Controller Design for Distributed Parameter Systems [1st ed.] 9783030349486, 9783030349493

Table of contents :
Front Matter ....Pages i-xii
Introduction (Kirsten A. Morris)....Pages 1-11
Infinite-Dimensional Systems Theory (Kirsten A. Morris)....Pages 13-69
Dynamics and Stability (Kirsten A. Morris)....Pages 71-102
Optimal Linear-Quadratic Controller Design (Kirsten A. Morris)....Pages 103-153
Disturbances (Kirsten A. Morris)....Pages 155-190
Estimation (Kirsten A. Morris)....Pages 191-218
Output Feedback Controller Design (Kirsten A. Morris)....Pages 219-252
Back Matter ....Pages 253-287

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Communications and Control Engineering

Kirsten A. Morris

Controller Design for Distributed Parameter Systems

Communications and Control Engineering Series Editors Alberto Isidori, Roma, Italy Jan H. van Schuppen, Amsterdam, The Netherlands Eduardo D. Sontag, Boston, USA Miroslav Krstic, La Jolla, USA

Communications and Control Engineering is a high-level academic monograph series publishing research in control and systems theory, control engineering and communications. It has worldwide distribution to engineers, researchers, educators (several of the titles in this series find use as advanced textbooks although that is not their primary purpose), and libraries. The series reflects the major technological and mathematical advances that have a great impact in the fields of communication and control. The range of areas to which control and systems theory is applied is broadening rapidly with particular growth being noticeable in the fields of finance and biologically-inspired control. Books in this series generally pull together many related research threads in more mature areas of the subject than the highly-specialised volumes of Lecture Notes in Control and Information Sciences. This series’s mathematical and control-theoretic emphasis is complemented by Advances in Industrial Control which provides a much more applied, engineering-oriented outlook. Indexed by SCOPUS and Engineering Index. Publishing Ethics: Researchers should conduct their research from research proposal to publication in line with best practices and codes of conduct of relevant professional bodies and/or national and international regulatory bodies. For more details on individual ethics matters please see: https://www.springer.com/gp/authors-editors/journal-author/journal-authorhelpdesk/publishing-ethics/14214

More information about this series at http://www.springer.com/series/61

Kirsten A. Morris

Controller Design for Distributed Parameter Systems

123

Kirsten A. Morris Department of Applied Mathematics University of Waterloo Waterloo, ON, Canada

ISSN 0178-5354 ISSN 2197-7119 (electronic) Communications and Control Engineering ISBN 978-3-030-34948-6 ISBN 978-3-030-34949-3 (eBook) https://doi.org/10.1007/978-3-030-34949-3 Mathematics Subject Classification (2010): 49M99, 93-01, 93C20, 93C95 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To Tristan and Gabriel.

Preface

Many systems have dynamics that depend on space as well as on time. Examples include diffusion, transmission lines, acoustic noise, and structural vibrations. Because the system state is distributed in space along a continuum, they are sometimes known as distributed parameter systems (DPS). This is different from lumped parameter systems, such as circuits, where there are a finite number of state variables and the dynamics only depend on time. Lumped parameter systems are modeled by ordinary differential equations and the state evolves on a finite-dimensional space, usually Rn . Because the dynamics of DPS depend on both time and space, the models are partial differential equations and the state evolves on an infinite-dimensional state space. For this reason, they are sometimes referred to as infinite-dimensional systems. The purpose of controller design for DPS is similar to that for any other system. Every controlled system must of course be stable. Beyond that, the goals are to improve the response in some well-defined manner, such as driving the system response to equilibrium optimally in defined sense, or tracking a desired reference signal. Another common goal is minimization of the system's response to disturbances. The related issue of estimation is of great interest for DPS since the distributed state generally needs to be calculated based on information from a finite number of sensors. There is now fairly extensive theory extending the common approaches to controller and estimation design for lumped systems to DPS. However, for most practical examples, controller synthesis based directly on the partial differential equation description is not feasible, since a closed-form expression for the solution is not available. Instead, a finite-dimensional approximation of the system is first obtained and controller design is based on this finite-dimensional approximation. The hope is that the controller has the desired effect on the original system. This is not always the case. However, conditions under which this practical approach to controller design works have been obtained and are described in this book. Another issue with control and estimator design for DPS is that the calculations often need to be done with high-order systems, particularly for problems in more than one space

vii

viii

Preface

dimension. Algorithms that are reliable for low-order systems sometimes cannot be used and an algorithm suitable for high-order systems is needed. Another issue for DPS is that because of the distributed nature of the system, there is often freedom on where to put the control hardware, the actuators and/or sensors. This freedom complicates the design process, but this freedom can be used to improve performance by careful placement of the hardware. This book is an introduction to the exciting and still growing subject of control and estimator design for distributed parameter systems. It is intended to provide a foundation and techniques for someone wishing to control or estimate DPS. There are rigorous statements of results, but no proofs. A number of examples are provided to illustrate the theory. To illustrate how the results can be used for systems in multiple space dimensions, some examples are in multiple space dimensions. The notes at the end of each chapter list references where proofs may be found. In addition, there are references to books for further reading on the topics covered in each chapter. I have made no attempt to provide an exhaustive bibliography of the extensive literature on control and systems theory for distributed parameter systems. Such an attempt would be doomed to failure. Because this book is an introduction, the class of models is restricted in several ways. First, all models are linear. Another restriction is to consider only systems with bounded control and observation. For many situations, such as control on the boundary of the region, idealized models lead to a state-space representation where the control operator B is unbounded on the state-space. More precisely, it is a bounded operator into a larger space than the state space. This can sometimes provide insight into the limitations of control but the analysis is more complicated than for systems with bounded control and observation. Including a more detailed model for the actuator often leads to a bounded operator. Similarly, modeling of a sensor often leads to a bounded observation operator. Reference is provided at the end of each chapter of generalizations to more general systems, when available. Systems modeled by delay differential equations also have a solution that evolves on an infinite-dimensional space. Although the physical situations are quite different, theory and controller/estimator design are quite similar to that of systems modeled by partial differential equations. However, the focus of this book is DPS, and all the examples in this book involve PDE models. A review of the control and estimation techniques used is provided. No previous knowledge of DPS is assumed. It is also not assumed that the reader has studied functional analysis. The required functional analysis is in Appendix A or covered as needed. This book arose from courses taught at the University of Waterloo to students with varied backgrounds from mathematics and engineering. If students have previous courses in functional analysis and control systems, most of the material in this book can be covered in one term. For students with no analysis background, a slower treatment of Chaps. 2 and 3 is needed, and perhaps also some of the material in the appendix. If the students have a good background in analysis and differential equations, but not in control, Chaps. 2 and 3 can be covered more quickly, but the later chapters, which cover control and estimator design, will need more attention.

Preface

ix

I thank the Canadian National Sciences and Engineering Research Council for ongoing financial support of my research, and the US Air force Office of Scientific Research for their support since 2010. Travel to learn from colleagues at conferences and at their institutions was funded by these agencies, and some of the material in this book arose from research funded by these agencies. I have learnt a lot from discussions and research with colleagues in both mathematics and engineering. In particular, I am indebted to M. Vidyasagar and Tom Banks for starting my education in distributed parameter systems and also to Ruth Curtain. I thank Amenda Chow, Yanni Guo, Chris Nielsen, Lassi Paunonen, and Hans Zwart for their very useful detailed comments and suggestions on a draft of this book. Waterloo, Canada

Kirsten A. Morris

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 11

2 Infinite-Dimensional Systems Theory 2.1 Riesz-Spectral Operators . . . . . . . 2.2 Lumer–Phillips Theorem . . . . . . . 2.3 Sesquilinear Forms . . . . . . . . . . . 2.4 Control . . . . . . . . . . . . . . . . . . . 2.5 Observation . . . . . . . . . . . . . . . . 2.6 Controllability and Observability . 2.7 Input/Output Maps . . . . . . . . . . . 2.8 Notes and References . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . .

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13 26 31 39 46 49 53 59 67 68

3 Dynamics and Stability . . . . . . . . 3.1 Sesquilinear Forms . . . . . . . . 3.2 Spectrum Determined Growth 3.3 Boundary Conditions . . . . . . 3.4 External Stability . . . . . . . . . 3.5 Notes and References . . . . . . References . . . . . . . . . . . . . . . . . .

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4 Optimal Linear-Quadratic Controller Design . . . . 4.1 Finite-Time Optimal Linear Quadratic Control . 4.2 Infinite-Time Optimal Linear Quadratic Control 4.3 Solving the Algebraic Riccati Equation . . . . . . 4.4 LQ-Optimal Actuator Location . . . . . . . . . . . . 4.5 Notes and References . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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103 104 107 123 130 150 152

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Contents

5 Disturbances . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Fixed Disturbance . . . . . . . . . . . . . . . . . 5.2 H2 -Optimal Actuator Location . . . . . . . 5.3 Unknown Disturbance . . . . . . . . . . . . . . 5.4 Solving H1 -Algebraic Riccati Equations 5.5 H1 -Optimal Actuator Location . . . . . . . 5.6 Notes and References . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

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155 156 160 170 177 183 189 190

6 Estimation . . . . . . . . . . . . . . . . . . . 6.1 Minimum Variance Estimation 6.2 Output Estimation . . . . . . . . . . 6.2.1 H2 -Output Estimation . 6.2.2 H1 -Output Estimation . 6.3 Optimal Sensor Location . . . . . 6.4 Notes and References . . . . . . . References . . . . . . . . . . . . . . . . . . .

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7 Output Feedback Controller Design . . . . 7.1 Dissipativity . . . . . . . . . . . . . . . . . . . 7.2 Final Value Theorem . . . . . . . . . . . . 7.3 Approximation of Control Systems . . 7.4 State-Space Based Controller Design . 7.5 H2 - and H1 -Controller Design . . . . . 7.5.1 H2 -Cost . . . . . . . . . . . . . . . . 7.5.2 H1 -Cost . . . . . . . . . . . . . . . . 7.6 Notes and References . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .

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Appendix: Functional Analysis and Operators . . . . . . . . . . . . . . . . . . . . . 253 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

Chapter 1

Introduction

In many systems the physical quantity of interest depends on several independent variables. For instance, the temperature of an object depends on both position and time, as do structural vibrations and the temperature and velocity of water in a lake. When the dynamics are affected by more than one independent variable, the equation modeling the dynamics involves partial derivatives and is thus a partial differential equation (PDE). A number of examples are illustrated in Figs. 1.1, 1.2, 1.3 and 1.4. Since the solution of the PDE is a physical quantity, such as temperature, that is distributed in space, these systems are often called distributed parameter systems (DPS). The state of a system modeled by an ordinary differential equation evolves on a finite-dimensional vector space, such as Rn . In contrast, the solution to a partial differential equation evolves on an infinite-dimensional space. For this reason, these systems are often called infinite-dimensional systems. The underlying distributed nature of the physical problem affects the dynamics and controller design. Although there are many similarities, the systems theory for infinite-dimensional systems differs in some important aspects from that for finite-dimensional systems. Some of the differences can be seen by looking at several relatively simple examples. Example 1.1 (Heat equation) The temperature z(x, t) at time t at position x from the left-hand end in a long thin bar of length  (Fig. 1.1), with constant thermal conductivity κ, mass density ρ, specific heat C p , initial temperature profile z(x, 0), is modelled by the partial differential equation C pρ

∂ 2 z(x, t) ∂z(x, t) =κ , ∂t ∂x 2 z(x, 0) = z 0 (x).

x ∈ (0, ),

© Springer Nature Switzerland AG 2020 K. A. Morris, Controller Design for Distributed Parameter Systems, Communications and Control Engineering, https://doi.org/10.1007/978-3-030-34949-3_1

t ≥0

(1.1)

1

2

1 Introduction

Fig. 1.1 Heat Flow in a Rod. The regulation of the temperature profile of a rod is the simplest example of a control system modelled by a partial differential equation

Fig. 1.2 Acoustic noise in a duct. A noise signal is produced by a loudspeaker placed at one end of the duct. In this photo a loudspeaker is mounted midway down the duct where it is used to control the noise signal. The pressure at the open end is measured by means of a microphone as shown in the photo. (Photo by courtesy of Prof. S. Lipshitz, University of Waterloo.) Fig. 1.3 A flexible beam is the simplest example of transverse vibrations in a structure. It has relevance to control of flexible robots and space structures. This photograph shows a beam controlled by means of a motor at one end. (Photo by courtesy of Prof. F. Golnaraghi, Simon Fraser University.)

This equation also models other types of diffusion, such as chemical diffusion and ∂2 neutron flux. In more than one space dimension, ∂x 2 becomes the Laplace operator ∇2. The boundary conditions at each end need to be specified. Suppose the temperature at both ends is fixed. Setting the temperature of the immersing medium to 0, the boundary conditions are

1 Introduction

3

Fig. 1.4 Vibrations in a plate occur due to various disturbances. In this apparatus the vibrations are controlled via the piezo-electric patches shown. (Photo by courtesy of Prof. M. Demetriou, Worcester Polytechnic Institute.)

Fig. 1.5 Lake Huron, Canada. Large lakes such as Lake Huron have temperature and currents that depend on depth as well as latitude and longitude. Determining these quantities based on a limited number of sensor readings is a challenging estimation problem

4

1 Introduction

z(0, t) = 0,

z(, t) = 0.

This equation may be solved using separation of variables. Define α2 = non-trivial solutions to

(1.2) κ . Cpρ

The

d 2 φ(x) dx2 φ(0) = 0, φ() = 0, λφ(x) = α2



are φk (x) =

x 2 αkπ 2 sin(kπ ), λk = −( ) k = 1, 2, . . .   

Because of the similarity to the linear algebra problem λφ = Aφ where φ is a vector and A a matrix, λk are called eigenvalues and φk are eigenfunctions. Defining   zk = z 0 (x)φk (x)d x, 0

the solution of (1.1), (1.2) is the Fourier series z(x, t) =

∞ 

z k φk (x)eλk t .

(1.3)

k=1

Since Reλk < 0 for all k, for all initial conditions z 0 satisfying  z 0  =



|z(x)| d x 2

 21

< ∞,

0

the solution converges to the zero function as time increases. Also, as the time t is taken smaller, for each initial condition z 0 the initial condition is recovered: lim z(x, t) = z 0 (x) t↓0

(where the arrow ↓ indicates only t > 0 is considered). In particular, consider an initial condition z 0 = φn . Then lim z(·, t) − z(·, 0)2 = lim eλn t φn (·) − φn (·)2 t↓0

t↓0

= lim 1 − e−( t↓0

= 0.

αnπ 2  ) t

(1.4)

1 Introduction

5

No single value of T will give a uniformly small error in z(·, t) − z(·, 0)2 for t < T for all initial conditions z 0 . This is different from the situation for ordinary differential equations where the solution is described by the matrix exponential.  The boundary conditions of partial differential equations are an important part of the model and affect the dynamics. Example 1.2 (Heat equation) (Example 1.1 cont.) Suppose instead of temperature being fixed at the ends as in (1.2), the rod is insulated at each end. Fourier’s law of heat conduction leads to boundary conditions κ

∂z (0, t) = 0, ∂x

κ

∂z (, t) = 0. ∂x

(1.5)

The solution to (1.1) with insulated boundary conditions (1.5) is still of the form (1.3) but now the eigenvalue problem has solution  φk (x) =

  x 2 αkπ 2 cos(kπ ), λk = − , k = 0, 1, . . . .   

The solution to the PDE (1.1) with boundary conditions (1.5) is z(x, t) =

∞ 

z 0 , φk φk (x)eλk t .

k=0

Because the first eigenvalue λ0 = 0,    2 lim z(x, t) = z 0 , φ0 = z 0 (x)d x t→∞  0 which in general is non-zero. This is different from the previous example (Example 1.1) where, for any initial condition, the temperature eventually converges to zero for large time. Changing the boundary conditions fundamentally alters the eigenvalues and hence the dynamics.  Example 1.3 (Wave equation) The pressure p(x, t) and velocity v(x, t) associated with acoustic waves in a duct of length , such as that shown in Fig. 1.2, are functions of space and time. Denoting the air density by ρ0 and the speed of sound by c, and neglecting non-linearities, leads to the following system of partial differential equations that describe the propagation of sound in a one-dimensional duct ∂v(x, t) 1 ∂ p(x, t) = −ρ0 , 0 < x < , t > 0 c2 ∂t ∂x ∂ p(x, t) ∂v(x, t) ρ0 =− , ∂t ∂x p(x, 0) = p0 (x), v(x, 0) = v0 (x).

(1.6) (1.7) (1.8)

6

1 Introduction

The system of two first-order equations (1.6), (1.7) are often combined to obtain the wave equation 2 ∂ 2 w(x, t) 2 ∂ w(x, t) = c , 0 μn > μn+1 . . . with corresponding modes (or eigenfunctions) φi (x). The deflections can be written in the form w(x, t) =

∞  i=1

for some coefficients wi (t).

wi (t)φi (x)

8

1 Introduction

In structures the higher frequency modes are heavily damped and the coefficients associated with them decay to zero quickly. This motivates assuming that the deflections w have the form N  wi (t)φi (x) w(x, t) = i=1

for some N. Consider then a simply supported Euler–Bernoulli beam with control localized around x = 0.2 and choose N = 3. The vibrations can be stabilized using linear-quadratic controller design, which yields a state feedback K . (Linear-quadratic controller design is discussed in detail in Chap. 4.) Suppose that the vibrations are measured at x = 0.9 and design an observer to estimate the state. The control signal in (1.12) is then u(t) = K zˆ (t) where zˆ (t) is the estimated state. This is a standard approach to controller design for finite-dimensional systems and the controller stabilizes the 3 mode model used in design (Fig. 1.6a). However, the actual system contains the influence of the higher modes. Figure 1.6b shows simulations with 6 modes used to simulate the system. Even though the initial condition is just the first mode, the higher modes are activated by the controller and the controlled system is not stable. (This example is discussed in more detail in Example 4.8.)  The phenomenon in the previous example, in which a controller designed using an approximation does not stabilize the original system, or even a higher order approximation, due to the effect of the neglected modes on the system dynamics, is often called spillover. A controller can activate higher modes that were not included in the controller design. This can lead to instability, particularly when these modes correspond to eigenvalues that are on or near the imaginary axis. Another important issue in controller and estimator design for DPS is the choice and location of the control hardware, the actuators and sensors. First, the choice of possible actuators and sensors is generally greater than for lumped systems. Secondly, there is generally a large number of possible locations for the hardware. Consider a flexible beam with localized control as in Example 1.4 but include damping in the model. Standard linear-quadratic controller design was done with the actuator placed at different locations. As shown in Fig. 1.7, appropriate location of the actuator can lead to better control of the vibrations, with smaller control effort. The actuator location can have a dramatic effect on controlled system performance. Similarly, sensor location can have an effect on estimator performance. To simplify the exposition, this book will consider only bounded B. Including a model for the actuator often changes a simple model with an unbounded actuator or sensor to one with bounded control or sensing. This is illustrated in Chap. 2. There are occasions where a model with boundary control and/or point sensing is most appropriate. One reason is that both the partial differential equation theory and the transfer function might be simpler, enabling a more complete analysis. Appropriate references at the end of a chapter will be given for extension to unbounded operators where available.

1 Introduction

9

0.05 0.04 0.03 0.02

y

0.01 0 -0.01 -0.02 -0.03 -0.04

0

5

10

15

t

(a) Simulation with 3 modes 0.06

0.04

y

0.02

0

-0.02

-0.04

-0.06

0

50

100

150

t

(b) Simulation with 6 modes Fig. 1.6 A linear-quadratic feedback controller K was designed for the beam in Example 1.4 using the first 3 modes (eigenfunctions). Simulation of the controlled system with (a) 3 modes and (b) 6 modes is shown. The initial condition in both cases is zero velocity, and deflection is the first mode. Figure (b) illustrates that the addition of additional modes to the model leads to instability in the controlled system

10

1 Introduction actuator location: 0.254 actuator location: 0.1

0.25 0.2 0.15

deflection

0.1 0.05 0 −0.05 −0.1 −0.15 −0.2 −0.25

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

time

(a) Controlled response 5 actuator location: 0.254 actuator location: 0.1

4 3 2

control

1 0 −1 −2 −3 −4 −5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

time

(b) Control signal Fig. 1.7 Control of vibrations in a simply supported beam. Choice of actuator location can lead to better vibration suppression with smaller control effort. (Reprinted from [5], © (2015), with permission from Elsevier.)

1 Introduction

11

There are rigorous statements of results, but instead of proofs, a number of examples are provided to illustrate the theory. The notes at the end of each chapter list references where the proofs may be found. There are references to books for further reading. Chapter 2 provides an introduction to systems theory for DPS. The required background on functional analysis in provided in Appendix A. Some readers may wish to read this appendix before starting Chap. 2; or alternatively refer to it as they work through Chap. 2. Chapter 3 covers dynamics and stability. Subsequent chapters cover the common controller and estimator design approaches, with a focus on using approximations in controller/estimator synthesis.

Notes and References Many books cover modelling of problems by PDEs, see for example, [1, 2]. The examples in this chapter are described in more detail in [3–6] and all are discussed later in this book. The term spillover for instability caused by neglecting higher order modes of a beam as in Example 1.4 was first used in [7]. How to avoid spillover is discussed in detail in later chapters of this book. There are many textbooks on linear finite-dimensional systems; one is [8]. An overview of the functional analysis used in this book is provided in the Appendix.

References 1. Guenther RB, Lee JW (1988) Partial differential equations of mathematical physics and integral equations. Prentice-Hall, Upper Saddle River 2. Towne DH (1988) Wave phenomena. Dover Publications, New York 3. Curtain RF, Morris KA (2009) Transfer functions of distributed parameter systems: a tutorial. Automatica 45(5):1101–1116 4. Morris KA (2010) Control of systems governed by partial differential equations. In: Levine WS (ed) Control handbook. CRC Press, Boca Raton 5. Yang SD, Morris KA (2015) Comparison of actuator placement criteria for control of structural vibrations. J Sound Vib 353:1–18 6. Zimmer BJ, Lipshitz SP, Morris KA, Vanderkooy J, Obasi EE (2003) An improved acoustic model for active noise control in a duct. ASME J Dyn Syst, Meas Control 125(3):382–395 7. Balas MJ (1978) Active control of flexible systems. J Optim Theory Appl 23(3):415–436 8. Morris KA (2001) An introduction to feedback controller design. Harcourt-Brace Ltd., San Diego

Chapter 2

Infinite-Dimensional Systems Theory

Every limit is a beginning as well as an ending. (George Eliot, Middlemarch)

Systems modelled by linear ordinary differential equations are generally written as a set of n first-order differential equations with solution taking values in Rn . Distributed parameter systems (DPS) are modelled by partial differential equations but can be written in a similar way. The main difference is that the state-space is no longer Rn , but an infinite-dimensional Hilbert space, and the matrix A is no longer a matrix, but an operator acting on this infinite-dimensional space. Consequently, systems with partial differential equation models are also called infinite-dimensional systems. The extension of the familiar state-space framework to distributed parameter systems is not entirely straightforward. Recall first the situation of a system of linear ordinary differential equations. Example 2.1 (Systems of ordinary differential equations) For a matrix A ∈ Rn×n and a vector of initial conditions z 0 ∈ Rn , consider the initial value problem z˙ (t) = Az(t), z(0) = z 0 ,

(2.1)

. In control systems theory, Rn is refereed to as the state-space. where z˙ (t) indicates dz dt Defining the matrix exponential S(t) = exp(At) =

∞  (At)k k=0

the solution z(t) ∈ Rn is

k!

,

z(t) = S(t)z 0 .

© Springer Nature Switzerland AG 2020 K. A. Morris, Controller Design for Distributed Parameter Systems, Communications and Control Engineering, https://doi.org/10.1007/978-3-030-34949-3_2

13

14

2 Infinite-Dimensional Systems Theory

Note that z(0) = S(0)z 0 , and z(t) − z(0) t S(t)z 0 − z 0 = lim t→0 t = AS(t)z 0 = Az(t).

z˙ (t) = lim

t→0

so this is indeed the solution to the differential equation. For any norm in Rn , there are real numbers M ≥ 1, and α determined by the real parts of the eigenvalues of A so that for all t, z(t) = S(t)z 0  ≤ Meαt z 0 . This implies that small changes in the initial condition lead to small changes in the solution z(t). Also, S(t) − I  =  =

∞  (At)k k=0 ∞  k=1

≤

k!

− I

(At)k  k!

∞  (At)k k=1

k!

= exp(At) − 1. Thus, z(t) − z(0) = S(t)z 0 − z 0  ≤ S(t) − I z 0  ≤ (exp(At) − 1)z 0 . The solution depends continuously on the initial condition with respect to time as well. From properties of the matrix exponential, S(0) = I, so setting t = 0 yields the initial condition, as required. Also S(t + s) = S(t)S(s)

2 Infinite-Dimensional Systems Theory

15

which implies that S(t + s)z 0 = S(t)z(s). This means that the solution obtained at time t + s with initial condition z(0) = z 0 equals that at time t + s starting with initial condition z(s) = S(s)z 0 , which is expected since the matrix A modeling the dynamics is independent of time.  Example 2.2 (Diffusion) Consider again the model of heat flow first introduced in 2 Example 1.1. For simplicity, introduce dimensionless variables x¯ = x , t¯ = α2t so that the problem is on the interval (0, 1) and the governing equation is, omitting the bar notation, ∂ 2 z(x, t) ∂z(x, t) = , x ∈ (0, 1), t ≥ 0, (2.2) ∂t ∂x 2 with boundary conditions z(0, t) = 0,

z(1, t) = 0,

and some initial temperature distribution z(x, 0) = z 0 (x), z 0 (x) ∈ L 2 (0, 1). This partial differential equation can be written in a form analogous to (2.1). The solution is not a vector in Rn , but a function in some Hilbert space of functions defined on (0, 1). Since the Fourier series solution is known to provide solutions on L 2 (0, 1), the linear space of square-integrable functions defined on (0, 1), this suggests defining an operator on Z = L 2 (0, 1). The norm on L 2 (0, 1) is (see A.12) 

1

z =

|z(x)| d x

 21

2

.

0

Letting z  indicate the derivative of a function with respect to the spatial variable x, consider the Hilbert space H2 (0, 1) = {z | z ∈ L 2 (0, 1), z  ∈ L 2 (0, 1), z  ∈ L 2 (0, 1)}, with inner product, letting ·, · indicate the usual L 2 (0, 1) inner product, w, z 2 = w, z + w  , z  + w  , z  . (This is an example of a Sobolev space; see Definition A.28.) Define also the set dom(A) = {z(x) ∈ H2 (0, 1) ; z(0) = z(1) = 0} and the operator A : dom(A) ⊂ L 2 (0, 1) → L 2 (0, 1);

16

2 Infinite-Dimensional Systems Theory

Az =

d2z . dx2

(Because A is defined in terms of a function that depends only on x, the ordinary, not partial, derivative notation is used.) The partial differential equation (2.2) can now be written in the concise operator form z˙ (t) = Az(t), z(0) = z 0 .

(2.3)

Notice that the boundary conditions are incorporated in the definition of dom(A). The domain of the operator A is an important part of its definition. It is known that the set of functions φn (x) =

√

2 sin(nπx), n = 1, 2 . . .

are an orthonormal basis for L 2 (0, 1); that is, φn , φn = 1, φn , φm = 0, n = m and every z 0 ∈ L 2 (0, 1) can be written z0 =

∞  z 0 , φn φn n=1

in the sense that lim z 0 −

N →∞

N  z 0 , φn φn  = 0.

(2.4)

n=1

The partial differential equation (2.2) or equivalently (2.3) has the Fourier series solution ∞  2 2 z(x, t) = z 0 , φn φn e−n π t . (2.5) n=1

The right-hand side of (2.5) defines a linear operator S(t) from the initial condition z 0 to the solution z(·, t) : S(t)z 0 =

∞  n=1

From Parseval’s Equality (A.10)

z 0 , φn φn e−n

π t

2 2

.

(2.6)

2 Infinite-Dimensional Systems Theory

17

S(t)z 0 2 =

∞ 

| z 0 , φn e−n

π t 2

2 2

|

(2.7)

| z 0 , φn |2

(2.8)

n=1

≤ e−2π

2

t

∞  n=1

= e−2π t z 0 2 . 2

(2.9)

Thus, S(t) is a bounded operator from L 2 (0, 1) to L 2 (0, 1) for each t. Some calculations show ∞  2 2 2 2 S(t + s)z 0 = z 0 , φn φn e−n π t e−n π s n=1 ∞  2 2 2 2 = z 0 e−n π s , φn φn e−n π t n=1 ∞  ∞  2 2 2 2 = z 0 , φm φm e−n π s , φn φn e−n π t n=1 m=1 ∞  2 2 = S(s)z 0 , φn φn e−n π t n=1

= S(t)S(s)z 0 . It will now be shown that also S(t)z 0 → z 0 as t → 0 for each z 0 . Since {φn }∞ n=1 is a basis for L 2 (0, 1), for any  > 0 there is N so that ∞ 

| z 0 , φn |2
0 consider the sequence zn =

|x| ≥

0

1 − n|x| |x|
0, real ω such that for all positive integers k and real s > ω, M (s I − A)−k  ≤ . (2.15) (s − ω)k In this case, S(t) ≤ Meωt . It is difficult to verify directly condition (2.15) of the Hille–Yosida theorem. Three common approaches to establishing that an operator A generates a C0 semigroup, and hence that the partial differential equation is well-posed, are given here. For some operators, the set of eigenfunctions form a basis for a Hilbert space, but perhaps not an orthogonal basis. For these operators, known as Riesz-spectral operators, a generalization of the Fourier series shows that the associated partial differential equation is well-posed. Another approach, the Lumer–Phillips Theorem does not require knowledge of the eigenfunctions and is useful in many applications. The third approach discussed here, closely related to the Lumer–Phillips Theorem, and useful in many applications, is to use sesquilinear forms. The use of sesquilinear forms is explained in a subsequent section. Many systems can be shown to be wellposed using more than one method. The following theorem can also be useful in establishing well-posedness of a model closely related to another model that is known to be well-posed. Theorem 2.18 Suppose that A with dom(A) generates a C0 -semigroup S(t) on Z satisfying S(t) ≤ Meωt for some real constants M, ω. Then for any B ∈ B(Z, Z), the operator A B = A + B, with domain dom(A B ) = dom(A) generates a C0 semigroup S B (t) on Z with bound S B (t) ≤ Me(ω+MB)t . If B = αI for some α ∈ R, the resulting semigroup satisfies S B (t) ≤ Me(ω+α)t .

26

2 Infinite-Dimensional Systems Theory

2.1 Riesz-Spectral Operators Definition 2.19 A set {φn }∞ n=1 ⊂ Z is a Riesz basis for a Hilbert space Z if 1. span{φn } = Z, 2. There exist constants m > 0, M > 0 such that for any positive integer N and scalars αn , n = 1 . . . N , m

N  n=1

|αn |2 ≤ 

N 

αn φn 2 ≤ M

n=1

N 

|αn |2 .

n=1

Lemma 2.20 Let {φn }∞ n=1 be a Riesz basis for a Hilbert space Z. Then there exists another sequence {ψn }∞ n=1 biorthogonal to {φn }; that is, φm , ψn = δnm . With the constants m, M from Definition 2.19, for every z ∈ Z

∞ 2 2 2 1. m ∞ n=1 | z, ψn | ≤ z ≤ M n=1 | z, ψn | , 2. z can be written uniquely ∞  z, ψn φn , z= n=1

3. {ψn }∞ n=1 is also a Riesz basis for Z:

1 ∞ 2 – M n=1 | z, φn |2 ≤ z2 ≤ m1 ∞ n=1 | z, φn | , – z can be written uniquely ∞  z, φn ψn . z= n=1

Clearly every orthonormal basis is a Riesz basis, but this class also includes bases that are related to an orthonormal basis by a bounded transformation with bounded inverse. Theorem 2.21 The set {φn }∞ n=1 is a Riesz basis for Z if and only if for an orthonormal basis {en } for Z there is a bounded linear operator T with bounded inverse such that T en = φn . Definition 2.22 The set of points λ ∈ C for which (λI − A)−1 exists and is a bounded operator on Z is called the resolvent set of A, ρ(A). All other λ are said to be in the spectrum of A, σ(A). If λI − A is not one-to-one, so that a non-trivial solution to λφ = Aφ exists, then λ is an eigenvalue and φ is an eigenfunction. In finite dimensions if (λI − A) is one-to-one then the inverse is a bounded operator on the entire space so that the spectrum contains only eigenvalues. In infinite dimensions, (λI − A) may be one-to-one so an inverse exists, but (s I − A)−1 may be unbounded and/or not defined on all of Z. Thus, the spectrum may contain points that are not eigenvalues. This is discussed more in Chap. 3.

2.1 Riesz-Spectral Operators

27

Definition 2.23 A linear closed operator A with eigenvalues λn and eigenfunctions φn on a Hilbert space Z is a Riesz-spectral operator if 1. each eigenvalue has multiplicity one, 2. for every λn , λm with n = m there is r > 0 so that the disc Br = {s ∈ C| |s − λn | < r } / Br , has boundary in ρ(A) and λm ∈ 3. its eigenfunctions {φn }∞ n=1 form a Riesz basis for Z. Theorem 2.24 Let A be a Riesz-spectral operator on a Hilbert space Z with eigen∞ values {λn }∞ n=1 and eigenfunctions {φn }n=1 . Then ∗ – The eigenvectors {ψn }∞ n=1 of the adjoint of A, A , (Definition A.55) corresponding ∞ to eigenvectors {λ¯ n }n=1 can be scaled so φn , ψm = δn,m and also they satisfy the assumptions of Lemma 2.20; – For z ∈ dom(A), A can be written

Az =

∞ 

λn z, ψn φn

n=1

dom(A) = {z ∈ Z|

∞ 

|λn z, ψn |2 < ∞}.

n=1

Theorem 2.25 A Riesz-spectral operator A on Z is the generator of a C0 -semigroup S(t) on Z if and only if supn Re λn < ∞. In this case, this semigroup can be written S(t)z =

∞ 

eλn t z, ψn φn , z ∈ Z.

n=1

The operator (2.6) in Example 2.2 is a simple example of Theorem 2.25 where ψn = φn . The following example is less straightforward. Example 2.26 (Damped Euler–Bernoulli Beam) The analysis of beam vibrations is useful for applications such as flexible links in robots and other long thin structures but also for understanding the dynamics of more complex structures. A simple beam is illustrated in Fig. 2.2. Consider the Euler–Bernoulli model for the transverse deflections in a beam and let w(x, t) denote the deflection of the beam from its rigid body

Fig. 2.2 Simply supported beam with length 1 and an applied force u.

u =0

=1

28

2 Infinite-Dimensional Systems Theory

motion at time t and position x. In the Kelvin–Voigt damping model, the moment is, for positive physical parameters E > 0, cd ≥ 0, m(x, t) = E

∂ 3 w(x, t) ∂ 2 w(x, t) + c . d ∂x 2 ∂x 2 ∂t

If there is no control u and viscous damping with parameter cv ≥ 0 is also included, the dynamics of the deflections are governed by the PDE ∂w ∂ 2 m ∂2w + + c = 0, x ∈ (0, 1). v ∂t 2 ∂t ∂x 2 Suppose that the beam is simply supported so that deflections w and moment m are zero at each end. This implies boundary conditions w(0, t) = 0, m(0, t) = 0, w(1, t) = 0, m(1, t) = 0. Define

  Hs (0, 1) = w ∈ H2 (0, 1); w(0) = w(1) = 0 .

The usual norm on H2 (0, 1) is  w2H2

1

=





1

|w (x)| d x + 2

0





1

|w (x)| d x + 2

0

|w(x)|2 d x.

0

For any w ∈ Hs (0, 1),  w(x) =

x

0



s





1

w (r )dr ds + ao x, ao = −

0

0



s

w  (r )dr ds.

(2.16)

0

Thus, the Cauchy–Schwarz Inequality can be used to show that there are constants c1 > 0, c2 > 0 such that for all w ∈ Hs (0, 1), 

1

 |w(x)|2 d x ≤ c2

0

1



|w  (x)|2 d x,

0

1

|w  (x)|2 d x ≤ c1

0

This implies that

and since trivially

 w2H2 ≥

0

1

1 0

 w2H2 ≤ (1 + c1 + c2 )



1

|w  (x)|2 d x

0

|w  (x)|2 d x,

|w  (x)|2 d x.

2.1 Riesz-Spectral Operators

29





1

|w  (x)|2 d x

0

defines a norm on Hs (0, 1) equivalent to the usual norm on H2 (0, 1) (Definition A.20). Define the inner product on Hs (0, 1) to be the simpler 

1

w1 , w2 Hs = E 0

w1 (x)w2 (x)d x,

of α. where α indicates the complex  conjugate  Define the state to be z = w w˙ with state space Z = Hs (0, 1) × L 2 (0, 1). The quantity z2Z is proportional to the sum of the potential and kinetic energies. Rewrite the PDE in state-space form as z˙ (t) = Az(t) where A

    d 2 w(x) d 2 v(x) v w , m(x) = E + cd = 2 d 2 v − d x 2 m − cv v dx dx2

dom(A) = {(w, v) ∈ Hs (0, 1) × Hs (0, 1) | m ∈ Hs (0, 1)} . Let φ ∈ Hs (0, 1), μ ∈ C, be the solutions to d 4φ = μφ, φ(0) = φ (0) = φ(1) = φ (1) = 0. dx4 The non-trivial solutions φn , μn are, for integers n = ±1, 2 . . . and arbitrary constants an , μn = (nπ)4 , φn (x) = an sin(nπx). Now consider λψ = Aψ, ψ ∈ dom(A)   or equivalently, writing ψ = ψ1 ψ2 , λψ1 = ψ2 , ψ1 (0) = ψ1 (1) = 0, λψ2 = −Eψ1I V − cd ψ2I V − cv ψ2 , ψ2 (0) = ψ2 (1) = 0. The non-trivial solutions are, for n = ±1, ±2, . . . and arbitrary bn ,  ψn (x) = bn where λ±n are the solutions to

1 φ (x) λn n

φn (x)

 (2.17)

30

2 Infinite-Dimensional Systems Theory

λ2n + (cd μn + cv )λn + Eμn = 0.

(2.18)

The constants an , bn are arbitrary. For simplicity, for now set an = 1, bn = 1. It is well known that {φn } = {sin nπx}, n = 1, 2, . . . form an orthogonal basis for L 2 (0, 1). They are also orthogonal in the Hs norm. Note that Hs (0, 1) ⊂ L 2 (0, 1). Also, for any w ∈ Hs (0, 1), w, φn Hs = −(nπ)2 w  , φn L 2 . Since {φn } are a basis for L 2 (0, 1), if w, φn Hs = 0 for all n, w  is identically zero (Theorem A.16). But w ∈ Hs (0, 1), and so w(0) = w(1) = 0 which implies that w is the zero function 0. Thus, {φn } is an orthogonal basis for Hs (0, 1). It will now be shown that {ψn }, n = −∞ . . . ∞, is a orthogonal basis for Z = Hs (0, 1) × L 2 (0, 1). It is straightforward to show that {ψn } is an orthogonal set. If for some z = (w, v) ∈ Z, z, ψn = 0 for all n then for all n = 1, 2, . . . 0 = z, λn ψn + λ−n ψ−n

    w 0

= , v (λn − λ−n ) sin(nπx) = v, (λn − λ−n ) sin(nπx) L 2 (0,1) and so v = 0. Similarly, 0 = z, ψn + ψ−n

   1 ( − w = , λn v = w, (

 1 ) sin(nπx) λ−n 0

1 1 − ) sin(nπx) Hs λn λ−n

and so also w = 0. Thus, if z, ψn = 0 for all ψn , z is the zero element. This implies that {ψn }, n = ±1, ±2, . . . is a basis for Z (Theorem A.16). By choosing the constant bn appropriately in (2.17), it will be an orthonormal basis. From (2.18), the eigenvalues are λ±n =

  1 −(cd μn + cv ) ± (cd μn + cv )2 − 4Eμn . 2

As long as E >> cd , and cv is small, as is the case in applications, for small values of |n|, the eigenvalues λn are in complex conjugate pairs. For large n, if cd = 0, they become real and there is a limit point at − cEd . This point is not an eigenvalue. Since for any eigenvalues λn , λm , it is possible to create a disc around λn that doesn’t include λm , the eigenvalues satisfy conditions (1) and (2) of Definition 2.23 and A is a Riesz-spectral operator.  For all n, Reλn ≤ 0 and so A generates a C0 -semigroup on Z.

2.2 Lumer–Phillips Theorem

31

2.2 Lumer–Phillips Theorem Many partial differential equations can be shown to be well-posed using the Lumer– Phillips Theorem, stated below. The sufficient conditions in the corollary to this theorem are particularly useful in that they are generally more easily verifiable than those in the Hille–Yosida Theorem. Definition 2.27 The C0 -semigroup S(t), t ≥ 0 is a contraction semigroup, or a contraction, if S(t) ≤ 1 for all t ≥ 0. Theorem 2.28 (Lumer–Phillips Theorem) Let A : dom(A) ⊂ Z → Z be a closed, densely defined operator on a Hilbert space Z. The operator A generates a contraction semigroup on Z if and only if for all real ω > 0 (ω I − A)z) ≥ ωz, for all z ∈ dom(A), (ω I − A∗ )z) ≥ ωz, for all z ∈ dom(A∗ ). Corollary 2.29 The operator A : dom(A) ⊂ Z → Z on a Hilbert space Z generates a contraction if it is closed, densely defined and Re Az, z ≤ 0, for all z ∈ dom(A), Re A∗ z, z ≤ 0, for all z ∈ dom(A∗ ). One advantage of working with the conditions in the Lumer–Phillips Theorem over the spectral approach of Theorem 2.25 is that it is not necessary to know the eigenvalues or eigenvectors of the generator. Example 2.30 (First-order PDE) A simple PDE is ∂z ∂z = , z(1, t) = 0, −1 < x < 1. ∂t ∂x

(2.19)

It was shown in Example 2.14 that the operator Az =

dz dx

with domain dom(A) = {z ∈ L 2 (−1, 1)|

dz ∈ L 2 (−1, 1), z(1) = 0} dx

is a closed operator on the Hilbert space L 2 (−1, 1). From Theorem A.30, C 1 ([−1, 1]) is dense in L 2 (−1, 1), and so H1 (−1, 1) is dense in L 2 (−1, 1). Changing the value of function in L 2 (−1, 1) at a finite number of points does not affect the L 2 -norm.

32

2 Infinite-Dimensional Systems Theory

(Functions in L 2 (−1, 1) that differ only at a finite number of points are regarded as equivalent in L 2 (−1, 1).) Thus, dom(A) is dense in L 2 (−1, 1). For any z ∈ dom(A) Re Az, z =

1 ( Az, z + z, Az ) = −z(−1)2 ≤ 0. 2

Similarly (see Example A.56 for the calculation of A∗ ) Re A∗ z, z =

 1 ∗ A z, z + z, A∗ z = −z(1)2 ≤ 0. 2

Thus, from the corollary to the Lumer–Phillips Theorem (Corollary 2.29), A generates a C0 -semigroup S(t) on L 2 (−1, 1) with S(t) ≤ 1. Example 2.10 suggests that the semigroup generated by A is  (S(t)z 0 )(x) =

z 0 (t + x) 0

−1 < x + t < 1 . x +t ≥1 

This conjecture can be verified.

Example 2.31 (Flexible beam) As in Example 2.26, consider a simply supported flexible beam such as that shown in Fig. 2.2. In this example, damping will be neglected to simplify the calculations, so cd = 0, cv = 0. Let w(x, t) denote the deflection of the beam from its rigid body motion at time t and position x. Use of the Euler–Bernoulli model for the transverse vibrations in a beam of length  leads to the partial differential equation ∂4w ∂2w + E 4 = 0, 2 ∂t ∂x

t ≥ 0, 0 < x < ,

where E > 0 is a material constant. For a simply supported beam the boundary conditions are w(0, t) = 0, wx x (0, t) = 0, w(, t) = 0, wx x (, t) = 0.

(2.20)

This system is second-order in time, so to put the system into standard state-space form, a product state space is needed. One approach is to define the state as z(t) = w(·, t) v(·, t) where v denotes the velocity. The total energy of the beam is the sum of the potential and kinetic energies, that is 1 2

   2    d w 2 2 E ( 2 ) dx + v(x) d x . dx 0 0

Therefore, as in Example 2.26, define Hs (0, ) = {w ∈ H2 (0, ), w(0) = 0, w() = 0}

2.2 Lumer–Phillips Theorem

33

with inner product





w1 , w2 s = 0

w1 (x)w2 (x)d x.

Define the state-space Z = Hs (0, ) × L 2 (0, ). A state-space formulation of the above partial differential equation is z˙ (t) = Az(t), where

⎡ A=⎣

with domain

0 −E

d4 dx4

I

⎤ ⎦,

0

dom(A) = {(w, v) ∈ (Hs (0, ) ∩ H4 (0, )) × Hs (0, )}.

The operator A with this domain is closed and densely defined on Z. Integrating by parts shows that for all z ∈ dom(A), Re Az, z = 0. Using integration by parts shows that that A∗ = −A with dom(A∗ ) = dom(A) and so Re A∗ y, y = 0. Therefore, by the Corollary to the Lumer–Phillips Theorem, A generates a contraction semigroup on Z. If the damping is non-zero, the conclusion is identical, except calculation of the adjoint operator A∗ is more complicated and A∗ = −A. ˙ the Lumer–Phillips Theorem can also Alternatively, defining m = Ew , v = w, be used to show that this model is well-posed on L 2 (0, ) × L 2 (0, ) with the state   z= mw . The fact that A generates a semigroup on Z was also shown in Example 2.26 by showing that the generator was a Riesz-spectral operator with eigenvalues all in the left-hand-plane.  A particular PDE is not well-posed on every Hilbert space. As the following example indicates, well-posedness is dependent upon the choice of state and statespace. The energy in the system is often a guide to the appropriate state space. Example 2.32 (Wave equation, Example 1.3 cont.) Consider the wave equation introduced in Example 1.3, 2 ∂2w 2∂ w = c , 0 < x < , (2.21) ∂t 2 ∂x 2 w(x, 0) = w0 (x), w(x, ˙ 0) = v0 (x).

34

2 Infinite-Dimensional Systems Theory

Consider boundary conditions ∂w ∂w (0, t) = 0, (1, t) = 0. ∂x ∂x

(2.22)

For pressure waves in a duct, since pressure p = −ρ0 c2 ∂w (see Example 1.3) this ∂x corresponds physically to the duct being open at each end. For a string, these boundary conditions mean that the strain at both ends is zero, which corresponds to clamping of the ends. For simplicity, normalize the variables by defining x¯ = 1 x and t¯ = ct . Dropping the bar notation, the equation becomes ∂2w ∂2w = , 0 < x < 1. 2 ∂t ∂x 2

(2.23)

Put the system into state space form by defining z=

  w . w˙

The obvious choice of state space is Z0 = L 2 (0, 1) × L 2 (0, 1) with the operator  0 I . A = d2 0 dx2 

(2.24)

Define the domain so that the boundary conditions are satisfied, and so that A : dom(A) ⊂ Z0 → Z0 : dom(A) = {(w, v) ∈ H2 (0, 1) × L 2 (0, 1); w  (0) = 0, w  (1) = 0}.

(2.25)

The PDE (2.23) is then written as z˙ (t) = Az(t), z(x, 0) =

  w0 (x) . v0 (x)

To investigate whether the solution to this equation defines a semigroup, for integer n, consider the initial condition w0 (x) =

√ 2 cos(nπx), v0 (x) = 0, 0 < x < 1.

The norm of the initial condition is √ z(x, 0) L 2 (0,1)×L 2 (0,1) = w0  L 2 (0,1) =  2 cos(nπx) L 2 (0,1) = 1.

2.2 Lumer–Phillips Theorem

35

For each n, the solution to the PDE (2.23) with such an initial condition is z n = T  wn vn where wn (x, t) =

√

√ 2 cos(nπt) cos(nπx), vn (x, t) = − 2nπ sin(nπt) cos(nπx).

At time t = 21 , and odd values of n, the solution has norm  z n (x, t) L 2 (0,1)×L 2 (0,1) = 

wn (·, 21 )



 L 2 (0,1)×L 2 (0,1) vn (·, 21 ) √ nπ = nπ 2 sin( ) cos(nπx) L 2 (0,1) 2 = nπ.

There is no constant M so that for all n z n (·, 21 ) ≤ M. z n (·, 0) The solution operator is not bounded from the initial condition to the state and so A doesn’t generate a C0 -semigroup on Z0 = L 2 (0, 1) × L 2 (0, 1). The problem is not with the wave equation, but with the choice of state space. An appropriate choice for this example, and for many other examples arising from applications, can be found by considering the energy in the system. The total energy at any time t is the sum of the potential and kinetic energies. Since the various parameters are normalized to 1, the total system energy is E(t) =

1 2



1

p(x, t)2 d x +

0

1 2



1

v(x, t)2 d x.

(2.26)

0

where p indicates pressure and v velocity. Since p = − ∂w , (2.26) suggests the state ∂x space Z = H1 (0, 1) × L 2 (0, 1). However, only derivatives of w appear in the PDE and boundary conditions. This means that shifting w by a constant does not affect the solution. The Lumer–Phillips Theorem cannot be used to show that A generates a C0 semigroup on this state space. Construct a state space where functions that differ by a constant are regarded as the same function. Define H¯ 1 (0, 1) = {w ∈ H1 (0, 1)} with inner product



1

w1 , w2 = 0

w1 (x)w2 (x)d x.

36

2 Infinite-Dimensional Systems Theory

This yields a norm if the set of functions with the same derivative are regarded as equivalent. The norm on H¯ 1 (0, 1) × L 2 (0, 1) is exactly 2E(t) and the generator is (2.24) but with domain dom(A) = {(w, v)|w ∈ (H2 (0, 1) ∩ H¯ 1 (0, 1)), v ∈ H¯ 1 (0, 1), w (0) = w (1) = 0}.

(2.27) With this choice of state space, the Lumer–Phillips Theorem (Theorem 2.28) can be used to show that A, regarded as an operator on H¯ 1 (0, 1) × L 2 (0, 1) with domain (2.27), generates a contraction semigroup on Z. Another approach is to note that the energy is a function of the pressure and velocity. A natural choice of state variable and state space is thus   p z˜ = , Z˜0 = L 2 (0, 1) × L 2 (0, 1). v The PDE is written in the original first order form (1.6), (1.7), ∂v(x, t) ∂ p(x, t) =− , ∂t ∂x ∂v(x, t) ∂ p(x, t) =− ∂t ∂x and the boundary conditions (2.22) written p(0, t) = 0,

p(1, t) = 0.

 T With state variable z˜ = p v , and state space Z0 , the generator is      ∂ 0 − ∂x p p , A˜ = ∂ v v − ∂x 0 ˜ = {( p, v) ∈ H1 (0, 1) × H1 (0, 1), p(0) = p(1) = 0}. dom( A) The Lumer–Phillips Theorem (Theorem 2.28) can be used to show generation of a contraction semigroup.  Example 2.33 (Wave equation, Example 2.32 cont.) Consider again the normalized wave equation (2.23) ∂2w ∂2w = , 0 < x < 1. ∂t 2 ∂x 2 Suppose instead of the boundary conditions (2.22), the deflections are fixed so that w(0, t) = 0, w(1, t) = 0.

2.2 Lumer–Phillips Theorem

37

With the state space H1 (0, 1) × L 2 (0, 1) A is defined by (2.24) as for the first set of boundary conditions but now dom(A) = {(w, v) ∈ H2 (0, 1) × H1 (0, 1), w(0) = w(1) = 0}.

(2.28)

Because of the boundary conditions, the domain of the generator A contains only functions that are 0 at x = 0, x = 1. Recall that the inner product on H1 (0, 1) is  w, v H1 =

1

 w(x)v(x)d x +

0

1

w  (x)v  (x)d x.

0

The difficulty is that if w(0) = 0, the approximating function will have a derivative that gets very large near 0 and so the H1 -norm becomes large. It is not possible to approximate an arbitrary element of H1 (0, 1) in the H1 (0, 1) norm by a function with w(0) = 0, w(1) = 0. Thus, the domain (2.28) is not dense in H1 (0, 1) × L 2 (0, 1). This means that A doesn’t generate at C0 -semigroup on this space. Instead, consider the state space H01 (0, 1) × L 2 (0, 1) where H01 (0, 1) = {z ∈ H1 (0, 1), z(0) = 0, z(1) = 0}. For every z ∈ H01 (0, 1),



x

z(x) =

z  (s)ds,

0

and so



1



1

|z(x)| d x ≤ 2

0

|z  (x)|2 d x.

0

Thus, 

1

|z  (x)|2 d x ≤

0

This implies that



1

|z  (x)|2 d x +



0

1



0



1

1

|z(x)|2 d x ≤ 2

|z  (x)|2 d x.

(2.29)

0

|z  (x)|2 d x

0

defines a norm on H01 (0, 1) equivalent to the usual norm on H1 (0, 1). Use then the simpler inner product  1 w  (x)v  (x)d x w, v 1 = 0

on H01 (0, 1). This simpler inner product and norm is generally used for H01 (0, 1). This leads to the state-space

38

2 Infinite-Dimensional Systems Theory

Z˜ = H01 (0, 1) × L 2 (0, 1) with norm equal to the energy 2E(t). The operator A is again defined by (2.24) and the domain is   dom(A) = {(w, v) ∈ H2 (0, 1) ∩ H01 (0, 1) × H01 (0, 1)}. With this choice of state space, and corresponding definitions of A and its domain, the Lumer–Phillips Theorem can again be used to show that A generates a C0 -semigroup ˜ on Z.  Example 2.32 illustrates that the choice of Hilbert space is not a trivial question for infinite-dimensional systems. However, the energy of the system is generally a good guide to choosing an appropriate space. Examples 2.32 and 2.33 also illustrate the boundary conditions are important in defining the generator A, its domain and the state space. Example 2.34 (Diffusion in Rn ) The simple case of heat flow on a rod (Example 2.2) generalizes to diffusion of a quantity z with diffusivity coefficient κ(x) on a bounded region  ⊂ Rn where 1 ≤ n ≤ 3 has a piecewise smooth boundary ∂. Applications include diffusion of not only temperature but also chemical diffusion and neutron flux. The variable z(x, t) is governed by the partial differential equation ∂z = ∇ · (κ(x)∇z). ∂t

(2.30)

Assume that the temperature (or the diffused quantity) is zero on the boundary z(x, t) = 0, x ∈ ∂. A state-space formulation of (2.30) on L 2 () is Az = ∇ · (κ∇z), dom(A) = {z ∈ H2 ()| z(x) = 0, x ∈ ∂} = H2 () ∩ H01 ().

(2.31) The domain dom(A) is dense in L 2 (). Also, it is well known that the elliptic problem Az = f, z ∈ dom(A) has a solution for every f ∈ L 2 () and so A is a closed operator. The Fourier series solution approach used in Example 2.2 can be extended to show that A generates a C0 −semigroup on L 2 (). Alternatively, the Lumer–Phillips Theorem can be used. This requires less calculation. Green’s Identity yields that for all z ∈ dom(A),

2.2 Lumer–Phillips Theorem

39

 Az, z =

∇ · (κ(x)∇z)zd x  = z∇z · nd x − ∇z · ∇zd x ∂   = − ∇z2 d x 



≤ 0.

Also, A∗ = A and dom(A∗ ) = dom(A), that is, A is a self-adjoint operator. Therefore, by the Lumer–Phillips Theorem, A generates a contraction semigroup  on L 2 (). Example 2.35 (Example 2.34 cont.) Suppose there is a source term so that the governing equation is ∂z = ∇ · (κ(x)∇z) − βz, β > 0 ∂t and the same boundary conditions z(x, t) = 0, x ∈ ∂. The state space is still L 2 () but the generator is now Aβ z = Az − βz with dom(Aβ ) = dom(A). Theorem 2.18 implies that Aα generates a C0 -semigroup  Sβ (t) on L 2 () with Sβ (t) ≤ e−βt .

2.3 Sesquilinear Forms In Example 2.34 Green’s Identity was used to write  Aw, v = −



∇w∇vd x ≤ 0.

(2.32)

Although the left hand side is only valid for w ∈ dom(A), the right-hand side is well-defined for all w, v ∈ H 1 (). This approach can be generalized. Definition 2.36 A sesquilinear form on a Hilbert space V is a function a : V × V → C, such that for all φ1 , φ2 , z 1 , z 2 ∈ V, and scalars α , β, a(φ1 + φ2 , z 1 ) = a(φ1 , z 1 )+a(φ2 , z 1 ), a(φ1 , z 1 + z 2 ) = a(φ1 , z 1 ) + a(φ1 , z 2 ) a(αφ1 , βz 1 ) = αβa(φ1 , z 1 ).

40

2 Infinite-Dimensional Systems Theory

A sesquilinear form is continuous if there is c1 > 0 so that for all φ, z ∈ V, |a(φ, z)| ≤ c1 φV zV . A sesquilinear form is symmetric if for all φ, z ∈ V, a(z, φ) = a(φ, z). Any inner product on a Hilbert space is a continuous symmetric sesquilinear form. Also, the form  a(φ, ψ) =



∇φ∇ψd x.

