Robust industrial control systems: optimal design approach for polynomial systems

Robust Industrial Control Systems......Page 3
Contents......Page 9
Preface......Page 21
Acknowledgements......Page 23
1.1 Introduction......Page 25
1.1.1 Optimality, Feedback and Robustness......Page 26
1.1.2 High-integrity and Fault-tolerant Control Systems......Page 27
1.1.3 Self-healing Control Systems......Page 28
1.1.6 Artificial Intelligence, Neural Networks and Fuzzy Control......Page 29
1.1.7 Discrete-time Systems......Page 31
1.2 The H2 and H-infinity Spaces and Norms......Page 32
1.3 Introduction to H-infinity Control Design......Page 33
1.3.1 Properties of H-infinity Robust Control Design......Page 35
1.3.2 Comparison of H-infinity and H2/LQG Controllers......Page 36
1.3.5 H-infinity Polynomial Systems Synthesis Theory......Page 37
1.4.1 State-space Solution of Discrete-time H-infinity Control Problem......Page 38
1.4.3 State-feedback Control Solution......Page 39
1.4.4 State-feedback Control Problem: Cross-product Costing Case......Page 42
1.4.5 State-space Solution of Discrete-time H-infinity Filtering Problem......Page 43
1.4.6 Bounded Real Lemma......Page 45
1.4.7 Output Feedback H-infinity Control Problem......Page 48
1.5.1 System Description......Page 53
1.5.3 Minimisation of the Performance Criterion......Page 55
1.5.4 Solution of the Diophantine Equations and Stability......Page 58
1.5.5 H2 /LQG Design Examples......Page 59
1.6 Benchmarking......Page 64
1.6.1 Restricted Structure Benchmarking......Page 65
1.6.2 Rules for Benchmark Cost Function Selection......Page 66
1.7 Condition Monitoring......Page 68
1.8 Combining H2, H-infinity and L1 Optimal Control Designs......Page 69
1.9 Linear Matrix Inequalities......Page 70
1.10 Concluding Remarks......Page 71
1.11 Problems......Page 72
1.12 References......Page 75
2.1 Introduction......Page 81
2.1.1 Industrial Controller Structures......Page 82
2.1.2 The 2½-DOF Structure......Page 83
2.2 Stochastic System Description......Page 84
2.2.2 System Equations......Page 86
2.2.3 Cost Function Weighting Terms......Page 87
2.3 Dual-criterion Cost-minimisation Problem......Page 88
2.3.1 Solution of the Dual-criterion Minimisation Problem......Page 90
2.3.2 Theorem Summarising LQG Controller......Page 95
2.3.3 Remarks on the Equations and Solution......Page 97
2.3.4 Design Guidelines......Page 100
2.3.5 Controller Implementation......Page 101
2.3.6 LQG Ship-steering Autopilot Application......Page 102
2.4.1 Youla Parameterisation......Page 106
2.4.2 Cost Function with Robust Weighting Function......Page 107
2.4.3 Solution of the Dual-criterion Problem with Robust Weighting......Page 108
2.4.4 Summary of H2 /LQG Synthesis Problem with Robust Weighting......Page 110
2.4.5 Comments on the Solution......Page 112
2.5.1 Standard System Model......Page 113
2.6 The Standard System Model Structure......Page 115
2.6.1 Polynomial System Models......Page 116
2.6.2 Reference Model......Page 117
2.6.3 Cost Function Signals to be Weighted......Page 118
2.7 Generalised H2 Optimal Control: Standard System Model......Page 119
2.7.1 Optimal Control Solution of the Standard System Model Problem......Page 120
2.7.2 Summary of H2 /LQG Controller for Standard System Results......Page 126
2.7.3 Remarks on the Solution......Page 128
2.9 Problems......Page 129
2.10 References......Page 133
3.1 Introduction......Page 137
3.1.1 Links Between LQG and H-infinity Solutions......Page 138
3.3 Lemma Linking H-infinity and LQG Control Problems......Page 139
3.4 Calculation of the H-infinity Optimal Controller......Page 140
3.4.2 Zero Measurement Noise Case......Page 141
3.4.3 Solution for the H-infinity Optimal Controller......Page 142
3.4.4 Stability Robustness of Mixed-sensitivity H-infinity Designs......Page 145
