Nonlinear systems tracking

467 Pages·2016·35.102 MB·English

by  Gruĭich, Li︠u︡bomir Tikhomira

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Nonlinear
Systems
Tracking
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Nonlinear
Systems
Tracking
LYUBOMIR T. GRUYITCH

Boca Raton London New York
CRC Press is an imprint of the
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Contents
I Preface xv
Systems, control, and computers xvii
0.1 Dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
0.2 Dynamical systems and computers . . . . . . . . . . . . . . . . . . . . . . . . xviii
0.3 Dynamical systems and control . . . . . . . . . . . . . . . . . . . . . . . . . . xviii
0.4 Control goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii
0.5 Tracking control tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix
On the book xxi
0.6 Goals of the book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi
0.7 The book structure and composition . . . . . . . . . . . . . . . . . . . . . . . xxiii
0.8 In gratitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiv
II SYSTEMS AND CONTROL BASIS 1
1 Introduction 3
1.1 Time and systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Time, physical variables, and systems . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Notational preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Classes of the systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Sets 13
2.1 Set basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Neighborhood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Continuity of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 Set contraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Systems 21
3.1 General dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Plants and control systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2.1 System description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2.2 Special form of the system. . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2.3 System basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2.4 Sets and systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2.5 System motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2.6 System generalized motions . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2.7 Existence and uniqueness of motions . . . . . . . . . . . . . . . . . . . 33
3.2.8 System advanced properties . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2.9 Special system class: IO systems . . . . . . . . . . . . . . . . . . . . . 36
3.2.10 Special system class: ISO systems . . . . . . . . . . . . . . . . . . . . 37
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3.3 Existence and solvability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.4 Fundamental control principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4 Desired regime 41
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2 Concept and definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5 Origins of time-varying models 53
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.2 Deviations and mathematical models . . . . . . . . . . . . . . . . . . . . . . . 54
III TRACKABILITY 59
6 Trackability concept 61
6.1 On system and control concepts . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6.2 Controllability and observability . . . . . . . . . . . . . . . . . . . . . . . . . 61
6.3 Disturbance rejection or compensation . . . . . . . . . . . . . . . . . . . . . . 62
6.4 New control concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6.5 Time and control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
7 Perfect trackability concepts 65
7.1 Perfect trackability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
7.2 Perfect natural trackability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
8 Perfect trackability criteria 69
8.1 Output space criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
8.2 State space criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
8.3 Both spaces criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
9 Perfect natural trackability criteria 77
9.1 Output space criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
9.2 State space criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
9.3 Both spaces criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
10 Imperfect trackability concepts 81
10.1 Introduction to (imperfect) trackability. . . . . . . . . . . . . . . . . . . . . . 81
10.2 Trackability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
10.3 Natural trackability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
10.4 Elementwise trackability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
10.5 Elementwise natural trackability . . . . . . . . . . . . . . . . . . . . . . . . . 88
11 Imperfect trackability criteria 91
11.1 Output space criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
11.2 State space criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
11.3 Both spaces criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
12 Imperfect natural trackability criteria 95
12.1 Output space criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
12.2 State space criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
12.3 Both spaces criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
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CONTENTS vii
IV PERFECT TRACKING 99
13 Tracking generally 101
13.1 Primary control goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
13.2 Tracking versus stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
14 Tracking concepts 105
14.1 Tracking characterization and space . . . . . . . . . . . . . . . . . . . . . . . 105