(2.33)

is a continuous symmetric sesquilinear form on H1 (). The notation V → Z means that V is a Hilbert space contained in another Hilbert space Z and dense in Z with respect to the norm on Z. That is, for every z ∈ Z, there is a sequence {vn } ⊂ V with limn→∞ z − vn Z = 0. Consider a sesquilinear form a continuous on V → Z. For any Hilbert space, the Riesz Representation Theorem (Theorem A.42) implies that any element of the dual space can be identified with an element of the space itself. This identification will be done for Z: we regard Z  = Z. However, the dual space of V will not be identified with V and this leads to the nested spaces V → Z = Z  → V  . For each z ∈ V,

(Az)(·) = −a(·, z)

(2.34)

defines a bounded linear operator from V to the scalars. In other words, it is an element of the dual space V  . For some z, this operator is actually bounded on the space Z; that is, there exists cz ≥ 0 so that for all φ ∈ V |(Az)(φ)| = |a(φ, z)| ≤ cz φZ . Since V → Z, the Riesz Representation Theorem (Theorem A.42) implies that there is z A ∈ Z depending on z such that − a(φ, z) = φ, z A Z ,

for all φ ∈ V.

(2.35)

An operator A on Z can then be defined as Az = z A ,

(2.36)

dom(A) = {z ∈ V | there exists z A ∈ Z, −a(φ, z) = φ, z A Z , for all φ ∈ V}.

2.3 Sesquilinear Forms

41

Note the minus sign on a when defining A. The adjoint operator of A, A∗ can be defined identically to A using the form a(φ, ¯ z) = a(z, φ). If the form a is symmetric then it follows immediately that A is a self-adjoint operator: A∗ = A. Definition 2.37 A sesquilinear form a on V → Z is V-coer cive if for all φ, z ∈ V and some 0 < c2 ≤ c1 , α ∈ R, |a(φ, z)| + α φ, z Z ≤ c1 φV zV , Re a(φ, φ) + α φ, φ Z ≥ c2 φ2V .

(2.37) (2.38)

If a is V-coercive then a(·, ·) + α ·, · Z defines an inner product on V with a norm equivalent to the original norm on V. Every V-coercive sesquilinear form is continuous, but not every sesquilinear form that is continuous on V is also coercive. Definition 2.38 An operator A on Z is V-coer cive if V → Z and it can be defined through a V-coercive sesquilinear form a as in (2.36). It can be shown that the coercivity of a implies that the operator A defined by (2.36) is closed and densely defined. Then the Lumer–Phillips Theorem and Theorem 2.18 imply the following result. Theorem 2.39 If a is a coercive sesquilinear form on V → Z satisfying the inequalities (2.37), (2.38), then the operator A defined by (2.36) is closed, dom(A) is dense in Z and A generates a C0 -semigroup S(t) on Z with bound S(t) ≤ eαt . Theorem 2.40 (Poincaré Inequality) Let  be a smooth bounded domain. Let λ1 indicate the smallest eigenvalue of −∇ 2 on  with boundary conditions z(0) = 0 on the boundary of . The smallest eigenvalue λ1 > 0 and for all z ∈ H01 (), ∇z2L 2 () ≥ λ1 z2L 2 () . Example 2.41 (Laplace operator) The form  a(φ, z) =



∇φ(x)∇z(x)d x

is a continuous symmetric sesquilinear form with respect to V = H01 () and V → L 2 (). Also, since H01 () contains only functions that are 0 on the boundary, the Poincaré Inequality (Theorem 2.40) implies that a(·, ·) defines an inner product on V with norm equivalent to the usual norm on H1 (). (See also (2.29) in Example 2.33 for details in the case of  = [0, 1].) For z ∈ H2 () ∩ V,  −a(φ, z) = φ(x)(∇ 2 z(x)d x. 

42

2 Infinite-Dimensional Systems Theory

Thus, a defines an operator Az = ∇ 2 z, dom(A) = H2 () ∩ H01 (). Theorem 2.39 implies that A generates a C0 -semigroup S(t) on L 2 () with S(t) ≤ 1. In Example 2.34 the same operator was analyzed using the Lumer–Phillips Theorem, and the same conclusion was obtained.  Second-order in time systems, such as Examples 2.31 and 2.32, can also be analyzed using sesquilinear forms. Consider sesquilinear forms ao and d, both continuous on Vo → Ho and also symmetric. They can be used to define a second-order differential equation on Vo : for any ψ ∈ Vo , ˙ = 0. ψ, w(t)

¨ Ho + ao (ψ, w(t)) + d(ψ, w(t))

(2.39)

In applications, ao is related to the stiffness of the system and ao (w, w) is generally the potential energy while d(v, v) ≥ 0 models dissipation. For each w ∈ Vo , (Ao w)(·) = a(·, w)

(2.40)

defines a bounded linear operator from Vo to the scalars: it is an element of the dual space Vo . Define similarly D. (In the context of a second-order system, Ao and D are defined without a minus sign on the associated sesquilinear form.) This leads to an abstract differential equation on Vo equivalent to (2.39): ˙ = 0. w(t) ¨ + Ao w(t) + Dw(t)

(2.41)

Definition 2.42 A second-order (Vo , Ho )-system where Vo → Ho , is an abstract second-order in time differential equation (2.39) or (2.41) defined through symmetric sesquilinear forms ao and d both continuous on Vo → Ho . For every w, v ∈ Vo ,

ao (·, w) + d(·, v) ∈ Vo .

This element of Vo can also be written Ao w + Dv. As discussed above for singleorder in time systems, for some w, v there is v A ∈ Ho so that for all ψ ∈ Vo , ao (ψ, w) + d(ψ, v) = (ψ, v A )Ho . Equivalently, Ao w + Dv ∈ Ho . The differential equation (2.41) can be written in state-space form (2.10) on the state space Z = Vo × Ho , as

2.3 Sesquilinear Forms

    0 I , dom(A) = wv ∈ Vo × Vo | Ao w + Dv ∈ Ho . −Ao −D

43

 A=

(2.42)

The operator A can also be defined directly through a single sesquilinear form a continuous on V = Vo × Vo . First, rewrite the second-order differential equation (2.39) as a system of first-order equations. For φ, ψ, w, v ∈ Vo , φ, w(t)

˙ Vo = φ, v(t) Vo ψ, v(t)

˙ Ho = −ao (ψ, w(t)) − d(ψ, v(t)). For (φ, ψ), (w, v) ∈ V define a((φ, ψ), (w, v)) = − φ, v Vo + ao (ψ, w) + d(ψ, v).

(2.43)

The form a is continuous on V = Vo × Vo , and (2.39) is equivalent to for any (φ, ψ) ∈ V, (φ, ψ), (w(t), ˙ v(t))

˙ Vo ×Ho = −a((φ, ψ), (w, v)). For some (w, v) ∈ V, a defines an element of Z = Vo × Ho . Trivially this is always true of the first component ·, v Vo . But for some (w, v) ∈ V, −ao (·, w) − d(·, v) (the same operator as −Ao w − Dv) defines an element of Ho , call this element −v A . The corresponding generator can be defined as     w v , dom(A) = {(w, v) ∈ Vo × Vo , −ao (·, w) − d(·, v) ∈ Ho }. A = v −v A (2.44) This operator A and its domain are identical to that in (2.42). Assumptions on the coercivity of ao and the non-negativity of d imply wellposedness of the associated differential equation. Theorem 2.43 Suppose that a second-order (Vo , Ho )-system (2.41) is defined through symmetric sesquilinear forms ao and d each continuous on Vo and also there is c > 0 such that ao (w, w) ≥ cw2Vo , d(v, v) ≥ 0, for all w, v ∈ Vo . Then the operator A defined in (2.42) (or (2.44)) is the generator of a contraction on the state space Z = Vo × Ho . The assumptions in Theorem 2.43 imply that ao defines a norm on Vo equivalent to the original norm on Vo . Example 2.44 (Euler–Bernoulli Beam with Kelvin–Voigt damping) As in Example 2.26, consider the Euler–Bernoulli model for the transverse deflections in a beam and let w(x, t) denote the deflection of the beam from its rigid body motion at time t and

44

2 Infinite-Dimensional Systems Theory

position x. With Kelvin–Voigt damping model, the moment is, for positive physical parameters E, cd , ∂3w ∂2w m(x) = E 2 + cd 2 ∂x ∂x ∂t and the dynamics of the deflections are described by ∂2 ∂2w + ∂t 2 ∂x 2

 E

∂3w ∂2w + c d ∂x 2 ∂x 2 ∂t

 = 0, x ∈ (0, 1).

(2.45)

Suppose now that the beam is clamped at the end x = 0, so deflection and strain are are 0 at x = 1. 0 there, and free at x = 1, so that the moment m and shear force dm dx (This is known as a cantilevered beam.) The boundary conditions are w(0, t) = ∂w (0, t) = ∂x m(1, t) = ∂m (1, t) = ∂x

0, 0, 0, 0.

(2.46)

Define Ho = L 2 (0, 1),   Vo = z ∈ H2 (0, 1); w(0) = w  (0) = 0 , and



1

w1 , w2 Vo = E 0

w1 (x)w2 (x)d x.

(2.47)

Note that for every w ∈ Vo , 

x

w(x) = 0



s

w  (r )dr ds.

0

Using the same argument as in (2.29) shows that 



1

|w  (x)|2 d x

0

defines a norm on Vo equivalent to the usual norm on H2 (0, 1). Thus, (2.47) defines an inner product on Vo . Defining Z = Vo × Ho , the quantity z2Z is proportional to the sum of the potential and kinetic energies. In Example 2.26, it was shown that for a similar model, with different boundary conditions, A is a Riesz-spectral operator and so generates a C0 -semigroup on Z. The Lumer–Phillips Theorem can also be used to show that A generates a contraction on Z.

2.3 Sesquilinear Forms

45

Alternatively, the model is in the form (2.41) with ao (w1 , w2 ) = w1 , w2 Vo , d(v1 , v2 ) =

Cd a(v1 , v2 ). E

The assumptions of Theorem 2.43 are satisfied and so A as defined in (2.42) generates a contraction on Z.  Example 2.45 (Vibrations in Rn ) Consider acoustic vibrations in a bounded connected region  ⊂ Rn with boundary . The region  has a piecewise smooth boundary , where  = 0 ∪ 1 and 0 , 1 are disjoint subsets of  with both 0 and 1 having positive measure. The partial differential equation describing the system is, letting n indicate the outward normal along the boundary, w(x, ¨ t) = ∇ 2 w(x, t), ˙ 0) = w1 , w(x, 0) = w0 , w(x, w(x, t) = 0, w(x, t) · n = 0,

(x, t) ∈  × (0, ∞), x ∈ , (x, t) ∈ 0 × (0, ∞), (x, t) ∈ 1 × (0, ∞) .

(2.48)

Define Ho = L 2 () and Vo = H1 0 () = {g ∈ H1 () | g|0 = 0}. The usual inner product on H1 () is 

 

∇ f (x)∇g(x)d x +



f (x)g(x)d x.

Since 0 has positive measure by assumption, the simpler inner product  f, g Vo =



∇ f (x)∇g(x)d x

(2.49)

can be used for Vo . This PDE can be put in the form of Theorem 2.43 with  ao (ψ, w) =



∇ψ(x) · ∇w(x)d x

and zero damping d. The operator (2.42) generates a contraction on the state space Vo × Ho . 

46

2 Infinite-Dimensional Systems Theory

2.4 Control Consider now a system where an external signal f is applied: dz = Az(t) + f (t), dt

z(0) = z 0 .

(2.50)

Definitions 2.6 and 2.8 are extended to cover systems with inputs. Definition 2.46 A function z(t) ∈ Z, t ≥ 0 is a classical solution, or strong solution of (2.50) on [0, T ] if z ∈ C 1 ([0, T ]; Z) and for all t ∈ [0, T ], z(t) ∈ dom(A) and (2.50) is satisfied. t Definition 2.47 If for t ∈ [0, T ], 0 z(s)ds ∈ dom(A), z(t) ∈ C([0, T ]; Z), f ∈ L 1 ([0, T ]; Z) and 

t

z(t) − z(0) = A



t

z(s)ds +

0

f (s)ds

0

then z(t) is a mild solution of (2.50). Theorem 2.48 If A with domain dom(A) generates a C0 semigroup S(t) on Z, z 0 ∈ dom(A) and f ∈ C 1 ([0, T ]; Z) then  z(t) = S(t)z 0 +

t

S(t − s) f (s)ds

(2.51)

0

is the unique classical solution of (2.50). Although a classical solution may exist under assumptions weaker than those in the preceding theorem, frequently in applications the classical solution does not exist. However, provided that A generates a C0 -semigroup and the forcing term f is integrable, the expression (2.51) is well-defined. Theorem 2.49 If A with domain dom(A) generates a C0 semigroup S(t) on Z, z 0 ∈ Z and f ∈ L 1 ([0, T ]; Z) then (2.51) is the unique mild solution of (2.50). In applications typically f ∈ L 2 ([0, T ]; Z) which implies f ∈ L 1 ([0, T ]; Z). Example 2.50 (Diffusion) (Example 2.2 cont.) Suppose that the temperature in a rod of length 1 with Dirichlet boundary conditions is controlled using an input flux u(t) with spatial distribution b(x) ∈ L 2 (0, 1) so that the governing equation is now ∂2z ∂z = + b(x)u(t), ∂t ∂x 2

0 < x < 1,

with boundary conditions z(0, t) = z(1, t) = 0. It was shown in Example 2.2 that the operator A : dom(A) ⊂ L 2 (0, 1) → L 2 (0, 1) defined by

2.4 Control

47

Az =

d2z , dom(A) = {z(x) ∈ H2 (0, 1) ; z(0) = z(1) = 0} dx2

generates a C0 -semigroup on L 2 (0, 1). Defining an operator B ∈ B(R, Z) by Bu = b(x)u leads to z˙ (t) = Az(t) + Bu(t) which is in the form (2.50) with f (t) = Bu(t).



Example 2.51 (Simply supported beam) (Example 2.31 cont.) Consider a simply supported beam with an applied force u(t) distributed along the beam according to the function b(x). Neglect damping and set ρ = 1, E = 1 and the   = 1 to simplify  ˙ t) with the equations. As in Example 2.31, define the state as z(t) = w(·, t) w(·, state-space Z = Hs (0, ) × L 2 (0, ). A state-space formulation of the model is d z(t) = Az(t) + Bu(t), dt where ⎡ A=⎣ ⎡ B=⎣

0

I

4 − ddx 4

0

⎤

⎤ ⎦ , dom(A) = {(w, v) ∈ Hs (0, ) × Hs (0, ); w  ∈ Hs (0, )},

0

⎦.

b(·) This is of the form (2.50) with f (t) = Bu(t).



The effect of the actuators leads to dz = Az(t) + Bu(t), dt

z(0) = z 0

where A with domain dom(A) generates a strongly continuous semigroup S(t) on a Hilbert space Z and B ∈ B(U, Z). If z 0 ∈ Z and u ∈ L 1 ([0, T ]; U), the mild solution is  t S(t − s)Bu(s)ds. z(t) = S(t)z 0 + 0

The map from control u(·) to the state z(T ) is, for T > 0, 

T

B(T )u =

S(T − s)Bu(s)ds.

0

Theorem 2.52 Suppose that S(t) is a C0 -semigroup on Z and B ∈ B(U, Z). For each t > 0, the controllability map

48

2 Infinite-Dimensional Systems Theory

 B(t)u =

t

S(t − s)Bu(s)ds

0

is a well-defined continuous linear operator from L 2 ([0, t]; U) to Z . For many situations, such as control on the boundary of the region, simple models lead to a state-space representation where B is unbounded on the state-space. More precisely, it is a bounded operator into a larger space than the state space. In this case the controllability map B(t) is not always continuous into the state space. This aspect of well-posedness needs to be established. However, including a model for the actuator generally leads to a model with bounded B. This is illustrated by the following example. Example 2.53 (Control of Acoustic Noise in a Duct) Consider the model of acoustic waves in a duct introduced in Example 1.3 with normalized variables. In terms of the pressure p and velocity v the governing equations are ∂ p(x, t) ∂v(x, t) =− , ∂t ∂x

(2.52)

∂ p(x, t) ∂v(x, t) =− . ∂t ∂x

(2.53)

If both ends of the duct are open boundary conditions p(0, t) = 0,

p(1, t) = 0

are reasonable. This example was put into state space form in Example 2.32 with state ( p, v) and state space Z = L 2 (0, 1) × L 2 (0, 1). Suppose that instead of an open end at x = 0, a loudspeaker acting as a source of noise is mounted at x = 0. A loudspeaker can be regarded as a velocity source f (t) which leads to the boundary condition v(0, t) = f (t).

(2.54)

If the system is now written in state-space form, the injection of velocity on the boundary leads to an operator B with range in a space larger than the usual energy space. This introduces theoretical, and sometimes also numerical, complexities. The boundary condition (2.54) implies that, when undriven, the loudspeaker acts as a perfectly rigid end with zero velocity. In fact, a loudspeaker has stiffness, mass and damping, even when undriven. So it will not act as a perfectly rigid end. The loudspeaker can be modeled as a mass–spring–damper system. Indicating the driving voltage of the loudspeaker by u(t), and the loudspeaker cone deflection by s the governing equations of the loudspeaker are described by the second-order ordinary differential equation

2.4 Control

49

m L s¨ (t) + d L s˙ (t) + k L s(t) = bu(t) − a L p(0, t),

(2.55)

where m L , d L , k L , b and a L are loudspeaker parameters. The loudspeaker is coupled to the duct by the pressure at the end p(0, t) and also by a L s˙ (t) = πa 2 v(0, t),

(2.56)

where a is the duct cross-sectional area. For simplicity, assume that boundary condition at the end x = 1 is still p(1, t) = 0. Thus, the model is the original partial differential equation and the boundary conditions (2.56), p(1, t) = 0. With state  T z = p v s s˙ the state-space is Z = L 2 (0, 1) × L 2 (0, 1) × R2 . Define aL s2 , p(1) = 0}, dom(A) = {( p, v, s1 , s2 ) ∈ H1 (0, 1) × H1 (0, 1), v(0) = πa 2 ⎡ ⎤ ⎤ ⎡ ∂ 0 − ∂x 0 0 0 ⎢0⎥ ⎢− ∂ 0 0 0 ⎥ ∂x ⎥, B = ⎢ ⎥ . A=⎢ ⎣0⎦ ⎣ 0 0 0 1 ⎦ b −k L −d L 0 0 mL mL mL This defines the state-space formulation z˙ (t) = Az(t) + Bu(t), for the wave equations (2.52), (2.53) coupled to (2.56) and loudspeaker driving voltage u is Note that the coupling of the loudspeaker and duct dynamics is entirely through the domain of the operator A. The Lumer–Phillips Theorem (Theorem 2.28) can be used to show that the operator A with this domain generates a contraction on Z. 

2.5 Observation Even for finite-dimensional systems, the entire state cannot generally be measured. Because the state is distributed over a region in space and only a finite number of sensors can be installed, measurement of the entire state is never possible for systems described by partial differential equations. Consider z˙ (t) = Az(t), z(0) = z 0 ,

50

2 Infinite-Dimensional Systems Theory

where A generates a C0 -semigroup on a Hilbert space Z and measurement y(t) = C z(t) + Du(t)

(2.57)

where C ∈ B(Z, Y) , D ∈ B(U, Y) and U, Y are Hilbert spaces. The expression (2.57) can also represent the cost in controller or estimator design. Note that it is assumed that C is a bounded operator from the state-space Z. Typically there are a finite number of measurements and Y = R p . For any T > 0, define the map from the state to the output as (C(T )z 0 ) = C S(·)z 0 . Theorem 2.54 Suppose that S(t) is a C0 -semigroup on Z and C ∈ B(Z, Y). For each t > 0, the observability map (C(T )z 0 ) = C S(·)z 0 , is a well-defined continuous linear operator from Z to L 2 ([0, T ]; Y). Observations at a point often lead to an operator that is not defined on the entire system. For example, the heat equation is well-posed on the state space L 2 (0, 1) but evaluation of a function at a point is not well-defined for every element of L 2 (0, 1). As for boundary control, this issue can often be addressed by a more accurate modelling of the sensor. Example 2.55 (Diffusion) (Example 2.2 cont.) The temperature in a rod of length 1 with Dirichlet boundary conditions is modelled by z˙ (t) = Az(t) where Az =

d2z , dom(A) = {z ∈ H2 (0, 1), z(0) = 0, z(1) = 0} dx2

generates a C0 -semigroup on L 2 (0, 1). Measurement of temperature at point y(t) = z(x0 , t) is not defined on all of the state space L 2 (0, 1). Evaluation is not even a closed operator on L 2 (0, 1). (See Example 2.15.) However, temperature measurement is a result of sensing over a small region of non-zero size. The simplest such model is to assume that the sensor measures the average temperature over a small region; that is for small δ > 0 and a parameter c0 ,  y(t) = c0

x0 +δ

z(x, t)d x. x0 −δ

(2.58)

2.5 Observation

51

The right-hand side defines a bounded linear operator from L 2 (0, 1) into the scalars. The Riesz Representation Theorem (Theorem A.42 ) states there is c ∈ L 2 (0, 1) so that  1 z(x)c(x)d x. Cz = 0

The required element of L 2 (0, 1) is easily seen to be the piecewise constant function  c(x) =

c0 |x − x0 | < δ . 0 else

The observation (2.58) can be written y(t) = z(·, t), c(·) , which defines a bounded operator from L 2 (0, 1) to C.



Example 2.56 (Measurement of Beam Vibrations) Consider a general second-order system (2.41) ˙ = 0. (2.59) w(t) ¨ + Ao w(t) + Dw(t) Under assumptions on Ao , D this model is well-posed on Z = Vo × Ho (Theorem  T 2.43) with state z = w w˙ . For example, a beam such as in Example 2.44, is of the form (2.59) with   Ho = L 2 (0, 1), Vo = Hs (0, 1) = w ∈ H2 (0, 1); w(0) = w(1) = 0 , and the state space Z = Vo × Ho . One measurement type for a beam is the position at 0 < x0 < 1: y(t) = w(x0 , t).   Defining Co w = w(x0 ), C p = Co 0 , y(t) = C p z(t). Rellich’s Theorem (Theorem A.35) implies that since [0, 1] ⊂ R, there is m > 0 so that for any w ∈ C([0, 1]), max |w(x)| ≤ m wH1 (0,1) .

0≤x≤1

Thus, the operator Co is bounded from Vo and so C p is bounded from the state space Z. Another common measurement is acceleration. Acceleration involves the second time derivative and cannot easily be written in terms of the states. One approach is to write

52

2 Infinite-Dimensional Systems Theory

  y(t) = 0 Co



0 I −Ao −D

     w(t) 0 + u(t) . w(t) ˙ Bo

However, unless very strong assumptions, not generally satisfied, are imposed on stiffness Ao and damping D, this does not lead to an observation map z(0) → y(t) that is bounded from the energy space Z. Small changes in the state z can lead to large changes in y. The model is ill-posed. However, accelerometers have dynamics. A common type is a micro-electromechanical system (MEMS) where a mass is suspended between two capacitors and the measured voltage is proportional to the mass position. Letting f (t) be the force applied by the structure to the accelerometer, and a the deflection of the accelerometer mass, and m, d and k accelerometer parameters, m a(t) ¨ + ka(t) + d a(t) ˙ = f (t). Using Hamilton’s principle to obtain a description for the dynamics of a structure with Co ∈ B(Vo , C), coupled to an accelerometer leads to     ˙ − Co w(t) ˙ = 0, m a(t)+k ¨ a(t) − Co w(t) + d a(t)    ∗ ∗ ρw(t) ¨ + Ao w(t) + Dw(t) ˙ + kCo Co w(t) − a(t) + dCo Co w(t) ˙ − a(t)) ˙ =0 y(t) = α(Co w(t) − a(t)), where Co indicates position measurement and α is a parameter. This model can be written √ in the standard second-order form (2.41) as, defining √ ˜ = ma(t), w(t) ˜ = ρw(t), a(t) 

     ¨˜ ˙˜ w(t) w(t) ˜ w(t) ˜ ˜ + Ao +D ˙ =0 ¨˜ a(t) ˜ a(t) a(t) ˜

where ˜o = A

˜ = D

1

ρ

(Ao + kCo∗ Co ) √−k C o ρm

1

ρ

(D + dCo∗ Co ) √−d C o ρm

√−k C ∗ ρm o k m √−d C ∗ ρm o d m

 ,



are operators on Vo × R. The fact that the generator associated with the original system generates a C0 -semigroup on Vo × Ho implies that this system is also well T posed with state z˜ = w˜ a˜ w˙˜ a˙˜ on the state space Z = Vo × R × Ho × R. The output operator   y(t) = αCo 0 − α 0 z˜ (t).

2.5 Observation

53

Provided that Co ∈ B(Vo , R), observation is now bounded on the natural (energy)  state-space Vo × R × Ho × R.

2.6 Controllability and Observability Consider z˙ (t) = Az(t), z(0) = z 0 , where A generates a C0 -semigroup S(t) on a Hilbert space Z and for C ∈ B(Z, Y) consider the output y(t) = C z(t) and the corresponding map from initial condition z 0 to output y (C(T )z 0 ) = C S(·)z 0 . From Theorem 2.54, for each T > 0, C(T ) is a bounded operator from Z to L 2 ([0, T ]; Y). Definition 2.57 The observation system (A, C) is exactly observable on [0, T ] if the initial state is uniquely and continuously determined by observations C z(t) over the interval [0, T ]. Definition 2.58 The observation system (A, C) is approximately observable on [0, T ] if the initial state is uniquely determined by observations C z(t) over the interval [0, T ]. Theorem 2.59 The observation system (A, C) is 1. exactly observable on [0, T ] if and only if there is γ > 0 so that for all z 0 ∈ Z, 

T

(C(T )z 0 )(s)2 ds ≥ γz 0 2 ;

0

2. approximately observable on [0, T ] if and only if for all z 0 = 0, 

T

(C(T )z 0 )(s)2 ds > 0,

0

or equivalently, ker (C(T )) = {0}. Approximate observability means that the kernel of the observability map is 0 so that the initial condition is determined uniquely by the observations. This implies that the inverse map (C(T ))−1 is defined on the range of C(T ). Exact observability

54

2 Infinite-Dimensional Systems Theory

means that the observability map C(T ) has a bounded inverse C(T )−1 . This implies that small changes in the output reflect a small change in the initial condition. Clearly exact observability implies approximate observability. For finitedimensional systems approximately observability and exact observability are equivalent. However, an infinite-dimensional system can be approximately but not exactly observable. This is illustrated by the following example. Example 2.60 (Heat flow on a rod) Consider heat flow on an interval as in Examples 1.1,2.2: ∂ 2 z(x, t) ∂z(x, t) = , x ∈ (0, 1), t ≥ 0, ∂t ∂x 2 with boundary conditions z(0, t) = 0, z(1, t) = 0, and some initial temperature distribution z(x, 0) = z 0 (x), z 0 (x) ∈ L 2 (0, 1). For any initial condition z 0 ∈ L 2 (0, 1), this equation has a unique solution on Z = L 2 (0, 1) described by the semigroup S(t)z 0 =

∞ 

z 0 , φn φn e−n

π t

2 2

,

n=1

√ where φn (x) = 2 sin(nπx). Consider the strong case where the entire state is measured. The observation operator C = I and y(t) = z(·, t). The observability map is (C(T )z 0 )(x, t) =

∞  2 2 z 0 , φn φn (x)e−n π t . n=1

The observability map will have a bounded inverse if and only if there is γ > 0 so that for all z 0 ∈ L 2 (0, 1), 

T

(C(T )z 0 )(s)2 ds ≥ γz 0 2 .

0

Consider the eigenfunctions as initial conditions. If z 0 = φm ,

(2.60)

2.6 Controllability and Observability

55

(C(T )φm )(s) = φm e−m and

(C(T )φm )(s)2 = e−2m

Thus,



T

(C(T )φm )(s)2 ds =

0

π s

2 2

π s

2 2

.

1 2 2 (1 − e−2m π T ). 2m 2 π 2

For no T does there exist a γ > 0 so that (2.60) holds for all z 0 . The system is not exactly observable on any time interval. However, in order for (C(T )z 0 ) to map to the zero function in L 2 (0, T ), from Parseval’s Equality (A.10), (C(T )z 0 )(s)2 =

∞ 

| z 0 , φn |2 e−2n

π s

2 2

=0

n=1

for all s which implies that z 0 is the zero function. The observation system is approximately observable.  This example illustrates that exact observability is a strong property for DPS. If the observation space Y is finite-dimensional and C ∈ B(Z, Y), then the system is never exactly observable. Theorem 2.61 If the observation space Y is finite-dimensional then the observation system (A, C) is not exactly observable on [0, T ] for any T > 0. Example 2.62 (Right shift) Consider the right shift semigroup  (S(t)z 0 )(x) =

0 0≤x 0,  (C z)(x) =

0 0 ≤ x ≤ t0 , z(x) x > t0 .

If T < t0 , it is clear that this observation is not approximately (or exactly) observable on [0, T ]. If T > t0 then for any z 0 ∈ Z, 

T

 (C S(s)z 0 )2 ds =

0

T



T



0

 =

0

∞

|C z 0 (x − s)|2 d xds

s ∞

max(t0 ,s)

|z 0 (x − s)|2 d xds

56

2 Infinite-Dimensional Systems Theory

 = 0

 =

0

T



∞

max(t0 −s,0) t0  ∞ t0 −s

|z 0 (r )|2 dr ds 

|z 0 (r )|2 dr ds +

T

t0



∞

|z 0 (r )|2 dr ds

0

≥ (T − t0 )z 0 2 . Therefore, for T > t0 the observation system is exactly observable and therefore it is also approximately observable.  This example illustrated that a system may be (exactly or approximately) observable on a given time interval, but not on smaller intervals. This is different from the situation for finite-dimensional systems. Consider now a control system (A, B) dz = Az(t) + Bu(t), dt

z(0) = z 0

(2.61)

where A with domain dom(A) generates a strongly continuous semigroup S(t) on a Hilbert space Z and B ∈ B(U, Z). There are various definitions of controllability for infinite-dimensional systems (2.61). Two common definitions of controllability are as follows. Definition 2.63 The control system (A, B) is exactly controllable on [0, T ] if for every z f ∈ Z there is u ∈ L 2 ([0, T ]; U) such that B(T )u = z f . Definition 2.64 The control system (A, B) is approximately controllable on [0, T ] if for every z f ∈ Z and  > 0 there is u ∈ L 2 ([0, T ]; U) such that B(T )u − z f  < . For an exactly controllable system the range of B(T ) equals the state space Z, while if the system is only approximately controllable the range of B(T ) is only dense in Z, that is, RangeB(T ) = Z. Clearly, every exactly controllable systems is approximately controllable. Theorem 2.65 For T > 0 and control system (A, B), the adjoint map B(T )∗ ∈ B(Z, L 2 ([0, T ]; U)) is (B(T )∗ z 0 )(t) = B∗ S(T − t)∗ z 0

(2.62)

where S(t)∗ is the semigroup generated by A∗ . Since the observability map C(T ) corresponds to the adjoint of the controllability map for (A∗ , C ∗ ) by Theorem 2.65, the following results are immediate. Theorem 2.66 The control system (A, B) is approximately controllable on [0, T ] if and only if the observation system (A∗ , B ∗ ) is approximately observable on [0, T ]; and similarly the control system (A, B) is exactly controllable on [0, T ] if and only if the observation system (A∗ , B ∗ ) is exactly observable on [0, T ].

2.6 Controllability and Observability

57

Theorem 2.67 The control system (A, B) is 1. exactly controllable on [0, T ] if and only if there is γ > 0 so that 

T

(B(T )∗ z 0 )(s)2 ds ≥ γz 0 2 ;

0

2. approximately controllable on [0, T ] if and only if for all z 0 = 0, 

T

(B(T )∗ z 0 )(s)2 ds > 0,

0

or equivalently, ker (B(T )∗ ) = {0}. As for exact observability, exact controllability is a strong property for DPS. Example 2.68 (Control of Diffusion, Example 2.60 cont.) Consider control of heat flow on an interval ∂z(x, t) ∂ 2 z(x, t) = + u(x, t), ∂t ∂x 2

x ∈ (0, 1),

t ≥ 0,

with boundary conditions z(0, t) = 0, z(1, t) = 0, and some initial temperature distribution z(x, 0) = z 0 (x), z 0 (x) ∈ L 2 (0, 1). The control operator B = I. Since the generator A is self-adjoint, this control system is the dual of the observation system in Example 2.60 and so it is not exactly controllable on any interval [0, T ]. It is approximately controllable on any [0, T ].  Theorem 2.69 If the control space U is finite-dimensional and B ∈ B(U, Z) then (A, B) is not exactly controllable on [0, T ] for any T > 0. Since in practice, most control systems involve only a finite number of controls, this means that DPS are generally not exactly controllable. Definition 2.70 For a control system (A, B) define the reachable subspace R = ∪T ≥0 Range B(T ). If R = Z then the control system is approximately controllable. Definition 2.71 For an observation system (A, C) define the unobservable subspace O = ∩T ≥0 ker C(T ). If O = {0} then the observation system is approximately observable.

58

2 Infinite-Dimensional Systems Theory

Approximate observability of a system means that if an observation is 0 on all intervals of time, the initial condition was zero. Similarly, if a system is approximately controllable, then for any state z f and  > 0 there is a time T > 0 so that the state can be steered to within  of z f in time T. Example 2.62 illustrates that a system may be exactly observable, but for some T > 0 may fail to be even approximately observable. Theorem 2.72 Suppose that A is a Riesz-spectral operator as defined in Theorem 2.25 with eigenfunctions {φn } and eigenfunctions {ψn } of A∗ . Assume that A generates a C0 -semigroup on Z and consider B ∈ B(U, Z), C ∈ B(Z, Y) where U and Y are both finite-dimensional Hilbert spaces. 1. The operator B can be written, for some integer m and bi ∈ Z, Bu =

m 

bi u i .

i=1

The control system is approximately controllable if and only if for each n,   rank b1 , ψn b2 , ψn · · · bm , ψn = 1. 2. The operator C can be written, for some integer p and ci ∈ Z, ⎤ z, c1

⎢ z, c2 ⎥ ⎥ ⎢ Cz = ⎢ . ⎥ . ⎣ .. ⎦ ⎡

z, c p

The observation system is approximately observable if and only if for each n,   rank φn , c1 φn , c2 · · · φn , c p = 1. Example 2.73 (Control and observation of temperature in a rod) Consider the example of control of temperature in a rod described √ in Example 2.50. The operator A is Riesz-spectral, in fact the eigenfunctions φn = 2 sin(nπx) form an orthonormal basis for Z = L 2 (0, 1) (see Example 2.2). The system is approximately controllable if and only if for all n,  1 b(x) sin(nπx)d x = 0. 0

For example, if b(x) = x 2 ,  0

1

b(x) sin(nπx)d x =

−n 2 π 2 (−1)n + 2((−1)n − 1) . n3 π3

2.6 Controllability and Observability

59

This is non-zero for all n and the system is approximately controllable. That is, the set ∪T ≥0 RangeB(T ) is dense in L 2 (0, 1). Consider on the other hand observation of the same diffusion model. In Example 2.55, there is observation of the average temperature over an interval and  c(x) =

c0 |x − x0 | < δ . 0 else

The system is approximately observable if and only if 

x0 +δ x0 −δ

sin(nπx)d x = 2

sin(nπx0 ) sin(nπδ)

= 0. nπ

The system is not approximately observable if the sensor is centered on a zero of one of the eigenfunctions sin(nπx). If x0 and δ are irrational, the system is approximately observable.  In summary, for infinite-dimensional systems it is important to distinguish exact from approximate controllability; and similarly to distinguish exact from approximate observability. Exact controllability is difficult to achieve with bounded control. Exact observability is also difficult with bounded observation. This is different from the finite-dimensional situation where approximate and exact controllability are equivalent. Also if a finite-dimensional system is controllable on an interval [0, T ], it is controllable on any other interval. On the other hand, an infinite-dimensional system may be approximately (or exactly) controllable for a given T > 0 but not for smaller times. A similar statement holds for observability.

2.7 Input/Output Maps The state-space realization for a general control system can be written z˙ (t) = Az(t) + Bu(t), z(0) = z 0 , y(t) = C z(t) + Du(t)

(2.63)

where A generates a C0 -semigroup S(t) on a Hilbert space Z, B ∈ B(U, Z), C ∈ B(Z, Y), D ∈ B(U, Y), and U, Y are both Hilbert spaces. Such a control system is often described as (A, B, C, D), or just (A, B, C) if the feedthrough operator D is zero, as is common. If also A is a Riesz-spectral operator (Definition 2.23) and U and Y are both finite-dimensional Hilbert spaces, (2.63) is a Riesz-spectral system. The output is 

t

y(t) = C S(t)z 0 + C 0

S(t − τ )Bu(τ )dτ + Du(t).

60

2 Infinite-Dimensional Systems Theory

Define the impulse response g(t) = C S(t)B + Dδ(t), t ≥ 0

(2.64)

with g(t) = 0, t < 0 and the operator from L 2 ([0, T ]; U) to L 2 ([0, T ]; Y)  (Gu) =

t

g(t − s)u(s)ds, 0 ≤ t ≤ T.

0

The operator G is the map from the input u to the output y and is called the input/output map : if z(0) = 0, y(t) = (Gu)(t). Definition 2.74 If w : [0, ∞) → W where W is a Hilbert space, satisfies e−αt w ∈ L 1 ([0, ∞); W) for some real α, it is Laplace transformable and the Laplace transform is  ∞

w(s) =

w(t)e−st dt, Res > α.

0

Definition 2.75 The Laplace transform G of the impulse response g is the transfer function of the system. If there is a single input and a single output then the transfer function is a complexvalued function. If there are multiple inputs and outputs, G will be a matrix with complex-valued entries. The number of rows in G is the number of outputs and the number of columns equals the number of inputs. More generally, G(s) ∈ B(U, Y) for s where G is defined. For any semigroup S(t) there is M ≥ 1, and real α so that S(t) ≤ Meαt , t ≥ 0. The corresponding impulse response g is therefore Laplace transformable on the half-plane Res > α. The transfer function of a system modelled by a system of ordinary differential equations is always rational with real coefficients; for example 2s+1 . Transfer functions of systems modelled by partial differential equations are s 2 +s+25 non-rational. The extension of the transfer function outside of Res > α is not-trivial. A representation valid for all s where (s I − A)−1 exists and is a bounded operator, that is s ∈ ρ(A) (Definition 2.22), exists. Theorem 2.76 The transfer function of (2.63) is G(s) = C(s I − A)−1 B + D, s ∈ ρ(A).

(2.65)

The transfer function is the unique solution to the following differential equation: for any u o ∈ U, s ∈ ρ(A), write sz o = Az o + Bu o , G(s)u o = C z o + Du o .

(2.66)

2.7 Input/Output Maps

61

Using the form (2.66) is generally simpler since (2.65) requires calculating the explicit inverse of s I − A and (2.66) is solving a differential equation. For systems in one space dimension with constant coefficients the calculation is generally feasible. The calculation of the transfer function of a DPS is illustrated in the following examples. Note that the transfer function is non-rational. Example 2.77 (First-order PDE) The transport equation on the interval [0, 1] with a a single control and observation described by b ∈ L 2 (0, 1) and c ∈ L 2 (0, 1) respectively, is ∂z ∂z (x, t)= (x, t) + b(x)u(t) t ≥ 0, x ∈ (0, 1) ∂t ∂x z(1, t) = 0 t ≥ 0, z(x, 0)=z 0 (x), x ∈ (0, 1) y(t)= z(·, t), c(·)

t ≥ 0.

The state-space is L 2 (0, 1), the generator A : dom(A) ⊂ L 2 (0, 1) → L 2 (0, 1) is Az = z  , dom(A) = {z ∈ H1 (0, 1) | z(1) = 0},

(2.67)

and Bu = bu, C z = z, c . The transfer function can be found by first solving the ordinary differential equation sz o (x) = z  (x) + b(x) z o (1) = 0 for z o (x) (which will depend on s as a parameter) and then calculating  G(s) =

1

z o (x)c(x)d x.

0

If b(x) = c(x) = x 2 , the transfer function is G(s) =

(120 + 120s + 60s 2 )e−s − 120 + 20s 3 − 15s 4 + 6s 5 . 30s 6

(2.68)

The Taylor series for the numerator has a leading term 53 s 6 and so the s 6 in the denominator cancels with the numerator. The transfer function is analytic in the entire complex plane and has no poles. 

62

2 Infinite-Dimensional Systems Theory

Example 2.78 (Diffusion with Neumann boundary conditions) Consider the problem of controlling the temperature profile in a rod of length 1 and dimensionless variables. Assume that the ends are insulated. With control distributed through some weight b(x) ∈ L 2 (0, 1), the temperature is governed by ∂z(x, t) ∂ 2 z(x, t) + b(x)u(t), = ∂t ∂x 2 ∂z (0, t) = 0, ∂x

x ∈ (0, 1),

t ≥ 0.

∂z (1, t) = 0. ∂x

The temperature sensor is modelled by 

1

y(t) =

c(x)z(x, t)d x.

(2.69)

0

The PDE in this example differs from that in Example 2.2 in the boundary conditions. This system can be written as an abstract control system (2.63) on the state space L 2 (0, 1) with Az =

d2z , dx2

dom(A) = {z ∈ H2 (0, 1), z  (0) = z  (1) = 0}

and Bu = b(x)u, and C is defined by (2.69). With b(x) = c(x) = x 2 , the transfer function is √ √ √ (−20s − 60 + 3s 2 )(1 − e−2 s ) + 60 s(1 + e−2 s ) √ G(s) = . 15s 3 (1 − e−2 s )

(2.70)

The denominator of G(s) has zeros at s = −n 2 π 2 , n ≥ 0. These correspond to the eigenvalues of the generator A.  Example 2.79 (Transfer function for wave equation) Consider vibrations w(x, t) in a string. Suppose that the ends are fixed with both control and observation described by the function b(x) ∈ L 2 (0, 1). The model is ∂ 2 w(x, t) ∂ 2 w(x, t) = + b(x)u(t), ∂t 2 ∂x 2 w(0, t) = 0, w(1, t) = 0,  1 ∂z(x, t) b(x)dx. y(t) = ∂t 0

(2.71) (2.72) (2.73)

2.7 Input/Output Maps

63

The transfer function can be calculated by solving d 2 w(x, s) + b(x)u(s), dx2 z(0, s) = 0, z(1, s) = 0,  1 ∂ w(x, s) y(s) = b(x)dx. ∂t 0

s 2 w(x, s) =

(2.74) (2.75) (2.76)

Equation (2.74) is a forced, linear ordinary differential equation in the variable x with boundary conditions (2.75). There are a number of methods available to solve linear ordinary equations. One method is to use variation of parameters. Another is to write the differential equation as a system of two first-order equations in x then either use an integrating factor or the matrix exponential to solve this system. This last method will be used. A first-order form of (2.74) is d dx





z dz dx

    z 0 1 0 = 2 − b(x)u(s). dz s 0 1 ! "# $ d x 

(2.77)

F(s)

The matrix exponential (with respect to x) of F in (2.77) is  cosh (sx) 1s sinh (sx) , E(x, s) = exp(F(s)x) = s sinh (sx) cosh (sx) 

and the general solution of the system of equations is, with arbitrary “initial conditions” at x = 0, z(0, s) = α and dd xz (0, s) = β, 

z(x) d z(x) dx



   x   α 0 = E(x, s) − E(ξ, s) b(x − ξ)u(s)d ξ. β 1 0

Using the boundary conditions (2.75) to eliminate α and β leads to z(x, s) =

u(s) s



   sinh(sx) 1 sinh (1 − ξ) s b(ξ) dξ... sinh (s) 0  x   − 0 sinh (1 − ξ) s b(ξ) dξ .

(2.78)

Substitution into (2.76), calculation of the integral and dividing by u(s) leads to the system transfer function. Defining    sinh (sx) 1 a(x, s) = sinh (1 − ξ) s b(ξ)d ξ... sinh(s) 0  x   − sinh (1 − ξ) s b(ξ)d ξ, 0

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2 Infinite-Dimensional Systems Theory

the transfer function is



1

G v (s) =

a(x, s) b(x) dx. 0



If b(x) =

1, 0 ≤ x ≤ 1/2, 0, 1/2 < x ≤ 1,

the transfer function is 2 cosh 1 + G v (s) = 2s

s 2

  − cosh2 2s − cosh(s) . s 2 sinh(s)

(2.79)

The poles are at j πk, for integers k. If instead of the velocity measurement (2.73) deflection is measured so that  y(t) =

1

w(x, t)b(x)dx,

(2.80)

0

the analysis is very similar. The calculation of the transfer function is identical up to (2.78). Then using (2.80) instead of (2.73) leads to the transfer function G p (s) = 1 G (s).  s v Transmission zeros and invariant zeros are defined similarly to the finitedimensional case. Definition 2.80 The invariant zeros of (A, B, C, D) are the set of all λ ∈ C such that      λI − A −B x 0 = (2.81) C D u 0 has a solution for some scalar u and non-zero x ∈ dom(A). Denote the set of invariant zeros of a system by inv(A, B, C, D). To avoid complication in the definition of transmission zeros in the multi-input– multi-output case, the remainder of this section will only consider the case where there is a single input and output. This means that there is b, c ∈ Z so that the control operator B and observation operator C can be written Bu = bu, C z = z, c . Definition 2.81 A complex number s ∈ ρ(A) is a transmission zero of (A, B, C, D) if G(s) = C(s I − A)−1 B + D = 0. Theorem 2.82 The number λ ∈ ρ(A) is a transmission zero if and only it is an invariant zero.

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65

Definition 2.83 The system (A, B, C, D) is minimal if inv(A, B, C, D) ∩ σ(A) = ∅. Theorem 2.84 Suppose that the system (A, B, C, D) is minimal and that σ(A) is non-empty and consists only of isolated eigenvalues with finite algebraic multiplicity. Then the transfer function G(s) of (A, B, C, D) is meromorphic on C\σ(A) and each λ ∈ σ(A) is a pole of G(s). Example 2.85 (First-order PDE) (Example 2.68 cont.) Consider the generator A on the state space Z = L 2 (0, 1) associated with this control system: Az = z  , dom(A) = {z ∈ H1 (0, 1) | z(1) = 0}. For any s ∈ C, f ∈ L 2 (0, 1) the boundary value problem sz(x) − z  (x) = f (x), z(1) = 0, has solution



1

z(x) =

f (r )es(x−r ) dr.

x

This defines a bounded operator on L 2 (0, 1) and so every s ∈ C is in the resolvent of the generator A defined in (2.67). The spectrum of A is empty. Theorem 2.84 implies that any transfer function associated with such a generator will have no poles. Indeed, the transfer function (2.68) was shown to have no poles.  Theorem 2.84 is further illustrated by the diffusion system in Example 2.78 where the poles of the transfer function (2.70) are the eigenvalues of the generator A, and also by the wave equation in Example 2.79 where the poles of the transfer function (2.79) are the eigenvalues of the generator A associated with the PDE (2.71), (2.72). Theorem 2.86 Suppose that the system (A, B, C, D) is minimal and that σ(A) is non-empty and consists only of isolated eigenvalues with finite algebraic multiplicity. Then the set of invariant zeros inv(A, B, C, D) is countable. Theorem 2.87 If A is Riesz-spectral and (A, B, C, D) is approximately observable and approximately controllable then the system is minimal. The term minimal is often used in finite-dimensional systems theory to refer to a system that is controllable and observable. However, for infinite-dimensional systems that are not Riesz-spectral observability and controllability does not always imply minimality. The invariant zeros can be characterized as the spectrum of an operator. The case where D is invertible is simplest and is presented first.

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2 Infinite-Dimensional Systems Theory

Theorem 2.88 The invariant zeros of (A, B, C, D) with D = 0 are the eigenvalues of the operator A∞ = A + B D −1 C with dom(A∞ ) = dom(A). Theorem 2.89 Suppose that (A, B, C, 0) is a minimal system on Z with a single input and output so that Bu = bu and C z = z, c for some b, c ∈ Z and also b, c = 0. For z ∈ dom(A∞ ) = dom(K ) = dom(A), define Kz = −

Az, c

, A∞ z = Az + B K z. b, c

(2.82)

Then (A + B K )(ker C ∩ dom(A)) ⊂ ker C and the invariant zeros of (A, B, C) are the eigenvalues of A∞ |ker C . Moreover, denoting by {μn } the invariant zeros of (A, B, C, 0), the eigenfunctions of A∞ restricted to the Hilbert space ker C, A∞ |ker C , are {(μn I − A)−1 b}. The kernel of C is ker C := {z ∈ Z | z, c = 0}. The operator K is unique up to addition of another operator K˜ where K˜ z = 0 for z ∈ ker C. If c ∈ dom(A∗ ), K is a bounded operator, but in general A∞ involves an unbounded perturbation and may not generate a C0 -semigroup. If c ∈ dom(A∗n ) for some integer n ≥ 1, the invariant zeros may also be characterized as the eigenvalues of an operator in many cases. It is convenient to introduce the notation, for any f ∈ Z, f ⊥ = {z ∈ Z| z, f = 0}. In particular, c⊥ = ker C. Theorem 2.90 Suppose that (A, B, C, 0) is a minimal system on Z with a single input and output so that Bu = bu and C z = z, c for some b, c ∈ Z and that an integer n ≥ 1 exists such that c ∈ D(A∗n ), and

b ∈ Z n−1

b, A∗n c = 0.

(2.84)

Define Z 0 = c⊥ = ker C, Z 1 = Z 0 ∩ (A∗ c)⊥ , . . . Z n = Z n−1 ∩ (A∗n c)⊥ . Defining K z = Az, a ,

a=

−An∗ c , b, An∗ c

(2.83)

D(K ) = dom(A),

2.7 Input/Output Maps

67

A∞ = A + B K , the invariant zeros of (A, B, C, 0) are the eigenvalues of A∞ | Z n . As in the case where b, c = 0, changing K on Z n does not change the conclusion of Theorem 2.90.

2.8 Notes and References A brief review of functional analysis is in Appendix A and can be found in more detail in a number of texts; [1] contains applications to control systems and [2, 3] include more general applications. A detailed introduction to infinite-dimensional systems theory that includes many examples is in [4]. For the reader wishing more detail on partial differential equations, good introductory textbooks on applied partial differential models are [5, 6]. The modelling of control of acoustic noise in a duct (Example 2.53) is in [7], Example 2.56 on acceleration measurement is from [8]. The theory of semigroups, including the Hille–Yosida Theorem, can be found in, for instance, [9]. Semigroup generation through sesquilinear forms is described in [10]. This book considers only Hilbert spaces as state-spaces, but the theory can be extended to Banach spaces. See for example, [11]. The use of a Banach space as a state space is particularly useful for delay-differential equations. Different versions of Theorem 2.43 can be found in a number of sources. The result was first proved in [12, 13]; see [14, 15] for the framework used here. Characterization of zeros for infinite-dimensional systems is not straightforward and only a few results are given here. More information on zeros, covered briefly in Sect. 2.7, can be found in [16–20]. The paper [16] contains some detail on the root locii of infinite-dimensional systems, which are not covered here. For many situations, such as control on the boundary of the region, typical models lead to a state-space representation where the input operator B is unbounded on the state-space. More precisely, it is a bounded operator into a larger space than the state space. This means that definition and boundedness of the controllability map into the state-space is not automatic and must be established. Similarly, point sensing leads to an output operator that is bounded from a space smaller than the state space. This complicates the state-space analysis and calculation of approximations. A complete treatment of observability and controllability that includes unbounded input/output operators is in [21]. Systems theory in an abstract setting for systems with unbounded control and observation is covered in [22]. Some systems theoretic points are sometimes clearer with boundary control and/or point sensing. In particular, the transfer function can be much easier to calculate. A number of examples as well as Example 2.79, which involves bounded control and observation, can be found in the tutorial paper [23]. Theory on relating transfer

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functions to boundary value problems is in [24]. The book [25] provides a detailed introduction to Laplace transforms in general spaces and [26] discusses the issue of defining transfer functions in detail. A introduction to infinite-dimensional systems using the port-Hamiltonian approach is provided in [27].

References 1. Banks HT (2012) A functional analysis framework for modeling, estimation and control in science and engineering. CRC Press, Boca Raton 2. Kreyszig E (1978) Introductory functional analysis with applications. Wiley, New Jersey 3. Naylor AW, Sell GR (1982) Linear operator theory in science and engineering. Springer, New York 4. Curtain RF, Zwart H (1995) An introduction to infinite-dimensional linear systems theory. Springer, Berlin 5. Guenther RB, Lee JW (1988) Partial differential equations of mathematical physics and integral equations. Prentice-Hall, Upper Saddle River 6. Trim DW (1990) Applied partial differential equations. PWS-Kent 7. Zimmer BJ, Lipshitz SP, Morris KA, Vanderkooy J, Obasi EE (2003) An improved acoustic model for active noise control in a duct. ASME J Dyn Syst, Meas Control 125(3):382–395 8. Jacob B, Morris KA (2012) Second-order systems with acceleration measurements. IEEE Trans Autom Control 57:690–700 9. Pazy A (1983) Semigroups of linear operators and applications to partial differential equations. Springer, Berlin 10. Showalter RE (1977) Hilbert space methods for partial differential equations. Pitman Publishing Ltd., London 11. Bensoussan A, DaPrato G, Delfour MC, Mitter SK (2007) Representation and control of infinite dimensional systems. Birkhauser, Basel 12. Banks HT, Ito K (1988) A unified framework for approximation in inverse problems for distributed parameter systems. Control Theory Adv Tech 4:73–90 13. Lasiecka I (1989) Stabilization of wave and plate equations with nonlinear dissipation on the boundary. J Differ Equ 79(2):340–381 14. Banks HT, Ito K, Wang Y (1995) Well-posedness for damped second-order systems with unbounded input operators. Differ Integr Equ 8:587–606 15. Chen S, Liu K, Liu Z (1998) Spectrum and stability for elastic systems with global or local Kelvin-Voigt damping. SIAM J Appl Math 59(2):651–668 16. Jacob B, Morris KA (2016) Root locii for systems defined on Hilbert spaces. IEEE Trans Autom Control 61(1):116–128 17. Morris KA, Rebarber RE (2007) Feedback invariance of SISO infinite-dimensional systems. Math Control, Signals Syst 19:313–335 18. Morris KA, Rebarber RE (2010) Zeros of SISO infinite-dimensional systems. Int J Control 83(12):2573–2579 19. Zwart H, Hof MB (1997) Zeros of infinite-dimensional systems. IMA J Math Control Inf 14:85–94 20. Zwart H (1990) A geometric theory of infinite-dimensional systems. Springer, Berlin 21. Tucsnak M, Weiss G (2009) Observation and control for operator semigroups. Birkhauser, Basel 22. Staffans O (2005) Well-posed linear systems. Cambridge University Press, Cambridge 23. Curtain RF, Morris KA (2009) Transfer functions of distributed parameter systems: a tutorial. Automatica 45(5):1101–1116 24. Cheng A, Morris KA (2003) Well-posedness of boundary control systems. SIAM J Control Optim 42(4):1244–1265

References

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25. Zemanian AH (1965) Distribution theory and transform analysis. Dover Publications, United States 26. Zwart H (2004) Transfer functions for infinite-dimensional systems. Syst Control Lett 52:247– 255 27. Jacob B, Zwart HJ (2012) Linear port-Hamiltonian systems on infinite-dimensional spaces. Operator theory: advances and applications, vol 223. Birkhäuser/Springer, Basel AG, Basel. (Linear operators and linear systems)

Chapter 3

Dynamics and Stability

But the future must be met, however stern and iron it be. (Elizabeth Gaskell, North and South)

Stability is an important property of all systems, whether natural or engineered. Various definitions are possible. The following two types of stability are appropriate for the study of linear DPS. Definition 3.1 The semigroup S(t) is exponentially stable if there is M ≥ 1, α > 0 such that S(t) ≤ Me−αt for all t ≥ 0. For exponentially stable semigroups, the corresponding Lyapunov equation has a solution. Theorem 3.2 Consider the control system (A, B) on Z where A generates an exponentially stable C0 -semigroup. The Lyapunov equation A∗ Lz + L Az + B B ∗ z = 0, for all z ∈ dom(A), has a unique self-adjoint positive semi-definite solution L ∈ B(Z, Z) and L : dom(A) → dom(A∗ ). The operator L is known as the controllability Gramian. The controllability Gramian L is positive definite, that is, for all non-zero z ∈ Z, Lz, z > 0, if and only if (A, B) is approximately controllable. The operator L is coercive, that is there is c > 0 such that for all z ∈ Z Lz, z ≥ cz2 , if and only if (A, B) is exactly controllable. Definition 3.3 The semigroup S(t) is asymptotically stable if for every z 0 ∈ Z, limt→0 S(t)z 0 = 0. © Springer Nature Switzerland AG 2020 K. A. Morris, Controller Design for Distributed Parameter Systems, Communications and Control Engineering, https://doi.org/10.1007/978-3-030-34949-3_3

71

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3 Dynamics and Stability

In the infinite-dimensional systems literature the term strongly stable is often used in place of asymptotically stable. Clearly, every exponentially stable system is asymptotically stable. For finitedimensional linear systems, asymptotic and exponential stability are equivalent: either type of stability holds if and only if all eigenvalues of A have negative real parts. It is possible for a linear infinite-dimensional system to be asymptotically stable but not exponentially stable. Example 3.4 (Generalized Fourier series) Consider a system with the semigroup written using a generalized Fourier series as in Example 2.11. Let {φn }∞ n=1 be an orthonormal basis for a separable Hilbert space Z, and define the C0 −semigroup S(t)z 0 =

∞ 

e− n t z 0 , φn φn . 1

n=1

For every z 0 ∈ Z, limt→∞ S(t)z 0  = 0 so the semigroup is asymptotically stable. However, 1 S(t)φn  = e− n t and so it is not exponentially stable.



