This book is an outgrowth of my longstanding interest in robust
control problems involving structured real parametric uncertainty.
At the risk of beginning on a controversial note, I believe that it
is fair to say that in the robust control field, most research is currently
concentrated in five areas. The popular labels for these areas
are H00 , μ, Kharitonov, Lyapunov and QFT. Some colleagues in the
field would insist on including L1 as a sixth area. In terms of the
five labels above, the takeoff point for this book is what I believe
to be one of the major milestones in the literature relevant to control
theory-a 1978 paper in a differential equations journal by the
Russian mathematician V. L. Kharitonov; see Kharitonov {1978a).
Kharitonov's paper began to receive attention in the control field
in 1983 and provided strong motivation for a decade of furious work
by researchers interested in robustness of systems with real parametric
uncertainty. I use the word "furious" above because at times, the
race for results got rather heated. On numerous occasions, the same
result appeared nearly simultaneously in two journals-by different
authors, of course. Given this explosive rate of publication, much
"smoke" has emerged. The uninitiated reader who wants to become
familiar with the new developments faces an enormous pile of papers
and may not know which ones to read first. My choice of material
for this text implicitly provides my perspective on this matter. One
of my main objectives is distillation-taking this large body of new
literature, picking out the most important results and simplifying
their explanation so as to minimize time investment associated with
learning the new techniques. In this regard, many of the proofs are
new and given here for the first time.
At the outset, the reader should be aware that the robust control
literature does not contain many results "linking" the different areas
of research. I am hoping, however, that my exposition will motivate
others to undertake efforts aimed at unification of the field; this
book is not the "grand unifier." My point of view is as follows: The
serious student of robust control might reasonably be expected to
take three or four courses in the area. In this sense, my hope is that
this book would be a strong competitor for being the text in one of
these courses.
After weighing the trade-offs between encyclopedic coverage and
pedagogy, I resisted the temptation to let the scope get too broad. I
opted to concentrate on trying to write a text which is "technically
tight" and yet does not require too high a level of technical sophistication
to read. My targeted reader is the beginning graduate student
who is familiar with just the basics such as Bode, Nyquist, root locus
and elementary state space analysis. For a one-semester course
of 13-15 weeks, I would recommend Chapters 1-11 and 14-16. An
ambitious instructor might also include selected results from Chapters
12, 13 and 17.
In many places throughout the text, I refer to the value set.
Once this rather simple concept is understood, it becomes possible
to unify most of the new technical developments emanating from
Kharitonov's Theorem. While we may have the illusion that we
have been bombarded with dozens of new "lines of proof" over the
last decade, the truth of the matter is that most of the seemingly
disparate new results can be easily understood with the help of one
simple idea-the value set. Granted, I am overstating my case a bit
here, but in spirit, I feel that my contention is correct.
At this point, I must note that I have avoided calling the value
set concept "new." Value sets arise in many fields, for example,
mathematics, economics and optimization. In fact, even in the control
literature, value sets appear as early as 1963 in the textbooks
of Horowitz and Zadeh and Desoer. What is new in this book is
the way the value set is used to unify a large body of literature on
robustness of control systems. In fact, one of the greatest challenges
in writing this book was taking existing results from the literature
and finding new ways to explain them using the value set.
To provide my personal perspective on how this research area
came into being, let me begin by noting that Kharitonov's Theorem
first came to my attention in 1982 at a workshop in Switzerland
organized by Jiiergen Ackermann. At that time, I remember sitting
next to Manfred Morari and listening to Andrej Olbrot exploit
Kharitonov's Theorem to prove a result on delay systems. Given
that Kharitonov's Theorem was published in 1978, my immediate
reaction to Olbrot's presentation was one of bewilderment. Despite
the fact that it was published in a Russian differential equations
journal, I could not understand how such an important result had
been unheralded in the control community for more than four years.
Immediately following the workshop in Switzerland, there was a period
of about six months which I spent working with Kris Hollot
and Ian Petersen expending considerable effort trying to decide if
Kharitonov's cryptic proof was correct. It was.
Apparently, Olbrot was aware of the importance of Kharitonov's
Theorem at least one year before the workshop in Switzerland. In
a 1981 letter from Olbrot to Ackermann (following a workshop in
Bielefeld), the theorem was stated precisely. In his letter, Olbrot also
recognized that this result had possible applications to "insensitive
stabilization."
