S.Skogestad and I.Postlethwaite, "Multivariable feedback control - analysis and design", Wiley, 1996, 570 pages. --------------------------------------------------------------------- Table of contents (extended compared to the one given in the book) --------------------------------------------------------------------- PREFACE -- ix 1 INTRODUCTION -- 1 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 -- 8 1.6 Notation -- 11 2 CLASSICAL FEEDBACK CONTROL -- 15 2.1 Frequency response -- 15 o Frequency-by-frequency sinusoids -- 16 2.2 Feedback control -- 21 2.2.1 One degree-of-freedom controller -- 21 2.2.2 Closed-loop transfer functions -- 22 2.2.3 Why feedback? -- 23 2.3 Closed-loop stability -- 24 2.4 Evaluating closed-loop performance -- 27 2.4.1 Typical closed-loop responses -- 27 2.4.2 Time domain performance -- 29 2.4.3 Frequency domain performance -- 30 o Gain and phase margins -- 31 o Maximum peak criteria -- 33 2.4.4 Relationship between time and frequency domain peaks -- 35 2.4.5 Bandwidth and crossover frequency -- 36 2.5 Controller design -- 39 2.6 Loop shaping -- 40 2.6.1 Trade-offs in terms of $L$ -- 40 2.6.2 Fundamentals of loop-shaping design -- 41 o Limitations imposed by RHP-zeros and time delays -- 45 2.6.3 Inverse-based controller design -- 45 2.6.4 Loop shaping for disturbance rejection -- 47 2.6.5 Two degrees-of-freedom design -- 51 o Loop shaping applied to a flexible structure -- 53 2.6.6 Conclusions on loop shaping -- 54 2.7 Shaping closed-loop transfer functions -- 54 2.7.1 The terms ${\cal H}_\infty$ and ${\cal H_2}$ -- 55 2.7.2 Weighted sensitivity -- 56 2.7.3 Stacked requirements: mixed sensitivity -- 58 2.8 Conclusion -- 62 3 INTRODUCTION TO MULTIVARIABLE CONTROL -- 63 3.1 Introduction -- 63 3.2 Transfer functions for MIMO systems -- 64 o Negative feedback control systems -- 66 3.3 Multivariable frequency response analysis -- 68 3.3.1 Obtaining the frequency response from $G(s)$ -- 68 3.3.2 Directions in multivariable systems -- 69 3.3.3 Eigenvalues are a poor measure of gain -- 71 3.3.4 Singular value decomposition -- 72 o Non-Square plants -- 77 o Use of the minimum singular value of the plant -- 77 3.3.5 Singular values for performance -- 77 3.4 Control of multivariable plants -- 79 3.4.1 Decoupling -- 80 3.4.2 Pre- and post-compensators and the SVD-controller -- 81 3.4.3 Diagonal controller (decentralized control) -- 81 3.4.4 What is the shape of the ``best'' feedback controller? -- 81 3.4.5 Multivariable controller synthesis -- 82 3.4.6 Summary of mixed-sensitivity ${\cal H}_\infty $ design ($S/KS$) -- 83 3.5 Introduction to multivariable RHP-zeros -- 84 3.6 Condition number and RGA -- 86 3.6.1 Condition number -- 87 3.6.2 Relative Gain Array (RGA) -- 88 3.7 Introduction to MIMO robustness -- 91 3.7.1 Motivating robustness example no. 1: Spinning Satellite -- 91 3.7.2 Motivating robustness example no. 2: Distillation Process -- 93 3.7.3 Robustness conclusions -- 97 3.8 General control problem formulation -- 98 3.8.1 Obtaining the generalized plant $P$ -- 99 3.8.2 Controller design: Including weights in $P$ -- 100 3.8.3 Partitioning the generalized plant $P$ -- 102 3.8.4 Analysis: Closing the loop to get $N$ -- 103 3.8.5 Generalized plant $P$: Further examples -- 104 3.8.6 Deriving $P$ from $N$ -- 107 3.8.7 Problems not covered by the general formulation -- 107 3.8.8 A general control configuration including model uncertainty -- 108 3.9 Additional exercises -- 110 3.10 Conclusion -- 112 4 ELEMENTS OF LINEAR SYSTEM THEORY -- 113 4.1 System