In Example 3.4 there is a sequence of eigenvalues λn with Reλn < 0 but supn Reλn = 0. This behavior sometimes occurs with lightly damped waves. The stability of a finite-dimensional system is determined entirely by the eigenvalues of the matrix A. The spectrum of an infinite-dimensional system can contain elements besides eigenvalues—for instance it is possible that the inverse of λI − A exists but is not a bounded operator. Definition 3.5 Let A : dom(A) ⊂ Z → Z be a linear operator on a Hilbert space Z. If λI − A is one-to-one and onto Z so that the inverse (λI − A)−1 exists and is a bounded operator defined on all of Z, then λ is in the resolvent set of A, ρ(A). Otherwise λ is in the spectrum of A, σ(A). The spectrum is classified as follows: – If λI − A is not one-to-one, so that its inverse does not exist, λ is in the point spectrum of A, denoted σ p (A). – If λI − A is one-to-one and the range of (λI − A) is dense in Z, but (λI − A)−1 is not a bounded operator, then λ is in the continuous spectrum of A, denoted σc (A). – If λI − A is one-to-one but the range of (λI − A) is not dense in Z, λ is in the residual spectrum, denoted σr (A). Elements of the point spectrum are often referred to as eigenvalues. Classification of the spectrum is illustrated in Fig. 3.1. Since continuity of a linear operator is equivalent to it being bounded, the terminology continuous spectrum for points where the inverse is not continuous at first seems strange. But as will be seen in examples, often (not always) the continuous spectrum appears as a continuous curve in the

3 Dynamics and Stability

73

Fig. 3.1 Classification of the spectrum of an operator

complex field. The residual spectrum includes all the points that don’t fit into the other categories; that is, residual points. The following example illustrates that the spectrum of an operator on a general Hilbert space can contain elements other than eigenvalues, and that these elements may affect stability. Example 3.6 Consider the operator on L 2 (0, ∞) Az = −

dz , dx

dom(A) = { f ∈ L 2 (0, ∞)| on every interval [0, b], there is g ∈ L 2 (0, ∞),  x f (x) = g(s)ds}. 0

The spectrum of A can be calculated. Consider first point spectrum. The only solution to dz = λz, z(0) = 0 − dx is the zero function and so A has no eigenvalues: the point spectrum is empty. For λ ∈ C, g ∈ L 2 (0, ∞), the inverse of (λI − A) is defined by the solution to λz +

dz = g, z(0) = 0 dx

(3.1)

when this solution exists. For any λ ∈ C, g ∈ L 2 (0, ∞) the solution to this differential equation is z(x) = e

−λx



x 0

eλτ g(τ )dτ

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3 Dynamics and Stability



x

=

e−λ(x−τ ) g(τ )dτ .

0

Defining

 h(x) =  z(x) =

∞

0 r 0,  z = 2

0

∞

 ∞ ∞ 1 |z(x)| d x ≤ |h(x − τ )||g(τ )|2 d xdτ |Reλ| 0 0  ∞ 1 |g(τ )|2 dτ = |Reλ|2 0 1 g2 . = |Reλ|2 2

Thus, for Reλ > 0, (3.2) defines a bounded operator from Z to Z and all Reλ > 0 are in the resolvent set of A. If Reλ < 0 then eλx ∈ Z = L 2 (0, ∞) and for z ∈ dom(A), dz eλx , (λI − A)z = eλx , λz + d x  ∞ dz = )d x eλx (λz + d x 0 ∞ d λx = (e z(x))d x d x 0 =0 since z ∈ dom(A) implies that z(0) = 0 and also lim x→∞ eλx z(x) = 0. Thus, the range of λI − A is orthogonal to the function eλx and the range is not dense. The set Reλ < 0 is in the residual spectrum. (It can be shown that Range (λI − A) is exactly the subspace of Z orthogonal to eλx ). If Reλ = 0, consider again (3.2). Set λ = 0. For any z ∈ dom(A)

3 Dynamics and Stability

75



∞

− 0

(Az)(x)d x

= lim N →∞

N 0

dz dx dx

= lim N →∞ z(N ) − z(0) = lim N →∞ z(N ).

Since z ∈ dom(A), it is on every compact interval the integral of a square integrable function. This implies that lim N →∞ z(N ) = 0. Thus, the range of A is contained in the set  ∞ 2 g(t)dt = 0}. S = {g ∈ L (0, ∞); 0

It can be shown after some calculation that S equals Range (A) and also that S is dense in Z. Furthermore, (3.2) defines the inverse map but it is not bounded on Z. It follows that 0 is in the continuous spectrum. The more general case of λ = j ω, ω ∈ R, but ω = 0 can be transformed into the λ = 0 case by multiplying (3.1) through by ej ωt and setting y(t) = ej ωt z(t), f (t) = ej ωt g(t) which reduces (3.1) to d y(x) = f (x) dx which is precisely the case λ = 0. Thus, the imaginary axis is in the continuous spectrum of A, σc (A). The operator A generates the right shift semigroup  (S(t)z 0 )(x) =

0 x t

on Z = L 2 (0, ∞). This can be verified to be the solution to the first-order partial differential equation ∂z ∂t

∂z = − ∂x , z(x, 0) = z 0 (x), 0 < x < ∞. z(0, t) = 0.

(3.3)

Clearly, S(t) ≤ 1. For positive integers n consider the sequence of functions ⎧ 0 ⎪ ⎪ ⎪ ⎪ ⎨ 3 (t − n + 1) f n (x) = 2 3 ⎪ ⎪ ⎪ 2 (n + 1 − t) ⎪ ⎩ 0

0≤ x c2 so that for all φ, z ∈ V, |a(φ, z)| + αφ, z Z ≤ c1 φV zV . Thus, A generates an exponentially semigroup S(t) on L 2 (0, 1) with bound S(t) ≤  e−αt . Example 3.11 (Diffusion in  ⊂ Rn ) Consider diffusion of a quantity z with smooth diffusivity coefficient κ(x) > 0 on a bounded region  ⊂ Rn where 1 ≤ n ≤ 3 has a smooth boundary ∂. The variable z(x, t) is governed by ∂z = ∇ · (κ(x)∇z). ∂t

(3.4)

3.1

Sesquilinear Forms

79

The boundary condition is z(x, t) = 0, x ∈ ∂.

(3.5)

In formal operator form, this model is z˙ (t) = Az(t), where (Az)(x) = ∇ · (κ(x)∇z)(x), dom(A) = H2 () ∩ H01 (). It was shown in Example 2.41 that defining  a(w, v) =

κ(x)∇w(x)∇v(x)d x, 

(3.6)

Aw, v = −a(w, v). The eigenvalues of the negative Laplace operator are the non-trivial solutions to λφ = −∇ 2 φ, φ(x) = 0, x ∈ ∂. Their values depend on the region . However, since φ(x) = 0 on the boundary, all the eigenvalues are positive. Let λ1 > 0 indicate the smallest eigenvalue. The Poincaré Inequality then implies a(φ, φ) − λ1 φ, φ L 2 () ≥ 0. As in Example 3.10, it can be shown that A generates an exponentially stable semi group S(t) on L 2 () with for any α > −λ1 , S(t) ≤ eαt . The stability of second-order in time systems depends on the damping. As in (2.41), consider second-order in time systems satisfying Definition 2.42. For Hilbert spaces Vo → Ho , let the stiffness operator Ao : dom(Ao ) ⊂ Vo → Ho be a selfadjoint operator on Ho that is defined through a symmetric sesquilinear form ao continuous on Vo .The damping operator D is also self-adjoint and can be defined through a continuous symmetric sesquilinear form on Vo . As for (2.39) ao and d define a second-order differential equation on Vo : for any ψ ∈ Vo , ˙ = 0. ψ, w(t) ¨ Ho + ao (ψ, w(t)) + d(ψ, w(t))

(3.7)

Let Ao w indicate the element of Vo defined by a(·, w) and similarly let Dv indicate the element of Vo defined by d(·, v). As in (2.42), the differential equation (3.7) can be written in standard first-order form with state z = (w, w) ˙ on the space Z = Vo × Ho . For every (w, v) ∈ Vo × Vo ao (·, w) + d(·, v)

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3 Dynamics and Stability

(or equivalently Ao w + Dv) is an element of Vo . For some (w, v) ∈ Vo × Vo there is an element v A ∈ Ho so that for all ψ ∈ Vo , ao (ψ, w) + d(ψ, v) = (ψ, v A )Ho . Define then    dom(A) = wv ∈ Vo × Vo | ao (ψ, w) + d(ψ, v) = (ψ, v A )Ho , for all ψ ∈ Vo .     w v . (3.8) A = v −v A Theorem 2.43 states that if in addition there is c > 0 such that ao (w, w) ≥ cw2Vo , d(v, v) ≥ 0, for all w, v ∈ Vo then the operator A defined in (3.8) is the generator of a contraction on the state space Z = Vo × Ho . Depending on the strength of the damping, the model may be further asymptotically or exponentially stable. In the following theorem, Ao indicates the operator on Ho with domain dom(Ao ) ⊂ Vo associated with the form ao . That is, for some w ∈ Vo there is w A ∈ Ho such that ao (φ, w) = φ, w A Ho ,

for all φ ∈ Vo .

Then dom(Ao ) = {w ∈ Vo | there exists w A ∈ Ho , ao (φ, w) = φ, w A Ho , for all φ ∈ Vo }, Ao w = w A .

Theorem 3.12 Consider the second-order system (3.7) defined through symmetric sesquilinear forms a0 and d each continuous on Vo → Ho such that there is c2 > 0 so a0 (w, w) ≥ c2 w2Vo and also d(v, v) ≥ 0 for all w, v ∈ Vo . 1. If A−1 o is a compact operator on Ho and for any eigenvector φ of Ao , d(φ, φ) > 0 then A defined in (3.8) generates an asymptotically stable semigroup on Z = Vo × Ho . 2. If there is cd > 0 so that for all v ∈ Vo , d(v, v) ≥ cd v2Ho

(3.9)

3.1

Sesquilinear Forms

81

then A as defined in (3.8) generates an exponentially stable semigroup S on Vo × Ho . Furthermore, there is M ≥ 0 such that, defining ω0 ≤ max{−

cd , −A−1 −1 }, 2

S(t) ≤ Meω0 t . 3. If in addition to (3.9), (a) A−1 o is a compact operator and (b) D extends to a bounded operator on H, then letting m d > 0 indicate the constant such that |d(v, v)| ≤ m d v2Ho and m v the constant so that vVo ≥ m v vHo for all v ∈ Vo , if m 2d m 2v < 4c2 , then cd ω0 ≤ − . 2 The conditions in item (3) in the above theorem remove the possibility that for large cd the system is over-damped. Definition 3.13 A Hilbert space V is compactly embedded or compact in another Hilbert space H if V is dense in H and the unit ball in the V norm is compact in the H-norm. Equivalently, every sequence {z k }∞ k=1 ⊂ V with z k V ≤ M for some M has a subsequence convergent in the H-norm. Using the notation of Theorem 3.12, for any w ∈ Vo A−1 o w ∈ dom(A). Since is compact in H follows if the unit ball dom(A) ⊂ Vo the assumption that A−1 o o in Vo in compact in the Ho -norm. For every domain , Rellich’s Theorem (Theorem A.35) states that for m ≥ 1, Hm () is compact in L 2 (). In many applications, dom(Ao ) ⊂ Hm () for some m, and so A−1 o is often a compact operator. Example 3.14 (Wave equation with distributed damping) The simple wave equation does not have any damping. More realistic models include damping. In this example two types of damping are considered. In one approach, for some function b(·) ∈ L 2 (0, 1) consider   ∂ 2 z(x, t) ∂z(·, t) ∂ 2 z(x, t) , b(·) b(x) − + = 0, ∂t 2 ∂t ∂x 2

0 0 describes elasticity and cd > 0 the damping. If the end x = 0 is clamped and the other end free, as in Example 2.44, the boundary conditions are ∂w |x=0 = 0, ∂x   3 ∂4w ∂ w = 0, E 3 + cd = 0. ∂x ∂t∂x 3 x=1

w(0, t) = 0, 

∂2w ∂3w E 2 + cd I ∂x ∂t∂x 2

 x=1

Define the state z = (w, w), ˙ Ho = L 2 (0, 1) and the Hilbert space Vo = {w ∈ H2 (0, 1); w(0) = w (0) = 0} with the inner product

 w1 , w2 Vo = E 0

1

w1

w2

d x.

This model is in the form (3.7) with a(φ, w) = φ, w Vo , d(φ, v) =

cd a(φ, v). E

It was shown in Example 2.44 that the model leads to a contraction semigroup S(t) on Vo × Ho . Furthermore, since a(w1 , w2 ) = w1 , w2 Vo and

84

3 Dynamics and Stability

d(v, v) =

cd cd a(v, v) = v2Vo ≥ cd v2Ho , E E

(3.12)

condition (2) (or (3)) in Theorem 3.12 implies that if cd > 0 the semigroup is expo nentially stable on Vo × Ho .

3.2 Spectrum Determined Growth For finite-dimensional systems, the eigenvalues of the matrix A entirely determine the growth or decay of the associated matrix exponential. Most importantly, if all the eigenvalues lie in the open left half plane, that is have negative real parts, the system is exponentially stable. It has already been noted that the spectrum of an operator on an infinite-dimensional system may contain elements that are not eigenvalues. (See Definition 3.5 and Example 3.6.) Furthermore, it is possible for the spectrum of a generator to be contained in the open left-hand-plane and the corresponding semigroup may fail to be even asymptotically stable. Reference to several counterexamples illustrating this are provided in the notes at the end of this chapter. This section is concerned with providing sufficient conditions for the generator’s spectrum to determine stability of the system. Definition 3.16 The growth bound ωo of a C0 -semigroup S(t) on a Hilbert space Z is ωo = inf{ω ∈ R | there exists M > 0 : S(t) ≤ Meωt , t ≥ 0}. Theorem 3.17 Suppose A : dom(A) ⊂ Z → Z generates a C0 -semigroup S(t) on a Hilbert space Z. The growth bound ωo of S(t) satisfies ωo ≥ sup Reλ. λ∈σ(A)

(3.13)

Definition 3.18 Suppose A : dom(A) ⊂ Z → Z generates a C0 -semigroup S(t) on a Hilbert space Z. If the growth bound ωo of S(t) satisfies ωo = sup Reλ, λ∈σ(A)

(3.14)

A is said to satisfy the Spectrum Determined Growth Assumption (SDGA) . Definition 3.19 If A generates an exponentially stable semigroup, that is, the growth bound is negative, then A is Hurwitz. Since the spectrum of A may contain points that are not eigenvalues, and the spectrum does not always determine stability the property of A being Hurwitz is considerably stronger than merely having all its eigenvalues with negative real parts.

3.2

Spectrum Determined Growth

85

However, the SDGA does hold for many semigroups that arise from PDEs. Some useful conditions are listed here. Consider first Riesz spectral operators (Definition 2.23). Theorem 3.20 If A is a Riesz-spectral operator and sup Reλn (A) < ∞ n

then the semigroup generated by A satisfies the SDGA. Theorem 3.21 Suppose that A is a Riesz-spectral operator on Z with supn Reλn (A) < ∞. If for any r > 0 there is an integer k so that for every point μ ∈ C, {s ∈ C| |s − μ| ≤ r } contains only k elements of the spectrum of A, then for any bounded operator D ∈ B(Z, Z) the operator A + D generates a C0 -semigroup on Z that satisfies the SDGA. Example 3.22 (Simply supported beam) Let w(x, t) denote the deflection of a simply supported beam from its rigid body motion at time t and position x. Normalizing the variables and including viscous damping with parameter cv leads to the partial differential equation ∂w ∂ 4 w ∂2w + + c = 0, 0 < x < 1, v ∂t 2 ∂t ∂x 4

(3.15)

with boundary conditions w(0, t) = 0, w

(0, t) = 0, w(1, t) = 0, w

(1, t) = 0. Defining

∂w(x, t) ∂ 2 w(x, t) , , v(x, t) = ∂x 2 ∂t   choose state z(t) = m(x, t) v(x, t) . Define the operator on L 2 (0, 1), m(x, t) =

∇2w =

d 2w , dom(∇ 2 ) = {w ∈ H2 (0, 1)| w(0) = 0, w(1) = 0} dx2

and the operator on Z = L 2 (0, 1) × L 2 (0, 1)  0 ∇2 , dom(A) = dom(∇ 2 ) × dom(∇ 2 ). A= −∇ 2 −cv I 

The PDE can be written in state space form on Z = L 2 (0, 1) × L 2 (0, 1) as

(3.16)

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3 Dynamics and Stability

z˙ (t) = Az(t). This example is very similar to Example 2.26 except that the Kelvin–Voigt damping in this case is zero, and there is a different choice of the state, leading to a different state-space. The generator A in this example can also be shown to be a Riesz-spectral operator. Solve the eigenvalue problem λz = Az, or, with m = w

, v = λw, d 4w = (λ2 + cv λ)w. dx4 Writing μn = nπ, n = 1, 2 . . . , the eigenvalues λn are the solutions of λ2n + cv λn + μ4n = 0, so that λ±n =

−cv 1 2 c − 4μ4n , n = 1, 2, . . . ± 2 2 v

and the eigenfunctions are, letting an be an arbitrary constant,  φn (x) = an

−μ2n sin(μn x) λn sin(μn x)



√ Since { 2 sin(μn x)} form a orthogonal basis for L 2 (0, 1), an argument identical to that in Example 2.26 shows that {φn } forms an orthogonal basis for Z = L 2 (0, 1) × L 2 (0, 1). The constants an can be chosen so that φn  = 1. Assuming that cv2 < 4π 4 , as is the case in applications, each λ has real part −c2 v < 0 and the system is exponentially stable.  In some situations a generator A, not necessarily Riesz-spectral, generates a C0 semigroup. In some of these cases the semigroup has stronger properties, including the SDGA. Definition 3.23 A C0 -semigroup S(t) is an analytic semigroup if for some α > 0 it can be continued from t ∈ R+ to an analytic function in t on the sector {t ∈ C| | arg(t) < α, t = 0}. Analytic semigroups have a number of nice properties, including the SDGA. Some are listed in the following theorem. Theorem 3.24 Suppose that A with domain dom(A) generates an analytic semigroup S(t) on Z and define ωo = supλ∈σ(A) Re(λ). Then

3.2

Spectrum Determined Growth

87

1. for all t > 0, z ∈ Z, S(t)z ∈ dom(A), 2. the spectrum of A is contained in a sector of C, that is, for some real c < 0, σ(A) ⊂ {λ ∈ C; Reλ − ωo ≤ c|Imλ|}; 3. For some M ≥ 1, T (t) ≤ Meωo t . Corollary 3.25 Suppose that A generates an analytic semigroup on Z. Then for any real k, A + k I generates an analytic semigroup on Z and for any bounded operator D ∈ B(Z, Z) the operator A + D generates a C0 -semigroup on Z that satisfies the SDGA. Theorem 3.24 implies that the SDGA holds for analytic semigroups. Furthermore, σ(A) is contained in a sector. Semigroups generated by a Riesz spectral operator with spectrum that satisfies the second property are analytic. However, not every Riesz-spectral operator generates an analytic semigroup and semigroups that are not necessarily generated by a Riesz-spectral operator may be analytic. In particular, an operator defined through a coercive sesquilinear form (Definition 2.37) generates an analytic semigroup. Theorem 3.26 If A is a V-coercive operator with V → Z then it generates an analytic C0 -semigroup on Z. A second-order in time system (3.7) with stiffness form ao and damping form d each continuous with respect to Hilbert spaces Vo → Ho can be defined as a first-order system through a sesquilinear form (see (2.43)) a((φ, ψ), (w, v)) = −φ, v Vo + ao (ψ, w) + d(ψ, v). This form is continuous with respect to Vo × Vo → Vo × Ho . If in addition to the assumptions of Theorem 2.43 guaranteeing that associated generator A is a contraction on Vo × Ho , d is Vo -coercive, then for any k > 0, a(·, ·) + k·, · Vo ×Ho is coercive with respect to Vo × Vo . Theorems 3.25 and 3.26 imply that A generates an analytic semigroup. Corollary 3.27 Consider a second-order (Vo , Ho ) system (Definition 2.42) defined through the positive definite stiffness form ao and non-negative damping form d. If there is cd > 0 so that for all v ∈ Vo , d(v, v) ≥ cd v2Vo then letting A indicate the generator, A generates an exponentially stable and analytic semigroup S(t) on Vo × Ho .

88

3 Dynamics and Stability

Example 3.28 (Beam vibrations with Kelvin–Voigt damping, Example 3.15 cont.) From (3.12) the damping in this second-order system satisfies d(v, v) =

Cd Cd a(v, v) = v2Vo E E

and so not only is the system exponentially stable, but the semigroup is analytic. 

3.3 Boundary Conditions The boundary conditions of a PDE affect the stability of the system. Example 3.29 (Diffusion) In the model of heat flow first introduced in Example 1.1, the PDE is ∂z(x, t) ∂ 2 z(x, t) = , ∂t ∂x 2

x ∈ (0, 1),

t ≥ 0, z(x, 0) = z 0 (x),

and the boundary conditions are z(0, t) = 0, z(1, t) = 0.

(3.17)

Defining the operator A : dom(A) ⊂ L 2 (0, 1) → L 2 (0, 1) Az =

d2z , dom(A) = {z ∈ H2 (0, 1) | z(0) = z(1) = 0}, dx2

this PDE can be written in state space form with state space L 2 (0, 1) as z˙ (t) = Az(t), z(x, 0) = z 0 (x). 2 2 The operator A is Riesz-spectral √ with eigenvalues −n π , n = 1, 2, . . . and orthonormal eigenfunctions φn (x) = 2 sin(nπx). (See Example 2.2.) Since supn Reλn = −π 2 Theorem 3.20 implies that the system is exponentially stable. Suppose that instead of temperature fixed at the ends (3.17), the ends are insulated. This leads to the boundary conditions

∂z (0, t) = 0, ∂x

∂z (1, t) = 0. ∂x

The state space is still L 2 (0, 1) but the generator is now Az =

∂2 z , dom(A) = {z ∈ H2 (0, 1) | z (0) = z (1) = 0} . ∂x 2

(3.18)

3.3

Boundary Conditions

89

The different boundary conditions lead to a different domain, and so a different oper2 2 ator. The operator is still Riesz-spectral, with √ eigenvalues λn = −n π , n = 0, 1, . . . and eigenfunctions ψ0 (x) = 1, ψn (x) = 2 cos(nπx), n = 1, 2, . . . Because of the 0 eigenvalue, the system is not exponentially stable. With Neumann boundary conditions the system is not even asymptotically stable. The solution to the differential equation is S(t)z 0 = z 0 , ψ0 ψ0 (x) +

∞ 

z 0 , ψn ψn (x)eλn t .

n=1

For large time, the solution converges to z 0 , ψ0 . This generalizes to diffusion on general domains. Example 3.11 with the state set to 0 on the boundary, (3.5), is exponentially stable. Suppose instead of the boundary condition (3.5), there is no flux across the boundary as in Example 2.34. That is, letting n indicate the outward normal along the boundary ∂, ∇z(x, t) · n(x) = 0, x ∈ ∂. In this example, V = H1 0, 1), H = L 2 (0, 1). The form a(v, v) ≥ 0 and so A generates a contraction. In this case, Az = ∇ 2 z, dom(A) = {z ∈ H2 (0, 1) | ∇z(x) · n(x) = 0, x ∈ ∂}. Every constant function z 0 ∈ dom(A) and Az 0 = 0 so 0 is an eigenvalue. Thus, the growth bound of the semigroup is 0. The system is not asymptotically stable.  Example 3.30 (Wave equation with boundary dissipation) Consider again the undamped wave equation on an interval ∂ 2 w(x, t) ∂ 2 w(x, t) − = 0, ∂t 2 ∂x 2

0 < x < 1.

(3.19)

In general, waves are not entirely transmitted or reflected on the boundary. Some energy is transmitted into the exterior and some is reflected back into the region. This dissipative behaviour can be modelled by a mixed boundary condition at x = 1. For simplicity a simple boundary condition at x = 0 is kept. Consider for real 0 < β < 1, the boundary conditions w(0, t) = 0,

∂w ∂w (1, t) + β (1, t) = 0. ∂x ∂t

As in Example 3.14, Ho = L 2 (0, 1), but now Vo = {w ∈ H1 (0, 1), w(0) = 0}

90

3 Dynamics and Stability

with inner product



1

v1 , v2 Vo = 0

v1 (x)v2 (x)d x.

For w ∈ H 2 (0, 1) ∩ Vo and φ ∈ Vo ,  −

1



1

(φ(x))w

(x)d x =

0

φ (x)w (x)d x − φ(1)w (1).

(3.20)

0

Define the sesquilinear form on Vo 

1

ao (φ, w) =

φ (x)w (x)d x.

0

Since ao (w, v) = w, v Vo , this form is continuous with respect to Vo and also Vo coercive. If w ∈ Vo also satisfies w (1) = 0, then, φ, −w

= ao (φ, w) where ·, · as usual indicates the inner product on L 2 (0, 1). This defines the stiffness operator Ao w = −

d 2w , dom(Ao ) = {w ∈ Vo ∩ H2 (0, 1); w (1) = 0}. dx2

Any solution to the PDE (3.19) satisfies for all φ ∈ Vo , φ, w ¨ − φ, w

= 0. Then (3.20) and the boundary condition at x = 1 yields ˙ = 0. φ, w ¨ + ao (φ, w) + βφ(1)w(1) This is a second-order in time system on Vo of the form (3.7), with damping d(φ, v) = βφ(1)v(1). Rellich’s Theorem (Theorem A.35) implies that d is a continuous sesquilinear form on Vo and it is clear that d is non-negative and symmetric. Since Vo is compact in Ho (Rellich’s Theorem A.35), A−1 o is a compact operator on Ho . The eigenfunctions φ of Ao are the solutions to −

∂2φ = μ2 φ, φ(0) = 0, ∂x 2

∂φ (1) = 0. ∂x

(3.21)

3.3

Boundary Conditions

91

These are, for integers n ≥ 1, 1 φn (x) = sin((n − )πx) 2 with eigenvalues μ2n = (n − 21 )2 π 2 . Since d(φn , φn ) = β, Theorem 3.12 implies that the system is asymptotically stable when β > 0. Consider v = φ1 + φ2 . Since φ1 (1) + φ2 (1) = 0, d(v, v) = 0 and Theorem 3.12 does not imply exponential stability. A state-space representation of the system will now be derived and shown to be Riesz-spectral. For (w, v) ∈ Vo × Vo , ao (·, w) + d(·, v) defines an element of Vo : for any φ ∈ Vo ,  ao (φ, w) + d(φ, v) =

1

φ (x)w (x)d x + βφ(1)v(1).

0

As discussed above, for some w, v this defines an element of Ho . If w ∈ H2 (0, 1), integrate the first term by parts yielding 

1

ao (φ, w) + d(φ, v) = −

φ(x)w

(x)d x + φ(1)(w (1) + βv(1)).

0

Thus, if w ∈ H2 (0, 1), and w (1) + βv(1) = 0, then ao (·, w) + d(·, v) = ·, (−w

) Ho The operator A defined by the forms a, d (see (3.8)) is  A=

 0 I , 2 d w 0 dx2

dom(A) = {(w, v) ∈ Vo × Vo |w ∈ H2 (0, 1), w (1) + βv(1) = 0}. The operator A is a Riesz-spectral operator. There are a few ways to show this. One method is to consider first the eigenfunctions of the operator Au for the PDE with no boundary dissipation, that is β = 0:     0 I w , dom(Au ) = {(w, v) ∈ Vo × Vo |w ∈ H2 (0, 1), w (1) = 0}. Au = d2w v 0 dx2 The non-trivial solutions to (3.21) are, for integers n ≥ 1, μn = (n − 21 )π and sin(μn x). The eigenvalues of Au are λu,n = j μn , λu,−n = −j μn , with eigenfunctions

92

3 Dynamics and Stability

 φn (x) = c˜n

   1 sin(μn x) sin(μn x) −j μ n , φ−n (x) = c˜−n , n = 1, 2, . . . sin(μn x) sin(μn x)

1 j μn

or equivalently,  φn (x) = cn





1 λu,n sinh(λu,n x) , sinh(λu,n x)

φ−n (x) = c−n



1 λu,−n sinh(λu,−n x) , sinh(λu,−n x)

n = 1, 2, . . .

where c˜n , cn are arbitrary constants. As for the Euler–Bernoulli beam analyzed in Example 2.26, {φn } form an orthogonal basis for Vo × Ho and if the constants an are chosen appropriately, the basis is orthonormal. It follows that Au is a Riesz-spectral operator. For the case where β = 0, the eigenvalues and eigenfunctions of A can be calculated in the same manner to be   + j (nπ − π2 ), n = 1, 2, . . . λn = − 21 ln 1+β 1−β   . − j (nπ − π2 ) λ−n = − 21 ln 1+β 1−β The eigenfunctions are, for n = 1, 2, . . . and arbitrary constants bn ,  ψn (x) = bn

  1  sinh(λ−n x) sinh(λn x) . , ψ−n (x) = b−n λ−n sinh(λn x) sinh(λ−n x)

1 λn

Using double angle formula for hyperbolic trigonometric functions, the eigenfunctions {ψn } can be shown to be related to the orthonormal basis {φn } by a bounded invertible transformation T : ψn = T φn . Theorem 2.21 then implies that A is a Riesz-spectral operator and so the stability is determined by the spectrum of A. Since 0 < β < 1, all the eigenvalues have negative real parts and in this case the associated semigroup is exponentially stable. If β = 1, all waves are transmitted at x = 1 and the solution consists only of a right travelling wave. Direct analysis of the solution shows that it converges to 0 in finite time and it is exponentially stable.  Example 3.31 (Vibrations on Rn ) This is a generalization of Example 3.30 to higher spatial dimensions. Consider a structurally damped plate on a domain  ⊂ Rn with smooth boundary . Deflections are held to 0 on part of the boundary, 0 , with partial dissipation on the other part of the boundary, 1 , where  = 0 ∪ 1 and 0 , 1 are disjoint open subsets of  with both 0 and 1 not empty. Assume the damping function β ∈ C(1 ) ∩ L 2 (1 ) and inf x∈1 β(x) > 0. Deflections are modelled by, letting n indicate the outward normal along the boundary

3.3

Boundary Conditions

93

w(x, ¨ t) = ∇ 2 w(x, t), ˙ 0) = w1 (x), w(x, 0) = w0 (x), w(x, w(x, t) = 0, ∇w(x) · n + β(x)w(x, ˙ t) = 0

x x x x

∈  × (0, ∞), ∈ , ∈ 0 × (0, ∞), ∈ 1 × (0, ∞).

(3.22)

Define Ho = L 2 () and Vo = H1 0 () = {g ∈ H1 () | g|0 = 0}. Since 0 is assumed not empty, the inner product on Vo is defined as (see (2.49) in Example 2.45)   f, g =



∇ f (x) · ∇g(x)d x.

It was shown in Example 2.45 that if β = 0, the associated semigroup is a contraction on Vo × Ho . Using the Divergence Theorem, for w ∈ H 2 () and φ ∈ H 1 (), 

 −



(φ(x))∇ 2 w(x)d x =





∇φ(x) · ∇w(x)d x −



(φ(x))∇w(x) · n(x)d x

for φ ∈ Vo = H10 (). If w also satisfies the boundary condition on 1 ,    2 − φ∇ wd x = ∇φ · ∇wd x + βφwd ˙ x. 



(3.23)

1

Define the sesquilinear form on Vo ,  ao (φ, w) =



∇φ · ∇wd x.

It is clear that ao is a Vo -coercive form; in fact it is precisely the Vo -inner product. An operator γ1 from a function in Vo to its values on the boundary 1 , denoted by γ1 (g) = g|1 can be defined. It is a Dirichlet trace operator. The operator γ1 is known to be a linear bounded operator from Vo to L 2 (1 ). Define also the sesquilinear form on Vo ,  d(φ, v) =

1

β(x)(γ1 φ(x))(γ1 v(x))d x.

Since γ1 ∈ B(Vo , L 2 (1 )), d is a sesquilinear form on Vo × Vo and also since β(x) > 0 for all x ∈ 1

94

3 Dynamics and Stability

d(v, v) ≥ 0, v ∈ Vo . Thus, use of the Divergence Theorem leads to rewriting the PDE and boundary conditions (3.22) as, for φ ∈ Vo , ˙ = 0. φ, w ¨ + ao (φ, w) + d(φ, w) This is an abstract second-order system of the form (3.7). The associated generator A (see (3.8)) generates a contraction on Z = Vo × Ho . (See Theorem 2.43.) Theorem 3.12 can be used to show that the boundary damping implies asymptotic stability. The operator Ao defined by ao needs to be defined. In general, if w ∈ Vo ∩ H2 (), (3.23) holds. If w also satisfies (∇w · n)|1 = 0, then the boundary term is 0. Defining Ao w = −∇ 2 w, dom(Ao ) = { f ∈ H 2 () ∩ Vo | (∇ f · n)|1 = 0}, (φ, Ao w) = ao (φ, w), w ∈ dom(Ao ). Since Rellich’s Theorem (Theorem A.35) implies that the unit ball in Vo = H1 0 () is compact in the L 2 ()-norm (Theorem A.35), A−1 o is a compact operator. Theorem 3.12 then implies that the system is asymptotically stable if for any eigenfunction φ of Ao , d(φ, φ) = 0. A function φ is an eigenfunction of Ao if it solves, for some real λ, ∇ 2 φ + λφ = 0, φ|0 = 0, (∇φ · n)|1 = 0. If d(φ, φ) = 0 then also φ = 0 on 1 and so ∇ 2 φ + λφ = 0, φ| = 0, (∇φ · n)|1 = 0 . From results in partial differential equations, for any λ, the only solution to this boundary value problem is the zero function. Thus, the system is asymptotically stable on Vo × Ho . The question of exponential stability is more complicated. For many geometries 0 , 1 , there does not exist c2 > 0 such that for all w ∈ V, d(w, w) ≥ c2 w2H . For some geometries the semigroup is not exponentially stable; references are cited in the notes at the end of this chapter. 

3.4

External Stability

95

3.4 External Stability Frequently analysis is done based only on knowledge of the external signals, the inputs and outputs. Consider a general control system (A, B, C, D) (2.63) z˙ (t) = Az(t) + Bu(t), z(0) = z 0 , y(t) = C z(t) + Du(t)

(3.24)

where A generates a C0 -semigroup S(t) on a Hilbert space Z, B ∈ B(U, Z), C ∈ B(Z, Y), D ∈ B(U, Y) , and U, Y are both Hilbert spaces. The following definition is the same as that commonly used for linear finite-dimensional systems. Definition 3.32 A system is externally stable or L 2 -stable if for every input u ∈ L 2 (0, ∞; U), and zero initial condition, the output y ∈ L 2 (0, ∞; Y). If a system is externally stable, the maximum ratio between the norm of the input and the norm of the output is called the L 2 -gain. Let C+ indicate the set of complex numbers with positive real part. For any separable Hilbert space X define the Hilbert space 1 H2 (X ) = { f : C → X | f is analytic and sup 2π x>0 +



∞ −∞

 f (x + j ω)2 dω < ∞}

with inner product  f, g = sup x>0

1 2π



∞

−∞

 f (x + j ω), g(x + j ω) dω.

If f and g are analytic on a region containing the closed right half plane Res ≥ 0, then  ∞ 1  f (j ω), g(j ω) dω  f, g = 2π −∞ and  f 2 =

1 2π



∞ −∞

 f (j ω)2 dω.

Theorem 3.33 (Paley–Wiener Theorem) If X is a separable Hilbert space then under the Laplace transform L 2 (0, ∞; X ) is isomorphic to H2 (X ). For any f, g ∈ f , g indicate the Laplace transforms of f and g respectively, L 2 (0, ∞; X ), letting  f , g H2 .  f, g L 2 =   y∈ Thus, a system is externally stable if and only if every  u ∈ H2 (U) maps to  H2 (Y) and there is a maximum ratio between the Laplace transforms of the inputs and outputs.

96

3 Dynamics and Stability

For separable Hilbert spaces U, Y, letting  ·  indicate the operator norm for elements of B(U, Y), define the complete normed space (that is, a Banach space) H∞ (B(U, Y)) = {G : C+ 0 → B(U, Y)| G is analytic; sup G(s) < ∞} Res>0

with norm G∞ = sup G(s). Res>0

If both U and Y are finite-dimensional, as typically is the case in applications, then the transfer function is a matrix. Matrices with entries in H∞ will be simply indicated by M(H∞ ). Definition 3.34 The singular values σi of a matrix M with n rows are σi =

 λi (M ∗ M), i = 1 . . . n.

It is clear from the definition that singular values are non-negative real numbers. If a matrix M is symmetric and non-negative, the singular values are equal to the eigenvalues. The operator norm of a matrix is the maximum of the singular values and hence the H∞ -norm of a matrix-valued function is   G∞ = sup σmax G(s) . Res>0

In the simplest case, U and Y are real (or complex numbers, the transfer function is a complex-valued function and H∞ = {G : C+ 0 → C | G is analytic and sup |G(s)| < ∞}. Res>0

Theorem 3.35 A linear system is externally stable if and only if its transfer function G ∈ H∞ (B(U, Y)). In this case, G∞ is the L 2 -gain of the system and we say that G is a stable transfer function. The definitions of internal and external stability for finite-dimensional systems generalize to infinite dimensions, as do the conditions under which these types of stability are equivalent. Definition 3.36 The system (A, B, C) is internally stable if A generates an exponentially stable semigroup S(t). Theorem 3.37 The system (3.24) is internally stable if and only if (s I − A)−1 ∈ H∞ (B(Z, Z)); or equivalently, {s ∈ C; Res ≥ 0} ⊂ ρ(A) and there exists M such that supRes>0 (s I − A)−1  ≤ M. Clearly every internally stable system is externally stable. As for finite-dimensional systems, for some control operators B and/or observation operators C a system that

3.4

External Stability

97

is not internally stable may be externally stable. Under conditions similar to those for finite-dimensional systems, external stability implies internal stability. Definition 3.38 The pair (A, B) is stabilizable if there exists K ∈ B(U, Z) such that A − B K generates an exponentially stable semigroup. Definition 3.39 The pair (C, A) is detectable if there exists F ∈ B(Y, Z) such that A − FC generates an exponentially stable semigroup. Definition 3.40 The system (A, B, C, D) is jointly stabilizable and detectable or simply stabilizable/detectable if it is both stabilizable and detectable. Theorem 3.41 A stabilizable and detectable system is internally stable if and only if it is externally stable. The stabilizability (or detectability) of a given system (A, B, C, D) is strongly affected by the spectrum of its generator A. For any real α and operator A, the spectrum of A, σ(A), can be decomposed into two parts: σ+,α (A) = σ(A) ∩ {s ∈ C; Res ≥ α}, σ−,α (A) = σ(A) ∩ {s ∈ C; Res < α}. If α = 0 then the above sets are simply written σ+ (A), σ− (A) respectively. Definition 3.42 The operator A on Z satisfies the spectrum decomposition assumption if σ+ (A) is bounded and separated from σ− (A) in such a way that a smooth, non-intersecting closed curve can be drawn so as to enclose an open set containing σ+ (A) in its interior. Theorem 3.43 Assume that B ∈ B(U, Z) is a compact operator and that (A, B) is stabilizable. This implies that there is  < 0 such that σ+, (A) contains only eigenvalues and the span of the eigenfunctions associated with each eigenvalue is finitedimensional. If in addition U is finite-dimensional then the spectrum decomposition assumption is satisfied and also σ+ (A) contains only a finite number of eigenvalues. Since commonly there are a finite number of actuators, the control space U is finitedimensional. Theorem 3.43 states that such systems can only be stabilized if (1) the unstable part of the spectrum only contains eigenvalues and also (2) the total span of all the associated eigenspaces is finite-dimensional. Similar statements hold for detectability. Theorem 3.44 If C ∈ B(Z, Y) is a compact operator and (A, C) is detectable then there is  < 0 such that σ+, (A) contains only eigenvalues and the span of the eigenfunctions associated with each eigenvalue is finite-dimensional. If in addition Y is finite-dimensional, then the spectrum decomposition assumption is satisfied and σ+ (A) contains only a finite number of eigenvalues.

98

3 Dynamics and Stability

Example 3.45 (Wave equation with distributed damping, Example 3.14 cont.) Consider again, for some b ∈ L 2 (0, 1), the wave equation with distributed damping (3.10),   ∂ 2 z(x, t) ∂ 2 z(x, t) ∂z(·, t) , b(·) b(x) − + = 0, ∂t 2 ∂t ∂x 2

0 ω, lim (s I − An )−1 z − (s I − A)−1 z = 0, for all z ∈ Z;

n→∞

– for each z ∈ Z, and each t1 , t2 , 0 ≤ t1 < t2 , lim sup Sn (t)z − S(t)z = 0.

n→∞ t1 ≤t≤t2

Theorem 4.4 Suppose that the systems (An , Bn ) satisfy Assumption (A1) and that U is finite-dimensional. Let z n indicate the state of (An , Bn ). Then for each initial condition z 0 ∈ Z, control u ∈ L 2 (0, t f ; U), lim z n (t f ) − z(t f ) = 0, lim z n (·) − z(·) L 2 (0,t f ;Z) = 0.

n→∞

n→∞

Thus, Assumption (A1) implies convergence of the response on finite time intervals, with a particular initial condition and control.

106

4 Optimal Linear-Quadratic Controller Design

With the approximating systems (4.4) the cost functional becomes, defining Cn = C|Zn , n, f = Pn  f Pn , t f Jn (u, Pn z 0 ; t f ) = n, f z(t), z(t) +

Cn z(t), Cn z(t) + u(t), Ru(t)dt. (4.5) 0

Strong convergence of Cn to C and n, f to  f follows from strong convergence of Pn to the identity operator. The cost functional (4.5) has the minimum cost n (0)Pn z 0 , Pn z 0  where n is the unique positive semi-definite solution to the differential Riccati equation dn (t) dt

+ A∗n n (t) + n (t)An − n (t)Bn R −1 Bn∗ n (t) + Cn∗ Cn = 0, (4.6) n (t f ) = Pn  f Pn ,

on the finite-dimensional space Zn . This system of ordinary differential equations can be solved using a number of computational methods, although special approaches may be needed when the order is very large. The finite-dimensional feedback control K n (t) = R −1 Bn∗ n (t) can be used to control the original system (4.1). The fundamental question is: what effect does K n (t) have on (4.1)? Since (t) is the solution to an optimization problem, convergence of the dual systems is needed in order ensure convergence of n (t) → (t). Assumption (A1*) (i) For each z ∈ Z, and all intervals of time [t1 , t2 ] sup Sn∗ (t)Pn z − S ∗ (t)z → 0;

t∈[t1 ,t2 ]

(ii) For all z ∈ Z, y ∈ Y, Cn∗ y − C ∗ y → 0 and Bn∗ Pn z − B ∗ z → 0. Theorem 4.5 If Assumptions (A1), (A1∗ ) are satisfied, then letting (t) indicate the solution to the infinite-dimensional differential Riccati equation (4.3), and similarly n (t) the solution to (4.6), for each z 0 ∈ Z, and each t, 0 < t < t f , lim n (t)Pn z 0 − (t)z 0 Z = 0,

n→∞

lim n (·)Pn z − (·)z L 2 ([0,t f ];Z) = 0.

n→∞

Define u o (t) = −K (t)z o (t), where z o (t) solves (4.1) with u(t) = u o (t), and similarly define z no (t) to be the solution of (4.4) with control u on (t) = −K n (t)z no (t). For each initial condition z 0 ∈ Z, lim u on (t) − u o (t) L 2 ([0,t f ];U ) = 0,

n→∞

lim z no (t) − z o (t) L 2 ([0,t f ];Z) = 0.

n→∞

4.1 Finite-Time Optimal Linear Quadratic Control

107

Furthermore, letting z u on indicate the solution of (4.1) with u(t) = u on (t), lim z u on (t f ) − z o (t f )Z = 0,

n→∞

lim z u on (·) − z o (·) L 2 ([0,t f ];Z) = 0,

n→∞

and lim J (u on , z 0 ; t f ) = J (u o , z 0 ; t f ) = (0)z 0 , z 0 .

n→∞

4.2 Infinite-Time Optimal Linear Quadratic Control The infinite-time linear-quadratic (LQ) controller design objective is to find a control u(t) so that the cost functional ∞ J (u, z 0 ) = C z(t), C z(t) + u(t), Ru(t)dt,

(4.7)

0

where R ∈ B(U, U) is a self-adjoint coercive operator control and C ∈ B(Z, Y) (with Hilbert space Y) is minimized, subject to z(t) being determined by (4.1). Definition 4.6 The system (4.1) with cost (4.7) is optimizable if for every z 0 ∈ Z there exists u ∈ L 2 (0, ∞; U) such that the cost is finite. As for finite-dimensional systems, with the additional assumption of optimizability, the solution to the infinite-time optimal control problem can be obtained from the finite-time problem by reversing the time variable in (4.3) so that a Riccati differential equation is solved with (0) =  f and taking t f → ∞. Theorem 4.7 If (4.1) with cost (4.7) is optimizable and (A, C) is detectable, then the cost function (4.7) has a minimum for every z 0 ∈ Z. Furthermore, there exists a self-adjoint positive semi-definite operator  ∈ B(Z, Z) such that min

u∈L 2 (0,∞;U )

J (u, z 0 ) = z 0 , z 0 .

Also, (dom(A)) ⊂ dom(A∗ ) and  is the unique positive semi-definite solution to the operator equation Az 1 , z 2  + z 1 , Az 2  + C z 1 , C z 2  − B ∗ z 1 , R −1 B ∗ z 2  = 0,

(4.8)

for all z 1 , z 2 ∈ dom(A). Defining K = R −1 B ∗ , the corresponding optimal control is u = −K z(t) and A − B K generates an exponentially stable semigroup.

108

4 Optimal Linear-Quadratic Controller Design

Theorem 4.7 implies that, as for finite-dimensional systems, optimizability is equivalent to exponential stabilizability. The Riccati operator equation (4.8) is equivalent to 

 A∗  + A − B R −1 B ∗  + C ∗ C z = 0,

for all z ∈ dom(A).

(4.9)

In practice, the solution to the operator Eq. (4.8) (or (4.9)) cannot be explicitly calculated and the control is calculated using a finite-dimensional approximation (4.4). Consider a sequence of finite-dimensional subspaces Zn ⊂ Z, let Pn indicate the orthogonal projection of Z → Zn and define the finite-dimensional approximating systems (4.4). Defining Cn = C|Zn , the cost functional becomes ∞ J (u, z 0 ) = Cn z(t), Cn z(t) + u(t), Ru(t)dt.

(4.10)

0

If (An , Bn ) is stabilizable and (An , Cn ) is detectable, then the cost functional has the minimum cost Pn z 0 , n Pn z 0  where n is the unique positive semi-definite solution to the algebraic Riccati equation A∗n n + n An − n Bn R −1 Bn∗ n + Cn∗ Cn = 0

(4.11)

on the finite-dimensional space Zn . The finite-dimensional feedback control K n = R −1 Bn∗ n is used to control the original system (4.1). Controllers are generally implemented in a feedback configuration and this places additional requirements on the approximation. High frequency noise, disturbances and unmodelled dynamics can destabilize the closed loop. Example 4.8 (Undamped simply supported beam) Consider a simply supported Euler–Bernoulli beam and let w(x, t) denote the deflection of the beam from its rigid body motion at time t and position x. The deflection is controlled by a force centered on the point r with width . With beam parameters normalized the PDE is ∂2w ∂4w + = br u(t), ∂t 2 ∂x 4

t ≥ 0, 0 < x < 1,

where, letting  indicate the width of the actuator and r its location,  br (x) =

1/, |r − x| < 0, |r − x| ≥

 2  2

The boundary conditions are w(0, t) = 0, w (0, t) = 0, w(1, t) = 0, w (1, t) = 0.

.

4.2 Infinite-Time Optimal Linear Quadratic Control

109

This is the same model asin Example 2.26, but with zero damping. Defining v(x, t) =  ∂w w v , let the state be z = . Define ∂t  Hs (0, 1) = w ∈ H2 (0, 1); w(0) = w(1) = 0 . The operator A





v w , = 4 v − dd xw4

dom(A) = {(w, v) ∈ Hs (0, 1) × Hs (0, 1)| w ∈ Hs (0, 1)} generates a contraction on Z = Hs (0, 1) × L 2 (0, 1). A state-space formulation on Z of the above partial differential equation problem is d x(t) = Ax(t) + B(r )u(t), dt ⎡

where

B(r ) = ⎣

0 br (·)

⎤ ⎦.

Since a closed form solution to the partial differential equation problem is not available, a linear-quadratic optimal control must be calculated using an approximation. The operator A is Riesz-spectral (see Example 2.26) so the eigenfunctions are a basis for the state-space. A truncation of the generalized Fourier series solution, often called a modal approximation, approximates the solution to the PDE. Solve first μφ = −

d 4φ , dx4

φ(0) = 0, φ (0) = 0, φ(1) = 0, φ (1) = 0. The solutions to this boundary value problem are φm (x) = sin(mπx) and the eigenvalues μm = −(mπ)4 . Thus the eigenvalues of A are λm = j (mπ)2 , λ−m = −j (mπ)2 , m = 1, 2, . . . , and the eigenfunctions of A are, for arbitrary constants {am },

110

4 Optimal Linear-Quadratic Controller Design

ψm (x) = am

 1 φ (x) λm m , φm (x)

m = ±1, ±2, . . . .

The problem with using {ψm } as a basis for an approximation is that it involves complex arithmetic. But, noting that φ−m (x) = −φm (x),

ψm + ψ−m = am

( λ1m −

 1 )φm λ−m ,

0

λm ψm + λ−m ψ−m

= am

 0 . (λm − λ−m )φm

Thus the following is also an orthogonal basis for the span of the first N eigenfunctions of A,



 φ 0 cm m , dm , m = 1, . . . N 0 φm where cm , dm are arbitrary constants, often chosen so each basis element has norm 1. For any positive integer n, define Vo,n to be the span of φi , i = 1, . . . n. Choose Zn = Vo,n × Vo,n and define Pn to be the projection onto Zn . Define An = Pn A Pn and Bn = Pn B. This approximation scheme satisfies Assumptions (A1) and (A1∗ ). In the simulations the actuator was at x = 0.2 with width  = 0.001. The state weight was C = I and the control weight R = 0.01. The first 3 modes were used to calculate a control K 3 that is L Q-optimal for that approximation. This is straightforward since the resulting state matrix An has dimension 6. In practice, the full state is not available and the control is implemented using an observer. Suppose that the deflections are measured at x = 0.9. A Luenberger observer was designed that placed the observer eigenvalues slightly to the left of the eigenvalues of A3 − B3 K 3 . The observer filter is  F3 = 382.7

−1054.

1569.

71.70

−427.3

∗ 919.9 .

Figures 1.6 show that the control works well when implemented with a system approximation also with the first 3 modes, so n = 3. However, increasing the number of modes in the approximate system in the simulations to n = 6 leads to instability, even when the initial condition is only the first mode. The controller activates the higher modes that were neglected in the controller design. Higher modes can also be activated by disturbances on a system. Consider now the sequence of controllers K n obtained for each modal approximation. Since there is only one control, and K n is a bounded linear operator, the Riesz Representation Theorem (Theorem A.42) means that any feedback controller K can be identified with an element of the state space (k p , kv ). (The first function k p affects the beam positions, kv affects the velocity.) Call these elements the controller gains. Figure 4.1 indicates that the controller gains are not converging. This suggests that even for large approximation order n, K n is not a controller suitable for the original system. In fact, since the model has an infinite number of imaginary eigenvalues, and the control operator B is bounded, it cannot be stabilized using a finite number of

4.2 Infinite-Time Optimal Linear Quadratic Control

111

0.4

n=3 n=6 n=12 n=18

0.35

0.3

w

0.25

0.2

0.15

0.1

0.05

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

(a) position feedback gains 250

n=3 n=6 n=12 n=18

200

150

v

100

50

0

-50

-100

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

(b) velocity feedback gains Fig. 4.1 Linear-quadratic optimal feedback for modal approximations of the simply supported beam in Example 1.4. Since there is only one control, each feedback operator K n can be identified with an element of the state space (k p , kv ). The upper figure shows k p ; the lower figure shows kv . Although k p appears to be converging, the other half of the feedback operator, kv , does not seem to be converging as the approximation order increases. This suggests that the optimal feedback for the approximations K n is not converging to a controller suitable for the original distributed parameter model

112

4 Optimal Linear-Quadratic Controller Design

control variables with bounded feedback (Theorem 3.43). Thus, there is no sequence of controllers that converge to a stabilizing controller for the original model.  Although Assumption (A1) guarantees convergence in finite time of the open loop systems, additional conditions are required in order to obtain convergence of designed controllers and of closed loops. The sequence of controllers K n , along with the associated performance must converge in some sense. Assumptions additional to those required in simulation are required. Definition 4.9 The control systems (An , Bn ) are uniformly stabilizable if there exists a sequence of feedback operators {K n } with K n  ≤ M1 for some constant M1 such that An − Bn K n generate SK n (t), SK n (t) ≤ M2 e−α2 t , M2 ≥ 1, α2 > 0. Definition 4.10 The control systems (An , Cn ) are uniformly detectable if there exists a sequence of operators {Fn } with Fn  ≤ M1 for some constant M1 such that An − Fn Cn generate SK n (t), SK n (t) ≤ M2 e−α2 t , M2 ≥ 1, α2 > 0. The approximating systems (An , Cn ) are uniformly detectable if and only if (A∗n , Cn∗ ) is uniformly stabilizable. Uniform stabilizability implies that there is an upper bound on the optimal cost over all approximation orders. Also, as for finite-time LQ control, the optimal control K z relates to an optimization problem involving the dual system. Thus, in order to obtain controller convergence additional assumptions involving the dual system are required. Theorem 4.11 If Assumptions (A1), (A1∗ ) are satisfied, (An , Bn ) is uniformly stabilizable and (An , Cn ) is uniformly detectable, then for each n, the finite-dimensional ARE (4.11) has a unique nonnegative solution n with supn n  < ∞. Letting  indicate the solution to the infinite-dimensional Riccati equation (4.8), for all z ∈ Z, lim n Pn z − z = 0.

n→∞

(4.12)

Defining K n = R −1 Bn∗ n , K = R −1 B ∗ , lim K n Pn z − K z = 0,

n→∞

(4.13)

and the cost with feedback K n z(t) converges to the optimal cost: lim J (−K n z(t), z 0 ) = z 0 , z 0 .

n→∞

Furthermore, there exists constants M1 ≥ 1, α1 > 0, independent of n, such that the semigroups Sn K (t) generated by An − Bn K n satisfy Sn K (t) ≤ M1 e−α1 t .

4.2 Infinite-Time Optimal Linear Quadratic Control

113

For sufficiently large n, the semigroups SK n (t) generated by A − B K n are uniformly exponentially stable; that is there exists M2 ≥ 1, α2 > 0, independent of n, such that SK n (t) ≤ M2 e−α2 t . If U is finite-dimensional, or more generally, B is a compact operator, then the convergence in (4.13) is uniform: limn→∞ K n Pn − K  = 0. Example 4.12 (Diffusion Example 2.2) ∂z ∂2z = + b(x)u(t), ∂t ∂x 2 z(0, t) = 0, z(1, t) = 0

0 < x < 1,

for some b ∈ L 2 (0, 1). This can be written as z˙ (t) = Az(t) + Bu(t) where A is as defined earlier, see Example 2.2, and Bu = b(x)u. √ The eigenvalues of A are λm = −m 2 π 2 . The eigenfunctions φm (x) = 2 sin(πmx), m = 1, 2, . . ., of A form an orthonormal basis for L 2 (0, 1). The operator A is a Riesz-spectral operator and thus satisfies the SDGA. The operator A generates an 2 exponentially stable semigroup S(t) with S(t) ≤ e−π t on the state-space L 2 (0, 1) and so the system is trivially stabilizable and detectable. Defining Zn = spanm=1,...n φm (x), and letting Pn be the projection onto Zn , for z n ∈ Zn , An z n = Az n =

n 

λm z, φm φm ,

Bn u = Pn Bu = u

m=1

n  m=1

1 Cn z n =

z n (x)c(x)d x 0

= z n , c =

n  m=1

The semigroup generated by An is

c, φm  z, φm .

b, φm φm ,

114

4 Optimal Linear-Quadratic Controller Design

Sn (t)z n =

n 

eλm t z n , φm φm ,

m=1

which is exponentially stable with Sn (t) ≤ e−π t . Assumption (A1) is clearly satisfied and the the approximations are uniformly exponentially stable, hence trivially uniformly stabilizable (and detectable).  2

This result generalizes to approximation using the eigenfunctions of general Riesz-spectral systems. Theorem 4.13 Suppose that (A, B, C) is a Riesz-spectral system. Then letting Pn indicate the projection onto the first n eigenfunctions, Zn the span of the first n eigenfunctions, and C|Zn indicates the restriction of the operator C to Zn , define the approximations An = Pn A Pn , Bn = Pn B, Cn = C|Zn . If the original system is stabilizable by K then the approximations are uniformly stabilizable by K |Zn . Similarly, if the original system is detectable then the approximations are uniformly detectable. This approximation method is sometimes referred to as modal truncation. It is very effective when the eigenfunctions are known. Often the eigenfunctions are not known. Another approximation method, such as a finite element approximation, is needed. For the important class of systems where the generator is defined through a sesquilinear form, an approximation, such as finite elements, can be defined using the sesquilinear form. Let V be a Hilbert space that is dense in Z. The notation ·, · indicates the inner product on Z, and ·, ·V indicates the inner product on V. The norm on Z is indicated by  ·  while the norm on V will be indicated by ·V . Consider a V-coercive sesquilinear form a : V × V → C so that for 0 < c2 ≤ c1 , real α (Definition 2.37) |a(φ, ψ)| + αφ, ψZ ≤ c1 φV ψV a(φ, φ) + αφ, φZ ≥ c2 φV for all φ, ψ ∈ V. An operator A is defined using this form by Az = z A , dom(A) = {z ∈ V | there exists z A ∈ Z, −a(φ, z) = φ, z A Z , for all φ ∈ V}. The V-coerciveness of a implies that A generates a C0 semigroup on Z and if α < 0, the semigroup is exponentially stable (Theorem 3.9). Consider a sequence of finite-dimensional subspaces Vn ⊂ V and define the approximating generator An through the same form a:

4.2 Infinite-Time Optimal Linear Quadratic Control

φn , An z n  = −a(φn , z n ), for all φn , z n ∈ Vn .