Kharitonov's name finally surfaced in the control journals in 1983
in Bialas (1983) and Barmish (1983). While my paper is frequently
cited for exposing the power of Kharitonov's Theorem in the "western
literature,'' the paper by Bialas had an equally important role.
Although Bialas' attempt to generalize from polynomials to matrices
turned out to be incorrect (for example, see Karl, Greschak and
Verghese (1984) for a counterexample), his paper served to stimulate
researchers to address the following question: To what extent
can Kharitonov's strong assumptions on the uncertainty structure
be relaxed? An important breakthrough in this direction was the
Edge Theorem of Bartlett, Hollot and Huang (1988). This added
fuel to the fire just as the flames were beginning to subside.
Author(s): B. Ross Barmish
Publisher: Macmillan Coll Div
Year: 1993
Language: English
Pages: 410
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Contents
Part I. Preliminaries 1
Chapter 1. A Global Overview 2
1.1 Introduction . . . . . . . 2
1.2 Robustness Problems . . 3
1.3 Some Historical Perspective . 5
1.4 Refinement of the Scope. . . 5
1.5 Kharitonov's Theorem: The Spark . 7
1.6 The Issue of Uncertainty Structure . 8
1. 7 The Mathematical Programming Approach 9
1.8 The Toolbox Philosophy . . . . . . . . . . 10
1.9 The Value Set Concept . . . . . . . . . . . 11
1.10 Mathematical Model Versus the True System . 12
1.11 Family Paradigm. . . . . . . . 13
1.12 Robustness Analysis Paradigm 13
1.13 Robustness Margin Paradigm. 14
1.14 Robust Synthesis Paradigm . . 15
1.15 Testing Sets and Computational Complexity 15
1.16 Conclusion . . . . . . . . . . . . . . . 16
Chapter 2. Notation for Uncertain Systems 19
2.1 Introduction . . . . . . . . . . . . . 19
2.2 Notation for Uncertain Parameters. . 19
2.3 Subscripts and Superscripts . . . . . . 21
2.4 Uncertainty Bounding Sets and Norms 21
2.5 Notation for Families . . . . . . . . . .
2.6 Uncertain Functions Versus Families . . . . . . 22
2. 7 Convention: Real Versus Complex Coefficients 23
2.8 Consolidation of Notation . . . . . . . . . . 23
2.9 Conclusion . . . . . . . . . . . . . . . . . . 27
Chapter 3. ·case Study: The Fiat Dedra Engine 29
3.1 Introduction . . . . . . . . . . . . 29
3.2 Control of the Fiat Dedra Engine . . . . 30
3.3 Discussion of the Engine Model . . . . . 30
3.4 Transfer Function Matrix for the Engine 33
3.5 Uncertain Parameters in the Engine Model 33
3.6 Discussion of the Controller Model . . . . . 34
3. 7 The Closed Loop Polynomial . . . . . . . . 36
3.8 Is Symbolic Computation Really Needed? . 37
3.9 Symbolic Computation with Transfer Functions 38
3.10 Conclusion . . . . . . . . . . . . . . . . . . . . . 39
Part II. From Robust Stability to the Value Set 41
Chapter 4. Robust Stability with a Single Parameter 42
4.1 Stability and Robust Stability . 42
4.2 Basic Definitions and Examples 43
4.3 Root Locus Analysis. . . . . . 44
4.4 Generalization of Root Locus . 45
4.5 Nyquist Analysis . . . . . . . . 47
4.6 The Invariant Degree Concept 47
4. 7 Eigenvalue Criteria for Robustness . 50
4.8 Machinery for the Proof . . . . . . . 52
4.9 Proof of the Theorem . . . . . . . . 54
4.10 Convex Combinations and Directions 54
4.11 The Theorem of Bialas . . . . . . . 55
4.12 The Matrix Case . . . . . . . . . . . 56
4.13 Introduction to Robust V-Stability 60
4.14 Robust V-Stability Generalizations 62
4.15 Extreme Point Results . . . . . . . 63
4.16 Conclusion . . . . . . . . . . . . . . 64
Chapter 5. The Spark: Kharitonov's Theorem 65
5.1 Introduction . . . . . . . . . . . . . 65
5.2 Independent Uncertainty Structures 65
5.3 Interval Polynomial Family 67
5.4 Shorthand Notation . . . . .
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
The Kharitonov Polynomials
Kharitonov's Theorem ....
