MANFRED MORARI ROBUST PROCESS CONTROL EVANGHELOS ZAFIRIOU Morari M,Zafiriou E. New Jersey:Prentice-Hall,1989
Morari M, Zafiriou E. New Jersey: Prentice-Hall, 1989
Contents PREFACE XV NOMENCLATURE xix 1 INTRODUCTION 1 1.1 The Evolution of Control Theory .. 1 1.2 Controller Parametrization:The IMC Structure.. 3 1.3 Robustness 4 1.4 Scope of Book.···.·. 4 1.5 Some Hints for the Reader....... 5 Part I:CONTINUOUS SINGLE-INPUT SINGLE-OUTPUT SYSTEMS 9 2 FUNDAMENTALS OF SISO FEEDBACK CONTROL 11 2.1 Definitions..·.··,······ 11 2.2 Formulation of Control Problem 13 2.2.1 Process[odel..·.·.··· 14 2.2.2 Model Uncertainty Description 15 2.2.3 Input Specification.,····,··· 19 2.2.4 Control Objectives... 21 2.3 Internal Stability........... 22 2.4 Nominal Performance.... 23 2.4.1 Sensitivity and Complementary Sensitivity Function 24 2.4.2 Two-Degree-of-Freedom Controller ·· 25 2.4.3 Asymptotic Properties of the Closed-Loop Response (Sys- tem Type).,·,,, 27 2.4.4 Linear Quadratic (H2-)Optimal Control... 28 2.4.5 Ho-Optimal Control...... 29 2.5 Robust Stability 31 2.6 Robust Performance 3 2.6.1 H2 Performance Objective....... 34 2.6.2 Hoo Performance Objective 35
CONTENTS y 16.3.3 Controllers··········· ..,444 lG.4 Results for Operating Point A...,··.· 445 16.4.1 Discussion of Controllers..·,·,·· 446 16.4.2 Conclusions..... ...451 16.5 Effect of Nonlinearity (Results for Operating Point C) ·、···,.452 l6.5.1 odelling...·..,··,············· 452 453 16.5.2-Analysis.······· 16.5.3 Logarithmic Versus Unscaled Compositions..... 453 16.5.4 Transition from Operating Point A to C...... 455 456 l6.6 Conclusions,....,,····· 456 lG.7 References,.,·· 459 Appendix 469 References 479 Index
CONTENTS VI 2.7 Summary,······· 36 2.8 References..... 38 3 THE SISO IMC STRUCTURE FOR STABLE SYSTEMS 39 3.1 IMC Structure,.·....,,········,·········· 39 3.2 Stability Conditions for IMC·.··.:· 41 3.2.1 Internal Stability.....·.·.··· 41 3.2.2 Relationship with“Classic Feedback” 42 3.2.3 Implementation:Classic Feedback Versus IMC 44 3.3 Performance of IMC·.,·,················ 45 3.3.1 Sensitivity and Complementary Sensitivity Function 45 3.3.2 Two-Degree-of-Freedom Controller..... 46 3.3.3 Asymptotic Properties of Closed-Loop Response (System Type)·········: 46 3.3.4 The Concept of "Perfect Control" 47 3.4 Outline of the IMC Design Procedure... 50 3.4.1 Basic Design Philosophy.····· 50 3.4.2Two-Degree-of-Freedom Design.··, 52 3.4.3 Design in the Presence of Measurement Device Dynamics .54 3.5 Summary·,··········· 55 55 3.6 References.....,.·.·. I SISO IMC DESIGN FOR STABLE SYSTEMS 57 4.1 Nominal Performance........ 57 4.1.1 H2-Optimal Control.... 58 4.1.2 Design for Specific Input Forms.. 60 4.1.3 Minimum Error Norm for Step Inputs ..... 61 4.2 The IMC Filter.,·,,,··········,······ 64 4.2.1 Filter Form...· 64 4.2.2 A Qualitative Interpretation of the Function of the Filter 65 l.3 Robust Stability·,·..········ 66 4.3.1 Norm-Bounded Uncertainty Regions...... 66 4.3.2 General.Uncertainty Regions·..· 68 t.+Robust Performance·,..········· 70 4.4.1 Norm-Bounded Uncertainty Regions 70 +.4.2 General Uncertainty Regions··,·..···.· 73 1.5 Summary of IMC Design Procedure... 74 4.6 Application:IMC Design for a First-Order Deadtime System 75 4.6.1 Deadtime Uncertainty,...··..··.·,········ 75 4.6.2 Three-Parameter Uncertainty······ 81