115

(4.14)

This type of approximation is generally referred to as a Galerkin approximation and includes popular approximation methods such as finite-element and spectral methods (or assumed modes) as well as the eigenfunction truncations discussed above. Since An is defined through the same form a that defines A it generates a semigroup with the same growth bound. This implies uniform stability, if the original system is exponentially stable, or more generally, uniform stabilizability. Theorem 4.14 Let Vn ⊂ V be a sequence of finite-dimensional subspaces such that for all z ∈ V there exists a sequence z n ∈ Vn with lim z n − zV = 0.

n→∞

(4.15)

Let Pn indicate the projection of Vn onto Z in the Z norm. Define the operator An through a V-coercive linear form a as in (4.14) and define Bn = Pn B, Cn = C|Vn . 1. Assumption (A1) is satisfied and for all n, Sn (t) ≤ eαt ; 2. If K ∈ B(Z, U) is such that A − B K generates an exponentially stable semigroup then the semigroups Sn K (t) generated by An − Bn K Pn are uniformly exponentially stable. In other words, there exists N , and M ≥ 1, ω > 0 independent of n, such that for all n > N (4.16) Sn K (t) ≤ Me−ωt . 3. If F ∈ B(Y, Z) is such that A − FC generates an exponentially stable semigroup then the semigroups Sn K (t) generated by An − Pn FCn are uniformly exponentially stable. In other words, there exists N , and M ≥ 1, ω > 0 independent of n, such that for all n > N (4.17) Sn F (t) ≤ Me−ωt . 4. The adjoint operator A∗ is defined through a(φ, z) = a(z, φ) and defining then A∗n analogously to (4.14) as φ, A∗n z n  = −a(z n , φn ), for all z n , φn ∈ Vn ,

(4.18)

the semigroups Sn∗ (t) generated by A∗n satisfy (A1∗ ). Thus, uniform stabilizability (or detectability) of Galerkin approximations follows naturally from stabilizability (or detectability) for problems where the generator is defined through a coercive sesquilinear form. Example 4.15 (Diffusion, Example 2.2) In Example 4.12 the heat equation was approximated using its eigenfunctions. Here a finite element approximation is used. Define V = H01 (0, 1), Z = L 2 (0, 1), 1 a(φ, ψ) = 0

φ (x)ψ (x)d x,

116

4 Optimal Linear-Quadratic Controller Design

and note that a(φ, ψ) = φ, ψV . It was shown in Example 3.10 that for any β > 0, π2 φ, ψZ ≤ φ, ψV , 1+β β π2 φ2Z ≥ φ2V . a(φ, φ) − 1+β 1+β

a(φ, ψ) −

(4.19)

It follows that the V-coercive operator associated with a, Az =

d2z , dom(A) = H2 (0, 1) ∩ H01 (0, 1) dx2

generates an exponentially stable C0 -semigroup S(t) on L 2 (0, 1). Theorem 3.9 implies that for any α such that 0 > α > −π 2 , S(t) ≤ eαt . For integer n define h = 1/n. Define Vn ⊂ V to be the span of the uniformly distributed linear splines, for i = 1, . . . n − 1, ⎧1 ⎨ h (x − (i − 1)h) (i − 1)h ≤ x ≤ i h φi (x) = h1 ((i + 1)h − x) i h ≤ x ≤ (i + 1)h ⎩ 0 else. Then (4.15) is satisfied with respect to V = H01 (0, 1). Define An to be the Galerkin approximation φ j , Aφi  = −a(φ j , φi ) = −φ j , φi 

i = 1, . . . n, j = 1, . . . n.

(4.20)

The form a is V-coercive with V = H01 (0, 1) and satisfies inequality (4.19). For any 0 > α > −π 2 , The semigroups Sn (t) generated by An satisfy Sn (t) ≤ eαt , the same bound as that of S(t). Hence the approximations satisfy Assumption (A1) and are also uniformly exponentially stable.  Theorem 4.14 holds for second-order systems defined through sesquilinear forms (Definition 2.42) if the damping is sufficiently strong. For Hilbert spaces Vo → Ho , let a stiffness operator Ao : dom(Ao ) ⊂ Ho → Ho be a self-adjoint, operator on Ho that can be defined through a Vo -coercive positive symmetric sesquilinear form ao . The damping D is assumed to be defined through a continuous symmetric non-negative sesquilinear form d on Vo . Formally, ˙ = 0. w(t) ¨ + Ao w(t) + D w(t)

(4.21)

The abstract form of the differential equation on Vo is ˙ = 0, for all ψ ∈ Vo . ψ, w(t) ¨ Ho + ao (ψ, w(t)) + d(ψ, w(t))

(4.22)

4.2 Infinite-Time Optimal Linear Quadratic Control

117

The differential equation (4.22) can be written in state-space form (2.10) on the state space Z = Vo × Ho . For any w, v ∈ Vo , a(·, w) + d(·, v) defines an element of Vo , which can be written Ao w + Dv. Define    0 I , dom(A) = wv ∈ Vo × Vo | Ao w + Dv ∈ Ho . A= −Ao −D

(4.23)

The operator A can also be defined directly through a single sesquilinear form a continuous on V = Vo × Vo . First, rewrite the second-order differential equation (4.22) as a system of first-order equations. For φ, ψ, w, v ∈ Vo , φ, w(t) ˙ Vo = φ, v(t)Vo ψ, v(t) ˙ Ho = −ao (ψ, w(t)) − d(ψ, v(t)). For (φ, ψ), (w, v) ∈ V define a((φ, ψ), (w, v)) = −φ, vVo + ao (ψ, w) + d(ψ, v).

(4.24)

The form a is continuous on V = Vo × Vo , and (4.22) is equivalent to (φ, ψ), (w(t), ˙ v(t)) ˙ Vo ×Ho = −a((φ, ψ), (w, v)). A typical Galerkin type approximation is to consider a sequence of finitedimensional subspaces Vo,n ⊂ Vo and use a to define the approximating generator An (φn , ψn ), An (wn , vn )) = −a((φn , ψn ), (wn , vn )), for all φn , ψn , wn , vn ∈ Vo,n . (4.25) If the damping is Vo -coercive then the sesquilinear form a defined in (4.24) is V-coercive, and a result similar to Theorem 4.14 is obtained. Theorem 4.16 Consider a second-order system (4.21) where d is Vo -coercive; that is, there is cd such that for all v ∈ Vo , d(v, v) ≥ cd v2Vo . Let Vo,n ⊂ Vo → Ho be a sequence of finite-dimensional subspaces such that for all v ∈ Vo there exists a sequence vn ∈ Von with lim vn − vVo = 0.

n→∞

(4.26)

Defining V = Vo × Vo , Vn = Vo,n × Vo,n , Z = Vo × Ho , let Pn indicate the projection of Vn onto Z in the Z-norm. Define the operator An through a sesquilinear form as in (4.25) and define Bn = Pn B, Cn = C|Vn . Then

118

4 Optimal Linear-Quadratic Controller Design

1. Assumption (A1) is satisfied and for all n, Sn (t) ≤ 1; 2. If K ∈ B(Z, U) is such that A − B K generates an exponentially stable semigroup then the semigroups Sn K (t) generated by An − Bn K Pn are uniformly exponentially stable. In other words, there exists N , ω > 0 such that for all n > N , Sn K (t) ≤ e−ωt .

(4.27)

3. If F ∈ B(Y, Z) is such that A − FC generates an exponentially stable semigroup then the semigroups Sn K (t) generated by An − Pn FCn are uniformly exponentially stable. In other words, there exists N , ω > 0 such that for all n > N , Sn F (t) ≤ e−ωt .

(4.28)

4. The adjoint operator A∗ is defined through a((w, v), (φ, ψ)) and the semigroups Sn∗ (t) generated by A∗n satisfy (A1∗ ). Example 4.17 (Cantilevered Euler–Bernoulli Beam) Consider control of the deflections of an Euler–Bernoulli beam of unit length rotating about a fixed hub. It is clamped at one end and free to vibrate at the other. Let w denote the deflection of the beam from its rigid body motion. Suppose a torque u(t) is applied at the hub. With hub inertia Ih , elasticity E, and viscous cv and Kelvin–Voigt damping cd parameters, the motion is modelled by the PDE ∂2 ∂w ∂2w + + c v ∂t 2 ∂t ∂x 2



∂2w ∂3w E 2 + cd 2 ∂x ∂x ∂t

 =

x u(t), Ih

0 < x < 1,

(4.29)

with boundary conditions ∂w |x=0 = 0, w(0, t) = 0, ∂x  

2

3 ∂3w ∂4w ∂ w ∂ w E 2 + cd = 0, E 3 + cd = 0. ∂x ∂t∂x 2 x=1 ∂x ∂t∂x 3 x=1

(4.30)

Define the state z = (w, w), ˙ let Ho = L 2 (0, 1), with the usual inner product, indicated by ·, ·, and define the Hilbert space Vo = {w ∈ H2 (0, 1); w(0) = with the inner product

∂w (0) = 0} ∂x

w1 , w2 Vo = Ew1 , w2 .

Define on Vo a(φ, ψ) = φ, ψVo , d(φ, ψ) =

cd cv φ, ψ + φ, ψVo . E E

4.2 Infinite-Time Optimal Linear Quadratic Control

119

The damping d is Vo -coercive and is identical to that in Example 3.15 except that now viscous damping is included. In Example 3.15 it was shown that if cd > 0 the solution to (4.29) with boundary conditions (4.30) defines an exponentially stable semigroup on Z = Vo × Ho . Hence, this model is also exponentially stable. n that Let Vo,n ⊂ Vo be a sequence of finite-dimensional spaces with basis {ei }i=1 satisfy the Vo -approximation property. A typical choice of basis for the approximations for structures is finite elements with cubic splines. Since the associated generator is a Riesz-spectral operator, modal truncations with the eigenfunctions are another possibility; see Example 4.8. Defining V = Vo × Vo , it follows that Vn = Vo,n × Vo,n satisfies the V-approximation property. Letting ei , i = 1 . . . n be a basis for Vo,n , define a basis for Vn :

  ei 0 , i = 1 . . . n. , 0 ei The approximation will have the structure z n (x, t) =

n  i=1

  



n ei (x) 0 + . wi (t) vi (t) 0 ei (x) i=1

Since the damping is Vo -coercive, Theorem 4.16 implies that the approximating systems are uniformly exponentially stabilizable, uniformly exponentially detectable, and also the approximating adjoint semigroups converge. In fact, Theorem 3.12 implies that this PDE is exponentially stable. Since the same sesquilinear form is used to define the approximations, they are also exponentially stable with the same decay rate.  The uniform stabilizability and detectability stated in Theorems 4.14 and 4.16 hold because V-coercivity of the associated sesquilinear form implies that the spectrum is contained in a sector (Theorem 3.24). This aids in approximation of the spectrum when the sesquilinear form is used to construct the approximation. For second-order systems where the damping is not strong enough for the spectrum to lie in a sector, obtaining uniformly stable approximations is sometimes not straightforward. This is illustrated by the next example. Example 4.18 (Wave equation with boundary damping, Example 3.30 cont.) Consider ∂2w ∂2w = , 0 0. Several values of R were considered. The operator A=

d2 , dom(A) = {z ∈ H2 (0, 1)| z (0) = z (1) = 0} dx2

generates an analytic semigroup on L 2 (0, 1) and the eigenfunctions form an approximation that satisfies the assumptions of Theorem 4.11. Furthermore, because the semigroup is analytic and (I − A)−1 is a compact operator, the solution to the ARE is a compact operator. Convergence of the optimal performance and locations is obtained after 3 eigenfunctions are used. The cost  (r ) for various actuator locations is in Fig. 4.6, with C = I and R = 0.01, R = 1. The result that the centre is the best location when R = 0.01 is counter-intuitive since the eigenfunctions used in the approximation are √

2 cos(πx), φ3 (x) =

1

1

0.8

0.8 normalized cost

normalized cost

φ1 (x) = 1, φ2 (x) =

0.6

0.4

√

2 cos(2πx)

0.6

0.4

0.2

0.2

0

0 0

0.1

0.2

0.3

0.4

0.5 r

0.6

0.7

(a) C = I, R = 0.01

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

r

(b) C = I, R = 1

 (" r ) Fig. 4.6 Normalized performance max  (" r ) versus actuator location for heat equation with different choices of weights in LQ cost function. For more expensive control, shown in (b), the actuator location does not affect the cost

4.4 LQ-Optimal Actuator Location

137

and φ2 (0.5) = 0. However, the optimal feedback is  K = 10

0

 −1.4 .

There is no attempt by the controller to control the second eigenfunction; for this problem it is more effective to focus effort on the other eigenfunctions, notably the first. For the second choice of weights, with more expensive control, the cost is not affected by the actuator location.  Consider now the situation where the initial condition is regarded as random. A different norm from the usual operator norm is appropriate. Definition 4.33 Let U be a separable Hilbert space with orthonormal basis {ek } and Y a Hilbert space. Operators M ∈ B(U, Y) for which ∞ 

(M ∗ M) 2 ek , ek  < ∞ 1

k=1

are said to be trace class. Theorem 4.34 If M is of trace class its trace traceM =

∞  Mek , ek  k=1

is independent of the choice of basis {ek }.  1 ∗ 2 Theorem 4.35 The quantity M1 = ∞ k=1 (M M) ek , ek  defines a norm, called the nuclear norm or trace norm on the linear subspace of trace class operators. Theorem 4.36 If M is self-adjoint and positive semi-definite then it is of trace class if and only if traceM < ∞. If so, M1 = traceM =

∞ 

λk (M),

k=1

where the eigenvalues λk of M are counted according to their multiplicity. If the initial condition is random, with zero mean and variance V then the 1 1 expected cost is (V 2 (r  )V 2 1 , or since,  is self-adjoint and positive semi1 1 definite, trace V 2 (r )V 2 . In order to use the trace norm the operator (r ) (or 1

1

V 2 (r )V 2 ) must have finite trace. Assume in order to simplify the exposition that the variance is unity. The performance for a particular r is μ(r ) = (r )1 and the optimal performance is

138

4 Optimal Linear-Quadratic Controller Design

 μ = infm (r )1 . r ∈

(4.54)

Theorem 4.37 If the input and output spaces, U and Y, are both finite-dimensional the positive semi-definite solution to the algebraic Riccati equation (4.8) , when it exists, is a trace class operator. Finite-dimensionality of U and Y also implies that the trace norm of the optimal cost is a continuous function of the actuator location. Theorem 4.38 Let B(r ) ∈ B(U, Z), r ∈ m , be a family of input operators such that for any r0 ∈ m , lim B(r ) − B(r0 ) = 0. r →r0

Assume that (A, B(r )) are all stabilizable and that (A, C) is detectable where C ∈ B(Z, Y). If U and Y are finite-dimensional, then the corresponding Riccati operators (r ) are continuous functions of r in the trace norm: lim (r ) − (r0 )1 = 0,

r →r0

and there exists an optimal actuator location  r such that μ. ( r )1 = infm (r )1 =  r ∈

The control and the optimal actuator locations will be calculated using an approximation. Consider an approximation (An , Bn , Cn ) on a finite-dimensional subspace μn analogously Zn ⊂ Z. Define the optimal cost for the approximating problems  to (4.54). Theorem 4.38 applies to these finite-dimensional problems. The performance measure μn (r ) defined by (4.50) is continuous with respect to r and the optimal performance  μn is well-defined. As long as U and Y are finite-dimensional, any approximation scheme that satisfies the usual assumptions for controller design will lead to a convergent sequence of actuator locations that are optimal in the trace norm. This is a consequence of approximation of the Riccati operator in the trace norm. Theorem 4.39 Assume that (A, B) is stabilizable and (A, C) is detectable, and that U and Y are finite-dimensional. Let (An , Bn , Cn ) be a sequence of approximations to (A, B, C) that satisfy Assumptions (A1), (A1∗ ) and are uniformly stabilizable and uniformly detectable. Then lim n Pn − 1 = 0.

n→∞

Theorem 4.40 Assume a family of control systems (A, B(r ), C) with finitedimensional input space U and finite-dimensional output space Y such that

4.4 LQ-Optimal Actuator Location

139

1. (A, B(r )) are stabilizable and (A, C) is detectable, 2. for any r0 ∈ , limr →r0 B(r ) − B(r0 ). Let (An , Bn (r ), Cn ) be a sequence of approximating systems with Bn (r ) = Pn B(r ), Cn = C|Zn , that satisfy Assumptions (A1), (A1∗ ) and are uniformly stabilizable for each (A, B(r )) and uniformly detectable. Letting " r be an optimal actuator location for (A, B(r ), C) with optimal cost " μ μn , it follows that and defining similarly " rn , " μn , " μ = lim " n→∞

and there exists a subsequence {" rm } of {" rn } such that rm )1 . " μ = lim  (" m→∞

Example 4.41 (Viscously Damped Beam, Example 4.29 cont.) Consider the same viscously damped beam system and control problem as in. Since there is only one control, choose again control weight R = 1. If the state weight C = I , the trace of n does not even converge for a fixed actuator location, as shown in Fig. 4.7. This indicates that the Riccati operator is not of trace class. trace(PN): Beam, viscous damping with actuator at x=0.5, C=I 1000 900 800 700 600 500 400 300 200 100 0

0

5

10

15

20

25

30

Fig. 4.7  n 1 for different approximations of the viscously damped beam with actuator at x = 0.5 and weights C = I , R = 1. No convergence is obtained. (©2011 IEEE. Reprinted, with permission, from [2])

140

4 Optimal Linear-Quadratic Controller Design Optimal performance: viscous damping, C=deflection at x=0.5

Optimal performance: viscous damping, C=deflection at x=0.5 0.05

0.05

0.045

0.045

0.04

0.04

0.035

0.035

0.03

0.03

0.025

0.025

0.02

0.02

0.015

0.015

0.01

0.01

0.005

0.005

0

0 0

5

10 Number of modes

(a) Optimal performance Π(r)1

15

0

5

10

15

Number of modes

(b) Optimal actuator location

Fig. 4.8 Optimal performance and actuator locations" rn with respect to LQ-cost and random initial condition for different approximations of the viscously damped beam, C = deflection at x = 0.5, R = 1. Since C is a compact operator, the calculations converge. (©2011 IEEE. Reprinted, with permission, from [2])

On the other hand, if the state weight C z(t) = w(0.5, t) where w is the first component of the state z, quite different results are obtained. The input and output spaces are both one-dimensional and so has finite trace. This leads to the sequence of optimal actuator locations and performance shown in Fig. 4.8. As predicted by the theory, the calculated optimal locations and costs converge.  Example 4.42 (Comparison of LQ and controllability criteria for actuator placement) Let w(x, t) denote the deflection of a beam from its rigid body motion at time t and position x and use a standard Euler–Bernoulli beam model. The deflection is controlled by applying a force u(t) with localized spatial distribution br (x) that varies with actuator location r. Normalizing the variables and including viscous damping and Kelvin–Voigt damping leads to the partial differential equation ∂w ∂5w ∂2w ∂4w + c + c + = br (x)u(t) 0 < x < 1, v d ∂t 2 ∂t ∂t∂x 4 ∂x 4

(4.55)

where cv > 0, cd > 0. The beam is simply supported and so the boundary conditions are 2 ∂3 w w(0, t) = 0, ∂∂xw2 (0, t) + cd ∂t∂x 2 (0, t) = 0, (4.56) 2 ∂3w w(1, t) = 0, ∂∂xw2 (1, t) + cd ∂t∂x 2 (1, t) = 0. Define Hs (0, 1) = {w ∈ H2 (0, 1), w(0) = 0, w(1) = 0} ˙ A state-space and the state-space Z = Hs (0, 1) × L 2 (0, 1) with state z = (w, w). formulation of Eq. (4.55) is

4.4 LQ-Optimal Actuator Location

141

dz (t) = Az(t) + B(r )u(t) dt

(4.57)

where

A=

 I , 4 −cv I − cd ddx 4

0 4

− ddx 4

dom(A) = {(w, v) ∈ ( Hs (0, 1) )2 ; (w + cd v) ∈ Hs (0, 1)}, 

0 . B(r ) = br (x)

(4.58) (4.59) (4.60)

This is the same model as Example 2.26. The operator A is Riesz-spectral and provided that at least one of cv or cd is positive, it generates an exponentially stable semigroup on Z. Since a closed form solution to the partial differential equation problem is not available, the optimal actuator location must be calculated using an approximation. Both finite-elements and modal methods can be used. Here, since the modes can be easily calculated, a modal approximation is used. Let {φi } indicate the eigenfunctions 4 of ∂∂xw4 with boundary conditions (4.56). For any positive integer n define Vo,n to be the span of φi , i = 1, . . . n. Choose Zn = Vo,n × Vo,n and define Pn to be the projection onto Zn . To avoid computation with complex numbers, the basis functions used in construction of the approximating system are not the eigenfunctions of A; see Example 4.29 for details. Define An to be the Galerkin approximation to A, or Pn A Pn and Bn := Pn B. This leads to a finite-dimensional control system of dimension 2n. Since the damping is coercive with respect to Hs (0, 1), both Theorems 4.13 and 4.16 imply that the approximations are uniformly exponentially stabilizable and detectable, as are the adjoint semigroups and so this scheme can be used for LQcontroller design. The damping parameters are set to cv = 0.1, cd = 0.0001. The actuator is localized at x = r with width  = 0.001, and  br (x) =

1 , 

|r − x| < 0, |r − x| ≥

 2  . 2

.

(4.61)

Controllability can be chosen as an actuator location criterion. Recall from Theorem 3.2 that the solution L c (r ) to the Lyapunov equation AL c (r ) + L c (r )A∗ + B(r )B(r )∗ = 0.

(4.62)

is positive definite, that is L c z, z > 0 for all z = 0, if and only if (A, B) is approximately controllable. If (A, B) is exactly controllable then L c (r ) is coercive; that is, there is m > 0 so that L c z, z ≥ mz2 for all z (Definition A.65). If a system is uncontrollable, then there are certain states to which the system cannot be controlled.

142

4 Optimal Linear-Quadratic Controller Design 0.18

0.16

controllability

0.14

0.12

0.1

0.08

0.06

0.04

0.02

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

actuator location

Fig. 4.9 Controllability c(r ) (see (4.63)) of the first 5 and 10 modes versus actuator location. When 5 modes (—) are used, the location r = 0.1 optimizes controllability. But with 10 modes (- - -) , the system with the actuator at this location is almost uncontrollable. (Reprinted from [3], ©(2015), with permission from Elsevier)

In actuator placement based on controllability actuators are placed at the point that is most controllable; more precisely, letting λmin (L c ) indicate the minimum eigenvalue of L c , the cost (4.63) c(r ) = λmin (L c (r )) is maximized. Controllability versus actuator location for a model with 5 modes and a model with 10 modes is shown in Fig. 4.9. The two curves are quite different. Points of high controllability for 5 modes can be almost uncontrollable for the model with 10 modes. The fundamental issue can be understood by examining the original partial differential equation. Since the range of the control operator is finite-dimensional, in this case, dimension one, the system cannot be exactly controllable. Theorem 3.2 then implies that for any bounded control operator B there is no m > 0 such that the controllability Gramian L c satisfies for all z ∈ Z, L c z, z ≥ mz2 .

(4.64)

The generator A is Riesz-spectral and so by Theorem 2.25 the system will be approximately controllable if and only if

4.4 LQ-Optimal Actuator Location

143

b, φn = b, sin(nπx) = 0 for all n. Calculating, 1 b, sin(nπx) = 

r + 2 sin(nπr ) sin(nπ) . sin(nπx)d x = nπ

r −

If either r or  is rational, for some n, b, sin(nπx) = 0 and the system is not approximately controllable. Even if b is selected so that the system is approximately controllable, since (4.64) does not hold the lower bound on the eigenvalues of L c is 0. This means that the controllability cost function (4.63) is always 0. As an illustration of this point, a plot of controllability versus the number of modes for a fixed actuator location is shown in Fig. 4.10. Thus, there are serious numerical issues associated with the use of controllability on the entire state-space as a criterion for actuator location. Also, even neglecting the issue of lack of exact controllability in the original PDE model, controllability is not generally the best measure of controlled system performance. Consider now the approximated system with 5 modes as the model. In Fig. 4.11, the response and control effort with the actuator placed at the LQ-optimal location, based on a random initial condition, versus those with the actuator at the location of optimal controllability for initial conditions

1.4

1.2

smallest eigenvalue

1

0.8

0.6

0.4

0.2

0 0

2

4

6

8

10

12

14

16

18

20

number of modes

Fig. 4.10 Controllability versus number of modes with  = 0.001 and actuator location r = 0.11. Regardless of the choice of actuator location, the controllability measure c(r ) converges to 0 as the number of modes increases. (Reprinted from [3], ©(2015), with permission from Elsevier)

144

4 Optimal Linear-Quadratic Controller Design 5

actuator location: 0.254 actuator location: 0.1

0.25

actuator location: 0.254 actuator location: 0.1

4 0.2 3

0.1

2

0.05

1 control

deflection

0.15

0 −0.05

0 −1

−0.1 −2 −0.15 −3

−0.2

−4

−0.25

−5 0

0.5

1

1.5

2

2.5 time

3

3.5

4

4.5

5

0

0.5

1.5

2

2.5 time

3

3.5

4

4.5

5

(b) z(0) = z0,1 , control signal

(a) z(0) = z0,1 , beam vibration at centre 5

actuator location: 0.254 actuator location: 0.1

0.25

1

actuator location: 0.254 actuator location: 0.1

4 0.2 3

0.1

2

0.05

1

0

control

deflection

0.15

−0.05

0 −1

−0.1 −2 −0.15 −3

−0.2

−4

−0.25

−5 0

0.5

1

1.5

2

2.5 time

3

3.5

4

4.5

5

(c) z(0) = z0,2 , beam vibration at centre

0

0.5

1

1.5

2

2.5 time

3

3.5

4

4.5

5

(d) z(0) = z0,2 , control signal

Fig. 4.11 Comparison of the response with the actuator placed at x = 0.254 (LQ-optimal actuator location, Q = I and R = 1 with random initial condition, covariance V = I ) to the response with the actuator at x = 0.1(location of maximum controllability). Both the deflection and control signal are smaller for an actuator placed at the LQ-optimal actuator location than at the location of optimal controllability. (Reprinted from [3], ©(2015), with permission from Elsevier)

⎤ ⎡ ⎤ 1.5 −1 ⎢1.4⎥ ⎢1⎥ ⎢ ⎥ ⎢ ⎥ ⎢1.3⎥ ⎢−1⎥ ⎢ ⎥ ⎢ ⎥ ⎢1.2⎥ ⎢1⎥ ⎢ ⎥ ⎢ ⎥ ⎢1.1⎥ ⎢ ⎥ ⎥ , z = ⎢−1⎥ =⎢ 0,2 ⎢1.0⎥ ⎢1⎥ ⎢ ⎥ ⎢ ⎥ ⎢0.9⎥ ⎢−1⎥ ⎢ ⎥ ⎢ ⎥ ⎢0.8⎥ ⎢1⎥ ⎢ ⎥ ⎢ ⎥ ⎣0.7⎦ ⎣−1⎦ 0.6 1 ⎡

z 0,1

are compared. Both the beam deflection and the control signal are smaller for an actuator placed at the LQ-optimal actuator location than at the location of optimal controllability.

4.4 LQ-Optimal Actuator Location

145

Optimizing controllability considers control to all states, most of which are not of interest. Using the same criterion for actuator location as for controller design significantly improves the control system performance. Figure 4.11 also shows that careful actuator placement can improve the controlled states’ trajectory with smaller control effort.  The L Q-optimal actuator location depends on whether the worst initial condition is considered, or whether the initial condition is considered random. Also, like the control signal, the optimal location depends on the choice of weights C ∗ C and R. This is illustrated by Example 4.32 and also the following example. Example 4.43 (Comparision of weights for LQ-optimal control of beam vibrations) Consider the same beam model as in Example 4.42, with the same modal approximation method, and 5 modes in the approximation. Plots of optimal linear-quadratic cost versus actuator location for full state weight C = I and various weights on the control R are shown in Fig. 4.12 (worst initial condition) and in Fig. 4.13 (random initial condition). As R decreases, that is the state cost increases compared to the control cost, the optimal actuator location moves towards the region between 0.1 and 0.2 (or 0.8 and 0.9) for the random initial condition case, and between 0.3 and 0.4 (or 0.6 and 0.7) for the worst initial condition case. As R increases, or in other words, the control cost increases compared to the state, for both approaches the optimal actuator location converges to the centre 0.5. (See Figs. 4.12b and 4.13b.) Comparision with the controllability plot (Fig. 4.9) reveals that the centre is a point of zero controllability. The even modes have nodes at the centre and are not affected by an actuator placed there. In general, the centre is a poor choice for the actuator location. However, the most significant mode, the first mode, has a peak at the centre, and so the centre is a good spot for controlling this mode. If control cost is significant the effort is put into controlling this important mode. To analyze control effort in more detail, at each actuator location the control signals with the optimal LQ state feedback controller and various initial conditions are calculated. Simulations were done for the initial conditions ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 1 1.5 −1 1 0 ⎢1⎥ ⎢1.4⎥ ⎢1⎥ ⎢0⎥ ⎢1⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢1⎥ ⎢1.3⎥ ⎢−1⎥ ⎢0⎥ ⎢0⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢1⎥ ⎢1.2⎥ ⎢1⎥ ⎢0⎥ ⎢0⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢1⎥ ⎢1.1⎥ ⎢−1⎥ ⎢0⎥ ⎢ ⎥ ⎥, z ⎢ ⎥, z = ⎢ ⎥, z ⎢ ⎥ , z = ⎢0⎥ . = = z 0,1 = ⎢ 0,2 0,3 0,4 0,5 ⎢1⎥ ⎢1⎥ ⎢1⎥ ⎢0⎥ ⎢0⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢1⎥ ⎢0.9⎥ ⎢−1⎥ ⎢0⎥ ⎢0⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢1⎥ ⎢0.8⎥ ⎢1⎥ ⎢0⎥ ⎢0⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣1⎦ ⎣0.7⎦ ⎣−1⎦ ⎣0⎦ ⎣0⎦ 1 0.6 1 0 0 For weighting matrices C = I and R = 0.01, and a minimum variance LQ-cost (with variance V = I ), a plot of the L 2 norm of the control signal versus actuator location

146

4 Optimal Linear-Quadratic Controller Design Q = I, R = 100 Q = I, R = 1 Q = I, R = 0.01 Q = I, R = 0.0001

linear−quadratic cost (relative)

0.25

0.2

0.15

0.1

0.05

0

0

0.1

0.2

0.3

0.4 0.5 0.6 actuator location

0.7

0.8

0.9

1

(a) C = I, R = 0.0001, 0.01, 1, 100 1 Q = I, R = 100 Q = I, R = 200 Q = I, R = 500 Q = I, R = 1000

0.9

linear−quadratic cost (relative)

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.1

0.2

0.3

0.4 0.5 0.6 actuator location

0.7

0.8

0.9

1

(b) C = I, R = 100, 200, 500, 1000 Fig. 4.12 Optimal linear-quadratic cost with respect to worst initial condition, λmax ( (r ), versus actuator location for different weights. To facilitate comparision, each curve has been shifted and scaled so that the minimum is 0 and the maximum is 1. (Reprinted from [3], ©(2015), with permission from Elsevier)

4.4 LQ-Optimal Actuator Location

147 Q = I, R = 100 Q = I, R = 1 Q = I, R = 0.01 Q = I, R = 0.0001

linear−quadratic cost (relative)

0.25

0.2

0.15

0.1

0.05

0 0

0.1

0.2

0.3

0.4 0.5 0.6 actuator location

0.7

0.8

0.9

1

(a) C = I, R = 0.0001, 0.01, 1, 100 1 Q = I, R = 100 Q = I, R = 200 Q = I, R = 500 Q = I, R = 1000

linear−quadratic cost (relative)

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.1

0.2

0.3

0.4 0.5 0.6 actuator location

0.7

0.8

0.9

1

(b) C = I, R = 100, 200, 500, 1000 Fig. 4.13 Optimal linear-quadratic cost with respect to a random initial condition with variance V = I, trace (r ), versus actuator location for different weights. To facilitate comparison, each curve has been shifted and scaled so that the minimum is 0 and the maximum is 1. (Reprinted from [3], ©(2015), with permission from Elsevier)

148

4 Optimal Linear-Quadratic Controller Design 30 x x x 25

x

L2 norm of control signal

x

0, 1 0, 2 0, 3 0, 4 0, 5

20

15

10

5

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

actuator location

Fig. 4.14 L 2 -norm of the control signal versus actuator location for different initial conditions. Cost is linear-quadratic, random initial condition, with covariance V = I and weights C = I , R = 0.01. (Reprinted from [3], ©(2015), with permission from Elsevier)

for each initial condition is shown in Fig. 4.14. Note that in general the cost is a nonconvex function of the actuator location. This needs to be considered in optimization. Another observation is that there is no apparent relationship between locations with high controllability and optimal actuator locations using linear-quadratic control. Also, at the LQ optimal actuator locations, the L 2 norm of the control signal is not larger as compared to other locations. Thus, better performance is achieved without increased controller cost.  LQ-optimal actuator location can also be considered for the finite-time problem. The theoretical results are similar to the infinite-time case. For computation using approximations, weaker assumptions suffice, as they do for the case of controller design without actuator location. Theorem 4.44 Consider the observation system (A, C) with state space Z and observation space Y. Let B(r ) ∈ B(U, Z) be a family of compact input operators such that for any r0 ∈ , lim B(r ) − B(r0 ) = 0. r →r0

The Riccati operators (r, t) solving the differential Riccati equation (4.3) for each r are continuous functions of r in the operator norm: lim  (r, t) − (r0 , t) = 0, 0 ≤ t ≤ t f

r →r0

4.4 LQ-Optimal Actuator Location

149

and there exists an optimal actuator location  r such that μ. ( r , 0) = inf (r, 0) =  r ∈

If U and Y are in addition finite-dimensional, and  f is a trace class operator then (r, t) are continuous functions of r in the trace norm: lim (r, t) − (r0 , t)1 = 0, 0 ≤ t ≤ t f

r →r0

and there exists a trace norm LQ-optimal actuator location  r such that μ. ( r , 0)1 = inf (r, 0))1 =  r ∈

Theorem 4.45 With the assumptions of Theorem 4.44 consider a projection Pn : Z → Zn and a sequence of approximations (An , Bn (r ), Cn ) with Bn (r ) = Pn B(r ) that satisfy Assumptions (A1) and (A1∗ ) for each r. If also C and  f are compact operators, then letting  r be a optimal actuator location for (A, B(r ), C) on the time rn , μn , it follows that interval [0, t f ] with optimal cost μ and defining similarly  μn → μ, rn } such that and there exists a subsequence { rm } of the optimal actuator locations { r.  rm →  Assume in addition that U and Y are finite-dimensional,  f is a trace class operator and lim Pn  f Pn −  f 1 = 0. n→∞

Then letting r be a trace norm LQ-optimal actuator location for (A, B(r ), C) on the time interval [0, t f ] with optimal cost μ and defining similarly  rn , μn , it follows that μn → μ, rn } such that and there exists a subsequence { rm } of the optimal actuator locations { r.  rm → 

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4 Optimal Linear-Quadratic Controller Design

4.5 Notes and References Linear quadratic control of finite-dimensional systems is covered in many textbooks; see for instance, [4]. The theory for linear-quadratic control of infinite-dimensional systems is provided in detail in [5, 6]. The latter reference includes results for unbounded control operators and also approximation theory. Results for convergence of optimal linear-quadratic controllers calculated using an approximation is now fairly complete. Theorem 4.5 is a special case of the results in [7] for time-varying systems. Theorem 4.11 on strong convergence uses in particular [8, 9]. Uniform convergence was shown in [2]. Other results on convergence with different assumptions can be found in [10]. A counter-example where the approximating controllers do not converge because of lack of convergence of the adjoint systems is in [11]. Theorem 4.11 has been extended to unbounded control operators B, with stronger results for systems governed by an analytic semigroup [6, 12]. In particular, Theorem 4.21 applies to operators B and C that may be unbounded [6]. The examples of non-compact solutions to AREs first appeared in [5, 13]. Theorem 4.37 is due to [14]. The assumption of uniform exponential stabilizability (and detectability) appears in many different contexts in the literature for controller and estimator design using approximations. Some early results are in [6, 8]. Theorem 4.14 generalizes a result in [8] and is given in [15]. Advanced linear algebra, such as Theorem 4.25 on approximation of a matrix by one of lower rank, can be found in a number of books; for instance, [16]. A tighter bound than that in Theorem 4.25 on the error in a low-rank approximation to a matrix that is applicable to all diagonalizable matrices exists [17]. The most basic iterative method to solve Lyapunov equations is Smith’s method [18]. Convergence of the iterations can be improved by careful choice of the iteration parameter p; see [19]. This method of successive substitution is unconditionally convergent, but the convergence rate is linear convergence. The ADI method [20, 21] improves Smith’s method by using a different parameter pi at each step. For a symmetric system matrix A the optimal parameters are easily calculated. For other matrices, these parameters can be estimated as in [22]. Alternatively, the calculation can be split into 2 real parts [23] but their presence increases computation time. As the spectrum of A flattens to the real axis the ADI parameters are closer to optimal. See [24] for an approach that relates these calculations to the original PDE. The modification to standard ADI, Cholesky-ADI [23, 25], is advantageous since it takes advantage of several key features of the large Lyapunov equations that arise in control of PDEs: (1) the matrices are often sparse and (2) the number of controls is typically much smaller than the order of the approximation so that B ∈ Rn×m where m  n. A bound on the rank of the solution to a Lyapunov equation if A is symmetric is in [26]. The usual direct method, Theorem 4.22, for solving an ARE is a Schur vectorbased technique due to [27]. The iterative Newton–Kleinman method was first introduced in [28]. The Schur direct method is implemented in a number of software libraries, as are Cholesky-ADI iterations and Newton–Kleinman solution of

4.5 Notes and References

151

an ARE. A modification of the Newton–Kleinman method was proposed in [29]. The modified Newton–Kleinman method is mathematically equivalent to the original method, involving just a reformulation of the iterations. The advantage of the modified Newton–Kleinman method lies in the fact that the Lyapunov equation to be solved at each step has a right-hand side with rank limited by the number of controls, while the rank of the right-hand side in the original formulation (4.42) has rank equal to that of the state weight C which could be quite large. This technique, implemented with a Cholesky-ADI Lyapunov solver can significantly reduce computation time; see [1, 30] for implementation of Newton–Kleinman methods on several PDE examples. Example 4.27 is analyzed in more detail in [1]. Less damping increases the error in an approximation to the Riccati solution (or the solution to the Lyapunov equation) by a solution of low rank. More iterations in the Lyapunov loop will be required as the damping parameter is decreased and the imaginary part of the eigenvalues becomes more significant. This may have consequences for control of systems such as plates and coupled acoustic-structures where the spectra have large imaginary parts relative to the real parts. Special methods may be needed to calculate controllers for these systems. Theoretical bounds for order reduction of a class of infinite-dimensional systems are given in [31, 32]. Controller design using approximations for systems that are not uniformly exponentially stabilizable or detectable is incomplete at this time. There are some results on approximation of algebraic Riccati equations for systems that are asymptotically stabilizable in a certain sense in [33, 34]. Given the slow convergence of iterations for lightly damped structures, stabilizing controller computation for such systems may not be straightforward. Systems where the real part of the eigenvalues does not decrease with frequency are often not straightforward to approximate for the purposes of control. This was illustrated by Example 4.18 in approximation of the wave equation with dissipation on the boundary. Using finite differences to approximate this equation leads to approximations that are not uniformly stable, and in fact are not even uniformly stabilizable [35]. A useful approach when theoretical results for controller convergence are not available is to check numerically (1) whether the stability margin of the original system is preserved and (2) calculate the controller for a series of approximation orders n 1 < n 2 < n 3 . . . and then implement these controllers with a higher-order approximate plant. The controlled system should have a uniform stability margin and convergent performance. This issue is discussed in more detail in Chap. 7. The results on LQ-optimal actuator location in Sect. 4.4 for the infinite-time cost can be found in [2]. The finite-time case, including time-varying systems, is covered in [36]. Comparisons using a simple beam of optimal location using controllability versus a linear-quadratic cost, and also with different weights, such as in Examples 4.42 and 4.43, are covered more completely in [3]. That paper also has similar comparisons for multiple actuators on a vibrating plate. For placement of a single actuator, checking all possible locations is often not excessively time-consuming. However, for placement of multiple actuators, use of

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an optimization algorithm is required. Since the problem is not convex, multiple calculations with different initial positions is advised in order to avoid locating a local minimum. In [37] an approach that makes the LQ-optimal actuator problem convex is described. An algorithm using this idea is described in detail in [38] and experimental implementation of the results is described. One of the reasons for the popularity of optimal linear-quadratic control is that the optimal control can be calculated as a feedback control. When the objective is to reduce the effect of an disturbance, H2 or H∞ optimal control, discussed in the next chapter can be used. These approaches also lead to feedback control laws. For more general optimal control of DPS, with particular relevance to problems where the underlying semigroup is analytic, see [39].

References 1. Morris KA, Navasca C (2010) Approximation of low rank solutions for linear quadratic feedback control of partial differential equations. Comp Opt App 46(1):93–111 2. Morris KA (2011) Linear quadratic optimal actuator location. IEEE Trans Autom Control 56:113–124 3. Yang SD, Morris KA (2015) Comparison of actuator placement criteria for control of structural vibrations. J Sound Vib 353:1–18 4. Morris KA (2001) An introduction to feedback controller design. Harcourt-Brace Ltd 5. Curtain RF, Zwart H (1995) An introduction to infinite-dimensional linear systems theory. Springer, Berlin 6. Lasiecka I, Triggiani R (2000) Control theory for partial differential equations: continuous and approximation theories. Cambridge University Press, Cambridge 7. Gibson JS (1979) The Riccati integral equations for optimal control problems on Hilbert spaces. SIAM J Control Optim 17(4):637–665 8. Banks HT, Kunisch K (1984) The linear regulator problem for parabolic systems. SIAM J Control Optim 22(5):684–698 9. Ito K (1987) Strong convergence and convergence rates of approximating solutions for algebraic Riccati equations in Hilbert spaces. In: Schappacher W, Kappel F, Kunisch K (eds) Distributed parameter systems. Springer, Berlin 10. De Santis A, Germani A, Jetto L (1993) Approximation of the algebraic Riccati equation in the Hilbert space of Hilbert-Schmidt operators. SIAM J Control Optim 31(4):847–874 11. Burns JA, Ito K, Propst G (1988) On non-convergence of adjoint semigroups for control systems with delays. SIAM J Control Optim 26(6):1442–1454 12. Banks HT, Ito K (1997) Approximation in LQR problems for infinite-dimensional systems with unbounded input operators. J Math Systems Estim Control 7(1):1–34 13. Burns JA, Sachs EW, Zietsman L (2008) Mesh independence of Kleinman-Newton iterations for Riccati equations in Hilbert space. SIAM J Control Optim 47(5):2663–2692 14. Curtain RF, Mikkola K, Sasane A (2007) The Hilbert-Schmidt property of feedback operators. J Math Anal Appl 329:1145–1160 15. Morris KA (1994) Design of finite-dimensional controllers for infinite-dimensional systems by approximation. J Math Syst Estim Control 4(2):1–30 16. Golub GH, van Loan CF (1989) Matrix computations. John Hopkins 17. Antoulas AC, Sorensen DC, Zhou Y (2002) On the decay rate of Hankel singular values and related issues. Syst Control Lett 46:323–342 18. Smith RA (1968) Matrix equation XA + BX = C. SIAM J Appl Math 16:198–201

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19. Russell DL (1979) Mathematics of finite dimensional control systems: theory and design. Marcel Dekker 20. Lu A, Wachpress EL (1991) Solution of Lyapunov equations by alternating direction implicit iteration. Comput Math Appl 21(9):43–58 21. Wachpress E (1988) Iterative solution of the Lyapunov matrix equation. Appl Math Lett 1:87–90 22. Ellner N, Wachpress EL (1991) Alternating direction implicit iteration for systems with complex spectra. SIAM J Numer Anal 28(3):859–870 23. Li JR, White J (2002) Low rank solution of Lyapunov equations. SIAM J Matrix Anal Appl 24:260–280 24. Opmeer MR, Reis T, Wollner W (2013) Finite-rank ADI iteration for operator Lyapunov equations. SIAM J Control Optim 51(5):4084–4117 25. Penzl T (2000) A cyclic low-rank smith method for large sparse Lyapunov equations. SIAM J Sci Comput 21(4):1401–1418 26. Penzl T (2000) Eigenvalue decay bounds for solutions of Lyapunov equations: the symmetric case. Syst Control Lett 40:139–144 27. Laub AJ (1979) A Schur method for solving algebraic Riccati equations. IEEE Trans Autom Control 24:913–921 28. Kleinman D (1968) On an iterative technique for Riccati equation computations. IEEE Trans Autom Control 13:114–115 29. Banks HT, Ito K (1991) A numerical algorithm for optimal feedback gains in high-dimensional linear quadratic regulator problems. SIAM J Control Optim 29(3):499–515 30. Grad JR, Morris KA (1996) Solving the linear quadratic control problem for infinitedimensional systems. Comput Math Appl 32(9):99–119 31. Guiver C, Opmeer MR (2013) Error bounds in the gap metric for dissipative balanced approximations. Linear Algebra Appl 439(12):3659–3698 32. Guiver C, Opmeer MR (2014) Model reduction by balanced truncation for systems with nuclear Hankel operators. SIAM J Control Optim 52(2):1366–1401 33. Oostveen JC, Curtain RF (1998) Riccati equations for strongly stabilizable bounded linear systems. Automatica 34(8):953–967 34. Oostveen JC, Curtain RF, Ito K (2000) An approximation theory for strongly stabilizing solutions to the operator LQ Riccati equation. SIAM J Control Optim 38(6):1909–1937 35. Peichl GH, Wang C (1997) On the uniform stabilizability and the margin of stabilizability of the finite-dimensional approximations of distributed parameter systems. J Math Syst Estim Control 7:277–304 36. Wu X, Jacob B, Elbern H (2015) Optimal control and observation locations for time-varying systems on a finite-time horizon. SIAM J Control Optim 54(1):291–316 37. Geromel JC (1989) Convex analysis and global optimization of joint actuator location and control problems. IEEE Trans Autom Control 34(7):711–720 38. Darivandi N, Morris K, Khajepour A (2013) An algorithm for LQ-optimal actuator location. Smart Mater Struct 22(3):035001 39. Tröltzsch F (2010) Optimal control of partial differential equations, volume 112 of graduate studies in mathematics. American Mathematical Society, Providence, RI. Theory, methods and applications, Translated from the 2005 German original by Jürgen Sprekels

Chapter 5

Disturbances

Surprises are foolish things. The pleasure is not enhanced, and the inconvenience is often considerable. (Jane Austen, Emma)

The LQ controller design objective, described in Chap. 4, is to find a control u(t) that minimizes a quadratic cost functional. The objective of the control is to minimize the effect of the initial condition on the cost functional. The model is z˙ (t) = Az(t) + B2 u(t), z(0) = z 0 where the operator A with domain dom(A) generates a strongly continuous semigroup S(t) on a separable Hilbert space Z, and B2 ∈ B(U, Z). However, in many systems, uncontrolled inputs ν(t), known as disturbance(s), are present. A controller can reduce the response to the disturbance(s). Set the initial condition to zero to simplify the discussion. The system being controlled is z˙ (t) = Az + B1 ν(t) + B2 u(t), z(0) = 0

(5.1)

where B1 ∈ B(V, Z) and V is a separable Hilbert space. The disturbance ν is generally due to uncontrolled inputs, but also modelling errors can be regarded as disturbances to the model. Define the performance output y1 (t) = C1 z(t) + D12 u(t)

(5.2)

where with Y a Hilbert space, C1 ∈ B(Z, Y), D12 ∈ B(U, Y). Throughout this chapter it will be assumed that

© Springer Nature Switzerland AG 2020 K. A. Morris, Controller Design for Distributed Parameter Systems, Communications and Control Engineering, https://doi.org/10.1007/978-3-030-34949-3_5

155

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5 Disturbances

∗ 1. R = D12 D12 has a bounded inverse so that the control cost is non-singular, and 2. to simplify the formulae, ∗ C1 = 0. (5.3) D12

One controller design objective is to find a control u to reduce the cost  y1 2 =

∞

y1 (t)2 dt.

(5.4)

0

The cost (5.4) is identical to the linear quadratic (LQ) cost  J (u, z 0 ) =

∞

C z(t), C z(t) + u(t), Ru(t)dt,

(5.5)

0

if C1 =

  C , 0

 D12 =

 0 . 1 R2

The difference between this problem and the LQ problem is that in LQ controller design the aim is to reduce the response to the initial condition z(0) with disturbance ν = 0 while now the objective is to reduce the response to the disturbance ν with z(0) = 0. The coerciveness of R is needed in order to ensure a non-singular control ∗ cost. The second assumption, that D12 C1 = 0 is made solely to simplify the formulae. This assumption will be dropped in Chap. 7 when output feedback is covered. The full information case is considered in this chapter. That is, the input to the controller is   z(t) (5.6) y2 (t) = v(t)     I 0 = z(t) + ν(t). 0 I

5.1 Fixed Disturbance Consider first a fixed disturbance ν ∈ L 2 (0, ∞; V). Unknown disturbances will be considered later in this chapter. Suppose the simplest case of V = C so there is a single scalar-valued disturbance and also there is a single cost so Y = C. If u ≡ 0, yˆ (s) = G(s)ν(s) ˆ where

G(s) = C1 (s I − A)−1 B1 .

5.1 Fixed Disturbance

157

Suppose that ν(s) ˆ = 1 for all s, that is, ν is an impulse. Such a disturbance is a signal with a uniform response over all frequencies but no stochastic effects. In this situation, yˆ (s) = G(s). By the Paley–Wiener Theorem (Theorem 3.33) y L 2 =  yˆ H2 = GH2 . The subscript 2 will henceforth be used for both the L 2 - and H2 -norms. The case of different disturbances ν is handled by absorbing the frequency content of ν into the system description (5.1) so the model has ν(s) ˆ = 1. To illustrate this idea, consider a control system z˙ p (t) = A p z(t) + Bd do (t) + Bu u(t),

z(0) = 0

(5.7)

where A p with domain D(A p ) generates a C0 -semigroup on the state space Z p , Bd ∈ B(C, Z), Bu ∈ B(U, Z) and U is a Hilbert space. Assume that the Laplace transform of the disturbance, dˆo (s), is rational and strictly proper. This means that it can be written for bo , co ∈ Rn , and matrix Ao ∈ Rn×n , dˆo (s) = co , (s I − Ao )−1 bo .

(5.8)

Thus for ν such that ν(s) ˆ = 1 for all s, do (t) is the output of the system z˙ o (t) = Ao z o (t) + bo ν(t), z o (0) = 0, do (t) = z o , co (t).

(5.9)

Define  the operators Bo ν = bo ν, and Co z o = z o , co . Define the augmented state z = z p z o , and state-space Z = Z p × Rn . The control system (5.7) with disturbance (5.8) can be written        z˙ p (t) 0 B A p Bd Co z˙ (t) = + ν(t) + u u(t). = Ao 0 0 z˙ o (t) Bo 

 

  



A

B1

B2

This is of the form (5.1) with the disturbance νˆ = 1. It will be assumed henceforth that this normalization has been done. Some additional framework is needed. Definition 5.1 A bounded linear operator M ∈ B(U, Y), the linear space of bounded linear operators between separable Hilbert spaces U and Y, is a Hilbert–Schmidt operator if ∞  Mek , Mek  < ∞ M H S = k=1

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5 Disturbances

where {ek } is any orthonormal basis for U. It can be shown that this definition is independent of the choice of basis. The quantity M H S is a norm, known as the Hilbert–Schmidt norm, on the linear space of Hilbert–Schmidt operators. Recall the definition of a trace class operator (Definition 4.33). Theorem 5.2 1. Every Hilbert–Schmidt operator is a compact operator. 2. Every trace class operator is a Hilbert–Schmidt operator and, letting M indicate the usual operator norm and M1 the trace norm, M ≤ M H S ≤ M1 . The H2 -norm of a transfer function G is G2 =

1 2π



∞

trace(G( jω)∗ G( jω))dω,

−∞

provided that G( jω) is a Hilbert–Schmidt operator. Theorem 5.3 If at least one of the following assumptions is satisfied: – B1 is a Hilbert–Schmidt operator, – C1 is a Hilbert–Schmidt operator, or – both B1 and C1 are trace class; then for every s ∈ ρ(A) the transfer function G(s) = C1 (s I − A)−1 B1 is a Hilbert– Schmidt operator. The above assumptions are sufficient conditions for a transfer function to be a Hilbert– Schmidt operator. However, there are generally a finite number of disturbances, control signals, and measurements. Since every finite-rank operator is trace class, in practice all the above assumptions are generally satisfied. As mentioned above for V = C, if ν(s) ˆ = 1 for all s, or equivalently, ν is an impulse, then G2 is the L 2 -norm of the output y2 . More generally, let {ek } be an orthonormal

∞ basis  for U, 2and denote the output with input δ(t)ek by yk . Then 1 ˆk (j ω)| dω. G22 = 2π k |y −∞ For the open-loop case the H2 -norm is the solution to a Lyapunov equation. Theorem 5.4 Consider the system (A, B, C) with transfer function G where either B and C are both trace class or one of B or C is a Hilbert–Schmidt operator. Let the controllability Gramian L c be the solution to the operator Lyapunov equation (AL c + L c A∗ + B B ∗ )z = 0, z ∈ dom(A∗ ),

(5.10)

and the observability Gramian L o the solution to (A∗ L o + L o A + C ∗ C)z = 0, z ∈ dom(A).

(5.11)

5.1 Fixed Disturbance

159

The H2 -norm of the system is G22 = trace(B ∗ L o B) = trace(C L c C ∗ ). If B is defined by Bu = bu for some b ∈ Z, then G22 = b, L o b. Consider now the problem of choosing the control to minimize the H2 -norm. The objective is to find a control signal u minimize the H2 -norm of ˆ + B2 u(s)) ˆ + D12 u(s). ˆ yˆ1 (s) = C1 (s I − A)−1 (B1 ν(s) This is equivalent to minimizing the norm of y1 ∈ L 2 (0, ∞; Y). Theorem 5.5 Consider the system (5.1), (5.2) satisfying at least one assumption in ∗ Theorem 5.3 and assume that (A, B2 ) is stabilizable. Defining R = D12 D12 , the H2 -optimal control is the state feedback u(t) = −R −1 B2∗ z(t) where  (dom(A)) ⊂ dom(A∗ ) and  ∈ B(Z, Z) is the positive semi-definite solution to the algebraic Riccati equation (ARE) (A∗  + A − B2 R −1 B2∗  + C1∗ C1 )z = 0 z ∈ dom(A).

(5.12)

The optimal cost, or closed-loop H2 -norm, is 

trace(B1∗ B1 ).

If there is a single disturbance, so B1 d can be written b1 d for some b1 ∈ Z, the optimal cost is  b1 , b1 . Note that the optimal control is found by solving the ARE (5.12) which does not involve the model of the disturbance operator B1 . Thus, just as the LQ-optimal feedback control is independent of the initial condition, the H2 -optimal control is independent of B1 . However, the initial condition z(0) affects the LQ-cost, and similarly, the disturbance operator B1 affects the H2 -cost. Finite-dimensional approximations are normally used to calculate an approximating optimal control. Since H2 -optimal control is calculated by solving an ARE identical to that for LQ-control, approximating control converges to the optimal control, stabilizes the system and provides sub-optimal performance under the same assumptions as for LQ control; see in particular Theorems 4.11, 4.21, 4.39.