3.4.5 One-block H-infinity Control Problems......Page 146
3.5 The GH-infinity Control Problem......Page 147
3.5.1 GH-infinity Cost Function Definition......Page 148
3.5.2 Youla Parameterised Form of the GH-infinity Controller......Page 150
3.5.3 Calculation of the GH-infinity Controller......Page 152
3.6.1 Structure of the Uncertain System......Page 160
3.6.2 Rational Uncertainty Structure......Page 161
3.6.3 Stability Lemma......Page 163
3.6.5 Design Procedure for Uncertain Systems......Page 164
3.8 Calculation of H-infinity Controller for the Standard System......Page 171
3.8.1 F-iteration Method of Solving the Robust Weighting Equation......Page 172
3.8.2 H2 /H-infinity Trade-off......Page 173
3.9 Probabilistic System Descriptions and H-infinity Control......Page 174
3.9.1 Uncertain System Model......Page 175
3.9.2 Cost Function Definition......Page 177
3.9.3 Uncertain System and Polynomial Equation Representation......Page 179
3.10 Concluding Remarks......Page 182
3.11 Problems......Page 183
3.12 References......Page 187
4.1 Introduction......Page 191
4.2 Multivariable System Description......Page 192
4.2.1 Multivariable Sensitivity Matrices and Signal Spectra......Page 194
4.3 LQG Optimal Control Problem and Solution......Page 195
4.3.1 Solution of the H2 /LQG Problem......Page 196
4.3.2 Solution of the Diophantine Equations......Page 199
4.4 Youla Parameterisation and Auxiliary Problem......Page 206
4.4.1 Youla Parameterisation for the Auxiliary Problem......Page 208
4.4.2 Summary of Multivariable Problem Results with Robust Weighting......Page 210
4.5 H2 /LQG Optimal Control Problem: Measurement Noise Case......Page 211
4.5.2 SIMO Predictive Optimal Control Problem......Page 214
4.6 The GLQG Optimal Control Problem......Page 220
4.6.1 Solution of the GLQG Problem......Page 221
4.6.2 Modified GLQG Cost Function and Youla Parameterisation......Page 223
4.7 Design of Automatic Voltage Regulators......Page 224
4.8.1 Introduction to Pseudo-state Methods......Page 234
4.8.2 Pseudo-state Discrete-time Plant Model......Page 235
4.8.3 Discrete Pseudo-state Feedback Optimal Control......Page 239
4.8.4 Solution of the Pseudo-state Feedback Control Problem......Page 241
4.8.5 Discrete Pseudo-state Estimation Problem......Page 246
4.8.6 Solution of the Discrete-time pseudo-state Estimation Problem......Page 248
4.8.7 Output Feedback Control Problem and Separation Principle......Page 254
4.8.8 Computational Example......Page 259
4.9 Concluding Remarks......Page 264
4.10 Problems......Page 265
4.11 References......Page 269
5.1 Introduction......Page 273
5.2 H-infinity Multivariable Controllers......Page 274
5.2.1 Derivation of the Weighting Filter W-sigma......Page 275
5.2.2 Robust Weighting Equation......Page 276
5.2.3 Calculation of the H-infinity Optimal Controller......Page 277
5.2.4 Superoptimality in H-infinity Design......Page 282
5.3.1 One-block Nehari Problems......Page 283
5.3.2 Categories of Nehari Problem......Page 284
5.3.3 Constraint on the Choice of Weights for Simplified Design......Page 285
5.3.4 GH-infinity Optimal Control Problem......Page 286
5.3.5 Final Remarks on LQG Embedding H-infinity Solution......Page 291
5.4 Suboptimal H-infinity Multivariable Controllers......Page 292
5.4.1 System Description and Game Problem......Page 293
5.4.3 Signals and Bounded Power Property......Page 295
5.4.4 System and Cost Weighting Function Definitions......Page 296
5.5 Polynomial System for Suboptimal H-infinity Control Problem......Page 297
5.5.2 Diophantine Equations for Causal and Noncausal Decomposition......Page 298
5.6.1 Discrete-time Game Problem......Page 299
5.6.2 Relationship Between the Game and H-infinity Problems......Page 300
5.6.4 Completing-the-squares......Page 301
5.6.5 Cost Index Terms......Page 302
5.6.7 Contour Integral Simplification......Page 303