vii CONTENTS 84 4.7 References...···· 85 5 SISO IMC DESIGN FOR UNSTABLE SYSTEMS 5.1 Parametrization of All Stabilizing Controllers...... 85 5.1.1 Conditions for Internal Stability 85 5.l.2 Controller parametrization.····· 87 5.2 Nominal Performance....,,······ 88 5.2.1H2-Optimal Controller.,.·:,,.··. 88 5.2.2 Design for Common Input Forms,····、·,·· 92 5.2.3 Minimum Error Norm for Step Inputs to Stable Systems.. 94 5.2.4Two-Degree-of-Freedom Controller··, 94 5.3 The IMC Filter·..··············· 96 5.3.1 Filter Form.··.,·· 96 5.3.2 Qualitative Interpretation of the Filter Function 99 5.4 Robust Stability....······ 101 5.4.1 Norm-Bounded Uncertainty Regions....... ...101 5.4.2 General Uncertainty Regions... .102 5.5 Robust Performance .. 102 5.6 Summary of the IMC Design Procedure 103 5.7 Applications,········· ..104 5.7.1 Distillation Column Base Level Control 104 5.7.2 NMP Unstable Systems..... 107 58 References.......·.···· ,.110 6 ISSUES IN SISO IMC DESIGN 113 6.1 Implications of IMC for Classic Feedback Controllers .··.113 6.l.1 General Relationships...,..·.,...·. 113 6.l.2 PID Settings for Simple Models..·..··· 114 6.1.3 PID Settings for a First-Order System with Deadtime ·.,121 6.1.4 Summary...·.········· ·········.125 6.2 IMC Interpretation of Smith Predictor Controller..........126 6.2.1 General Relationships ....126 6.2.2 Some Myths about the Tuning of Smith-Predictor Controllers128 6.2.3 Robust Tuning of Smith-Predictor Controller for First- Order System with Deadtime.,·.··...·....,..l30 6.2.4 Summary··········· 131 6.3 Feedforward Control....... 131 6.3.1 Objectives and Structure 131 6.3.2 Design.······ 132 6.3.3 Summary..... ,,135
vui CONTENTS 6.4 Cascade Control........ ···.135 6.4.1 Objectives,Structure,and Design ...135 6.4.2 Implementation.,·.·· .·.,138 6.4.3 Summary,....··. .,..138 6.5 References ...... .··,139 Part II:SAMPLED DATA SINGLE-INPUT SINGLE-OUTPUT SYSTEMS 141, 7 FUNDAMENTALS OF SAMPLED-DATA SYSTEMS CON- TROL 143 7.1 Sampled-Data Feedback Structure... ·..143 7.2 IMC Structure,...··...,· 147 7.3 Formulation of Control Problem ...147 7.3.1 Process Model····,··· .149 7.3.2 Model Uncertainty Description ,149 7.4 Internal Stability··,.,.,.····,··· ...150 7.5 Nominal Performance.......... 151 7.5.1 Sensitivity and Complementary Sensitivity Function....151 7.5.2 Asymptotic Properties of Closed-Loop Response ,...,.153 7.5.3 Limitations on Achievable Performance.... 155 7.5.4 Discrete Linear Quadratic (H2-)Optimal Control...... 158 7.5.5 Hoo Performance Objective 159 7.6 Robust Stability········· 160 7.7 Robust Performance....... 161 7.7.1 H2 Performance Objective.. 161 7.7.2 Hoo Performance Objective 162 i.3 Summary··..······ .162 79 References...·········· 164 SISO IMC DESIGN FOR STABLE SAMPLED-DATA SYS- TEMS 165 s.1 Nominal Performance··...,. ...165 8.1.1 H;-Optimal Control... ..165 8.1.2 Design of the IMC Controller g(=) ..168 s.The Discrete IMC Filter,··,.·· 170 3 Robust Stability..··· ..174 8.3.1 Filter Design·,· .....174 8.3.2 Effect of Sampling ..176 Robust Performance... 、.176