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5 Disturbances

5.2 H2 -Optimal Actuator Location Suppose that there are M actuators with locations that can be varied over some compact set  ⊂ Rn . Parametrize the actuator locations by r and indicate the dependence of the input operator B2 on the locations by the notation B2 (r ). Letting  M indicate vectors of length M with each component in , r ∈  M . The H2 -optimal actuator location problem is thus to find the actuator location rˆ that minimizes the cost trace(B1∗ (r )B1 ). Theorem 5.6 Consider the system (5.1), (5.2) satisfying at least one assumption in Theorem 5.3. If for any r, ro ∈  M , lim B2 (r ) − B2 (ro ) = 0

r →ro

(A, B2 (r )) is stabilizable for all r ∈  M and (A, C1 ) is detectable, then there exists an optimal actuator location rˆ ∈  M such that trace(B1∗ (ˆr )B1 ) = inf trace(B1∗ (r )B1 ). r ∈ M

(5.13)

Consider a single disturbance so that B1 d = b1 d for some b1 ∈ Z. If the spatial distribution of the disturbance, b1 , is not known then the objective is to find the actuator location that minimizes the H2 -cost over possible disturbance distributions: The problem becomes that of choosing the actuator location r to minimize the closed loop response to the worst spatial disturbance distribution; that is inf sup b1 , (r )b1 

r ∈ M b1 ∈Z

where (r ) indicates the solution to the ARE (5.12) with actuator location r . Corollary 5.7 With the same assumptions as Theorem 5.6, and a scalar disturbance (V = C), there is rˆ ∈  M , φ ∈ Z such that φ, (ˆr )φ = (ˆr ) = inf sup b1 , (r )b1  r ∈ M b1 ∈H

(5.14)

and this cost is achieved when φ is the eigenfunction φ L corresponding to the largest eigenvalue of (ˆr ). Thus, the worst spatial disturbance is obtained by B1 u = φ L u. Thus, if B2 (r ) is a continuous function of r , and there is a single disturbance, both the optimal actuator location problem with a known spatial disturbance distribution b1 and the problem where b1 and r are to be determined, are well-posed optimization problems. When the disturbance shape B1 is known, the problem is of minimizing the weighted trace of the Riccati operator over the actuator locations r . If b1 is unknown,

5.2 H2 -Optimal Actuator Location

161

the problem is to minimize (r ) over the actuator locations r . Both problems are discussed in the context of linear-quadratic optimal actuator location in Sect. 4.4. In practice, the control is calculated using an approximation to the solution (5.12). As in previous sections, let {Zn } be a family of finite-dimensional subspaces of the state space Z and Pn the orthogonal projection of Z onto Zn . The space Zn is equipped with the norm inherited from Z. Consider a sequence of operators An ∈ B(Zn , Zn ) and define B2n (r ) = Pn B2 (r ), B1n = Pn B1 . This leads to a sequence of approximations to the system (5.1), (5.2) z˙ (t) = An z(t) + B1n d(t) + B2n (r )u(t).

(5.15)

Defining C1n = C1 |Zn , the cost (5.4) becomes 

∞

C1n z(t) + D12 u(t)2 dt.

0

If (An , B2n ) is stabilizable and (An , C1n ) is detectable, then the minimum cost is ∗ n B1n ), where n is the unique positive semi-definite solution to the algetrace(B1n braic Riccati equation ∗ ∗ n + C1n C1n = 0, A∗n n + n An − n B2n R −1 B2n

on the finite-dimensional space Zn . (For simplicity of notation the dependence of B2 , B2n and n on r will not always be indicated.) Define μ to be the optimal H2 -cost, μ = inf B1 , (r )B1  = B1 , (ˆr )B1  r ∈ M

where rˆ is an optimal actuator location and define μn , rˆn similarly. The following theorem shows that strong convergence of n →  implies convergence of the cost: μn → μ. Convergence of the optimal actuator location and controllers is also implied. That is, performance arbitrarily close to optimal can be achieved with the approximating actuator locations and controllers. The assumptions on the approximation are the same as required for other controller design results: assumptions (A1) and (A1∗ ) and uniform stabilizability and detectability (Definitions 4.9, 4.10). Theorem 5.8 Consider the control system (5.1), (5.2) satisfying at least one assumption in Theorem 5.3 and assume also that for any r, ro ∈  M , lim B2 (r ) − B2 (ro ) = 0.

r →ro

If the approximation scheme satisfies assumptions (A1), (A1∗ ), and is both uniformly stabilizable and uniformly detectable then

162

5 Disturbances ∗ inf trace(B1∗ (r )B1 ) = lim inf trace(B1n n (r )B1n ).

r ∈ M

n→∞ r ∈ M

Also, the sequence of approximating actuator locations rˆm has a convergent subsequence. Any convergent subsequence has the property that μ = lim trace(B1∗ (ˆrm )B1 ); m→∞

(5.16)

and the corresponding controllers converge. Now consider the situation where there is a single disturbance ν(t) but the disturbance distribution is unknown so that the actuator location should minimize inf (r ).

r ∈ M

Theorem 5.9 Consider the control system (5.1), (5.2). Assume that for any r, ro ∈ M , lim B2 (r ) − B2 (ro ) = 0, r →ro

and that either (i) C1 is a compact operator or (ii) A generates an analytic semigroup and for some real α (αI − A)−1 is compact. Assume also that the approximation scheme satisfies assumptions (A1), (A1∗ ), and is both uniformly stabilizable and detectable. Consider the problem of minimizing the H2 -cost over all possible disturbance functions inf sup b1 , (r )b1 . r ∈ M b1 ∈Z

Let rˆ be the optimal actuator location for the original problem with optimal cost μ and define similarly μn , rˆn . Then – μ = limn→∞ μn , – the sequence of approximating actuator locations rˆm has a convergent subsequence and any convergent subsequence has the property that μ = lim (ˆrm ), m→∞

and also the corresponding controllers converge. Example 5.10 (Cantilevered beam) As in Example 4.17, but with a different actuator, consider an Euler–Bernoulli model for a cantilevered beam of length  and deflections w(x, t) from its rigid body motion. The deflection is controlled by a force u(t) centered on the point r with width δ. The spatial part of the disturbance is some b1 ∈ L 2 (0, ). The partial differential equation is ρ

∂2w ∂w ∂5w ∂4w + c + c + E = br u(t) + b1 ν(t), v d ∂t 2 ∂t ∂t∂x 4 ∂x 4

5.2 H2 -Optimal Actuator Location

163

for t ≥ 0, 0 < x < , where  br (x) =

1/δ

|r − x|


δ 2 δ 2

,

with boundary conditions ∂w |x=0 = 0, ∂x   3 ∂4w ∂ w = 0, E 3 + cd = 0. ∂x ∂t∂x 3 x=

w(0, t) = 0,  E

∂2w ∂3w + cd 2 ∂x ∂t∂x 2

 x=

Let Ho = L 2 (0, ) with the usual inner product w1 , w2  and Vo = {w ∈ H2 (0, ); w(0) = 0, w (0) = 0}, with inner product

w1 , w2 Vo = Ew1

, w2

.

˙ is The state-space formulation on Z = Vo × Ho with state z = (w, w) z˙ (t) = Az(t) + B1 v(t) + B2 (r )u(t), where A is as defined in Example 3.15 and  B1 =

   0 0 , B2 (r ) = . b1 br

It was shown in Example 3.15 that this model is well-posed and exponentially stable on Z. There is only one control so the control weight is chosen to be 1; that is D12

  0 = . 1

Consider state weights of the form   Co , C1 = 0

(5.17)

where Co ∈ B(Vo , Y). An obvious choice is Co = I where I here indicates the map from Vo into H. Another choice is the tip displacement. Defining C z(t) = w(, t), the state weight Co = C in (5.17). Rellich’s Theorem (Theorem A.35) implies that C ∈ B(Vo , Y).

164

5 Disturbances

Since the semigroup generated by A is exponentially stable, and B2 (r ) has finite rank, the assumptions of Theorem 5.6 are satisfied. Thus, the cost (r ) depends continuously on the actuator location and there exists an optimal actuator location for the problem of an unknown disturbance location and also for a fixed disturbance location. As in Example 4.17 a Galerkin method with cubic splines φi as the basis functions is used to approximate the PDE. This was shown in Example 4.17 to define approximations that satisfy assumptions (A1), (A1∗ ) and are uniformly exponentially stable. This implies that the calculated optimal cost and actuator locations will converge to the true optimal cost and location for both fixed and unknown disturbance locations respectively. The optimal actuator location was calculated for six different spatial disturbance functions b1 . The various functions are shown in Fig. 5.1. Formally, defining β1 = 1.875, s1 = 0.7341, β2 = 4.694, s2 = 1.0185,   2   , μ = 4 , σ = 12 exp − 21 x−μ , σ     1 x−μ 2   b12 (x) = , μ = 2, σ = 8, exp − 2 σ     2 x−μ  b13 (x) = , μ = 3 exp − 21 σ , σ = 12 , 4 b14 (x) = cosh(β1 x) − cos(β1 x) − s1 (sinh(β1 x) − sin(β1 x)) , b15 (x) = cosh(β2 x) − cos(β2 x) − s2 (sinh(β2 x) − sin(β2 x)) , b16 (x) = 1.48. b11 (x) =

√1 2πσ √1 2πσ √1 2πσ

Beam parameters were ρ = 0.093 kg/m, cd = 6.49 × 10−5 Nsm2 , cv = 0.0013 Nsm2 , E = 0.491 Nm2 ,  = 0.4573 m and δ = 0.01 for the computer simulations. The beam was approximated using 80 cubic splines.

Fig. 5.1 Spatial disturbance functions. The mark ◦ denotes the maximum value of each b1 j (x), j = 1, . . . , 6. (©2015 IEEE. Reprinted, with permission, from [1].)

5

5 b11(ξ)

4

2

2

1

1 0

0.2

0.4

0.6

0.8

1

5

0

0

0.4

0.2

0.6

0.8

1

5 b13(ξ)

4

b14(ξ)

4

3

3

2

2

1

1

0

12

3

3

0

b (ξ)

4

0

0.2

0.4

0.6

0.8

1

0

0

0.2

0.4

0.6

0.8

1

0.8 0.6 0.4 Spatial variable ξ/L

1

4 1.5 2 1

0

−4

0

0.2

0.8 0.6 0.4 Spatial variable ξ/L

b (ξ)

0.5

b15(ξ)

−2

1

0

16

0

0.2

5.2 H2 -Optimal Actuator Location

165

Table 5.1 Spatial distribution of disturbances and corresponding optimal actuator location with different state weights. The optimal location is affected by the state weight and the spatial distribution of the disturbance. (©2015 IEEE. Reprinted, with permission, from [1, Tables I, II]) Disturbance shape max b1 j (x) rˆ , Co = I rˆ , Co = C b11 b12 b13 b14 b15 b16

0.25 0.50 0.75 1.0 0.47 [0, ]

Fig. 5.2 Eigenfunctions corresponding to largest eigenvector of (r ) where r is the actuator location. This is the worst spatial disturbance. The maximum does not always occur at r. (©2015 IEEE. Reprinted, with permission, from [1].)

0.26 0.75 0.75 0.75 0.47 0.65

0.51 0.99 0.99 0.99 0.45 0.99

0.5 b (ξ) corresponding to ξ=0.5091L

0

1

−0.5 −1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.5

0.6

0.7

0.8

0.9

1

0.9

1

0.5 0 −0.5 b (ξ) corresponding to ξ=0.9950L

−1 −1.5

1

0

0.1

0.2

0.3

0.4

0.5 b1(ξ) corresponding to ξ=0.4475L

0 −0.5 −1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Spatial variable ξ/L

Table 5.1 shows the optimal actuator location for each disturbance b1 j , for Co = I and Co = C . If the state weight Co is a multiple of the identity operator, the best actuator location is generally at a point where the disturbance function has a maximum. On the other hand, if only the tip position affects the cost, Co = C , the optimal actuator location is very strongly influenced by the weight on the tip displacement. The ARE was solved with Co = C and the actuator placed at points optimal for various disturbance functions: r = 0.5091, 0.9950 and r = 0.4475. The spatial distributions representing the worst spatial distributions are obtained from the largest eigenvector of the Riccati matrix (r )). They are shown in Fig. 5.2. Figure 5.3 shows the time evolution of the norm of the state and tip displacement of the closed loop system with disturbance shape 300b15 (x). The optimal case has the actuator placed at r = 0.4701, and the non-optimal case used an actuator placed at r = 0.7162. The temporal component of the disturbance was an impulse so that ν(s) ˆ = 1 variance in both the optimal and non-optimal cases. The performance is

166

5 Disturbances Evolution of beam energy norm

Evolution of beam tip displacement

10 non−optimal optimal

9

0.5 non−optimal optimal

0.4

8

0.3

7

0.2

6

0.1

5

0

4

−0.1

3

−0.2

2

−0.3

1

−0.4

0

−0.5 0

0.1

0.2

0.3

0.4

0.5

Time (sec)

(a) L2 -norm of the controlled state.

0.6

0

0.1

0.2

0.3

0.4

0.5

0.6

Time (sec)

(b) Beam tip displacement of the controlled system.

Fig. 5.3 Evolution of the controlled system with d1 = 300b15 , Co = I and actuator locations r = 0.4701 (optimal) and r = 0.7162 (non-optimal). (©2015 IEEE. Reprinted, with permission, from [1].) Fig. 5.4 Domain . With origin at the bottom left corner,  is a 4 × 4 units square with a circle of radius 0.4 units centered at (3, 1) removed. (©2013 IEEE. Reprinted, with permission, from [2].)

far superior with the actuator optimally placed than with the actuator placed at a non-optimal location.  Example 5.11 (Diffusion on an irregular region in R2 ) Consider the heat diffusion problem on a two-dimensional irregular shape  shown in Fig. 5.4. This region has no particular application. It was chosen because it is irregular enough that eigenfunctions are not known and intuition to obtain control and actuator placement is difficult. Let z(x, y, t) denote the temperature at the point (x, y) ∈  at time t and assume that z(x, y, t) = 0 on the boundary ∂. The conductivity is κ(x, y). The temperature is controlled by applying a heat source u(t) on a patch with spatial distribution

5.2 H2 -Optimal Actuator Location

167

br (x, y) ∈ L 2 (0, 1). The spatial distribution of the disturbance ν(t) is described by b1 (x, y) ∈ L 2 (0, 1). This leads to the PDE, for (x, y) ∈  ∂z (x, y, t) = ∇ · (κ(x, y)∇z(x, y, t)) + br (x, y)u(t) + b1 (x, y)ν(t), ∂t

(5.18)

with boundary condition z(x, y, t) = 0, (x, y) ∈ ∂. Letting (r x , r y , δ) indicate a square centered at (r x , r y ) with sides of length 2δ and writing r = (r x , r y ), define the square actuator patch  br (x, y) =

1 2δ

0

(x, y) ∈ (r x , r y , δ) . otherwise

The disturbance d(x, y) is of the same shape as the actuator: d(x, y) = 10br (x, y). Various locations of a disturbance will be considered. The state-space formulation of (5.18) on Z = L 2 () is z˙ (t) = Az(t) + B1 ν(t) + B2 u(t), where Az = ∇ · (κ∇z), dom(A) = {z ∈ H2 () | z(x, y) = 0, (x, y) ∈ ∂}, B1 ν = d(x, y)ν, B2 u = br (x, y)u. Since there is a single control signal, the control weight is chosen to be 1. Consider a cost based on the entire state z ∈ L 2 () so that y(t) =

    I 0 z(t) + u(t). 0 1

A finite element method with linear splines {φi } as the basis for the finitedimensional subspace is used for approximating (5.18). The system (5.18) is exponentially stable and since the generator A is defined through a H01 ()-coercive sesquilinear form, Theorem 4.14 implies that the same stability is preserved by the approximations. Furthermore, assumptions (A1) and (A1∗ ) are satisfied. This approximation scheme can be used for controller design and optimal actuator location. The predicted optimal cost and actuator locations converge to the true optimal cost and location respectively for both fixed and unknown disturbance locations.

168

5 Disturbances

Fig. 5.5 Disturbance location and H2 -optimal actuator location with conductivity κ1 . The symbol Di indicates a disturbance and Ai is the corresponding optimal H2 -optimal actuator location. (©2015 IEEE. Reprinted, with permission, from [1].)

In the simulations, 479 elements were used. The width of both the actuator and disturbance was δ = 0.2. Set the conductivity to κ1 (x, y) = 3(3 − x)2 e−(2−x)

2

−(2−y)2 )

+ 0.01.

(5.19)

Figure 5.5 shows the disturbance location and resulting optimal actuator location for four different disturbance locations, plotted over a graph of the conductivity coefficient on the domain . The results suggest that the optimal actuator location is the disturbance location when the disturbance is located in a region of low conductivity, but when the disturbance is located in a region of higher conductivity, the actuator tends to shift towards a location where the conductivity is lower. In particular, with the disturbance D4, which is centered at (1, 2), the optimal actuator and disturbance are not collocated. Now increase the weighting on the state so that     10I 0 y˜ (t) = z(t) + u(t). 0 1

(5.20)

5.2 H2 -Optimal Actuator Location

169

Fig. 5.6 Disturbance location and H2 -optimal actuator location with conductivity κ2 . The symbol Di indicates a disturbance and Ai is the corresponding optimal H2 -optimal actuator location. (©2015 IEEE. Reprinted, with permission, from [1]

With this cost and disturbance D4, the optimal actuator location is collocated at the disturbance location. This suggests that if the state weighting is large enough, the optimal actuator location will coincide with the disturbance location. To investigate further the role of the conductivity in actuator location, consider a model that is identical except that κ2 (x, y) =

1 ((x − 2)2 + (y − 3)2 ) + 0.01. 3

(5.21)

Figure 5.6 shows the disturbance location and resulting optimal actuator location with this conductivity, using state measurement y I (t), plotted over a graph of the conductivity coefficient (5.21) on the domain . It is again seen that the optimal actuator location coincides with the disturbance location when the disturbance is located in a region of low conductivity. These results suggest that, although in general collocation of the actuator with the disturbance is advantageous, the actuator is more effective in regions with low conductivity. 

170

5 Disturbances

5.3 Unknown Disturbance Sometimes the disturbance signal ν(t) in (5.1) is not known, other than that it belongs to L 2 (0, ∞; V). Consider the same class of cost functions (5.4). As for H2 -control, frequency weighting in the disturbance can be absorbed into the system description (5.1). Recall (see Theorem 3.35) that a system is externally stable if and only if its transfer function belongs to H∞ (U, Y) (or M(H∞ ) if both input and output spaces are finite-dimensional). Moreover, for a stable system with input ν and transfer function G, ˆ 2  yˆ 2 ≤ G∞ ν and for any tolerance δ > 0 there is ν ∈ L 2 (0, ∞; V) so that ˆ 2 −  yˆ 2 < δν ˆ 2. G∞ ν Thus, in order to reduce the effect of disturbances on the cost function, or measured output, the controlled transfer function should have H∞ -norm as small as possible. The fixed attenuation H∞ control problem for attenuation γ of (5.1), (5.4) is to construct a stabilizing controller with transfer function H so that the closed loop ˆ = H (s)ˆz (s) is externally stable and satisfies the bound system G yv with u(s) G yv ∞ < γ.

(5.22)

Definition 5.12 A system is stabilizable with attenuation γ, if for each disturbance ν ∈ L 2 (0, ∞; V), there exists a control u ∈ L 2 (0, ∞, U) so that the closed loop is stable with gain less than γ; that is (5.22) is satisfied. Definition 5.13 The state feedback K ∈ L(Z, U ) is said to be γ - admissible if it is stabilizing and the linear feedback u(t) = −K z(t) is such that the attenuation bound (5.22) is achieved. Theorem 5.14 Assume that (A, B2 ) is stabilizable and (A, C1 ) is detectable. For any non-zero attenuation γ the following statements are equivalent: 1. There exists a γ-admissible state feedback. 2. The system is stabilizable with attenuation γ. 3. There exists a positive semi-definite, self-adjoint operator  ∈ B(Z, Z) with dom(A) ⊂ dom(A∗ ) satisfying for all z ∈ dom(A), the H∞ -Riccati operator equation, 

 1 ∗ −1 ∗ ∗ A  +  A + ( 2  B1 B1  −  B2 R B2 ) + C1 C1 z = 0, γ ∗

(5.23)

5.3 Unknown Disturbance

171

and A + ( γ12 B1 B1∗ − B2 R −1 B2∗ ) generates an exponentially stable semigroup on Z. Moreover, a γ-admissible state feedback is K = R −1 B2∗ . Some values of attenuation γ may not be achievable. But, if a system is stabilizable with attenuation γ, this attenuation can be achieved with state feedback. Moreover, this state feedback can be calculated by solving an algebraic Riccati equation. This result is identical to that for finite-dimensional systems except that the H∞ -ARE (5.23) is an operator equation on a Hilbert space. Both H2 and H∞ controllers, as well as LQ, are found by solving an ARE, but the quadratic term of the H∞ -ARE, 1  B1 B1∗  −  B2 R −1 B2∗ , γ2 is in general sign indefinite. As γ becomes large, this middle term becomes negative semi-definite. In the limit as γ is large, that is, the desired attenuation becomes insignificant, the optimal H2 -controller is obtained. In most cases the operator Eq. (5.23) cannot be solved exactly. Instead, a finitedimensional approximation is used and the resulting finite-dimensional approximating ARE solved to find a controller. Let Zn be a finite-dimensional subspace of Z and Pn the orthogonal projection of Z onto Zn . Consider a sequence of operators An ∈ L(Zn , Zn ), B1n = Pn B1 , B2n = Pn B2 , C1n = C1 |Zn . Assumptions similar to those sufficient for convergence of an optimal linear quadratic controller (Theorem 4.11) are required. Definitions of assumptions (A1) , (A1∗ ), uniform stabilizability and detectability are in sections 4.1 and 4.2. Theorem 5.15 Assume that B1 and B2 are compact and that a sequence of approximations satisfy (A1) and (A1∗ ), (An , B2n ) are uniformly stabilizable, and (An , C1n ) are uniformly detectable. If the original problem is stabilizable with attenuation γ then there is No such that for n > No , the approximations are stabilizable with attenuation γ and the Riccati equation A∗n n + n An + n



 1 ∗ −1 ∗ ∗ B B − B R B 1n 2n 1n 2n n + C 1n C 1n = 0, γ2

(5.24)

has a nonnegative, self-adjoint solution n such that K n = R −1 (B2n )∗ n is a γadmissible state feedback for the approximating system. Furthermore, – There exist positive constants M1 and ω1 such that for all n > No the semigroup ∗ ∗ n − B2n R −1 B2n n satisfies Sn2 (t) ≤ Sn2 (t) generated by An + γ12 B1n B1n −ω1 t . M1 e – There exist positive constants M2 and ω2 such that for all n > No , the semigroup Sn K (t) generated by An + B2n K n satisfies Sn K (t) ≤ M2 e−ω2 t .

172

5 Disturbances

Moreover, for all z ∈ Z, n Pn z → z as n → ∞ and K n converges to K = R −1 B2∗  in norm. For n sufficiently large, K n Pn is a γ-admissible state feedback for the infinite-dimensional system. Example 5.16 (Disturbance rejection in cantilevered flexible beam) (Example 5.10 cont.) Consider again a cantilevered Euler–Bernoulli beam with Kelvin–Voigt damping. Let w(x, t) denote the deflection of the beam from its rigid body motion at time t and position x. The deflection is controlled by applying a torque at the clamped end (x = 0). The disturbance ν(t) induces a uniformly distributed load ρd ν(t) so b1 = ρd . With state z(t) = (w(·, t), ∂t∂ w(·, t)), the system is well-posed and exponentially stable on the state space Z = Vo × L 2 (0, ), where Vo = {w ∈ H2 (0, )| w(0) = 0, w (0) = 0}, with inner product

w1 , w2 Vo = Ew1

, w2

.

Consider the same objective as in Example 5.10, to reduce the effect of disturbances on the tip position:   w(, t) . C1 z(t) = 0 For the approximations, again use cubic spline Galerkin approximation as in Example 5.10. For this PDE, the method satisfies assumptions (A1) and (A1∗ ), and is also uniformly stabilizable/detectable. (See Example 4.17.) Solutions to the finite-dimensional Riccati equations (5.24) converge strongly to that of the operator Riccati equation (5.23). The corresponding series of finite-dimensional H∞ -AREs (5.24) were solved with γ = 2.3. The physical parameters used in the simulations are ρ = 3.0 kg/m, E = 25 Nm2 , cv = 0.0010 Ns/m2 , cd = 0.010 Ns/m2 ,  = 7.0 m, Ih = 39 kgm2 , ρd = 0.12 1/m. (Solution of finite-dimensional H∞ -AREs is discussed in the next section.) Figure 5.7 displays the convergence of the feedback gains. Since Zn is a product space, the first and second components of the gains are displayed separately as displacement and velocity gains respectively. Figure 5.8 compares the open and closed loop responses of the flexible beam, for the approximation with 10 elements. In Fig. 5.8a the response w(, t) to a disturbance consisting of a 100 second pulse is shown. The feedback controller leads to a closed loop system which is able to almost entirely reject this disturbance. Figure 5.8b compares the open and closed loop responses to the periodic disturbance sin(ω1 t) where ω is the first resonant frequency: ω1 = mini |Im(λi (A10 )|. The resonance in the open loop is removed by the controller.  A question that naturally arises is the existence of a state-feedback that minimizes the attenuation γ.

5.3 Unknown Disturbance

173 Displacement Gains

4

(m)

3 2 1 0

0

1

2

3

4

5

6

7

5

6

7

r (m) Velocity Gains 10

(m)

8 6 4 2 0

0

1

4

3

2

r (m)

3

30

2

20

1

10 (m)

(m)

Fig. 5.7 H∞ -state feedback for flexible beam. Feedback gains for: 2 elements *, 4 elements …, 6 elements _._., 8 elements _ _, 10 elements, __ are plotted. As predicted by the theory, the feedback operators are converging. ([3], ©1998 Society for Industrial and Applied Mathematics. Reprinted with permission. All rights reserved.)

0

0

-1

-10

-2

-20

-3

0

50

100 time (s)

150

200

(a) disturbance d(t) = 1, t ≤ 100s

-30

0

50

100 time (s)

150

200

(b) disturbance d(t) = sin(ω1 t) where ω1 is first natural frequency of beam

Fig. 5.8 H∞ -state feedback for flexible beam. Open (..) and closed loop (–) responses to different disturbances. ([3] ©1998 Society for Industrial and Applied Mathematics. Reprinted with permission. All rights reserved.)

174

5 Disturbances

Definition 5.17 The optimal H∞ -control problem is to calculate γˆ = inf γ over all K ∈ B(Z, U ) such that the system (5.1), (5.4) is stabilizable with attenuation γ. The infimum γˆ is called the optimal H∞ -disturbance attenuation. An important difference between H∞ and H2 (or LQ) controller design is that the optimal attenuation must usually be found iteratively. Since the calculations are done with approximations, let γˆ n indicate the corresponding optimal disturbance attenuation for the approximating problems. Theorem 5.15 implies that lim supn→∞ γˆ n ≤ γˆ but in addition the optimal disturbance attenuation converges. Corollary 5.18 With the same assumptions as Theorem 5.15, it follows that ˆ lim γˆ n = γ.

n→∞

One exception to the requirement of iterative solution of an H∞ -ARE to find the optimal attenuation occurs when the generator A is self-adjoint and negative definite. Theorem 5.19 Assume that (A, B2 ) is stabilizable, A is self-adjoint and there is c > 0 such that for all z ∈ dom(A), Az, z ≤ −cz2 , C1 ∈ B(Z, Z × U) , D12 ∈ B(U, Z × U) are defined by     I 0 C1 = , D12 = 0 D where R = D ∗ D is coercive. Then the optimal H∞ -disturbance attenuation is γˆ = B1∗ (A2 + B2 R −1 B2∗ )−1 B1  2

1

and it is achieved by the state feedback u(t) = R −1 B2∗ A−1 z(t). Generally this optimal controller needs to be approximated, but in some cases a closed form representation is possible. Example 5.20 (State feedback for temperature regulation in a rod with variable conductivity) The following PDE models heat propagation in a rod of length  with variable conductivity and uniformly distributed disturbance ν(t) ∈ L 2 (0, ∞; R) and controlled input u(t) ∈ L 2 (0, ∞; R), ∂z ∂ (x, t) = ∂t ∂x



∂ k(x) ∂x

 z(x, t) + u(t) + v(t), 0 < x < , t ≥ 0.

(5.25)

5.3 Unknown Disturbance

175

The variable z(x, t) is the temperature at time t in position x. Assume Dirichlet boundary conditions z(0, t) = 0, z(, t) = 0. The goal is to design a state feedback control law u = K z such that the effect of the disturbance on the temperature and the control input is optimally attenuated. Define the operator on L 2 (0, ) ∂ A= ∂x



∂ k(x) ∂x

 , dom(A) ={z ∈ H2 (0, ) | z(0) = 0, z() = 0},

(5.26)

and (B1 ν)(x, t) = ν(t),

B2 = B1 .

(5.27)

The state-space realization of (5.25) on the state space L 2 (0, ) is z˙ (t) = Az(t) + B1 ν(t) + B2 u(t).   I C1 = , 0

With cost

D12

  0 = . 1

Theorem 5.19 implies that the optimal state feedback is K opt = B2∗ A−1 . The adjoint of B2 is, for any f ∈ L 2 (0, ), B2∗ f =





f (x)d x.

0

Calculation of A−1 requires finding a solution f ∈ dom(A) to A f = z, z ∈ L 2 (0, ). This can be written as the boundary value problem ∂ ∂x

  ∂f κ(x) = z(x), ∂x

f (0) = 0, f () = 0.

Choosing κ(x) = x 2 + 1, the solution to this boundary value problem is, defining the Green’s function,

176

5 Disturbances

 ⎧ ⎨ arctan(s) − 1 arctan(x) if 0 < x < s, arctan()  G(x, s) =  ⎩ arctan(x) − 1 arctan(s) if s < x < . arctan() (A−1 z)(x) =





G(x, s)z(s)ds. 0

Defining  k(s) =



G(x, s) d x   arctan(s) 1 2 2 ln(s + 1) − ln( + 1) , = 2 arctan() 0

the optimal control is u(t) = B ∗ A−1 z(x, t)   k(s)z(s, t) ds. = 0

The feedback gain k(s) is plotted in Fig. 5.9 with  = 1.



0 -0.01 -0.02 -0.03 -0.04 -0.05 -0.06 -0.07 -0.08 -0.09 -0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 5.9 Plot of H∞ -optimal feedback gain k(s) for diffusion on rod of length  = 1 with conductivity κ(x) = x 2 + 1

5.4 Solving H∞ -Algebraic Riccati Equations

177

5.4 Solving H∞ -Algebraic Riccati Equations In this section it is assumed that the PDE has already been approximated. Together with any model(s) for the actuator and sensor dynamics this leads to a system of ODE’s on Rn : E z˙ (t) = Az(t) + B1 ν(t) + B2 u(t), y1 (t) = C1 z(t) + D12 u(t)

(5.28) (5.29)

where z ∈ Rn is the state variable, u ∈ Rm is the control variable, ν ∈ Rq is the disturbance and y1 ∈ R p is the cost variable. Here, E, A ∈ Rn×n , B1 ∈ Rn×q , B2 ∈ ∗ D12 is invertible; this is Rn×m , C1 ∈ R p×n , D12 ∈ R p×m . It is assumed that D12 implied by full column rank of matrix D12 and ensures a non-singular map from the control u to the cost y1 . The presence of a matrix E on the right hand side that is not the identity occurs naturally in finite-element approximations and many other approximations. If E is non-singular, it can in theory be inverted to obtain a standard state-space form. However, the matrices in (5.28) are generally sparse. Inversion of E destroys this sparsity and should be avoided. The following definitions refer to a matrix pair, E, A ∈ Rn×n . Definition 5.21 The set of eigenvalues of the matrix pair (E, A) is λ(A, E) = {z ∈ C| det (A − z E) = 0}. Definition 5.22 The matrix pair (A, E) is Hurwitz, if max1≤i≤n Re(λi ) < 0, where λi ∈ λ(A, E), i = 1, . . . , n are the eigenvalues of the matrix pair (A, E). Definition 5.23 Consider the system (5.28), (5.29). The system is stabilizable if there exists K ∈ Rm×n such that the matrix pair (A − B K , E) is Hurwitz. Similarly, the system is detectable if there exists F ∈ Rn× p such that the matrix pair (A − FC, E) is Hurwitz. If E is non-singular the above definitions are equivalent to those for the standard system (AE −1 , B, C E −1 ). Theorem 5.14 for standard state-space systems is written below using the generalized system notation (5.28), (5.29). This will facilitate discussion of an iterative algorithm later in this section.     ∗ D12 C1 = R 0 , where R is non-singular, and Theorem 5.24 Assume that D12 also assum E is non-singular, (E, A, B1 ) is stabilizable and (E, A, C1 ) is detectable. For given attenuation γ > 0, there exists a stabilizing controller for (5.28)–(5.29) so that G yv ∞ < γ if and only if there exists a symmetric matrix  ≥ 0 such that  solves the H∞ -ARE   1 ∗ ∗ ∗ ∗ −1 ∗ A E + E A + E  B1 B1 − B2 R B2 E + C1∗ C1 = 0, (5.30) γ2

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5 Disturbances

and (A + ( γ12 B1 B1∗ − B2 R −1 B2∗ )E, E) is Hurwitz. If so, one such control is u = −K z, where K = R −1 B2∗ E

(5.31)

and also the pair (A − B2 K , E) is Hurwitz. The orthogonality assumption C1∗ D12 simplifies the formulae but is not necessary. This assumption is removed in Chap. 7. As for LQ-control, the H∞ -ARE (5.30) can often be solved by computing the stable invariant subspace spanned by Schur vectors. The H∞ -ARE (5.30) should be rewritten as  −1  ∗ ∗ B1 −γ I 0 B1 B2 E + C1∗ C1 = 0 A E + E A − E  B2 0 I  

   

B˜ ∗ ∗

∗

∗



B˜

R˜ −1

and then use the Schur vector approach, Theorem 4.22. Although in LQ control R˜ > 0 so that the quadratic term E ∗  B˜ R˜ −1 B˜ ∗ E ≥ 0, this is not necessary for solution of the ARE using this algorithm. This algorithm is coded in many commercial and publicly available software packages. An iterative algorithm may be needed to solve (5.30). The Schur method works well for many problems, but may fail or provide incorrect results for large-order systems. As for LQ-control, this is a particular issue for problems modelling vibrations. As mentioned in Sect. 4.3 this issue can sometimes be alleviated by using a realization z˜ based on the system energy so that ˜z (t)2 is proportional to the system’s energy. Also, when γ is close to the optimal attenuation, there can be difficulties in using the Schur vector approach to solving the H∞ -ARE (5.30) even for relatively low-order systems. The following theorem defines an iterative approach to solving H∞ -AREs.     ∗ D12 C1 = R 0 , R is non-singular, E is nonTheorem 5.25 Assume that D12 singular, (E, A, B1 ) is stabilizable and (E, A, C1 ) is detectable. Assume a stabilizing solution  ≥ 0 for (5.30) exists. For any self-adjoint matrix  define 1 ˜ F() = E ∗ A + A∗ E + E ∗ ( 2 B1 B1∗ − B2 R −1 B2∗ )E + C1∗ C1 . γ

(5.32)

Define 0 to be the zero matrix. For each k ≥ 0, define Ak = A +

1 B1 B1∗ k E − B2 B2∗ k E. γ2

Calculate the unique stabilizing solution Z k ≥ 0 of

(5.33)

5.4 Solving H∞ -Algebraic Riccati Equations

179

˜ k ), 0 = E ∗ Z k Ak + A∗ Z k E − E ∗ Z k B2 B2∗ Z k E + F(

(5.34)

k+1 = k + Z k .

(5.35)

and define

Then 1. for every k ≥ 0, (E, A + γ12 B1 B1∗ k E, B2 ) is stabilizable, 2.  = limk→∞ k exists with  ≥ 0. Furthermore,  is the unique stabilizing solution of the H∞ -ARE (5.30). Note that if there exists a k ≥ 0 such that (E, A + γ12 B1 B1∗ k E, B2 ) is not stabilizable, then there does not exist a stabilizing solution  ≥ 0 to the H∞ -ARE (5.30). A particular implementation of the approach defined by Theorem 5.25 is described in Algorithm 5.1. The two main computational issues in the implementation of the algorithm are solution of the intermediate LQ-AREs (step 2) and checking stabilizability (step 5). There are a number of methods suitable for solving LQ-AREs. The most popular, Schur and Newton–Kleinman, are discussed in Sect. 4.3. If the Kleinman method is used to solve the intermediate LQ-AREs then a stabilizing feedback must be chosen at every intermediate step. Clearly, if (E, A + γ12 B1 B1∗ k E, B2 ) is stabilizable, then so is (E, A + γ12 B1 B1∗ k E − B2 B2∗ k E, B2 ). If K is the calculated stabilizing feedback for (E, A + γ12 B1 B1∗ k E, B2 ) then, K˜ = K − B2∗ k E

(5.36)

is a stabilizing feedback for (E, A + γ12 B1 B1∗ k E − B2 B2∗ k E, B2 ). Thus, the stabilizing feedback calculated in step 6 can be used to initialize the iterations. The stabilizability of (E, A + γ12 B1 B1∗ k E, B2 ) needs to be verified in step 5. Explicit inversion of E should be avoided in order to prevent a possible loss of accuracy when the condition number of E is large and also to avoid “fill-in” of sparse matrices. The algorithm for stabilization of descriptor systems in Algorithm 5.2 separates the stable and unstable parts of the spectrum using orthogonal similarity transformations. In Step 4 of Algorithm 5.2, a Lyapunov equation is solved to calculate the stabilizing feedback K 2 for the unstable spectrum in the next step. The sparse matrix E is not inverted while building the stabilizing feedback for the entire system. Thus, the sparse structure of the original system (E, A) is preserved. Stabilizability of the system is verified by checking the controllability of the low-order unstable part (E 22 , A22 , B2 ). Furthermore, a stabilizing feedback K for the complete system (E, A + γ12 B1 B1∗ k E, B2 ) is constructed using the feedback for the unstable modes. This algorithm is suitable for systems with less than 20 unstable eigenvalues.

180

5 Disturbances

Algorithm 5.1 Solution to H∞ -ARE (5.30). Given: E, A, B2 , B2 , C1 , D12 as in (5.28)–(5.29),  > 0, γ > 0. Calculate: Solution  ≥ 0 of (5.30), if it exists. Set k = 0, 0 = 0. Step 1: Set Ak = A + γ12 B1 B1∗ k E − B2 B2∗ k E. Step 2: Construct the unique real symmetric stabilizing solution Z k ≥ 0 that satisfies ˜ k) 0 = E T Z k Ak + AkT Z k E − E T Z k B2 B2∗ Z k E + F( where F˜ is defined in (5.32) . Step 3: Set k+1 = k + Z k . Step 4: If σmax ( γ1 B1∗ Z k E)2 < , then set  = k+1 and stop. If not, go to step 5.

Step 5: If (E, A + γ12 B1 B1∗ k+1 E, B2 ) is stabilizable, then let k = k + 1 and go to step 1 . If not, a positive semi-definite stabilizing solution of (5.30) does not exist, stop.

Algorithm 5.2 Stabilizability (step 5 of Algorithm 5.1) Given: A, E ∈ R n×n and B2 ∈ R n×m . Calculate: K ∈ R m×n such that (E, A + B2 K ) is stable. If (E, A, B2 ) is not stabilizable, no solution. Step 1 : If (E, A) is stable, K = 0 and stop. Else go to Step 2. Step 2 : Reduce the pair (E, A) by an orthogonal similarity transformation, to the ordered generalized real Schur form (GRSF)     E 11 E 12 A11 A12 QE Z = , Q AZ = , 0 E 22 0 A22 where Q and Z are orthogonal matrices such that (E 11 , A11 ) corresponds to the stable spectrum ∗ , B ∗ ]∗ partitioned and (E 22 , A22 ) corresponds to the unstable spectrum. Compute Q B2 = [B21 22 conformally with the above matrices. Step 3: If (E 22 , A22 , B2 ) is controllable, then (E, A, B) is stabilizable and go to Step 4. Else, no solution and stop. Step 4: Solve the Lyapunov equation ∗ ∗ A22 Y E 22 + E 22 Y A∗22 − 2B22 B22 =0 ∗ )−1 ; K = [0, K ]Z ∗ . Step 5 : K 2 = −B2∗ (Y E 22 2

Except for a special case described in Theorem 5.19 the optimal attenuation γ and corresponding control law must be calculated iteratively. The simplest method is to first obtain γl such that no stabilizing solution to the ARE exists and γu so that a stabilizing solution does exist. These are upper and lower bounds for the optimal attenuation. One way to choose γu is to calculate a stabilizing feedback K by solving a LQ (or H2 ) ARE and then calculate the H∞ -norm of the resulting closed loop. Then often γ2u - attenuation is not possible and γl = γ2u . The H∞ -ARE is then solved with γ = 21 (γu + γl ), the value of γu or γl is updated accordingly and the procedure repeated.

5.4 Solving H∞ -Algebraic Riccati Equations

181

Both the Schur and iterative algorithms give good results for diffusion examples, although the iterative method is sometimes faster for large-order systems. Also, for such problems the direct method implied by Theorem 5.19 can sometimes be used. For large-order problems where A is non-self-adjoint, the iterative algorithm is generally more accurate. Example 5.26 (Simply supported beam) Let w(x, t) denote the deflection of a simply supported beam from its rigid body motion at time t and position x. The deflection is controlled by applying a force u(t) around the point r = 0.5 with distribution b(x) =

1

, |r − x| < 2δ 0, |r − x| ≥ 2δ . δ

The exogenous disturbance ν(t) induces a distributed load d(x)ν(t). Normalizing the variables and including viscous damping with parameter cv leads to the partial differential equation ∂w ∂ 4 w ∂2w + + c = b(x)u(t) + d(x)ν(t), 0 < x < 1. v ∂t 2 ∂t ∂x 4

(5.37)

The boundary conditions are w(0, t) = 0, w

(0, t) = 0, w(1, t) = 0, w

(1, t) = 0.

(5.38)

Defining

∂w(x, t) ∂ 2 w(x, t) , v(x, t) = , 2 ∂x ∂t   choose state z(t) = m(x, t) v(x, t) . Define the operator on L 2 (0, 1), m(x, t) =

∇2w =

d 2w , dom(∇ 2 ) = {w ∈ H2 (0, 1)| w(0) = 0, w(1) = 0} dx2

and the operator on Z = L 2 (0, 1) × L 2 (0, 1) 

 0 ∇2 A= , dom(A) = dom(∇ 2 ) × dom(∇ 2 ). −∇ 2 −cv I The PDE can be written in state space form on Z = L 2 (0, 1) × L 2 (0, 1) as z˙ (t) = Az(t). It was shown in Example 4.29 that A is a Riesz-spectral operator on Z = L 2 (0, 1) × L 2 (0, 1) with an orthonormal set of eigenfunctions. Since A is a

182

5 Disturbances

Riesz-spectral operator the Spectrum Determined Growth Assumption applies (Theorem 3.20). The eigenvalues of A are λ±m

 cv c2 = − ± j (mπ)4 − v , m ≥ 1. 2 4

As long as cv < 4π 2 , as is the case in practice, all the eigenvalues have real part − c2v and so A generates an exponentially stable semigroup on Z. Using the eigenfunctions as a basis for a Galerkin approximation yields approximations that satisfy Assumptions (A1) and (A1∗ ). The approximations have the same decay rate as the original system and so are uniformly exponentially stable. In computer simulations, cv = 0.1, the actuator width δ = 0.001 and the disturbance distribution is collocated with the actuator: d = 10 × b0.5 with the same width δ = 0.001. Consider the first N eigenfunctions, define μm = mπ, and  N be the diagonal matrix with mth entry μ2m . The finite-dimensional matrices defining the approximation have the structure, letting ψm (x) = sin(μm x) √  [b N ]m = 2

1

br (x)ψm (x)d x, m = 1, . . . , N

0

√  1 b0.5 (x)ψm (x)d x, m = 1, . . . , N , [d N ]m = 10 2 0       0 0 0 N , [B1 ] = , , [B2 ] = [A] = − N −cv I bN dN     I 0 [E] = I, [C] = , [D12 ] = . 0 1 The entries in matrix [A] have the same magnitude. If the realization  off-diagonal  w, w˙ was used instead, also with a eigenfunction approximation, the A matrix is 

 0 I . −2N −cv I

The largest entry in 2N is (N π)4 which is usually much larger than 1. This choice of state has poorer properties for computation. Both the Schur and iterative methods were used to design an H∞ -controller. In the implementation of the iterative method, the intermediate sequence of LQ-AREs can be solved more quickly with the Schur method than the Kleinman method for the relatively low-order systems in this example. The tolerance for Algorithm 5.1 is  = 10−12 . With attenuation γ = 10, both solution methods work well on different approximations. The closed-loop plant with controllers calculated by both methods are stable and achieved the specified attenuation. The optimal H∞ -attenuation

5.4 Solving H∞ -Algebraic Riccati Equations

183

γˆ = 9.95 computed using both methods was accurate with a tolerance of 0.05, using different approximations of the beam. However, even for this low-order example where the system order was less than 100, for values of γ near optimal attenuation γ, ˆ the Schur method yielded normalized error on the order of 1% while the iterative method had normalized errors below  10−10 %.

5.5 H∞ -Optimal Actuator Location Consider now the situation where there are m actuators with locations that can be varied over some compact set . The location of the actuators can be selected, along with the controller, in order to maximize disturbance attenuation. As for other actuator location objectives, parametrize the actuator location by r ∈ m so r is a vector of length m with components in  denote the dependence of the corresponding control operator B2 with respect to the actuator location by B2 (r ). The system of Eqs. (5.1), (5.2) become z˙ (t) = Az + B1 ν(t) + B2 (r )u(t), z(0) = 0 y1 (t) = C1 z(t) + D12 u(t)

(5.39)

with all other definitions and assumptions as in the rest of this chapter. The optimal attenuation with actuators at r is denoted by γ(r ˆ ). Definition 5.27 The H∞ -optimal cost μ over all possible locations is ˆ ). μ = infm γ(r r ∈

(5.40)

Also, if it exists, a location rˆ ∈ m that satisfies ˆ ) rˆ = arg infm γ(r r ∈

(5.41)

is called an H∞ -optimal actuator location. The H∞ -performance γ(r ˆ ) is continuous with respect to actuator location under natural assumptions. Theorem 5.28 Consider a family of systems (5.39), r ∈ m such that 1. (A, B2 (r )), r ∈ m , are stabilizable and the pair (A, C1 ) is detectable. 2. The family of input operators B2 (r ) ∈ B(U, Z), r ∈ m are continuous functions of r in the operator norm, that is for any r0 ∈ m , lim  B2 (r ) − B2 (r0 ) = 0.

r →r0

(5.42)

184

5 Disturbances

3. The disturbance operator B1 is compact. If the system with actuators at r0 is stabilizable with attenuation γ(r0 ) then there is δ > 0 such that for all r − r0  < δ the systems (5.39) are stabilizable with attenuation γ(r0 ) and – limr →r0 γ(r ˆ ) = γ(r ˆ 0 ), – there exists an optimal actuator location rˆ such that ˆ ) = μ, γ(ˆ ˆ r ) = infm γ(r r ∈

– a sequence of state feedback operators K (r ) ∈ B(Z, U ) can be chosen that are γ(r0 )-admissible at r and converge strongly to an operator K (r0 ) that is γ(r0 )admissible at r0 . If B2 (r ), r ∈ m are compact operators then {K (r )} converges uniformly. In practice, the control and optimal actuator locations are calculated using an approximation. Let Zn be a family of finite-dimensional subspaces of Z and Pn the orthogonal projection of Z onto Zn . The space Zn is equipped with the norm inherited from Z. Consider approximating systems on Zn , An , B2n = Pn B2 , B1n = Pn B1 , and C1n = C1 |Zn . Theorem 5.29 Consider a family of systems (5.39), r ∈ m satisfying the same assumptions as Theorem 5.28 and in addition assume that B2 is a compact operator. Choose an approximation scheme that satisfies (A1), (A1∗ ) and is both uniformly stabilizable with respect to (A, B2 ) and uniformly detectable with respect to (A, C1 ). Let rˆ be an optimal actuator location for the original problem with optimal cost μ and defining similarly rˆ N μ N , for the approximations. Then μ = lim μ N , N →∞

and there exists a subsequence {ˆr M } of {ˆr N } such that ˆ r M ). μ = lim γ(ˆ M→∞

In the case of linear-quadratic optimal control, the actuator location is chosen to minimize  where  is the solution to the LQ algebraic Riccati equation. Compactness of the cost operator C is generally required for the use of approximations; see Theorem 4.30 and Example 4.29. This is because uniform convergence of the LQRiccati operator is required for continuity of the performance measure (r ) and for convergence of LQ-optimal actuator locations. However, for H∞ -performance strong convergence of the Riccati operator is enough to ensure convergence of optimal attenuation and so C1 does not need to be compact.

5.5 H∞ -Optimal Actuator Location

185

Example 5.30 (Simply supported beam with disturbances) Consider again a simply supported beam as in Example 5.26 but include Kelvin–Voigt damping and 2 disturbances ν1 (t), ν2 (t) distributed according to functions f 1 (x), f 2 (x) respectively. Normalizing the variables leads to the partial differential equation on 0 < x < 1 ∂2w ∂w ∂5w ∂4w + c = br (x)u(t) + f 1 (x)ν1 (t) + f 2 (x)ν2 (t). + c + v d ∂t 2 ∂t ∂x 4 ∂t ∂x 4 (5.43)

w(0, t) = 0, cd ∂ ∂xw(0,t) + 2 ∂t 3

w(1, t) = 0,

3 cd ∂ ∂xw(1,t) 2 ∂t

+

∂ 2 w(0,t) ∂x 2 ∂ 2 w(1,t) ∂x 2

= 0, = 0.

˙ as in Example 5.26, the system is exponentially stable Choosing the state (w

, w) on the state-space Z = L 2 (0, 1) × L 2 (0, 1). An obvious choice of cost is the norm of the state. There is only one control, choose control weight 1. This leads to   I , C1 = 0

D12

  0 = . 1

In the calculations, cv = 0.1, cd = 0.0001. The 2 disturbances are localized centered at x = 0.25 and x = 0.75 respectively with width δ = 0.001 and so f 1 = 10 × b0.25 ,

f 2 = 10 × b0.75 .

This is a Riesz-spectral system (see Example 2.26) and since the eigenfunctions are easily calculated, they will be used. As usual, the modification explained in Example 4.29 (and in Example 4.8) is used to avoid complex arithmetic. The optimal actuator locations and cost converged with fewer than 5 eigenfunctions. The H∞ -optimal cost and the corresponding actuator location calculated for different approximations converge to the exact cost μ = 14.036 and an optimal actuator location rˆ = 0.75. It is natural to expect that the optimal actuator location might fall at the center x = 0.5 since the 2 disturbances are symmetrical with respect to the center of the beam. However, as shown in Fig. 5.11 the H∞ -optimal actuator location for this problem is either of the two disturbance locations. The optimal attenuation with an actuator at the centre r = 0.5 is γ(r ˆ ) = 110.54 and the degradation in performance at this location over an optimal location is 686%. The obvious choice of the centre of the beam is in fact a poor choice of actuator location. Now consider placing two actuators on the simply supported beam and center 2 disturbances at x = 0.4 and x = 0.9 respectively, each with width δ = 0.001. The PDE (5.43) is changed slightly to

186

5 Disturbances

12

1

11

0.9

1st Actuator 2nd Actuator

0.8

10

0.7 9

0.6

8

0.5

7

0.4 0.3

6

0.2 5

0.1

4

0 0

10

5 Modes

(a) Optimal H∞ -performance

15

0

5

10

15

Modes

(b) Optimal actuator locations

Fig. 5.10 Convergence of H∞ -performance and corresponding optimal actuator locations for different approximations of the K–V damped beam with f 1 = 10 b0.4 , f 2 = 10 b0.9 , C = I . (©2013 IEEE. Reprinted, with permission, from [2].)

∂2w ∂w ∂5w ∂4w + c = br1 u 1 (t) + br2 u 2 (t) + c + v d ∂t 2 ∂t ∂x 4 ∂t ∂x 4 + 10b0.4 (x)ν1 (t) + b0.9 (x)ν2 (t). Since there are two controls, choose control weight R = I2×2 . The parameters used for this problem are otherwise the same. The H∞ -optimal cost for different approximations converged to the exact cost μ = 8.5 with optimal actuator locations are rˆ1 = 0.58 and rˆ2 = 0.23 (Fig. 5.10). These actuator locations are not the same as the disturbance locations. As shown in Fig. 5.12, there are multiple local minima on this problem. However, there are no local minima at disturbance locations. If, instead of placing the actuators at the optimal location, the actuators are collocated with the disturbances, that is r1 = 0.4 and ˆ 1 , r2 ) = 10. The degradation in r2 = 0.9, then the H∞ -performance is γ(r performance is 17.5%.  Example 5.31 (Diffusion on an irregular region in R2 , Example 5.11 cont.) Consider again diffusion with variable conductivity κ1 (x, y) on a irregular region  ⊂ R2 (Fig. 5.4). The same finite element approximation is used. Consider first a single, evenly distributed disturbance so that d(x, y) = 1. The optimal H∞ -cost calculated over  converged with increasing approximation order to μ = 15 as shown in Fig. 5.13. The corresponding H∞ -optimal actuator location converged to rˆ = (3.1, 3.35). If the actuator was placed instead at r = (1.5, 3), then ˆ ) = 28 which is almost twice the optimal H∞ -cost. the H∞ -cost increases to γ(r Optimal actuator location is not the center of the geometry due to the irregularity in the domain and the variable conductivity coefficient. Now consider a different disturbance. Let the effect of disturbance be concentrated in a region of high conductivity (a square patch centered at (2, 1.5) with

5.5 H∞ -Optimal Actuator Location

187

200 Optimal attenuation for N = 15

180 160 140 120 100 80 60 40 20 0

0

0.2

0.4

0.6

0.8

1

Actuator location(r)

Fig. 5.11 Variation of H∞ -cost with respect to actuator location for a viscously damped beam with 2 disturbances and 1 actuator f 1 = 10 b0.25 , f 2 = 10 b0.75 , C = I. (©2013 IEEE. Reprinted, with permission, from [2].)

Fig. 5.12 Variation of H∞ -cost with respect to the locations of 2 actuators on the beam with 2 disturbances f 1 = 10 f 0.4 , d2 = 10 b0.9 , C = I. (©2013 IEEE. Reprinted, with permission, from [2].)

188

5 Disturbances 25

Optimal attenuation

20

15

10

5

0

200

300

400

500

600

700

Order

Fig. 5.13 Optimal H∞ -cost of diffusion (5.18) over different approximations. Triangular elements are used. The system order denotes the number of nodes. (©2013 IEEE. Reprinted, with permission, from [2]

half-width 0.2). The optimal actuator location is (2.2, 3.1), which is in a region of low conductivity. If instead, the disturbance is in a low conductivity region, a square patch centered at (3, 3) with half-width 0.2, then the optimal actuator location falls close to the same location as the disturbance. Suppose now there are 2 disturbances contained within square patches centered at (0.5, 0.5) and (0.5, 3), each with half-width 0.2. Consider the problem of placing 2 actuators. The control weight R = I is chosen. Optimal actuator locations are rˆ1 = (0.52, 0.48) and rˆ2 = (1.8, 3.13) and the optimal cost is μ = 9.22. One of the actuators is close to the disturbance that is in a region of low conductivity. The optimal location for the other actuator is far from the other disturbance. If the actuators are placed at the same location as the disturbances r1 = (0.5, 0.5), r2 = (0.5, 3) then the ˆ 1 , r2 ) = 10.02; a degradation of 8.6%. H∞ -cost increases to γ(r The results for optimal actuator location for attenuation in the diffusion examples suggest that optimal locations depend on the shape of the domain, location of the disturbance(s) and conductivity. For instance, from this example and the results for H2 -optimal control in Example 5.11 it appears that the actuator should be placed in a region of low conductivity. 

5.6 Notes and References

189

5.6 Notes and References Theorems 5.4 and 5.5 extend the analogous results for finite-dimensional systems (see for example [4–6]) to infinite-dimensional systems, although the assumption used in those references that the control is a feedback control is not required. These theorems hold under weaker assumptions than given here. The key point is that the input/output operator should be Hilbert–Schmidt. However, in practice, there are generally a finite number of disturbances, control signals, and outputs. Since every finite-rank operator is Hilbert–Schmidt operator, typically both B1 are B2 are Hilbert–Schmidt operators. See [1] for details and the approximation theory, as well as results on H2 -optimal actuator location. Theorem 5.14 generalizes the solution for H∞ -control of finite-dimensional systems [7] to infinite-dimensional systems. The finite-dimensional theory is covered in the textbooks [4–6]. The result holds for some systems with unbounded control and/or observation operators [8, 9]. The approximation theory for full-information H∞ -control, in particular Theorem 5.15, Corollary 5.18 and Example 5.16 is in [3]. The fact that the quadratic term in the H∞ -ARE is sometimes not negative semidefinite, and the need to find optimal attenuation through repeated solution of the H∞ -ARE, makes computation of optimal H∞ -controllers more challenging than for LQ- and H2 -control. The Schur method is described in [10] and implemented in a number of commercial software packages. It is shown in [11, 12] through examples that the Schur method is not always successful in solving an H∞ -ARE. For large-order systems, and sometimes for smaller systems near optimal attenuation, an iterative algorithm may be needed. The algorithm described here, based on Theorem 5.25, was originally described in [12]. It has second-order convergence. Details on the implementation for large-order descriptor systems can be found in [11]. Either the Schur or the iterative method can be used for many problems of small or moderate order. For large-order problems, even when the Schur method can be used, an iterative method may be faster. The iterative algorithm is a better choice for problems where the Hamiltonian may have eigenvalues near the imaginary axis. This difficulty typically arises at attenuation near optimal attenuation and in second-order in time systems. This is illustrated by Example 5.26, given in more detail in [11], and by other examples in [12]. Algorithm 5.2 describes the algorithm for stabilization of descriptor systems proposed in [13]. This algorithm separates the stable and unstable parts of the spectrum using orthogonal similarity transformations. For negative self-adjoint generators, the optimal attenuation and feedback control can be found without iteration. Theorem 5.19 and Example 5.20 are in [14]. See [15, 16] for details on Green’s functions as a tool for solving differential equations. H2 -optimal actuator location, including Examples 5.10 and 5.11, is covered in more detail than in Sect. 5.2 in [1]. Optimal H2 -actuator location can be done with the same algorithm for LQ-optimal location described in [17]. Conditions for optimal H∞ -actuator location, as well as Examples 5.30 and 5.31 are in [2]. An algorithm for

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5 Disturbances

H∞ -optimal actuator location is also described in [2]. Examples 5.11 and 5.31 indicate that H2 - and H∞ criteria yield similar predictions of optimal actuator locations. This is also the case for the vibrating beam example analysed in [18].

