Observability and controllability of general linear systems.

351 Pages·2018·3.524 MB·English

by  LYUBOMIR T. GRUYITCH

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Control of Linear Systems
Observability and Controllability
of General Linear Systems
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Control of Linear Systems
Observability and Controllability
of General Linear Systems
Lyubomir T. Gruyitch
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CRC Press
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Contents
Preface xi
I SYSTEM CLASSES 1
1 Introduction 3
1.1 Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Notational preliminaries . . . . . . . . . . . . . . . . . . . . 6
1.3 Compact, simple, and elegant calculus . . . . . . . . . . . . 8
1.4 System behavior . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2 IO systems 19
2.1 IO system mathematical model . . . . . . . . . . . . . . . . 19
2.1.1 Time domain . . . . . . . . . . . . . . . . . . . . . 19
2.1.2 Complex domain . . . . . . . . . . . . . . . . . . . 22
2.2 IO plant desired regime . . . . . . . . . . . . . . . . . . . . . 28
2.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3 ISO systems 31
3.1 ISO system mathematical model . . . . . . . . . . . . . . . 31
3.1.1 Time domain . . . . . . . . . . . . . . . . . . . . . 31
3.1.2 Complex domain . . . . . . . . . . . . . . . . . . . 33
3.2 ISO plant desired regime . . . . . . . . . . . . . . . . . . . . 38
3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4 EISO systems 41
4.1 EISO system mathematical model . . . . . . . . . . . . . . . 41
4.1.1 Time domain . . . . . . . . . . . . . . . . . . . . . 41
4.1.2 Complex domain . . . . . . . . . . . . . . . . . . . 47
v
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4.2 EISO plant desired regime . . . . . . . . . . . . . . . . . . . 62
4.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5 HISO systems 65
5.1 HISO system mathematical model. . . . . . . . . . . . . . . 65
5.1.1 Time domain . . . . . . . . . . . . . . . . . . . . . 65
5.1.2 Complex domain . . . . . . . . . . . . . . . . . . . 66
5.2 The HISO plant desired regime . . . . . . . . . . . . . . . . 74
5.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6 IIO systems 77
6.1 IIO system mathematical model . . . . . . . . . . . . . . . . 77
6.1.1 Time domain . . . . . . . . . . . . . . . . . . . . . 77
6.1.2 Complex domain . . . . . . . . . . . . . . . . . . . 79
6.2 IIO plant desired regime . . . . . . . . . . . . . . . . . . . . 90
6.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
II OBSERVABILITY 93
7 Mathematical preliminaries 95
7.1 Linear independence and matrices . . . . . . . . . . . . . . . 95
7.2 Matrix range, null space, and rank . . . . . . . . . . . . . . 97
7.3 Linear independence and scalar functions . . . . . . . . . . . 104
7.4 Linear independence and matrix functions . . . . . . . . . . 107
7.5 Polynomial matrices. Matrix polynomials . . . . . . . . . . . 120
7.6 Rational matrices . . . . . . . . . . . . . . . . . . . . . . . . 125
8 Observability and stability 133
8.1 Observability and system regime . . . . . . . . . . . . . . . . 133
8.2 Observability definition in general . . . . . . . . . . . . . . . 134
8.3 Observability criterion in general . . . . . . . . . . . . . . . 136
8.4 Observability and stability . . . . . . . . . . . . . . . . . . . 139
8.4.1 System stability . . . . . . . . . . . . . . . . . . . . 139
8.4.2 System stability and observability . . . . . . . . . . 141
9 Various systems observability 143
9.1 IO systems observability . . . . . . . . . . . . . . . . . . . . 143
9.2 ISO systems observability . . . . . . . . . . . . . . . . . . . 147
9.3 EISO systems observability . . . . . . . . . . . . . . . . . . 149
9.4 HISO systems observability . . . . . . . . . . . . . . . . . . 150
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9.5 IIO systems observability . . . . . . . . . . . . . . . . . . . . 152
III CONTROLLABILITY 157
10 Controllability fundamentals 159
10.1 Controllability and system regime . . . . . . . . . . . . . . . 159
10.2 Controllability concepts . . . . . . . . . . . . . . . . . . . . . 159
10.3 Controllability definitions in general . . . . . . . . . . . . . . 160
10.4 General state controllability criteria . . . . . . . . . . . . . . 162
10.5 General output controllability criteria . . . . . . . . . . . . . 165
11 Various systems controllability 171
