A Course in Robust Control Theory - A Convex Approach

398 Pages·2005·2.012 MB·English

by  Dullerud G.E., Paganini F.

A Course in Robust
Control Theory
a convex approach
Geir E. Dullerud Fernando G. Paganini
University of Illinois University of California
Urbana-Champaign Los Angeles
Contents
0 Introduction 1
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(cid:12)nite matrices 41
1.3.3 Singular value decomposition . . . . . . . . . . . 45
1.4 Linear Matrix Inequalities . . . . . . . . . . . . . . . . . 47
ii Contents
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 L^2 and the Fourier transform . . . . . 113
3.3.2 The spaces H2 and H2? 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 and H . . . . . . . . . . . . . . . . . 124
1
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 gramiansand 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 Parametrizationof 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 for H2 control . . . . . . . . . . . . . . . . . 190
6.2 Riccati equation and Hamiltonian matrix . . . . . . . . . 192
6.3 Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
6.4 State feedback H2 synthesis via LMIs . . . . . . . . . . . 202
6.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
7 H Synthesis 208
1
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 su(cid:14)ciency . . . . 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
iv Contents
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 on L2-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 PerformanceAnalysis . . . . . . . . . . . 282
9.3 Robust Controller Synthesis . . . . . . . . . . . . . . . . 284
9.3.1 Robust synthesis against (cid:1)a;c . . . . . . . . . . 285
9.3.2 Robust synthesis against (cid:1)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 appropriateIQC . . . . . . . . 308
10.2 Robust H2 Performance Analysis . . . . . . . . . . . . . 310
10.2.1 Frequencydomainmethodsandtheirinterpretation 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-diagonaloperators . . . . . . . . . . . . . 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 (cid:12)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 and Lp spaces . . . . . . . . . . . . 357
B Proofs of Strict Separation 359
C (cid:22)-Simple Structures 365
Contents v
C.1 The case of (cid:1)1;1 . . . . . . . . . . . . . . . . . . . . . . 366
C.2 The case of (cid:1)0;3 . . . . . . . . . . . . . . . . . . . . . . 370
References 375
Preface
Research in robust control theory has been one of the most active areas of
mainstream systems theory since the late 70s. This research activity has
been at the con(cid:13)uence of dynamical systems theory, functional analysis,
matrixanalysis,numericalmethods,complexitytheory,andengineeringap-
plications.Thedisciplinehasinvolvedinteractionsbetweendiverseresearch
groups including pure mathematicians, applied mathematicians, computer
scientists and engineers, and during its development there has been a sur-
prisingly close connection between pure theory and tangible engineering
application. By now this research e(cid:11)ort has produced a rather extensive
set of approaches using a wide variety of mathematical techniques, and
applications of robust control theory are spreading to areas as diverse as
control of (cid:13)uids, power networks, and the investigation of feedback mech-
anisms in biology. During the 90s the theory has seen major advances and
achievedanewmaturity,centeredaroundthenotionofconvexity.Thisem-
phasisis two-fold.On onehand, the methods of convexprogramminghave
been introduced to the (cid:12)eld and released a wave of computational meth-
ods which, interestingly, have impact beyond the study of control theory.
Simultaneously a new understanding has developed on the computational
complexity implications of uncertainty modeling; in particular it has be-
comeclearthatonemustgobeyondthetimeinvariantstructuretodescribe
uncertainty in terms amenable to convex robustness analysis.
Our broad goal in this book is to give a graduate-level course on ro-
bust control theory that emphasizes these new developments, but at the
same time conveys the main principles and ubiquitous tools at the heart
of the subject. This courseis intended asan introduction to robustcontrol
Contents vii
theory, and begins at the level of basic systems theory, but ends having
introduced the issues and machinery of current active research. Thus the
pedagogicalobjectivesofthebookare(1)tointroduceacoherentanduni-
(cid:12)ed framework for studying robust control theory; (2) to provide students
with the control-theoretic background required to read and contribute to
the research literature; (3) the presentation of the main ideas and demon-
strations of the major results of robust control theory. We therefore hope