References 1. Morris KA, Demetriou MA, Yang SD (2015) Using H2 -control performance metrics for infinite-dimensional systems. IEEE Trans Autom Control 60(2):450–462 2. Kasinathan D, Morris KA (2013) H∞ -optimal actuator location. IEEE Trans Autom Control 58(10):2522–2535 3. Ito K, Morris KA (1998) An approximation theory for solutions to operator Riccati equations for H∞ control. SIAM J Control Optim 36(1):82–99 4. Colaneri P, Geromel JC, Locatelli A (1997) Control theory and design. Academic 5. Morris KA (2001) An introduction to feedback controller design. Harcourt-Brace Ltd 6. Zhou K, Doyle JC, Glover K (1996) Robust and optimal control. Prentice-Hall, Englewood Cliffs, NJ 7. Khargonekar PP, Petersen IR, Rotea MA (1988) H∞ -control with state feedback. IEEE Trans Autom Control 33(8):786–788 8. Bensoussan A, Bernhard P (1993) On the standard problem of H∞ -optimal control for infinitedimensional systems. In: Identification and control in systems governed by partial differential equations. SIAM, pp 117–140 9. Keulen BV (1993) H ∞ − control for distributed parameter systems: a state-space approach. Birkhauser, Boston 10. Arnold WF, Laub AJ (1984) Generalized eigenproblem algorithms and software for algebraic Riccati equations. Proc IEEE 72(12):1746–1754 11. Kasinathan D, Morris KA, Yang S (2014) Solution of large descriptor H∞ algebraic Riccati equations. J Comput Sci 5(3):517–526 12. Lanzon A, Feng Y, Anderson BDO, Rotkowitz M (2008) Computing the positive stabilizing solution to algebraic Riccati equations with an indefinite quadratic term via a recursive method. IEEE Trans Auto Control 53(10):2280–2291 13. Varga A (1995) On stabilization methods of descriptor systems. Syst Control Lett 24:133–138 14. Bergeling C, Morris KA, Rantzer A (2020) Direct method for optimal H∞ control of infinitedimensional systems, Automatica. 15. Naylor AW, Sell GR (1982) Linear operator theory in science and engineering. Springer, New York 16. Zaitsev VF, Polyanin AD (2002) Handbook of exact solutions for ordinary differential equations. CRC Press 17. Darivandi N, Morris K, Khajepour A (2013) An algorithm for LQ-optimal actuator location. Smart Mater Struct 22(3):035001 18. Yang SD, Morris KA (2015) Comparison of actuator placement criteria for control of structural vibrations. J Sound Vib 353:1–18

Chapter 6

Estimation

A man falling into dark waters seeks a momentary footing even on sliding stones. (George Eliot, Silas Marner)

Even for systems with ordinary differential equation models, the full set of states is generally not measured. This issue is more serious for distributed parameter systems since the state (for instance, temperature in a heating problem) cannot be measured everywhere. The problem is to find an estimate of the state based only on the measurements y(t) and the model. Estimation is of interest in reconstructing the state from measurements. It can also be combined with state feedback to construct a controller; this is discussed in Chap. 7. Consider a system z˙ (t) = Az(t) + Bd ν(t), z(0) = z 0 .

(6.1)

where z is the state, d indicates a disturbance, A with domain D(A) generates a strongly continuous semigroup S(t) on a separable Hilbert space Z and Bd ∈ B(W, Z) where W is a separable Hilbert space. The measurements are y(t) = C z(t) + Dη(t)

(6.2)

where C ∈ B(Z, Y), D ∈ B(Y, Y) and η accounts for sensor errors. It will be assumed throughout this chapter that there are a finite number of measurements, so Y is finite-dimensional.

© Springer Nature Switzerland AG 2020 K. A. Morris, Controller Design for Distributed Parameter Systems, Communications and Control Engineering, https://doi.org/10.1007/978-3-030-34949-3_6

191

192

6 Estimation

Definition 6.1 The function z˜ (t) estimates z(t) if lim z˜ (t) = z(t)

t→∞

(6.3)

for any initial condition z˜ (0). An obvious choice of estimator is to reproduce the system model (6.1): z˙˜ (t) = A˜z (t) + Bd ν(t) z˜ (0) = z 0 . However, generally z 0 is not known, nor is the disturbance ν. Also, this scheme does not use the information from the measurements y(t). It makes sense to feedback the error in the estimated output to the estimator. Let F ∈ B(Y, Z) be some operator. d z˜ (t) = A˜z (t) + Bd ν(t) + F(y(t) − C z˜ (t)) dt = (A − FC)˜z (t) + Bd ν(t) + F y(t)

(6.4)

If (A, C) is detectable, then F can be chosen so that (6.4) yields an estimate of z(t). This result is entirely analogous to the corresponding result for finite-dimensional systems. Theorem 6.2 The system (6.4) is an estimator for z(t) in (6.1) if A − FC generates an asymptotically stable semigroup. There are a number of ways to choose F. Common estimator objectives are discussed in the following sections.

6.1 Minimum Variance Estimation The most famous class of estimators for finite-dimensional systems, Kalman filters, extend to infinite-dimensional systems. A Kalman filter provides an optimal estimate if the external signals d and n are random noise with properties as defined below. For ζ1 , ζ2 ∈ Z, define ζ1 ◦ ζ2 ∈ B(Z, Z) by (ζ1 ◦ ζ2 )h = ζ1 ζ2 , h, for all h ∈ Z. Definition 6.3 If a Z-valued random variable ζ is integrable on [0, t1 ], t1 > 0 then its expectation is  1 t1 ζ(s)ds. E(ζ) = t1 0

6.1 Minimum Variance Estimation

193

The covariance of a Z-valued random variable ζ with E(ζ2 ) < ∞ is Cov(ζ) = E{(ζ − Eζ) ◦ (ζ − Eζ)} and it is a self-adjoint positive trace class operator. In the following definition, it is assumed that the space W is a separable Hilbert space. Definition 6.4 The signal w(·) is an W-valued Wiener process on [0, t1 ] if it is an Wvalued process on [0, t1 ] such that w(t) − w(s) ∈ L 2 ([0, t1 ]; W) for all s, t ∈ [0, t1 ], and – E(w(t) − w(s)) = 0, – Cov(w(t) − w(s)) = (t − s)Q where Q ∈ B(W, W) is a self-adjoint positive trace class operator. – w(s4 ) − w(s3 ) and w(s2 ) − w(s1 ) are independent if 0 ≤ s1 ≤ s2 ≤ s3 ≤ s4 ≤ t1 . – w is Gaussian; that is, letting {ei } be an orthonormal basis for W, w, ei  is Gaussian for all i. Assume that – ν(t) is a Wiener process with incremental covariance Q on a separable Hilbert space W, – η(t) is a Wiener process with incremental covariance Ro on the finite-dimensional Hilbert space Y, – the initial state z 0 is a Z-valued Gaussian random variable, with zero mean value and covariance given by the trace class operator 0 . – Ro is a positive operator, R = D Ro D ∗ is invertible, and – ν(t), η(t), z 0 are mutually uncorrelated. Because of considering the disturbances as random processes, a formal definition of the solution to (6.1) is complicated. It can most simply be considered as an integral process  t

z(t) = S(t)z 0 +

S(t − s)Bd ν(s)ds

(6.5)

C z(s)ds + Dη(t), t ≥ 0.

(6.6)

0



with measurement y(t) =

t

0

The objective is to find an estimate z˜ (t) of the state z(t) for each t ≥ 0, based on the measured output {y(s) : 0 ≤ s ≤ t}. More precisely, define ⎧ ⎫ ⎪ ⎨  : (s) ∈ B(Y , Z ), s ∈ [0, t], ⎪ ⎬ (·)y, z is measurable on B2 ([0, t]; B(Y, Z)) = [0, t] for any y ∈ Y , z ∈ Z , ⎪. ⎪ ⎩ ⎭ t 2 and

0

(s) dt < ∞

194

6 Estimation



An estimate of the form z˜ (t) =

t

(t, s)y(s)ds,

(6.7)

0

where (t, ·) ∈ B2 ([0, t]; B(Y, Z)) is sought. As for finite-time linear quadratic control, the solution to this optimal estimation problem involves a time-varying operator and the solution involves a mild evolution operator (Definition 4.1). Theorem 6.5 (Finite-time Kalman filter) Let [0, t1 ] be a finite time interval, and (t) ∈ B(Z, Z) the unique self-adjoint, strongly continuous solution to the differential Riccati equation (DRE). Then for all z ∈ dom(A∗ ), t ∈ [0, t1 ], (t)dom(A∗ ) ⊂ dom(A) and ˙ (t)z = [A(t) + (t)A∗ − (t)C ∗ R −1 C(t) + Bd Q Bd ∗ ]z, (0) = 0 .

(6.8)

Let S p (t, ·) indicate the mild evolution operator generated by A − (·)C ∗ R −1 C. The estimate  t S p (t, s)(s)C ∗ R −1 y(s)ds

z˜ opt (t) =

(6.9)

0

is the unique optimal estimate for z(t) in that for each h ∈ Z, E{(z(t) − z˜ opt (t), h2 } = min E{(z(t) − z˜ (t), h2 }, z˜

where the minimum is taken over all estimates z˜ (t) of the form (6.7). Also E{z(t) − z˜ opt (t)2 } = min E{z(t) − z˜ (t)2 } = (t)1 z˜

(6.10)

and the error covariance E{(z(t) − z˜ opt (t)) ◦ (z(t) − z˜ opt (t))} = (t).

(6.11)

√ Theorem 6.6 If (A, Bd Q) is exponentially stabilizable and (A, C) is exponentially detectable, then the algebraic Riccati equation (ARE) 

A + A∗ − C ∗ R −1 C + Bd Q Bd ∗ z = 0,

(6.12)

for all z ∈ dom(A∗ ), has a unique nonnegative solution, ss ∈ B(Z, Z), ss dom(A∗ ) ⊂ dom(A), such that A − ss C ∗ R −1 C generates an exponentially stable C0 -semigroup and for all z ∈ Z, letting (t) indicate the solution to the differential Riccati equation (6.8), lim (t)z = ss z. t→∞

6.1 Minimum Variance Estimation

195

Furthermore, (t) converges in trace norm to ss , that is, lim (t) − ss 1 = 0.

t→∞

(6.13)

Let T p (t) be the exponentially stable C0 -semigroup generated by A − ss C ∗ R −1 C on Z. The steady-state Kalman filter is characterized by  z˜ (t) =

t

T p (t − s)ss C ∗ R −1 dy(s).

(6.14)

0

Furthermore, ss 1 = lim E{z(t) − z˜ opt (t)2 } t→∞

= min lim E{z(t) − z˜ (t)2 }, z˜

t→∞

where the minimum is taken over all estimates z˜ (t) of the form (6.7) such that the limit limt→∞ E{z(t) − z˜ (t)2 } exists. Because the disturbance ν is not known, the steady-state error is in general not zero. The solution to the filtering ARE (6.12) is the same as the solution to the LQARE (4.8) with A → A∗ , B → C ∗ , C ∗ C → Bd Q Bd ∗ . Just as controllability and observability are dual concepts, the problems of designing an optimal state feedback and an optimal estimator are dual. The assumptions on the nature of the noise processes are reasonable in many instances. Unfortunately, in practice, the noise covariances Q and R are rarely known. As for systems modelled by ordinary differential equations, the solution of the ARE can be used to design a non-statistical estimator. This is known as Linear Quadratic Estimator (LQE). The operators Q and R play a role similar to the control and weighting parameters in the dual control problem. Adjustments in the weights can be used to design an acceptable filter. Increasing R relative to Q means that the measurements are less reliable than the model. Typically the settling time of the estimator will be longer, which corresponds to a smaller bandwidth in the frequency domain. Suppose that in an initial design, the filter bandwidth is too high. This means the settling time is very fast. Increasing R with respect to Q means that the plant output noise is more significant. The resulting estimator will have a longer settling time and make less use of the measurements. Conversely, if the settling time of the estimator is too long, decreasing R relative to Q will tend to increase the bandwidth. Computation of the infinite-time Kalman filter is the dual of calculating a linear quadratic optimal control, and the same approach can be used. Suppose the approximation lies in some subspace Zn of the state-space Z, with an orthogonal projection Pn : Z → Zn where for each z ∈ Z, limn→∞ Pn z − z = 0. The space Zn is equipped with the norm inherited from Z. Define Cn = C|Zn ( the restriction of Cn to Zn ) and define An ∈ B(Zn , Zn ) using some method.

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6 Estimation

Assumptions (A1), (A1∗ ), stated on pp. 105, 106, in the context of system (6.1), (6.2) are Assumption (A1) (i) For each z ∈ Z, and all intervals of time [t1 , t2 ] lim sup Sn (t)Pn z − S(t)z = 0.

n→∞ t∈[t ,t ] 1 2

(ii) For all ν ∈ W, Bd n ν − Bd ν → 0, Assumption (A1∗ ) (i) For each z ∈ Z, and all intervals of time [t1 , t2 ] sup Sn∗ (t)Pn z − S ∗ (t)z → 0;

t∈[t1 ,t2 ]

(ii) For all z ∈ Z, y ∈ Y, Cn∗ y − C ∗ y → 0, Bd ∗n Pn z − Bd ∗ z → 0. The strong convergence of Cn to C is also required but this is implied by strong convergence of Pn to the identity. Theorem 6.7 Suppose that assumptions (A1), (A1∗ ) are satisfied – If W is finite-dimensional, then the positive semi-definite solution n (t) to the finite-dimensional differential Riccati equation, ˙ n (t) = An n (t) + n (t)A∗n − n (t)Cn∗ R −1 Cn (t) + Bd n Q Bd ∗n  (0) = Pn 0 Pn , (6.15) converges in trace norm to the solution to (6.8): lim n (s) − (s)1 = 0, 0 ≤ s ≤ t.

n→∞

√ – If (An , Bd n Q) is uniformly stabilizable and (An , C2n ) is uniformly detectable, then for each n, the finite-dimensional ARE ∗ R −1 C2n  + Bd n Q Bd ∗n = 0 An  + A∗n − C2n

has a unique nonnegative solution ss,n with sup ss,n  < ∞ and for all z ∈ Z, lim ss,n z = z.

n→∞

There exist constants M1 ≥ 1, α1 > 0, independent of n, such that the semigroups Sn,F (t) generated by An − Fn Cn satisfy Sn,F (t) ≤ M1 e−α1 t .

6.1 Minimum Variance Estimation

197

For sufficiently large n, the semigroups S Fn (t) generated by A − Fn C are uniformly exponentially stable; that is there exists M2 ≥ 1, α2 > 0, independent of n, such that S Fn (t) ≤ M2 e−α2 t . If additionally W is finite-dimensional then lim ss,n Pn − ss 1 = 0.

n→∞

(6.16)

6.2 Output Estimation Let v indicate all external disturbances and write the system as z˙ (t) = Az(t) + B1 v(t), z(0) = 0 y2 (t) = C2 z(t) + D21 v(t)

(6.17)

where z is the state, A with domain D(A) generates a strongly continuous semigroup S(t) on a Hilbert space Z, and B1 ∈ B(U1 , Z), C2 ∈ B(Z, Y2 ), D21 ∈ B(U1 , Y2 ). It is assumed that U1 and Y2 are separable Hilbert spaces; in practice they are generally finite-dimensional. The disturbance term v is due to uncontrolled inputs, such as process noise and sensor noise, but sometimes modelling errors are regarded as disturbances to the model. The initial condition is set to zero in order to focus on the disturbance. Instead of aiming to estimate all the states, consider the more general problem of estimating a linear combination of the states C1 z. Define for C1 ∈ B(Z, Y1 ) where Y1 is a separable Hilbert space, y1 (t) = C1 z(t) − u(t).

(6.18)

The goal is to find an estimate z(t) of the state z(t), based only on the measurement y2 and the model (6.1) so that C1 (z(t) − z˜ (t)) is small. Finding an estimate for C1 z(t) using only the measurements y2 (t) can be formulated as finding u(t) so that y1 (t) is as small as possible. Then u(t) will be an estimate of C1 z(t). Equations (6.17), (6.18) define a output estimation system z˙ (t) = Az(t) + B1 v(t), z(0) = 0 y1 (t) = C1 z(t) − u(t), y2 (t) = C2 z(t) + D21 v(t).

(OE)

The system (6.1), (6.2) defined above and discussed in the previous section on Kalman filtering can be written in this format. Define disturbance signals v1 , v2

198

6 Estimation 1

1

2 so ν = Q  2 v1 where v1 has v1 2 = 1, and write similarly η = Ro v2 . Defining v v = 1 , (6.1), (6.2) can be rewritten v2

  1 z˙ (t) = Az(t) + Bd Q 2 0 v(t)    B1

  1 y2 (t) = C2 z(t) + 0 D Ro2 v(t).   

(6.19)

D21

    1 1 Defining B1 = Bd Q 2 0 , D21 = 0 D Ro2 , C1 = I, this is in the form (OE). ∗ It will be assumed that R = D21 D21 has a bounded inverse; in other words, R is coercive. It will also be assumed in this chapter that 

   D21 R ∗ D21 = B1 0

(6.20)

in order to simplify the formulae. This orthogonality assumption is essentially assuming process and sensor noise are decoupled, as they are for a Kalman filter. This assumption will be dropped in Chap. 7. Depending on whether there is a single fixed disturbance or the disturbance is unknown, a different optimal estimator is obtained. However, in both cases the estimator will have the form, as for a Kalman estimator, z˙ e (t) = (A − FC2 )z e (t) + F y2 (t), z e (0) = 0, u(t) = C1 z e (t).

(6.21)

The only difference is that the filter F may be different.

6.2.1 H2 -Output Estimation Let {ek }∞ k=1 be an orthonormal basis for the Hilbert space of disturbances U1 . If each component of the disturbance has vk (s) = 1, then consider, as in Sect. 5.1, G v,y1 2

(6.22)

where G v,y1 indicates the transfer function from v to y1 . If the disturbance does not have a uniform frequency response then a model for the disturbance can be absorbed into the plant model, as in (5.9), so that the L 2 -norm of the output is (6.22). The output estimation problem is to find a system with transfer function H so that with

6.2 Output Estimation

199

u (s) = H (s) y2 (s),

(6.23)

G v,y1 2 is minimized. Theorem 6.8 Assume (A, C2 ) is detectable. Assume also that one of the following conditions holds: C1 is a Hilbert–Schmidt operator, B1 is a Hilbert–Schmidt operator, or both B1 and C1 are trace class. The best estimate of C1 z in the H2 -norm yields estimation error  trace(C1 C1∗ ) (6.24) where  ∈ B(Z, Z), dom(A∗ ) ⊂ dom(A), is the positive semi-definite solution of  (6.25) A∗ + A − C2∗ R −1 C2  + B1 B1∗ z = 0 for all z ∈ dom(A∗ ), such that A − C2∗ R −1 C2 is Hurwitz. The optimal estimate u is defined by (6.21) with F = C2∗ R −1 . As in the case of H2 -control, the cost operator C1 affects the cost (6.24) but not the solution of the ARE (6.25) and hence not the filter F. There is a strong connection between an H2 estimator and a Kalman filter. With the definition of the system in (6.19), Eqs. (6.1), (6.2) are in the form (OE). The optimal H2 -estimator is the same as the Kalman filter. The difference is in viewpoint. The Kalman filter provides the minimum error in response to a random initial condition and Gaussian process and sensor noise. The H2 -estimator provides the minimum error with zero initial condition and a single disturbance with uniform spectral density. The original infinite-dimensional model cannot generally be used for estimator design, and some finite-dimensional approximation is used. Suppose the approximation lies in some finite-dimensional subspace Zn of the state-space Z, with an orthogonal projection Pn : Z → Zn where for each z ∈ Z, limn→∞ Pn z − z = 0. The space Zn is equipped with the norm inherited from Z. Define B1n = Pn B1 , and define An ∈ B(Zn , Zn ) using some method. Indicate the restriction of C1 to Zn by C1 |Zn and define C1n = C1 |Zn , C2n = C2 |Zn . Let Sn indicate the semigroup (a matrix exponential) generated by An . This leads to an estimator of the form z˙e (t) = (An − Fn C2n )z e (t) + Fn y2 (t), z e (0) = 0, u(t) = C1n z e (t)

(6.26)

where Fn is designed using the approximating system. Sufficient conditions for this estimator to provide a good estimate of the original system for large enough approximation order are similar to those for state feedback design. Theorem 6.9 If assumptions (A1), (A1∗ ) are satisfied, (An , B1 n) is uniformly stabilizable and (An , C2n ) is uniformly detectable, then for each n, the finite-dimensional ARE ∗ ∗ R −1 C2n n + B1n B1n =0 n A∗n + An n n C2n

200

6 Estimation

has a unique nonnegative solution n with sup n  < ∞. Define F = C2∗ R −1 ∗ where  solves (6.25) and Fn = n C2n R −1 . 1. There exist constants M1 ≥ 1, α1 > 0, independent of n, such that the semigroups Sn F (t) generated by An − Fn C2n satisfy Sn F (t) ≤ M1 e−α1 t . 2. For all z ∈ Z,

lim n Pn z − z = 0

n→∞

and for all y ∈ Y, limn→∞ Fn y − F y = 0. 3. If one of the following conditions holds: C1 is a Hilbert–Schmidt operator, B1 is a Hilbert–Schmidt operator, or both B1 and C1 are trace class, then the H2 estimation error converges; that is, lim



n→∞

∗ trace(C1n n C1n )=



trace(C1 C1∗ )

(6.27)

4. If C1 is a compact operator and either C2 or  is a compact operator then Fn − F → 0 and also the H2 -error of the infinite-dimensional system (OE) with the finite-dimensional estimator (6.26) converges to the optimal H∞ -error as n becomes large.

6.2.2 H∞ -Output Estimation In many situations, the disturbance v is not known. Then a reasonable objective is to find an estimate of the output with error that is small over all disturbances. The problem is for some desired error γ > 0, find a system with transfer function H so that with u (s) = H (s) y2 (s); sup v∈L 2 (0,∞;V) v≤1

 y1 2 < γ.

(6.28)

This corresponds to the H∞ -norm of the transfer function from v to y1 being less than γ and so this approach is referred to as H∞ -estimation. Theorem 6.10 Assume that (A, B1 ) is stabilizable and (A, C2 ) is detectable. There is an estimate of C1 z so that the error satisfies (6.28) if and only if there is a positive semi-definite solution  ∈ B(Z, Z), dom(A∗ ) ⊂ dom(A), of (A∗ + A + 



 1 ∗ ∗ −1 ∗ C C − C R C 1 2  + B1 B1 )z = 0 2 γ2 1

(6.29)

6.2 Output Estimation

201

for all z ∈ dom(A∗ ), such that A + 



1 ∗ C C γ2 1 1

 − C2∗ R −1 C2 is Hurwitz. If so, the

estimator is (6.21) with F = C2∗ R −1 where  solves (6.29). Also, A − FC2 generates an exponentially stable C0 -semigroup. Unlike H2 -estimation, or minimum variance estimation, the definition of the output C1 z affects the design of an H∞ -estimator. In order to calculate the minimum H∞ -error, an iterative process to find the smallest possible error γ is usually needed. A Kalman filter or H2 estimator is designed to minimize the error variance for a single disturbance, while the H∞ -estimator bounds the error over all disturbances. In this respect, an H∞ -estimator is more robust. As the desired error γ → ∞, an H2 -estimator (or Kalman filter) is obtained. In practice an approximation is used in design of an estimator for a DPS. Provided that a suitable approximation scheme is used, estimation error arbitrarily close to that possible with the full infinite-dimensional estimator can be obtained. Again, suppose the approximations lie in some finite-dimensional subspace Zn of the state-space Z, with an orthogonal projection Pn : Z → Zn where for each z ∈ Z, limn→∞ Pn z − z = 0. The space Zn is equipped with the norm inherited from Z. Define B1n = Pn B1 , C1n = C1 |Zn , C2n = C2 |Zn and define An ∈ B(Zn , Zn ) using some method. Let Sn indicate the semigroup generated by An . Theorem 6.11 Assume a sequence of approximations satisfy (A1), (A1∗ ), (An , B1n ) are uniformly stabilizable and (An , C2n ) are uniformly detectable and that C1 and C2 are compact operators. If the original problem is stabilizable with attenuation γ then for sufficiently large n the Riccati equation n A∗n

 + An n + n

 1 ∗ ∗ −1 ∗ C C1n − C2n R C2n n + B1n B1n = 0, γ 2 1n

(6.30)

has a nonnegative, self-adjoint solution n such that ∗ ∗ C1n − C2n R −1 C2n ) is uni1. the semigroup Sn2 (t) generated by An + n ( γ12 C1n formly exponentially stable; that is there exist positive constants M1 and ω1 independent of n with Sn2 (t) ≤ M1 e−ω1 t ; and also defining ∗ R −1 , Fn = n C2n

the semigroups Sn K (t) generated by An + Fn C2n are uniformly exponentially stable; that is there exists M2 , ω2 > 0 independent of n with Sn K (t) ≤ M2 e−ω2 t . 2. Letting  indicate the solution to (6.29), for all z ∈ Z, and defining F = C2∗ R −1 , lim n Pn z − z, Fn − F. n→∞

For sufficiently large n, Fn provides estimation error less than γ when used γn for the in the estimator (6.21). Moreover, the optimal H∞ -estimation error approximating system converges to the optimal estimation error γ for (OE); that is,

202

6 Estimation

lim γn = γ.

n→∞

3. The H∞ -error of the infinite-dimensional system (OE) with the finite-dimensional estimator (6.26), with γ > γ o , converges to γ as n becomes large. Several numerical algorithms for solving H∞ -AREs are described in Sect. 5.4. Example 6.12 (Estimation of centre position of simply supported beam) Consider a simply supported beam, as in Example 4.42. Let w(x, t) denote the deflection of a simply supported beam of length 1 from its rigid body motion at time t and position x. Define the localized function around the point r, 0 < r < 1, br (x) =

 10

, |r − x| < 2 0, |r − x| ≥ 2 . 

The exogenous disturbance induces a load b.25 (x)d(t). Normalizing the variables and including viscous damping with parameter cv leads to the partial differential equation ∂4w ∂2w ∂w ∂5w + c + + c = b.25 (x)d(t), 0 < x < 1. v d ∂t 2 ∂t ∂x 4 ∂t ∂x 4 The boundary conditions are w(0, t) = 0, w(1, t) = 0,

∂2 w (0, t) ∂x 2 ∂2 w (1, t) ∂x 2

+ cd ∂x∂ 2w∂t (0, t) = 0 , 3

+ cd ∂x∂ 2w∂t (1, t) = 0. 3

  ˙ t) , and define Choose the state z(t) = w(x, t) w(x, Hs (0, 1) = {w ∈ H2 (0, 1)| w(0) = 0, w(1) = 0}. The state-space realization of the system on the state space Hs (0, 1) × L 2 (0, 1) is z˙ (t) = A P z(t) + B P1 d(t) where

    z2 z1  = , AP  z2 − z 1 + cd z 2 − cv z 2 dom(A) = {(z 1 , z 2 ) ∈ Hs (0, 1) × Hs (0, 1)| z 1 + cd z 2 ∈ Hs (0, 1)}, 

B P1

 0 = . b.25 (x)

6.2 Output Estimation

203

As shown in Example 2.26, all the eigenvalues of A P have negative real parts and A P is a Riesz-spectral operator. Theorem 3.20 implies that A P generates an exponentially stable semigroup on Hs (0, 1) × L 2 (0, 1). The scalar disturbance d(t) is assumed to be frequency dependent and is described by

= (s I − Ao )−1 bo , co  d(s) where bo , co are vectors and Ao is a matrix. Writing Bo u = bo u and Co z = z, co , v1 (s) = 1 for all s, has realization the overall plant description with a disturbance v1 ,  A=

 A P B P1 Co , 0 Ao

 B1 =

 0 . Bo

The deflections are measured near the tip, at x = 0.9. Including the effect of sensor noise η(t), the measurement y2 (t) is, defining C p ∈ B(Z, R) by CP

  w = w(xo ), v

  y2 (t) = C P 0 z(t) + η(t). Thus, setting v2 = η,

  C2 = C P 0 ,

  D21 = 0 1 .

An estimate of the deflections at the centre is wanted; this defines C1 and hence y1 . In the simulations, √   Ao = −20 , bo = co = 20. The first 3 eigenfunctions of A P were used for the calculations. Figures 6.1, 6.2, 6.3 and 6.4 compare the predictions of the optimal H2 -estimator and H∞ -estimators to the true state with several different disturbances. Including the dynamics of the process noise yields much better error (see Figs. 6.1 and 6.2), which is not surprising. For the single disturbance used in the H2 -estimator, the performance of the H2 - and H∞ -estimators are similar for this example, although for a different system H2 may be better. For a different disturbance, H∞ design yields a much smaller error, which reflects the fact that H∞ -estimation is designed to bound the error over all disturbances. 

204

6 Estimation 0.3

true H2/Kalman filter (no process noise dynamics) 0.2

0.1

0

-0.1

-0.2

-0.3

0

1

2

3

4

5

6

time (s)

Fig. 6.1 Comparison of exact and estimated position H2 /Kalman filter. The process disturbance dynamics are not included in the estimator design true and H2

0.3

true H2/Kalman estimate 0.2

0.1

0

-0.1

-0.2

-0.3

0

1

2

3

4

5

6

time (s)

Fig. 6.2 Comparison of exact and estimated position H2 /Kalman filter. The process disturbance dynamics are included in the estimator design

6.2 Output Estimation

205

0.14

H2 estimator error H-infinity estimator error

0.12

0.1

0.08

0.06

0.04

0.02

0

0

1

2

3

4

5

6

time (s)

Fig. 6.3 Comparision of H2 and H∞ estimator error in estimating position with a single process disturbance included in both estimator designs. The estimator errors are comparable, although the H∞ design yields a slightly smaller error for this example 0.35

H2 estimator error H-infinity estimator error

0.3

0.25

0.2

0.15

0.1

0.05

0

0

2

4

6

8

10

12

14

16

18

20

time (s)

Fig. 6.4 Comparision of H2 and H∞ estimator errors. External process disturbance ν is sin(10t) instead of δ(t). The H∞ design is designed to yield a smaller error over all disturbances, and so yields a significantly smaller error than the H2 estimator

206

6 Estimation

6.3 Optimal Sensor Location As for controller design, sensors should be placed using the same criterion used to design the estimator. Since observability of (A, C) is equivalent to controllability of (A∗ , C ∗ ) identical concerns as illustrated in Example 4.42 apply to the use of observability as a cost in placement of sensors. Also, this will not always lead to the best estimator performance, even when the finite-dimensional model is regarded as exact. Suppose the estimator is designed using a Kalman filter over the infinite-time interval. Theorem 6.6 shows that ||ss ||1 is the minimum steady-state estimate error variance. Since for an infinite-time Kalman filter, this is the criterion used in estimator design, sensors should also be chosen to minimize ||ss ||1 . The value of ||ss ||1 is dependent on the observation operator C, and thus on the number of sensors, as well as on the sensor noise covariance R. Consider sensing with a different observation operator C˜ ∈ B(Z, Y) such that the ˜ is exponentially detectable, with noise covariance R˜ ∈ B(Y, Y) where pair (A, C) ˜ ˜ ss be the unique nonnegative solution to the ARE R is positive definite. Let  ˜ ss C˜ ∗ R˜ −1 C˜  ˜ ss + Bd Q Bd ∗ = 0. ˜ ss + ss A∗ −  A

(6.31)

˜ ss to ss , the solution to (6.12) for estimation with The objective is to compare  the observation operator C and noise covariance R. A similar question arises in estimation over a finite time interval. Theorem 6.13 Assume the spaces W and Y are finite-dimensional and consider two sets of sensors, – C ∈ B(Z, Y), D ∈ B(Y, Y) with sensor noise covariance Ro ˜ D˜ ∈ B(Y, ˜ Y) ˜ with sensor noise covariance R˜ o – C˜ ∈ B(Z, Y), ˜ Let (t) be the unique solution to the DRE (6.8) and (t) the unique solution to ∗ ˜ ˜ (6.8) with C replaced by C and R = D Ro D replaced by R = D˜ R˜ o D˜ ∗ . If C ∗ R −1 C ≥ ˜ then for any t1 > 0, C˜ ∗ R˜ −1 C, ˜ 1 )1 . (t1 )1 ≤ (t √ ˜ are exponenIf (A, Bd Q) is exponentially stabilizable and (A, C) and (A, C) ˜ ss tially detectable, let ss be the unique nonnegative solution to the ARE (6.12) and  ∗ −1 ∗ ˜ −1 ˜ ˜ the non-negative solution to the corresponding ARE (6.31). If C R C ≥ C R C, then ˜ ss ||1 . ||ss ||1 ≤ || This result is not surprising: if the number of sensors is increased, and/or the sensor noise covariance is reduced, then the estimation error is reduced. It does have implications for sensor selection. Suppose that m identical sensors are used. The measurement operator C ∈ B(Z, Rm ) can be written as

6.3 Optimal Sensor Location

207

C = (C(1), C(2), . . . , C(m))T , where C( j) ∈ B(Z, R), j = 1, 2, . . . , m, represents the measurement operator of the j-th sensor. Let v j (t) ( j = 1, 2, . . . , m) represent the noise in the j-th sensor. Assume the noises v1 (t), v2 (t), . . . , vm (t) are mutually independent real-valued Wiener processes with variance r j ∈ R+ . If the sensor is of high quality, then r j ∈ R+ is small, while r j is larger for a low-quality sensor. If m sensors are used, each with variance r0 then the covariance of the measurement noise v(t) is R = diag(r0 , r0 . . . r0 ) = r0 Im , where Im represents the m-dimensional identity matrix. The following result is a straightforward consequence of Theorem 6.13. Corollary 6.14 Let ss be the Riccati operator that solves (6.12) for m identical ˜ ss be the operator obtained for m sensors sensors with variance r0 , and similarly let  with variance r˜0 . If r˜0 ≤ r0 then ˜ ss 1 ≤ ss 1 .  Also, comparing the performance of identical sensors, if m 1 > m sensors are used ˜ ss be the Riccati operator with m 1 sensors, then the estimate is improved. Letting  and ss the Riccati operator with m sensors, then ˜ ss 1 ≤ ss 1 .  Of course, the performance of different sensor types and different number of sensors can only be compared if they are properly placed. Optimal sensor placement follows by duality from results for LQ-optimal actuator location (Sect. 4.4). Suppose m sensors with corresponding measurements y(t) = (y1 (t), y2 (t), . . . , ym (t))T ∈ Rm , t ≥ 0 are available. The sensors lie within some compact set  ⊂ Rq . Generally in applications  is a region in space so q ≤ 3. Denoting the location of the m sensors by l := (l1 , l2 , . . . , lm ) ∈ m ⊂ Rq×m ; the output operator C is parameterized by the sensor location: C = C(l). The estimator is designed to minimize the estimation error. The location of the sensors should be chosen to also minimize the estimation error.

208

6 Estimation

Theorem 6.15 Assume W is finite-dimensional. Let C(l) ∈ B(Z, Y), l ∈ m , be a family of output operators such that for any l 0 ∈ m , lim ||C(l) − C(l 0 )|| = 0.

l→l 0

For the finite-time problem, the solutions (l, t) to (6.8) for each l are continuous in trace norm: lim ||(l, s) − (l 0 , s)||1 = 0, 0 ≤ s ≤ t l→l 0

and there exists an optimal sensor location  l such that ||( l, t)||1 = minm ||(l, t)||1 . l∈

√ For estimation on the infinite-time interval, if (A, Bd Q) is exponentially stabilizable and (A, C(l)) is exponentially detectable, then the corresponding Riccati operators ss = ss (l) are continuous with respect to l in the nuclear norm: lim ||ss (l) − ss (l 0 )||1 = 0,

l→l 0

and there exists an optimal sensor location  l such that ||ss ( l)||1 = minm ||ss (l)||1 . l∈

For a DPS, the ARE (6.12) cannot be solved exactly. For n ≥ 1, let Zn be an ndimensional subspace of Z with inner product inherited from Z and Pn ∈ B(Z, Zn ) the orthogonal projection of Z onto Zn . Denote the original system (6.5) by (A, Bd , C). Approximate the system (A, Bd , C) by a sequence (An , Bd n , Cn ), with An ∈ B(Zn , Zn ), Cn = C |Zn ∈ B(Zn , Y) (the restriction of C to Zn ), and Bd n = Pn Bd ∈ B(W, Zn ). Since Zn is also a Hilbert space, the previous theorems apply to the approximate system. √ If (An , Cn ) is exponentially detectable and (An , Bdn Q) is exponentially stabilizable, then by Theorem 6.6, the finite-dimensional ARE An  + A∗n − Cn∗ R −1 Cn  + Bd n Q Bd ∗n = 0,

(6.32)

has a unique nonnegative solution ss,n ∈ B(Zn ). The existence of an optimal sensor location vector for a finite-dimensional problem is guaranteed by Theorem 6.15. Convergence of the approximate locations follow using duality from the corresponding results for linear-quadratic optimal control (Theorem 4.45). As for the infinite-time control problem, for steady-state minimum variance estimation using approximations, assumptions additional to those required for the finite-time problem are needed.

6.3 Optimal Sensor Location

209

Theorem 6.16 Consider (A, Bd , C(l)) where W is finite-dimensional and C(l) ∈ B(Z, Y), l ∈ m , is a family of output operators such that for any l 0 ∈ m , lim ||C(l) − C(l 0 )|| = 0.

l→l 0

Consider approximations (An , Bd n , Cn (l)) that satisfy assumptions (A1), (A1∗ ). – Let l n be a sequence of optimal sensor locations for the approximations so that l n , t)||1 = minm ||n (l, t)||1 . ||n ( l∈

l n } such that limk→∞ l where l is an l nk = There exists a subsequence { l n k } of { optimal sensor location; that is, ||( l, t)||1 = minm ||(l, t)||1 l∈

and also

l n , t)||1 . ||( l, t)||1 = lim ||n ( n→∞

Moreover, any convergent subsequence of { l n }∞ n=1 converges to an optimal sensor location. √ – Assume, in addition, that (A, Bd Q) is exponentially stabilizable, for each l (A, C(l)) is exponentially detectable and that the approximations are uniformly detectable for each l and uniformly stabilizable. Then there exists a subsequence

∞

{ l n k }∞ k=1 of {l n }n=1 such that lim k→∞ l n k = l where l is an optimal sensor location; that is, l )||1 = minm ||ss (l)||1 ||ss ( l∈

and also

l )||1 = lim ||ss,n ( l n )||1 . ||ss ( n→∞

Moreover, any convergent subsequence of { l n }∞ n=1 converges to an optimal sensor location. Example 6.17 (Diffusion with Neumann boundary conditions, Example 3.29 cont.) Diffusion on a rod is considered: ∂z ∂2z = α 2 + g(x)ν(t), 0 ≤ x ≤ 1, t ≥ 0, ∂t ∂x ∂z ∂z (0, t) = 0, (1, t) = 0, ∂x ∂x z(x, 0) = z 0 (x),

(6.33)

210

6 Estimation

where α is constant diffusivity, g(x) ∈ L 2 (0, 1) models the shape of the spatially distributed disturbance, and d(t) is assumed to be a real-valued white Gaussian noise with variance Q. The state space is Z = L 2 (0, 1), A=α

d2 , dom(A) = {z ∈ H2 (0, 1)| z  (0) = z  (1) = 0} ⊂ Z, dx2

and for Bd ∈ B(C, L 2 (0, 1)) is the operator defined by Bd d = g(x)d 2

for any scalar d. The operator A = α ddx 2 with this domain is a Riesz spectral operaλ j := −α j 2 π 2 , j ≥ 0 and orthonormal eigentor on Z = L 2 (0, 1) with eigenvalues √ functions φ0 (x) = 1, φ j (x) = 2 cos( jπx) for j ≥ 1. The operator A generates a contraction semigroup but because of the 0 eigenvalue it is not asymptotically sta1 ble. (See Example 3.29.) Assume that 0 g(x)d x = 0 so that the system (A, Bd ) is stabilizable by Theorem 3.46. Suppose that there are m identical sensors. Each sensor measures the average temperature over an interval of length  > 0. For  j ∈ (0, 1), define  c j (x) = Then

1/, |x −  j | ≤ 0, otherwise

 2

.

(6.34)

⎡ 1 ⎤ 0 c1 (x)z(t, x)d x  ⎢ 1 c (x)z(t, x)d x ⎥ 2 ⎥. 0 C z(t) = ⎢ ⎣ ⎦ ... 1 0 cm (x)z(t, x)d x

1 Since 0 c j (x)d x > 0, Theorem 3.46 implies that the control system is detectable if there is at least one sensor. Let n j (t), j = 1, 2, . . . , m, represent the noise in the j-th sensor. The noises n 1 (t), n 2 (t), . . . , n m (t) are mutually independent real-valued Wiener processes, each with variance r0 ∈ R+ . If the sensors are of high quality, then r0 ∈ R+ is small, while r0 is larger for low-quality sensors. The covariance of the measurement noise vector η(t) is R = diag(r0 , r0 . . . r0 ) ∈ Rm×m . Galerkin approximations obtained with the first n eigenfunctions of A lead to a sequence of approximations that satisfy assumptions (A1), (A1∗ ) because the eigenfunctions are a basis for L 2 (0, 1). Theorem 4.13 also implies that the approximations are uniformly stabilizable and detectable. Uniform stabilizability can also be estab1 lished directly by choosing k0 so k0 0 g(x)d x > 1 and setting

6.3 Optimal Sensor Location

211

  K = k0 0 . . . . Uniform detectability can be similarly established. The process noise space W = R. The approximating systems satisfy the assumptions of Theorems 6.7, 6.15, 6.16. Most importantly, letting Pn indicate the projection onto the space spanned by the first n eigenfunctions, and ss the solution to the ARE, lim ss,n Pn − ss 1 = 0.

n→∞

For the simulations, process noise was concentrated at the centre as shown in Fig. 6.8a: g(x) = sech(x − 0.5). Diffusivity was α = 0.1, approximation order n = 10 and process noise variance Q = 1. The sensors had width of  = 0.04. Several different values of sensor variance were considered. If the sensor noise variance is large, the estimation error variance ss  did not vary with sensor location in this example, see Fig. 6.5. For a single sensor with variance R = 0.01, the best location for the sensor is at the centre; see Figs. 6.5 and 6.6. Although the second eigenfunction has a zero at the centre, concentration of the disturbance at the centre makes this location advantageous for sensing. In Fig. 6.7, the estimation error with 20 evenly distributed sensors each with variance 2 is compared with a single optimally placed sensor with variance 0.01. A larger number of sensors compensates for their lower accuracy.  Example 6.18 (Simply supported Euler–Bernoulli beam) Consider an Euler–Bernoulli beam of length 1, with Kelvin–Voigt damping. Let w(t, x) denote the deflection of the beam at time t and position x. The beam deflection is described by the partial differential equation ∂5w ∂2w ∂4w = g(x)ν(t), t ≥ 0, 0 < x < 1, + + c d ∂t 2 ∂x 4 ∂x 4 ∂t where cd is the damping parameter, g(x) ∈ L 2 (0, 1) models the shape of the spatially distributed disturbance, and ν(t) is a real-valued white Gaussian noise with variance Q. Assume simply supported boundary conditions w(t, 0) = 0, w(t, 1) = 0,

∂ 3 w(t, 0) ∂ 2 w(t, 0) = 0, + cd 2 ∂x ∂x 2 ∂t ∂ 2 w(t, 1) ∂ 3 w(t, 1) = 0. + c d ∂x 2 ∂x 2 ∂t

Write Hs (0, 1) = {w ∈ H2 (0, 1)| w(0) = w(1) = 0}

212

6 Estimation 1 R= .01 R=2

0.99

norm. variance

0.98

0.97

0.96

0.95

0.94

0.93

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x 1 Fig. 6.5 Estimation of one-dimensional diffusion. Normalized minimum error variance max(s) s (s)1 as a function of sensor location for sensors with variance R = 2 and R = 0.01. For large sensor variance, the location is not important

and define the state space Z = Hs (0, 1) × L 2 (0, 1) and state z := (w, w). ˙ The operator   0 1 , A= 4 4 ∂ ∂ − ∂x 4 −cd ∂x 4 dom(A) = {z = (w, v) ∈ Hs (0, 1) × Hs (0, 1)| w  ∈ Hs (0, 1), v  ∈ Hs (0, 1)}, 

and Bd =

 0 . g(x)

It was shown in Example 4.42 that the operator A generates an exponentially stable semigroup on Z. Each sensor measures average deflection over a small interval of length  > 0, centered at 0 <  j < 1. Defining

6.3 Optimal Sensor Location

213

1.5 sensor at x=0.5 sensor at x=0.01

1 0.5 0 -0.5 -1 -1.5 -2 -2.5 -3

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

time (s) Fig. 6.6 Estimation of one-dimensional diffusion. Average estimation error over time for sensor with variance R = 0.01. One is with optimal estimator, sensor optimally placed at x = 0.5; the other with the optimal estimator with the sensor at x = 0.01. Initial condition is (1, 1, . . .). Both locations can be used in a design of an estimator. However, the estimation error is smaller with a properly placed sensor

 c j (x) :=

1/, |x −  j | ≤ 0, otherwise

 2

,

with m sensors centered at x = 1 , 2 , . . . m , ⎤ ⎡ 1 0 c1 (x)w(t, x)d x  ⎢ 1 c (x)w(t, x)d x ⎥ 2 ⎥, 0 C z(t) = ⎢ ⎦ ⎣ ... 1 0 cm (x)w(t, x)d x and the measurement is 

t

y(t) = 0

C(l)z(s)ds + η(t),

214

6 Estimation

1.2 1 sensor, R=0.01 20 sensors, R=2

1

0.8

0.6

0.4

0.2

0

-0.2

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

time (s) Fig. 6.7 Estimation of one-dimensional diffusion. Comparision of average estimation error over time with single accurate sensor or with 20 inaccurate sensors. Initial condition is (1, 1, . . .). Using more sensors can compensate for lower accuracy

where η(t) has covariance R = diag(r0 , r0 . . . r0 ) ∈ Rm×m . A modal approximation was used. This yields a sequence of approximations that satisfies all the standard assumptions for controller design and actuator placement; see Example 4.42. For simulations, the parameter values are cd = 0.0001,  = 0.02, 30 modes were used, and the initial condition ˙ 0) = 0, 0 < x < 1. w(x, 0) = 0.25 − (x − 0.5)2 , w(x, The disturbance g(x) = sech(100(x − 0.2)), (see Fig. 6.8 (b)) was used, and process noise variance Q = 1.

6.3 Optimal Sensor Location

215

1

1 0.9

0.98

0.8 0.7

0.96

0.6 0.94

0.5 0.4

0.92

0.3 0.2

0.9

0.1 0

0.88 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

(a) g(x) = sech(x − 0.5)

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

(b) g(x) = sech(100(x − 0.2))

Fig. 6.8 Spatial distribution of disturbances g(x) 0.3

actual state R0=0.002; m=1

deflection at x=0.5

0.2

R0=0.02; m=15

0.1 0 −0.1 −0.2 −0.3 −0.4 0

2

4

6

8

10 t

12

14

16

18

20

Fig. 6.9 Estimation with a spatially localized disturbance on a simply supported beam. Comparison of the actual state at x = 0.5, with the estimate using a single optimally placed sensor with noise variance r0 = 0.02 and 15 optimally placed sensors each with r0 = 0.2. The estimate with a large number of noisy sensors is better than that obtained with one more accurate sensor. (©IEEE 2017. Reprinted, with permission, from [1].)

The optimal location for the single sensor is l = 0.49; the optimal location l for 15 sensors is  l =(0.41, 0.43, 0.45, 0.47, 0.49, 0.51, 0.53, 0.55, 0.57, 0.59, 0.61, 0.63, 0.65, 0.67, 0.69). Comparisons of the actual system state with estimates made by a single sensor with r0 = 0.002 and by 15 sensors with r0 = 0.02 are displayed in Fig. 6.9. The estimation errors are similar. Increasing the number of sensors can compensate for poor sensor quality if they are properly placed. 

216

6 Estimation

In Examples 6.17, 6.18 if more than a few modes are used in the approximation optimal observability of the approximating systems over the set of possible sensor locations is almost 0, reflecting the fact that the original model is at best only approximately observable. Other criteria for sensor location can be used of course. If an H2 -criterion is used, the computation is identical to that of a Kalman filter. However, the cost function to be minimized is trace(C1 C1∗ ) which in general will lead to a different optimal location. For an H∞ -criterion, iteration on the error γ is generally needed to find the optimal attenuation and hence the optimal location. The problem is exactly dual to that of optimal actuator location in the full information problem. Theorem 6.19 Consider a family of systems (OE), r ∈ m such that 1. (A, C2 (l)), l ∈ m , are detectable and the pair (A, B1 ) is stabilizable. 2. The family of operators C2 (l) ∈ B(Z, Y), l ∈ m are continuous functions of l in the operator norm, that is for any l 0 ∈ m , lim  C2 (l) − C2 (l 0 ) = 0.

l→l 0

3. The operator C1 is compact. If for error γ > 0 the system with sensors at l 0 the ARE (6.29) with C2 (l 0 ) has a stabilizing solution then there is δ > 0 such that for all l − l 0  < δ the ARE with C2 (l) and the same value of γ has a stabilizing solution. Indicating the optimal attenuation at , by γ (), – lim l→l 0 γ (l) = γ (l 0 ), – there exists an optimal sensor location l such that γ (l) = μ.

γ ( l) = infm l∈

Theorem 6.20 Consider a family of approximating systems, l ∈ m satisfying for each  the assumptions of Theorem 6.11. Let l be an optimal sensor location for the N original problem with optimal H∞ -estimation error μ and defining similarly l μN , for the approximations. Then μ = lim μ N , N →∞

M N and there exists a subsequence { l } of { l } such that

μ = lim γ ( l ). M

M→∞

6.4 Notes and References

217

6.4 Notes and References Theory for the infinite-dimensional Kalman filter, Theorem 6.5, is described in detail in [2, chap. 6], see also [3]. Theorem 6.9 is in [1]. The infinite-dimensional theory is a generalization of that for finite dimensions. Kalman filtering for finite-dimensional systems can be found in a number of books; see for example, [4–6]. The relationship between design for what is known as the full control problem and the output estimation problem has been used in the development of an output feedback controller for H∞ control (covered in Chap. 7). The full control problem is used to find a good estimate of K z where K is the state feedback that solves the full information problem. It is assumed that an intermediate system is exponentially stable, which is reasonable in output feedback but may not hold for estimation alone. See [7] for the finite-dimensional result or textbooks [8–10]. The extension to infinitedimensional systems is in [11]. Estimation in the H∞ context without control was shown for finite-dimensional systems in [12]; see also the tutorial article [13]. In [14] H∞ -output estimation without control on a Hilbert space is described, with an outline of the proof. The full proof as well as the approximation framework is in [15]. The first results on the theory of optimal sensor location are for finite-time Kalman filters, see [16] and also [3]. These are refined and extended to time-varying systems along with a rigorous development of the numerics in [17]. The assumption that the covariance 0 of the initial condition is trace class is important for convergence of approximations in finite-time minimum variance estimation; a counter-example is provided in [17]. Minimum variance estimation in steady-state including Example 6.18 is in [1]. Theorem 6.13 is based on results concerning comparison of solutions to AREs in [18] and can be found in [1]. Sensors can also be placed using the Fisher information matrix, see [19]. Just as estimator design is dual to controller design, optimal sensor placement is dual to optimal actuator location and the same algorithms can be used. As for optimal actuator placement, optimal sensor placement generally depends on the cost function being optimized. Even for a specific type of controller or estimator objective, changing the weights may change the controller (or estimator) and the optimal actuator (or sensor) locations.

References 1. Zhang M, Morris KA (2018) Sensor choice for minimum error variance estimation. IEEE Trans Autom Control 63(2):315–330 2. Curtain RF, Pritchard AJ (1978) Infinite-dimensional linear systems theory, vol 8. Lecture notes in control and information sciences. Springer, Berlin-New York 3. Omatu S, Seinfeld JH (1989) Distributed parameter systems: theory and applications. Oxford Science Publications 4. Chui CK, Chen G (2009) Kalman filtering with real-time applications. Springer

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6 Estimation

5. Crassidis JL, Junkins JL (2012) Optimal estimation of dynamic systems, Applied mathematics and nonlinear science series, 2nd ed, vol 24. Chapman and Hall/CRC, Boca Raton 6. Grewal Mohinder S (2015) Kalman filtering: theory and practice using MATLAB, 4th edn. Wiley, Hoboken, New Jersey 7. Doyle JC, Glover K, Khargonekar P, Francis B (1989) State-space solutions to standard H2 and H∞ control problems. IEEE Trans Autom Control 34(8):831–847 8. Colaneri P, Geromel JC, Locatelli A (1997) Control theory and design. Academic Press 9. Morris KA (2001) An introduction to feedback controller design. Harcourt-Brace Ltd 10. Zhou K, Doyle JC, Glover K (1996) Robust and optimal control. Prentice-Hall, Englewood Cliffs, NJ 11. Keulen BV (1993) H ∞ − Control for distributed parameter systems: a state-space approach. Birkhauser, Boston 12. Nagpal KM, Khargonekar PP (1991) Filtering and smoothing in an H∞ setting. IEEE Trans Autom Control 36(2):152–166 13. Shaked U, Theodor Y (1992) H∞ -estimation: a tutorial. In: Proceedings of the 31st IEEE conference on decision and control. IEEE 14. Ichikawa A (1996) H∞ -control and filtering with initial uncertainty for infinite dimensional systems. Int J Robust Nonlinear Control 6:431–452 15. Morris KA (2020) Output estimation for infinite-dimensional systems with disturbances 16. Curtain RF, Ichikawa A (1978) Optimal location of sensors for filtering for distributed systems, Lecture notes in control and information sciences, vol 1. Springer, Berlin, pp 236–255 17. Wu X, Jacob B, Elbern H (2015) Optimal control and observation locations for time-varying systems on a finite-time horizon. SIAM J Control Optim 54(1):291–316 18. Curtain RF, Rodman L (1990) Comparison theorems for infinite-dimensional Riccati equations. Syst Control Lett 15(2):153–159 19. Uci´nski D (2005) Optimal measurement methods for distributed parameter system identification. Systems and Control Series. CRC Press, Boca Raton, FL

Chapter 7

Output Feedback Controller Design

We all have our best guides within us, if only we would listen. (Jane Austen, Mansfield Park)

For DPS, as for most other types of systems, the measured output y does not include all the states z. Consider a general control system (A, B, C, D) (3.24) z˙ (t) = Az(t) + Bu(t), z(0) = z 0 , y(t) = C z(t) + Du(t)

(7.1)

where A generates a C0 -semigroup S(t) on a Hilbert space Z, B ∈ B(U, Z), C ∈ B(Z, Y), D ∈ B(U, Y), and U, Y are both Hilbert spaces. It will be assumed that U and Y are finite-dimensional and thus may be regarded as Rm and R p respectively for integers m, p. Indicate the transfer function of (7.1) by G. There are basically two ways of approaching controller synthesis using the output y. One relies primarily on the input/output description of the system; the other on the state-space description. If the input/output approach is taken, care needs to be taken to ensure that internal stability is maintained. Equivalence between internal and external stability if the sub-systems are stabilizable and detectable (Theorem 3.41) justifies the use of controller design based on system input/output behaviour. On the other hand, in the state-space approach the fact that the overall design goal often relates to the input/output behaviour needs to be considered. This is true for both finite-dimensional and infinite-dimensional systems. Several common approaches to input/output controller design are first described in the context of DPS. State-space based output controller design will then be covered.

© Springer Nature Switzerland AG 2020 K. A. Morris, Controller Design for Distributed Parameter Systems, Communications and Control Engineering, https://doi.org/10.1007/978-3-030-34949-3_7

219

220

7 Output Feedback Controller Design

7.1 Dissipativity The classical results on passivity, and more generally dissipativity, extend to infinitedimensions. The definition and treatment is very similar to that for finite-dimensional systems. Definition 7.1 Let P be a self-adjoint positive semi-definite operator in B(Z, Z); and let Q ∈ R p× p , S ∈ R p×m and R ∈ Rm×m be constant matrices, with Q and R symmetric. Define the supply rate p(u, y) = y, Qy + y, Su + Su, y + u, Ru. The system (7.1) is (Q, S, R)-dissipative if for all t > 0, u ∈ L 2 (0, t; Rm ), 

t

p(u(τ ), y(τ ))dτ ≥ z(t), Pz(t) − z(0), Pz(0).