CONTENTS ix 8.4.1 Filter Design ...177 8.4.2 Sampling Time Selection ..178 8.4.3 Example.····· 178 8.5 Summary ...181 8.6 References..················ ,..182 9 SISO DESIGN FOR UNSTABLE SAMPLED-DATA SYSTEMS183 9.1 Parametrization of All Stabilizing Controllers ..... .....183 9.l.1 Internal Stability....··.... ..183 9.1.2 Controller Parametrization ..184 9.2 Nominal Performance···..,· ,.184 9.2.1 H2-Optimal Controller.... ·..185 9.2.2 Design of the IMC Controller g(z) ..188 9.2.3Anti-aliasing Prefilter..,·.·· 。。 189 9.2.4 Design for Common Input Forms. 190 9.2.5 Integral Squared Error (ISE)for Step Inputs to Stable Sys- 9.3 The Discrete IMC Filter........ 193 9.3.1 Filter Form..... 193 9.3.2 Qualitative Interpretation of the Filter Function ···...195 9.4 Robust Stability.·····,···········, ...198 9.5 Robust Performance.·· 198 9.6 Summary of the IMC Design Procedure ..199 9.7 Application:Distillation Column Base Level Control 200 98 References....·················· 201 Part III:CONTINUOUS MULTI-INPUT MULTI-OUTPUT SYSTEMS 203 10 FUNDAMENTALS OF MIMO FEEDBACK CONTROL 205 10.1 Definitions and Basic Principles..... 205 10.1.1 Modeling.·.···.· 205 10.1.2 Poles..... 206 10.1.3 Zeros.·. ..207 10.1.4 Vector and Matrix Norms 208 10.1.5 Singular Values and the Singular Value Decomposition... 211 10.1.6 Norms on Function Spaces 215 l0.2 Classic Feedback.·.····.,·,. 217 10.2.1 Definitions.. 217 10.2.2 Multivariable Nyquist Criterion.... ..217
CONTENTS 10.2.3 Internal Stability 221 10.2.4 Small Gain Theorem .. 221 10.3 Formulation of Control Problem 222 10.3.1 Process Model 222 10.3.2 Model Uncertainty Description 223 l0.3.3 Input Specifications...···. 225 10.3.4 Control Objectives.···.,. 226 10.4 Nominal Performance......... 227 10.4.1 Sensitivity and Complementary Sensitivity Function 227 10.4.2 Asymptotic Properties of Closed-Loop Response (System Type)······::· 228 10.4.3 Linear Quadratic(H2-)Optimal Control.... 228 l0.4.4 Ho-Optimal Control,··.·.·..·,· 230 10.5 Summary············ 232 l0.6 References.,,,··.······ 233 11 ROBUST STABILITY AND PERFORMANCE 235 11.1 Robust Stability for Unstructured Uncertainty 236 11.1.1 Uncertainty Description ........ 236 11.1.2 General Robust Stability Theorem 237 11.1.3 Multiplicative Output Uncertainty 。。。 239 11.1.4 Multiplicative Input Uncertainty .... 240 11.1.5 Inverse Multiplicative Output Uncertainty 242 11.1.6 Example:Input Uncertainty for Distillation Column ·...242 11.1.7 Integral Control and Robust Stability 244 11.2 Robust Stability for Structured Uncertainty 246 11.2.1 Uncertainty Description ....... 246 11.2.2 Structured Singular Value...... 248 11.2.3 Simultaneous Multiplicative Input and Output Uncertainty 252 11.2.4 Batch Reactor:Simultaneous Parametric and Unstruc- tured Uncertainty.,,..·,,···,.·,, 254 11.2.5 Independent Uncertainty in the Transfer Matrix Elements.257 11.2.6 Condition Number and Relative Gain Array as Sensitivity easures...·.············· 258 1l.3 Robust Performance.·......·. 262 11.3.1 Hoo-Performance Objective... 262 11.3.2 Multiplicative Output Uncertainty 264 11.3.3 Multiplicative Input Uncertainty 266 11.3.4 H2-Performance Objective ..... 268 11.3.5 Application:High-Purity Distillation...... 271
