Ineq. (15) has a solution P > 0}.

(17)

Clearly, the convergence of the observer (13) can be achieved for all nonlinearities F bounded by (11) with γ < γmax . If the convergence can be achieved for any Lipschitz constant γ, we obtain γmax = ∞. Later on in section 3 we will show that (17) is the best bound in the sense of Theorem 2.1.

2.4 Computation of the observer gain In this section we will discuss the actual computation of the observer gain, which will be helpful for practical applications. In order to design a convergent observer (13) we have to solve (15) simultaneously w.r.t. P > 0 and L. In the given form (15) the inequality is nonlinear. We can remove the

Improving High Gain Observer Design for Nonlinear Systems

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63

bilinear terms PLC and C T L T P with the substitution Y = PL,

(18)

A T P + PA − C T Y T − YC + γ2 PBB T P + D T D < 0.

(19)

which results in

The quadratic term PBB T P can be removed using Schur’s complement [4]. To get a solution of (19) we have to solve the linear matrix inequality (LMI) 

P> 0 −A T P−PA+C T Y T +YC−D T D γB T P

γPB I

(20a)

 >0

(20b)

simultaneously for P and Y. Due to (18), the observer gain L is obtained by L = P−1 Y .

(21)

Note that the LMI (20) can be solved numerically with standard software such as Matlab [12] and Scilab [25]. The LMI (20) is solvable if and only if the Riccati inequality (15) has a symmetric positive definite solution [39, 4], i.e., both formulations are equivalent. Combining the numerical solution of (20) with a bisection algorithm we can also calculate the bound γmax from (17), provided γmax < ∞.

3 Stability radius 3.1 Basic definition The concept of the stability radius of a linear system was introducted by Hinrichsen and Pritchard [15]. Consider a linear system x˙ = A0 x + Bu

(22a)

y = Dx

(22b)

with a stable matrix A 0 and appropriate matrices B and D. An output feedback u = −Ey

(23)

with a gain matrix E can be interpreted as a structured perturbation of the orginal linear system (22). The closed-loop system x˙ = (A0 − BED)x

(24)

64 | K. Röbenack

is stable for E = 0 since A0 is stable by assumption. Since the eigenvalues of A0 − BED depend continuously on the entries of E, system (24) remains stable for sufficiently small entries in E. The stability radius is a measure of the maximum value of E until the system loses its stability. More precisely, the stability radius is defined as rC (A0 , B, D) = inf {E : A0 − BED is unstable}, E

(25)

where  ·  is the operator norm induced by the euclidean vector norm. Remark The stability radius defined by (25) is the so-called complex stability radius, because condition (25) has to be fulfilled for all complex matrices E of appropriate dimension. Note that the real stability radius rR is defined in the same manner by (25) with the restriction to real matrices E, see [16]. However, the complex stability radius is advantageous for the treatment of nonlinear systems. The stability radius can alternatively be characterized as follows: Consider the transfer function of system (22) given by G(s) = D(sI − A0 )−1 B,

(26)

which is assumed to be stable. Hence, the H∞ -norm is well-defined. It was shown in [15, Prop. 2.1] that the stability radius (25) can be obtained from (26) by rC (A0 , B, D) =

G(s)−1 ∞

∞

for for

G(s) = 0, G(s) ≡ 0.

(27)

Remark For linear feedback, the stability property follows immediately from definition (25): The closed-loop system (24) is (asymptotically and even exponentially) stable for any output feedback (23) with E < rC . Remark Consider a nonlinear and possibly time-varying output feedback u = N(y, t)

with

N(0, t) = 0,

(28)

which is assumed to be bounded by a Lipschitz constant γ. The origin x = 0 of the resulting closed-loop system (see Fig. 1) is asympototically stable for γ < rC , see [15, Prop. 5.1/5.2].

A0 B

D N

Fig. 1. Structure of the nonlinear control system discussed in Remark 3.1.

Improving High Gain Observer Design for Nonlinear Systems |

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An important result from [15] is the characterization of the stability radius in terms of a parametrized algebraic Riccati equation. The results from [15, Section 3] can be summarized as follows: Theorem Let A0 be stable and γ > 0. The Riccati inequality A0T P + PA0 + γ2 PBB T P + D T D < 0

(29)

has a symmetric positive definite solution P if and only if γ < rC (A0 , B, C).

(30)

3.2 Application to observer design The stability radius (25) depends on the three matrices A, B, D of system (22). We want to apply the concept of the stability radius to the observer design problem for system (12). The convergence of the observer (13) is ensured by the asymptotic stability of the error dynamics (14). Due to the Lipschitz property (11) of F, we can treat the difference of the function values of the nonlinearity F for the original as well as estimated state in (14) as a (single) time-varying nonlinearity depending only on z˜ ∈ im D, but having the same Lipschitz constant: ∆F(˜z , t) := F(z(t), u(t)) − F(ˆz (t), u(t)). The resulting (simplified) structure of the error dynamics is sketched in Fig. 2.

L – A0 B

C D

ΔF

Fig. 2. Essential structure of the error dynamics (14).

Comparing the structures sketched in Fig. 1 and 2, we set A0 = A − LC.

(31)

The matrix A 0 is stable through an appropriate choice of the observer gain L. From Remark 3.1 we conclude that the observer converges asymptotically for any

66 | K. Röbenack

stabilizing matrix L if γ < rC (A − LC, B, D).

(32)

Taking Eq. (27) into account, we see that condition (32) is equivalent to .−1 . . . γ < .D(sI − (A − LC))−1 B. .

(33)

∞

Now, we are able to formulate sufficient and necessary conditions for the stabilizability of the observer’s error dynamics: Theorem Consider system (12) with a detectable pair (A, C), a nonlinearity F bounded by (11) with the Lipschitz constant γ > 0, and γmax defined by (17). Then, we have: 1. If γ < γmax , there exists a gain matrix L such that the error dynamics (14) of the observer (13) is asymptotically stable. 2. If γ > γmax , then for any gain matrix L there exists a complex matrix E of appropriate dimensions such that the function F(x, u) = Ex

(34)

has the Lipschitz constant γ and the error dynamcis (14) is unstable. Proof: If γ < γmax , Theorem 2.3 ensures the asymptotic stability of the error dynamics (14). Now, let γ > γmax . By definition (17), there exists no positive definite solution P of (15). Taking (31), the Riccati inequality (29) coincides with (15). Since (29) has no positive definite solution P, Theorem 3.1 implies γ > rC (A0 , B, C). In other words, for any L we have γ > rC (A − LC, B, C). The Lipschitz constant γ of (34) means that E = γ. Since γ > rC (A − LC, B, C), the definition (25) implies the existence of a matrix E with E  = γ such that the closed-loop linear system (24) is unstable. Clearly, system (24) corresponds directly to the error dynamics (14) with (34), which is hence also unstable. This means, that the upper bound γmax on the nonlinearity’s Lipschitz constant γ is sharp. Taking (32) and (33) into account, we can characterize the bound γmax in terms of the stability radius γmax = sup rC (A − LC, B, D)

(35)

L

as well as in terms of the H∞ -norm .−1 . . . γmax = sup .D(sI − (A − LC))−1 B. . L

∞

(36)

Improving High Gain Observer Design for Nonlinear Systems

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Equation (36) might be helpful for a symbolic computation of L. To maximize the value on the right-hand side of (36) we have to minimize the H∞ -norm of G(s) := D(sI − (A − LC))−1 B.

(37)

In particular, we have γmax = ∞ if G(s)∞ → 0 for appropriate L. One might ask whether there exists a finite gain L with G(s) = 0. Since  · ∞ is a norm, we have G(s)∞ = 0 if and only if G(s) ≡ 0. This situation can be characterized as follows: Corollary The error dynamics (14) of the observer (13) of system (12) can be stabilized asymptotically for any Lipschitz countinuous nonlinearity F regardless of the actual Lipschitz constant if and only if the observer gain L can be chosen such that D[sI − (A − LC)]−1 B ≡ 0.

4 High gain design using the observability canonical form In many practical applications, high gain design is carried out in the observability canonical form [13, 6], where the pair (A, C) is in Brunovsky form [5] ⎛ ⎞ 0 1 ··· 0 ⎜ . . ⎟ ⎜ .. . . . . . ... ⎟ ⎜ ⎟ A=⎜ (38a) ⎟ .. ⎜ .. ⎟ . 1 ⎠ ⎝ . 0 ··· ··· 0   C = 1 0 ··· 0 , (38b) and the influence of the nonlinearities is described by B = e n with the nth unit vector e n , and D = I. The gain vector ⎛ ⎞ ε−1 p n−1 ⎜ ⎟ .. ⎟ Lε = ⎜ . ⎝ ⎠ ε−n p0 with the parameter ε ∈ (0, 1] results in the characteristic polynomial ρ ε (s) = det[sI − (A − L ε C)] p p1 p s + n0 = s n + n−1 s n−1 + · · · + n−1 ε ε ε  s1   sn  = s− ··· s − ε ε

(39)

68 | K. Röbenack

of the linear part with the roots s1 , . . . , s n for ε = 1. The transfer function (37) of the linear part has the form ⎛ ⎞ 1 ⎜ ⎟ s + p n−1 ε−1 ⎟ 1 ⎜ 2 −1 −2 ⎟ . G ε (s) = ε ⎜ (40) ⎜ ⎟ s + p ε s + p ε n−1 n−2 ρ (s) ⎝ ⎠ .. . In the beginning we consider the case ε = 1. Clearly, the gain vector L1 will be chosen such that the characteristic polynomial (39) is a Hurwitz polynomial, i.e., the roots s1 , . . . , s n are in the open complex left half-plane. Moreover, the components of the transfer function are strictly proper. This implies that the kth component of the transfer function (40) has a finite maximum gain μ = G1k ∞ = sup |G1k (jω)| < ∞. ω≥0

Now we consider the case 0 < ε 1. For ε → 0, the roots of the characteristic polynomial (39) are moved uniformly to the left. The kth component of the transfer function (41) can be written as G εk (jω) = =

k−2 + · · · + pεn−k+1 (jω)k−1 + p n−1 k−1 ε (jω) , , jω − sε1 · · · jω − sεn

(jωε)k−1 + p n−1 (jωε)k−2 + · · · n−k+1 ε (jωε − s1 ) · · · (jωε − s n )

(41)

= G1k (jωε) ε n−k+1 . This implies G εk ∞ = sup |G εk (jω)| ω≥0

= sup |G1k (jωε)| ε n−k+1 ω≥0

= sup |G1k (jω)| ε n−k+1 ω≥0

= μ ε n−k+1 → 0 for ε → 0 and k = 1, . . . , n. If the norm of every component of the transfer function (41) goes to zero, we also have  G ε ∞ → 0

for

ε → 0,

G ε −1 ∞ →∞

for

ε → 0.

or equivalently,

Improving High Gain Observer Design for Nonlinear Systems

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This implies γmax = ∞ according to (36). In other words, the observation error can be stabilized for an arbitrary Lipschitz nonlinearity provided the eigenvalues of the linear part are moved sufficiently far to the left-hand side in a geometric fashion. This offers an alternative proof of the stability results already established in [13, 6]. A similar result can be obtained for the observer canonical form [32, 31].

5 Example We consider the following example taken from [28, Section 6] with       −2 3 0 A= , C = 0 1 , Φ(x, u) = 3 1 κ sin(x1 ) + u

(42)

with κ > 0. First, let us consider (42) as a system of the form (1), i.e., we use the approach from [29] omitting any structural information concerning the nonlinearity. This corresponds to the structured form (12) with B = D = I. In order to determine the maximum admissible Lipschitz constant we have to minimize the H∞ -norm of the transfer function G(s) = [sI − (A − LC)]−1 . The bisection method in connection with an LMI solver yields the minimum value 0.2773505 of the norm and therefore γmax ≈ 3.6055459. This is confirmed by the numerical results in [9, 27] and guarantees the stability for κ = γ = 1.5 as used in [28]. Now, we use the full structured nonlinearity Φ(x, u) = BF(Dx, u) with B = (0, 1)T , D = (1, 0) and F(x1 , u) = κ sin x1 + u. The associated scalar transfer function (37) of the linear part is G(s) = D[sI − (A − LC)]−1 B =

3 − l1 , s2 + (l2 + 1)s + 2l1 + 2l2 − 11

where L = (l1 , l2 )T . We have to find gain entries l1 and l2 such that the H∞ -norm of the transfer function is minimized. The numerator becomes zero for l1 = 3. The resulting denominator polynomial s2 + (l2 + 1)s + 2l2 − 2 is stable for l2 > 1, e.g. for l2 = 2. With the observer gain L = (3, 2)T we have G∞ = 0 and γmax = ∞. Hence, the observer can be stabilized for any Lipschitz constant. This extends the results given in [9, 27]. To verify these theoretical results, we carried out a numerical simulation using the computer algebra system WxMaxima. System (12) and observer (13) with (42) and κ = 25 were simulated employing the stabilizing observer state feedback u = −(11, 13) xˆ + 10 sin(2πt) with an additional excitation similar to [27, 9]. With this feedback, the controller eigenvalues of the linear part are placed at −6 and −8. Furthermore, we used the above-mentioned observer gain L = (3, 2)T , and the initial values x(0) = (1.6, 2)T and xˆ (0) = (0, 0)T . The simulation results shown in Fig. 3 indicate that observer trajectories converge to the trajectories of the original

70 | K. Röbenack

system. In addition, Fig. 4 shows the eudlidean norm of the observation error along the simulated trajectory, which confirms the convergence.

5

x1 x2 x1 x2

4 3 2 1 0 –1 0

0.5

1

1.5 Time t

2

2.5

3

2.5

3

Fig. 3. Simulation results of the example system (42).

4 3.5 3 2.5 2 1.5 1 0.5 0

0

0.5

1

1.5 Time t

2

Fig. 4. Euclidean norm ˜x (t) of the observation error x˜ (t) of the simulated example system (42).

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6 Conclusion In this contribution we considered the problem of high gain observer design for Lipschitz nonlinear systems. We showed that it is advantageous to take the detailed structure of the nonlinearity into account. In particular, we characterized the case when the high gain observer can be designed such that convergence of the error dynamics is guaranteed without explicit knownledge of the actual Lipschitz constant. Acknowledgment: The author is indepted to Dr. Lutz Gröll (Karlsruhe Institute of Technology, Institute for Applied Computer Science) and Dr. Ralf Bartholomäus (Fraunhofer Institute for Transportation and Infrastructure Systems, Dresden) for interesting discussions which contributed to the paper. Moreover, the author would like to thank his former student, Paul Schulze, for his work.

Bibliography [1] [2] [3] [4] [5] [6] [7] [8] [9]

[10] [11]

[12] [13]

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[14] M. L. J. Hautus. Stabilization, controllability, and observability for linear autonomous systems. Ned. Akad. Wetenschappen, Proc. Ser. A, 73:448–455, 1970. [15] D. Hinrichsen and A. J. Pritchard. Stability radius for structured perturbations and the algebraic Riccati equations. Systems & Control Letters, 8:105–133, 1986. [16] D. Hinrichsen and A. J. Pritchard. A note on some differences between real and complex stability radii. Systems & Control Letters, 14:401–408, 1990. [17] K. Kalsi, J. Lian, S. Hui, and S. Żak. Sliding-mode observers for systems with unknown inputs: A high-gain approach. Automatica, 46(2):347–353, 2010. [18] C. Kravaris, J. Hahn, and Y. Chu. Advances and selected recent developments in state and parameter estimation. Computers & Chemical Engineering, 51:111–123, 2013. [19] G. Kreisselmeier. Adaptive observers with exponential rate of convergence. IEEE Trans. on Automatic Control, 22(1):2–8, 1977. [20] A. J. Krener and A. Isidori. Linearization by output injection and nonlinear observers. Systems & Control Letters, 3:47–52, 1983. [21] D. G. Luenberger. Observing the state of a linear system. IEEE Trans. Mil. Electronics, ME-8(2):74–80, 1964. [22] D. G. Luenberger. Observers for multivariable systems. IEEE Trans. on Automatic Control, AC-11(2):190–197, 1966. [23] E. A. Misawa and J. K. Hedrick. Nonlinear observers – a state-of-the art survey. Journal of Dynamic Systems, Measurement, and Control, 111:344–352, 1989. [24] H. Nijmeijer and T. I. Fossen, editors. New Directions in Nonlinear Observer Design, volume 244 of Lecture Notes in Control and Information Science. Springer-Verlag, London, 1999. [25] R. Nikoukhah, F. Delebecque, and L. El Ghaoui. LMITOOL: a package for LMI optimization in Scilab User’s Guide. Rapport technique 170, INRIA, 1995. [26] A. M. Pertew, H. J. Marquez, and Q. Zhao. Dynamic observers for nonlinear Lipschitz systems. In Preprints of the 16th IFAC World Congress, Prague, Czech Republic, July 3-8, 2005. [27] A. M. Pertew, H. J. Marquez, and Q. Zhao. H∞ observer design for Lipschitz nonlinear systems. IEEE Trans. on Automatic Control, 51(7):1211–1216, 2006. [28] S. Raghavan and J. K. Hedrick. Observer design for a class of nonlinear systems. Int. J. Control, 59(2):515–528, 1994. [29] R. Rajamani. Observers for Lipschitz nonlinear systems. IEEE Trans. on Automatic Control, 43(3):397–401, 1998. [30] K. Röbenack. An approximation of normal form observer design: Convergence and computation. In Preprint 6th IFAC-Symposium on Nonlinear Control Systems (NOLCOS 2004), pages 965–970, Stuttgart, 2004. [31] K. Röbenack. Structure matters – some notes on high gain observer design for nonlinear systems. In Proc. of the 9th International Multi-Conference on Systems, Signals and Devices, Chemnitz, Germany, 2012. [32] K. Röbenack and A. F. Lynch. High-gain nonlinear observer design using the observer canonical form. IET Control Theory & Applications, 1(6):1574–1579, 2007. [33] E. Rocha-Cózatl and J. Moreno. Passivity and unknown input observers for nonlinear systems. In 15th Triennial World Congress of the International Federation of Automatic Control Barcelona, 21–26 July 2002, 2002. [34] J. Schaffner and M. Zeitz. Variants of nonlinear normal form observer design. In Nijmeijer and Fossen [24], pages 161–180. [35] S. K. Spurgeon. Sliding mode obersers: a survey. Int. J. Systems Science, 39(8):751–764, 2008. [36] F. E. Thau. Observing the state of nonlinear dynamical systems. Int. J. Control, 17(3):471–479, 1973.

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[37] M. Vidyasagar. On the stabilization of nonlinear systems using state detection. IEEE Trans. on Automatic Control, 25:504–509, 1980. [38] B. L. Walcott, M. J. Corless, and S. H. Żak. Comparative study of non-linear state-observation techniques. Int. J. Control, 45(6):2109–2132, 1987. [39] J. C. Willems. Least squares strationary optimal control and the algebraic Riccati equation. IEEE Trans. on Automatic Control, 16:621–634, 1971. [40] S. H. Żak. On the stabilization and observation of nonlinear/uncertain dynamic systems. IEEE Trans. on Automatic Control, 35(5):604–607, 1990. [41] F. Zhu and Z. Han. A note on observers for Lipschitz nonlinear systems. IEEE Trans. on Automatic Control, 47(10):1751–1754, 2002.

Biography Klaus Röbenack received his Dipl.-Ing. and Dr.-Ing. degrees in electrical engineering from the Technische Universität Dresden in 1993 and 1999, respectively. Moreover, he received the Dipl.-Math. degree in 2002 and the postdoctoral qualification (habilitation) in 2005. He is head of the Institute of Control Theory at Technische Universität Dresden since 2009. His research interests include nonlinear control, observer design, descriptor systems and scientific computing.