(7.2)

0

The quantity z 0 , Pz 0  is called the storage function. A (Q, S, R)-dissipative system is passive if Q = 0, S = I , R = 0, input passive if for some  > 0, Q = 0, S = I , R = −I and output passive if for some  > 0, Q = −I , S = I , R = 0. It is an immediate consequence of Definition 3.32 and Theorem 3.35 that the dissipation inequality (7.2) holds with S = 0, Q = −I and R = γ 2 I for all t > 0 if and only if the system is externally stable with L 2 -gain γ and G∞ ≤ γ. The physical idea behind dissipativity is that z(t), Pz(t) is the stored energy in the system at time t, and the supply rate p is rate of energy transferred in/out of the system. This viewpoint often provides a guide to a suitable storage function and supply rate. However, the concept can be generalized beyond an energy interpretation. There is an algebraic characterization of dissipativity in terms of the state-space realization. Theorem 7.2 The system (7.1) is (Q, S, R) dissipative with storage function z, Pz if and only if for all u o ∈ U, z o ∈ dom(A),     z zo , M o ≤ 0, Az o , Pz o  + Pz o , Az o  + uo uo   P B − C∗S − C∗ Q D −C ∗ QC . M= ∗ B P − S ∗ C − D ∗ QC −R − D ∗ Q D − S ∗ D − D ∗ S Corollary 7.3 The system (A, B, C, D) is passive with storage function P if and only if for all z o ∈ dom(A), Az o , Pz o  + Pz o , Az o  ≤ 0 and also C = B ∗ P, D + D ∗ ≥ 0.

7.1 Dissipativity

221

In particular, if ReAz o , z o  ≤ 0, C = B ∗ and D = 0 a system is passive. Note that this also implies that the semigroup generated by A is a contraction. Passivity of a system can also be characterized in terms of its transfer function. Definition 7.4 A transfer function G with U = Y is positive real if it is analytic for Res > 0 and G(s) + G(s)∗ ≥ 0 for Res > 0. Corollary 7.5 Suppose that U = Y, A generates a bounded semigroup; that is the growth bound of the semigroup is 0, and the system (A, B, C, D) is passive. Then the transfer function is positive real. Characterization of systems in terms of dissipativity can be used to conclude external stability. The proofs are based entirely on the input/output behaviour and so generalize without difficulty to DPS. Theorem 7.6 Consider a plant and controller arranged in the standard feedback configuration (Fig. 7.1). Suppose the controller  H is (Q 1 , S1 , R1 )-dissipative and the plant  P is (Q 2 , S2 , R2 )-dissipative. Define  Q = diag(Q 1 , Q 2 ), S = diag(S1 , S2 ), R = diag(R1 , R2 ), H =

 0 I , −I 0

Q˜ = S H + H ∗ S ∗ − H ∗ R H − Q. If Q˜ is positive definite, the connected system is externally stable. This approach to stability often leads to conservative results: a system may be stable even when the conditions are not satisfied. However, it is useful in that it relies only on the structural form of the model and so stability is not dependent on particular parameter values. The small gain and passivity theorems can be viewed as special cases of Theorem 7.6. Corollary 7.7 (Small Gain Theorem) If plant and controller are both externally stable, with L 2 -gains γ P and γ H respectively, and γ P γ H < 1 then the closed loop is externally stable. Corollary 7.8 (Passivity Theorem) If both plant and controller are passive, and at least one of the two systems is externally stable, then the closed loop is externally stable and also passive. Suppose that there is only one input and output: U = Y = R. Such systems are said to be single-input–single-output (SISO). The most popular class of controllers for SISO systems are PI controllers, that is those for which, for constants k p , ki  u(t) = k p e1 (t) + ki 0

t

e1 (τ )dτ

(7.3)

222

7 Output Feedback Controller Design

where e1 is as in Fig. 7.1. Equivalently, the controller has transfer function kp +

ki . s

If k p > 0 any PI controller is positive real and hence passive. The following result is thus a direct consequence of Corollary 7.8. Corollary 7.9 Suppose that (7.1) is passive (or equivalently that G(s) is positive real) and externally stable. Then for any PI controller (7.3), the closed loop system is stable and passive. Example 7.10 (Controlled wave equation) (Example 2.79 cont.) Recall the controlled undamped wave equation ∂ 2 w(x, t) ∂ 2 w(x, t) = + b(x)u(t), ∂t 2 ∂x 2 w(0, t) = 0, w(1, t) = 0,  1 ∂w(x, t) y(t) = b(x)dx. ∂t 0

(7.4)

Suppose, as illustration, that  b(x) =

1, 0 ≤ x ≤ 1/2, 0, 1/2 < x ≤ 1.

(7.5)

The transfer function is G(s) =

1 2s

2 cosh( 2s )−cosh2 ( 2s )−cosh(s)) +( . s 2 sinh(s)

There are several ways to show that this system is passive. 1. This system is well-posed on state-space Z = H01 (0, 1) × L 2 (0, 1) with  A=

 0 I , 2 d 0 dx2

dom(A) = {(w, v) ∈ H2 (0, 1) × H01 (0, 1) |w(0) = 0, w(1) = 0}, Bu = bu, C z = z, b, D = 0. (See Example 2.33.) A straightforward calculation shows that for all z o ∈ dom(A), ReAz o , z o  = 0 and also C = B ∗ . Corollary 7.3 implies that the system is passive with storage function I.

7.1 Dissipativity

223

2. Another way to show that this system is passive is to show that the transfer function is real. The residue of a meromorphic function G at s = p is Res( p) = [G(s)(s − p)] |s= p if this is well-defined. If G(s) is defined at s = p then the residue is zero but when G has a pole at s = p the residue will in general be non-zero. By calculating the residues at each pole of G a partial fraction expansion can be computed: G(s) =

∞  Res(j kπ) Res(−j kπ) 1 + + 2s k=1 s − j kπ s + j kπ ∞

 Res(j kπ) 1 + 2s = . 2s s 2 + k 2 π2 k=1 The residue at j kπ is

ν4r = 0, ν4r +1

Res(j kπ) = Res(−j kπ) = νk , 1 4 1 = , ν4r +2 = ,ν4r +3 = . 2 2 2 2 (4r + 1) π (4r + 2) π (4r + 3)2 π 2

Thus, G(s) + G(s) = G(s) + G(¯s ) ≥ 0 and hence the system is passive. The system is passive, but not stable because of poles on the imaginary axis. It can be stabilized by any stable passive controller. In particular, the feedback u(t) = −k p y(t) for any positive k p will stabilize the system. Furthermore, with a passive controller, the controlled system is also passive. Corollary 7.8 implies that the controlled system is externally stable, but says nothing about internal stability. The uncontrolled system, (7.4) with u(t) ≡ 0, has an infinite number of eigenvalues on the imaginary axis. Since B has finite-dimensional range, the span of b, Theorem 3.43 implies that the system is not stabilizable and hence the controlled system is not internally stable. The system (7.4) with u(t) = ky(t), k > 0 is similar to the system in Example 3.14. If b, φn  = 0 for all 2 eigenvalues φn of the operator ddx 2 with boundary conditions φ(0) = 0, φ(1) = 0, then the √ controlled system is also asymptotically stable. The eigenfunctions are ). With the choice of b (7.5), φn = 2 sin( nπ 2 √ b, φn  =

nπ 2 sin( ) nπ 2

which is zero for even values of n. Hence, the controlled system is not even asymptotically stable with this choice of b. 

224

7 Output Feedback Controller Design

7.2 Final Value Theorem Designing a controller so that the output tracks a step, or some other reference signal, is a common controller design objective. Theorem 7.11 (Final Value Theorem) If yˆ (s) is the Laplace transform of an integrable function such that lim y(t) exists, then t→∞

lim y(t) = lim s yˆ (s)

t→∞

s→0

where s on the right-hand side is real. Let r be the reference input or desired response and y the actual response. In Fig. 7.1, the signal e1 (t) = y(t) − r (t) is the tracking error. The controller design objectives are a stable closed loop and lim t→∞ e1 (t) = 0. For system transfer function G and controller transfer function H, define Sˆ = (1 + G H )−1 , eˆ = Sˆ rˆ . Consider first a step input:

 r (t) =

r0 0

(7.6)

t ≥0 t < 0.

An application of this type of problem is a temperature-control thermostat in a room: when the setting is changed, the room temperature should eventually change to the new setting. The Laplace transform of a unit step is rˆ (s) = 1s . Therefore, using (7.6) and the Final Value Theorem, Theorem 7.11, the steady-state response to a unit step is ˆ 1 = lim S(s). ˆ lim estep (t) = lim s S(s) t→∞ s→0 s→0 s ˆ The steady-state error has magnitude | S(0)|. The closed loop tracks a step if and only ˆ if S(0) = 0. Another common reference input is a ramp: for some constant r0 ,

Fig. 7.1 Standard feedback configuration

7.2 Final Value Theorem

225

 r (t) =

r0 t t ≥ 0 . 0 t 0 and b ∈ L 2 (0, 1) ∂2w ∂ 2 w(x, t) ∂w(·, t) , b(·)b(x) = +  + b(x)u(t), ∂t 2 ∂t ∂x 2 w(0, t) = 0, w(1, t) = 0. Consider observation

 y(t) =

1

b(x) 0

0 < x < 1,

(7.8) (7.9)

∂w(x, t) dx ∂t

(This is a similar model to Example 7.10 except that now damping is included.) With state-space Z = H01 (0, 1) × L 2 (0, 1) the state-space equations are, defining , v = ∂w ∂t d dt







   w 0 =A + u(t), v b   w y(t) = C v w v

where  A

w v



 =

d w 2

2

 v , − v, bb

 dx dom(A) = H01 (0, 1) ∩ H2 (0, 1) × H01 (0, 1),

7.3 Approximation of Control Systems

 C

w v

229

 = v, b.

It was shown in Examples 3.14, 3.45 that all the eigenvalues λn of A have nonpositive real parts. Also, for certain choices of b, A generates an asymptotically stable semigroup. However, the system is not exponentially stable. Since there is a single control variable the system is not exponentially stabilizable and no sequence of approximations is uniformly stabilizable. Suppose that  1, 0 < x < 21 , b(x) = 0, 21 < x < 1 Defining

s 

s  s sinh (s) + 2 cosh − 3 cosh2 + 1, 2 2 2



s  s ε − 3 cosh2 + 1), D(s) = s(s + ) sinh (s) + ε(2 cosh 2 2 2

N (s) =

the transfer function is G(s) =

N (s) . D(s)

With this choice of b, the system is not asymptotically stable. However, the function G was shown in Example 3.45 to be in H∞ , so the system is externally stable. Furthermore, G(s) = 0. lim |s|→∞,s>0

Thus, there are sequences of rational functions {G n } so that lim G n (s) − G(s)∞ = 0.

n→∞

One possibility is to note that N (s) and D(s) are analytic on all of C, and so each function has a product expansion of the form, for complex numbers {z k }, { pk },

∞  s 1− , N (s) = zk k=1

∞  s D(s) = 1− . pk k=1

G(s) can be approximated by the rational function n−1

k=1 (1

−

k=1 (1

−

G n (s) = n

s ) zk s . ) pk

230

7 Output Feedback Controller Design

It can be shown that lim G − G n ∞ = 0.

n→∞

The functions G n are rational and therefore have finite-dimensional state-space realizations. Thus, there are finite-dimensional approximations of G that converge in the gap topology.  Example 7.17 (Tracking temperature with PI controller) (Example 7.12 cont.) Consider the heat equation on an interval with collocated control and observation: ∂2z ∂z = + b(x)u(t), ∂t ∂x 2 z(0, t) = 0, z(1, t) = 0,  1 y(t) = b(x)z(x, t)d x

0 < x < 1,

0

for some b ∈ L 2 (0, 1). This is a special case of the model described in Example 7.12, in one space dimension and with constant diffusivity. The operator A generates stable semigroup on L 2 (0, 1). The √ an exponentially ∞ eigenfunctions {φi (x) = 2 sin(πi x)}i=1 of A form an orthonormal basis for L 2 (0, 1). The approximations (An , Bn , Cn ) obtained with the projection onto the first n eigenfunctions satisfy assumption (A1) and are uniformly exponentially stable. (See Example 4.12.) Thus, they converge to the original system in the gap topology (Theorem 7.15). This is illustrated in Fig. 7.2 for control localized at  ≤ r ≤ 1 −  with spatial distribution 5 , |x − r | < , b(x) =  0 else. with r = 0.1,  = 0.01. Any PI controller H (s) = k p +

ki s

with k p ≥ 0, will lead to a stable closed loop, and if ki = 0 the controlled system will asymptotically track a constant reference signal. Use the first-order approximation obtained with n = 1 to tune the parameters. With n = 1 the matrices describing the approximation are, defining √ sin(π) , b1 = 10 2 sin(πr ) π     A1 = −π 2 , B1 = b1 = C1 .

7.3 Approximation of Control Systems

231 Bode Diagram 20

1.2 n=1 n=5 n=10 n=20

0 Magnitude (dB)

1

temp. (C)

0.8

-20

-40

-60 0

0.6

n=1 n=5 n=10 Phase (deg)

0.4

0.2

0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

n=20

-45

-90 10-1

100

time (s)

101

102

103

104

105

Frequency (rad/s)

(b) Frequency Response

(a) Step Response

Fig. 7.2 Convergence of modal approximations for heat equation with z(0, t) = z(1, t) = 0 and localized control, collocated with observation so C = B ∗

The transfer function is G(s) =

b12 . s + π2

The characteristic polynomial of the closed loop is s 2 + (π 2 + k p b12 )s + ki b12 . This is a second-order system. The settling time of the system is determined by π 2 + k p b12 if the system is not overdamped. Suppose that the settling time of the uncontrolled system is satisfactory and set k p = 0. For little overshoot, the system should have  2 (ki b12 ) ≈ π 2 . π Solving for ki yields ki = 4b 2. 1 Simulations of the closed loops obtained with this simple controller implemented with different orders of approximating system are shown in Fig. 7.3. Although the response changes with increasing approximation order until convergence is obtained, the system remains stable and the performance is satisfactory.  4

232

7 Output Feedback Controller Design 1.2 n=1 n=5 n=10 n=20

1

temp. (C)

0.8

0.6

0.4

0.2

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time (s)

Fig. 7.3 Step response of modal approximations for controlled heat equation with localized control, collocated with observation so C = B ∗ . The control is pure integral control. Note the step response is different for different approximations, but stable, and converges as approximation order is increased

7.4 State-Space Based Controller Design State-space based controller design is very popular, particularly for multi-input– multi-output systems. Consider a standard control system (7.1). A common procedure for controller design is to first design a state feedback controller: u(t) = −K z(t). Then, since the full state is not available, an estimator is designed to obtain an estimate of the state using knowledge of the output y and the input u. The controller is formed by using the state estimate as input to the state feedback controller. The general structure of a controller designed by placing state feedback in series with an estimator is as follows. Choosing F so that A − FC is Hurwitz, consider the system z˙ H (t) = (A − FC)z H (t) + Bu(t) + F y(t),

z H (0) = z H o .

(7.10)

If A − FC is Hurwitz, then z H is an estimate of the state of the original system. This motivates defining the controller output to be y H (t) = −K z H (t). In practice, due to disturbances such as sensor noise, indicated by r in Fig. 7.1, the controller input is

7.4 State-Space Based Controller Design

233

e1 = −y + r. Similarly, due to disturbances on the plant and also actuator error, indicated by d in Fig. 7.1, the input to the plant is not u but e2 = d + u, The plant is thus

z˙ (t) = Az(t) + Be2 (t) y(t) = C z(t) + De2 (t).

(7.11)

z˙ H (t) = (A − FC − B K )z H (t) − Fe1 (t) y H (t) = −K z H (t).

(7.12)

The controller is

This control system is in the standard feedback framework, shown in Fig. 7.1. The following theorem states that in general, the dynamics of the output feedback controller are those of the estimator plus those of state feedback. Also, the map from the external disturbance to the output is identical to that obtained with state feedback. No assumptions are made on how the state feedback K and filtering F operators are determined. Theorem 7.18 Consider the closed loop system (7.11), (7.12) with external inputs   d, n, outputs y and y H and state z(t) z H (t) . The closed-loop spectrum is the union of the spectrums of of A − FC and A − B K and also λ(Acls ) = λ(A − B K ) ∪ λ(A − FC). Assume that K , F are chosen so that the semigroup SK (t) generated by A − B K and the semigroup S F (t) generated by A − FC are both exponentially stable. Let M ≥ 1, α F > 0 and α K > 0 be constants such that SK (t) ≤ Me−α K t and S F (t) ≤ Me−α F t . For any α < min(α F , α K ), the closed loop semigroup is internally exponentially stable with decay rate α. Furthermore, the map from d to z and the map from d to y are identical to those of full state feedback. A classical method for design of an output feedback controller is to choose constant state feedback to be a linear quadratic regulator and then to use a well known estimator design, the Kalman filter. Such a controller is known as a Linear-QuadraticGaussian (LQG) controller. Suppose that d, n are Wiener processes (Definition 6.4) u a control variable, and consider the noisy system

234

7 Output Feedback Controller Design

 z(t) = T (t)z 0 +

t



t

T (t − s)Bu(s) ds +

0



with measurement

T (t − s)Bd(s) ds

(7.13)

0

y(t) =

t

C z(s)ds + Dn(t), t ≥ 0,

(7.14)

0

where B ∈ B(W, Z), W is a separable Hilbert spaces. As in Sect. 6.1 assume – d(t) is a Wiener process of covariance Q e on a separable Hilbert space W, – n(t) is a Wiener process of covariance Ro on the finite-dimensional Hilbert space Y, – the initial state z 0 is a Z-valued Gaussian random variable, with zero mean value and covariance 0 . – Ro is a positive operator, Re = D Ro D ∗ is invertible, – and d(t), n(t), z 0 are mutually uncorrelated. Let Q c ∈ B(Z, Z) be a positive semi-definite operator weighting the state and Rc ∈ B(U, U) a positive definite coercive operator weighting the control. The objective is to find the control signal u that minimizes the finite-time linear quadratic cost, with positive semi-definite weight  f ∈ B(Z, Z) on the final state,  J (u) = E

T

Q c z(t), z(t) + Rc u(t), u(t)dt + E f z(T ), z(T )dt, (7.15)

0

subject to (7.13), (7.14). If the full state z is available for control, the optimal control is u(t) = −K (t)z(t) where K (t) is defined through the solution to a differential Riccati equation (4.3). A different differential Riccati equation, (6.8), gives the optimal—optimal in that the variance of the error is minimized—estimate zˆ (t) of z(t). The following theorem states that the optimal controller is u(t) = −K zˆ (t), and that it is found by solving the optimal linear quadratic state feedback problem and the minimum variance problem separately. Theorem 7.19 Consider the problem of minimizing J (u) defined in (7.15) over u ∈ L 2 (0, T ; U) that depends causally on (7.14); that is, u(s) depends on y(t), t ≤ s. Let e (t) be the positive semi-definite solution to the estimation differential Riccati equation (6.8) with Bd Q Bd∗ replaced by B Q e B ∗ , R replaced by Re and define F(t) = e (t)C ∗ Re−1 . Let c (t) be the positive semi-definite solution to the control differential Riccati equation (4.3), as defined in Theorem 4.2, with C ∗ C replaced by Q c , R replaced by Rc . Defining K (t) = Rc−1 B ∗ c (t), the optimal control u minimizing J is d zˆ (t) = Aˆz (t) + F(t)(y(t) − C zˆ (t)) + Bu(t), zˆ (0) = E(z(0)), dt u(t) = −K (t)ˆz (t).

7.4 State-Space Based Controller Design

235

Note that, as for the case where the entire state is available for control, studied in Chap. 4, the optimal output control is also a feedback controller. The necessity to estimate the state introduces dynamics into the controller. Once an approximation scheme that converges in the gap topology is found (typically, by finding one that satisfies (A1) and is uniformly stabilizable), the next step is controller design. For control of DPS, typically both the state feedback and the estimator are designed using a finite-dimensional approximation (An , Bn , Cn ) with the finite-dimensional state space Zn . Suppose Fn ∈ B(Y, Zn ) is found so that all the eigenvalues of An − Fn Cn have negative real parts, and similarly K n ∈ B(Zn , U) is such that all the eigenvalues of An − Bn K n have negative real parts. Referring to Fig. 7.1, the resulting finite-dimensional controller is z˙ H (t) = (An − Fn Cn − Bn K n )z H (t) − Fn e1 (t) y H (t) = −K n z H (t).

(7.16)

Theorem 7.18 implies that this controller stabilizes (An , Bn , Cn ). The sequence of controllers designed using the approximations should converge to a controller for the original infinite-dimensional system (A, B, C) that yields the required performance. For this to happen, the controller sequence must converge in some sense. For controller convergence, an assumption similar to that of uniform stabilizability is required. Theorem 7.20 Assume that (A1) holds and that the operators K n , Fn used to define the sequence of controllers (7.16) satisfy the following assumptions 1. the semigroups generated by An − Bn K n are uniformly exponentially stable, 2. there exists K ∈ B(U, Z) such that for some subsequence {K n k }, lim K n k Pn k z = K z f or all z ∈ Z,

n k →∞

3. the semigroups generated by An − Fn Cn are uniformly exponentially stable, 4. there exists F ∈ B(Y, Z) such that for some subsequence {Fn k }, lim Fn k y = F y f or all y ∈ Y.

n k →∞

Indicating the controller transfer function by Hn , the controllers converge in gap and so for sufficiently large n, the output feedback controllers (7.16) stabilize the infinitedimensional system (2.63), (2.57). Furthermore, the closed loop systems (G, Hn ) and also (G n , Hn ) converge uniformly to the closed system (G, H ) where H indicates the transfer function of the infinite-dimensional controller (7.12). The key to the above theorem is that both the sequence of approximating plants and corresponding controllers converge in gap. This implies that closed loops (G, Hn ) converge to (G, H ) and that the controllers stabilize the original system for large enough n. Also, since (G n , Hn ) → (G, H ), simulations will predict the behaviour of the infinite-dimensional system.

236

7 Output Feedback Controller Design

For the situation where it is not known if a given sequence of approximations converges in the gap topology, or if the corresponding sequence of controllers converge, simulations can be used to check if at least it appears that such convergence exists. The closed loops (G n , Hn ) should appear to converge, and also using a large order approximation G N of G, (G N , Hn ) should also converge. In earlier chapters conditions for suitable convergence of state feedback controllers and of estimators designed different approaches were established. These results will now be used to synthesize output feedback controllers. More precise versions of Theorem 7.18 provide that performance converges with both H2 - and H∞ -controller design.

7.5 H2 - and H∞ -Controller Design Consider the system dz = Az(t) + B1 v(t) + B2 u(t), dt

z(0) = 0

(7.17)

with disturbances v and controls u indicated separately, and cost y1 (t) = C1 z(t) + D12 u(t).

(7.18)

It is not assumed that the full state is measured and so the measured signal, or input to the controller, is (7.19) y2 (t) = C2 z(t) + D21 v(t) where C2 ∈ B(Z, Y) for some finite-dimensional Hilbert space Y and D21 ∈ B(V, Y). Since the focus is on reducing the response to the disturbance, the initial condition z(0) is set to zero. Assume that v(t) ∈ L 2 (0, ∞; V) where V is a separable Hilbert space and that B1 ∈ B(V, Z) is a compact operator. (This last assumption follows automatically if V is finite-dimensional.) It is assumed that there are a finite number of controls so that B2 ∈ B(U, Z) where U is a finite-dimensional Hilbert space. Let G denote the transfer function from d to y1 and let H denote the controller transfer function: u(s) ˆ = H (s) yˆ2 (s). The map from the disturbance v to the cost y1 is ˆ yˆ1 = C1 (s I − A)−1 (B1 vˆ + B2 u) = C1 (s I − A)−1 (B1 vˆ + B2 H yˆ2 ). Using (7.17) and (7.19) to eliminate y2 , and defining

7.5 H2 - and H∞ -Controller Design

237

G 11 (s) = C1 (s I − A)−1 B1 , G 12 (s) = C1 (s I − A)−1 B2 + D12 , −1 G 21 (s) = C2 (s I − A) B1 + D21 , G 22 (s) = C2 (s I − A)−1 B2 , leads to the transfer function F(G, H ) = G 11 (s) + G 12 (s)H (s)(I − G 22 (s)H (s))−1 G 21 (s) from the disturbance d to the cost y1 . If     ∗ C1 D12 = 0 I , D12



   B1 0 ∗ D21 = , D21 I

(7.20)

the cost y1 has norm 

∞



∞

y1 (t)2 dt =

0

C1 z(t)2 + u(t)2 dt

0

which is the linear quadratic cost (4.7) with normalized control weight R = I . The difference is that here the effect of the disturbance d on the cost is considered, instead of the initial condition z(0). Also, as illustrated by Example 7.32 below, the system definition (7.17), (7.18), (7.19) can include robustness and other performance constraints. The special case where y2 (t) =

    I 0 z(t) + v(t), 0 I

known as full information, was described in Chap. 5. The solution for both H2 - and H∞ -controller design in this special case is a state feedback control of the form K z(t). If only y2 is available to the controller, and not all of the state z, the controller design problem is more complicated than the full information case. The solution is to find an estimate K zˆ of K z using only the measured output (7.19), and then to establish the performance of the controller with K zˆ . The following assumptions will be used. ∗ D12 is coercive. This ensures a non-singular penalty y1 on the control (H1a) R = D12 u. ∗ is coercive. This assumption is dual to (H1a) and it relates to (H1b) Re = D21 D21 effect of the disturbance on the controller input. It ensures that the effect of the exogenous input v on y2 is non-singular. (H2a) (A, B2 ) is stabilizable. (H2b) (A, C2 ) is detectable. ∗ ∗ C1 , (I − D12 R −1 D12 )C1 ) is detectable. (H3a) (A − B2 R −1 D12 ∗ ∗ −1 −1 (H3b) (A − B1 D21 Re C2 , B1 (I − D21 Re D21 )) is stabilizable.

238

7 Output Feedback Controller Design

Assumptions (H2a) and (H2b) ensure existence of an internally stabilizing closed loop exists. Assumptions (H3a) and (H3b) ensure that the closed loop is stabilizable and detectable, and so if the closed loop is externally stable, then it is internally stable. Existence of a controller with the required disturbance attenuation can be established with weaker assumptions than listed here; see notes at the end of this chapter.

7.5.1 H2 -Cost In this controller design approach, the disturbance v is a sum of impulses, or equivalently, uniform spectral density. More precisely, let {ek }∞ k=1 be an orthonorby yk . Then G22 = mal basis for V, and denote the output with input δ(t)e k ∞  1 2 ˆk (j ω)| dω. The case of a different disturbance is handled by absorbing k |y 2π −∞ the frequency content of v into the system description (7.17), (7.18), (7.19) as in Sect. 5.1. It will be assumed here that this normalization has already been done. The optimal H2 -controller is exactly the optimal H2 -estimate with the H2 -optimal state feedback. As for LQG controller design, the two design problems are entirely decoupled. Theorem 7.21 Consider the system (7.17), (7.18), (7.19) and assume that in addition to the assumptions (H1)–(H3) at least one of the following assumptions is satisfied: – B1 is a Hilbert–Schmidt operator, – C1 is a Hilbert–Schmidt operator, or – both B1 and C1 are trace class. ∗ ∗ Define Ac = A − B2 R −1 D12 C1 , C1c = (I − D12 R −1 D12 )C1 and let  ∈ B(Z, Z), ∗ dom(A) ⊂ dom(A ), be the positive semi-definite solution to ∗ C1c )z = 0, (A∗c  + Ac − B2 R −1 B2∗  + C1c

for all z ∈ dom(A). ∗ ∗ Re−1 C2 , B1e = B1 (I − D21 Re−1 D21 ), let  ∈ B(Z, Z), Define Ae = A − B1 D21 ∗ ∗ dom(A ) ⊂ dom(A ), be the positive semi-definite solution of ∗ )z = 0, ( A∗e + Ae  − C2∗ Re−1 C2  + B1e B1e

for all z ∈ dom(A∗ ). Define the H2 -optimal state feedback ∗ C1 K = R −1 B2∗  + R −1 D12

and the H2 -optimal filter ∗ Re−1 . F = C2∗ Re−1 + B1 D21

(7.21)

7.5 H2 - and H∞ -Controller Design

239

The H2 -optimal controller is z˙ e (t) = (A − B2 K − FC2 )z e (t) + F y2 (t), z e (0) = 0, u(t) = −K z e (t). The optimal cost is



trace(B1∗ B1 ) + trace(K  K ∗ ).

(7.22)

(7.23)

If approximations are used to calculate an approximate H2 -optimal controller, convergence of the controller and performance as the approximation order increases follow immediately from Theorems 4.11, 6.9 and 7.20 if the assumptions of those theorems are satisfied. Theorem 7.22 Assume the assumptions of Theorem 7.21 hold for the infinitedimensional system  (7.17), (7.18), (7.19) and consider the approximations   C1n . Assume that for the approximating systems, An , B1n B2n , C2n – (A1) and (A1∗ ) hold, ∗ ∗ Re−1 C2n , B1n (I − D21 Re−1 D21 )) are uniformly sta– (An , B2n ) and (An − B1n D21 bilizable, and ∗ ∗ C1n , (I − D12 R −1 D12 )C1n ) and (An , C2n ) are uniformly – (An − B2n R −1 D12 detectable. Then the finite-dimensional controllers calculated by solving the finite-dimensional versions of (7.21), (7.21) converge in the gap topology to the infinite-dimensional controller (7.22).

7.5.2 H∞ -Cost In H∞ -control the objective is to reduce the cost y1 2 over all disturbances. More precisely; find, for the desired disturbance attenuation γ > 0, a stabilizing controller so that, for all v ∈ L 2 (0, ∞; V),  0

∞

 y1 (t)2 < γ 2

∞

v(t)2 dt.

0

If such a controller is found, the controlled system will have L 2 -gain less than γ. Definition 7.23 The system (7.17), (7.18), (7.19) is stabilizable with attenuation γ if there is a stabilizing controller with transfer function H so that F(G, H )∞ < γ.

240

7 Output Feedback Controller Design

The control and estimation problems for H∞ -control, unlike LQ and H2 , are not entirely decoupled. A stronger assumption than (H1) will be made initially solely to obtain simpler formulae and clarify the development. Relaxation of this assumption involves routine manipulations and the general result will be given later. Strengthen assumption (H1) to ∗ [D12 C1 ] = (S1a) D12  [R 0] where R is coercive and D21 Re ∗ where Re is coercive. D21 = (S1b) B1 0

As long as assumption (H1) holds, a simple transformation will put any system in the form where (S1) is satisfied. With assumption (S1), (H2) is unchanged and (H3) is simplified. The assumptions are now (S2a) (S2b) (S3a) (S3b)

(A, B2 ) is stabilizable. (A, C2 ) is detectable. (A, C1 ) is detectable and (A, B1 ) is stabilizable.

Theorem 7.24 Assume the system (7.17), (7.18), (7.19) satisfies assumptions (S1)– (S3). The system is stabilizable with attenuation γ > 0 if and only if the following two conditions are satisfied: 1. There exists a positive semi-definite operator  ∈ B(Z, Z), dom(A) ⊂ dom(A∗ ), satisfying the Riccati equation 

A∗  + A + 



1 ∗ −1 ∗ ∗ B B − B R B 1 1 2 2  + C 1 C 1 z = 0, 2 γ

(7.24)

for all z ∈ dom(A), such that A + ( γ12 B1 B1 ∗ − B2 R −1 B2∗ ) generates an exponentially stable semigroup on Z; ˜ = dom(A), 2. Define K = R −1 B2∗ , A˜ = A + γ12 B1 B1 ∗  with domain dom( A) ∗ ∗ ˜ dom( A ) = dom(A ). There exists a positive semi-definite operator ∗ ˜ ∈ B(Z, Z), dom(A ˜  ) ⊂ dom(A), satisfying the Riccati equation 

˜ + ˜ A˜ ∗ +  ˜ A˜ 



1 ∗ ∗ −1 ˜ + B1 B1 ∗ z = 0, K K − C R C 2  2 e 2 γ

˜ for all z ∈ dom(A∗ ), such that A˜ +  nentially stable semigroup on X .

1 γ2

(7.25)

 K ∗ K − C2∗ Re−1 C2 generates an expo-

˜ 2∗ Re−1 . The controller with stateIf both conditions are satisfied, define F = C space description z˙ c (t) = ( A˜ − B2 K − FC2 )z c (t) + F y2 (t) u(t) = −K z c (t)

(7.26)

7.5 H2 - and H∞ -Controller Design

241

stabilizes the system (7.17), (7.18), (7.19) with attenuation γ. Condition (1) in Theorem 7.24 is the same Riccati equation solved to obtain the full-information state feedback controller. Condition (2) leads to an estimate of −K z(t). Unlike H2 control, the design of the estimator is coupled to that of the state feedback control. Also, it is possible that a solution may not exist for a particular attenuation level γ. To compute an H∞ -controller using a finite-dimensional approximation, define a sequence of approximations on finite-dimensional spaces Z N , as for the full information case, with the of C2n = C2 |Zn . Indicate the approximating operators  addition   C1n  by An , B1n B2n , . C2n Strong convergence of solutions n to finite-dimensional Riccati equations approximating (7.24) will follow from Theorem 5.15 and a straightforward duality argument if (A1) and (A1∗ ) hold, along with assumptions on uniform stabilizability ˜n and uniform detectability. A similar approach leads to convergence of solutions  to the Riccati equations approximating (7.25). Theorem 7.25 Assume (S1)–(S3) hold for the infinite-dimensional system (7.17), (7.18), (7.19), B1 , B2 , C2 are all compact operators and for the approximating systems, – (A1) and (A1∗ ) hold, – (An , B1n ) and (An , B2n ) are uniformly stabilizable, – (An , C1n ) and (An , C2n ) are uniformly detectable. Let γ > 0 be such that the infinite-dimensional problem is stabilizable with attenuation γ. For sufficiently large n the approximations are stabilizable with attenuation γ. For such n, the following statements hold. 1. The Riccati equation A∗n n

+ n An + n

1 ∗ −1 ∗ ∗ B1n B1n − B2n R B2n n + C1n C1n = 0, γ2

has a positive semi-definite solution n and K n = R −1 (B2n )∗ n is a γ-admissible state feedback for the approximating system. There exists M2 , ω2 > 0 such that for all n > No , the semigroups Sn K (t) generated by An + B2n K n satisfy Sn K (t) ≤ M2 e−ω2 t , and for all z ∈ Z, n Pn z → z where  solves (7.24) as n → ∞ and K n converges to K = R −1 B2∗  in norm. 2. The Riccati equation ˜n + ˜ n A˜ ∗n +  ˜n A˜ n 



1 ∗ ∗ −1 ˜ n + B1n B1 ∗n = 0 K K n − C2n Re C2n  γ2 n

˜ n and there exist positive constants M3 and ω3 has a positive semi-definite solution  ∗ ˜ n K n∗ K n −  ˜ n C2n such that the semigroups S˜n2 (t) generated by A˜ n + γ12  Re−1 C2n

242

7 Output Feedback Controller Design

˜ n Pn z →  ˜ z where satisfy  S˜n2 (t) ≤ M3 e−ω3 t . Moreover, for each z ∈ Z,  ∗ ˜ solves (7.25) as n → ∞ and Fn =  ˜ n C2n ˜ 2∗ Re−1 in  Re−1 converges to F = C norm. Theorem 7.26 Let the assumptions of Theorem 7.25 hold. Defining Acn = An +

1 B1n B1 ∗n  − B2n K n − Fn C2n , γ2

the controllers z˙ c (t) = Acn z c (t) + Fn y2 (t)

(7.27)

u(t) = −K n z c (t) converge to the infinite-dimensional controller (7.26) in the gap topology. Thus, there is N so that the finite-dimensional controllers (7.27) with n > N stabilize the infinite-dimensional system and provide γ-attenuation. Letting γˆ indicate the optimal disturbance attenuation for the output feedback problem (7.17), (7.18), (7.19), and similarly indicating the optimal attenuation for the approximating problems by γˆ n , ˆ lim γˆ n = γ.

n→∞

Condition (2) is more often rewritten as two equivalent conditions. A definition is first needed. Definition 7.27 The spectral radius r of an operator M ∈ B(Z, Z) where Z is a Hilbert space is 1 r (M) = lim M k  k . k→∞

Theorem 7.28 For M ∈ B(Z, Z) r (M) = sup |λ(M)|. λ∈σ(M)

If L , M ∈ B(Z, Z) are self-adjoint operators then r (L M) = r (M L). Thus, the spectral radius is the largest magnitude of elements in the spectrum of a bounded operator. Theorem 7.29 If the assumptions of Theorem 7.24 hold, and condition (1) is satisfied, then condition (2) holds if and only if both of the following two conditions hold: (2a) there exists a positive semi-definite operator  ∈ B(Z, Z), dom(A∗ ) ⊂ dom(A), satisfying the Riccati equation

7.5 H2 - and H∞ -Controller Design



A +  A∗ + 



243



1 ∗ ∗ −1 ∗  + B z = 0, C C − C R C B 1 2 1 1 2 e γ2 1

(7.28)

for all z ∈ dom(A∗ ), such that A + ( γ12 C1∗ Re−1 C1 − C2∗ C2 ) generates an exponentially stable semigroup on Z, and (2b) The spectral radius, (7.29) r () < γ 2 . Also, if conditions (1), (2a) and (2b) hold, then (I − γ12 ) has a bounded inverse, ˜ solves (7.25). ˜ = (I − 12 )−1  = (I − 12 )−1 where  and  γ

γ

There is a computational advantage to replacing condition (2) by conditions (2a) and (2b). The Riccati equation in (2) is coupled to the solution of (1) while the Riccati equation in (2a) is independent of the solution of (1). Thus, any error in calculating  does not affect calculation of . If the simplifying orthogonality assumptions (S1) do not hold, the result is similar but as for H2 -controller design, additional terms are present in the Riccati equations and the controller. The result is stated in the second form commonly used in controller synthesis, with two uncoupled Riccati equations (7.24), (7.28) and a third condition (7.29) on the spectral radius of the product of both solutions. ∗ C1 , Theorem 7.30 Assume that (H1)–(H3) hold. Define Ac = A − B2 R −1 D12 ∗ ∗ −1 ∗ −1 −1 C1c = (I − D12 R D12 )C1 and Ae = A− B1 D21 Re C2 , B1e = B1 (I − D21 Re D21 ). There exists a stabilizing controller with transfer function H such that F(G, H )∞ < γ if and only if the following three conditions all hold:

1. The algebraic Riccati equation 

A∗c  + Ac + (

1 ∗ B1 B1∗ − B2 R −1 B2∗ ) + C1c C1c z = 0, 2 γ

(7.30)

for all z ∈ dom(A), has a self-adjoint positive semi-definite solution  ∈ B(Z, Z), dom(A) ⊂ dom(A∗ ), such that Ac + ( γ12 B1 B1∗ − B2 R −1 B2∗ ) generates an exponentially stable semigroup on Z; 2. The algebraic Riccati equation 

Ae  +  A∗e + (

1 ∗ ∗ z = 0, C C1 − C2∗ Re−1 C2 ) + B1e B1e γ2 1

(7.31)

for all z ∈ dom(A∗ ), has a self-adjoint positive semi-definite solution  ∈ B(Z, Z), dom(A∗ ) ⊂ dom(A), such that Ae + ( γ12 C1∗ C1 − C2∗ Re−1 C2 ) generates an exponentially stable semigroup on Z; and 3. r () < γ 2 . Moreover, if the above conditions are satisfied, then defining E = (I − γ12 ), K = ∗ ∗ C1 and F = E −1 C2∗ Re−1 + E −1 B1 D21 Re−1 , the controller with R −1 B2∗  + R −1 D12 state-space description

244

7 Output Feedback Controller Design

1 1 B1 B1 ∗  − B2 K − F(C2 + 2 D21 B1∗ ))z e (t) + F y2 (t) 2 γ γ u(t) = −K z e (t) (7.32)

z˙ e (t) = (A +

stabilizes the system (7.17), (7.18), (7.19) with attenuation γ. Typically an approximation on a finite-dimensional state-space n is used. As  Z   C1n . Strong conearlier, indicate the approximating operators by An , B1n B2n , C2n vergence of n →  and of n →  does not imply convergence (or even existence) of the inverse operator (I − γ12 n n )−1 so controller convergence cannot be shown ˜ n to the estimation Riccati equation (7.25) this way. Convergence of the solution  does hold, and this implies convergence of the approximating controllers. Theorem 7.31 Assume (H1)–(H3) hold for the infinite-dimensional system (7.17), (7.18), (7.19) and consider the approximating systems 

(An , B1n

  C B2 , 1n ). C2n 

Assume that for the approximating systems – (A1) and (A1∗ ) hold, ∗ ∗ Re−1 C2n , B1n (I − D21 Re−1 D21 )) are uniformly sta– (An , B2n ) and (An − B1n D21 bilizable, and ∗ ∗ C1n , (I − D12 R −1 D12 )C1n ) and (An , C2n ) are uniformly – (An − B2n R −1 D12 detectable. Let γ > 0 be such that the infinite-dimensional problem is stabilizable with attenuation γ. For sufficiently large n approximations to the full-information problem are stabilizable with attenuation γ. Also, the finite-dimensional controllers corresponding to (7.32) converge in the gap topology to the infinite-dimensional controller (7.32). For sufficiently large n the finite-dimensional controllers stabilize the infinite-dimensional system and provide γ-attenuation. Let γˆ indicate the optimal disturbance attenuation for the output feedback problem (7.17), (7.18), (7.19), and similarly indicate the optimal attenuation for the approximating problems by γˆ n . Then ˆ lim γˆ n = γ.

n→∞

Example 7.32 Consider a simply supported beam, as in Example 2.26. Let w(x, t) denote the deflection of a simply supported beam from its rigid body motion at time t and position x. The deflection is controlled by applying a force u(t) around the point r = 0.25 with distribution b(x) =

 10

, |r − x| < 2 0, |r − x| ≥ 2 . 

7.5 H2 - and H∞ -Controller Design

245

The exogenous disturbance v(t) induces a load d(x)ν(t). Normalizing the variables and including viscous damping with parameter cv leads to the partial differential equation ∂4w ∂2w ∂w ∂5w + c + + c = b(x)u(t) + d(x)ν(t), 0 < x < 1. v d ∂t 2 ∂t ∂x 4 ∂t ∂x 4 The boundary conditions are ∂2 w (0, t) ∂x 2 ∂2 w (1, t) 2 ∂x

w(0, t) = 0, w(1, t) = 0,

+ cd ∂x∂ 2w∂t (0, t) = 0 , 3 + cd ∂x∂ 2w∂t (1, t) = 0. 3

  ˙ t) , and define Choose the state z(t) = w(x, t) w(x, Hs (0, 1) = {w ∈ H2 (0, 1)| w(0) = 0, w(1) = 0}. The state-space realization of the system on the state space Hs (0, 1) × L 2 (0, 1) is z˙ (t) = A P z(t) + B P1 ν(t) + B P2 u(t) where

    z2 z1  = , AP  z2 − z 1 + cd z 2 − cv z 2 dom(A) = {(z 1 , z 2 ) ∈ Hs (0, 1) × Hs (0, 1)| z 1 + cd z 2 ∈ Hs (0, 1)}, 

B P1

 0 = , d(x)



B P2

 0 = . b(x)

The operator B P1 models the effect of the disturbance on the beam and B P2 models the effect of the controlled input. As shown in Example 2.26, all the eigenvalues of A P have negative real parts and A p is a Riesz-spectral operator. Theorem 3.20 implies that A P generates an exponentially stable semigroup on Hs (0, 1) × L 2 (0, 1). The objective in the controller design will be to reduce the response to low frequency disturbances, particularly near the first resonant mode, and also be robustly stable with respect to modeling errors and sensor noise. Positions are measured at a point xo . Define C p ∈ B(Z, R) by   z C P 1 = z 1 (xo ). z2 Including the effect of sensor noise n(t), measurement of the position at the point xo leads to the controller input

246

7 Output Feedback Controller Design

y2 (t) = C P z(t) + η(t). The effect of sensor noise on the measurements is described in the frequency domain by the signal η(s) ˆ = Wn (s) where Wn has realization as an internally stable finitedimensional system (A Wn , BWn , C Wn , DWn ). Assume that DWn > 0 to account for the fact that sensor noise and modelling errors are non-zero at high frequencies. This weight also improves robust stability to account for modelling and parameter error, as well as the effect of neglecting higher modes in the model used in controller synthesis. (See the references at the end of this chapter for detail on robust H∞ controller design.) The primary aim of the controller is to reject the effect of disturbances ν on the position at xo . More specifically, the controlled system at this point should not respond strongly to disturbances at frequencies up to and including the first natural frequency. A related goal is tracking of low frequency reference inputs. Let y p indicate the position and choose a weighting function W1 that is large at low frequencies and small at large frequencies. The goal is to reduce W1 C p zˆ H2 over all disturbances, or equivalently, the H∞ -norm of the transfer function from vˆ to W1 C p zˆ . Choosing W1 to be large only at low frequencies means that only the effect of the disturbance on low frequencies of the output are weighted. Indicate a state space realization of this weight by the internally stable finite-dimensional realization (A W1 , BW1 , C W1 , 0). A non-singular cost on the controller effort is needed. This also adds additional robustness in the controller design. Set a frequency independent weight δ > 0 on the controller output u. The total cost to be reduced is   W1 C P z . y1 = δu This leads to an augmented system of the form (7.17), (7.18), (7.19) with ⎡

⎤ AP 0 0 A = ⎣ B W1 C P A W1 0 ⎦ , 0 ⎤ 0 A Wn⎡ ⎡ ⎤ B P1 0 B P2 B1 = ⎣ 0 0 ⎦ , B2 = ⎣ 0 ⎦ , 0  0 BWn   0 C W1 0 , C 2 = C P 0 C Wn , C1 = 0 0  0   0 , D21 = 0 DWn . D12 = δ Let n 1 be the dimension of A W1 and n 2 that of A Wn . The state space of this augmented system is Z = Hs (0, 1) × L 2 (0, 1) × Rn 1 × Rn 2 . Since A p generates an exponentially stable semigroup on Hs (0, 1) × L 2 (0, 1) and the additional systems are finitedimensional, A generates an C0 -semigroup on Z. Exponential stability of this system

7.5 H2 - and H∞ -Controller Design

247

can be shown using the stability of the sub-systems and the fact that A is lower block triangular. ∗ D12 = Now check that the assumptions (H1)–(H3) are satisfied. First, R = D12 ∗ 2 2 δ = 0 and Re = D21 D21 = Dn = 0 so the assumptions (H1) hold. The inclusion of a control cost δ and non-zero sensor noise at high frequencies ensured that these assumptions are satisfied. The operator A generates an exponentially stable semigroup so assumptions (H2), stabilizability of (A, B2 ) and detectability of (A, C2 ) follow trivially. Now consider assumptions (H3)       C W1 0 0 0 0 ∗ D12 C1 = 0 δ D12 δ 0000   2 = δ 0000 . Thus D12 and C1 are orthogonal. Detectability of (A, C1 ), assumption (H3a) or simplified assumption (S3a), follows from stability of A. On the other hand, 



D21 ∗ D21 B1

⎤ 0 D Wn   ⎢ B P1 0 ⎥ 0 ⎥ =⎢ ⎣ 0 0 ⎦ Dn 0 B Wn ⎡ ⎤ 2 DW n ⎢ 0 ⎥ ⎥ =⎢ ⎣ 0 ⎦ B Wn D Wn ⎡

The simplifying orthogonality assumption on B1 and D21 does not hold. ⎤ AP 0 0 ∗ ⎦, 0 A − B1 D21 Re−1 C2 = ⎣ BW 1 C P A W1 −1 −1 −Bn DWn C p 0 A Wn − BWn DWn C Wn ⎡ ⎤ B p1 0 ∗ B1 (I − D21 Re−1 D21 ) = ⎣ 0 0⎦ . 0 0 ⎡

−1 C Wn Since A p and A W1 are both Hurwitz, Assumption (H3) holds if A Wn − BWn DW n is Hurwitz. If Wn is stable and passive, this assumption will hold, although passivity of Wn is not required. In computer simulations, cv = 0.1, cd = 0.001, the actuator width was set to  = 0.001, the disturbance distribution is collocated with the actuator: d = b0.25 . Measurements are at the centre: xo = 0.5.

248

7 Output Feedback Controller Design

The first eigenvalue of the beam has a frequency π 2 rad/s. In order to reduce the effect of disturbances up to this frequency, choose W1 to be a first-order system with a pole beyond this: α W1 (s) = s 1 + 20 √ √ and a realization ([−20], [ 20], [α 20], 0). The value of α will be increased as much as possible, in order to improve disturbance rejection. The sensor noise is modelled by 0.01 . Wn (s) = 0.1 + s + 100 with realization ([−100], [0.01], [1], [.1]). This noise model leads to a system that satisfies Assumption (H3). The value of δ = 0.1. Uniformly stable approximations of the augmented plant can be obtained by using the eigenfunctions of the beam, see Theorem 4.13 and Example 4.29 for details. A model with only the first mode of the beam was used to design the controller. In calculations, it was found that α = 10 yielded a solution to the H∞ -output controller design problem with γ = 0.97. The closed loop is stable with good rejection of a disturbance sin(10t), even though it is very close to the first natural frequency. The calculated controller also stabilized the model with 3 modes. This is illustrated in Fig. 7.4. Furthermore, the designed controller provides good damping of vibrations even when used with the model with 3 modes; see Fig. 7.5. For the best performance, the best model should of course be used in controller design. However, this example illustrates that spillover of neglected higher modes and the resulting instability can be avoided.  5

5 uncontrolled controlled

3

3 2 1 0 -1 -2

2 1 0 -1 -2

-3

-3

-4

-4

-5

uncontrolled controlled

4

mid-point deflection

mid-point deflection

4

0

10

20

30

40

50 60 time

70

80

90

100

(a) nominal model (1 mode)

-5

0

10

20

30

40

50 60 time

70

80

90

100

(b) model with 3 modes

Fig. 7.4 H∞ -output feedback controller design for a lightly-damped simply-supported beam. The figure shows the deflections at the mid-point of the beam with a disturbance sin(10t) near the beam’s natural frequency. The closed loop does not respond strongly to the disturbance. The controller was designed using only 1 mode but stabilizes the higher order approximation and provides good performance with the higher order model

7.6 Notes and References

249

0.3 uncontrolled controlled

mid-point deflection

0.25

0.2

0.15

0.1

0.05

0

0

10

20

30

40

50

60

70

80

90

100

time

Fig. 7.5 H∞ -output feedback controller design for a lightly-damped simply-supported beam. The figure shows the deflections at the mid-point in response to a step input. Vibrations are attenuated. The controller was designed using only 1 mode but stabilizes the approximation with 3 modes and also provides good performance with the higher-order model

7.6 Notes and References Theorem 7.11, the Final Value Theorem, is well known for finite-dimensional systems, see for example [1]. A detailed general treatment is in [2, Chap. 8]. Some results on PI-control for irrational transfer functions can be found in [3–5]. The internal model principle and the state-space based approach to regulation are covered in [6, 7]. The approach has been extended to systems with unbounded control and observation [8–11]. Dissipativity is a broad collection of stability results that depend only on the input/output behaviour of the system. The theory was first described in [12, 13] and these papers as well as [14] are a good introduction. Theorem 7.6 can be generalized to multiple system interconnections [15]. Generalizations of many other well-known finite-dimensional stability results such as the Nyquist stability criterion exist, see [16]. The idea can be used for structures with position control that are not dissipative in the sense of Definition 7.1 [17]. There is a close relationship between Riccati inequalities and dissipativity that is not explored here [18–20]. The book [21] focuses on finite-dimensional systems but has some DPS material. Port-Hamiltonian systems is a framework for physical modelling of passive systems [22]. Dissipativity is attractive as a design approach because it is robust to parameter uncertainties and also applies to nonlinear systems. But the stability results tend to be conservative. Also, this approach to stability does not provide an algorithm for controller synthesis. For a high-performance controlled system, a model of the system needs to be used.

250

7 Output Feedback Controller Design

The separation principle for finite-time LQG controller design, Theorem 7.19, is in [23]. H∞ -controller design has been successfully developed as a method for robust control and disturbance rejection. There are many books that explain how H∞ controller design can be used to obtain robust stability and performance objectives for finite-dimensional systems; see, for example, [1, 24]. The calculation of an output feedback controller for finite-dimensional systems by solution of two Riccati equations was first shown in [25]. State-space based H2 and H∞ -output controller design for infinite-dimensional systems, in particular Theorem 7.24, is established in [26, 27]. The result applies to some systems with unbounded control and observation operators. Approximation of H∞ -output feedback control was shown in [28]. Direct H∞ -controller synthesis for DPS can be accomplished, without AREs, if a closed-form representation of the transfer function is available [29]. The power of this method is that no approximation is used in the controller design. The controller may be approximated later, but determining the degradation in performance is simpler than when an approximation to the system is used in controller design. Although the transfer function can generally only be written out for problems in one-space dimension with uniform parameters, and boundary control, this approach is also useful in obtaining performance limitations and testing algorithms. The idea of measuring the distance between systems using the gap between their graphs was first introduced for systems in [30]. It is based on the gap topology for operators [31]. The gap between systems can be quantified using several possible norms [32, 33]. The use of the gap topology in determining whether an approximation is suitable for controller design is first described in [34], convergence of state-space based controllers designed using approximations is in [35] and there is an overview in the tutorial paper [36]. A approach using the gap topology to show non-convergence of approximations of an undamped beam is in [37]. Uniform exponential stabilizability (and detectability), although as illustrated by Example 7.16, is not necessary for convergence in the gap topology, appears in many different contexts in literature on controller and estimator design. Its importance to linear-quadratic controller design and H∞ -control, as well as to estimation using both approaches was established in earlier chapters. The analytic function theory used for the expansion in Example 7.16 can be found, for instance, [38, Chap. 9]. Techniques appropriate for control of process systems that include stochastic effects are covered in [39] and for control of possibly nonlinear reaction–diffusion systems see [40]. Optimal actuator location with linear quadratic cost was covered in Chap. 4 and for H2 and H∞ costs in Chap. 5. Optimal sensor location with various costs was described in Chap. 6. An obvious next step for output control of DPS is to optimize both actuator and sensor locations, along with the controller design. Since in LQ control, the estimator design is entirely decoupled from the state feedback, optimal actuator and sensor location are also decoupled. For H2 -design, the estimator and control design are also decoupled. However, the actuators and sensors are coupled in the overall cost (7.23). Numerical results in [41] on flow control indicate that optimal placement with an H2 -cost is sometimes coupled in practice. Since for H∞ output

7.6 Notes and References

251

feedback, the control and estimation are coupled, this will also be true for sensor and actuator placement. This chapter and earlier chapters provide a number of results that guarantee stability and convergent performance of the closed loop when an approximation to a PDE model is used to design a controller. Of course, there are numerous systems for which, at the present moment, no checkable sufficient conditions for this convergence exist. In these situations, a useful strategy is to check whether the various conditions at least appear to be satisfied. First, if the original system is exponentially stable, as is common in applications, the approximations should ideally also be exponentially stable with the same decay margin, as in Example 4.18. This test is not possible for systems that are only asymptotically stable. Another check is to plot the approximate system time response and transfer function, for increasing approximation order. The most important test is to simulate the designed controller implemented with approximate plants of increasing order. The controlled system should be stable, converge with increasing plant order, and also performance not be too much worse than that obtained with the nominal plant. This approach is illustrated in Examples 7.17 and 7.32; although theoretical results guaranteed convergence for these systems.