CONTENTS xi 11.4 Robustness Conditions in Terms of Specific Transfer Matrices...275 11.4.1 How to find the LFT..... 278 11.4.2 New Properties of 281 11.4.3 xamples.,.... 285 11.5 Summary ..... 287 11.6 References ..... 290 12 MIMO IMC DESIGN 293 12.1 IMC Structure·,. ..293 12.2 Conditions for Internal Stability ..295 12.3 Parametrization of All Stabilizing Controllers 296 12.4 Asymptotic Properties of Closed-Loop Response 297 12.5 Outline of the IMC Design Procedure 298 12.6 Nominal Performance........... 301 12.6.1 Assumptions············ 301 12.6.2 H2-Optimal Control for a Specific Input 302 12.6.3 H2-Optimal Control for a Set of Inputs 306 l2.6.4 Algorithm for“【nner-Outer”Factorization. 309 12.7 Robust Stability and Performance........ 310 12.7.1 Filter Structure,.··.、,····· 311 12.7.2 General Interconnection Structure with Filter 。 313 12.7.3 Robust Control:Ho Performance Objective.. 315 12.7.4 Robust Control:H2-Type Performance Objective .,318 l2.8 Application:High-Purity Distillation,···. 320 12.9 Summary··.············ 324 12.10Discussion and References ... 325 13 PERFORMANCE LIMITATIONS FOR MIMO SYSTEMS 327 13.1 Effect of Plant Gain 328 13.1.1 Constraints on Manipulated Variable...... 328 l3.l.2 Disturbance Condition Number.·...,··,,· 329 13.1.3 Implications of Ka for Closed-Loop Performance... 330 13.1.4 Decomposition of d along Singular Vectors...... 332 13.1.5 Summary 338 l3.2 NMP Characteristics.,,·· 339 13.2.1 Zero Direction 339 13.2.2 Implications of Zero Direction for Achievable Performance. 342 13.2.3 Summary·························· 343 13.3 Sensitivity to Model Uncertainty .. 344 13.3.1 Sensitivity to Diagonal Input Uncertainty .. 344
1 CONTENTS 13.3.2 Sensitivity with Different Controller Structures....... 346 l3.3.3“Vorst-Case”Uncertainty.,.... ..,348 13.3.4 Example,:.······· .·.349 13.3.5 Summary 351 l3.4 References..,,·,·,, 356 1 DECENTRALIZED CONTROL 359 14.1 Motivation 359 14.2 Definitions........ 361 14.3 Necessary Conditions for Controllability 362 14.3.1 Results 362 14.3.2 Proofs... 365 14.4 Stability Conditions -Interaction Measures 367 14.4.1 Necessary and Sufficient Stability Conditions 368 14.4.2 Sufficient Stability Conditions...... ....370 14.4.3 Diagonal Dominance Interaction Measures 372 14.4.4 Generalized Diagonal Dominance Interaction Measures...373 l4.4.5Theh-Interaction Measure····,···· 374 14.4.6 Interaction Measures for 2 x 2 Systems 375 l4.4.7 Examples..················· ...376 14.5 Robust Performance Conditions.... .,.378 14.5.1 Sufficient Conditions for Robust Performance ...379 l4.5.2 Design Procedure.,·· ...382 14.5.3 Example...... 3S3 l+.6 Summary.··· 388 l+.References...· ..389 Part IV:SAMPLED DATA MULTI-INPUT MULTI-OUTPUT SYSTEMS 391 15 MIMO Sampled-Data Systems 393 15.1 Fundamentals of MIMO Sampled-Data Systems...... 393 15.1.1 Sampled-Data Feedback ... 393 15.1.2 Poles and Zeros ...... 395 15.1.3 Internal Stability ..... 396 15.1.4 IMC Structure ... 397 15.1.5 Model Uncertainty Description 397 15.2 Nominal Internal Stability ..... 399 15.2.1 IMC Structure.. 399 15.2.2 Feedback Structure .. 400