A course in robust control theory 3

15 1 Preliminaries in Finite Dimensional Space 18 1.1 Linear spaces and mappings.. 347 11.2.5 Periodic systems and nite dimensional conditions 349 A Some Basic Measure Theory 352 A.1 Set

F ernando G Paganini

University of California Los Angeles

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Contents

0.1 System representations 2

0.1.1 Block diagrams 2

0.1.2 Nonlinear equations and linear decompositions 4

0.2 Robust control problems and uncertainty 9

0.2.1 Stabilization 9

0.2.2 Disturbances and commands 12

0.2.3 Unmodeled dynamics 15

1 Preliminaries in Finite Dimensional Space 18 1.1 Linear spaces and mappings 18

1.1.1 Vector spaces 19

1.1.2 Subspaces 21

1.1.3 Bases, spans, and linear independence 22

1.1.4 Mappings and matrix representations 24

1.1.5 Change of basis and invariance 28

1.2 Subsets and Convexity 30

1.2.1 Some basic topology 31

1.2.2 Convex sets 32

1.3 Matrix Theory 38

1.3.1 Eigenvalues and Jordan form 39

1.3.2 Self-adjoint, unitary and positive de nite matrices 41 1.3.3 Singular value decomposition 45

1.4 Linear Matrix Inequalities 47

1.5 Exercises 53

2 State Space System Theory 57 2.1 The autonomous system 58

2.2 Controllability 61

2.2.1 Reachability 61

2.2.2 Properties of controllability 66

2.2.3 Stabilizability and the PBH test 69

2.2.4 Controllability from a single input 72

2.3 Eigenvalue assignment 74

2.3.1 Single input case 74

2.3.2 Multi input case 75

2.4 Observability 77

2.4.1 The unobservable subspace 78

2.4.2 Observers 81

2.4.3 Observer-Based Controllers 83

2.5 Minimal realizations 84

2.6 Transfer functions and state space 87

2.6.1 Real-rational matrices and state space realizations 89 2.6.2 Minimality 92

2.7 Exercises 93

3 Linear Analysis 97 3.1 Normed and inner product spaces 98

3.1.1 Complete spaces 101

3.2 Operators 103

3.2.1 Banach algebras 107

3.2.2 Some elements of spectral theory 110

3.3 Frequency domain spaces: signals 113

3.3.1 The space ^L2and the Fourier transform 113

3.3.2 The spacesH2andH? 2 and the Laplace transform 115 3.3.3 Summarizing the big picture 119

3.4 Frequency domain spaces: operators 120

3.4.1 Time invariance and multiplication operators 121

3.4.2 Causality with time invariance 122

3.4.3 Causality andH1 124

3.5 Exercises 127

4 Model realizations and reduction 131 4.1 Lyapunov equations and inequalities 131

4.2 Observability operator and gramian 134

4.3 Controllability operator and gramian 137

4.4 Balanced realizations 140

4.5 Hankel operators 143

4.6 Model reduction 147

Contents iii

4.6.1 Limitations 148

4.6.2 Balanced truncation 151

4.6.3 Inner transfer functions 154

4.6.4 Bound for the balanced truncation error 155

4.7 Generalized gramians and truncations 160

4.8 Exercises 162

5 Stabilizing Controllers 167 5.1 System Stability 169

5.2 Stabilization 172

5.2.1 Static state feedback stabilization via LMIs 173

5.2.2 An LMI characterization of the stabilization prob-lem 174

5.3 Parametrization of stabilizing controllers 175

5.3.1 Coprime factorization 176

5.3.2 Controller Parametrization 179

5.3.3 Closed-loop maps for the general system 183

5.4 Exercises 184

6 H2 Optimal Control 188 6.1 Motivation forH2 control 190

6.2 Riccati equation and Hamiltonian matrix 192

6.3 Synthesis 196

6.4 State feedbackH2 synthesis via LMIs 202

6.5 Exercises 205

7 H1 Synthesis 208 7.1 Two important matrix inequalities 209

7.1.1 The KYP Lemma 212

7.2 Synthesis 215

7.3 Controller reconstruction 222

7.4 Exercises 222

8 Uncertain Systems 227 8.1 Uncertainty modeling and well-connectedness 229

8.2 Arbitrary block-structured uncertainty 234

8.2.1 A scaled small-gain test and its suciency 236

8.2.2 Necessity of the scaled small-gain test 239

8.3 The Structured Singular Value 245

8.4 Time invariant uncertainty 248

8.4.1 Analysis of time invariant uncertainty 249

8.4.2 The matrix structured singular value and its upper bound 257

8.5 Exercises 262

9 Feedback Control of Uncertain Systems 270

9.1 Stability of feedback loops 273

9.1.1 L2-extended and stability guarantees 274

9.1.2 Causality and maps onL2-extended 277

9.2 Robust stability and performance 280

9.2.1 Robust stability under arbitrary structured uncer-tainty 281

9.2.2 Robust stability under LTI uncertainty 281

9.2.3 Robust Performance Analysis 282

9.3 Robust Controller Synthesis 284

9.3.1 Robust synthesis against
a;c 285

9.3.2 Robust synthesis against
TI 289

9.3.3 D-K iteration: a synthesis heuristic 293

9.4 Exercises 295

10 Further Topics: Analysis 298 10.1 Analysis via Integral Quadratic Constraints 298

10.1.1 Analysis results 303

10.1.2 The search for an appropriate IQC 308

10.2 RobustH2 Performance Analysis 310

10.2.1 Frequency domain methods and their interpretation 311 10.2.2 State-Space Bounds Involving Causality 316

10.2.3 Comparisons 320

10.2.4 Conclusion 321

11 Further Topics: Synthesis 323 11.1 Linear parameter varying and multidimensional systems 324 11.1.1 LPV synthesis 327

11.1.2 Realization theory for multidimensional systems 333 11.2 A Framework for Time Varying Systems: Synthesis and Analysis 337