5.6.8 Optimal Control Law Calculation......Page 304
5.6.9 Expression for H0 JH0......Page 305
5.6.10 Saddle-point Solution......Page 306
5.6.11 Expression for the Minimum Cost......Page 307
5.7 Suboptimal H-infinity State-feedback Control Problem......Page 308
5.7.1 Remarks on the Solution......Page 309
5.8 Relationship Between Polynomial and State-space Results......Page 311
5.8.1 J-spectral Factorisation Using Riccati Equation......Page 312
5.8.2 Relationship between the Polynomial and State-space Equations......Page 314
5.9.1 Final Remarks on the Suboptimal H-infinity Solution......Page 315
5.10 Problems......Page 316
5.11 References......Page 319
6.1 Introduction......Page 323
6.1.1 The Control Design Problem......Page 324
6.1.2 Justification for H-infinity Control Design......Page 326
6.1.3 Dynamic Cost Function Weightings......Page 327
6.1.4 Properties of Sensitivity Functions for Discrete-time Systems......Page 328
6.2 Avoiding Impractical H-infinity Designs......Page 330
6.2.1 Equalising H-infinity Solutions and Implications for Multivariable Design......Page 331
6.3.2 H2 LQG Optimal Control Problem......Page 332
6.3.3 H-infinity Optimal Control Problem......Page 334
6.3.4 Cancellation of Minimum-phase Plant Zeros......Page 335
6.3.5 Cancellation of Stable Plant Poles......Page 336
6.4.1 Controller Poles and Zeros due to Weightings......Page 338
6.4.2 Poles of the Closed-loop System......Page 339
6.5.1 Stability Criterion and Cost Function Weighting Selection......Page 340
6.5.2 Influence of the Choice of Weights on the Sensitivity Functions......Page 341
6.5.3 Use of Constant Cost Weightings in H-infinity Design......Page 343
6.5.4 Poor Robustness due to Unrealistic Weightings......Page 344
6.6.1 Singular Value Approximations......Page 348
6.6.2 Robustness and Loop Shaping......Page 350
6.6.3 Stability and Performance Boundaries......Page 351
6.6.4 Robust Design for Systems in Standard Model Form......Page 352
6.6.5 Structured Singular Values......Page 354
6.7.1 Steps in a H-infinity Design Procedure......Page 355
6.7.2 Cost Function Weighting Selection for Scalar Systems......Page 356
6.8 Mutivariable Robust Control Design Problem......Page 358
6.8.1 Problems in Multivariable Control......Page 359
6.8.2 Poles and Zeros of Multivariable Systems......Page 360
6.9.1 Selection of Weights in Multivariable Problems......Page 361
6.9.2 Multivariable Submarine Motion Control......Page 362
6.9.3 Multivariable Submarine Control Design Results......Page 364
6.9.4 Speed of Response and Interaction......Page 367
6.10 Restricted Structure and Multiple Model Control......Page 370
6.10.1 Feedforward and Feedback Polynomial System Plant......Page 371
6.10.2 H2 /LQG Restricted Structure Optimal Control Problem......Page 374
6.10.3 Numerical Algorithm for Single- and Multi-model Systems......Page 386
6.10.4 Hot Strip Finishing Mill Tension Control......Page 394
6.10.6 Restricted Structure Benchmarking......Page 403
6.11 Concluding Remarks......Page 405
6.12 Problems......Page 406
6.13 References......Page 408
7.1 Introduction......Page 413
7.2 Signal Proces sing Syste m Descri ption......Page 414
7.2.1 Summary of Estimation Problem Assumptions......Page 415
7.2.4 Polynomial Matrix Descriptions......Page 416
7.3 The Standard H-infiinity Optimal Estimation Problem......Page 417
7.3.2 Estim ation Error Power Spectrum: Complet ion of Squares......Page 418
7.3.3 Wiener Filtering Solution......Page 419
7.3.5 Optimal Estimator when Signa l Model Stable......Page 420
7.3.6 Optimal Estimator when Signal Model can be Unstable......Page 423
7.3.7 Optimal Estimator when Signal Model can be Unstable......Page 428
7.4.1 State Estimation Problem......Page 432
7.4.2 Output Filtering and Prediction......Page 433
7.4.3 Deconvolution Estimation......Page 434
