S.Skogestad and I.Postlethwaite, "Multivariable feedback control - analysis and design", 2nd Edition, Wiley, 2005, 590 pages. --------------------------------------------------------------------- Table of contents (extended compared to the one given in the book) --------------------------------------------------------------------- 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 -- 7 1.6 Notation -- 10 2 CLASSICAL FEEDBACK CONTROL -- 15 2.1 Frequency response -- 15 o Frequency-by-frequency sinusoids -- 16 2.2 Feedback control -- 20 2.2.1 One degree-of-freedom controller -- 20 2.2.2 Closed-loop transfer functions -- 21 2.2.3 Two degrees-of-freedom and feedforward control -- 23 2.2.4 Why feedback? -- 24 2.2.5 High-gain feedback -- 24 2.3 Closed-loop stability -- 26 2.4 Evaluating closed-loop performance -- 28 2.4.1 Typical closed-loop responses -- 28 2.4.2 Time domain performance -- 30 2.4.3 Frequency domain performance -- 32 o Gain and phase margins -- 32 o Maximum peak criteria -- 35 2.4.4 Relationship between time and frequency domain peaks -- 37 2.4.5 Bandwidth and crossover frequency -- 38 2.5 Controller design -- 40 2.6 Loop shaping -- 42 2.6.1 Trade-offs in terms of $L$ -- 42 2.6.2 Fundamentals of loop-shaping design -- 43 o Limitations imposed by RHP-zeros and time delays -- 45 2.6.3 Inverse-based controller design -- 46 2.6.4 Loop shaping for disturbance rejection -- 48 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 IMC design procedure and PID control for stable plants -- 54 2.8 Shaping closed-loop transfer functions -- 59 2.8.1 The terms ${\@mathcal H _\infty $\ and ${\@mathcal H _2$ -- 60 2.8.2 Weighted sensitivity -- 60 2.8.3 Stacked requirements: mixed sensitivity -- 62 2.9 Conclusion -- 65 3 INTRODUCTION TO MULTIVARIABLE CONTROL -- 67 3.1 Introduction -- 67 3.2 Transfer functions for MIMO systems -- 68 o Negative feedback control systems -- 69 3.3 Multivariable frequency response analysis -- 71 3.3.1 Obtaining the frequency response from $G(s)$ -- 71 3.3.2 Directions in multivariable systems -- 73 3.3.3 Eigenvalues are a poor measure of gain -- 75 3.3.4 Singular value decomposition -- 75 o Non-square plant -- 79 3.3.5 Singular values for performance -- 80 3.3.6 Condition number -- 81 3.4 Relative gain array (RGA) -- 82 3.4.1 Original interpretation: RGA as an interaction measure -- 83 3.4.2 Examples: RGA -- 85 3.4.3 RGA number and iterative RGA -- 87 3.4.4 Summary of algebraic properties of the RGA -- 88 3.4.5 Summary of control properties of the RGA -- 89 3.5 Control of multivariable plants -- 91 3.5.1 Diagonal controller (decentralized control) -- 91 3.5.2 Two-step compensator design approach -- 91 3.5.3 Decoupling -- 92 3.5.4 Pre- and post-compensators and the SVD controller -- 93 3.5.5 What is the shape of the ``best'' feedback controller? -- 93 3.5.6 Multivariable controller synthesis -- 94 3.5.7 Summary of mixed-sensitivity ${\@mathcal H _\infty $\ synthesis ($S/KS$) -- 94 3.6 Introduction to multivariable RHP-zeros -- 95 3.7 Introduction to MIMO robustness -- 98 3.7.1 Motivating robustness example no. 1: spinning satellite -- 98 3.7.2 Motivating robustness example no. 2: distillation process -- 100 3.7.3 Robustness conclusions -- 103 3.8 General control problem formulation -- 104 3.8.1 Obtaining the generalized plant $P$ -- 105 3.8.2 Controller design: including weights in $P$ -- 106 3.8.3 Partitioning the generalized plant $P$ -- 108 3.8.4 Analysis: closing the loop to get $N$ -- 108 3.8.5 Generalized plant $P$: further examples -- 109 3.8.6 Deriving $P$ from $N$ -- 111 3.8.7 Problems not covered by the general formulation -- 112 3.8.8 A general control configuration including model uncertainty -- 113 3.9 Additional exercises -- 115 3.10 Conclusion -- 117 4 ELEMENTS OF LINEAR SYSTEM THEORY -- 119 4.1 System descriptions -- 119 4.1.1 State-space representation -- 119 4.1.2 Impulse response representation -- 121 4.1.3 Transfer function representation -- Laplace transforms -- 121 4.1.4 Frequency response -- 122 4.1.5 Coprime