Symbolic Methods in Control System Analysis and Design

Symbolic Methods in control system analysis and design......Page 4
Contents......Page 6
Foreword......Page 14
Contributors......Page 18
Part I: System modelling......Page 20
1.1 Introduction......Page 22
1.2.1 System elements and operations......Page 23
1.3.1 Elemental equations......Page 24
1.3.3 Multidimensional transfer functions......Page 25
1.4 Application of symbolic computation......Page 26
1.5 Example......Page 27
1.5.1 Manual procedure......Page 28
1.6 Nonlinear system response......Page 30
1.6.1 Time-domain response......Page 31
1.6.2 Frequency-domain response......Page 32
1.7.1 Time-domain response......Page 33
1.7.2 Frequency-domain response......Page 34
1.8 Graphical user interface......Page 35
1.9 Example......Page 38
1.10 References......Page 40
2.1 Introduction......Page 42
2.2 Modelling with bond graphs......Page 43
2.2.1 Variables......Page 44
2.2.3 Junctions......Page 45
2.2.4 Comaonents......Page 46
2.2.7 Causality......Page 47
2.3 Hierarchical bond graphs......Page 48
2.4 Representations, languages and software......Page 50
2.5.1.1 Subsystems......Page 52
2.5.1.2 System Generator......Page 55
2.5.1.3 System DC......Page 56
2.5.1.4 System Load......Page 57
2.5.1.6 System PSU......Page 58
2.5.1.7 System Shaft......Page 59
2.5.2 System structure......Page 60
2.5.3 System ordinary differential equations......Page 61
2.5.4 System state matrices......Page 63
2.5.5 System frequency response......Page 65
2.5.6 Numerical parameters and initial states......Page 67
2.5.7 System response......Page 68
2.7 References......Page 69
3.1.1 General comments......Page 72
3.1.2 Multibody dynamics and computer algebra......Page 73
3.2 Dynamic formulations and modelling approaches......Page 75
3.2.1 Bond graphs......Page 77
3.3 Customised dynamics programs: type and application......Page 78
3.3.2 Symbolic methods......Page 79
3.3.3 Applications......Page 80
3.4.1 Early dynamics programs......Page 81
3.4.2 Pioneering symbolic dynamics programs......Page 82
3.4.3 Second-generation symbolic programs......Page 83
3.4.4 Continuing use of numeric programs......Page 84
3.4.6 Modern programs......Page 85
3.5 Summary......Page 86
3.5.1 Concluding remarks......Page 91
3.6 References......Page 93
Part II: System analysis......Page 98
4.1 Introduction......Page 100
4.2 Notation and preliminaries......Page 101
4.2.2 Boundary crossing and zero exclusion......Page 103
4.3 Real parameter stability margin......Page 104
4.3.1 l2 real parametric stability margin......Page 107
4.3.2 l2 stability margin for time-delay systems......Page 111
4.4 Extremal results in RPRCT......Page 113
4.4.1 Kharitonov's theorem......Page 114
4.4.2 The edge theorem......Page 117
4.4.3 The generalised Kharitonov theorem......Page 118
4.4.3.1 Construction of the extremal subset......Page 120
4.5 Frequency-domain results in RPRGT......Page 125
4.5.1 Frequency-domain properties......Page 127
4.5.2 Closed-loop transfer functions......Page 128
4.6 Concluding remarks......Page 130
4.8 References......Page 132
5.2 Differential algebra: motivating examples......Page 134
5.3 The basic ideas of differential algebra......Page 136
5.3.2 Reducedness......Page 137
5.3.3 Autoreduced sets......Page 138
5.3.5 Ritt's algorithm......Page 139
5.4 Real algebra: motivating example......Page 140
5.5 Constructive methods in real algebra......Page 141
5.5.1 Cylindrical algebraic decompostion......Page 142
5.5.2 Quantifier elimination......Page 145
5.5.3.1 Stationarisable points......Page 146
5.5.3.2 Stability......Page 148
5.5.3.3 Curve following......Page 150
5.7 References......Page 152
6.1 Introduction......Page 154
6.2 Models with numerical inaccuracies and classification of algebraic computation problems......Page 156
6.3.1.1 The SVD criterion......Page 159
6.3.2 Compound matrices......Page 160
6.3.3 Selection of a best uncorrupted base for a numerically dependent set......Page 161
6.4 Almost zeros of a set of polynomials......Page 165
6.5 GGD method based on ERES......Page 168
6.5.1 The ERES method: theoretical and numerical issues......Page 169
6.6.2 Reduction of the original set......Page 172
6.6.3 Determination of the associated pencil......Page 173
6.7.1 Fundamental definitions and properties of LCM of polynomials......Page 175
6.7.2 Factorisation and almost factorisation of polynomials......Page 176
6.8 Applications, examples and numerical performance of methods......Page 180
6.8.1 Performance of the ERES and MP methods......Page 184
6.8.2 Performance of the LCM method......Page 185
6.10 References......Page 186
7.1 Introduction......Page 188
7.2 Direct Nyquist array design method......Page 189
7.3 Uncertain parametric systems and polynomial families......Page 193
7.4 From classical to robust control......Page 195
7.5 Robust direct Nyquist array (RDNA)......Page 198