the book will be of value to mathematical researchers and computer sci-
entists wishing to learn about robust control theory, graduate students
planningtodoresearchinthearea,andengineeringpractitionersrequiring
advancedcontroltechniques.Thebookismeanttofeatureconvexmethods
and the viewpoint gained from a general operator theory setting, however
rather than be purist we have endeavored to give a balanced course which
(cid:12)ts these themes in with the established landscape of robust control the-
ory. The e(cid:11)ect of this intention on the book is that as it progresses these
themesareincreasinglyemphasized,whereasmoreconventionaltechniques
appear less frequently. The current research literature in robust control
theory is vast and so we have not attempted to cover all topics, but have
insteadselectedthosethatwebelievearecentralandmost e(cid:11)ectivelyform
a launching point for further study of the (cid:12)eld.
The text is written to comprise a two-quarter or two-semester gradu-
ate course in applied mathematics or engineering. The material presented
has been successfully taught in this capacity during the past few years by
the authors at Caltech, University of Waterloo, University of Illinois, and
UCLA. For students with background in state space methods a serious
approachat asubset of the materialcan be achievedin one semester.Stu-
dents are assumed to have familiarity with linear algebra, and otherwise
only advanced calculus and basic complex analysis are strictly required.
After an introduction and a preliminary technical chapter, the course
begins with athoroughintroduction tostate space systems theory.It then
moves on to cover open-loop systems issues using the newly introduced
concept of a norm. Following this the simplest closed-loop synthesis issue
isaddressed,thatofstabilization.Thentherearetwochaptersonsynthesis
whichcovertheH2andH formulations.Nextopen-loopuncertainsystem
1
models are introduced; this chapter gives a comprehensive treatment of
structured uncertainty using perturbations that are either time invariant
orarbitrary.Theresultsonopen-loopuncertainsystemsarethenappliedto
feedback control in the following chapter where both closed-loop analysis
and synthesis are addressed. The (cid:12)nal two chapters are devoted to the
presentation of four advanced topics in a more descriptive manner. In the
preliminarychapter of the book some basic ideas from convex analysisare
presented as is the important concept of a linear matrix inequality (LMI).
Linear matrix inequalities are perhaps the major analytical tool used in
thistext,andcombinedwiththeoperatortheoryframeworkpresentedlater
viii Contents
provide a powerful perspective. A more detailed summary of the chapters
is given below.
Chapter 1 Preliminaries in Finite Dimensional Space
Elementary linear algebra is (cid:12)rst reviewed, and a short summary of basic
concepts from convex analysis are provided. A selection of matrix theory
topicsispresentedincludingJordanformandsingularvaluedecomposition.
The chapter ends with a section on linear matrix inequalities.
Chapter 2 State Space Systems Theory
Thischapterintroducesthebasicstatespacemodel.Controllability,reach-
abilityandstabilizabilityarethencoveredinconjunctionwithvarioustests
for these properties. Eigenvalue assignment is discussed in full for multi-
inputsystems.Theconceptofobservabilityisthenpresentedtogetherwith
an introduction to observers and observerbased controllers. Minimal real-
izations are treated, and the connection between state space realizations
and transfer functions is made.
Chapter 3 Linear Analysis
The major objective of this chapter is to introduce the operator theory
needed in the sequel. It begins by introducing normed and inner product
spaces, and then the operators on Hilbert space. The focus of the text is
systems with L2 inputs, and so a number of related function spaces are
presented, including the H2 and H spaces. The various connections of
1
these spaces with time invariance and causality are introduced.
Chapter 4 Model Realization and Reduction
The chapter initiates the quantitative study of systems using norms as
the system measure. The open-loop characteristics of systems are exam-
ined using the controllabilty and observability gramians, and a geometric
motivation is given for balanced realizations. Hankel operators and singu-
lar values are then discussed, followed by model reduction using balanced
truncation.
Chapter 5 Stabilizing Controllers
The concepts of closed-loop well-posedness and stability are de(cid:12)ned. An
LMI solution to (cid:12)nding a stabilizing controller is stated. Following this
thequestionof parametrizingall stabilizingcontrollersispursued,and the
important idea of a coprime factorization is brought in. With this tool a
complete controller parametrization is given.