Machinery for the Proof . . .
Proof of Kharitonov's Theorem.
Formula for the Robustness Margin
Robust Stability Testing via Graphics
Overbounding via Interval Polynomials
Conclusion ............... .
68
69
70
77
79
79
81
83
Chapter 6. Embellishments of Kharitonov's Theorem 86
6.1 Introduction . . . . . . . . . . . . 86
6.2 Low Order Interval Polynomials . . . 87
6.3 Extensions with Degree Dropping . . 88
6.4 Interval Plants with Unity Feedback . 89
6.5 Frequency Sweeping Function H(w) 91
6.6 Robustness Margin Geometry . . . . 94
6.7 The Tsypkin-Polyak Function . . . . 96
6.8 Complex Coefficients and Transformations . 102
6.9 Kharitonov's Theorem with Complex Coefficients . 104
6.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . 107
Chapter 7. The Value Set Concept 109
7.1 Introduction . . . . . . . . . . 109
7.2 The Value Set . . . . . . . . . 110
7.3 The Zero Exclusion Condition . 112
7.4 Zero Exclusion Condition for Robust V-Stability . . 114
7.5 Boundary Sweeping Functions . 116
7.6 More General Value Sets . 118
7.7 Conclusion . . . . . . . . . . . . 121
Part III. The Polyhedral Theory
Chapter 8. Polytopes of Polynomials
8.1 Introduction ......... .
8.2 Affine Linear Uncertainty Structures .
8.3 A Primer on Polytopes and Polygons
8.4 Introduction to Polytopes of Polynomials
8.5 Generators for a Polytope of Polynomials
8.6 Polygonal Value Sets .......... .
8. 7 Value Set for a Polytope of Polynomials .
8.8 Improvement over Rectangular Bounds
8.9 Conclusion ................
Chapter 9. The Edge Theorem 149
9.1 Introduction . . . . . . . . . . . . . . . . . . . 149
9.2 Lack of Extreme Point Results for Polytopes . 150
9.3 Heuristics Underlying the Edge Theorem . 150
9.4 The Edge Theorem . . . . . . . . . . 152
9.5 Fiat Dedra Engine Revisited . . . . . 154
9.6 Root Version of the Edge Theorem. . 160
9.7 Conclusion . . . . . . . . . . . . . . . 162
Chapter 10. Distinguished Edges 164
10.1 Introduction . . 164
10.2 Parallelotopes . . . . . . . . 165
10.3 Parpolygons . . . . . . . . . 167
10.4 Setup with an Interval Plant . 167
10.5 The Thirty-Two Edge Theorem . 168
10.6 Octagonality of the Value Set . . . 169
10. 7 Proof of Thirty-Two Edge Theorem . 175
10.8 Conclusion . . . . . . . . . . . . . . 176
Chapter 11. The Sixteen Plant Theorem 1 78
11.1 Introduction . . . . . . . . . . 178
11.2 Setup with an Interval Plant . 179
11.3 Sixteen Distinguished Plants . 180
11.4 The Sixteen Plant Theorem. . 182
11.5 Controller Synthesis Technique . . 182
11.6 Machinery for Proof of Sixteen Plant Theorem . 186
11.7 Proof of the Sixteen Plant Theorem . 192
11.8 Conclusion . . . . . . . . . . . . . . . . . . . . . 193
Chapter 12. Rantzer's Growth Condition 196
12.1 Introduction . . . . . . . . . . 196
12.2 Convex Directions . . . . . . . . . . 197
12.3 Rantzer's Growth Condition . . . . 200
12.4 Machinery for Rantzer's Growth Condition . 202
12.5 Proof of the Theorem . . . . . . . . . . . . . 212
12.6 Diamond Families: An Illustrative Application . 214
12. 7 Conclusion . . . . . . . . . . . . . . . . . . . . . 216
Chapter 13. Schur Stability and Kharitonov Regions 218
13.l Introduction . . . . . . . . . . . . . . 218
13.2 Low Order Coefficient Uncertainty . . . 219
13.3 Low Order Polynomials . . . . . . . . . 223
13.4 Weak and Strong Kharitonov Regions