descriptions -- 113 4.1.1 State-space representation -- 113 4.1.2 Impulse response representation -- 115 4.1.3 Transfer function representation - Laplace transforms -- 115 4.1.4 Frequency response -- 116 4.1.5 Coprime factorization -- 116 4.1.6 More on state-space realizations -- 118 4.2 State controllability and state observability -- 122 4.3 Stability -- 127 4.4 Poles -- 127 4.4.1 Poles and stability -- 128 4.4.2 Poles from state-space realizations -- 128 4.4.3 Poles from transfer functions -- 128 4.5 Zeros -- 130 4.5.1 Zeros from state-space realizations -- 131 4.5.2 Zeros from transfer functions -- 131 4.6 More on poles and zeros -- 132 4.6.1 Directions of poles and zeros -- 132 4.6.2 Remarks on poles and zeros -- 134 4.7 Internal stability of feedback systems -- 137 4.7.1 Implications of the internal stability requirement -- 140 4.8 Stabilizing controllers -- 142 4.8.1 Stable plants -- 142 4.8.2 Unstable plants -- 143 4.9 Stability analysis in the frequency domain -- 144 4.9.1 Open and closed-loop characteristic polynomials -- 144 o Relationship between characteristic polynomials -- 145 4.9.2 MIMO Nyquist stability criteria -- 146 4.9.3 Eigenvalue loci -- 149 4.9.4 Small gain theorem -- 149 4.10 System norms -- 151 4.10.1 ${\cal H}_2$\ norm -- 152 4.10.2 ${\cal H}_\infty $\ norm -- 153 4.10.3 Difference between the ${\cal H}_2$\ and ${\cal H}_\infty $\ norms -- 153 4.10.4 Hankel norm -- 155 4.11 Conclusion -- 157 5 LIMITATIONS ON PERFORMANCE IN SISO SYSTEMS -- 159 5.1 Input-Output Controllability -- 159 5.1.1 Input-output controllability analysis -- 161 5.1.2 Scaling and performance -- 161 5.1.3 Remarks on the term controllability -- 162 5.2 Perfect control and plant inversion -- 163 5.3 Constraints on $S$ and $T$ -- 164 5.3.1 $S$ plus $T$ is one -- 164 5.3.2 The waterbed effects (sensitivity integrals) -- 165 o Pole excess of two: First waterbed formula -- 165 o RHP-zeros: Second waterbed formula -- 167 5.3.3 Interpolation constraints -- 170 5.3.4 Sensitivity peaks -- 170 5.4 Ideal ISE optimal control -- 172 5.5 Limitations imposed by time delays -- 173 5.6 Limitations imposed by RHP-zeros -- 174 5.6.1 Inverse response -- 174 5.6.2 High-gain instability -- 175 5.6.3 Bandwidth limitation I -- 175 5.6.4 Bandwidth limitation II -- 177 o Performance at low frequencies -- 177 o Performance at high frequencies -- 178 5.6.5 Limitations at low or high frequencies -- 179 5.6.6 Remarks on the effects of RHP-zeros -- 181 5.7 Non-causal controllers -- 182 5.8 Limitations imposed by RHP-poles -- 184 5.9 Combined RHP-poles and RHP-zeros -- 185 5.10 Performance requirements imposed by disturbances and commands -- 187 5.11 Limitations imposed by input constraints -- 189 5.11.1 Inputs for perfect control -- 190 5.11.2 Inputs for acceptable control -- 191 5.11.3 Unstable plant and input constraints -- 192 5.12 Limitations imposed by phase lag -- 193 5.13 Limitations imposed by uncertainty -- 195 5.13.1 Feedforward control -- 195 5.13.2 Feedback control -- 195 5.14 Controllability analysis with feedback control -- 196 5.15 Controllability analysis with feedforward control -- 200 5.16 Applications of controllability analysis -- 201 5.16.1 First-order delay process -- 201 5.16.2 Application: Room heating -- 203 5.16.3 Application: Neutralization process -- 205 5.16.4 Additional exercises -- 211 5.17 Conclusion -- 212 6 LIMITATIONS ON PERFORMANCE IN MIMO SYSTEMS -- 213 6.1 Introduction -- 213 6.2 Constraints on $S$ and $T$ -- 214 6.2.1 $S$ plus $T$ is the identity matrix -- 214 6.2.2 Sensitivity