11.1 IO system state controllability . . . . . . . . . . . . . . . . . 171
11.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . 171
11.1.2 Criteria . . . . . . . . . . . . . . . . . . . . . . . . . 172
11.2 IO system output controllability . . . . . . . . . . . . . . . . 185
11.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . 185
11.2.2 Criteria . . . . . . . . . . . . . . . . . . . . . . . . . 185
11.3 ISO system state controllability . . . . . . . . . . . . . . . . 195
11.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . 195
11.3.2 Criterion . . . . . . . . . . . . . . . . . . . . . . . . 196
11.4 ISO system output controllability . . . . . . . . . . . . . . . 197
11.4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . 197
11.4.2 Criteria . . . . . . . . . . . . . . . . . . . . . . . . . 197
11.5 EISO system state controllability . . . . . . . . . . . . . . . 201
11.5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . 201
11.5.2 Criterion . . . . . . . . . . . . . . . . . . . . . . . . 201
11.6 EISO system output controllability . . . . . . . . . . . . . . 203
11.6.1 Definition . . . . . . . . . . . . . . . . . . . . . . . 203
11.6.2 Criteria . . . . . . . . . . . . . . . . . . . . . . . . . 203
11.7 HISO system state controllability . . . . . . . . . . . . . . . 206
11.7.1 Definition . . . . . . . . . . . . . . . . . . . . . . . 206
11.7.2 Criterion . . . . . . . . . . . . . . . . . . . . . . . . 207
11.8 HISO system output controllability . . . . . . . . . . . . . . 210
11.8.1 Definition . . . . . . . . . . . . . . . . . . . . . . . 210
11.8.2 Criteria . . . . . . . . . . . . . . . . . . . . . . . . . 210
11.9 IIO system state controllability . . . . . . . . . . . . . . . . 214
11.9.1 Definition . . . . . . . . . . . . . . . . . . . . . . . 214
11.9.2 Criterion . . . . . . . . . . . . . . . . . . . . . . . . 216
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11.10 IIO system output controllability . . . . . . . . . . . . . . . 222
11.10.1 Definition . . . . . . . . . . . . . . . . . . . . . . . 222
11.10.2 Criteria . . . . . . . . . . . . . . . . . . . . . . . . . 222
IV APPENDIX 227
A Notation 229
A.1 Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . 229
A.2 Indexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
A.2.1 SUBSCRIPTS . . . . . . . . . . . . . . . . . . . . . 230
A.2.2 SUPERSCRIPT . . . . . . . . . . . . . . . . . . . . 230
A.3 Letters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
A.3.1 CALLIGRAPHIC LETTERS . . . . . . . . . . . . 231
A.3.2 FRAKTUR LETTERS . . . . . . . . . . . . . . . . 231
A.3.3 GREEK LETTERS . . . . . . . . . . . . . . . . . . 233
A.3.4 ROMAN LETTERS . . . . . . . . . . . . . . . . . 234
A.4 Name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
A.5 Symbols, vectors, sets and matrices . . . . . . . . . . . . . . 240
A.6 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
B Example 243
B.1 IO system example . . . . . . . . . . . . . . . . . . . . . . . 243
C Transformations 249
C.1 Transformation of IO into ISO system . . . . . . . . . . . . 249
C.2 ISO and EISO forms of IIO system . . . . . . . . . . . . . . 251
D Proofs 255
D.1 Proof of Lemma 97 . . . . . . . . . . . . . . . . . . . . . . . 255
D.2 Proof of Theorem 110 . . . . . . . . . . . . . . . . . . . . . . 256
D.3 Proof of Theorem 121 . . . . . . . . . . . . . . . . . . . . . . 258
D.4 Proof of Theorem 128 . . . . . . . . . . . . . . . . . . . . . . 270
D.5 Proof of Theorem 130 . . . . . . . . . . . . . . . . . . . . . . 273
D.6 Proof of Theorem 141 . . . . . . . . . . . . . . . . . . . . . . 276
D.7 Proof of Theorem 147 . . . . . . . . . . . . . . . . . . . . . . 277
D.8 Proof of Theorem 148 . . . . . . . . . . . . . . . . . . . . . . 279
D.9 Proof of Theorem 149 . . . . . . . . . . . . . . . . . . . . . . 280
D.10 Proof of Theorem 161 . . . . . . . . . . . . . . . . . . . . . . 282
D.11 Proof of Theorem 163 . . . . . . . . . . . . . . . . . . . . . . 294
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D.12 Proof of Theorem 173 . . . . . . . . . . . . . . . . . . . . . . 304
D.13 Proof of Theorem 181 . . . . . . . . . . . . . . . . . . . . . . 307
D.14 Proof of Theorem 185 . . . . . . . . . . . . . . . . . . . . . . 309
D.15 Proof of Theorem 215 . . . . . . . . . . . . . . . . . . . . . . 314
V INDEX 327
Author Index 329
Subject Index 331
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