13.5 Characterization of Weak Kharitonov Regions . 225
13.6 Machinery for Proof of the Theorem . . 228
13.7 Proof of the Theorem . 233
13.8 Conclusion . . . . . . . . . . . . . . . . 234
Chapter 14. Multilinear Uncertainty Structures 237
14.1 Introduction . . . . . . . . . . . . . . . . . 237
14.2 More Complicated Uncertainty Structures. . 238
14.3 Multilinear and Polynomic Uncertainty . . 238
14.4 Interval Matrix Family . . . . . . . . . . . 242
14.5 Lack of Extreme Point and Edge Results . 244
14.6 The Mapping Theorem . . 246
14.7 Geometric Interpretation . . . . . . . . . . 247
14.8 Value Set Interpretation. . . . . . . . . . . 248
14.9 Machinery for Proof of the Mapping Theorem . 250
14.10 Proof of the Mapping Theorem . . 252
14.11 Conclusion . . . . . . . . . . . . . . . . . . . . . 253
Part IV. The Spherical Theory 257
Chapter 15. Spherical Polynomial Families 258
15.1 Introduction . . . . . . . . . . . 258
15.2 Boxes Versus Spheres . . . . . . 259
15.3 Spherical Polynomial Families . 260
15.4 Lumping . . . . . . . . . . . . . 263
15.5 The Soh-Berger-Dabke Theorem. . 267
15.6 The Value Set for a Spherical Polynomial Family. . 270
15. 7 Proof of the Soh-Berger-Dabke Theorem . 272
15.8 Overbounding via a Spherical Family . . . . . 273
15.9 Conclusion . . . . . . . . . . . . . . . . . . . . 276
Chapter 16. Embellishments for Spherical Families 278
16.l Introduction . . . . . . . . . . . . . . . . . . . 278
16.2 The Spectral Set . . . . . . . . . . . . . . . . . 279
16.3 Formula and Theorem of Barmish and Tempo . 280
16.4 Affine Linear Uncertainty Structures . . . . . 284
16.5 The Testing Function for Robust Stability . 285
16.6 Machinery for Proof of the Theorem . . 289
16.7 Proof of the Theorem . 292
16.8 Some Refinements . 293
16.9 Conclusion
PartV. Some Happenings at the Frontier 297
Chapter 17. An Introduction to Guardian Maps 298
17.1 Introduction . . . . . . . . . . . . . . . . . 298
17.2 Overview . . . . . . . . . . . . . . . . . . . 299
17.3 Topological Preliminaries for Guardian Maps . 300
17.4 The Guardian Map . . . . . . . . . . . . . 302
17.5 Some Useful Guardian Maps . . . . . . . . 303
17.6 Families with One Uncertain Parameter . . 305
17. 7 Polynomic Determinants . . . . . . . . . 306
17.8 The Theorem of Saydy, Tits and Abed . 307
17.9 Schur Stability . . . . . . . . . . . . 311
17.10 Conclusion . . . . . . . . . . . . . . 312
Chapter 18. The Arc Convexity Theorem 314
18.1 Introduction . . . . . . . . . . . . . 314
18.2 Definitions for Frequency Response Arcs . 315
18.3 The Arc Convexity Theorem . . . . . . 316
18.4 Machinery 1: Chords and Derivatives . 317
18.5 Ma.Chinery 2: Flow . . . . . . . . . . . . 319
18.6 Machinery 3: The Forbidden Cone . . . 321
18. 7 Proof of the Arc Convexity Theorem . 323
18.8 Robustness Connections . . 324
18.9 Conclusion . . . . . . . . . . . . . . . . 328
Chapter 19. Five Easy Problems 330
19.1 Introduction . . . . . . . . . . . . . . . . . . 330
19.2 Problem Area 1: Generating Mechanisms . . 331
19.3 Problem Area 2: Conditioning of Margins . . 338
19.4 Problem Area 3: Parametric Lyapunov Theory . . 342
19.5 Problem Area 4: Polytopes of Matrices . . 34 7
19.6 Problem Area 5: Robust Performance . . . . . 351
19. 7 Conclusion . . . . . . . . . . . . . . . . . . . . 355
Appendix A. Symbolic Computation for Fiat Dedra 358
Bibliography 361
Index