M. Kchaou and A. Toumi

Fuzzy Network-Based Control for a Class of T–S Fuzzy Systems with Limited Communication

Abstract: In this paper we investigate the problem of designing fuzzy network-based controllers for continuous-time Takagi-Sugeno T–S fuzzy systems. Network-induced delays, data packet dropout, and measurement quantization are simultaneously considered for Network signal-transmission. A stability criterion is developed using a Lyapunov-Krasovskii functional and sufficient conditions for the existence of quantized state and observer-based feedbacks are derived. Finally, two examples are given to illustrate the application of the proposed design techniques. Keywords: T–S fuzzy systems, Networked control systems (NCSs), delay, packet dropout, quantization, LMI.

1 Introduction A Networked Control System (NCS) is a control system in which the sensors and actuators are connected to a feedback controller via a shared communication medium. The advantages of NCSs, including low cost, simple installation and maintenance, reduced system wiring and increased agility, have made researchers pay much attention. Inevitably, the communication network which connects sensors, actuators, and controllers introduces various issues, such as time delay, packet dropout, scheduling of packet transmission, limited communication capacity, data rate constraint, and so on. The aforementioned communication phenomena will induce inaccurate system information or limited effective information which can degrade the performance of control systems and can even destabilize the systems [4, 13, 14, 16]. Due to the bandwidth limitation, the real communication networks are not able to send data with infinite precision. Thus the length of each packet is finite and the effect of signal quantization must be considered into controller design. For several studies, the quantization error is treated as an uncertainty which depend on the quantization density [3, 7, 12, 17].

M. Kchaou and A. Toumi: University of Sfax, National School of Engineering of Sfax, Laboratory of Sciences and Techniques of Automatic control & computer engineering (Lab-STA), Sfax, Tunisia, email: [email protected], [email protected]

De Gruyter Oldenbourg, ASSD – Advances in Systems, Signals and Devices, Volume 1, 2016, pp. 75–93. DOI: 10.1515/9783110448375-006

76 | M. Kchaou and A. Toumi

Since most physical systems and processes in the real world are nonlinear, fuzzy logic has been introduced as an effective approach to deal with nonlinear control systems [10]. There are growing interests in the T–S fuzzy-model-based control of complex nonlinear systems because it provides a basis for the systematic stability analysis and controller design of T–S fuzzy control systems in view of powerful conventional control theory [1, 5, 6],. The present article investigates the networked-based feedback control for a class of nonlinear systems subject to limited communication capacity which can described by T–S fuzzy models. The physical plant and the controller are assumed to be in continuous time, and the measurement is sampled and transmitted over communication networks periodically. The sampling behaviour, together with the signal transmission delay and data packet dropout, is dealt with delay system approach, and the measurement quantization is treated using a sector bound method. The main contributions of this paper are summarized as follows. – A new approach is proposed to deal within a unified framework the effect of measurement quantization, signal transmission delay, and data packet dropout which appear typically in a network environment. – Two schemes of control design, including state and observer-based output feedbacks are presented using the parallel distributed compensation (PDC) concept. Two examples are provided to illustrate the effectiveness of the proposed method.

2 System description and preliminaries A typical NCS model with network-induced delays is shown in Fig. 1, where τ sc is sensor-to-controller delay and τ ca is the controller-to-actuator delay. It is assumed that the controller computational delay can be absorbed into either τ sc or τ ca . The nonlinear plant can be described by the following T-S fuzzy model: Plant rule i : , , IF θ1 (t) is F i1 and · · · θ s (t) is F is THEN: ˙ = A i x(t) + B i u(t) x(t) y(t) = C2 x(t)

(1)

where θ j (t) and F ij (i = 1, · · · , r, j = 1, · · · , s) are, respectively, the premise variables and the fuzzy sets . x(t) ∈ R n is the state vector; u(t) ∈ R m is the control input vector, y(t) ∈ R p is the measured output and r is the number of IF-THEN rules. A i and B i are constant matrices with appropriate dimensions.

T–S Fuzzy Systems with Limited Communication

|

77

Fig. 1. Framework of networked control system.

By using the commonly used gravity center defuzzification, product inference and singleton fuzzifier, the T-S fuzzy systems can be inferred as ˙ = x(t)

r 

# $ h i [θ(t)] A i x(t) + B i u(t)

(2)

i=1

where h i [θ(t)] =

υ i [θ(t)]

r

,

υ i [θ(t)]

υ i (θ(t)) =

s /

F ij [θ j (t)]

(3)

j=1

i=1

h i [θ(t)] is the weighted average for each rule, representing the normalized membership grade, and satisfies h i [θ(t)] ≥ 0,

r 

h i [θ(t)] = 1, t ≥ 0

(4)

i=1

The following assumptions, which are common for NCSs research in the open literature [9, 18], are also made in this work: 1. The sensors are clock driven, the controller and actuators are event driven. 2. Data, either from measurement or for control, are transmitted with a single packet. 3. The real input u(t), realized through a zero-order hold, is a piecewise constant function.

78 | M. Kchaou and A. Toumi

4. For the case of out-of-order packet sequences, the time stamping technique is applied to choose the latest message. In this work, we consider several cases where a logarithmic quantizer is more appropriate. The set of quantized levels is described as 1 0 U = u i , u i = ρ i u0 , i = 0, ±1, ±2, · · · ∪ {0}, u0 > 0

(5)

where the parameter 0 < ρ < 1 is called the quantization density, and the logarithmic quantizer q(.) is defined as ⎧ 1 i 1 i ⎪ ⎪ u if ρ u0 < ν ≤ ρ u0 ⎪ ⎨ i 1+δ 1−δ (6) q(ν) = 0 if ν = 0 ⎪ ⎪ ⎪ ⎩ − q(−ν) if ν < 0 1−ρ . 1+ρ It follows from [2] that quantized feedback control is equivalent to robust control with sector bounded uncertainty, and q(ν) can be expressed as

where δ =

q(ν) = [I + ∆(t)]ν

(7)

∆(t) = diag {∆1 (t), ∆2 (t), · · · ∆ n (t)},

(8)

where

∆ j (t) ≤ δ, j = 1, 2, · · · , n

(9)

In the squeal, we introduce the following lemmas that are essential for the proofs of our results Lemma 1[11] For any matrices D, E, F(t) with appropriate dimensions, and any scalar ε > 0, if F T (t)F(t) ≤ I then Sym[DF(t)E] ≤ εDD T + ε−1 E T E

(10)

Lemma 2[8] For any vectors ψ1 , ψ2 , matrices S, R and real scalars α1 ≥ 0, α2 ≥ 0, satisfying   R S ≥ 0, α1 + α2 = 1, ψ i = 0 if α i = 0 (i = 1, 2) ST R we have  T  ψ R 1 T 1 T − ψ1 Rψ1 − ψ2 Rψ2 ≤ − 1 α1 α2 ψ2 ST

S R



ψ1 ψ2



T–S Fuzzy Systems with Limited Communication

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3 Main results In this section, the attention is focused on the design of both state feedback and output feedback controller under the condition that the signal is transmitted through a channel with limited bandwidth, network-induced delays and data packet dropouts constraints.

3.1 State feedback design Assume that the state of system (1) is measurable and will be quantized before it is transmitted to the controller through communication network. To realize the state feedback control for system (1) subject to quantization, the following parallel distributed compensation scheme is constructed u(t k ) =

r 

h i [θ(t k − τ k )]K i x(t k − τ k )

(11)

i=1

where K i = K i (I + ∆(t)). From the BOZ, the input signal is u(t) =

r 

h i [θ(t − τ k )]K i x(t − τ k ),

t k ≤ t ≤ t k+1

(12)

i=1

Then the closed-loop system can be written as (for t k ≤ t ≤ t k+1 ): ˙ = x(t)

r  r 

  h i [θ(t)]h j [θ(t − τ k )] A i x(t) + B i K j x(t − τ k )

(13)

i=1 j=1

–

Network-induced delay: Network-induced delays always exist when the data transmits through a network, and obviously, it has both lower and upper bounds. Therefore, a plausible representation of delay would be non-differentiable interval time-varying function. A natural assumption on τ k can be made as 0 < τm ≤ τk ≤ τM

–

(14)

Packet dropouts: The effect of data packet dropouts in the communication channel can be described as the ZOH is not updating during the time interval of this event, which is referred as vacant sampling. Hence, the effect of one packet dropout in the transmission is just a case that one sampling period delay is induced in the updating interval of ZOH. t k+1 − t k = (σ k+1 + 1)h + τ k+1 − τ k

(15)

80 | M. Kchaou and A. Toumi

where σ k+1 is the number of accumulated packet dropouts in this period. Let us represent η(t) = t − t k + τ k , t k ≤ t ≤ t k+1 , then τ m ≤ τ k ≤ η(t) ≤ (σ + 1)h + τ k+1

(16)

where σ denotes the maximum number of packet dropouts in the updating periods, η1 = τ m and η2 = (σ + 1)h + τ M . Thus we get η1 ≤ η(t) ≤ η2 Since

∞

(17)

[t k , t k+1 ) = [0, ∞), we have

k=0

⎧   r r ⎪ ⎪ ˙ = h i [θ(t)]h j [θ(t − τ k )] A i x(t) + B i K j x[t − η(t)] ⎨ x(t) i=1 j=1

(18)

 ⎪ ⎪ ⎩ x(t) = ϕ(t), t ∈ t − η , t 0 2 0

where ϕ(t) is a given initial condition sequence. Therefore, it is natural and necessary to assume that the functions h i (θ(t) are well defined for all t ∈ [−η2 , 0], and satisfy the th following properties h i [θ(t − τ k )] ≥ 0,

r 

h i [θ(t − τ k )] = 1

(19)

i=1

Defining A=

r 

h i [θ(t)]A i ,

i=1

Aη =

r  r 

h i [θ(t)]h j [θ(t − τ k )]B i K j

(20)

i=1 j=1

then closed-loop system (18) can be described by ˙ x(t) x(t)

= Ax(t) + A η [I + ∆(t)]x[t  − η(t)], = ϕ(t), t ∈ t0 − η2 , t0

(21)

The controller design is based on the following preliminary result given by the following lemma. Lemma 3 For given scalars η1 > 0, η2 > 0 , μ1 , μ2 , ε > 0 and quantization density ρ > 0, closed-loop system (21) is asymptotically stable if there exist positive matrices P, Q1 , Q2 , Z1 and Z2 , and matrices S and G, with appropriate dimensions, such that the following conditions hold ⎡ ⎤ T Λ Γ ε∆ ⎢ ⎥ (22) ⎣ * −εI 0 ⎦0 Z2

Z2 *

Z1 Λ23 −Q1 − Z1 − Z2 * *

|

0 Λ24 S −Z2 − Q2 * T

81

(23)

⎤ Λ15 Λ25 ⎥ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎦ Λ55

∆= 0

,

δI

0

0

Λ11 = Sym(AG) + Q1 + Q2 − Z1 ,

Λ12 = Aη G + μ1 AT G,

Λ22 = −2Z2 + Sym(S) + μ1 Sym(Aη G),

Λ23 = Z2 − S T ,

T

0

 0 ,

Λ24 = Z2 − S,

Λ15 = P − G + μ2 A G,

Λ25 = −μ1 G + μ2 ATη G,

Λ55 = (η2 − η1 )2 Z2 + η21 Z1 − μ2 Sym(G),

Λ16 = G T Aη

Proof: Construct the following Lyapunov-Krasovskii function: V(x t ) = V1 (t) + V2 (t) + V3 (t)

(24)

T

V1 (t) = x (t)Px(t) t V2 (t) =

(25)

x T (s)Q1 x(s)ds +

t−η1

t

x T (s)Q2 x(s)ds

(26)

t−η2

0  t V3 (t) = η1

−η1 t

T

˙ x˙ (s)Z1 x(s)dsdθ + ηr

−η1 t+θ

˙ x˙ T (s)Z2 x(s)dsdθ

(27)

−η2 t+θ

with η r = η2 − η1 . Taking the time derivative of V(x t ) along the trajectories of (21) yields ˙ V˙ 1 (t) = 2x T (t)P x(t)

(28)

T

T

T

V˙ 2 (t) ≤ x (t)[Q1 + Q2 ]x(t) − x (t − η1 )Q1 x(t − η1 ) − x (t − η2 )Q2 x(t − η2 )

(29)

t−η t  1   T ˙V3 (t) = x˙ T (t) η21 Z1 + η2r Z2 x(t) ˙ − η1 ˙ ˙ x˙ (s)Z1 x(s)ds − ηr x˙ T (s)Z2 x(s)ds

(30)

t−η1

t−η2

Denoting ψ1 = x(t) − x(t − η1 ), ψ2 = x(t − η1 ) − x[t − η(t)] and ψ3 = x(t − η(t)) − x(t − η2 ), by Jensen inequality, one can obtain t −η1 t−η1

˙ x˙ T (s)Z1 x(s)ds ≤ −ψ1T Z1 ψ1

(31)

82 | M. Kchaou and A. Toumi

and t−η  1

−η r

t−η(t) 

T

˙ x˙ (s)Z2 x(s)ds = −η r

t−η2

t−η  1

T

˙ x˙ (s)Z2 x(s)ds − ηr t−η2

˙ x˙ T (s)Z2 x(s)ds

t−η(t)

ηr ηr ψ T Z2 ψ2 − ψ T Z2 ψ3 ≤− η(t) − η1 2 η2 − η(t) 3

(32)

According to Lemma 2, we have t−η  1

−η r t−η2



ψ ˙ x˙ (s)Z2 x(s)ds ≤− 2 ψ3

T 

T

Z2 *

S Z2



ψ2 ψ3

 (33)

From (21) we may construct for any appropriately dimensional matrix G the following equation:   T ˙ 2 x T (t)G T + μ1 x T [t − η(t)]G T + μ2 x(t)G   ˙ + Ax(t) + Aη [I + ∆(t)][t − η(t)] = 0 (34) × −x(t) T

Defining ξ (t) = x T (t) x T (t − η(t)) x T (t − η1 ) x T (t − η2 ) x˙ T (t) . A combination of (28), (33) and (34) leads to   ˙ t ) ≤ ξ T (t) Λ + Sym[Γ∆(t)] ξ (t) V(x

(35)

By (22) in Lemma 3 and according to Lemma 1, we readily obtain: −1 ˙ t ) < 0. Thus, according Λ + Sym[Γ∆(t)∆ ∆] = Λ + Sym[Γ∆(t)] < 0, which means V(x ˙ t ) < αx t . We to Lyapunov stability theory, there exist a scalar α > 0 such that V(x conclude that the system (21) is asymptotically stable.  The objective now is how to determine the gain matrices K i such that the feedback closed-loop system is asymptotically stable. Theorem 1 Consider system (2). For some given scalars η1 > 0, η2 > 0 , μ1 , μ2 , ε ij > 0, 1 ≤ i < j ≤ r and quantization density ρ > 0, the closed-loop system (18) is asymptotically ˜ 2 > 0, Z˜ 1 > 0, Z˜ 2 > 0, S, ˜ G˜ and Y i , ˜ 1 > 0, Q stable, if there exist matrices P˜ > 0, Q satisfying Ψ˜ ii < 0, Ψ˜ ij + Ψ˜ ji < 0,   Z˜ 2 S˜ >0 * Z˜ 2

(36) (37) (38)

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where ⎡

Λ˜ ij ⎢ Ψ˜ ij = ⎣ * *

T Γ˜ 1ij = Λ˜ 16

Γ˜ 1ij ε ij Γ˜ 2ij *

T μ1 Λ˜ 16

⎡

Λ˜ 11 ⎢ T ε ij G˜ ∆ ⎢ * ⎥ ⎢ 0 ⎦ , Λ˜ ij = ⎢ * ⎢ ⎣ * −ε ij I * ⎤ T

0

0

T μ2 Λ˜ 16

T

,

Λ˜ 12 Λ˜ 22 * * *

Z˜ 1 ˜Λ23 Λ˜ 33 * *

0 Λ˜ 24 S˜ Λ˜ 44 *

⎤ Λ˜ 15 Λ˜ 25 ⎥ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎦ Λ˜ 55

˜ Γ˜ 2ij = λ2 I − λSym(G),

˜ +Q ˜1+Q ˜ 2 − Z˜ 1 , Λ˜ 11 = Sym(A i G)

Λ˜ 12 = B i Y j + μ1 G˜ T A Ti ,

˜ + μ 1 Sym(B i Y i ), Λ˜ 22 = −2Z˜ 2 + Sym(S)

Λ˜ 23 = Z˜ 2 − S˜ T

Λ˜ 15 = P˜ − G˜ + μ 2 G˜ T A Ti , ˜ 1 − Z˜ 1 − Z˜ 2 , Λ˜ 33 = −Q

Λ˜ 25 = −μ1 G˜ + μ2 (B i Y j )T , ˜ 2, Λ˜ 44 = −Z˜ 2 − Q

˜ Λ˜ 55 = (η2 − η1 )2 Z˜ 2 + η21 Z˜ 1 − μ2 Sym(G)

Λ˜ 16 = B i Y j

˜ Λ˜ 24 = Z˜ 2 − S,

Moreover, the stabilizing feedback gains in (11) are given by K i = Y i G˜ −1

(39)

Proof: First, by noticing that (G˜ − λI)T (G˜ T − λI) ≥ 0, we have ˜ −G˜ T G˜ ≤ λ2 I − λSym(G)

(40)

Under conditions of the Theorem 1, it follows from Λ˜ 55 < 0 that G˜ is nonsingular. Define ˜ S˜ = G˜ T S G, ˜ Q ˜ 1 = G˜ T Q1 G, ˜ Q ˜ 2 = G˜ T Q2 G, ˜ Z˜ 1 = G˜ T Z1 G, ˜ Z˜ 2 = G˜ T Z2 G, ˜ G = G˜ −1 , P˜ = G˜ T P G, ˜ Y i = K i G. Taking account of (40) and performing a congruence transformation to (36) and (38) by diag(G, G, G, G, G, G, I) and diag(G, G), respectively, the conditions (22)–(23) hold considering (4) and notation (20). Thus there exists a fuzzy controller (11) such that the closed-loop system in (18) is asymptotically stable according to Lemma 3. 