factorization -- 122 4.1.6 More on state-space realizations -- 125 4.2 State controllability and state observability -- 127 4.3 Stability -- 134 4.4 Poles -- 135 4.4.1 Poles and stability -- 135 4.4.2 Poles from state-space realizations -- 135 4.4.3 Poles from transfer functions -- 135 4.4.4 Pole vectors and directions -- 137 4.5 Zeros -- 138 4.5.1 Zeros from state-space realizations -- 138 4.5.2 Zeros from transfer functions -- 139 4.5.3 Zero directions -- 140 4.6 Some important remarks on poles and zeros -- 141 4.7 Internal stability of feedback systems -- 144 4.7.1 Implications of the internal stability requirement -- 146 4.8 Stabilizing controllers -- 148 4.8.1 Stable plants -- 148 4.8.2 Unstable plants -- 149 4.9 Stability analysis in the frequency domain -- 150 4.9.1 Open- and closed-loop characteristic polynomials -- 151 o Relationship between characteristic polynomials -- 151 4.9.2 MIMO Nyquist stability criteria -- 152 4.9.3 Eigenvalue loci -- 154 4.9.4 Small-gain theorem -- 155 4.10 System norms -- 156 4.10.1 ${\@mathcal H _2$\ norm -- 157 4.10.2 ${\@mathcal H _\infty $\ norm -- 158 4.10.3 Difference between the ${\@mathcal H _2$\ and ${\@mathcal H _\infty $\ norms -- 159 4.10.4 Hankel norm -- 160 4.11 Conclusion -- 162 5 LIMITATIONS ON PERFORMANCE IN SISO SYSTEMS -- 163 5.1 Input--output controllability -- 163 5.1.1 Input--output controllability analysis -- 164 5.1.2 Scaling and performance -- 165 5.1.3 Remarks on the term controllability -- 166 5.2 Fundamental limitations on sensitivity -- 167 5.2.1 $S$ plus $T$ is one -- 167 5.2.2 Interpolation constraints -- 167 5.2.3 The waterbed effects (sensitivity integrals) -- 167 o Pole excess of two: first waterbed formula -- 168 o RHP-zeros: second waterbed formula -- 169 5.3 Fundamental limitations: bounds on peaks -- 172 5.3.1 Minimum peaks for $S$ and $T$ -- 172 5.3.2 Minimum peaks for other closed-loop transfer functions -- 175 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 5.7.1 Time response: inverse response and undershoot -- 184 5.7.2 High-gain instability -- 184 5.7.3 Frequency response: bandwidth limitation -- 185 5.7.4 RHP-zeros and non-causal controllers -- 189 5.7.5 LHP-zeros -- 191 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 5.11.1 Inputs for perfect control -- 200 5.11.2 Inputs for acceptable control -- 201 5.11.3 Inputs for stabilization -- 201 5.12 Limitations imposed by uncertainty -- 203 5.12.1 Feedforward control and uncertainty -- 203 5.12.2 Feedback control and uncertainty -- 204 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 5.15.1 First-order delay process -- 210 5.15.2 Application: room heating -- 211 5.15.3 Application: neutralization process -- 213 5.15.4 Additional exercises -- 218 5.16 Conclusion -- 219 6 LIMITATIONS ON PERFORMANCE IN MIMO SYSTEMS -- 221 6.1 Introduction -- 221 6.2 Fundamental limitations on sensitivity -- 222 6.2.1 $S$ plus $T$ is the identity matrix -- 222 6.2.2 Interpolation constraints -- 223 6.2.3 Sensitivity integrals -- 223 6.3 Fundamental limitations: bounds on peaks -- 223 6.3.1 Minimum peaks for $S$ and $T$ -- 224 6.3.2 Minimum peaks for other closed-loop transfer functions -- 229 6.4 Functional controllability -- 232 6.5 Limitations imposed by time delays -- 233 6.6 Limitations imposed by RHP-zeros -- 235 6.6.1 Moving the effect of a RHP-zero to a specific output -- 236 6.7 Limitations imposed by unstable (RHP) poles -- 238 6.8 Performance requirements imposed by disturbances -- 238 6.9 Limitations imposed by input constraints -- 240 6.9.1 Inputs for perfect control -- 240 6.9.2 Inputs for acceptable control -- 241 6.9.3 Inputs for stabilization -- 241 6.10 Limitations imposed by uncertainty -- 242 6.10.1 Input and output uncertainty -- 242 6.10.2 Effect of uncertainty on feedforward control -- 243 6.10.3 Uncertainty and the benefits of feedback -- 246 6.10.4 Effect of uncertainty on feedback sensitivity peak -- 247 o Upper bound on $\mathaccent "7016\relax \sigma (S')$ for output uncertainty -- 248 o Upper bounds