integrals -- 215 6.2.3 Interpolation constraints -- 215 6.2.4 Sensitivity peaks -- 216 6.3 Functional controllability -- 218 6.4 Limitations imposed by time delays -- 220 6.5 Limitations imposed by RHP-zeros -- 221 6.5.1 Moving the effect of a RHP-zero to a specific output -- 221 6.6 Limitations imposed by RHP-poles -- 224 6.7 RHP-poles combined with RHP-zeros -- 224 6.8 Performance requirements imposed by disturbances -- 226 6.9 Limitations imposed by input constraints -- 228 6.9.1 Inputs for perfect control -- 229 6.9.2 Inputs for acceptable control -- 230 o Exact conditions -- 230 o Approximate conditions in terms of the SVD -- 231 6.10 Limitations imposed by uncertainty -- 234 6.10.1 Input and output uncertainty -- 234 6.10.2 Effect of uncertainty on feedforward control -- 235 6.10.3 Uncertainty and the benefits of feedback -- 236 6.10.4 Uncertainty and the sensitivity peak -- 237 o Upper bound on $\sigma (S')$ for output uncertainty -- 238 o Upper bounds on $\sigma (S')$ for input uncertainty -- 238 o Lower bound on $\sigma (S')$ for input uncertainty -- 239 o Conclusions on input uncertainty and feedback control -- 244 6.10.5 Element-by-element uncertainty -- 244 6.10.6 Steady-state condition for integral control -- 245 6.11 Input-output controllability -- 246 6.11.1 Controllability analysis procedure -- 246 6.11.2 Plant design changes -- 248 6.11.3 Additional exercises -- 249 6.12 Conclusion -- 252 7 UNCERTAINTY AND ROBUSTNESS FOR SISO SYSTEMS -- 253 7.1 Introduction to robustness -- 253 7.2 Representing uncertainty -- 255 7.3 Parametric uncertainty -- 257 7.4 Representing uncertainty in the frequency domain -- 259 7.4.1 Uncertainty regions -- 259 7.4.2 Representing uncertainty regions by complex perturbations -- 260 7.4.3 Obtaining the weight for complex uncertainty -- 262 7.4.4 Choice of nominal model -- 265 7.4.5 Neglected dynamics represented as uncertainty -- 266 7.4.6 Unmodelled dynamics uncertainty -- 268 7.5 SISO Robust stability -- 270 7.5.1 RS with multiplicative uncertainty -- 270 7.5.2 Comparison with gain margin -- 274 7.5.3 RS with inverse multiplicative uncertainty -- 275 7.6 SISO Robust performance -- 276 7.6.1 SISO nominal performance in the Nyquist plot -- 276 7.6.2 Robust performance -- 277 7.6.3 The relationship between NP, RS and RP -- 281 7.6.4 The similarity between RS and RP -- 282 7.7 Examples of parametric uncertainty -- 283 7.7.1 Parametric pole uncertainty -- 283 7.7.2 Parametric zero uncertainty -- 284 7.7.3 Parametric state-space uncertainty -- 285 7.8 Additional exercises -- 289 7.9 Conclusion -- 290 8 ROBUST STABILITY AND PERFORMANCE ANALYSIS -- 291 8.1 General control configuration with uncertainty -- 291 8.2 Representing uncertainty -- 294 8.2.1 Differences between SISO and MIMO systems -- 294 8.2.2 Parametric uncertainty -- 295 8.2.3 Unstructured uncertainty -- 295 o Lumping uncertainty into a single perturbation -- 296 o Moving uncertainty from the input to the output -- 297 8.2.4 Diagonal uncertainty -- 299 8.3 Obtaining $P$, $N$ and $M$ -- 301 8.4 Definitions of robust stability and robust performance -- 303 8.5 Robust stability of the $M\Delta $-structure -- 304 8.6 RS for complex unstructured uncertainty -- 306 8.6.1 Application of the unstructured RS-condition -- 307 8.6.2 RS for coprime factor uncertainty -- 308 8.7 RS with structured uncertainty: Motivation -- 309 8.8 The structured singular value -- 311 8.8.1 Remarks on the definition of $\mu $ -- 313 8.8.2 Properties of $\mu $ for real and complex $\Delta $ -- 313 8.8.3 $\mu $ for complex $\Delta $ -- 314 o Properties of $\mu $ for complex perturbations -- 314 8.9 Robust stability with structured uncertainty -- 319 8.9.1 What do $\mu \not =1$ and skewed-$\mu $ mean? -- 321 8.10 Robust performance -- 322 8.10.1 Testing RP using $\mu $ -- 322 o Block diagram proof of Theorem 8.7 -- 323 o Algebraic proof of Theorem 8.7 -- 325 8.10.2 Summary of $\mu $-conditions for NP, RS and RP -- 325 8.10.3 Worst-case performance and skewed-$\mu $ -- 326 8.11 Application: RP with input uncertainty -- 326 8.11.1 Interconnection matrix -- 328 8.11.2 RP with input uncertainty for SISO system -- 328 8.11.3 Robust performance for $2\times 2$ distillation process -- 329 8.11.4 $\mu $ and the condition number -- 332 o Worst-case performance (any controller) -- 333 8.11.5 Comparison with output uncertainty -- 334 8.12 $\mu $-synthesis and $DK$-iteration -- 335 8.12.1 $DK$-iteration -- 335 8.12.2 Adjusting the performance weight -- 336 8.12.3 Fixed structure controller -- 337 8.12.4 Example: $\mu $-synthesis with $DK$-iteration -- 337 o Analysis of $\mu $-``optimal'' controller $K_3$ -- 340 8.13 Further remarks on $\mu $ -- 344 8.13.1 Further justification for the upper bound on $\mu $ -- 344 8.13.2 Real perturbations and the mixed $\mu $ problem -- 344 8.13.3 Computational complexity -- 345 8.13.4 Discrete case -- 345 8.13.5 Relationship to linear matrix inequalities (LMIs) -- 346 8.14 Conclusion -- 347 o Practical $\mu $-analysis -- 348 9 CONTROLLER DESIGN -- 349 9.1 Trade-offs in MIMO feedback design -- 349 9.2 LQG control -- 352 9.2.1 Traditional LQG and LQR problems -- 353 9.2.2 Robustness properties -- 357 9.2.3 Loop transfer recovery (LTR) procedures -- 361 9.3 ${\cal H}_2$ and ${\cal H}_\infty$ control -- 362 9.3.1 General control problem formulation -- 362 9.3.2 ${\cal H}_2$ optimal control -- 365 9.3.3 LQG: a special ${\cal H}_2$ optimal controller -- 365 9.3.4 ${\cal H}_\infty $ optimal control -- 366 9.3.5 Mixed-sensitivity ${\cal H}_\infty $ control -- 369 9.3.6 Signal-based ${\cal H}_\infty $ control -- 373 9.4 ${\cal H}_\infty $ loop-shaping design -- 376 9.4.1 Robust stabilization -- 376 9.4.2 A systematic ${\cal H}_\infty $ loop-shaping design procedure -- 380 9.4.3 Two degrees-of-freedom controllers -- 385 9.4.4 Observer-based structure for ${\cal H}_\infty $ loop-shaping controllers -- 390 9.4.5 Implementation issues -- 393 9.5 Conclusion -- 396 10 CONTROL STRUCTURE DESIGN -- 397 10.1 Introduction -- 397 10.2 Optimization and control -- 399 10.3 Selection of controlled outputs -- 402 o Measurement selection for indirect control -- 406 10.4 Selection of manipulations and measurements -- 408 10.5 RGA for non-square plant -- 410 10.6 Control configuration elements -- 413 10.6.1 Cascade control systems -- 415 10.6.2 Cascade control: Extra measurements -- 415 10.6.3 Cascade control: Extra inputs -- 418 10.6.4 Extra inputs and outputs (local feedback) -- 420 10.6.5 Selectors -- 420 10.6.6 Why use cascade and decentralized control? -- 420 10.7 Hierarchical and partial control -- 422 10.7.1 Partial control -- 422 10.7.2 Hierarchical control and sequential design -- 424 o Sequential design of cascade control systems -- 428 10.7.3 ``True'' partial control -- 428 10.8 Decentralized feedback control -- 432 10.8.1 RGA as interaction measure for decentralized control -- 434 10.8.2 Stability of decentralized control systems -- 435 o Sufficient conditions for stability -- 435 o Necessary steady-state conditions for stability -- 438 10.8.3 The RGA and right-half plane zeros -- 441 10.8.4 Performance of decentralized control systems -- 441 10.8.5 Summary: Controllability analysis for decentralized control -- 443 10.8.6 Sequential design of decentralized controllers -- 447 10.8.7 Conclusions on decentralized control -- 448 10.9 Conclusion -- 448 11 MODEL REDUCTION -- 449 11.1 Introduction -- 449 11.2 Truncation and residualization -- 450 11.2.1 Truncation -- 450 11.2.2 Residualization -- 451 11.3 Balanced realizations -- 451 11.4 Balanced truncation and balanced residualization -- 452 11.5 Optimal Hankel norm approximation -- 454 11.6 Two practical examples -- 456 11.6.1 Reduction of a gas turbine aero-engine model -- 456 11.6.2 Reduction of an aero-engine controller -- 459 11.7 Reduction of unstable models -- 465 11.7.1 Stable part model reduction -- 465 11.7.2 Coprime factor model reduction -- 465 11.8 Model reduction using MATLAB -- 466 11.9 Conclusion -- 467 12 CASE STUDIES -- 469 12.1 Introduction -- 469 12.2 Helicopter control -- 470 12.2.1 Problem description -- 470 12.2.2 The helicopter model -- 471 12.2.3 ${\cal H}_\infty $ mixed-sensitivity design -- 474 12.2.4 Disturbance rejection design -- 475 12.2.5 Comparison of disturbance rejection properties of the two designs -- 477 12.2.6 Conclusions -- 479 12.3 Aero-engine control -- 480 12.3.1 Problem description -- 480 12.3.2 Control structure design: output selection -- 481 12.3.3 A two degrees-of-freedom ${\cal H}_\infty $ loop-shaping design -- 486 12.3.4 Analysis and simulation results -- 488 12.3.5 Conclusions -- 489 12.4 Distillation process -- 490 12.4.1 Idealized $LV$-model -- 490 12.4.2 Detailed $LV$-model -- 494 12.4.3 Idealized $DV$-model -- 495 12.4.4 Further distillation case studies -- 496 12.5 Conclusion -- 496 APPENDIX A: MATRIX THEORY AND NORMS -- 497 A.1 Basics -- 497 A.1.1 Some useful matrix identities -- 498 A.1.2 Some determinant identities -- 499 A.2 Eigenvalues and eigenvectors -- 500 A.2.1 Eigenvalue properties -- 501 A.2.2 Eigenvalues of the state matrix -- 502 A.2.3 Eigenvalues of transfer functions -- 502 A.3 Singular Value Decomposition -- 503 A.3.1 Rank -- 504 A.3.2 Singular values of a $2\times 2$ matrix -- 504 A.3.3 SVD of a matrix inverse -- 505 A.3.4 Singular value inequalities -- 505 A.3.5 SVD as a sum of rank $1$ matrices -- 506 A.3.6 Singularity of matrix $A+E$ -- 507 A.3.7 Economy-size SVD -- 507 A.3.8 Pseudo-inverse (Generalized inverse) -- 507 o Principal component regression (PCR) -- 508 A.3.9 Condition number -- 508 A.4 Relative Gain Array -- 510 A.4.1 Properties of the RGA -- 510 A.4.2 RGA of a non-square matrix -- 512 A.4.3 Computing the RGA with MATLAB -- 513 A.5 Norms -- 514 A.5.1 Vector norms -- 515 A.5.2 Matrix norms -- 516 o Induced matrix norms -- 517 o Implications of the multiplicative property -- 519 A.5.3 The spectral radius -- 520 A.5.4 Some matrix norm relationships -- 520 A.5.5 Matrix and vector norms in MATLAB -- 521 A.5.6 Signal norms -- 522 A.5.7 Signal interpretation of various system norms -- 524 A.6 Factorization of the sensitivity function -- 526 A.6.1 Output perturbations -- 526 A.6.2 Input perturbations -- 527 A.6.3 Stability conditions -- 527 A.7 Linear fractional transformations -- 528 A.7.1 Interconnection of LFTs -- 529 A.7.2 Relationship between $F_l$ and $F_u$. -- 530 A.7.3 Inverse of LFTs -- 530 A.7.4 LFT in terms of the inverse parameter -- 530 A.7.5 Generalized LFT: The matrix star product -- 531 APPENDIX B: PROJECT WORK and SAMPLE EXAM -- 533 B.1 Project work -- 533 B.2 Sample exam -- 534 BIBLIOGRAPHY -- 539 INDEX -- 548