3.2 Observer-based feedback design Suppose that the estimated state will be quantized before it is transmitted to the controller through network. Then, the following output feedback controller is considered: ⎧ r # $  ⎪ ⎪ ⎪ xˆ˙ (t) = h i (θ(t)) A i xˆ (t) + B i u(t) + L i C2 (x(t) − xˆ (t)) ⎪ ⎪ ⎨ i=1 (41) r ⎪  ⎪ ⎪ ⎪ h i (θ(t − τ k ))K i (I + ∆(t))ˆx (t − η(t)) ⎪ ⎩ u(t) = i=1

84 | M. Kchaou and A. Toumi

Denoting e(t) = x(t) − xˆ (t), appending (2) and (41) we obtain the augmented model ¯ η (I + ∆(t))ξ (t − η(t)) ξ˙ (t) = Aξ (t) + A

(42)

with    0 ∆(t) 0 A= , , ∆(t) = A i − L i C2 0 ∆(t) i=1   r  r  K −B K B i j i j ¯η= h i (θ(t))h j (θ(t − τ k )) A 0 0 r 



A h i (θ(t)) i 0

i=1 j=1

Next, we focus our attention on the fuzzy observer-based control for system (2), which is summarized in the following theorem. Theorem 2 Consider system (2). For given scalars η1 > 0, η2 > 0 , μ1 , μ2 , ε ij > 0, 1 ≤ i < j ≤ r and quantization density ρ > 0, there exists a fuzzy observer (41) such that ˜ 1 > 0, closed-loop system (42) is asymptotically stable, if there exist matrices P˜ > 0, Q 2 2 2 ˜ G11 , G22 , G21 , G, F i , and Y i , satisfying (38) and ˜ 2 > 0, Z˜ 1 > 0, Z˜ 2 > 0, S, Q ˜ ii < 0, Φ ˜ ij + Φ ˜ ji < 0, Φ

(43) (44)

where ⎡

Λ˜ ij ⎢ ˜ Φ ij = ⎣ * *

Γ˜ 1ij ε ij Γ˜ 2ij *

⎤ T

ε ij G˜ T ∆ ⎥ 0 ⎦, −ε ij I

⎡

Λ˜ 11 ⎢ * ⎢ ⎢ Λ˜ ij = ⎢ * ⎢ ⎣ * *

Λ˜ 12 Λ˜ 22 * * *

0 Λ˜ 23 Λ˜ 33 * *

0 Λ˜ 24 S˜ ˜Λ44 *

⎤ Λ˜ 15 Λ˜ 25 ⎥ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎦ Λ˜ 55

˜1+Q ˜ 2 − Z˜ 1 , Λ˜ 11 = Sym(Aij ) + Q

˜ Γ˜ 2ij = λ2 I − λSym(G),

Λ˜ 12 = Bij + μ1 ATij ,

Λ˜ 15 = P˜ − G˜ + μ2 ATij ,

˜ + μ1 Sym(Bij ), Λ˜ 22 = −2Z˜ 2 + Sym(S)   Ai G Ai G Aij = , 0 A i G − F i C2   2 11 G 0 2= G 2 21 G 2 22 , G

Λ˜ 25 = −μ1 G˜ + μ2 (Bij )T , Λ˜ 16 = Bij   Bi Yj 0 Bij = 0 0   G G T 2 , G˜ = . G = V GV 0 G

Then the desired controller and observer gains are given by K i = Y i G−1 and 2 −1 S−1 U T , respectively, where U, V and S0 come from SVD decomposition L i = F i US0 G 11 0 of C2 .

T–S Fuzzy Systems with Limited Communication

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85

Proof: Under conditions of the theorem, it follows from Λ˜ 55 < 0 that G˜ is nonsingular. 2 11 S−1 U T = L i G 2 and Y i = K i G. Thus, G and G are also nonsingular. Setting F i = L i US0 G 0 Assume that rank(C2 ) = p, then the SVD decomposition exists such that U T C2 V = [S0 0] and we have

 T T 2 11 S−1 2 2 = US0 G GC 0 U U S0 0 V  

 2 11 0 G T T = U S0 0 V V 2 2 22 V = C2 G G21 G Then, the augmented matrices can be written as r  r 

˜ h i [θ(t)]h j [θ(t − τ k )])Aij = AG,

i=1 j=1 r  r 

¯ η G˜ h i [θ(t)]h j [θ(t − τ k )]Bij = A

i=1 j=1

Following the similar lines in the proof of Theorem 1, we conclude that the fuzzy observer (41) exists which garanties for closed-loop system (42) to be asymptotically stable. 

4 Numerical examples Example 1 Consider the following nonlinear mass-spring system [15] x˙ 1 = x2 x˙ 2 = −0.01x1 − 0.67x31 + u

(45)

Choose fuzzy membership function as h1 (x1 ) = 1 − x21 and h2 (x1 ) = 1 − h1 (x1 ) where

 x1 ∈ −1, 1 . The (TS) fuzzy model with the following parameters can be used to model the aforementioned nonlinear system.       0 1 0 1 0 , A2 = , B1 = B2 = . A1 = −0.01 0 −0.68 0 1 The network-related parameters are assumed: h = 20ms, the maximum delay η2 = 0.2s, the maximum number of data packet dropouts σ = 5, the quantizer parameters ρ = 0.75. Setting ε11 = 51, ε21 = 3.5 and ε22 = 51. With λ = 1.01, Theorem 1 produces

86 | M. Kchaou and A. Toumi

a set of feasible solutions to corresponding LMIs.   0.7933 −0.1425 G= −0.5824 1.7392



K1 = −0.0483 −3.3713 , K2 = 0.4137

 −3.2689 .

0.13 0.12 0.11 0.1 0.09 0.08 0.07 0.06 0

0.5

1

1.5 t

2

2.5

3

1.5 t

2

2.5

3

5.5 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 0

0.5

1

Fig. 2. Networks-induced delays and data packet dropout for example 1.

T–S Fuzzy Systems with Limited Communication

|

87

x1 x2

0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1.2 0

2

4

6 t

8

10

12

Fig. 3. Closed-loop responses of the system.

 Choosing an initial condition x0 = −1, −0.9 , the closed-loop behavior of the total system with the above fuzzy controller are plotted in Figs. 3 and 4 which confirm that the trajectories of the states converging asymptotically to zero.

u

4 3.5 3 2.5 2 1.5 1 0.5 0 −0.5 −1 0

1

2

Fig. 4. Control input u(t).

3

4 t

5

6

7

8

88 | M. Kchaou and A. Toumi

Example 1 We consider the following problem of balancing an inverted pendulum on a cart. The dynamic equations of motion of the pendulum are given as ⎧ ⎪ ⎪ x˙ 1 = x2 ⎪ ⎪ ⎨ g sin x1 − amlx22 sin 2x1 − a u cos x1 x˙ 2 = (46) 4 2 ⎪ ⎪ 3 l − aml cos x 1 ⎪ ⎪ ⎩ y(t) = x1 + x2 x1 denotes the angle of the pendulum from the vertical axis, and x2 is the angular velocity, g = 9.8m/s2 is the gravity constant, m is the mass of the pendulum, 2l is the length of the pendulum, a = 1/(m + M), M is the mass of the cart, and u is the force applied to the cart. In this simulation, the pendulum parameters are chosen as m = 2kg, M = 8kg, and 2l = 1m. The T–S fuzzy model of this system is given by the following parameters:         0 1 0 1 0 0 , A2 = , B1 = , B2 = , A1 = 17.2941 0 12.6305 0 −0.1765 −0.0779

 C2 = 1 0 We use the following membership functions h1 (x1 ) = 1 −

1 1 + exp[−7(x1 − 4π )]

h2 (x1 ) = 1 − h1 (x1 ).

The network-related parameters are assumed: h = 1ms, the maximum delay η2 = 10ms, the maximum number of data packet dropouts σ = 3, the quantizer parameters ρ = 0.85. Setting ε11 = 31.5, ε21 = 20 and ε22 = 32. With λ = 1.05, Theorem 2 produces a set of feasible solutions to corresponding LMIs.   5.6255 0 2 G= , 4.0471 8.4848

 K1 = 163.2782 58.8965 ,   547.4328 L1 = , 549.8290



 4.4905 −3.4572 G= , −5.6750 52.1590

 K2 = 272.3248 116.3252 ,   546.0797 L2 = . 553.4501

 For simulation, the initial condition is assumed to be x0 = 3π , 0 and the network-induced delays and the data packets dropouts are generated randomly and shown in Fig. 5. The state responses of the NCS with quantization and control input

T–S Fuzzy Systems with Limited Communication

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89

are depicted in Figs. 6 and 7 which we can see that all the states component and their estimations converge to zero. The simulation results are in accordance with the analysis and support the effectiveness of the developed design strategy.

x 10

−3

10 9 8 7 6 5 4 0

0.02

0.04

0.06

0.08

0.1 t

0.12

0.14

0.16

0.18

0.2

3 2.5 2 1.5 1 0.5 0 0

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 t

Fig. 5. Networks-induced delays and data packet drop-out.

90 | M. Kchaou and A. Toumi x1 xˆ 1

1

0.8

0.6

0.4

0.2

0 0

0.5

1

1.5

2

2.5 t

3

3.5

4

4.5

5

x2 xˆ 2

0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 0

0.5

1

1.5

2

2.5 t

Fig. 6. Closed-loop responses of the system.

3

3.5

4

4.5

5

T–S Fuzzy Systems with Limited Communication

250

|

91

u

200

150

100

50

0 0

0.5

1

1.5

2

2.5 t

3

3.5

4

4.5

5

Fig. 7. Control input u(t).

5 Conclusion The problem of control for fuzzy-model-based nonlinear systems with communication constraints is discussed in this paper. A controller design method under unreliable transmission is proposed, which considers quantization, network-induced delays and packet dropouts simultaneously. A novel interval-delay technique has been established for stability analysis and control synthesis for networked control systems. The simulation results have shown that the proposed strategy yields good system performance while maintaining the closed-loop stability. The future work will be associated with the following directions: – To propose new controllers which can guarantee the stability of the system with some desired performance. – Due to fact that networked-induced delay is stochastic; the stochastic system with incomplete information could be investigated. – To develop results for networked control system within the fault-tolerant control framework.

92 | M. Kchaou and A. Toumi

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M. Kchaou, A. Toumi, and M. Souissi. Robust reliable guaranteed cost piecewise fuzzy control for discrete-time nonlinear systems with time-varying delay and actuator failures. Int. J. of General Systems, 40(5):531–558, 2011. M. Fu and L. Xie. The sector bound approach to quantized feedback control. IEEE Transactions on Automatic Control, 50(11):1698–1711, 2005. H. Gao and T. Chen. H∞ estimation for uncertain systems with limited communication capacity. IEEE Trans. on Automatic Control, 52(11):2070–2084, 2007. D. Huang and S.K Nguang. Robust disturbance attenuation for uncertain networked control systems with random time delays. IET Control Theory Appl, 2:1008–1023, 2008. M. Kchaou, M. Souissi, and A. Toumi. A new approach to non-fragile H∞ observer-based control for discrete-time fuzzy systems. Int. J. of Systems Science, pages 1–12, 2010. M. Kchaou, A. Toumi, and M. Souissi. Delay-dependent H∞ resilient output fuzzy control for nonlinear discrete-time systems with time-delay. Int. J. of Uncertainty, Fuzziness and Knowledge-Based Systems, 19:229–250, 2011. M.S. Mahmoud. Improved networked-control systems approach with communication constraint. IMA J. of Mathematical Control and Information, page doi:10.1093/imamci/dnr039, 2011. P. Park, J.W. Ko, and C. Jeong. Reciprocally convex approach to stability of systems with time-varying delays. Automatica, 47(1):235–238, 2011. C. Peng, Y. Tian, and M.O. Tadé. State feedback controller design of networked control systems with interval time-varying delay and nonlinearity. Int. J. of Robust and Nonlinear Control, 18:1285–1301, 2008. T. Takagi and M. Sugeno. Fuzzy identification of system and its applications to modeling and control. IEEE Trans. Systems, Man & Cybernetics, 15:116–132, 1985. Y. Wang, L. Xie, and C.E. De Souza. Robust control of a class of uncertain nonlinear systems. Systems and Control Letters, 19:139–149, 1992. J. Wu, H.R. Karimi, and P. Shi. Observer-based stabilization of stochastic systems with limited communication. Mathematical Problems in Engineering, doi:10.1155/2012/781542:1–17, 2012. L. Wu, J. Lam, X. Yao, and J. Xiong. Robust guaranteed cost control of discrete-time networked control systems. Optimal Control Applications and Methods, 32:95–112, 2010. H. Yang, Y. Xia, and P. Shi. Stabilization of networked control systems with nonuniform random sampling periods. Int. J. of Robust and Nonlinear Control, 21:501–526, 2010. H. Zhang, M. Li, J. Yang, and D. Yang. Fuzzy model-based robust networked control for a class of nonlinear systems. IEEE Trans. on systems, Man and Cybernetics-Part A: Systems and humans, 39(2):437–447, 2009. H. Zhang, Y. Shi, and A.S. Mehr. Robust weighted H ∞ filtering for networked systems with intermittent measurements of multiple sensors. Int. J. of Adaptative Control and Signal Processing, 25:313–330, 2011. H. Zhang, H. Yan, F. Yang, and Q. Chen. Quantized control design for impulsive fuzzy networked systems. IEEE Trans. on Fuzzy Systems, 19(6):1153–1162, 2011. H. Zhang, J. Yang, and C. Su. T-S fuzzy-model-based robust H∞ design for networked control systems with uncertainties. IEEE Trans. on Industrial Informatics, pages 289–301, Novembre 2007.

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Biographies Mourad kchaou He received his Mastery of Sciences from (ENSET/Tunisia) in 1993. In 2004, he obtained his Master degree and the Doctoral Thesis in 2009 in Automatic and Industrial Computing from Engineering National School of Sfax (ENIS). In 2013, he has obtained the University Habilitation (HDR) from (ENIS). He is now an Associate Professor in the Electronic Engineering department of High Institute of Applied Sciences and Technology of Sousse. His main research interests include fuzzy control, time-delay systems, descriptor systems, with particular attention paid to nonlinear systems represented by multiple-models. Ahmed Toumi He received his Electrical Engineering Diploma from the Engineering National School of Sfax/Tunisia (ENIS) and the DEA (Master) in Instrumentation and Measurement from University of Bordeaux-1/France in 1981. He obtained in 1985 and 2000, respectively, the Doctoral Thesis in Physical Sciences from the University of Tunis, and the University Habilitation (HDR) in Electrical Engineering (Automatic Control) from (ENIS). He is now a Professor on Automatic Control, and the Team Leader of Research in Control of Industrial Processes at ENIS. Since 2002, he was the co-President of the international conference on Sciences and Techniques of Automatic control and computer engineering (STA) which has taken place in a number of tourist cities of Tunisia. His main research area concerns Process modeling, Stability, Delay systems, Singular systems, Fuzzy logic control, Robust control.

S. Bouallègue, R. Madiouni and J. Haggège

Particle Swarm Optimization-Based Approach for Digital RST Controller Design

Abstract: In this paper, an improved Particle Swarm Optimization (PSO) based approach, is proposed and successfully applied for digital RST controllers’ synthesis within a real-time framework. The robust two-degrees-of-freedom RST controller design is formulated as an optimization-based problem and solved by a constrained PSO algorithm. It is a systematic PSO-tuned control strategy to deal with the complexity of the well known classical poles placement and shaping of the sensitivity functions problems. A comparison, with the standard Genetic Algorithm Optimization (GAO) meta-heuristic, is investigated in order to show the superiority and the effectiveness of the developed PSO-based method. Simulation and experimental results, for an electrical DC drive benchmark, show the advantages of the proposed PSO-tuned RST control structure in terms of performance and robustness. Keywords: Digital RST controller, Particle Swarm Optimization, Genetic Algorithm Optimization, DC drive benchmark, real-time experimentation.

1 Introduction In the last decades, the robust control theory has reached a remarkable level of maturity. Many synthesis methods have been developed, allowing to set up a unified framework for the design and analysis of robust control laws [16, 24, 27]. The canonical RST structure-based controller design is a robust and effective control strategy that is widely used in practical and industrial applications [1, 2, 11, 14, 15, 16, 20, 21]. In the RST design formalism, the most useful method to synthesize same digital controller is based on the well known closed-loop poles placement [14, 15, 16]. In this design case, the Sylvester’s method can be used to determine the search parameters of the RST controller by resolving a polynomial equation which will ensure the desired closed-loop poles. The major drawback of this synthesis technique is the choice of the closed-loop poles that is usually difficult and becomes more complicated with the complexity of the controlled plants. Up to now, there has not been a clear and systematic procedure to guide the same choice. To overcome this problem, several techniques have been proposed in the literature.

S. Bouallègue, R. Madiouni and J. Haggège: Research Laboratory in Automatic Control (LA.R.A), National Engineering School of Tunis (ENIT), Tunisia, email: soufi[email protected] De Gruyter Oldenbourg, ASSD – Advances in Systems, Signals and Devices, Volume 1, 2016, pp. 95–114. DOI: 10.1515/9783110448375-007

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The two classical methods, based on pole placement combined with the sensitivity functions shaping, have been proposed by Landau and Karimi [15]. The first approach combines the pole placement and the calibration of the sensitivity functions, using the fixed parts in the controller. It adjusts iteratively the sensitivity functions in the frequency-domain where it is necessary and used robustness templates to obtain the placement of the poles. In the second method, the synthesis problem has been transferred to H∞ optimal control via a new interpretation of the weighting filter, as the inverse of the desired sensitivity function. In this approach, weighting filter selection is carried out automatically by an optimization program. These iterative and trial-error-based methods are not suitable for the complex systems. In [20, 21], Rotella et al. proposed a new approach for digital RST controller synthesis based on the flatness property of some dynamical systems. This design, that shows high performance in terms of tracking, has been applied to a variety of plants as given in [1, 2, 20, 21]. However, the dynamics of the obtained flatness-based RST controller depends on the choice, usually delicate, of the coefficients of the well known tracking polynomial. In [11], another RST controller design approach, using convex optimization is proposed by Galdos et al. The performance specifications given as the infinity norm of the weighted sensitivity functions are represented as convex constraints in the Nyquist diagram. Unfortunately, this method is time-consuming, difficult to implement and becomes ineffective for non-convex problems. View of these difficulties, proposing a systematic and easy procedure for the RST synthesis problem is an important and interesting task in this area. The meta-heuristic-based optimization theory can provide a suitable solution to deal with this difficult and NP-hard problem. The evolution of the powerful numerical calculation tools and the diversity of the meta-heuristic methods, such as the recent Particle Swarm Optimization technique [9, 13], can argued this orientation. The two-degrees-of-freedom RST synthesis problem can be reformulated as a constrained optimization problem which can be efficiently solved. In this paper, a new approach based on an improved Particle Swarm Optimization (PSO) technique is proposed for this problem. The RST control design problem is formulated as a constrained optimization problem, which is efficiently solved by a developed PSO algorithm [4, 5, 6, 7, 8]. Different optimization criteria, such as the IAE and ISE index, are considered under various control constraints. Convergence conditions of the proposed PSO algorithm are analytically guaranteed and verified in order to give a set of PSO algorithm parameters [4, 5, 22, 25]. The main contribution of this paper consists of proposing a systematic and less complicated method of synthesis and tuning of digital RST controllers. The rest of this paper is organized as follows. In Section 2, the considered RST structure is presented and its optimization-based synthesis problem is formulated. In Section 3, a constrained and improved PSO algorithm, that is used to solve the formulated RST synthesis problem, is proposed and implemented. Section 4 is dedicated to applying the proposed PSO-based control approach for an electrical DC

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drive benchmark within a real-time framework. All PSO-based simulation results are compared, and are discussed, with those obtained by the classical Genetic Algorithms Optimization (GAO)-based approach in order to show the effectiveness and superiority of the proposed strategy.

2 Control problem formulation In this section, the digital RST controller synthesis problem is formulated as a constrained optimization problem which will be resolved by the means of the developed PSO algorithm.

2.1 Digital RST controller structure In this study, the discrete-time model of the plant to be controlled is described in the time-domain by the following transfer operator: ,   y q−d B q−1 , (1) H q−1 = k = uk A q−1 where q−1 is the backward shift operator, d is the integer number of sampling period contained in the time-delay of the plant, k is the normalized discrete time and corresponds to the discrete time divided by the sampling period T s , u k and y k are the discrete plant input and output, respectively. , , The polynomials A q−1 and B q−1 , assumed that do not have common factors (co-prime polynomials), are respectively the denominator and the numerator of the transfer function, defined as:   (2) A q−1 = 1 + a1 q−1 + . . . + a n A q−n A   (3) B q−1 = b1 q−1 + b2 q−2 + . . . + b n B q−n B The canonical structure of the RST digital controller is represented in Fig. 1. This structure has two-degrees-of-freedom. The digital polynomials R and S are designed in order to achieve the desired regulation performance, and the digital polynomial T is designed afterwards in order to achieve the desired tracking performance [14, 16]. This structure allows the achievement of different levels of performance in tracking and regulation.