7.5.1 Robust generalised dominance (RGD) stability theorem......Page 202
7.5.2 Robust fundamental dominance (RFD) stability theorem......Page 204
7.6 Illustrative examples......Page 206
7.7 Conclusions......Page 214
7.8 Acknowledgments......Page 216
7.9 References......Page 217
7.10.1 Appendix A......Page 218
7.10.2 Appendix B......Page 219
Part III: Design and synthesis methods......Page 220
8.2 Robust classical controller design using RPRGT......Page 222
8.3 Linear programming approach to design......Page 229
8.3.1 Fixed-order pole assignment and robust stabilisation......Page 232
8.3.2 Robust pole assignment......Page 235
8.4 Conclusions and future directions......Page 244
8.6 References......Page 245
9.1 Introduction......Page 246
9.2 Dynamic sliding mode control......Page 247
9.3 Design method......Page 250
9.3.1 Direct sliding mode control......Page 251
9.3.2 Indirect sliding mode control......Page 253
9.3.3 Robust design method......Page 254
9.4 Mathematica implementation......Page 256
9.5 Design examples......Page 257
9.6 Conclusion......Page 260
9.8 References......Page 261
9.9.1 Appendix A......Page 262
9.9.2 Appendix B......Page 264
9.9.3 Appendix C......Page 266
10.1 Introduction......Page 270
10.2.1 Dyadic methods......Page 273
10.2.2 Full-rank methods......Page 277
10.3.1 Static feedback......Page 281
10.3.2 Dynamic feedback......Page 286
10.5 References......Page 290
11.1 Introduction......Page 292
11.2 Classification of computational problems......Page 294
11.3 Groebner basis computation......Page 296
11.4 The solution of the cover problem via Groebner basis computation......Page 298
11.5 Echelon form and canonical forms of descriptor systems......Page 303
11.6 Symbolic methods for global linearisation of pole assignment maps......Page 308
11.8 References......Page 311
Part IV: Nonlinear systems......Page 314
12.1 Introduction......Page 316
12.2.1 Problem setup......Page 319
12.2.2 Basic notions......Page 321
12.3 Computer algebra implementation......Page 326
12.4 Modelling......Page 327
12.4.1 Systematic modelling......Page 328
12.4.2 System characterisation......Page 331
12.5 Analysis......Page 332
12.6.1 Input-output exact linearisation......Page 333
12.6.2 State-space exact linearisation......Page 334
12.9 References......Page 337
13.1 Introduction......Page 340
13.2.1 Outline of spatial branch-and-bound algorithm......Page 341
13.2.2 Upper and lower bounds for spatial branch-and-bound algorithms......Page 342
13.2.3 Underestimators and over estimators of nonconvex functions......Page 343
13.3.1 The standard form for a nonlinear programming problem......Page 344
75.5.2 Basic ideas of symbolic reformulation algorithm......Page 345
13.3.3 An automatic symbolic reformulation algorithm......Page 346
13.3.4 Implementation of the spatial branch-and-bound algorithm......Page 348
13.4.1 Konno's bilinear program......Page 349
13.4.2 The doughnut slice problem......Page 350
13.4.3 Haverlfs pooling problem......Page 351
13.4.4 A six-hump camel back problem......Page 352
13.4.5 MIMO control system diagonal dominance problem......Page 353
13.5 Concluding remarks......Page 355
13.6 References......Page 356
14.1 Introduction......Page 358
14.2 Quantifier elimination......Page 359
14.3.1 Bernstein transformation of a polynomial......Page 360
14.3.3 Selection of the sweep direction......Page 362
14.4 Approximation of the solution set......Page 363
14.5 Algorithm......Page 364
14.6 Examples......Page 365
14.7 Conclusions......Page 369
14.9 References......Page 370
15.1 Background—design and control of nonlinear systems......Page 372
15.2.1 Linearisation of nonlinear systems......Page 374
15.2.2 State-space and transfer-function models......Page 375
15.2.3 State-space operations......Page 378
15.2.4 Transfer-function operations......Page 380
15.2.5 The linear quadratic Gaussian problem......Page 381
15.2.5.1 Algebraic Riccati equation......Page 382
15.2.5.2 Optimal state feedback......Page 383
15.2.5.3 Optimal stochastic state estimator......Page 384
15.3.1 Controls package description......Page 385
15.3.2 Basic analysis tools......Page 386
15.3.3.3 ShowSystem and GetResults......Page 389
15.3.4.2 AdaptiveTracking......Page 390
15.3.5.2 Simulate......Page 391
15.3.6 Application: adaptive control of a conical magnetic bearing......Page 392
15.3.6.1 Model......Page 393
15.3.6.2 Control system design......Page 395
15.3.7 Computing the local zero dynamics......Page 396
15.4 Design of variable structure systems......Page 397
15.4.1.2 Reaching......Page 398
15.4.1.3 Chattering reduction......Page 399
15.4.1.4 Example 7. A rotor with friction......Page 400
15.4.2.1 Sliding surf ace computations......Page 401
15.4.2.3 Example 7 (continued)......Page 402
15.5 Conclusions......Page 404
15.6 References......Page 405
Index......Page 408