14.1.1 Output tracking and output space . . . . . . . . . . . . . . . . . . . . 105
14.1.2 Output tracking and state space . . . . . . . . . . . . . . . . . . . . . 105
14.2 Various tracking concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
15 Perfect tracking concept 109
15.1 On perfect tracking generally . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
15.2 Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
15.2.1 Perfect tracking and the target set of the plant . . . . . . . . . . . . . 110
15.2.2 Perfect tracking and the desired regime of the plant . . . . . . . . . . 111
15.2.3 Perfect tracking and the nominal control of the plant . . . . . . . . . . 112
V IMPERFECT TRACKING: STABLE TRACKING 113
16 Output space definitions 115
16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
16.2 Definitions of L-tracking properties . . . . . . . . . . . . . . . . . . . . . . . . 117
17 State space definitions 127
17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
17.2 Definitions of L-tracking properties . . . . . . . . . . . . . . . . . . . . . . . . 127
18 Set tracking 137
18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
18.2 Definitions of set tracking properties . . . . . . . . . . . . . . . . . . . . . . . 137
VI CRITERIA FOR STABLE TRACKING 143
19 Introduction 145
19.1 Lyapunov methods and methodologies . . . . . . . . . . . . . . . . . . . . . . 145
19.2 Tracking accuracy and definitions . . . . . . . . . . . . . . . . . . . . . . . . . 146
19.3 Suitable mathematical models . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
20 Comparison and (semi)definite functions 149
20.1 Comparison functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
20.2 Semidefinite functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
20.3 Definite functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
20.4 Decrescent functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
20.5 Time-invariant vector definite functions . . . . . . . . . . . . . . . . . . . . . 156
20.6 Time-varying vector definite functions . . . . . . . . . . . . . . . . . . . . . . 158
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21 Sets and functions 163
21.1 Positive definite function induces sets. . . . . . . . . . . . . . . . . . . . . . . 163
21.2 Set invariance relative to a function. . . . . . . . . . . . . . . . . . . . . . . . 168
21.3 Semidefinite functions and time-varying sets . . . . . . . . . . . . . . . . . . . 170
21.4 Definite functions relative to time-varying sets . . . . . . . . . . . . . . . . . 171
21.5 Decrescent functions and time-varying sets . . . . . . . . . . . . . . . . . . . 174
21.6 Families F of time-varying sets . . . . . . . . . . . . . . . . . . . . . . . . . . 175
22 Outline of the Lyapunov method 179
22.1 Physical origin of the Lyapunov method . . . . . . . . . . . . . . . . . . . . . 179
22.2 Lyapunov method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
22.2.1 Essence of the Lyapunov method . . . . . . . . . . . . . . . . . . . . . 181
22.2.2 Application in the state space . . . . . . . . . . . . . . . . . . . . . . . 181
22.2.3 Application in the output space . . . . . . . . . . . . . . . . . . . . . . 183
22.3 Lyapunov theorems for nonlinear systems . . . . . . . . . . . . . . . . . . . . 186
22.4 Lyapunov original methodologies . . . . . . . . . . . . . . . . . . . . . . . . . 186
23 Lyapunov method extended to tracking 189
23.1 Criteria: Asymptotically contractive sets . . . . . . . . . . . . . . . . . . . . . 189
23.2 Criteria: Noncontractive time-varying sets . . . . . . . . . . . . . . . . . . . . 192
24 CLM: Motion and set tracking 197
24.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
24.2 Systems smooth properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
24.3 Systems and generating functions . . . . . . . . . . . . . . . . . . . . . . . . . 199
24.4 Criteria: Systems with continuous motions . . . . . . . . . . . . . . . . . . . . 203
24.5 Criteria: Systems with differentiable motions . . . . . . . . . . . . . . . . . . 208
25 Time-varying set tracking 213
25.1 Time-varying set and motion tracking . . . . . . . . . . . . . . . . . . . . . . 213
25.2 Conditions for stable tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
25.3 Conditions for exponential tracking . . . . . . . . . . . . . . . . . . . . . . . . 216
VII FINITE REACHABILITYTIME TRACKING 219
26 Output space definitions 221
26.1 Finite scalar reachability time . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
26.2 Finite vector reachability time . . . . . . . . . . . . . . . . . . . . . . . . . . 226
27 State space definitions 237
27.1 Finite scalar reachability time tracking . . . . . . . . . . . . . . . . . . . . . . 237
27.2 Elementwise state FVRT tracking . . . . . . . . . . . . . . . . . . . . . . . . 241
28 Criteria on contractive sets 247
28.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
28.2 Stable tracking with FSRT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
28.3 Stable tracking with FVRT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
29 Criteria on noncontractive sets 251
29.1 Stable tracking with FSRT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
29.2 Stable tracking with FVRT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
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30 FRT tracking control synthesis 253
30.1 Internal dynamics space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
30.1.1 Derivative of V(.) along motions . . . . . . . . . . . . . . . . . . . . . 253
30.1.2 Asymptotically contractive sets . . . . . . . . . . . . . . . . . . . . . . 254