98 | S. Bouallègue et al. d

y* k

u

k+ d+ 1

r

Bm Am

T (q

–1

)

+

1 / S (q

–1

k

+

–d

q B A

)

_

R (q

)

k

+

+ –1

y

+ n

Fig. 1. Canonical RST structure of a digital controller.

The classical RST control law, obtained by this structure of polynomial controller, is given by [14, 15, 16]:       (4) S q−1 u k = T q−1 y*k+d+1 − R q−1 y k , - , , The polynomials R q−1 , S q−1 and T q−1 have the form:   R q−1 = r0 + r1 q−1 + . . . + r n R q−n R   S q−1 = s0 + s1 q−1 + . . . + s n S q−n S   T q−1 = t0 + t1 q−1 + . . . + t n T q−n T

(5) (6) (7)

The desired tracking trajectory y*k+d+1 may be generated by the following tracking reference model: , B m q−1 * , - rk y k+d+1 = (8) A m q−1 where r k is the reference trajectory.

2.2 Robustness and performance constraints For different reasons of robustness and performance specifications of the proposed PSO-tuned RST controller, i.e., disturbance rejection, noise attenuation, etc. The , , polynomials R q−1 and S q−1 generally contain some fixed parts which are specified before solving the formulated PSO-tuned RST design optimization problem. , In [14, 15, 16], it is shown that the pre-specified polynomial H R q−1 is used to

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, eliminate the high frequency noises on the input signal and the H S q−1 pre-specified polynomial to allow the rejection of static disturbance on the output signal.   (9) H S q−1 = 1 − q−1   (10) H R q−1 = 1 + q−1 While taking into account the pre-specified parts H S and H R , the new polynomials of the PSO-tuned RST controller can be seen as in equations (11) and (12).       (11) Sˆ q −1 = H S q−1 S q−1       (12) Rˆ q−1 = H R q−1 R q−1 In order to assure a delay margin ∆τ = T s , the modulus of the output sensitivity , function S yp q−1 must lie between upper and lower templates as follow [15, 16]:    −1  W 

inf

 −1            = 1 − 1 − q−1  < S yp q−1  < W −1 

sup

 −1   = 1 + 1 − q−1 

(13)

Regarding the robust stability, the chosen modulus margin ∆M defines the maximum value of the modulus of the output sensitivity function as follow [15, 16]:     −1  = −∆M (14)  S yp q max

2.3 Optimization problem formulation In this section, the PSO-tuned RST design is formulated as a constrained optimization problem which is solved using the proposed PSO-based technique. Such a constrained optimization problem can be mathematically described as: ⎧ f (x) ⎪ ⎨ minimize x∈D (15) subject to ⎪ ⎩ g l (x) ≤ 0; ∀l = 1, . . . , n con 0 1 where f : Rm → R the cost function, D = x ∈ Rm ; xmin ≤ x ≤ xmax the initial search space, which is supposed contain the desired design parameters, and g l : Rm → R the problem’s constraints. The optimization-based RST synthesis problem consists in finding the optimal

T decision variables x* = x*1 , x*2 , . . . , x*m ∈ Rm , which representing the RST controller parameters. These decision variables minimize one of the cost functions, such as the Integral of Squared Error (ISE) and the Integral of Absolute Error (IAE) criteria, defined

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respectively as follows: ∞ f ISE (x) =

e (t, x)2 dt

(16)

|e (t, x )| dt

(17)

0 ∞

f IAE (x) = 0

where x = [s0 , s1 , . . . , s n S , r0 , r1 , . . . , r n R ]T ∈ Rm . The equations (13) and (14), as well as other types of constraints, such as overshoot, settling and rise times of step response, can be used as constraints for the formulated PSO-tuned RST optimization problem. In this design case, we denote that only the regulation performance is considered, , , , i.e., R q −1 and S q−1 polynomials are optimized. Hence, the digital filter T q−1 can be designed afterwards as follows [16]:   P (1) T q−1 = t0 = B (1)

(18)

3 Improved particle swarm optimization method In this section, the proposed PSO technique is presented and a constrained PSO algorithm is developed and implemented. Analytical convergence conditions for such an algorithm are established and analyzed.

3.1 Overview The PSO technique is an evolutionary computation method developed in 1995 by J. Kennedy and R. Eberhart [9, 13]. This recent meta-heuristic technique is inspired by the swarming or collaborative behaviour of biological populations. The cooperation and the exchange of information between population individuals allow to solve various complex optimization problems [10, 17, 19, 23]. Without any regularity on the cost function to be optimized, the recourse to this stochastic and global optimization technique is justified by the empirical evidence of its superiority in solving a variety of non-linear, non-convex and non-smooth problems. In comparison with other meta-heuristics, this optimization technique is a simple concept, easy to implement, and a computationally efficient algorithm [10, 19, 23]. The convergence and parameters selection of the PSO algorithm were proved using several advanced theoretical analysis [22, 25]. Its stochastic behaviour allows to overcome the local minima problem.

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PSO has been enormously successful in several and various industrial domains [17]. It has been used across a wide range of engineering applications which can be summarized as domains in robotics, image and signal processing, electronic circuits design, communication networks, but more especially the domain of control system design [4, 5, 6, 7, 8, 18, 26].

3.2 Improved PSO algorithm The basic PSO algorithm uses a swarm consisting of n p particles (i.e. x1 , x2 , . . . , x n p ), randomly distributed in the considered initial search space, to find an optimal solution x* = arg min f (x) ∈ Rm of a generic optimization problem (15). Each particle, that represents a potential solution, is characterised by a position and a velocity given T T   i,2 i,m i,1 i,2 i,m i , x , . . . , x and v = v , v , . . . , v where (i, k) ∈ [[1, n p ]] × by x ik = x i,1 k k k k k k k . 1, k [[ max ]] At each algorithm iteration, the i th particle position, x i ∈ Rm , evolves based on the following update rules: x ik+1 = x ik + v ik+1







i i p ik − x ik + c2 r2,k p gk − x ik v ik+1 = w k+1 v ik + c1 r1,k



(19) (20)

where w k+1 the inertia factor, c1 , c2 the cognitive and the social scaling factors i i , r2,k random numbers uniformly distributed in the interval [[0, 1]], respectively, r1,k i p k the best previously obtained position of the i th particle and p gk the best obtained position in the entire swarm at the current iteration k. The principle of a particle displacement in the swarm is graphically shown in the Fig. 2, for a two dimensional design space.

x ki +1

c 2r2i,k ( pkg – x ki ) pki

v ki +1 pkg wv ki

c 1r1 ,ik ( pki – x ki )

x ki v ki Fig. 2. Particle position and velocity update.

102 | S. Bouallègue et al.

In order to improve the exploration and exploitation capacities of the proposed PSO algorithm, we choose for the inertia factor a linear evolution with respect to the algorithm iteration as given in [23]:  w max − w min k (21) w k+1 = wmax − kmax where wmax = 0.9 and wmin = 0.4 represent the maximum and minimum inertia factor values, respectively, kmax is the maximum iteration number. Similarly to other meta-heuristic methods, the PSO algorithm is originally formulated as an unconstrained optimizer. Several techniques have been proposed to deal with constraints. For this purpose, let us consider the generic constrained optimization problem given by equation (15). One useful approach is by increasing the cost function of problem with penalties proportional to the degree of constraint infeasibility. In this paper, the following external static penalty technique is used: φ (x) = f (x) +

n con 



ρ l max 0, g l (x)2

(22)

l=1

where ρ l is a prescribed scaling penalty parameter, chosen as ρ l = 1015 , and n con is the number of problem constraints g l (x). Finally, the basic PSO algorithm can be summarized in the following steps: 1. Defining all PSO algorithm parameters such as swarm size n p , maximum and minimum inertia factor values, cognitive c1 and social c2 scaling factors, . . . 2. Initializing the n p particles with randomly chosen positions x0i and velocities v0i in search space D. Evaluating the initial population and determining p0i and p0g . 3. Incrementing the iteration number k. For each particle, applying the update equations (19) and( 20), and evaluating the corresponding fitness values φ ik =   φ x ik :

if φ ik ≤ pbest ik then pbest ik = φ ik and p ik = x ik if φ ik ≤ gbest k then gbest k = φ ik and p gk = x ik where pbest ik and gbest k represent the best previous fitness of the i th particle and the entire swarm, respectively. 4. If the termination #  criterion  $is satisfied, the algorithm terminates with the solution x* = arg min φ x ik , ∀i, k . Otherwise, go to step 3. x ik

3.3 Convergence analysis of the PSO algorithm In this part, the proposed PSO algorithm is analyzed based on [22, 25] results. Theoretical conditions for convergence algorithm and parameters choice are established.

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Let us replace the velocity update equation (20) into the position update equation (19) to get the following expression:   i i i i x ik + wv ik + c1 r1,k − c2 r2,k p ik + c2 r2,k p gk (23) x ik+1 = 1 − c1 r1,k A similar re-arrangement of the velocity term (23) leads to:   i i i i x ik + wv ik + c1 r1,k + c2 r2,k p ik + c2 r2,k p gk v ik+1 = − c1 r1,k

(24)

The obtained equations (23) and (24) can be combined and written in matrix form as:  ⎤   ⎡      i i i i i i i w r + c r 1 − c 1 2 x r c r c p x k+1 1,k 2,k 1 2 k 1,k 2,k k   ⎦ =⎣ + (25) i i i i v ik+1 v ik c1 r1,k c2 r2,k p gk − c1 r1,k w + c2 r2,k The above expression can be considered as a state-space representation of a discrete-time dynamic linear system, given by: yˆ k+1 = Mˆy k + N uˆ k

(26)

where yˆ k is the state vector, uˆ k the external input system, M and N the dynamic and input matrices respectively, defined as:     x ik p ik ; uˆ k = ; yˆ k = v ik p gk  ⎤ ⎡  (27)   i i i i w + c2 r2,k 1 − c1 r1,k c r c r 1 1,k 2 2,k   ⎦;N = M=⎣ i i i i c1 r1,k c2 r2,k − c1 r1,k w + c2 r2,k For a given particle, the convergent behaviour can be maintained while assuming that the external input is constant, as there is no external excitation in the dynamic system. In such a case, as the iterations go to infinity the updated positions and velocities become constants from the k th to the (k + 1)th iteration, given the following equilibrium state:  ⎤ ⎡      i i i i w + c2 r2,k − c1 r1,k c2 r2,k x ik c1 r1,k p ik   ⎦ ⎣ yˆ k+1 − yˆ k = + i i i i v ik c1 r1,k c2 r2,k p gk − c1 r1,k w−1 + c2 r2,k   0 = (28) 0 which is true only when: x ik = p ik = p gk , v ik = 0

(29)

104 | S. Bouallègue et al.

Therefore, we obtain an equilibrium point, for which all particles tend to converge as algorithm iteration progresses, given by:

T yˆ eq = p gk , 0

(30)

So, the dynamic behavior of the i th particle can be analyzed using the eigenvalues derived from the dynamic matrix formulation (26) and (27), solutions of the following characteristic polynomial:   i i λ+w=0 (31) − c2 r2,k λ2 − 1 + w − c1 r1,k The following necessary and sufficient conditions for stability of the considered discrete-time dynamic system (26) are obtained while applying the classical Jury criterion: |w| < 1 i i c1 r1,k + c2 r2,k >0

w+1−

i i c1 r1,k +c2 r2,k 2

(32) >0

i i , r2,k ∈ [[0, 1]], the above stability conditions are equivalent to the Assuming that r1,k following set of parameter selection heuristics which guarantee convergence for the PSO algorithm:

0 < c1 + c2 < 4 c1 + c2 −1 < w < 1 2

(33)

While these heuristics provide useful selection parameter bounds, an analysis of the effect of the different parameter settings is achieved and verified by some numerical simulations to determine the effect of such parameters in the PSO algorithm convergence performance.

4 Real-time control application In this section, the proposed PSO-tuned RST control approach is experimentally applied to an electrical DC drive benchmark within a real-time framework.

4.1 Plant model description The considered benchmark is a 250 watts electrical DC drive supplied by an AC-DC power converter. The machine’s speed rotation is 3000 rpm at 180 volts DC armature voltage. The developed real-time application acquires speed of the DC drive and

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105

generates control signal for thyristors of AC-DC power converter (PWM signal). This is achieved using a data acquisition and control system based on a PC computer and a multi-functions PCI-1710 board which is compatible with MATLAB/SIMULINK [3]. A complete description of the hardware and software control environments for this benchmark can be found in [3]. The model’s parameters are obtained by experimental identification procedures and they are summarized in Tab. 1 with their associated uncertainty bounds [1]. The discrete-time model is obtained by sampling of the continuous transfer function with a period : H (q) =

0.005q + 0.0004 q2 − 1.457q + 0.4735

(34)

Tab. 1. Identified DC drive model parameters. Parameters

Nominal values

Uncertainty bounds

G0 τm τe

0.05 300 ms 14 ms

±50% ±50% ±50%

4.2 Simulation results In this case of PSO-tuned RST controller design, the constrained optimization problem, to be solved by the means of our improved PSO algorithm, is described as follows: ⎧ minimize f (x) ⎪ ⎪ ⎪ x=[s1 ,r0 ,r1 ]T ∈R3 ⎪ ⎪ ⎪ ⎨ subject to   -  , g1 (x) = S yp q−1 , x  − W −1 sup ≤ 0 ⎪    ⎪ - , ⎪ ⎪ ⎪ g2 (x) = W −1 inf − S yp q−1 , x  ≤ 0 ⎪ , −1 - ⎩ + ∆M = 0 g3 (x) = S yp q , x 

(35)

max

The swarm size parameter algorithm is generally problem-dependent in the PSO framework. However, Eberhart and Shi [10] as well as Poli et al. [19] show that it is often empirically set with relation to the dimensionality and perceived difficulty of a considered optimization problem. They suggest that swarm size values in the range 20-50 are quite common. For this purpose, we tested the proposed PSO algorithm with different values of the swarm size n p in this range, as shown in Fig. 3. Globally, all the found results are close to each other. But, best values of the fitness are obtained while using a swarm size equal to 30.

106 | S. Bouallègue et al. 0.7 swarm size=20; (IAE=0.5183) swarm size=30; (IAE=0.5091) swarm size=40; (IAE=0.5257) swarm size=50; (IAE=0.5146)

Cost function value

0.65

0.6

0.55

0.5

0.45

0

10

20

30

40

50

Iteration

Fig. 3. Convergence of the improved PSO algorithm under swarm size variation.

In order to get some statistical data on the quality of optimization results, it is necessary to run the algorithm several times. We run the algorithm 20 times and feasible solutions are found in 95 % of trials and within an acceptable CPU computation time. The obtained optimization results are summarized in Tabs. 2 and 3. A comparison to the Genetic Algorithm Optimization (GAO)-based method [12], is considered in solving the formulated optimization problem. The performance comparison of PSO- and GAO-based approaches is achieved in the same conditions. Indeed, the population size, used in the GAO algorithm, is set as 30 individuals and the maximum generation number is 50. The GAO parameters, used for MATLAB simulations, are chosen as the Stochastic Uniform selection and the Gaussian mutation methods. The Elite Count is set as 2 and the Crossover Fraction as 0.8. The algorithm stops when the number of generations reaches the specified value for the maximum generation.

Tab. 2. Optimization results from 20 trials of problem (35). Cost function

Algorithm

Best

Mean

Worst

Std. dev.

ISE criterion ISE criterion IAE criterion IAE criterion

PSO GAO PSO GAO

0.5116 0.5189 0.5091 0.5264

0.5886 0.5699 0.5662 0.5857

0.6534 0.6892 0.6044 0.6771

0.0640 0.0680 0.0500 0.0610

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| 107

Tab. 3. Obtained parameters of the PSO-tuned RST controller: Mean case. Cost function

Algorithm

s0

s*1

r0*

r1*

ISE criterion ISE criterion IAE criterion IAE criterion

PSO GAO PSO GAO

1 1 1 1

0.5276 0.0260 0.5415 0.0375

0.0403 0.0350 0.0512 0.0466

0.4588 0.4622 0.4497 0.4738

According to the statistical analysis in Tab. 2, we can conclude that the proposed PSO-tuned RST approach produces better results in comparison with the standard GAO-based. Using a Pentium IV, 1.73 GHz and MATLAB 7.7.0, the CPU computation times are about 320 and 470 seconds for the PSO- and GAO-based methods, respectively. The fact that the algorithm’s convergence always takes place in the same region of the design space, whatever the initial population, that indicates that the algorithm succeeds in finding an area of interesting research space to explore. Performance on convergence properties, in term of iteration number, is compared as shown in Figs. 4 and 5. While using the PSO-based method, we succeed to obtain the best solution within only about 25 iterations. However, the GAO-based method finds the same result after 46 iterations. The quality of the obtained optimal solution, the fast convergence as well as the simple software implementation is better than those of the GAO-based approach. The robustness of the proposed PSO algorithm convergence, under variation of the cognitive, social and inertia factor parameters, is analysed on the basis of numerical simulations as shown in Figs. 6 and 7. The PSO algorithm convergence is guaranteed within the established domains given by equation (33).

0.9

Cost function value

0.85 0.8 0.75 0.7 0.65 0.6 IAE=0.5662 0.55

5

10

15

20

25

30

35

40

45

50

Iteration

Fig. 4. Convergence property of the PSO-tuned RST algorithm.

108 | S. Bouallègue et al. 1.8 1.6

Cost function value

1.4 1.2 1 0.8 IAE=0.5857

0.6 0.4

5

10

15

20

25 30 Genaration

35

40

45

50

Fig. 5. Convergence property of the GAO-tuned RST algorithm.

As they are given in [14, 15, 16], the robustness of the designed RST controller can be guaranteed while observing Figs. 8 and 9. Indeed, the module of the output sensitivity functions S yp remains inside the predefined template and the one of the input sensitivity function S up presents attenuation in high frequencies. This result leads also to obtain high time-domain performances of the proposed PSO-tuned RST controller structure. These simulation results show that the effective speed of DC motor tracks a desired trajectory with high performance. The tracking error is very small in the transient regime and equal to zero in steady-state.

0.9

c1=1, c2=1.19 c1=0, c2=3 c1=3, c2=0 c1=2, c2=2 c1=1.19, c2=1.19

0.85

Cost function value

0.8 0.75 0.7 0.65 0.6 0.55 0.5

5

10

15

20

25 30 Iteration

35

40

45

50

Fig. 6. Robustness of the PSO algorithm under variations of cognitive and social coefficients.

PSO-Based Approach for Digital RST Control Design

1 wmax=0.9,wmin=0.4 wmax=0.9,wmin=0.1 wmax=0.75,wmin=0.25 wmax=0.95,wmin=0.2 wmax=0.6,wmin=0.3

0.95 0.9

Cost function value

0.85 0.8 0.75 0.7 0.65 0.6 0.55

5

10

15

20

25 30 Iteration

35

40

45

50

Fig. 7. Robustness of the PSO algorithm under variations of the inertia factor.

50 robustness upper template robustness lower template s1 =0.527, r0 =0.040 r1 =0.459

40

Syp magnitude (dB)

30 20

Modulus margin

10

Delay margin

0 –10 –20 –30 –40 –50 0

0.1

0.2 0.3 Normalized frequency f/fs

0.4

Fig. 8. Output sensitivity function: static disturbance rejection.