7.4.4 Robust Weighting Function W-sigma......Page 437
7.4.5 Extensions of the Estimator Capabilities......Page 438
7.5 Strip Thickness Estimation from Roll Force Measurements......Page 439
7.5.2 Continuous-time Dynamic Mill Model......Page 440
7.6 Strip Thickness Estimation Using Force Measurments......Page 442
7.7 Strip Thickness Estimation Using X-Ray Gauge Measurements......Page 445
7.8 Strip Thickness Estimation Using Gauge Measurements......Page 446
7.9 Time-varying and Nonstationary Filtering......Page 450
7.9.1 Linear Multichannel Estimation Problem......Page 452
7.9.2 Output Estimation Problem......Page 455
7.9.3 Relationship to the Kalman Filtering Problem......Page 459
7.10 Conclusions......Page 464
7.11 Problems......Page 465
7.12 References......Page 466
8.1 Introduction......Page 469
8.1.1 The H-infinity Filtering Problem......Page 470
8.1.2 Smoothing Filters......Page 471
8.2.1 Relationship Between H2 and H-infinity Minimisation Problems......Page 472
8.2.2 Solution Strategy and Weightings......Page 473
8.2.3 Derivation of the Weighting Filter W-lamda......Page 474
8.2.4 Robustness Weighting Diophantine Equation......Page 475
8.2.5 H-infinity Optimal Estimator for the Generalised System Model......Page 476
8.3 H-infinity Deconvolution Filtering Problem......Page 477
8.3.1 Deconvolution System Description......Page 478
8.3.2 Solution of the H-infinity Deconvolution Estimation Problem......Page 479
8.4.1 Discrete -time System and Signal Source Descriptions......Page 481
8.4.2 Duality and the Game Problem......Page 483
8.4.3 Results for the Suboptimal H-infinity Filtering Problem......Page 484
8.4.4 Remarks on the Solution......Page 486
8.6 Final Remarks on the Suboptimal H-infinity Filtering Problem......Page 487
8.7 Problems......Page 488
8.8 References......Page 489
9.1 Introduction......Page 493
9.2 Wind Turbine Power Control Systems......Page 494
9.2.1 Definition of Wind Turbine Transfer Functions......Page 496
9.2.2 Weighting Function Definitions......Page 498
9.2.3 Numerical Results for Wind Turbine Example......Page 500
9.2.4 Wind Turbine Feedback Controller Cancellation Properties......Page 505
9.2.5 Role of the Ideal-response Models in Design......Page 507
9.2.8 Wind Turbine Condition Monitoring......Page 508
9.3.1 System Models......Page 509
9.3.2 Design Requirements and Specification......Page 511
9.3.3 Flight Control System: Time and Frequency Responses......Page 514
9.3.4 Flight Control System Design Including Flexible Modes......Page 518
9.3.5 LQG Flight Control Study Design Results......Page 519
9.3.6 Classical and LQG Controller Design......Page 521
9.4 Thickness Control Systems Design Using Force Feedback......Page 524
9.4.3 Continuous-time Mill Models......Page 526
9.4.4 Definition of the Polynomial Models for the Standard System......Page 527
9.4.5 Cost Function Definition......Page 528
9.4.6 BUR Eccentricity Problem Results......Page 530
9.5 Thickness Control Using Gauge Measurement......Page 534
9.5.1 Transport Delay in Thickness Measurement......Page 536
9.5.3 Choice of Cost Function Weightings for Gauge Feedback Control Problem......Page 540
9.5.4 Degree of Stability......Page 541
9.6 Ship Roll Stabilisation......Page 542
9.6.1 Fin Control Unit......Page 543
9.6.2 Speed Adaptation......Page 544
9.6.4 Weighting Selection for LQG Roll Stabilisation Design......Page 545
9.6.5 Frequency Responses......Page 546
9.6.6 Advantages of the Optimal System in Comparison with Classical Methods......Page 548
9.7 Concluding Remarks......Page 549
9.9 References......Page 550
10.1 Introduction......Page 553
10.1.3 Flight Control Systems......Page 554
10.2 H-infiinity Flight Control Systems Design......Page 556
10.2.2 Definition of Cost Function Weightings......Page 558