30.1.3 Noncontractive time-varying sets . . . . . . . . . . . . . . . . . . . . . 256
30.2 Output space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
30.2.1 Derivative of V(.) along the output behavior . . . . . . . . . . . . . . 256
30.2.2 Asymptotically contractive sets . . . . . . . . . . . . . . . . . . . . . . 258
30.2.3 Noncontractive time-varying sets . . . . . . . . . . . . . . . . . . . . . 260
VIII REQUIRED TRACKING QUALITY AND CONTROL
SYNTHESIS 263
31 Natural tracking control concept 265
32 Tracking quality: Output space 267
32.1 Output space tracking operator . . . . . . . . . . . . . . . . . . . . . . . . . . 267
32.2 Tracking operator properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
32.3 Reference output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
32.4 Tracking algorithm and initial conditions . . . . . . . . . . . . . . . . . . . . 279
33 Tracking algorithms: Output space 281
33.1 Matrix notation meaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
33.2 Examples of tracking algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 281
34 NTC synthesis: Output space 289
34.1 General NTC theorem: Output space . . . . . . . . . . . . . . . . . . . . . . 289
34.2 NTC synthesis: Output space . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
35 Tracking quality: State space 301
35.1 State space tracking operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
35.2 Tracking operator properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
35.3 Reference state vector R . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
R
35.4 Tracking algorithm and initial conditions . . . . . . . . . . . . . . . . . . . . 309
36 Tracking algorithms: State space 311
36.1 Matrix notation meaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
36.2 Examples of tracking algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 311
37 NTC synthesis: State space 319
37.1 General NTC theorem: State space . . . . . . . . . . . . . . . . . . . . . . . . 319
37.2 NTC synthesis in the state space . . . . . . . . . . . . . . . . . . . . . . . . . 320
38 Tracking quality: Both spaces 329
38.1 Both spaces (BS) tracking operator. . . . . . . . . . . . . . . . . . . . . . . . 329
38.2 Tracking operator properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
38.3 The reference BS vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
38.4 Tracking algorithm and initial conditions . . . . . . . . . . . . . . . . . . . . 335
39 Tracking algorithms: Both spaces 337
39.1 Matrix notation meaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
39.2 Examples of tracking algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 337
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40 NTC synthesis: Both spaces 343
40.1 General NTC theorem: Both spaces (BS) . . . . . . . . . . . . . . . . . . . . 343
40.2 NTC synthesis in both spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
IX CONCLUSION 351
41 Systems, control, tracking, trackability 353
41.1 Perturbed systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
41.2 Control goal: Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
41.3 Tracking demands trackability . . . . . . . . . . . . . . . . . . . . . . . . . . . 354
42 Lyapunov theory and tracking 355
42.1 Lyapunov theory extended to tracking . . . . . . . . . . . . . . . . . . . . . . 355
42.2 Consistent Lyapunov methodology: Tracking . . . . . . . . . . . . . . . . . . 356
42.3 Lyapunov control synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
43 High quality tracking: Control synthesis 359
43.1 Tracking with finite reachability time . . . . . . . . . . . . . . . . . . . . . . . 359
43.2 Demanded tracking quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
43.3 Trackability theory and tracking theory importance . . . . . . . . . . . . . . . 360
X APPENDICES 361
A Notation 363
A.1 Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
A.2 Indexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
A.2.1 Subscripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
A.2.2 Superscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
A.3 Letters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
A.3.1 Calligraphic letters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
A.3.2 Fraktur letters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
A.3.3 Greek letters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
A.3.4 Roman letters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
A.4 Names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
A.5 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
A.6 Matrix and vector notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378
A.7 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
A.8 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381

B Dini derivatives 383
B.1 Definitions of derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
B.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386
C Proofs for Part III 389
C.1 Proof of Theorem 89 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
C.2 Proof of Theorem 152 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390
C.3 Proof of Theorem 157 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394
C.4 Proof of Theorem 160 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
C.5 Proof of Theorem 162 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400
C.6 Proof of Theorem 164 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402
C.7 Proof of Theorem 165 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402
© 2016 Taylor & Francis Group, LLC
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