0.5

| 109

110 | S. Bouallègue et al. 50

Sup magnitude (dB)

0

–50

High frequency noises attenuation

–100

–150

0

0.1

0.2 0.3 Normalized frequency f/fs

0.4

0.5

Fig. 9. Input sensitivity function: high frequency noises elimination.

4.3 Real-time controller implementation The real-time implementation of the PSO-tuned RST controller is achieved using the multi-functions data acquisition PCI-1710 associated with the MATLAB/SIMULINK environment, as described and used in [3]. The proposed experimental setup schematic for the DC drive benchmark is shown in Fig. 10.

Interfacing of AC-DC power converter control and galvanic insulation

PC

Gate impulses

Thyristors gate drive circuit

vs

Th1 is

.

Th2 ω ν

~ D1

.

.

DC motor

Load

D3

D2 DT

Tachometer Data Acquisition and Control System

Interfacing of acquirement and adaptation of speed DC motor

Fig. 10. Designed experimental schematic setup for the RST-based controlled DC motor.

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| 111

The practical implementation of the proposed PSO-tuned RST control approach, for the above described DC drive plant, leads to the experimental results in Figs. 11 and 12. These results always remain well satisfactory for a digital RST controller with a less complicated synthesis and tuning methods.

Controlled speed variation (1000 rpm)

3 External load disturbance rejection

2

1

0 0

10

20

30

40

50

60

70

Acquisition time (sec) Fig. 11. Experimental result: controlled DC drive speed variation.

Speed tracking error (1000 rpm)

0.5

External load disturbance rejection 0.25

0

–0.25

–0.5 0

10

20

30

40

50

Acquisition time (sec)

Fig. 12. Experimental result: DC drive speed tracking error.

60

70

112 | S. Bouallègue et al.

5 Conclusion The synthesis and tuning problem of digital RST controllers, using a new improved PSO-based technique, is proposed and successfully applied to an electrical DC drive speed control. The performance comparison, with the standard GAO-based method, shows the efficiency and superiority of the proposed PSO-based approach in terms of the obtained solution qualities, the convergence speed and the simple software implementation of its algorithm. The convergence of the proposed PSO algorithm is guaranteed within an established analytical domain. The obtained simulation and experimental results show the efficiency in terms of performance, robustness and less complexity of the proposed RST control approach which can easily be applied to the industrial motor control field. A multi-objective formulation of the considered PSO-tuned RST control problem, under various frequency-domain robustness and performance constraints presents our future research.

Bibliography [1]

M. Ayadi, J. Haggège, S. Bouallègue, and M. Benrejeb. A Digital Flatness-based Control System of a DC Motor. Studies in Informatics and Control, 17(2):201–214, 2008. [2] M. Ayadi, N. Langlois, M. Benrejeb, and H. Chafouk. Flatness-based Robust Adaptive Polynomial Controller for a Diesel Engine Model. Trans. on Systems, Signals and Devices, 2(1):71–90, 2006. [3] S. Bouallègue, J. Haggège, and M. Benrejeb. On a robust real-time H∞ controller design for an electrical drive. Int. J. of Modelling, Identification and Control, 15(2):89–96, 2012. [4] S. Bouallègue, J. Haggège, and M. Benrejeb. A New Method for Tuning PID-Type Fuzzy Controllers Using Particle Swarm Optimization. Chapter 6: 139–162, in Fuzzy Controllers-Recent Advances in Theory and Applications book (edited by Sohail Iqbal), InTech Education and Publishing, Rijeka, Croatia, 2012. [5] S. Bouallègue, J. Haggège, M. Ayadi, and M. Benrejeb. PID-type fuzzy logic controller tuning based on particle swarm optimization. Engineering Applications of Artificial Intelligence, 25:484–493, 2012. [6] S. Bouallègue, J. Haggège, and M. Benrejeb. Particle Swarm Optimization-Based Fixed-Structure H∞ Control Design. Int. J. of Control, Automation, and Systems, 9(2):258–266, 2011. [7] S. Bouallègue, J. Haggège, and M. Benrejeb. Structured Loop- Shaping H∞ Controller Design using Particle Swarm Optimization. In the 2010 IEEE International Conference on Systems, Man, and Cybernetics, Istanbul-Turkey, 2010. [8] S. Bouallègue, J. Haggège, and M. Benrejeb. Structured Mixed-Sensitivity H∞ Design using Particle Swarm Optimization. 7th IEEE Int. Multi-Conf. on Systems, Signals and Devices, Amman-Jordan, 2010. [9] R. Eberhart and J. Kennedy. A New Optimizer Using Particle Swarm Theory. 6th the Int. Symp. on Micro Machine and Human Science, :39–43, Nagoya, 1995. [10] R. Eberhart and Y. Shi. Particle Swarm Optimization: Developments, Applications and Resources. the IEEE Congress on Evolutionary Computation, :81–86, Seoul-Korea, 2001.

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[11] G. Galdos, A. Karimi, and R. Longchamp. RST Controller Design by Convex Optimization Using Frequency-Domain Data. 18th IFAC World Congress, :11429–11434, Milano, 2011. [12] D. E. Goldberg. Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley, 1989. [13] J. Kennedy and R. Eberhart. Particle Swarm Optimization. IEEE Int. Joint Conf. on Neural Networks, :1942–1948, Perth-Australia, 1995. [14] I. D. Landau. The RST Digital controller design and applications. Control Engineering Practice, 6:155–165, 1998. [15] I. D. Landau and A. Karimi. Robust Digital Control using Pole Placement with Sensitivity function shaping method. Int. J. of Robust and Nonlinear Control, 8:191–210, 1998. [16] I. D. Landau, R. Lozano, and M. M’saad. Adaptive Control, Springer-Verlag, London, 1998. [17] A. Lazinica (editor). Particle Swarm Optimization. In-Tech, Rijeka, Croatia, 2009. [18] I. Maruta, T. Sugie, and T.-H. Kim. Synthesis of fixed-structure robust controllers using the distributed particle swarm optimizer with cyclic-network topology. American Control Conf., San Francisco, :3716–3721, 2011. [19] R. Poli, J. Kennedy, and T. Blackwell. Particle Swarm Optimization: An Overview. Swarm Intelligence, Springer, 1:33–57, 2007. [20] F. Rotella, F. Carillo, and M. Ayadi. Digital flatness based robust controller applied to a thermal process. IEEE Int. Conf. on Control Applications, :936–941, Mexico, 2001. [21] F. Rotella, F. Carillo, and M. Ayadi: Polynomial controller design based on flatness. First IFAC-IEEE Symp. on System Structure and Control, :936–941, Prague, 2001. [22] E. P. Ruben and B. Kamran. Particle Swarm Optimization in Structural Design. In F. T. S. Chan and M. K. Tiwari (Eds.) book Swarm Intelligence: Focus on Ant and Particle Swarm Optimization, :373–394, InTech Education and Publishing, Vienna, 2007. [23] Y. Shi and R. Eberhart. Empirical study of particle swarm optimization. Congress on Evolutionary Computation, :1945–1950, Washington, 1999. [24] S. Skogestad and I. Postlethwaite. Multivariable feedback Control: Analysis and Design. John Wiley & Sons, New York, 1996. [25] F. Van den Bergh. An Analysis of Particle Swarm Optimizers. PhD Thesis, University of Pretoria, Pretoria, South Africa, 2006. [26] M. Yagoubi and G. Sandou. Particle Swarm Optimization for the Design of H∞ Static Output Feedbacks. J. of Mechanics Engineering and Automation, 2:221–228, 2012. [27] K. Zhou, J. C. Doyle and K. Glover. Robust and Optimal Control. Prentice Hall, New Jersey, 1996.

Biographies Soufiene Bouallègue was born in 1982 in Nafta, Tunisia. He graduated from the National School of Engineers of Tunis (ENIT) in 2006 and received the PhD degree in Electrical Engineering in 2010. He is currently Associate Professor of Electrical Engineering at the High Institute of Industrial Systems of Gabès (ISSIG). His research interests are in the area of meta-heuristic optimization, fuzzy and structured H∞ controls, and embedded control systems design.

114 | S. Bouallègue et al. Riadh Madiouni has obtained in 2009 his Master in Computer Sciences, in the Faculty of Sciences of Bizerte, and the Automatic and Signal Processing master’s degree, at the National School of Engineers of Tunis (ENIT) in 2012. He is currently preparing a PhD Thesis in the LARA (ENIT) and LiSSi (University of Paris-Est) laboratories. His research focuses on the multi-objective particle swarm optimization and robust control design.

Joseph Haggège was born in 1975 in Tunis, Tunisia. He graduated from National School of Engineers of Tunis in 1998. He received the PhD degree in Electrical Engineering 2003 and the Habilitation in 2010. He is currently Senior Lecturer at the National School of Engineers of Tunis. His research interests are in the area of heuristic optimization, embedded systems and robust digital control.

D. Galdeano, A. Chemori, S. Krut and P. Fraisse

Optimal Pattern Generator for Dynamic Walking in humanoid Robotics

Abstract: This paper deals with an optimal Zero Moment Point (ZMP) based pattern generator for stable dynamic walking in humanoid robotics. The proposed method is based on a Three-Mass Linear Inverted Pendulum Model (3MLIPM), used as a simplified model of the biped robot. The 3MLIPM considers the biped robot as a three point masses and two-link system. A ZMP based performance index is then used in an optimization problem whose solution gives the best values of the model’s parameters w.r.t. dynamic walking stability. Numerical simulations are presented to show the effectiveness of the proposed optimal pattern generator for the case of the biped walking robot SHERPA. Keywords: Biped walking robot, Pattern generation, Three-mass linear inverted pendulum model, Optimization.

1 Introduction A humanoid robot is a robot with an appearance based on that of the human body. Humanoids walking is a very challenging field of research due to the complexity of synthesizing stable gaits for these systems. Indeed, within this field, humanoid robot control needs sophisticated control schemes to deal with their complexity including: – the high-order nonlinear dynamics, – the variable structure model according to the different phases of the walking cycle, – the contact constraints with the ground that should be managed, – the hybrid character of the dynamics due to rigid impacts between the robot’s foot and the ground, – the stability during walking that should be ensured. Most of the proposed control schemes in the literature are based on the use of some reference trajectories that should be tracked in real-time. This fact shows the importance of a pattern generator in humanoid walking control. Several types of pattern generators have then been proposed in the literature. However, those who

D. Galdeano, A. Chemori, S. Krut and P. Fraisse: LIRMM, Université Montpellier 2, LIRMM, Montpellier, France, emails: [email protected], [email protected], [email protected], [email protected]. De Gruyter Oldenbourg, ASSD – Advances in Systems, Signals and Devices, Volume 1, 2016, pp. 115–139. DOI: 10.1515/9783110448375-008

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guarantee an a-priori walking stability are often based on one of the following stabilization criteria: – The Center Of Mass (COM) – The Zero Moment Point (ZMP) – The Foot Rotation Indicator (FRI) The COM [1] is the mean location of all masses of the robot links. It is usually used as a static stability criterion. The ZMP [2, 3] is the point of junction between the center of the vertical reaction forces and the ground. It is the most used dynamic stability criterion. The FRI [4] is a point on the foot/ground-contact surface where the net ground-reaction force would have to act to keep the foot stationary. It is an indication of postural stability and, in case of instability, indicates how the robot will fall. It is used as a dynamic stability criterion.

2 Related works In the literature several methods have been proposed for generating walking trajectories in humanoid robotics. A method based on a motion capture of human walking has been proposed in [5]. This method can gives human-like motions, however its main drawback lies in the captured data that can be hard to adapt to the humanoid robot. Another method of trajectory generation for stable dynamic walking is proposed in [6]. It consists in using a 3rd order spline function to generate feet and hip trajectories. The foot trajectories can be adapted to the ground variations to generate a stable dynamic walking on a rough terrain. In [7] another pattern generation method has been proposed. It consists in using Fourier series to generate stable walking, with an iterative procedure to guarantee the stability. The main drawback of such a method lies in the computation time that does not enable a real-time implementation. Another interesting method is the so called Inverted Pendulum Model (IPM) [8, 9] that considers the robot as a single point mass and massless legs. This method simplifies the dynamics of the robot to an inverted pendulum with a point mass linked by a telescopic leg to a spherical ground/leg joint. The Linear Inverted Pendulum Model (LIPM) [10, 11] is an extension of the IPM where the height of the torso is considered to be constant leading to a more natural movement. The IPM ignores the dynamics of the legs since it considers the stance leg as an inverted pendulum with a point mass. This fact can involve a loss of walking stability since the considered model is considerably simplified and consequently not sufficiently accurate. For instance, this is the case when the robot has heavy legs or

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has no torso (case of our robot SHERPA). In order to approximate the robot with a more accurate model, it would be necessary to consider the whole-body dynamics by using a preview control to correct the error between the IPM and the real robot as in [12] or to use a multiple point mass model instead of considering a one point mass model. For instance, a Two-Mass Inverted Pendulum Model (TMIPM) and Multiple Mass Inverted Pendulum Model (MMIPM) extending the IPM were proposed in [13]. In the MMIPM, the considered model is composed by more than one mass. These masses are located at the hip and along the swinging leg in the case of a two-mass model. The Gravity-Compensated Inverted Pendulum Model (GCIPM) [14] uses also one mass to represent the body of the robot and an additional mass to represent the swing leg. A three-mass model using the concept of ZMP has been studied in [15]; however, it doesn’t use any LIPM to generate trajectories and the torso of the biped robot moves up and down. A generation of walking trajectories using a three point mass model to calculates center of mass trajectories from footstep locations has been proposed in [16]. This approach use offline optimization of some free geometrical parameters like the trunk angular motion w.r.t. the speed of the robot. These parameters are then used in a real-time fast planning to compute the reference torque patterns to apply on the robot. Another three-mass model using the inverted pendulum concept is proposed in [17]. However, it has one important drawback related to the geometrical parameters, such as the position and the location of masses, which have been chosen arbitrarily. One good idea would then be to tune these parameters at their best values in order to enhance the walking stability of the robot, and this is the scope of the present paper. The proposed solution is an extension of the method proposed in [17] to deal with dynamic stability of walking in the generated trajectories, as well as changes in direction during walking. As in [17], the robot’s dynamics will be approximated by a Three-Mass Linear Inverted Pendulum. The hip and feet trajectories are generated by the movement of masses. The walking movement of the robot is generated in the sagittal and frontal planes separately. The joint trajectories are computed using inverse kinematics of the biped robot. In order to ensure a stable dynamic walking, the parameters of the model must be well tuned. The best way to perform such a tuning is through an optimization of the generated joints’ trajectories to minimize the ZMP excursion within the footprint which will increase the stability margins. The second contribution of this work is to propose online change of walking direction, where new optimal trajectories are generated to ensure a-priori stability during walking while turning. This paper is organized as follows: in the next section, the prototype of our demonstrator SHERPA is introduced. Section 4 introduces the simplified model that will be used in the generation of the reference trajectories and how it was used in the pattern generator proposed in [17]. In section 5, the proposed extension of this

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method is presented, where our contributions are highlighted. Numerical simulations are presented and discussed in section 6, where the effectiveness of the proposed method is shown and compared with the original one for the case of SHERPA robot. The paper ends with some concluding remarks.

3 Description of SHERPA prototype The SHERPA walking robot (cf. Fig. 3) is a French biped robot developed at the LIRMM laboratory [18] within the framework of the national project ANR-06-BLAN-0244 SHERPA. The name of this robot comes from the so-called Sherpa, who are members of a people of Tibetan stock living in the Nepalese Himalayas, and who often serve as porters on mountain-climbing expeditions. Indeed, this robot is built to carry loads while walking in a human environment [19].

q1

q7 q2 Hip

q3

q8 q9

Knee q10

q4

q12

q6 Anckle q5

Fig. 1. SHERPA kinematics model.

q11

Fig. 2. CAD model of SHERPA.

Fig. 3. The SHERPA robot.

SHERPA is composed of a hip linking two legs together. Each leg has six degrees of freedom (dof), and the robot is equiped with 12 actuators, which is enough to reproduce a human gait. These dofs are distributed on the different articulations of the robot as follows: three dofs at the hip, one dof at the knee and two dofs at the ankle as it is illustrated on Fig. 1. The geometric parameters of the robot are summarized in Tab. 1.

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Tab. 1. Geometric parameters of SHERPA biped robot. Description

Parameter

Value

Length of thigh Width of the hip Total weight of the robot

Lthigh Ls mass

0.5033 m 0.31 m 47.1 kg

This robot differs basically from the other walking robots by its actuation system, which is transparent (backdrivable, with low inertia), and organized in modules (as illustrated in Figs. 4–5). Each actuation module includes two actuators acting in parallel on two dofs simultaneously. The mechanical transmission of these modules is such that when the two actuators work together on the joint, they cause the movement of the first dof; and when they act in opposite directions they cause the movement of the second dof. These modules are equipped with custom-made electric motors. The transmission of movement is based on the use of a ball screw transforming the rotation of the hollow shaft electrical actuator in a translational movement. This last one is then transmitted to pulleys using cables to produce the desired rotational movement. This basic principle of motion transmission is illustrated in Fig. 6.

2 DOF Path Change X2 Actuator Module Frame Joint Module

Secondary Rotation Axis

Main Rotation Axis

Ball Bearing Screw

Cable Return Module

Fig. 4. CAD view of an actuator module.

This technology gives the robot’s actuators remarkable characteristics such as no backlash, low friction, reversibility of the chain of transmission and low inertia. The kinematic model of SHERPA robot is created through the link representation proposed in [20]. This model uses the twelve joints’ values as well as the position and

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orientation of the hip to generate the Cartesian position of each articulation in the operational space.

Fig. 5. View of a real manufactured module.

Stroke R

F TB

Θ1=140°

T

Θ2=60°

Fig. 6. Cable/Pulley transmission system used in actuation module of SHERPA.

The forward kinematics model writes: X = f (q)

(1)

where X ∈ R39×1 includes Cartesian positions of articulations and the hip’s center position, q ∈ R18×1 includes a 12 × 1 vector of joints positions, and a 6 × 1 vector of the position and orientation of the hip.

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The inverse kinematic model is obtained by solving an analytical equation to find the articular positions expressed in terms of operational space positions of the hip and feet. The inverse kinematic model can then be written as: q = g(X r )

(2)

where X r ∈ R9×1 represents the operational space position of the hip (3 × 1) and the feet (6 × 1). The orientation of the hip and the feet are kept constant and equal to zero. The computation of the ZMP and CoM of the robot is based on the formalism presented in [20], in which the ZMP evaluation uses the angular momentum with the overall dynamic model of the robot.

4 Three-mass linear inverted pendulum model The three-mass Linear Inverted Pendulum Model (3MLIPM) as introduced in [17] simplifies the biped robot to a three-link system (as shown in Fig. 7) with a point mass on each link. The three masses represent the torso and the two legs, unlike the single mass model (as in IPM or LIPM) where a unique mass is located at the hip of the robot. The three links are connected together at the hip. This model is more accurate than the single mass model, especially for biped robots without torso, where the position of the CoM can be very different from the hip position. From Fig. 7, the equations of moment applied on the supporting ankle can be formulated as follows: τx =

3 

m i (g + z¨ i )y i −

i=1

τy =

3  i=1

3 

m i y¨ i z i

(3)

m i (g + z¨ i )x i

(4)

i=1

m i x¨ i z i −

3  i=1

where τ x and τ y are the torques applied on the ankle, m i represents the value of the i th point mass, (x i , y i , z i ) are the Cartesian positions of the i th point mass (i = 1, 2 or 3 represent respectively the stance leg, the torso and the swing leg), (¨x i , y¨ i , z¨ i ) are their corresponding accelerations. For simplification, the height of the masses is assumed to be constant and the torque applied on the supporting ankle is assumed to be zero [17]. With these

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assumptions, (3) and (4) can be simplified into: 3 

m i y¨ i z i =

i=1

3 

m i gy i and

i=1

3  i=1

m i x¨ i z i =

3 

m i gx i

(5)

i=1

These equations are decoupled, so the movement can be generated in sagittal and lateral planes separately.