10.2.3 Generalised LQG and H-infinity Controller Time- and Frequency-responses......Page 559
10.2.4 Introducing a Measurement Noise Model......Page 564
10.3 H-infinity Gauge Control System Design Using Force Feedback......Page 567
10.3.1 Thickness Control System Frequency- and Time-responses......Page 570
10.3.2 Mismatched Eccentricity Model and Robustness......Page 575
10.3.3 Thickness Profile Control......Page 576
10.4.1 Forces and Moments......Page 578
10.4.2 Depth Control......Page 579
10.4.3 Sea-state and Sea Current Disturbances......Page 580
10.4.4 Submarine Motion Dynamics......Page 582
10.4.5 Submarine Depth and Pitch Control Design......Page 585
10.4.6 Submarine Depth-keeping Controllers......Page 586
10.4.7 Submarine Model Responses......Page 587
10.4.8 Model Tuning......Page 592
10.4.9 Summary of the Output and Input Disturbance Models......Page 595
10.4.10 Submarine Depth and Pitch Control......Page 596
10.4.11 Summary of the Selected Weighting Terms......Page 597
10.4.12 Scalar Design and Responses: Depth Control......Page 598
10.4.13 Scalar Design and Responses: Pitch Control......Page 602
10.5 H-infinity Control of Remotely Operated Underwater Vehicles......Page 604
10.5.1 Design of ROV Controllers......Page 608
10.6.1 H-infinity Fin Roll Stabilisation System Design......Page 609
10.6.2 H-infinity Ship Track-keeping Control......Page 612
10.7 Concluding Remarks......Page 615
10.8 Problems......Page 616
10.9 References......Page 617
11.1 Introduction......Page 619
11.2 Optimal Control of Time-varying Linear Systems......Page 620
11.2.1 Linear Time-varying and Adjoint Operators......Page 621
11.2.2 The Quadratic Cost Index......Page 622
11.2.3 Solution of the Time-varying Linear Quadratic Control Problem......Page 623
11.3 Modelling and Control of Nonlinear Systems......Page 626
11.3.1 Nonlinear Systems Modelling......Page 627
11.3.2 Hard Nonlinearities......Page 628
11.3.4 Feedback Linearisation......Page 629
11.4 NLQG Compensation and Control......Page 631
11.4.1 Nonlinear Control Example......Page 632
11.4.2 Polynomial Versions of Plant Transfer-function Operators......Page 633
11.4.3 Use of Time-varying Cost Function Weighting......Page 634
11.4.4 The NLQG Algorithm and Properties......Page 635
11.5 NLQG Example with Input and Output Nonlinearities......Page 636
11.5.2 Simulation Results......Page 637
11.5.3 Frequency-domain Results......Page 638
11.5.4 Improving NLQG Control Using Future Change Information......Page 644
11.6 Nonlinear Generalised Minimum Variance Control......Page 646
11.6.1 Nonlinear System Description......Page 647
11.6.2 Nonlinear and Linear Subsystem Models......Page 649
11.7 Nonlinear Generalised Minimum Variance Problem......Page 651
11.7.1 Solution of the Nonlinear Feedback/Feedforward Control Problem......Page 653
11.7.3 Diophantine Equations......Page 654
11.7.4 Optimisation......Page 656
11.7.5 Alternative Control Solution and Stability......Page 658
11.7.7 Simplifying the Controller......Page 660
11.7.8 Effect of Bias or Steady-state Levels......Page 661
11.8 Nonlinear GMV Control Problem......Page 663
11.9 Nonlinear Smith Predictor......Page 668
11.9.1 Weighting Selection Based on an Existing Controller......Page 671
11.10 Concluding Remarks......Page 672
11.11 References......Page 673
Notation......Page 677
A1.1 Vectors......Page 678
A1.2 Matrices......Page 679
A1.2.1 Matrix Inverse Relationships......Page 681
A1.2.2 Matrix Singular Value Relationships......Page 682
A1.2.3 Matrix Norm Relationships......Page 683
A1.3 Polynomial Matrices......Page 685
A1.3.1 Polynomial Equations......Page 686
A1.4 Transfer-function Matrices......Page 687
A1.4.1 Adjoint, All-pass and Inner Functions......Page 688
A1.5 Vector and Normed Spaces......Page 689
A1.5.1 Hardy Spaces and Norms......Page 691
A1.6 References......Page 693