Multivariable Feedback Control

Professor Sigurd Skogestad, Norwegian University of Science and Technology (NTNU) Head of the Department of Chemical Engineering. Author of more than 100 journal publications and 150 conference publications. He was awarded "Innstilling to the King" for his Siv.Ing. degree in 1979, a Fullbright fellowship in 1983, received the Ted Peterson Award from AIChE in 1989, the George S. Axelby Outstanding Paper Award from IEEE in 1990, and the O. Hugo Schuck Best Paper Award from the American Automatic Control Council in 1992. Professor Ian Postlethwaite, University of Leicester, UK Head of Engineering Department, Fellow of the Institute of Electrical and Electronics Engineers, Fellow of the Institution of Electrical Engineers, and a Fellow of the Institute of Measurement and Control. In 1991 he received the IEE FC Williams Premium, in 2001 the Sir Harold Hartley Medal of the InstMC and in 2002 the Best Paper Prize for an article published in the IFAC Journal of Control Engineering Practice over the period 1999-2002.

1 Introduction1 1.1 The process of control system design 1 1.2 The control problem 2 1.3 Transfer functions 3 1.4 Scaling 5 1.5 Deriving linear models 7 1.6 Notation 10 2 Classical Feedback Control 15 2.1 Frequency response 15 2.2 Feedback control 20 2.3 Closed-loop stability 26 2.4 Evaluating closed-loop performance 28 2.5 Controller design 40 2.6 Loop shaping 42 2.7 IMC design procedure and PID control for stable plants 54 2.8 Shaping closed-loop transfer functions 59 2.9 Conclusion 65 3 Introduction to Multivariable Control 67 3.1 Introduction 67 3.2 Transfer functions for MIMO systems 68 3.3 Multivariable frequency response analysis 71 3.4 Relative gain array (RGA) 82 3.5 Control of multivariable plants 91 3.6 Introduction to multivariable RHP-zeros 96 3.7 Introduction to MIMO robustness 98 3.8 General control problem formulation 104 3.9 Additional exercises 115 3.10 Conclusion 117 4 Elements of Linear System Theory 119 4.1 System descriptions 119 4.2 State controllability and state observability 127 4.3 Stability 134 4.4 Poles 135 4.5 Zeros 138 4.6 Some important remarks on poles and zeros 141 4.7 Internal stability of feedback systems 144 4.8 Stabilizing controllers 148 4.9 Stability analysis in the frequency domain 150 4.10 System norms 156 4.11 Conclusion 162 5 Limitations on Performance In Siso Systems 163 5.1 Inputoutput controllability 163 5.2 Fundamental limitations on sensitivity 167 5.3 Fundamental limitations: bounds on peaks 172 5.4 Perfect control and plant inversion 180 5.5 Ideal ISE optimal control 181 5.6 Limitations imposed by time delays 182 5.7 Limitations imposed by RHP-zeros 183 5.8 Limitations imposed by phase lag 191 5.9 Limitations imposed by unstable (RHP) poles 192 5.10 Performance requirements imposed by disturbances and commands 198 5.11 Limitations imposed by input constraints 199 5.12 Limitations imposed by uncertainty 203 5.13 Summary: controllability analysis with feedback control 206 5.14 Summary: controllability analysis with feedforward control 209 5.15 Applications of controllability analysis 210 5.16 Conclusion 219 6 Limitations on Performance In Mimo Systems 221 6.1 Introduction 221 6.2 Fundamental limitations on sensitivity 222 6.3 Fundamental limitations: bounds on peaks 223 6.4 Functional controllability 232 6.5 Limitations imposed by time delays 233 6.6 Limitations imposed by RHP-zeros 235 6.7 Limitations imposed by unstable (RHP) poles 238 6.8 Performance requirements imposed by disturbance s238 6.9 Limitations imposed by input constraints 240 6.10 Limitations imposed by uncertainty 242 6.11 MIMO inputoutput controllability 253 6.12 Conclusion 258 7 Uncertainty And Robustness for Siso Systems 259 7.1 Introduction to robustness 259 7.2 Representing uncertainty 260 7.3 Parametric uncertainty 262 7.4 Representing uncertainty in the frequency domain 265 7.5 SISO robust stability 274 7.6 SISO robust performance 281 7.7 Additional exercises 287 7.8 Conclusion 288 8 Robust Stability And Performance Analysis For Mimo Systems 289 8.1 General control configuration with uncertainty 289 8.2 Representing uncertainty 290 8.3 Obtaining P, N and M 298 8.4 Definitions of robust stability and robust performance 299 8.5 Robust stability of the M -structure 301 8.6 Robust stability for complex unstructured uncertainty 302 8.7 Robust stability with structured uncertainty: motivation 305 8.8 The structured singular value 306 8.9 Robust stability with structured uncertainty 313 8.10 Robust performance 316 8.11 Application: robust performance with input uncertainty 320 8.12 -synthesis and DK-iteration 328 8.13 Further remarks on 336 8.14 Conclusion 338 9 Controller Design 341 9.1 Trade-offs in MIMO feedback design 341 9.2 LQG control 344 9.3 H2 and H control 352 9.4 H loop-shaping design 364