Z m2

m1 m3 Y X

Fig. 7. Graphical representation of the Three-Mass Linear Inverted Pendulum Model.

4.1 Assumptions and notations The following assumptions are considered to simplify the calculations [17]: – The ground is flat and horizontal. – The height variation of each mass can be neglected. – The double support phase is considered instantaneous. – There is no energy loss during impact. – The standing foot has no overturn when touching and leaving the ground. – The swing foot is parallel to the ground. – The torso is upright. Besides, the following notations are used: – 2T: The whole stepping cycle. – 2D s : The steeping length. – L s : The width of a step.

4.2 Motion study in the sagittal plane The movement in the sagittal plane and the associated notations are illustrated in Fig. 8.

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With the above assumptions, heights and masses can be normalized as follows: z2 = γz1 , z3 = z1 , x2 = λx1 , m2 = km1 , m3 = m1

(6)

where γ, λ and k are the parameters extracted from the robot’s model. Considering (6), equation (5) can be simplified into: b x¨ 1 + d x¨ 3 = ax1 + x3 z a = kλ + 1, b = 1 (1 + kγλ), g

(7) z d= 1 g

(8)

Z

X2 Z2 m2

m2

γZ1 λZ1

Hip

Knee

X3 Z3

X1 Z1 m1

m3

m3

Z1 λX1

Ds

0

Ds

X

Fig. 8. Illustration of movement in the sagittal plane.

In order to generate walking gaits, a trajectory for the swing foot (mass m3 ) is necessary. If the trajectory of this last one is known, then those of the two other masses can be computed. According to [17], a sinusoidal function is chosen to represent the trajectory of the mass m3 , as follows: x3 = A cos(ωt + ϕ)

(9)

where parameters ϕ and A are given in the sequel by equation (13). Here, ω is not the step frequency, it is a parameter that must be chosen to keep the foot position and velocity positive in order to ensure a forward movement of the foot during the step.

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Using (7) and (9), x1 can be expressed by: x1 = C1 cos(ωt + ϕ) + C2 e

√a b

t

√a

+ C3 e−

b

t

(10)

With the above assumptions given in section 4.1, initial conditions can be fixed as: Ds x1 (0) = − , 2λ  T = 0, x1 2

Ds x3 (0) = − (2λ − 1) 2λ  T x3 =0 2

(11) (12)

The equations’ coefficients can then be identified using initial conditions: ϕ=−

π T − ω, 2 2

A=−

D s (2λ − 1) 2λ cos ϕ

√a A(1 + dω2 ) b tC C = −e 3 2 a + bω2 Ds 1 √a − − C1 cos ϕ C2 = t 2λ 1−e b

C1 = −

(13) (14) (15)

With all trajectories of point masses computed, the trajectories of the hip and ankles can then be determined using geometrical constraints from Fig. 8. x st (t) = 0 for 0 ≤ t ≤ T, x h (t) = λx1 (t) for 0 ≤ t ≤ T z st (t) = 0 for 0 ≤ t ≤ T, z h (t) = λz1

for 0 ≤ t ≤ T

(16)

where x st , z st represent Cartesian positions of the stance foot along x and z axis respectively. x h , z h are the Cartesian positions of the hip along x and z axis respectively. During the step, the vertical position of the swing foot needs to be higher than the floor. A sinusoidal shape can then be an appropriate function for the swing foot trajectory along the z axis. However, due to the previous assumptions, the influence of this change of height is neglected in computations. λx3 (t) − λx h (t) for 0 ≤ t ≤ T λ−1 3 √  2π 2H s π for 0 ≤ t ≤ T 1 + sin t− z sw (t) = 2 T 2

x sw (t) =

(17)

where x sw , z sw are the positions of the swing foot along the x and z axis respectively, and H s is the maximal height of the swing foot during walking.

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4.3 Motion study in frontal plane The movement in the frontal plane and the associated notations are illustrated in Fig. 9. The 3MLIPM equations are now derived in the frontal plane. u y¨ 1 − vy1 = w

(18)

Where the coefficients u, v and w are given by:  Ls z1 u = (2 + kλγ) , v = (2 + kλ), w = L s + k g 2 These equations are solved considering the following initial conditions:  T = 0, y1 (T) = 0 y1 (0) = 0, y˙ 1 2

(19)

(20)

The trajectory of the mass m1 along the y axis is then computed and expressed by: y1 (t) = C1 e

√v u

t

√v

+ C2 e−

u

t

−

w v

(21)

where: √v

√v w 1−e uT √ v , C2 = e u t C1 C1 = v 1 − e2 u T

Z

m2

Hip

Knee

m3

m1

0

Ls

Y

Fig. 9. Illustration of movement in the frontal plane.

(22)

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Once the trajectory of the mass m1 is computed, those of m2 and m3 can be easily determined using geometrical constraints illustrated on Fig. 9. From these trajectories and the inverse kinematics of the robot, the trajectories of all the joints of the robot can then be computed.

4.4 Adaptation of the general model for SHERPA robot The SHERPA robot is a biped robot without a torso (cf. Fig. 3). In order to simplify the model, the torso point mass is set on the hip. This modification results in γ = λ. It is worth to note that the basic principle of the proposed method in [17] and summarized above uses some geometrical parameters of the robot (such as m1 , m2 , m3 and z1 , z2 , z3 ) but doesn’t give a method to calculate them. In order to take into account the dynamic stability of the resulting walking trajectories, we propose to tune these parameters using optimization to enhance the walking performance. The idea of such contribution is detailed in the following section.

5 Reference trajectories optimization The proposed pattern generator in [17] computes joint reference trajectories using arbitrary defined parameters. This can be improved by choosing the best values of some of them (namely m1 and z1 ) to ensure walking stability. Since we are interested in dynamic walking, the position of the Zero Moment Point (ZMP) [2] is then used as an indicator of stability.

5.1 Dynamic stability margins The stability margins are defined by the distance to the limit of stability (i.e. the boundary of the footprint), they are illustrated in Fig. 10.

Footprint

ZMP trajectory y My1

β

x

Mx1 My2

Mx2

α Fig. 10. ZMP displacement inside the footprint and stability margins.

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The dashed interior rectangle explicits the boundaries of the ZMP displacements. Based on Fig. 10, the stability margins can be mathematically expressed as follows: α α +min(dZMP x (t)), Mx2 = −max(dZMP x (t)) 2 2 β β My1 = +min(dZMP y (t)), My2 = −max(dZMP y (t)) 2 2 ∀ t ∈ [ts i , ts f ]

Mx1 =

Mx = min(Mx1 , Mx2 ), My = min(My1 , My2 )

(23)

were dZMP x (t) and dZMP y (t) are the deviations of the ZMP trajectory with respect to the center of the stance foot along x axis and y axis respectively, ts i and ts f are the time instants of landing and lift-off of the stance foot respectively. The duration of the step is T = ts f − ts i .

5.2 Optimization w.r.t. dynamic walking stability The first main contribution of this paper is to improve the pattern generator proposed in [17] by considering an optimization criterion in order to find the best values of z1 and m1 to enhance the dynamic walking stability. In order to do that, the ZMP corresponding to the generated joint trajectories should be computed with the overall dynamics of the robot and compared with the desired ZMP, set to the center of the stance foot. This chosen ZMP desired position corresponds to a maximum stability margins. The following objective function is then proposed to be optimized w.r.t. the parameters z1 and m1 :   4 zˆ 1 1 1 (24) = arg ⎡ min ⎤ max (x zmp − x dzmp )2 + (y zmp − y dzmp )2 α β ˆ1 m z ⎣

1

m1

⎦

where x zmp and y zmp are the positions of the computed ZMP, x dzmp and y dzmp are those of the desired one. z1 and m1 are the optimization parameters, α and β are the length and the width of the foot (i.e. along x axis and y axis respectively, cf. Fig. 10). By minimizing the maximum normalized deviation of the ZMP trajectory as cost function, the objective function computes the mass distribution to ensure the largest stability margins.

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5.3 Change of direction and stability optimization One key feature of a pattern generator is to allow the biped robot to change the walking direction using the kinematic model like in [21], or the dynamic model as in [22] with a two point mass system model. As proposed in [17], the original 3MLIPM pattern generator is designed to make the robot walking only in a straight line. Our second contribution is then to modify it to allow a change of direction while walking. The change of direction is obtained through the application of a rotation at the hip of the stance leg, as follows: Ω=−

πt R cos 2 T

(25)

with Ω is the angle of rotation and R is its amplitude. The change of direction alters the stability of the biped robot. Therefore, the proposed optimization criterion (24) is used again to improve dynamic walking stability. The solution of this optimization problem gives the best values for z1 and m1 which allow a better dynamic stability of the robot walking and turning.

6 Simulation results A simulator for SHERPA biped robot was developed using the Graphical User Interface of Matlab™software. Its graphical interface is shown in Fig. 11. This interface enables the tuning of some parameters of the robot as well as those of the optimization criterion. A graphical animation of the robot or the three mass model can also be displayed in the simulator to show the obtained movements. The parameters z1 and m1 are computed by optimization, the other parameters are constant and summarized in Tab. 2.

Fig. 11. View of the Graphical User Interface of SHERPA biped robot simulator.

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Tab. 2. Parameters of the 3 MLIPM pattern generator. Description

Parameter

Value

Weight of one leg Weight of the torso Height of the mass m1 Height of the hip Step time Step length Calculus variable Calculus variable

m1 m2 z1 z2 T Ds ω λ=γ

Optimized mass − 2m1 Optimized 0.95(Lshin + Lthigh ) 0.7 s 0.4 m π/(5T ) z2 /z1

Using the developed simulator, four simulation scenarios are proposed to validate the proposed optimal pattern generator, namely: – Simulation 1: optimal trajectories generation for straight walking, – Simulation 2: trajectories generation for walking with change of direction without optimisation, – Simulation 3: trajectories generation for walking with a change of direction and optimization. – Simulation 4: Effects of walking parameters These simulations will be detailed and commented in the following.

6.1 Simulation 1: straight walking The optimization criterion given in (24) is used to find the best values for z1 and m1 using the fminsearch algorithm proposed within Matlab software. The optimization algorithm uses a simplex search method described in [23]. The obtained optimization results are coherent with their physical meanings: the masses are found to be positive and the positions of the three masses are inside the convex envelope of the robot. The obtained optimal solution is summarized in Tab. 3, and the simulation results are illustrated, for six walking steps, through curves of Figs. 12–18.

Tab. 3. Resulting optimized parameters. Parameter z1 m1

Without optimization 0.6 m 6 kg

With optimization 0.2598 m 0.4442 kg

130 | D. Galdeano et al.

Figures 12 and 13 represent respectively the evolution of the joints’ positions and velocities, where it can clearly be seen that the obtained trajectories are periodic. Furthermore the trajectories of one leg are symmetrical w.r.t. those of the other one. position: q1 [rad] and q7 [rad]

0.1 0 –0.1

Left leg position: q2 [rad] and q8 [rad]

1 0 –1

position: q3 [rad] and q9 [rad]

1 0 –1

position: q4 [rad] and q10 [rad]

2 1 0

position: q5 [rad] and q11 [rad]

0 –0.5 –1 0.1 0 –0.1 0

Right leg

position: q6 [rad] and q12 [rad]

0.5

1

1.5

2 time[s]

2.5

3

3.5

4

Fig. 12. Joints’ positions generated by the proposed optimal pattern generator. velocity: 1 [rad/s] and 7 [rad/s]

0.5 0 –0.5

velocity: 2 [rad/s] and 8 [rad/s]

5 0 –5

velocity: 3 [rad/s] and 9 [rad/s]

1 0 –1

velocity: 4 [rad/s] and 10 [rad/s]

5 0 –5

velocity: 5 [rad/s] and 11 [rad/s]

5 0 –5 0.5 0 –0.5 0

Right leg Left leg

velocity: 6 [rad/s] and 12 [rad/s]

0.5

1

1.5

2 time[s]

2.5

3

3.5

Fig. 13. Joints’ velocities generated by the proposed optimal pattern generator.

4

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Figures 14, 15 and 16 represent the evolution of the ZMP and the CoM positions with respect to the footprints of the biped robot on the ground generated respectively by the linear inverted pendulum model (LIPM), the original (3MLIPM) pattern generator proposed in [17] and the optimal one proposed in this paper.

Y [m]

ZMP 0.2 0.1 0 CoM –0.1 –0.2 –0.5

0

0.5

1 X [m]

1.5

2

2.5

1.5

2

2.5

2

2.5

Fig. 14. Evolution of ZMP and CoM trajectories with the LIPM.

Y [m]

ZMP 0.2 0.1 0 CoM –0.1 –0.2 –0.5

0

0.5

1 X [m]

Fig. 15. Evolution of ZMP and CoM trajectories with the original 3MLIPM.

Y [m]

ZMP 0.2 0.1 0 CoM –0.1 –0.2 –0.5

0

0.5

1 X [m]

1.5

Fig. 16. Evolution of ZMP and CoM trajectories with the proposed optimal 3MLIPM.

The ZMP position calculation is issued from an angular momentum evaluation based on Kajita’s formulation [20]. This computation is not base on the simplified three-mass model; indeed, it uses the dynamic model of the biped robot to produce a realistic ZMP evaluation. For the LIPM as well as the original 3MLIPM, the ZMP moves inside the footprint with a big variation, however when the optimal parameters are used, the ZMP is more concentrated in the center of the footprint and the stability margins are better.

132 | D. Galdeano et al.

Therefore, the walking dynamic stability is clearly improved with the proposed optimal 3MLIPM pattern generator due to the increase of the stability margins.

CoM position: X [m]

X[m]

2 1.5 1

Y [m]

0.5 0 0

0.5

1

1.5

0.25 0.2 0.15 0.1 0.05 0 –0.05 –0.1 –0.15 –0.2 –0.25 0

0.5

1

1.5

2 2.5 time[s] CoM position: Y [m]

2 time[s]

2.5

3

3.5

4

3

3.5

4

Fig. 17. CoM evolution versus time with optimization.

Figure 17 displays the evolution of the position of the CoM along the x and y axes during walking. The obtained trajectories are cyclic along the y axis with a reduced amplitude thanks to the optimal parameters. Figure 18 shows the trajectories of movement of the swing foot for one step.

6.2 Simulation 2: walking with a change of direction The objective of this simulation is to evaluate the stability margins of the pattern generator proposed in [17] in case of a change of direction during walking. The obtained result is depicted in Fig. 19 where it is worth to note that the change of direction during walking has not been yet considered in the original 3MLIPM, therefore a simple simulation shows a loss of stability during walking while turning. Indeed, when the robot changes the walking direction, the position of the ZMP is moving within footprints, being sometimes outside of the stance footprint, therefore the robot becomes unstable. Consequently, a computation of optimal parameters in this scenario will be necessary to improve the dynamic walking stability. This is the objective of the next simulation scenario.

Z [m]

Optimal Pattern Generator for Dynamic Walking

Foot position

0.1 0.08 0.06 0.04 0.02 0 –0.4

–0.3

–0.1

0 X[m]

1.15

2 Ẍ[m/s ]

Ẋ[m/s]

–0.2

Foot velocity: Ẋ (t)

1.2 1.1 1.05 0

0.1 0.2 0.3 0.4

0.5 0.6

0.1

Ż[m/s]

0.3

0.4

0.1 0.2 0.3 0.4

0.5 0.6

Foot acceleration: Z¨ (t) Z¨[m/s2]

0.1 0.2 0.3 0.4

0.2

Foot acceleration: Ẍ (t)

1 0.5 0 –0.5 –1 0

Foot velocity: Ż (t) 0.4 0.2 0 –0.2 –0.4 0

133

|

–0.5 –1 –1.5 –2 0

0.5 0.6

time[s]

0.1 0.2 0.3 0.4 time[s]

0.5 0.6

Fig. 18. Evolution of the swing foot trajectories.

1

0.5 Y [m]

ZMP

0

–0.5 –0.5

CoM

0

0.5

1

1.5

X [m] Fig. 19. Evolution of ZMP and CoM trajectories with change of direction.

2

2.5

134 | D. Galdeano et al.

6.3 Simulation 3: change of direction with optimization In the proposed optimal pattern generator, the optimization process computes the best values of the parameters z1 and m1 and uses them to generate the joints’ trajectories. Figures 20 and 21 represent respectively the evolution of the new joints’ positions and velocities for this scenario. Theses trajectories allow a rotation of the biped robot during walking. Figure 22 represents the evolution of the ZMP and the CoM positions as well as the footprints of the biped robot on the ground level when turning with an angle of 20 degrees at each walking step. With the proposed optimization, as illustrated in Fig. 22, the robot remains stable during walking while turning since it keeps the ZMP always inside the footprint of the supporting leg. According to the obtained results, from simulation 1 and 3, it is clearly shown that the proposed optimal pattern generator has significantly improved the dynamic stability of the biped robot for both scenarios: straight walking and change of direction. position: q1 [rad] and q7 [rad]

0.1 0 –0.1

Left leg position: q2 [rad] and q8 [rad]

1 0 –1

position: q3 [rad] and q9 [rad]

0.1 0 –0.1

position: q4 [rad] and q10 [rad]

2 1 0

position: q5 [rad] and q11 [rad]

0 –0.5 –1 0.1 0 –0.1 0

Right leg

position: q6 [rad] and q12 [rad]

0.5

1

1.5

2 time[s]

2.5

3

3.5

4

Fig. 20. Joints’ positions with change of direction generated by the proposed optimal pattern generator.

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6.4 Simulation 4: effects of walking parameters The optimal values summarized in Tab. 3 are used to compute stability margins Mx and My for straight walking with different step lengths and the objective is to analyze the sensitivity of the model w.r.t. step length variation. The optimal values of mass distribution have been calculated with a step length of 0.4 m. These values are then used with different step lengths Ds. The corresponding stability margins are summarized in Tab. 4. It is worth to note that the modification of the step length don’t induce a loose of stability during walking. The optimal mass distribution make the proposed pattern generator less sensitive to step length variations. velocity: 1 [rad/s] and 7 [rad/s]

0.5 0 –0.5

velocity: 2 [rad/s] and 8 [rad/s]

5 0 –5

velocity: 3 [rad/s] and 9 [rad/s]

0.5 0 –0.5

velocity: 4 [rad/s] and 10 [rad/s]

5 0 –5

velocity: 5 [rad/s] and 11 [rad/s]

5 0 –5 0.5 0 –0.5 0

Right leg Left leg

velocity: 6 [rad/s] and 12 [rad/s]

0.5

1

1.5

2 time[s]

2.5

3

3.5

4

Fig. 21. Joints’ velocities with change of direction generated by the proposed optimal pattern generator.

1

Y [m]

0.5 ZMP 0 CoM

–0.5 –0.5

0

0.5

1 X [m]

1.5

2

2.5

Fig. 22. Evolution of ZMP and CoM trajectories in case of change of direction generated by the proposed optimal pattern generator.

136 | D. Galdeano et al.

Tab. 4. Step length influence on stability margins Ds

0.1

0.2

0.3

0.4

0.5

Stability Mx My

stable 0.105 0.028

stable 0.108 0.029

stable 0.110 0.030

stable 0.112 0.030

stable 0.115 0.029

Y [m]

ZMP 0.2 0.1 0 CoM –0.1 –0.2 –0.5

0

0.5

1 X [m]

1.5

2

2.5

Fig. 23. Evolution of ZMP and CoM trajectories with Ds = 0.4.