11.2.1 Block-diagonal operators 338

11.2.2 The system function 340

11.2.3 Evaluating the`2 induced norm 344

11.2.4 LTV synthesis 347

11.2.5 Periodic systems and nite dimensional conditions 349 A Some Basic Measure Theory 352 A.1 Sets of zero measure 352

A.2 Terminology 355

A.3 Comments on norms andLp spaces 357

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Introduction

In this course we will explore and study a mathematical approach aimeddirectly at dealing with complex physical systems that are coupled in feed-back The general methodology we study has analytical applications toboth human-engineered systems and systems that arise in nature, and thecontext of our course will be its use for feedback control

The direction we will take is based on two related observations aboutmodels for complex physical systems The rst is that analytical or com-putational models which closely describe physical systems are dicult orimpossible to precisely characterize and simulate The second is that amodel, no matter how detailed, is never a completely accurate represen-tation of a real physical system The rst observation means that we areforced to use simpli ed system models for reasons of tractability; the lat-ter simply states that models are innately inaccurate In this course bothaspects will be termed system uncertainty, and our main objective is todevelop systematic techniques and tools for the design and analysis of sys-tems which are uncertain The predominant idea that is used to contendwith such uncertainty or unpredictability is feedback compensation

There are several ways in which systems can be uncertain, and in thiscourse we will target the main three:

 The initial conditions of a system may not be accurately speci ed orcompletely known

 Systems experience disturbances from their environment, and systemcommands are typically not known a priori

 Uncertainty in the accuracy of a system model itself is a centralsource Any dynamical model of a system will neglect some physi-cal phenomena, and this means that any analytical control approachbased solely on this model will neglect some regimes of operation.

In short: the major objective of feedback control is to minimize the eectssubject to the constraint of not having a complete representation of the sys-tem This is a formidable challenge in that predictable behavior is expectedfrom a controlled system, and yet the strategies used to achieve this must

do so using an inexact system model The term robust in the title of thiscourse refers to the fact that the methods we pursue will be expected tooperate in an uncertain environment with respect to the system dynamics

The mathematical tools and models we use will be primarily linear, vated mainly by the requirement of computability of our methods; howeverthe theory we develop is directly aimed at the control of complex nonlinearsystems In this introductory chapter we will devote some space to discuss,

moti-at an informal level, the interplay between linear and nonlinear aspects inthis approach

The purpose of this chapter is to provide some context and motivationfor the mathematical work and problems we will encounter in the course

For this reason we do not provide many technical details here, however itmight be informative to refer back to this chapter periodically during thecourse

In this section we introduce the notion of a block diagram for representingsystems, and most importantly for specifying their interconnections

We use the symbolP to denote a system that maps an input function

u(t) to an output function y(t) This relationship is denoted by

y=P(u):

Figure 1 illustrates this relationship The direction of the arrows indicatewhether a function is an input or an output of the systemP The details

0.1 System representations 3

y

Figure 1 Basic block diagram

of howP constructs y from the input u is not depicted in the diagram,instead the bene t of using such block diagrams is that interconnections ofsystems can be readily visualized

Consider the so-called cascade interconnection of the two subsystems

This interconnection represents the equations

ofP1

P

Q

wz

y

u

Another type of interconnection involves feedback In the gure above

we have such an arrangement HereP has inputs given by the ordered pair(w; u) and the outputs (z; y) The system Q has input y and output u.This block diagram therefore pictorially represents the equations

(z; y) =P(w; u)

y=Q(y):

Since part of the output ofP is an input to Q, and conversely the output

ofQis an input toP, these systems are coupled in feedback

We will now move on to discussing the basic modeling concept of thiscourse and in doing so will immediately make use of block diagrams.

0.1.2 Nonlinear equations and linear decompositions

We have just introduced the idea of representing a system as an output mapping, and did not concern ourselves with how such a mappingmight be de ned We will now outline the main idea behind the modelingframework used in this course, which is to represent a complex system as

input-a combininput-ation of input-a perturbinput-ation input-and input-a simpler system We will illustrinput-atethis by studying two important cases

Isolating nonlinearitiesThe ... spectral theory 110

3. 3 Frequency domain spaces: signals 1 13

3. 3.1 The space ^L2and the Fourier transform 1 13

3. 3.2 The spacesH2andH?... study a mathematical approach aimeddirectly at dealing with complex physical systems that are coupled in feed-back The general methodology we study has analytical applications toboth human-engineered... decomposition of a system into a linear partand a static nonlinearity The motivation for this is so that later we canreplace the nonlinearity using objects more amenable to analysis

To start consider