References 1. Morris KA (2001) An introduction to feedback controller design. Harcourt-Brace Ltd., San Diego 2. Zemanian AH (1965) Distribution theory and transform analysis. Dover Publications, United States 3. Logemann H, Zwart H (1992) On robust PI-control of infinite-dimensional systems. SIAM J Control Optim 30(3):573–593 4. Logemann HL (1991) Circle criteria, small-gain conditions and internal stability for infinitedimensional systems. Automatica 27:677–690 5. Logemann HL, Ryan EP, Townley S (1999) Integral control of linear systems with actuator nonlinearities: lower bounds for the maximal regulating gain. IEEE Trans Autom Control 44:1315–1319 6. Byrnes CI, Laukó IG, Gilliam DS, Shubov VI (2000) Output regulation for linear distributed parameter systems. IEEE Trans Autom Control 45(12):2236–2252 7. Rebarber R, Weiss G (2003) Internal model-based tracking and disturbance rejection for stable well-posed systems. Automatica 39(9):1555–1569 8. Humaloja J-P, Paunonen L (2018) Robust regulation of infinite-dimensional port-Hamiltonian systems. IEEE Trans Autom Control 63(5):1480–1486 9. Paunonen L (2016) Controller design for robust output regulation of regular linear systems. IEEE Trans Autom Control 61(10):2974–2986 10. Paunonen L, Pohjolainen S (2014) The internal model principle for systems with unbounded control and observation. SIAM J Control Optim 52(6):3967–4000 11. Natarajan V, Gilliam DS, Weiss G (2014) The state feedback regulator problem for regular linear systems. IEEE Trans Autom Control 59(10):2708–2723 12. Willems JC (1972) Dissipative dynamical systems. I. General theory. Arch Rational Mech Anal 45:321–351 13. Willems JC (1972) Dissipative dynamical systems. II. Linear systems with quadratic supply rates. Arch Rational Mech Anal 45:352–393 14. Desoer CA, Vidyasagar M (1975) Feedback systems: input-output properties. Academic, New York

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15. Moylan PJ, Hill DJ (1978) Stability criteria for large-scale systems. IEEE Trans Autom Control 23(2):143–149 16. Curtain RF, Zwart H (1995) An introduction to infinite-dimensional linear systems theory. Springer, Berlin 17. Morris KA, Juang J-N (1994) Dissipative controller designs for second-order dynamic systems. IEEE Trans Autom Control 39(5):1056–1063 18. Curtain RF (1996) The Kalman-Yakubovich-Popov lemma for Pritchard-Salamon systems. Syst Control Lett 27(1):67–72 19. Curtain RF (1996) Corrections to: “The Kalman-Yakubovich-Popov lemma for PritchardSalamon systems. Syst Control Lett 27(1):67–72 20. Oostveen JC, Curtain RF (1998) Riccati equations for strongly stabilizable bounded linear systems. Automatica 34(8):953–967 21. Brogliato B, Lozano R, Maschke B, Egeland O (2007) Dissipative systems analysis and control. Commun Control Eng Ser, 2nd edn. Springer Ltd., Berlin 22. Duindam V, Macchelli A, Stramigioli S, Bruyninckx H (2009) Modeling and control of complex physical systems: the port-hamiltonian approach. Springer, Berlin 23. Curtain RF, Ichikawa A (1978) Optimal location of sensors for filtering for distributed systems. Lecture notes in control and informatics science, vol 1. Springer, Berlin, pp 236–255 24. Zhou K, Doyle JC, Glover K (1996) Robust and optimal control. Prentice-Hall, Englewood Cliffs 25. Doyle JC, Glover K, Khargonekar P, Francis B (1989) State-space solutions to standard H2 and H∞ control problems. IEEE Trans Autom Control 34(8):831–847 26. Keulen BV (1993) H ∞ − control for distributed parameter systems: a state-space approach. Birkhauser, Boston 27. Bensoussan A, Bernhard P (1993) On the standard problem of H∞ -optimal control for infinite dimensional systems. In: Banks HT, Fabiano R, Ito K (eds) Identification and control in systems governed by partial differential equations. SIAM, pp 117–140 28. Morris KA (2001) H∞ output feedback control of infinite-dimensional systems via approximation. Syst Control Lett 44(3):211–217 29. Ozbay H, Gümmssoy ¨ S, Kashima K, Yamamoto Y (2018) Frequency domain techniques for H∞ control of distributed parameter systems. SIAM 30. Zames G, El-Sakkary A (1980) Unstable systems and feedback: the gap metric. In: Proceedings of the Allerton Conference (1980), pp 380–385 31. Kato T (1976) Perturbation theory for linear operators. Springer, Berlin 32. Vidyasagar M (1985) Control system synthesis: a factorization approach. MIT Press, Cambridge 33. Zhu SQ (1989) Graph topology and gap topology for unstable systems. IEEE Trans Autom Control 34:848–855 34. Morris KA (1994) Design of finite-dimensional controllers for infinite-dimensional systems by approximation. J Math Syst, Estimat Control 4(2):1–30 35. Morris KA (1994) Convergence of controllers designed using state-space methods. IEEE Trans Autom Control 39(10):2100–2104 36. Morris KA (2010) Control of systems governed by partial differential equations. In: Levine WS (ed) Control handbook. CRC Press, Boca Raton 37. Morris KA, Vidyasagar M (1990) A comparison of different models for beam vibrations from the standpoint of controller design. ASME J Dyn Syst, Meas Control 112:349–356 38. Antimirov MY, Kolyshkin AA, Vaillancourt R (1998) Complex variables. Academic, Cambridge 39. Christofides PD, Armaou A, Lou Y, Varshney A (2009) Control and optimization of multiscale process systems. Control engineering, Birkhäuser Boston Inc, Boston 40. Christofides PD (2001) Nonlinear and robust control of PDE systems. Systems and control: foundations and applications. Birkhäuser Boston, Inc., Boston. (Methods and applications to transport-reaction processes) 41. Chen KK, Rowley CW (2011) H2 -optimal actuator and sensor placement in the linearised complex Ginzburg-Landau system. J Fluid Mech 681:241–260

Appendix

Functional Analysis and Operators

A.1

Linear Space

Definition A.1 A real linear space Z is a set {z 1 , z 2 , . . .} in which operations addition ( + ) and scalar multiplication by real numbers are defined. The following axioms are satisfied for all z i ∈ Z and all scalars α, β ∈ R: 1 2 3 4 5. 6. 7 8 9 10

z1 + z2 ∈ Z αz ∈ Z z1 + z2 = z2 + z1 (z 1 + z 2 ) + z 3 = z 1 + (z 2 + z 3 ) There exists 0 ∈ Z with the property that z + 0 = z for all z ∈ Z For every z ∈ Z there is z˜ ∈ Z such that z + z˜ = 0 (αβ)z = (βα)z (α + β)z = (β + α)z α(z 1 + z 2 ) = αz 1 + αz 2 1z = z

The element 0 is known as the zero element. The scalars can also be the complex numbers, in which case it is a complex linear space. (Linear spaces are sometimes referred to as vector spaces.) The following can all be verified to be real linear spaces: – Rn , vectors (z 1 , z 2 , . . . , z n ) of length n with real-valued entries, – C[a, b], the set of real-valued continuous functions on an interval [a, b]; while the following spaces are complex linear spaces – Cn , vectors (z 1 , z 2 , . . . , z n ) of length n with complex-valued entries, © Springer Nature Switzerland AG 2020 K. A. Morris, Controller Design for Distributed Parameter Systems, Communications and Control Engineering, https://doi.org/10.1007/978-3-030-34949-3

253

254

Appendix: Functional Analysis and Operators

– L 2 (a, b), the set of all scalar-valued functions f defined on (a, b) with 

b

| f (x)|2 d x < ∞

a

and scalar multiplication by complex numbers. (This definition implies that f is well enough behaved that the above integral is well-defined.)

A.2

Inner Products and Norms

Consider solving a problem such as 

1

k(s, t) f (s)ds = g(t)

(A.1)

0

for the function f. The kernel of the integral, k, and the right-hand side g are known continuous functions. This has a passing similarity to a matrix equation A f = g, where matrix A ∈ Rn×n and vector g ∈ Rn are known while f ∈ Rn is to be calculated. We know how to solve matrix equations, so trying to reformulate (A.1) as a matrix problem is tempting. √ The following is a basic result in Fourier series. The symbol j = −1 while α indicates the complex conjugate of α. Theorem A.2 For integers n = 0, ±1, ±2, . . . define φn (t) = ej 2πnt = cos(2nπt) + j sin(2nπt). 1



0, n = m , 1, n = m  1 1 2. for any f ∈ L 2 (0, 1), 0 f (t)φn (t)dt = 0 for all n implies 0 | f (t)|2 dt = 0.

1.

0

φn (t)φm (t)dt =

This suggests writing f (t) =

∞ 

f i φi (t)

(A.2)

i=−∞

1 where f i = 0 f (t)φi (t)dt. Multiplying each side of the above equation by φm and integrating over [0, 1] yields, making use of property (1) above,  0

1

f (t)φm (t)dt = f m .

(A.3)

Appendix: Functional Analysis and Operators

255

∞ Thus, for every continuous function f there corresponds an infinite vector { f i }i=−∞ . Note that



1

 | f (t)|2 dt =

0

1

∞ 

0 i=−∞  1  ∞

=

f i φi (t) f i φi (t)dt 

1

| f i |2

i=−∞ ∞ 

=

f m φm (t)dt

m=−∞

0 i=−∞ ∞ 

=

∞ 

f i φi (t)

φi (t)φi (t)dt

0

| f i |2 .

i=−∞

The expansion (A.2) can be done for any function for which 

1

| f (t)|2 dt < ∞ .

0

In what sense is (A.2) valid? Define { f i } as in (A.3) and consider the error function h(t) = f (t) − lim

N 

N →∞

f i φi (t).

i=−N

For any φm , 

1



1

h(t)φm (t)dt = f m −

0

lim

0

N →∞

N 

f i φi (t)φm (t)dt

i=−N

= fm − fm = 0. Property (2) in Theorem A.2 then implies statement (A.2) in the sense that the error 1 h satisfies 0 |h(t)|2 dt = 0. For notational convenience, renumber the indices: 0, −1, 1, −2, −2, . . . Defining f = { f 0 , f −1 , f 1 , f −2 , f 2 . . .}, f = (

∞ 

1

| f i |2 ) 2 .

(A.4)

i=1

The linear space of vectors of infinite length for which

∞ i=1

| f i |2 < ∞ is called 2 .

256

Appendix: Functional Analysis and Operators

Returning to (A.1), assume g, f ∈ L 2 (0, 1). Replacing f, g by their Fourier series (A.2), 

1

k(s, t) 0

f j φ j (s)ds =

j=1

∞   j=1

∞ 

1

∞ 

g j φ j (t)

j=1

k(s, t)φ j (s)ds f j =

0

∞ 

g j φ j (t).

j=1

Multiply each side by φi (t) and integrate with respect to t over [0, 1], 

∞  1 1 0

j=1

0

1

1

∞   j=1

0

 0



∞ 1

k(s, t)φ j (s)dsφi (t)dt f j = 0

g j φ j (t)φi (t)dt

j=1

k(s, t)φ j (s)dsφi (t)dt f j = gi . 

[A]ij

Thus, defining ⎡ ⎤ g1 ⎢g2 ⎥ g = ⎣ ⎦, .. .

⎡ ⎤ f1  1 1 ⎢ f2 ⎥ f = ⎣ ⎦ , [A]i j = k(s, t)φ j (s)dsφi (t)dt, .. 0 0 . ∞  [A]i j f j = gi j=1

or A f = g. An approximation to f can be calculated by solving the first N × N sub-block of the N , if this system of linear equations has a solution. Then, infinite-matrix A for { f i }i=1 f N (t) =

N 

f i φi (t)

i=1

is an approximate solution to the integral equation (A.1). It needs to be shown that the error in approximating f by f N is small in some sense. This approach to solving an integral Eq. A.1 motivates the definition of an infinite “dot” product for infinite products for vectors in 2 :

Appendix: Functional Analysis and Operators

257 ∞ 

f ·g=

f i gi .

(A.5)

i=1

Also, f ·g =

∞ 

f i gi

i=1

=

∞ 





∞ 1 0



∞ 

g j φ j (t)dt

j=1

f i φi (t)

i=1 1

=

φi (t)

0

i=1

=

1

fi

∞ 

g j φ j (t)dt

j=1

f (t)g(t)dt.

0

This suggests defining a “dot product” on L 2 (0, 1) by   f, g =

1

f (t)g(t)dt.

(A.6)

0

These scalar products are special cases of what is generally known as an inner product. Definition A.3 Let X be a complex linear space. A function ·, · : X × X → C is an inner product if, for all x, y, z ∈ X , α ∈ C, 1. 2. 3. 4.

x + y, z = x, z + y, z αx, y = αx, y x, y = y, x x, x > 0 if x = 0

The definition for a real linear space is identical, except that the scalars are real numbers and property (3) reduces to x, y = y, x . An inner product space is a linear space together with an inner product. Examples of inner product spaces include 2 with the inner product (A.5), L 2 (0, 1) with the inner product (A.6). Some other inner product spaces are – continuously differentiable functions on [a, b] with the inner product   f, g = 0

1



1

f (t)g(t)dt +

f (t)g (t)dt

0

– continuous functions defined on some closed region  ⊂ R3 with

258

Appendix: Functional Analysis and Operators

  f, g =



f (x)g(x)d x.

For vectors in Rn , the Euclidean inner product defines the length of the vector, x = (

n 

1

|xi |2 ) 2 .

i=1

This concept extends to general inner product spaces. Definition A.4 A real-valued function  ·  : Z → R on a linear space Z is a norm if for all y, z ∈ Z, α ∈ R, 1. 2. 3. 4.

z ≥ 0 (non-negative) z = 0 if and only z = 0 (strictly positive) αz = |α|z(homogeneous) y + z ≤ y + z (triangle inequality) For any inner product space, define 1

 f  =  f, f 2 .

(A.7)

It is clear that the first three properties of a norm are satisfied. The last property, the triangle inequality, can be verified using the following result, known as the Cauchy– Schwarz Inequality. Theorem A.5 (Cauchy–Schwarz Inequality) For any x, y in an inner product space X, |x, y | ≤ xy where  ·  is defined in (A.7). Theorem A.6 Every inner product defines a norm. Although every inner product defines a norm, not every norm is derived from an inner product. For example, for a vector x in Rn , x∞ = max |xi | 1≤i≤n

is a norm, but there is no corresponding inner product. The following result is now straightforward. Theorem A.7 (Pythogoreas) Let x, y be elements of any inner product space. x + y2 = x2 + y2 if and only if x, y = 0.

Appendix: Functional Analysis and Operators

259

Definition A.8 Elements f, g in an inner product space are orthogonal if  f, g = 0. Definition A.9 A linear space Z with a mapping  ·  : Z → R that satisfies the definition of a norm (A.4) is called a normed linear space. As noted above, every inner product defines a norm, so all inner product spaces are normed linear spaces. The linear space Rn with the norm  · ∞ is an example of a normed linear space. Other examples of normed linear spaces are – real-valued continuous functions defined on a closed bounded set  ⊂ Rn with norm  f  = max | f (x)|, x∈

– L 1 (), the linear space of all functions integrable on  ⊂ Rn , with norm   f 1 =

A.3



| f (x)|d x.

Linear Independence and Bases

Definition A.10 A set of elements z i , i = 1 . . . n in a linear space Z is linearly independent if a1 z 1 + a2 z 2 · · · + an z n = 0 implies a1 = a2 · · · = an = 0. If a set is not linearly independent, then it is linearly dependent. Definition A.11 A linear space Z is finite-dimensional if there is an integer n such that Z contains a linearly independent set of n elements, while any set of n + 1 or more elements is linearly dependent. The number n is the dimension of Z. If dim n is called a basis of Z. Z = n a linearly independent set of n elements {φi }i=1 Theorem A.12 Let {φi } be a basis for an n-dimensional linear space Z. For every z ∈ Z there is a unique set of n scalars {ai } such that z = a 1 φ1 + · · · a n φn . Thus, each element of a finite-dimensional space corresponds to a vector in Rn through its coefficients in some basis. If a space is not finite-dimensional, then by definition it has an infinite set of linearly independent elements. Such spaces are said to be infinite-dimensional. Consider 2 and the set of elements φn where all the coefficients are 0 except the nth.

260

Appendix: Functional Analysis and Operators

This is an infinite set, and no φn can be written as a linear combination of other elements from this set. Thus, 2 is infinite-dimensional. Similarly, considering {φn }∞ n=−∞ 2 defined in Theorem A.2, or alternatively the set of functions {sin(nπx)}∞ n=1 , L (0, 1) is infinite-dimensional. Bases can also be defined for infinite-dimensional spaces. Definition A.13 A set {φα } in a inner product space Z is a basis if z, φα = 0 for all φn implies z = 0, the zero element. Definition A.14 An inner product linear space with a countable basis is said to be separable. Definition A.15 If for a set {φn }, φn , φm = 0 for n = m and φn  = 1, the set is orthonormal. Any set of vectors in Rn can be replaced by a set of orthonormal vectors with the same span using Gram–Schmidt orthogonalization. Similarly, given any countable linearly independent set {wn }∞ n=1 in an inner product space, Gram–Schmidt orthogonalization can be used to construct an orthonormal set with the same span: 1 w1 , w1  φ2 = α2 (w2 − w2 , φ1 φ1 ), α2 chosen so φ2  = 1, .. . n  wn+1 , φk φk ), αn+1 chosen so φn+1  = 1 φn+1 = αn+1 (wn+1 − φ1 =

k=1

.. . Although this is straightforward in theory, this process may have numerical problems in practice, particularly if some of the basis elements are close to collinear; that is wm , wn is not small for m = n. Theorem A.2 states that φn (t) = ej 2πnt = cos(2nπx) + j sin(2nπx), n = 0, ±1, ±2, . . . is a (countable) orthonormal basis for L 2 (0, 1). Any function f ∈ L 2 (0, 1) can be expanded in terms of {φn } as f (t) =

∞ 

f n φi (x)

(A.8)

n=−∞

1 where f n = 0 f (t)φn (t)dt. (See A.2 above.) Here equality is understood to hold in the sense that N  f n φn 2 = 0 lim  f − N →∞

n=−N

Appendix: Functional Analysis and Operators

261

where  · 2 is the usual norm on L 2 (0, 1). The expansion (A.8) generalizes to any countable orthonormal basis for a inner product space in a straightforward way. Let f be an element of a inner product space with inner product ·, · , corresponding norm  ·  and an orthonormal basis {φn }∞ n=1 . Then ∞  f =  f, φn φn n=1

in the sense that lim  f −

N →∞

N 

φn  = 0.

n=−N

By definition, any inner product space with a countable basis is separable. Most inner product spaces of interest in applications are separable. Consider the following examples. – Define the set of elements {φn }∞ n=1 in the space 2 by the nth entry is 1 and all other entries are zero; that is, (φn )n = 1, (φn ) j = 0, j = n. Clearly this set is countable and also an orthonormal basis for 2 and so 2 is separable. 2 – Defining φn (t) = e j2πnt , consider the set {φn }∞ n=−∞ on L (0, 1). Since for any square integrable function f , 

1

f (t)φn (t)dt = 0

0

1 for all n implies 0 | f (t)|2 dt = 0 (Theorem A.2) this set is a basis for L 2 (0, 1) and L 2 (0, 1) is a separable space. – For any domain , L 2 () is separable. The following theorem makes the above discussion of generalizing Fourier series precise. Theorem A.16 Let {φn }∞ n=1 be an orthonormal set in an inner product space Z. The following properties are equivalent. 1. {φn }∞ n=1 is an orthonormal basis for Z. 2. z, φn = 0 for all φn implies that z = 0. 3. For any z ∈ Z, ∞  z, φn φn . z= n=1

(A.9)

262

Appendix: Functional Analysis and Operators

4. For any y, z ∈ Z, y, z =

∞  y, φn z, φn .

(A.10)

n=1

The series (A.9) is sometimes called a generalized Fourier series, or more simply, a Fourier series, for z. The equality (A.10) is Parseval’s Equality. Setting z = y in Parseval’s Equality implies that for any z ∈ Z, z2 =

∞ 

|z, φn |2 .

n=1

A.4

Convergence and Completeness

Consider a sequence {z n } ⊂ Z where Z is a linear space. If Z is the real numbers, this sequence is said to converge to some z 0 if for every  > 0, there is N so that |z n − z 0 | <  for all n > N . That is, by going far enough in the sequence it is possible to get arbitrarily close to z 0 . This idea is extended to vectors in Rn by defining distance between 2 vectors, using the Euclidean norm, or any other norm on Rn . In the previous section, a norm was defined on a general linear space (Definition A.4). This generalizes the concept of “closeness” and so allows limits of sequences in any normed space to be defined. Definition A.17 Let {z n } ⊂ Z where Z is a normed space. The statement lim z n = z 0

n→∞

(A.11)

means that z 0 ∈ Z and for every  > 0 there is N so that z n − z 0  < , for all n > N . By definition, for a convergent sequence, the limit z 0 can be approximated arbitrarily closely by terms in the sequence {z n }, by taking n large enough. Example A.18 On R, the familiar sequence { n1 }∞ n=1 converges to 0.



Example A.19 Define the normed space consisting of continuously differentiable functions defined on [0, 1] along with the norm  z = 2

z(x)2 d x. 0

Consider the sequence

1

Appendix: Functional Analysis and Operators

263

1 sin(nπx). n

z n (x) = Since

z n 2 = z n , z n =

1 , 2n 2

the sequence converges to the zero function 0. Now consider the same linear space, continuously differentiable functions defined on [0, 1], but now create a different normed space using the norm 

1

z1 =

 z(x) d x + 2

0

With this norm, z n 21 =

1

z (x)2 d x.

0

1 π2 + . 2 2n 2

The sequence {z n } does not converge to the zero function 0 in the norm  · 1 .



Definition A.20 A norm  · a on a linear space Z is equivalent to another norm  · b on Z if there are real numbers m > 0, M > 0 such that for all z ∈ Z, mza ≤ zb ≤ Mza . If two norms are equivalent, then convergence in one norm implies convergence in the other and vice versa. Theorem A.21 Every norm on a finite-dimensional space Z is equivalent to any other norm on Z. Thus, in Rn , or any other finite-dimensional normed linear space, if a sequence converges in one norm, then it converges in any other norm. The fact that in a general normed space different norms may not be equivalent is illustrated by Example A.19. Theorem A.12 can be used to show even further, that by mapping a basis in a given finite-dimensional space of dimension n to a basis of vectors for Rn , every finite-dimensional space is essentially the same as Rn . It is useful to have a test for convergence that does not require already knowing the limit L. Definition A.22 A sequence {z k } is Cauchy if for all  > 0 there is N so that for all k > j > N, z k − z j  < . Example A.23 Letting ln indicate the natural logarithm function, consider the sequence xk = ln k, k = 1, 2, . . . . For any k ≥ 1,

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xk+1 − xk = ln(k + 1) − ln k k+1 ). = ln( k Since limk→∞ k+1 = 1 and ln is a continuous function, limk→∞ xk+1 − xk = 0. But k ln k → ∞ as k → ∞ and the sequence diverges. Note that for any k, x2k − xk = ln (2k) − ln k = ln 2. For any  < ln 2 there is no N so that |xk − x j | <  for all k, j > N . The sequence is not Cauchy.  The previous example illustrates that a sequence is not necessarily Cauchy if z k is near z k+1 for large k. A Cauchy sequence has its elements essentially “clumping” as n → ∞. A sequence must be Cauchy in order to converge. Theorem A.24 If a sequence is convergent then it is Cauchy. Let Q indicate the normed linear space of rational numbers with absolute value as the norm. There are many Cauchy sequences in Q that converge to an irrational number. For example, – 3, 3.1, 3.14, 3.141, 3.1415, 3.14159 . . . converges to π; – xk = kj=0 1j! converges to e. Thus, not every Cauchy sequence in Q converges. To get convergence of Cauchy sequences of rational numbers, the “holes” need to be filled, which yields the real numbers R. Real numbers as limits of sequence of rational numbers is the fundamental definition of the real numbers. This gives us a different way to think about R. Any real number can be regarded as the limit of a Cauchy sequence of rational numbers. That is, any real number can be approximated by a rational number to arbitrary accuracy. The number e can be regarded as the limit of any of the following sequences:   1 (1 + )n , n 2., 2.7, 2.71, 2.718, . . . , n 

1/i! .

i=1

Definition A.25 Let (Z,  · ) be a normed space. If every Cauchy sequence converges to an element of Z , the space is complete. A complete normed space is called a Banach space and a complete inner product space is called a Hilbert space. As mentioned above, R is complete: every Cauchy sequence of real numbers converges to a real number. The space of rational numbers is not complete.

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265

Example A.26 Consider the inner product space of continuous functions on [−1, 1] with the inner product  1 y(x)z(x)d x. y, z = −1

Define the sequence of continuous functions ⎧ ⎨0 z n (x) = nx ⎩ 1

−1 ≤ x < 0 0 ≤ x < n1 1 ≤ x ≤ 1. n

Consider any 2 terms in the sequence, z m and z n where n = m + p, p > 0.  z n − z m 2 =

1 n

 (nx − mx)2 d x +

0

1 m 1 n

(1 − mx)2 d x

p2 3m(m + p)2 1 . ≤ 3m

=

Thus, for any  > 0, z m+ p − z m  <  for all m > N where N is chosen so 3N1 2 < . By definition, the sequence is Cauchy. However, {z n } does not converge to any continuous function. It is straightforward to show, defining  z 0 (x) =

−1 ≤ x < 0 , 0≤x ≤1

0 1

that limn→∞ z n − z 0  = 0.



In order to guarantee convergence of Cauchy sequences of continuous functions in the above norm, the linear space needs to be extended to include the limits of Cauchy sequences, just as the space of rational functions is extended to form the real numbers. The notation S for a set S in a normed linear space X indicates the closure of the set in the norm on X . For any closed bounded set  of Rn with piecewise continuously differentiable boundary indicate continuous functions on  by C() and define the inner product  y, z =

y(x)z(x) d x

(A.12)



with induced norm  · . The complete space formed by the limits of all Cauchy sequences of continuous functions is called L 2 (). Any function in L 2 () can be approximated to arbitrary accuracy by a continuous function, with the error measured in the norm (A.12).

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Two functions y, z ∈ L 2 () are regarded as the same function if y − z = 0. For instance, the function that is everywhere 0, the limit of the sequence ⎧ ⎨ 1 + nx z n (x) = 1 − nx ⎩ 0 

and g(x) =

1 0

− n1 ≤ x ≤ 0 0 < x ≤ n1 else x = 0.5 x = 0.5

are all regarded as the zero function in L 2 (0, 1). The space L 2 ()  can also be 1defined to be the set of all measurable functions for which  f  = (  | f (x)|2 d x) 2 < ∞ where the integral here is is the Lebesgue integral. The Lebesgue integral is a generalization of the Riemann integral and is needed because the space of Riemann integrable functions is not complete in the norm  · 2 . Definition A.27 A bounded region  ⊂ Rn is a domain if it is a connected open set and its boundary ∂ can be locally represented by Lipshitz continuous functions. That, is, for any x ∈ ∂, there exists a neighborhood of x, G, such that G ∩ ∂ is the graph of a Lipschitz continuous function. Furthermore,  is locally on one side of ∂. Any polygon in Rn is a domain, and furthermore any bounded connected open set in Rn with a piecewise continuously differentiable boundary is a domain. Definition A.28 For any domain  the Sobolev space Hm () is the set of all functions z defined on  such that for every multi-index α with |α| ≤ m, the mixed partial derivative ∂ |α| z Dα z = ∂x1α1 . . . ∂xnαn is in L 2 (). In other words,   Hm () = z ∈ L 2 () : D α z ∈ L 2 () for all |α| ≤ m . The corresponding inner product is w, z =

  |α|≤m



(D α w)(x)(D α z)(x) d x.

Definition A.29 Let Z be a Hilbert space. The subspace W ⊂ Z is dense in Z, written W = Z, or W → Z, if for every z ∈ Z and  > 0, there is w ∈ W such that w − z < .

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267

The statement that W is dense in Z means that every element of Z can be approximated arbitrarily closely by an element of W. Let  ⊂ Rn be a domain and consider functions defined on the closure of , , that are m-times differentiable, with continuous derivatives. Indicate the space of such functions by C m (). The notation C ∞ () indicates functions for which all derivatives exist and are continuous. Theorem A.30 The spaces C m () and C ∞ () are dense in Hm (). Definition A.31 A set B ⊂ Z where Z is a normed linear space is compact if every sequence in B has a sub-sequence that is convergent in the norm on Z. Theorem A.32 A finite-dimensional set B ⊂ Z is compact if and only if it is closed and bounded. Although it is always true that every compact set is closed and bounded, the converse statement is false for infinite-dimensional sets, as the following example illustrates. Example A.33 Consider the Hilbert space L 2 (0, 1) and the closed and bounded set B = {z ∈ L 2 (0, 1)| z ≤ 1}. This set contains z n (t) = sin(nπt). For any n, m, n = m, since {sin(nπt)} is an orthogonal set on L 2 (0, 1), 

1

z n − z m 2 =

| sin(nπt) − sin(mπt)|2 dt  1  1 = | sin(nπt)|2 dt + | sin(mπt)|2 dt 0

0

0

=1 and so {z n } doesn’t have a convergent subsequence.



Definition A.34 Consider normed linear spaces W ⊂ Z. The space W is compact in Z if the unit ball in W, BW = {w ∈ W| wW ≤ 1} is a compact set in the Z-norm. Equivalently, every set that is bounded in the W-norm contains a subsequence that is convergent in the Z-norm. Theorem A.35 (Rellich’s Theorem) Let  be a domain in R N . For any integer m ≥ 1, Hm () is compact in L 2 (). Also, if N2 < m then Hm () is compact in C().

268

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Appendix: Functional Analysis and Operators

Operators

An operator is a mapping from one linear space to another (possibly the same) linear space. There are many commonly used operators. For instance, – The function x 2 + y 2 is an operator from R2 to R. – Denote the linear space of integrable functions defined on [0, 1] with norm 1 1 0 | f (x)|d x by L (0, 1). Consider integration 

t

(F z)(t) =

z(r )dr

0

as a map from L 1 (0, 1) to L 1 (0, 1). Integration can also be defined as an operator from C([0, 1]) to C 1 ([0, 1]), the space of continuously differentiable functions, and of course between many other normed spaces. Integration can be used to define a wide class of operators. For a function k ∈ C([0, 1] × [0, 1]), define the operator from C([(0, 1)] to itself  (K z)(t) =

1

k(r, t)z(r )dr.

(A.13)

0

(Any operator of the form (A.13) is known as a Volterra operator.) Integration is a special case of (A.13) with  1r ≤t k(r, t) = 0 r > t. Definition A.36 For linear spaces Y and Z consider an operator T : Y → Z. The operator T is linear if, for any y1 , y2 ∈ Y and any scalar α, T (y1 + αy2 ) = T y1 + αT y2 . Setting y2 = 0 in the above definition shows that T (0) = 0 for any linear operator. It is straightforward to verify that any operator of the form (A.13) is a linear operator. Definition A.37 An operator T : Y → Z where Y and Z are normed linear spaces is bounded if there is a constant c such that for all y ∈ Y T y ≤ cy. Example A.38 Define the operator T to be integration from C([0, 1]) to itself with norm on C([0, 1])  f  = max | f (t)|. t∈[0,1]

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269

For any f ∈ C([0, 1]), 

t

T f  = max | f (r )dr | t∈[0,1] 0  t | f (r )|dr ≤ max t∈[0,1] 0

≤ max | f (r )| r ∈[0,1]

=  f . 

Thus, T is a bounded linear operator.

Denote the linear space of real-valued p-times continuously differentiable functions on  ⊂ Rn by C p (). Example A.39 Consider now differentiation as an operator D from C 1 ([0, 1]) to C([0, 1]) with norm  f  = max | f (t)| t∈[0,1]

on both spaces. Like integration, differentiation is a linear operator. Consider the functions sin(nπx) for any positive integer n. For all n,  sin(nπx) = 1 but D sin(nπx) = nπ. There is no constant M so that D f  ≤ M f  for all f ∈ C 1 ([0, 1]). On the other hand, if the norm  f 1 = max | f (t)| + | f (t)| t∈[0,1]

is used to make C 1 ([0, 1]) a normed linear space, then the differentiation operator  D : C 1 [0, 1] → C([0, 1]) is bounded. The idea of continuity for functions from R to R generalizes to operators on normed linear spaces. Definition A.40 Consider an operator T : Y → Z where Y, Z are normed linear spaces. The operator T is continuous at y0 ∈ Y if for any  > 0 there is δ > 0 such that if y − y0 Y < δ then T y − T y0 Z < . The following theorem states that a linear operator is continuous if and only if it is bounded.

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Theorem A.41 Consider any T : Y → Z. If T is a linear operator then the following properties are equivalent: 1. T is a bounded operator. 2. T is continuous at the zero element in 0 ∈ Y. 3. T is continuous at all y ∈ Y. This theorem and Example A.39 imply that differentiation is a discontinuous operator when the usual norms are used. Discontinuity means that small errors in the data can lead to large errors in the calculated derivative. This is reflected in the difficulty of computations that involve derivatives. Define the dual space Y to be the linear space of all bounded linear operators from a Banach space to the scalars. Every element y0 of a Hilbert space Y defines a scalar-valued bounded linear operator through the inner product: T y = y, y0 . The converse statement is also true: any bounded linear operator from a Hilbert space to a scalar value can be uniquely defined using an element of the Hilbert space. It is stated below for a complex inner product; the result for a Hilbert space with a different field of scalars is identical. Theorem A.42 (Riesz Representation Theorem) Let T ∈ Y . There exists a unique y0 ∈ Y so that for all y ∈ Y, T y = y, y0 . Thus, for a Hilbert space we may identify its dual space with itself; this is sometimes written Y = Y. Definition A.43 For a bounded linear operator T : Y → Z where Y and Z are normed linear spaces, the operator norm of T is the smallest M such that for all y∈Y T yZ ≤ MyY and is indicated by T . Equivalently, T  = sup y=0

T yZ = sup T yZ . yY yY =1

(A.14)

The sum of 2 linear operators is linear, and multiplying a linear operator by a scalar creates another linear operator. Definition A.44 The space of all bounded linear operators between two (not necessarily different) normed linear spaces Y and Z is itself a linear space, denoted B(Y, Z). The operator norm is a norm on this linear space. For any bounded linear operators S ∈ B(X , Y), T ∈ B(Y, Z) and any x ∈ X ,

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271

T Sx ≤ T  Sx ≤ T  S x, T S ≤ T  S. Thus, an operator norm has the special property of being sub-multiplicative This is not true of a general norm. Definition A.45 For Y, Z normed linear spaces, consider a sequence {Tn } ⊂ B(Y, Z) and T ∈ B(Y, Z). – If for all z ∈ Z, limn→∞ Tn z − T z = 0 then {Tn } converges strongly to T . – If limn→∞ Tn − T  = 0 then {Tn } converges uniformly or in operator norm to T . Example A.46 For any k ∈ C([0, 1] × [0, 1]), define the Volterra operator T : C([0, 1]) → C([0, 1]) by 

1

(T z)(t) =

k(s, t)z(s)ds. 0

By approximating the kernel k by a simpler function we may obtain an operator that is simpler to work with. For example, suppose k(s, t) = cos(st) and define kn to be the truncated Taylor series: 1 1 (−1)n (st)2n . kn (st) = 1 − (st)2 + (st)4 · · · + 2 4! (2n)! Use kn to define an approximation to T :  (Tn z)(t) =

1



1

kn (st)z(s)ds =

0

0

1 z(s)ds − t 2 2



1

s 2 z(s)ds + · · · .

0

Taylor’s Remainder Theorem can be used to bound Tn z − T z as follows. For any z ∈ C([0, 1]),  1 Tn z − T z = max | [kn (s, t) − k(s, t)]z(s)ds| 0≤t≤1 0  1 ≤ max |kn (s, t) − k(s, t)|dsz 0≤t≤1 0

≤

1 z (2n + 1)!

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by Taylor’s Remainder Theorem. Thus, T − Tn  ≤

1 (2n + 1)!

and limn→∞ T − Tn  = 0. The operator Tn uniformly approximates T . For any desired error , there is N so that all operators Tn with n > N , will provide the uniform error Tn z − T z < z for all functions z ∈ C([0, 1]). This sequence of operators converges uniformly, or in operator norm.  Strong convergence and uniform convergence of a sequence of operators are different. The definition of the operator norm can be used to show that uniform convergence implies strong convergence. But strong convergence does not imply uniform convergence. This is illustrated by this example. Example A.47 (Riemann Sums) Define simple integration T : C([0, 1]) → R, 

1

Tz =

z(x)d x 0

and also the Riemann sums Tn : C([0, 1]) → R, Tm z =

m−1  i=0

1 i z( ). m m

For each z, limm→∞ |Tm z − T z| = 0. By definition of the Riemann integral, the sequence of Riemann sums Tn converges strongly to the integral operator T. Consider for integer m, z m (x) = | sin(mπx)|, the integral of z m is  T zm = =

1

| sin(mπx)|d x

0 m   i=1



=m

i m i−1 m

1 m

sin(mπx)d x 0

= m( =

2 . π

| sin(mπx)|d x

2 ) mπ

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273

But, Tm z m =

m−1  i=0

1 | sin(π)| m

= 0. Thus for any 
.

This illustrates that for some functions z, a large number of points are needed to get a satisfactory error when approximating the integral by a Riemann sum. Thus, the Riemann sums {Tn } do not converge uniformly, although they do converge strongly.  Definition A.48 For any linear operator T mapping a linear space Y to a linear space Z, define the range of T : Range (T ) = {T y| y ∈ Y} ⊂ Z. Definition A.49 An operator T ∈ B(Y, Z) is finite-rank if the range of T , Range (T ), lies in a finite-dimensional subspace of X . Clearly, every operator into Rn is finite-rank, but finite-rank operators can exist in infinite-dimensional spaces. A simple example is B ∈ B(R, L 2 (0, 1)) defined by for some b(x) ∈ L 2 (0, 1), Bu = b(x)u which has one-dimensional range. Definition A.50 Let T ∈ B(X , Y) be a bounded linear operator between Banach spaces X and Y. Indicating the unit ball in X by B, if the closure of T (B) is a compact set in Y, then T is a compact operator. Clearly, every bounded finite-rank operator is a compact operator. There is a close link between more general compact operators and finite-rank operators. Theorem A.51 Let T ∈ B(X , X ) be a bounded linear operator on a Banach space X . If there exists a sequence of finite-rank operators Tn ∈ B(X , X ) so that limn→∞ T − Tn  = 0, then T is a compact operator or compact. If X is a Hilbert space, then T is a compact operator if and only if T is the uniform limit of a sequence of finite rank operators.

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Example A.52 Not every operator on an infinite-dimensional space is a compact operator. For example, consider the identity operator I on a separable infinitedimensional Hilbert space Z such as L 2 (a, b) with orthonormal basis {en }. Let {Tn } be any sequence of finite-rank operators. It is not difficult to find sequences that converge strongly to the identity; for instance Tn z =

n  z, en en . k=1

However, for any finite-rank operator Tn , it is possible to select em that is not in the range of Tn and so Tn em − em  = em  = 1. Thus, no sequence of finite rank operators converges uniformly to I and the identity is not a compact operator.  Example A.53 For any k ∈ C([0, 1] × [0, 1]), define the Volterra operator T : C([0, 1]) → C([0, 1]) by  (T z)(t) =

1

k(s, t)z(s)ds. 0

It was shown in Example A.46 that this class of operators can be uniformly approximated by a sequence of finite-rank operators. Thus, these operators are compact.  Theorem A.54 Let T be a compact operator and Sn a sequence of bounded operators on a Banach space X strongly convergent to a bounded operator S. Then lim T Sn − T S = 0,

n→∞

lim Sn T − ST  = 0.

n→∞

The following definition is a generalization of the complex conjugate transpose of a matrix (or the simple transpose on a real space) to general Hilbert spaces. Definition A.55 Consider a linear operator A : dom(A) ⊂ Z → Y where Z, Y are Hilbert spaces. The domain dom(A∗ ) of the adjoint operator A∗ : Y → Z is the set of all y ∈ Y where exists z˜ ∈ Z so that for all z ∈ dom(A), Az, y Y = z, z˜ Z . Furthermore, A∗ is defined by A∗ y = z˜ so the above expression can be written Az, y Y = z, A∗ y Z . If A is a matrix with real entries, then A∗ is simply the transpose of A; for a matrix with complex entries, A∗ is the complex conjugate transpose. For general operators the adjoint needs to be calculated. For differential operators, the calculation of the adjoint is generally done using “integration-by-parts” or the multi-dimensional generalization.

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275

Example A.56 Define a derivative operator A : dom(A) ⊂ L 2 (−1, 1) → L 2 (−1, 1), as Az =

∂z ∂z , dom(A) = {z ∈ L 2 (−1, 1)| ∈ L 2 (−1, 1), z(1) = 0}. ∂x ∂x

The domain of A can also be written dom(A) = {z ∈ H1 (−1, 1)| z(1) = 0}. Consider any z ∈ dom(A) and y ∈ L 2 (−1, 1). Formally integrating by parts so z is without an operation in the integral,  Az, y = −z(−1)y(−1) −

1

−1

z(x)y (x)d x.

In order for the previous step to be justified, y should be differentiable in some sense. Also, in order for there to exist z˜ ∈ L 2 (−1, 1) so that for all z ∈ dom(A),  z, z˜ =

1

−1

z(x)˜z (x)d x 

= −z(−1)y(−1) +

1

−1

z(x)(−y (x))d x,

also y(−1) = 0. Thus, dom(A∗ ) = {y ∈ L 2 (−1, 1)| or

∂y ∈ L 2 (−1, 1), y(−1) = 0}, ∂x

dom(A∗ ) = {y ∈ H1 (−1, 1)| y(−1) = 0},

and for y ∈ dom A∗ ,

z˜ = A∗ y = −y . 

The following definition generalizes the definition of a symmetric matrix on Rn , or Hermitian matrix on Cn to a general linear space. Definition A.57 If for a linear operator A : dom(A) ⊂ Z → Z, its adjoint operator has dom(A∗ ) = dom(A) and A∗ z = Az for all z ∈ dom(A), the operator A is selfadjoint.

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Example A.58 Consider the operator A : L 2 (−1, 1) → L 2 (−1, 1) defined by Az =

d2z dx2

with domain dom(A) = {z ∈ H2 (−1, 1) ; z(−1) = z(1) = 0}. Consider any z ∈ dom(A) and y ∈ L 2 (−1, 1) smooth enough so that formal integration by parts twice is justified. Integrate by parts twice to obtain Az, y = z (1)y(1) − z (−1)y(−1) +



1

−1

z(x)y

(x)d x.

In order that there be z˜ ∈ L 2 (−1, 1) so Az, y = z, z˜ , for all z ∈ dom(A) , y(1) = y(−1) = 0 and y ∈ H2 (−1, 1). In this case, set z˜ = y

and then z˜ = A∗ y = y

, dom(A∗ ) = {y ∈ H2 (−1, 1) ; y(−1) = y(1) = 0}. Both the definition of the operator and its domain equal that of A and so the operator A is self-adjoint.  Note that if A is not defined on the entire space, the definition of dom(A) and similarly dom(A∗ ) are an important part of the definition of the operator. The following definitions are straightforward extensions of the corresponding concepts in finite dimensions. Definition A.59 Let A : dom(A) ⊂ Z → Z be a self-adjoint linear operator on a Hilbert space Z. If for all non-zero z ∈ dom(A), – – – –

Az, z > 0, A is positive definite, Az, z ≥ 0, A is positive semi-definite. Az, z < 0, A is negative definite, Az, z ≤ 0, A is negative semi-definite.

Appendix: Functional Analysis and Operators

A.6

277

Inverses

Definition A.60 A mapping F : X → Y where X , Y are linear spaces is invertible if there exists a mapping G : Y → X such that G F and F G are the identity mappings on X and Y respectively. The mapping G is said to be an inverse of F and is usually written F −1 . Let Range (F) denote the range of an operator, F : X → Y : Range (F) = {y ∈ Y| there exists x ∈ X ; | F x = y}. Theorem A.61 A mapping F : X → Y where X , Y are linear spaces is invertible if and only if it is one-to-one and Range (F) = Y. What if F : X → Y is one-to-one but not onto Y? In this case F : X → Range (F) is one-one and onto. It has an inverse defined on Y = Range (F). Consider now only linear maps. A linear map L : X → Y where X , Y are linear spaces will have an inverse defined on its range if and only if it is one-to-one. That is, L x1 = L x2 implies that x1 = x2 . Since L is linear, this is equivalent to L x = 0 implies that x = 0. Define the nullspace of L N (L) = {x ∈ X |L x = 0}. Theorem A.62 A linear operator has an inverse defined on its range if and only if N (L) = 0. Theorem A.63 If L : dom(A) ⊂ X → X is positive definite or negative definite then it has an inverse defined on its range. The question of whether an inverse, when it exists, is bounded is important. Example A.64 Consider L : C([0, 1]) → C([0, 1]) defined by  (L x)(t) =

t

x(s)ds. 0

This is clearly a linear operator. With the usual norm on C([0, 1]),  f  = max | f (x)|. 0≤x≤1

|(L x)(t)| ≤ max |x(s)| 0≤s≤1

L x ≤ x and so the operator is bounded. It is also one-to-one:

t 0

x(s)ds = 0 for all t implies

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Appendix: Functional Analysis and Operators

x(t) = 0. Thus this operator has an inverse defined on its range. For y ∈ Range (L), L −1 y(t) = y (t). This operator is defined on Range (L) = C 1 [0, 1], but it is not bounded.



Definition A.65 For an operator L ∈ B(Y, Z) where Y, Z are normed linear spaces, if there exists m > 0 so that L y ≥ my for all y ∈ Y,

(A.15)

then L is coercive. Example A.66 L : C[0, 1] → C[0, 1]  (L x)(t) =

t

x(s)ds. 0

(Example A.64 cont.) Consider xn (t) = sin(nπt). Each xn has xn  = 1, L xn =

−1 cos(nπt) nπ

and with the usual norm on C([0, 1]) L xn  =

1 . nπ

There is no constant m so that (A.15) is satisfied. This operator is one-to-one but not coercive. This is reflected in the fact, shown in Example A.64, that the inverse operator is not bounded.  Example A.67 Define L : 2 → 2 by ⎡ a1 . . . ⎢ 0 a2 . . . ⎢ L = ⎢ 0 0 a3 . . . ⎣ where supk |ak | < ∞. Noting that for any z ∈ 2 [Lz]k = ak z k and defining M = supk |ak |

⎤

..

⎥ ⎥ ⎥ ⎦ .

Appendix: Functional Analysis and Operators

Lz2 =

279 ∞ 

|ak z k |2

k=1 ∞ 

≤ M2

|z k |2

k=1 2

≤ M z . 2

Thus, L is bounded with L ≤ M. If all ak > 0 then for any non-zero z ∈ 2 , Lz, z > 0 and L is positive definite. The nullspace of L contains only the zero element and so L is invertible. Formally, L −1 is defined on Range (L) by [L −1 y]k = or

⎡1 a1

L

−1

1 yk ak ⎤

...

⎢0 1 ⎢ a2 = ⎢0 0 ⎣

... ...

1 a3

..

⎥ ⎥ ⎥. ⎦ .

For y ∈ Range (L), by definition of the range of an operator, there is x ∈ 2 such that yk = ak xk and so

∞ ∞ ∞    1 (L −1 y)2k = | yk |2 = |xk |2 < ∞. a k k=1 k=1 k=1

The inverse operator is a bounded operator if and only if supk | a1k | < ∞, and in this case the inverse can be defined for all y ∈ 2 . In other words, if there is m > 0 so that 1 sup | | = inf |ak | = m > 0 k ak k then L is a coercive operator. Its inverse is a bounded operator defined on all of 2 and 1 . L −1  = inf k |ak | 

280

Appendix: Functional Analysis and Operators

Note that on an infinite-dimensional space, not every positive definite operator is coercive. Every coercive operator has a bounded inverse. Theorem A.68 The operator L : Y → Z where Y, Z are normed linear spaces has a bounded inverse defined on Range (L) if and only if L is a coercive operator. In this case, letting m indicate the constant in Definition A.65, for all y ∈ Range (L), L −1 y ≤

1 y. m

(A.16)

A bounded linear operator defined on a subset of a normed linear space can be extended to the entire space when the domain of definition is dense in the space. Theorem A.69 (Extension Theorem) Consider a linear operator A : dom(A) ⊂ X → Y where dom(A) is a dense linear subspace of a Banach space X and Y is also a Banach space. If A is a bounded operator on dom(A) then there is a unique, bounded extension of A, Ae , to all of X and Ae  = A. Thus, if a bounded linear operator is defined only on a dense subspace of a Banach space, this restriction is artificial. Similarly, unbounded operators are typically only defined on a subspace. For example, differentiation on L 2 (0, 1) is only defined for differentiable functions. Consider T ∈ B(X , X ) where X is a Banach space. In some cases there is a formula for the inverse of I − T. An example of such an operator is  z(t) − 

a

b

k(s, t)z(s)ds = y(t) 

Tz

which can be regarded as z − Tz = y with X = C[a, b]. For real numbers x −1

(1 − x)

=

∞ 

xn

n=0

if |x| < 1. This formula extends to bounded operators on Banach spaces. Theorem A.70 (Neumann Series) Consider T ∈ B(X , X ) where X is a Banach space and T  < 1. Then (I − T )−1 exists and (I − T )−1 =

∞  n=0

T n.

Appendix: Functional Analysis and Operators

281

Also, (I − T )−1  ≤

1 1 − T 

and defining SN =

N 

T n,

n=0

(I − T )−1 − S N  ≤

T  N +1 . 1 − T 

Theorem A.71 Let X be a Banach space and T, S and S −1 be bounded linear operators from X to X . If 1 (A.17) T − S < −1 , S  then T −1 exists, is a bounded operator, and T −1 − S −1  ≤

S −1 2 T − S . 1 − S −1 T − S

The above result implies that for an operator S, if S −1 exists and is bounded, then for all operators T close in operator norm to S, T −1 also exists and is a bounded operator.

A.7

Example: Sampling Theorem

The following application illustrates many of the concepts in this appendix. Most music today is recorded and played digitally. This requires sampling a continuous time function f (t) at a discrete set of time points and saving the samples { f n }. This set of samples is then used to reconstruct the original signal (music) when it is played back. Why is it possible to listen to sound reconstructed from samples without discernible distortion? What are the limitations? In order to provide a precise answer on how fast a signal needs to be sampled, and also to show that it can be reconstructed from samples, it’s useful to use the Fourier transform. Define the operator F : L 2 (−∞, ∞) → L 2 (−∞, ∞) known as the Fourier transform,   z(ω) = [F z](ω) =

∞ −∞

z(t)e−j 2πωt dt.

(A.18)

282

Appendix: Functional Analysis and Operators

This operator has an inverse, defined by 

−1

z](t) = z(t) = [F 

∞

−∞

 z(ω)ej 2πωt dω.

The variable t can be regarded as time (in seconds) while ω as frequency (in Hz). In order to distinguish one space from the other, indicate the space for the time version of the signal by Ht and the transform, or frequency space, by Hω and write F : Ht → Hω . A simple example is  z(t) =

e−t t ≥ 0 0 t 21 , ⎩1 , |ω| = 21 . 2 The inverse Fourier transform of  r is sinc(πt) =

 sin(πt) 1

πt

, t = 0 t = 0.

(A.19)

Since the human ear does not respond to frequencies above about 20,000 Hz and electronic devices also only respond to frequencies within a certain range, consider now functions with Fourier transform that is only non-zero on a finite-interval [−σ, σ]. Such functions are said to be band-limited. This is a useful idealization, but it is an idealization. Although the Fourier transform of continuous integrable functions is small outside of some range, it is not exactly 0. Define the linear subspace of Hω consisting of functions that are only non-zero on the interval [−σ, σ], that is band-limited, by σ = { z ∈ Hω ,  z(ω) = 0, |ω| > σ} M σ and the set of inverse Fourier transforms of functions in M Mσ = {z ∈ Ht ;  z(ω) = 0, |ω| > σ}. The functions n (ω) = √1 exp(j 2πn ω + σ ), n = 0, ±1, ±2, . . . φ 2σ 2σ

Appendix: Functional Analysis and Operators

283

are an orthonormal basis for L 2 (−σ, σ). (See Theorem A.2 with with (0, 1) rescaled to (−σ, σ).) Noting that exp(j πn) is a constant with magnitude 1, and extending n (ω) = 0 for |ω| > σ, leads to the n to all of the real line by setting φ the functions φ basis 1 ω ω  en (ω) = √ exp(j πn ) r ( ), n = 0, ±1, ±2, . . . (A.20) σ 2σ 2σ σ can be written σ . Any  z∈M for M ∞ 

 z=

 z, en  en

(A.21)

n=−∞

where

1  z, en = √ 2σ



σ

−σ

 z(ω) exp(

−j πnω )dω. σ

σ is band-limited the inverse transform of  Since  z∈M z is  σ z(t) =  z(ω) exp(j 2πωt)dω. −σ

Comparing (A.22) and (A.23) implies 1 −n ).  z, en = √ z( 2σ 2σ Thus, from (A.20) and (A.21), and defining  gn (ω) =

 z(ω) = =

∞  n=−∞ ∞  n=−∞

ω ω 1 exp(j πn ) r ( ), 2σ σ 2σ

z(

ω ω −n 1 ) exp(j πn ) r( ) 2σ 2σ σ 2σ

z(

−n ) gn (ω). 2σ

Using (A.19),  −1  gn (t) = sinc (π(n + 2σt)) , F  and defining T =

1 , 2σ

(A.22)

(A.23)

284

Appendix: Functional Analysis and Operators

z(t) =

∞  n=−∞

z(−nT ) sinc(π(n +

t )). T

(A.24)

Equation (A.24) implies that a band limited signal is completely defined by its sam1 ples as long as the samples are taken frequently enough, that is at the rate T = 2σ — twice as fast as the fastest frequency present. This is known as the Nyquist rate. For example, frequencies up to 20,000 Hz (the limit of most people’s hearing) in a sampled signal can be reproduced in theory, if the samples are taken at least 40,000 Hz. Errors in processing mean that a slightly larger rate yields better results.

A.8

Notes and References

Some of the definitions and theorems in this appendix are more restrictive than necessary in order to simplify the exposition and focus on the main points needed. There are many good introductions to functional analysis and operator theory. The books [1, 2] are two that focus on applications. For a treatment that focuses in particular on control systems, see [3]. Theorem A.54 is from [4, Theorem 9.19] which focuses on spectral theory.

References 1. Kreyszig E (1978) Introductory functional analysis with applications. Wiley, New Jersey 2. Naylor AW, Sell GR (1982) Linear operator theory in science and engineering. Springer, New York 3. Banks HT (2012) A functional analysis framework for modeling, estimation and control in science and engineering. CRC Press, Boca Raton 4. Hislop PD, Sigal IM (1996) Introduction to spectral theory with applications to Schrödinger operators, vol 113. Applied mathematical sciences. Springer, New York

Index

Symbols A∗ , 274 C0 -semigroup, 18 G(M, ω), 105 L 2 -gain, 96 M(H∞ ), 96 Pn , 104 H01 (0, 1), 37 Range , 273 Y , 270 α, 254 →, 40 H∞ (B(U , Y )), 96 H2 (X ), 95 j , 254 C+ , 95 S, 25, 265 L 1 (), 259 L 2 (), 265 L 2 (a, b),, 254 L 2 -gain, 95 L 2 -stable, 95 0, 253 H∞ -ARE, 171 H∞ -Riccati, 170 B(Y , Z ), 270 Hm (), 266 V -coercive, 41

A A1, 105 A1∗ , 106 adjoint operator, 274 Alternating Direction Implicit (ADI), 126

analytic semigroup, 86 approximately controllable, 57 approximately controllable on [0, T ], 56 approximately observable, 57 approximately observable on [0, T ], 53 assumed modes, 115 asymptotically stable, 71, 80

B Banach space, 264 band-limited, 282 basis, 259, 260 bounded, 268

C cantilevered, 44 Cauchy–Schwarz Inequality, 258 Cholesky-ADI, 127 classical solution, 19, 46 closed, 22 closed operator, 22 coercive, 103, 198, 278 compact, 81, 267, 273 compact operator, 273 compactly embedded, 81 complete, 264 continuous spectrum, 72 contraction, 31 controllability Gramian, 71, 158 controllability map, 47

© Springer Nature Switzerland AG 2020 K. A. Morris, Controller Design for Distributed Parameter Systems, Communications and Control Engineering, https://doi.org/10.1007/978-3-030-34949-3

285

286 controllable, 56 covariance, 193

D dense, 266 detectable, 97, 177 differential Riccati equation, 104 dimension, 259 Dirichlet trace operator, 93 dissipative, 220 Distributed Parameter Systems, 1, 13 domain, 266 dual space, 270

E eigenfunction, 4, 26 eigenvalue, 4, 26, 72, 177 evolution operator, 104 exactly controllable on [0, T ], 56 exactly observable on [0, T ], 53 expectation, 192 exponentially stable, 71, 81 externally, 223 externally stable, 95, 100, 220

F Final Value Theorem, 224 finite-dimensional , 259 finite-rank, 273 fixed attenuation H∞ control problem, 170 G gap, 227, 228, 230 graph, 22, 226 growth bound, 84

H Hilbert, 264 Hilbert-Schmidt operator, 157 Hille–Yosida Theorem, 25 Hurwitz, 84, 177

I infinitesimal generator, 19 inner product, 257 inner product space, 257 input passive, 220 Internal Model Principle, 225

Index internally stable, 96 invariant zeros of (A, B, C, D), 64 invertible, 277

J jointly stabilizable and detectable, 97

K Kalman filter, 192, 194, 195

L Laplace transform, 60, 95 Laplace transformable, 60 limit, 262 linear, 268 linear space, 253 linearly dependent, 259 linearly independent, 259 Lyapunov equation, 71

M matrix pair, 177 mild solution, 19, 46 minimal, 65 mixed finite element method, 121 modal truncation, 114

N negative definite, 276 negative semi-definite, 276 Neumann Series, 280 norm, 258 normed linear space, 259 nuclear norm, 137 nullspace, 277

O observability Gramian, 158 observability map, 50 observable, 53 operator norm, 270, 271 optimal H∞ -disturbance attenuation, 174 optimizable, 107 orthogonal, 259 orthonormal, 260 output passive, 220

Index P Paley–Wiener Theorem, 95 Parseval’s Equality, 262 passive, 220 Passivity Theorem, 221 Poincaré Inequality, 41 point spectrum, 72 positive definite, 276 positive real, 221 positive semi-definite, 276

R range, 277 reachable subspace, 57 Rellich’s Theorem, 267 residual spectrum, 72 resolvent set, 26, 72 Riesz basis, 26 Riesz-spectral, 27, 59

S Schur Method, 124 second-order (Vo , Ho )-system, 42 self-adjoint, 275 separable, 260 sesquilinear form, 39 single-input-single-output, 221 singular values, 96 Small Gain Theorem, 221 Sobolev space, 15, 266 spectral radius, 242 spectrum, 26, 72 spectrum decomposition assumption, 97 Spectrum Determined Growth Assumption (SDGA), 84 stabilizable, 97, 177 stabilizable with attenuation γ, 170 stabilizable/detectable, 97

287 stable transfer function, 96 state space, 18 storage function, 220 strong solution, 19, 46 strongly, 271 strongly continuous semigroup of operators, 18 strongly stable, 72 supply rate, 220

T trace, 137 trace class, 137, 158 trace norm, 137 tracking error, 224 transfer function, 60 transmission zero of (A, B, C, D), 64 U uniformly, 271 uniformly detectable, 112 uniformly stabilizable, 112 unobservable subspace, 57

V vector spaces, 253 Volterra, 268 Volterra operator, 271, 274

W well-posed, 19 Wiener, 233

Z zero element, 253