Y [m]

ZMP 0.2 0.15 0.1 0.05 CoM 0 –0.05 –0.1 –0.15 –0.2 –0.2

0

0.2

0.4

0.6 X [m]

0.8

1

1.2

1.4

Fig. 24. Evolution of ZMP and CoM trajectories with Ds = 0.2.

7 Conclusions and future work The objective of this work was to generate stable dynamic walking for SHERPA biped robot. To deal with this problem, the original pattern generator proposed in [17] based on a Three-Mass Linear Inverted Pendulum Model has been extended with a ZMP-based optimization to improve dynamic walking stability. The optimization use a ZMP based performance index in an optimization problem whose solution gives the best values of the model’s parameters w.r.t. dynamic walking stability computed using a whole body model. The case of change of direction during walking has also been studied and stability in this case has been enhanced thanks to the optimization.

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Several numerical simulations have been presented to show the effectiveness of the proposed optimal pattern generator for the case of the biped walking robot SHERPA. Future works can include real-time implementation of the proposed method on the prototype of SHERPA biped robot and the creation of a motion database to be able to change the walking speed of the robot in real-time. Acknowledgments: This research was supported by the French National Research Agency, within the project R2A2 (ANR-09-SEGI-011).

Bibliography [1] [2] [3]

[4] [5]

[6]

[7]

[8]

[9] [10]

[11] [12]

[13]

S. Ma, T. Tomiyama, and H. Wada. Omnidirectional static walking of a quadruped robot. IEEE Trans. on Robotics, 21(2):152–161, 2005. M. Vukobratovic and B. Borovac. Zero-moment point-thirty five years of its life. Int. J. of Humanoid Robotics, 1(1):157–173, 2004. P. Sardain and G. Bessonnet. Forces acting on a biped robot. Center of pressure-zero moment point. IEEE Trans. on Systems, Man and Cybernetics, Part A: Systems and Humans, 34(5):630–637, 2004. A. Goswami. Postural stability of biped robots and the foot-rotation indicator (FRI) point. Int. J. of Robotics Research, 18(6):523–533, 1999. K. Harada, K. Miura, M. Morisawa, K. Kaneko, S. Nakaoka, F. Kanehiro, T. Tsuji, and S. Kajita. Toward human-like walking pattern generator. IEEE/RSJ Int. Conf. on Intelligent robots and systems (IROS’09), (Albuquerque, NM, USA), :1071–1077, 2009. Q. Huang, S. Kajita, N. Koyachi, K. Kaneko, K. Yokoi, H. Arai, K. Komoriya, and K. Tanie. A high stability, smooth walking pattern for a biped robot. IEEE Int. Conf. on Robotics and Automation (ICRA’99), :65–71, Detroit, Michigan, USA, 1999. J. Yamaguchi, E. Soga, S. Inoue, and A. Takanishi. Development of a bipedal humanoid robot-control method of whole body cooperative dynamic biped walking. IEEE Int. Conf. on Robotics and Automation (ICRA’99), 1:368–374, Detroit, Michigan, USA, 1999. S. Kajita, F. Kanehiro, K. Kaneko, K. Fujiwara, K. Yokoi, and H. Hirukawa. Biped walking pattern generation by a simple three-dimensional inverted pendulum model. Advanced Robotics, 17(2):131–147, 2003. Z. Tang and M. Er. Humanoid 3D Gait Generation Based on Inverted Pendulum Model. IEEE 22nd Int. Symp. on Intelligent Control (ISIC’07), :339–344, Singapore, 2007. S. Kajita, F. Kanehiro, K. Kaneko, K. Yokoi, and H. Hirukawa. The 3D Linear Inverted Pendulum Mode: A simple modeling for a biped walking pattern generation. IEEE/RSJ Int. Conf. on Intelligent Robots and Systems (IROS’01), :239–246, Maui, Hawaii, USA, 2001. S. Feng and Z. Sun. A simple trajectory generation method for biped walking. 10th Int. Conf. on Control, Automation, Robotics and Vision (ICARCV’08), :2078–2082, 2 Hanoi, Vietnam, 008. S. Kajita, F. Kanehiro, K. Kaneko, K. Fujiwara, K. Harada, K. Yokoi, and H. Hirukawa. Biped walking pattern generation by using preview control of zero-moment point. IEEE Int. Conf. on Robotics and Automation (ICRA’03), 2:1620–1626, 2003. A. Albert and W. Gerth. Analytic Path Planning Algorithms for Bipedal Robots without a Trunk. J. of Intelligent and Robotic Systems, 36(2):109–127, 2003.

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[14] J. Park and K. Kim. Biped robot walking using gravity-compensated inverted pendulum mode and computed torque control. IEEE Int. Conf. on Robotics and Automation (ICRA’98), :3528–3533, Leuven, Belgium, 1998. [15] J. Kanniah, Z. Lwin, D. Kumar, and N. Fatt. A ZMP management scheme for trajectory control of biped robots using a three mass model. Proceedings of the 2nd Int. Conf. on Autonomous Robots and Agents (ICARA’04), :458–463, Palmerston North, New Zealand, 2004. [16] T. Buschmann, S. Lohmeier, M. Bachmayer, H. Ulbrich, and F. Pfeiffer. A collocation method for real-time walking pattern generation. IEEE/RAS Int. Conf. on Humanoid Robots, :1–6, 2007. [17] S. Feng and Z. Sun. Biped robot walking using three-mass linear inverted pendulum model. Proceedings of the Int. Conf. on Intelligent Robotics and Applications (ICIRA’08), :371–380, Springer-Verlag, Wuhan, China, 2008. [18] http://www.lirmm.fr [Online]. [19] I. M. C. Olaru, S. Krut, and F. Pierrot. Novel mechanical design of biped robot sherpa using 2 dof cable differential modular joints. Proceedings of the 2009 IEEE/RSJ Int. Conf. on Intelligent Robots and Systems (IROS’09), :4463–4468, St. Louis, MO, USA, 2009. [20] S. Kajita, H. Hirukawa, K. Harada, and K. Yokoi. Introduction à la commande des robots humanoides. Translated in French by Sakka, S. Springer, 2009. [21] S. Kajita, F. Kanehiro, K. Kaneko, K. Fujiwara, K. Yokoi, and H. Hirukawa. A realtime pattern generator for biped walking. IEEE Int. Conf. on Robotics and Automation, ICRA’02, 1:31–37, IEEE, 2002. [22] J. T. Kim and J. H. Park. Quick change of walking direction of biped robot with foot slip in single-support phase. 11th IEEE-RAS Int. Conf. on Humanoid Robots (Humanoids’11), :339–344, 2011. [23] J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright. Convergence properties of the Nelder–Mead simplex method in low dimensions. SIAM J. on Optimization, 9(1):112–147, 1998.

Biographies David Galdeano received the M.S. degree in Robotics and Automation from the University Montpelier 2, France, in 2010. Currently he is a Ph.D. student in the Laboratory of Informatics, Robotics, and Microelectronics (LIRMM). His current research interests include humanoid robotics, whole-body posture control.

Ahmed Chemori received his MSc and PhD degrees respectively in 2001 and 2005, both in automatic control from the Grenoble Institute of Technology. He has been a Post-doctoral fellow with the Automatic control laboratory of Grenoble in 2006. He is currently a tenured research scientist in Automation and Robotics at the Montpelier Laboratory of Informatics, Robotics, and Micro-electronics. His research interests include nonlinear, adaptive and predictive control and their applications in humanoid robotics, underactuated systems, parallel robots, and underwater vehicles.

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Sébastien Krut received the M.S. degree in mechanical engineering from the Pierre and Marie Curie University, Paris, France, in 2000 and the Ph.D. degree in automatic control from the Montpelier University of Sciences, Montpelier, France, in 2003. He has been a Post-doctoral fellow with the Joint Japanese-French Robotics Laboratory (JRL) in Tsukuba, Japan in 2004. He is currently a tenured research scientist in Robotics for the French National Centre for Scientific Research (CNRS), at the Montpelier Laboratory of Computer Science, Microelectronics and Robotics (LIRMM), Montpelier, France. His research interests include design and control of robotic systems.

Philippe Fraisse received M.Sc degree in Electrical Engineering from Ecole Normale Superieure de Cachan in 1988. He received Ph.D. degree in Automatic Control in 1994. He is currently Professor at the University of Montpelier, France. He is the head of robotics department (LIRMM) and co-chair of French National Workgroup (GDR Robotique) working on Humanoid Robotics (GT7). He is also member of JRL-France scientific board (Japanese- French joint Laboratory for Robotics, AIST-JRL) and member of IEEE. His research interests include modeling and control applied to robotic and rehabilitation fields, including humanoid robotics, robotics for rehabilitation.

Ulrich Vogl

Optimal Targeting in Chaos Control, a Discrete Hamiltonian Approach

Abstract: One of the most interesting paradigms of chaos control is the possibility of switching a system between different unstable periodic orbits (UPOs) with effectively zero control energy. We give a robust method to find finite-time optimal transient trajectories, and show how to stabilize both, UPOs and transients, within the same LQ-framework. The method is quite general, and can also be used to drive a system from static stationary points to an UPO. To illustrate our approach we apply it to the controlled logistic map, and also to an experimental driven-pendulum setup. Keywords: Chaos Control, nonlinear optimal control, targeting, time-dependent LQ, discrete Hamiltonian, driven pendulum.

1 Introduction In recent years much progress has been made in controlling nonlinear and especially chaotic systems, and a variety of algorithms have been developed to find and stabilize periodic solutions in chaotic systems [1, 2, 3, 4, 5]. Indeed, two of the most important paradigms in chaos control are stabilizing unstable periodic orbits (UPOs) embedded in an attractor and steering the system between different UPOs. The latter is of special interest, since it is well known [6], that any strange attractor has countable infinite UPOs embedded. If the UPOs are contained in the same attractor (i.e. the same invariant set of the dynamic flow), the energy needed to switch the system between these UPOs vanishes exponentially with the time N allowed for the transient trajectory. In this paper we give a time-discrete algorithm, which yields for any given finite number of time steps N an optimal control signal u(k) which switches the system between different UPOs. The method is universal and can e.g. also be used to find minimal-energy trajectories connecting static states outside an attractor with an UPO embedded in. This kind of transients are obviously a necessity in any practical or experimental setup. We can also connect UPOs embedded in different attractors, and find our way back to static stationary points. Since the method

Ulrich Vogl: Faculty of Electrical Engineering & Information Technology, OTH Amberg-Weiden, University of Applied Sciences, Amberg, Germany, email: [email protected]. De Gruyter Oldenbourg, ASSD – Advances in Systems, Signals and Devices, Volume 1, 2016, pp. 141–156. DOI: 10.1515/9783110448375-009

142 | U. Vogl

does not relay on chaos, also optimal trajectories for non-chaotic systems can be found. Once a transient to a desired UPO has been found, the latter, and in most cases also the transient itself, need to be stabilized. To this end we apply a time-dependent LQ- (linear-quadratic) approach which will briefly be outlined in section III. Finally, we apply our method to a periodically driven non-linear pendulum experiment. Short video sequences, demonstrating the practicability of the method in an experimental setup, can be watched at [7].

2 Optimal trajectories 2.1 Problem definition In most modern applications control algorithms will be implemented in a digital, time-discrete fashion. Thus we assume that we can describe our physical system with a set of (nonlinear) difference equations. The goal is to find control signals and transient trajectories between two states in a finite time N, which leads to a system of difference equations with boundary conditions. The problem can thus be stated as: x(k + 1) = f(x(k), u(k), k),

k = 0, 1, . . . N − 1

(1)

x(0) = a, x(N) = b. Here x(k) ∈ Rn is a n−dimensional state vector, u(k) ∈ Rm is the external control vector, and k is the (discrete) time index. The vector function f is a mapping of Rn × Rm × R → Rn . As stated, we seek an optimal control sequence u(k), k = 0, 1, . . . N −1, in the sense that the following weighted energy functional:    N−1  S R u(k) 1  T T u(k) , x(k) (2) J= 2 RT T x(k) k=0

is minimized. Here S, T and R are weighting matrices, T and S are supposed to be symmetric. Furthermore, S must be positive definite (thus non-singular) and T − RT S−1 R positive semi-definite. In a first approach, one might choose S = 1m×m , and R = T = 0. In this case, only the external control signal u(k) is used for the optimality criterion (minimal total drive energy). Note, that in special cases it could be even useful to allow for time- dependence in the above weighting matrices.

Optimal Targeting in Chaos Control

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143

2.2 Hamiltonian approach As in the time-continuous case (see e.g. [8]) we start defining the Lagrangian functional L = J+

N−1 

pT (k) · {f(x(k), u(k), k) − x(k + 1)},

(3)

k=0

where the components of p(k) are Lagrange-multipliers. The next step is to perform a Legendre-transform by defining the Hamiltonian function. In this context p(k) is considered as new, independent system state vector, and is referred to as conjugate variable or (generalized) impulse. The Hamiltonian function is then defined by H(x(k), p(k), u(k), k) ≡ H(k) = J + pT (k) · f(x(k), u(k), k),

(4)

Thus we have L=

N−1 

H(x(k), p(k), u(k), k) − pT (k) · x(k + 1).

(5)

k=0

Necessary conditions for an optimal trajectory are ∂H(k) ∂L = − p(k − 1) = 0 ∂x(k) ∂x(k) ∂H(k) ∂L = − x(k + 1) = 0 ∂p(k) ∂p(k) ∂H(k) ∂L = = 0. ∂u(k) ∂u(k)

(6a) (6b) (6c)

Since S is invertible, u(k) can readily be calculated from (6c) to yield u(k) = −S−1 pT (k) ·

∂f(x(k), u(k), k) − S−1 Rx(k). ∂u(k)

(7)

If the system is linear in the external control u(k), the right hand side does not depend on u(k) any more. In all other cases (7) can either be solved analytically or numerically. For the relaxation method described below, implicit differentiation can be applied to ∂f(x(k), u(k), k) yield the Jacobians Jf (k) = , thus avoiding the problem for explicitly ∂u(k) solving (7) for u(k). The result can be used in (6a) and (6b) to form the canonical 2n -dimensional system p(k − 1) = f1 (x(k), p(k), k),

k = 1, 2, . . . N

x(k + 1) = f2 (x(k), p(k), k),

k = 0, 1, . . . N − 1

x(0) = a, x(N) = b.

(8)

144 | U. Vogl Here the functions f1 and f2 are maps from R2n × R → Rn and are given as ∂H = ∂x(k)   pT (k) · f(x(k), u(k), k)

f1 (x(k), p(k), k) = ∂ ∂x(k)

∂f u=−S−1 pT · ∂u −S−1 Rx

(9) ,

 f2 (x(k), p(k), k) = f(x(k), u(k), k)u=−S−1 pT · ∂f −S−1 Rx . ∂u

In (8) we have in total 2nN equations and also 2nN unknowns: n · (N − 1) for the state trajectory (x(1), x(2), . . . x(N − 1)) and n(N + 1) unknown impulses (p(0), p(1), . . . p(N)). Note that this formulation of the canonical Hamiltonian equations (8) comprise one iteration forward and one backward in time.

2.3 Solution methods 2.3.1 Shooting If in (8) f1 is invertible in the variable p(k), we can define a totally forward directed difference equation system. To this end we define p ′ (k + 1) = p(k) and solve f1 for p ′ (k + 1) :     F1 (x(k), p ′ (k), k) p ′ (k + 1) z(k + 1) ≡ = = F(z(k), k) (10) x(k + 1) F2 (x(k), p ′ (k), k) Thus we end up with an iteration of a 2n-dimensional system z(k + 1) = F(z(k), k). To find a solution which fits the boundary conditions x(0) = a, and x(N) = b, we use the , ′start vector z0 = z(0) = pa0 , where p ′0 is an unknown variable. After N−fold iteration , p′ N F(N) (z0 ) := F(F(F(· . . . z(0), N − 2), N − 1), N) = x(N) we can read off the final state x(N). Thus the remaining problem is finding roots of the nonlinear equation in the unknown p′0   ′ (N) p 0 − b = 0. (11) (0 1 ) · F a For higher dimensional nonlinear systems however, solving this equation can become numerically very sensitive. After several iterations (N large), the function F(N) can become extremely complicated and we encounter a high sensitivity to initial conditions p′0 . This effect is well known for chaotic systems indeed. As an aside, it is worth noting that the flux of (10) is symplectic. This means, that the Jacobian JF (k) has, for all k, the property: JTF I JF = I,

(12)

Optimal Targeting in Chaos Control

where JF (k) and I are the following 2n × 2n matrices ⎛ ⎞ ∂F1 ∂F1  ⎜ ∂p ′ (k) ∂x(k) ⎟ ⎟, I = 0 JF (k) = ⎜ ⎝ ∂F2 ∂F2 ⎠ −1 ∂p ′ (k) ∂x(k)

|

145

 1 0

Note that I2 = −1n×n . Equation (12) implies for example, that det(JF ) = 1 for all k, thus it is always invertible. In fact, solving symplectic flux equations is well investigated [9].

2.3.2 Relaxation method A much more robust method for solving (8) is the following relaxation-based method. It is especially suitable for finding optimal transients in chaotic systems, where the above-mentioned sensitivity to initial conditions are encountered. In fact, we can extend the so called “targeting method” [10], allowing state transients with minimal energy (in fact, the energy will exponentially vanish with transient time) of the control signal within a given finite transient time N. We start out with linearizing (8) around a supposedly known start-trajectory: p(k − 1) + η(k − 1) = f1 (k) + J11 (k)ξ (k) + J12 (k)η(k), k = 1, 2, . . . N and

(13)

x(k + 1) + ξ (k + 1) = f2 (k) + J21 (k)ξ (k) + J22 (k)η(k), k = 0, 1, 2, . . . N − 1,

(14)

where the Jacobians are given here as ∂f1 , J12 (k) = ∂x(k) ∂f2 , J22 (k) = J21 (k) = ∂x(k)

J11 (k) =

∂f1 , ∂p(k) ∂f2 . ∂p(k)

(15)

Note, that with the indices chosen here, we have the desirable feature that the Jacobian is symmetric, i.e. JT (k) = J(k). The set of equations for the unknowns η(k) and ξ (k) can be casted in matrix form as follows: For k = 1, 2, 3, . . . N − 1 we have ⎛ ⎞  η(k − 1)    ⎟ p(k − 1) − f1 (k) −1 J11 (k) J12 (k) 0 ⎜ ⎜ ξ (k) ⎟ . ⎜ ⎟= T 0 J12 x(k + 1) − f2 (k) (k) J22 (k) −1 ⎝ η(k) ⎠ ξ (k + 1)

146 | U. Vogl

At the boundaries k = 0 and k = N we have ⎛ ⎞   ξ (0)   x(0) − a −1 0 0 ⎜ ⎟ and ⎝η(0)⎠ = T x(1) − f2 (0) (0) J22 (0) −1 J12 ξ (1) ⎛ ⎞   η(N − 1)   p(N − 1) − f1 (N) −1 J11 (N) J12 (N) ⎜ ⎟ ⎝ ξ (N) ⎠ = 0 −1 0 x(N) − b η(N)

(16)

(17)

respectively. Stacking the matrices above for k = 0, 1, . . . N into an almost block-diagonal matrix Z, the vector of unknowns to a vector ζ and the right hand vector (error vector) to e, we end up with a matrix equation of the form Z ζ = e.

(18)

Equation (8) is then solved by the following algorithm: 1. “Guess” some initial trajectory x(0) (k), p(0) (k). In most cases a very crude guess will be sufficient. 2. Calculate the Jacobians according to (15) and form the matrix Z(0) . Calculate the r.h.s error vector e(0) . 3. Solve the linear equation Z(0) ζ (0) = e(0) . Due to symmetry and the diagonal-dominated structure of Z this is in general no problem and can be done using standard linear algebra packages. 4. Calculate the relative residual error . . . . . . −1 . . . . . . ε(0) = .ζ (0) . .x(0) (k). + .p(0) (k). . (19) 2

5.

2

If the residual error is smaller than a given limit (e.g. 10 times machine precision), end the algorithm. Else, calculate the update parameter γ according to γ=

6.

2

ε(0) > β . ε(0) ≤ β

α 1

(20)

For practical implementation the values α = 0.4 and β = 0.05 have proved good performance. Update the state- and impulse vectors according to x(1) (k) = x(0) (k) + γξ (0) , (1)

(0)

p (k) = p (k) + γη

(0)

.

Then continue the iteration with step 2), using these updated vectors.

(21)

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This algorithm shows very robust behavior, even in complicated situations and for large N. Reducing the update gradient to a factor of α in (20, 21) if the error is large, avoids updating into the wrong direction, if the trajectory is still far away from a converged solution. Of course, the algorithm may be refined further, e.g. using a continuous behavior of γ(ε). However, numerical experiments have shown, that this does not improve convergence dramatically. Similar methods are used for solving differential boundary value equations, cf. e.g. [11].

2.3.3 Optimal targeting The true advantage of the relaxation algorithm emerges in chaos control applications. Here the initial trajectory can be used from a targeting procedure (x(0) (k) = xTarget (k) and u(0) (k) ≡ 0). We can choose p(0) (k) ≡ 0, letting the system do most of the control work by itself. The latter choice is motivated by (7), since we are interested in solutions with energy J almost zero (if the mixing weight R in (2) is small or zero). Small corrections necessary are then achieved by only very few iterations of the relaxation algorithm. To find xTarget (k) we iterate (1) with u(0) (k) ≡ 0 N− times, starting at a slightly displaced initial state x(0) (k) = a + ε a . The norm function: . . . . (22) ε b (ε a ) = .b − f (N) (a + ε a , 0, k). 2

measures the difference of x(0) (N) to the desired state b, and will typically show very sharp notches as function of ε a , see also Fig. 5. Other methods for finding targeting solutions use random-seed initial states, which are spread in the vicinity of a [13]. Iterating (1) with such a “notch-solution” as initial state x(0) = a + ε a,notch gives us a good candidate for xTarget (k). In fact, as shown in the application section, this extended targeting method yields trajectories with the property of exponentially decaying energy of the control signal as function of length N: N  . . .u(k).2 ∼ e−N . 2

(23)

k=0

3 Stabilizing trajectories Since most trajectories we are interested in are non-stable, implying (at least one) positive Lyapunov exponent, we need to stabilize them. This is especially important in the case of chaos control and targeting, where we want to connect UPOs embedded in the same attractor. Of course the optimal transients discussed above are also very

148 | U. Vogl close to that strange attractor, and in fact xTransient (k) converges to the attractor-set in the limit N → ∞. The method described below relies on the standard LQ (linear-quadratic) approach [12] and can be seen as a generalization of the OGY method [14] well known to the chaos-control community. The only difference to the standard textbook result is that we have to deal with a time-variant system here. We briefly outline this method here for completeness. The first step is linearizing (1) around the desired full trajectories (UPOs and transients) x0 (k) and u0 (k), resulting in a time-dependent linear difference equation: ξ (k + 1) = A(k)ξ (k) + B(k)η(k).

(24)

Here ξ (k) = x(k) − x0 (k) ∈ Rn is the n-dimensional state vector deviation, η(k) = u(k) − u0 (k) ∈ Rm is the external control vector deviation. Note that for UPOs we have u0 (k) ≡ 0. The time-dependent matrices A(k) and B(k) are the Jacobians of f with respect to x and u. We define a linear (L) control law η(k) = −K(k)ξ (k),

(25)

where the control coefficients K(k) to stabilize the system are found by minimizing the quadratic (Q) functional 1 T ξ (k)Uξ (k) + η(k)Vη(k), 2 ∞

I=

(26)

k=0

where U and V are positive definite weighting-matrices of states and control signal, respectively. A solution to this problem is provided by the Riccati backward iteration P(k) = U + AT (k)P(k + 1)A(k) − AT (k)P(k + 1)B(k)

−1 × V + BT (k)P(k + 1)B(k) BT (k)P(k + 1)A(k)

−1 K(k) = U + BT (k)P(k + 1)B(k) BT (k)P(k + 1)A(k)

(27) (28)

To compute e.g. the control coefficients for a UPO1 -transient-UPO2 scenario, we create the total trajectory by concatenating the parts from UPOs and transients. In addition, UPO2 will be appended several times, to ensure the iteration converges to a periodic solution. The backward iteration is then started with P(M) = U, where M is the last index in the full trajectory described above.

Optimal Targeting in Chaos Control

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4 Applications 4.1 Logistic map In the following we illustrate our method for a very simple yet nontrivial example, the controlled logistic map [15]: x(k + 1) = r x(k)(1 − x(k)) + u(k).

(29)

States x and control u are scalar signals and 1 < r ≤ 4 is a model parameter. Generally we have |u(k)| 1 and care must be taken that 0 < x(k) < 1 for all times. The logistic map is well known to exhibit chaotic behavior for r > r∞ ≈ 3.57. The functional (2) is chosen as 1 u(k)2 . 2 N

J := Ju (N) =

(30)

k=0

Equation (7) reduces to u(k) = −p(k) and the canonical equations (8) read here p(k − 1) = r p(k)(1 − 2x(k)) and

(31)

x(k + 1) = r x(k)(1 − x(k)) − p(k). The symmetric Jacobian (15) is  J(k) =

−2rp(k) r(1 − 2x(k))

r(1 − 2x(k)) −1

 .

(32)

Periodic orbits (UPOs) in the chaotic regime and optimal transients are easily found using the relaxation method. In Fig. 1 we display a rather long transient (N = 30) from a period p = 2 UPO to an (instable) stationary state. Since we are in the chaotic regime here (r = 3.7) the control signal has a very low amplitude (u x) (scale ∼ 10−7 ). It is also remarkable, that the action of the control signal u(k) is not immediately visible in the state signal x(k). It takes a delay of about 25 time steps, until a significant impact becomes evident in the graph.

150 | U. Vogl

Optimal Targeting Control from p= 2 −> p= 1, r= 3.7

x(k) 0.8 0.6 0.4

k 10 x 10

2

20

30

40

50

60

50

60

Control sequence u(k)

−7

u(k)

0 −2 −4

k 10

20

30

40

Fig. 1. Transition from a period p = 2 orbit to an (instable) stationary point with r = 3.7 (chaotic regime). State x(k) (upper) and control signal u(k) (lower graph). Note that the control signal is to be multiplied by10−7 .

The same setup has been repeated for the non-chaotic case with r = 3.1, cf. Fig. 2. The behavior of the control signal is quite different, and the scale of the control signal u(k) is much larger (∼ 5 · 10−3 ) here. Yet the desired target state x(N) = b ≈ 0.67 is exactly reached within finite time. Optimal Targeting Control from p= 2 −> p= 1, r= 3.1

x(k) 0.75 0.7 0.65 0.6

k 10 x 10

5

20

30

40

50

60

Control sequence u(k)

−3

u(k)

0 −5

k 10

20

30

40

50

60

Fig. 2. Transition from a period p = 2 orbit to an (instable) stationary point in a non-chaotic scenario (r = 3.1). State x(k) (upper) and control signal u(k) (lower graph). The the control signal is to be multiplied by 10−3 here.

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In Fig. 3 the r.m.s of the energy Ju (N)1/2 needed for the transient control signal in dependence of the transition length N is displayed on a logarithmic scale. Exponential decay of control energy as in (23) is clearly seen for the chaotic scenario r = 3.7. In the non-chaotic case r = 3.1 the control energy flattens out and increasing N does not yield any energy savings.

Total control energy Ju

Norm of control signal vs. transition length N

10

10

10

−2

r = 3.1 r = 3.7 −4

−6

5

10

15

20

25

30

Length of transient N Fig. 3. Norm of control signal vs. transition length N of transient control signal for the chaotic (r = 3.7) and the non-chaotic regime (r = 3.1).

4.2 Driven nonlinear pendulum 4.2.1 System description Next we consider a rigid pendulum hanging (with low friction) on a sled, which is driven by a stepper motor, cf. Fig. 4. Length of the pendulum is l ≈ 1.2m and sampling period is T0 = 1/(20Hz). Of course, the steps for the motor are interpolated (factor 213 ), which is achieved by an interpolation filter realized with a fix-point signal processor. The angle ϕ of the pendulum is measured by an high-resolution optical sensor, the corresponding signal ϕ(kT0 ) is returned to the signal processor, which in turn can drive the sled position s(kT0 ) via the motor. s Sled 0

φ l

m

Fig. 4. Experimental driven pendulum setup.

152 | U. Vogl

The time-continuous equation of motion for the system reads ¨ + ρ(ϕ) ˙ + ω2p sin(ϕ) = −ω2p cos(ϕ) s¨ , ϕ g

(33)

where s is the position of the sled (the acceleration s¨ will be used as the proper control signal u) and ω p is the eigenfrequency of the pendulum for small angles ϕ. ˙ resembles a nonlinear (yet continuous) friction function, which The function ρ(ϕ) ˙ The latter simulates combines a Stribeck type friction [16] with a quadratic term in ϕ. air friction and becomes increasingly important for orbits with fast rotating pendulum and high-speed transients. For a more complete description also the air friction induced by the speed of the sled can be considered, but this is neglected here for simplicity. Using standard techniques, this continuous equation of motion can be transformed into a system of difference equations compatible to (1): x(k + 1) = f(x1 (k), x2 (k), x3 (k), x4 (k), u(k), k) = ⎡ c 1 sin(πx1 ) + c2 cos(πx1 )(u + d) + x1 + x2 − ρ(x1 ) ⎢ c sin(πx ) + c cos(πx )(u + d) + x − ρ(x ) ⎢ 1 1 2 1 2 1 ⎢ ⎣ u + x3 x3 + x4

⎤ ⎥ ⎥ ⎥ ⎦

(34)

The state vector is n = 4-dimensional and the control signal u is scalar (m = 1). The state variables x(k) are connected to ϕ(k) and s(k) by x1 (k) = ϕ(k), x 2 (k) = ϕ(k − 1), x3 (k) = v(k) = v(k − 1) + u(k − 1) where v(k) = s(k) − s(k − 1) is the speed of the sled and x4 (k) = s(k) = s(k − 1) + v(k − 1) is its position (we let T 0 ≡ 1 to simplify notation). Basically, the third and fourth component of f doubly integrate the sled acceleration u(k) to the sled position s(k), which is the variable being directly accessible by the control output of the signal processor. With d(k) = d0 sin(2παk) we realize a periodically driven system with frequency α/T0 and amplitude d0 , rendering the difference equations non-autonomous. The nonlinear pendulum has only 2 degrees of ˙ which is not sufficient to create chaotic motion. With the external freedom (ϕ and ϕ), (periodic) force, however, we obtain a third degree of freedom. It is well known [17] that 3 degrees of freedom in non-linear systems suffice to produce chaotic motion. The parameters c1 , c2 and also the parameters describing the friction function ρ are fitted by experimental measurements.

4.2.2 Periodic orbits Calculation of (8) from (34) and applying the relaxation method for state transitions is straightforward. We can create all kinds of optimal transients, e.g. to drive the pendulum into a static inverse state (d0 ≡ 0, ϕ = π, ϕ˙ = 0) starting from the stable state (ϕ = 0, ϕ˙ = 0). It is more interesting, however, to drive the system into the chaotic

Optimal Targeting in Chaos Control

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regime, which becomes manifest for a wide range of frequencies α and amplitudes d0 . In this case the flux of the motion is part of a strange attractor, which hosts an abundant number of (unstable) periodic orbits. These orbits, having the property x(N + k) = x(k), can be found by fixing α = 1/N and searching for the roots of Φ defined by: Φ(x) = f (N) (x, 0, k) − x = 0 where u(k) ≡ 0

(35)

and f (N) (x) = f(f(f(. . . f(x, 0, k) . . .). Since x is in general higher dimensional (4-dimensional in this example) an analytical treatment of (35) is not possible. A practical method for finding UPOs is based on a monte-carlo method: randomly select a (physically reasonable) state x0 , which is then to be used as initial trial solution for a standard Gauss-Newton algorithm. If no convergence appears, randomly select another x0. To drive the system from the stable state (ϕ = 0, ϕ˙ = 0) into an UPO, simply select a =(0, 0, 0, 0)T as initial state and use any state on the UPO for b. The transient can then be found easily using the relaxation method.

4.2.3 Optimal targeting To connect two UPOs with (almost) no expense of energy we can proceed as follows: select a as state on UPO1 , b as state on UPO2 . Then use (22) to find a prototype transient (u ≡ 0) between the orbits. Here we vary the initial state a in the x1 −direction only, i.e. x(0) = a + ε a e1 with e1 the unit vector in x1 −direction. If the UPOs are contained in the same attractor, ε b will show distinct notches. An example of this function for a left/right alteration in rotation of the pendulum is shown in Fig. 5. As described before, min{ε b } will yield a starting trajectory x(0) (k) = xTarget (k), u(0) (k) ≡ 0 for the relaxation algorithm. Few iterations lead here to an optimized finite-time orbit-connecting control signal u(k).

10 log(εb) 0 −20

εb(εa)

−40 −60

εa

−80 −14

−12

−10

−8

−6

−4

−2

0

2

4

x 10

Fig. 5. Example of function (22) for the periodically driven pendulum. Note the logarithmic (dB) scale.

−5

154 | U. Vogl

angular speed for left−>right transition 0.1 0.05 0

ɸ(k)−ɸ(k−1)

−0.05

time k

−0.1 20

1

x 10

40

60

80

100

120

140

160

optimal targeting control signal u(k)

−4

u(k)

0

time k

−1 20

40

60

80

100

120

140

160

standard targeting control signal u(k) 0.02

u(k)

0.01 0 −0.01

time k

−0.02 20

40

60

80

100

120

140

160

Fig. 6. Comparison between standard targeting method and optimal targeting for a left/right rotation transition of the pendulum. Angular speed of pendulum (upper), optimal targeting control acceleration u(k) (middle), standard targeting of the same trajectory (lower). Note the different scales.

We give an example for an (almost) energy-free left/right rotation alteration. The additional acceleration u(k), needed using the standard targeting approach, is compared to our optimal targeting method (Fig. 6). Clearly, standard targeting works with two “wing beats of the butterfly”, the first is needed to leave UPO1 , the second to lock into the new UPO2 . In between u is exactly zero. In contrast, the energy of our optimal trajectory is distributed over the whole transient, which minimizes the overall energy needed. In this example we have an energy gain by a factor J u (uStandard )/J u (uOptimal ) ≈ 800 for the same length of the transient N = 70.

Optimal Targeting in Chaos Control

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This left/right rotation switch (among other UPO switches) of the pendulum can be watched on [7]. The additional acceleration u(k) needed to perform the change in rotation is so small, that it can not be observed by eye. In the video initial and finite UPOs are illuminated by green or blue LEDs, whereas during the transient phase red LEDs are activated. The sled is continuously and periodically driven with fixed frequency α and amplitude d(k).

5 Conclusion Based on quadratic optimality criteria we discussed a method, to actuate any time-discrete system dynamics into a defined state within a finite time. When applied to nonlinear systems containing one ore more chaotic attractors, we can find optimal “almost zero” control signals, which connect different periodic orbits contained in the same attractor. In our application to the chaotic pendulum we could gain a factor of about 800 in energy saving compared to the standard targeting procedure given the same transient time. Moreover, optimal transients can be found also outside the attractor. This allows for connecting non-chaotic to chaotic regimes, and of course any other transition, e.g. between static points. A time dependent LQ-approach to stabilize UPOs and transients has also been discussed. Application to an experimental pendulum setup has been lined out, and many of the UPOs and trajectories can be seen as video clip on [7]. Using the Hamiltonian approach to linear systems yields very efficient algorithms, which can be implemented under hard real time conditions. Work to apply this idea to fast laser scan-heads is currently in progress. Acknowledgment: The author would like to thank G. Mandel for fruitful discussions and all kinds of technical support.

Bibliography [1] [2] [3] [4] [5]

E. Schïll and H. G. Schuster. Handbook of Chaos Control. 2nd ed., Wiley-VCH, 2008. I. Harrington and J. E. S. Socolar. Design and robustness of delayed feedback controllers for discrete systems. Physical Review E, 69:056207, 2004. A. L. Fradkov and R. J. Evans. Control of Chaos: Methods and applications in engineering. Annual Review Control 29(1):33–56, 2005. P. L. Ramazza, U. Bortolozzo, and S. Boccaletti. Experimental synchronization of spatiotemporal chaos in nonlinear optics. Physical Review E, 73(3):6213, 2006. N. Sakamoto. Floquet-based chaos control for continuous systems with stability analysis. Physical Letters A, 356:316–323, 2006.

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[17]

R. Badii, E. Brun, M. Finardi, L. Flepp, R. Holzner, J. Parisi, C. Reyl, and J. Simonet. Progress in the analysis of experimental chaos through periodic orbits. Review Mod. Phys., 66:1389, 1994. U. Vogl. Videos of Chaotic Pendulum. HAW-AW, Amberg, Germany. [Online], 2011. Available: http://www.haw-aw.de/vogl/alias/targeting. V. I. Arnold. Mathematical Methods of Classical Mechanics. Springer-Verlag, 1989. C. D. Ahlbrandt and A. C. Peterson. Discrete Hamiltonian Systems. Kluwer Academic Publishers, 1996. T. Shinbrot, W. Ditto, C. Grebogi, E. Ott, M. Spano, and J. Yorke. Using the sensitive dependence of chaos (the “butterfly effect”) to direct trajectories in an experimental chaotic system. Physical Review Letters, 68:2863–2866, 1992. P. P. Eggleton. The evolution of low mass stars. Monthly Notices of the Royal Astronomical Society. 151:351–364, 1971. P. Lancester and L. Rodman. Algebraic Riccati Equations. Oxford Science Publications, Clarendon Press, Oxford, 1995. E. E. N. Macau and C. Grebogi. Driving trajectories in complex systems. Physical Review E 57:5337–5346, 1998. E. Ott, C. Grebogi, and J. A. Yorke. Controlling chaos. Physical Review Letters, 64(11):1196–1199, 1990. R. M. May. Simple Mathematical Models with very Complicated Dynamics. Nature, 26:459, 1976 R. Stribeck. The basic properties of sliding and rolling bearings, (Die wesentlichen Eigenschaften der Gleit- und Rollenlager). Zeitschrift des Vereins Deutscher Ingenieure, 36(46):1341–1348, 2002. E. Ott. Chaos in Dynamical Systems. Cambridge University Press, 1993.

Biography Ulrich Vogl studied theoretical physics at the University of Regensburg, Germany, and the University of Colorado at Boulder, USA. After receiving his Ph.D. in 1990, he was responsible for developing and implementing signal processing algorithms for GSM base stations with Siemens Mobile Networks in Munich. Since 1995 he is professor for Digital Signal Processing at the University of Applied Sciences OTH in Amberg, Germany. His current interests include signal processing aspects in non-linear control theory, as well as real-time implementations of the corresponding algorithms. He is member of the IEEE signal processing society.