on $\mathaccent "7016\relax \sigma (S')$ for input uncertainty -- 248 o Lower bound on $\mathaccent "7016\relax \sigma (S')$ for input uncertainty (including diagonal input uncertainty) -- 248 o Conclusions on input uncertainty and feedback control -- 251 6.10.5 Element-by-element uncertainty -- 251 6.10.6 Steady-state condition for integral control -- 252 6.11 MIMO input--output controllability -- 253 6.11.1 Controllability analysis procedure -- 253 6.11.2 Plant design changes -- 255 6.11.3 Additional exercises -- 256 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 7.4.1 Uncertainty regions -- 265 7.4.2 Representing uncertainty regions by complex perturbations -- 267 7.4.3 Obtaining the weight for complex uncertainty -- 268 7.4.4 Choice of nominal model -- 270 7.4.5 Neglected dynamics represented as uncertainty -- 271 7.4.6 Unmodelled dynamics uncertainty -- 273 7.5 SISO robust stability -- 274 7.5.1 RS with multiplicative uncertainty -- 275 7.5.2 Comparison with gain margin -- 279 7.5.3 RS with inverse multiplicative uncertainty -- 279 7.6 SISO robust performance -- 281 7.6.1 SISO nominal performance in the Nyquist plot -- 281 7.6.2 Robust performance -- 281 7.6.3 The relationship between NP, RS and RP -- 285 7.6.4 The similarity between RS and RP -- 286 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 8.2.1 Differences between SISO and MIMO systems -- 292 8.2.2 Parametric uncertainty -- 292 8.2.3 Unstructured uncertainty -- 293 o Lumping uncertainty into a single perturbation -- 294 o Moving uncertainty from the input to the output -- 295 8.2.4 Diagonal uncertainty -- 296 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\Delta $-structure -- 301 8.6 Robust stability for complex unstructured uncertainty -- 302 8.6.1 Application of the unstructured RS condition -- 303 8.6.2 RS for coprime factor uncertainty -- 304 8.7 Robust stability with structured uncertainty: motivation -- 305 8.8 The structured singular value -- 306 8.8.1 Remarks on the definition of $\mu $ -- 308 8.8.2 Properties of $\mu $ for real and complex $\Delta $ -- 308 8.8.3 $\mu $ for complex $\Delta $ -- 309 o Properties of $\mu $ for complex perturbations -- 309 8.9 Robust stability with structured uncertainty -- 313 8.9.1 What do $\mu \not =1$ and skewed-$\mu $ mean? -- 316 8.10 Robust performance -- 316 8.10.1 Testing RP using $\mu $ -- 316 o Block diagram proof of Theorem {8.7\hbox { -- 317 o Algebraic proof of Theorem {8.7\hbox { -- 319 8.10.2 Summary of $\mu $-conditions for NP, RS and RP -- 319 8.10.3 Worst-case performance and skewed-$\mu $ -- 320 8.11 Application: robust performance with input uncertainty -- 320 8.11.1 Interconnection matrix -- 321 8.11.2 RP with input uncertainty for SISO system -- 322 8.11.3 RP for $2\times 2$ distillation process -- 322 8.11.4 RP and the condition number -- 324 o Worst-case performance (any controller) -- 326 8.11.5 Comparison with output uncertainty -- 327 8.12 $\mu $-synthesis and $DK$-iteration -- 328 8.12.1 $DK$-iteration -- 328 8.12.2 Adjusting the performance weight -- 329 8.12.3 Fixed structure controller -- 329 8.12.4 Example: $\mu $-synthesis with $DK$-iteration -- 330 o Analysis of $\mu $-``optimal'' controller $K_3$ -- 332 8.13 Further remarks on $\mu $ -- 336 8.13.1 Further justification for the upper bound on $\mu $ -- 336 8.13.2 Real perturbations and the mixed $\mu $-problem -- 336 8.13.3 Computational complexity -- 336 8.13.4 Discrete case -- 337 8.14 Conclusion -- 338 o Practical $\mu $-analysis -- 339 9 CONTROLLER DESIGN -- 341 9.1 Trade-offs in MIMO feedback design -- 341 9.2 LQG control -- 344 9.2.1 Traditional LQG and LQR problems -- 344 9.2.2 Robustness properties -- 349 9.2.3 Loop transfer recovery (LTR) procedures -- 351 9.3 ${\cal H _{2 $ and ${\cal H _{\infty $ control -- 352 9.3.1 General control problem formulation -- 353 9.3.2 ${\cal H _{2 $ optimal control -- 355 9.3.3 LQG: a special ${\cal H _{2 $ optimal controller -- 356 9.3.4 ${\cal H _{\infty $ optimal control -- 357 9.3.5 Mixed-sensitivity ${\cal H _{\infty $ control -- 359 9.3.6 Signal-based ${\cal H _{\infty $ control -- 362 9.4 ${\cal H _{\infty $ loop-shaping design -- 364 9.4.1 Robust stabilization -- 365 9.4.2 A systematic ${\cal H _{\infty $ loop-shaping design procedure -- 368 9.4.3 Two degrees-of-freedom controllers -- 372 9.4.4 Observer-based structure for ${\cal H _{\infty $ loop-shaping controllers -- 376 9.4.5 Implementation issues -- 380 9.5 Conclusion -- 381 10 CONTROL STRUCTURE DESIGN -- 383 10.1 Introduction -- 383 10.2 Optimal operation and control -- 385 10.3 Selection of primary controlled outputs -- 388 10.3.1 Self-optimizing control -- 390 10.3.2 Selecting controlled outputs: local analysis -- 392 10.3.3 Selecting controlled outputs: maximum scaled gain method -- 394 10.3.4 Selecting controlled outputs: exact local method -- 396 10.3.5 Selecting controlled outputs: direct evaluation of cost -- 396 10.3.6 Selecting controlled outputs: measurement combinations -- 397 10.3.7 Selecting controlled outputs: examples -- 398 10.3.8 Selection of controlled variables: summary -- 402 10.4 Regulatory control layer -- 403 10.4.1 Objectives of regulatory control -- 403 10.4.2 Selection of variables for regulatory control -- 404 10.4.3 Stabilization: pole vectors -- 411 10.4.4 Local disturbance rejection: partial control -- 414 o 1. Cascade control system -- 415 o 2. Sequentially designed decentralized control system -- 417 o 3. Indirect control -- 417 o Optimal ``stabilizing" control in terms of minimizing drift -- 418 10.5 Control configuration elements -- 419 10.5.1 Why use simplified control configurations? -- 421 10.5.2 Cascade control systems -- 422 10.5.3 Extra measurements: cascade control -- 423 10.5.4 Extra inputs -- 426 10.5.5 Extra inputs and outputs -- 427 10.5.6 Selectors -- 428 10.6 Decentralized feedback control -- 428 10.6.1 Introduction -- 428 10.6.2 Introductory examples -- 430 10.6.3 Notation and factorization of sensitivity function -- 436 10.6.4 Stability of decentralized control systems -- 438 10.6.5 Integrity and negative RGA elements -- 442 10.6.6 RHP-zeros and RGA: reasons for avoiding negative RGA elements with sequential design -- 445 10.6.7 Performance of decentralized control systems -- 447 10.6.8 Summary: pairing selection and controllability analysis for decentralized control -- 448 10.6.9 Independent design -- 449 10.6.10 Sequential design -- 450 10.6.11 Conclusions on decentralized control -- 453 10.7 Conclusion -- 453 11 MODEL REDUCTION -- 455 11.1 Introduction -- 455 11.2 Truncation and residualization -- 456 11.2.1 Truncation -- 456 11.2.2 Residualization -- 456 11.3 Balanced realizations -- 457 11.4 Balanced truncation and balanced residualization -- 458 11.5 Optimal Hankel norm approximation -- 459 11.6 Reduction of unstable models -- 462 11.6.1 Stable part model reduction -- 462 11.6.2 Coprime factor model reduction -- 462 11.7 Model reduction using Matlab -- 462 11.8 Two practical examples -- 463 11.8.1 Reduction of a gas turbine aero-engine model -- 463 11.8.2 Reduction of an aero-engine controller -- 466 11.9 Conclusion -- 471 12 LINEAR MATRIX INEQUALITIES -- 473 12.1 Introduction to LMI problems -- 473 12.1.1 Fundamental LMI properties -- 474 12.1.2 Systems of LMIs -- 475 12.2 Types of LMI problems -- 476 12.2.1 LMI feasibility problems -- 476 12.2.2 Linear objective minimization problems -- 477 12.2.3 Generalized eigenvalue problems -- 477 12.3 Tricks in LMI problems -- 479 12.3.1 Change of variables -- 480 12.3.2 Congruence transformation -- 481 12.3.3 Schur complement -- 481 12.3.4 The S-procedure -- 482 12.3.5 The projection lemma and Finsler's lemma -- 483 12.4 Case study: anti-windup compensator synthesis -- 484 12.4.1 Representing anti-windup compensators -- 485 12.4.2 Lyapunov stability -- 487 12.4.3 $\@mathcal {L _{2 $ gain -- 487 12.4.4 Sector boundedness -- 487 12.4.5 Full-order anti-windup compensators -- 488 12.4.6 Anti-windup synthesis -- 488 12.5 Conclusion -- 490 13 CASE STUDIES -- 491 13.1 Introduction -- 491 13.2 Helicopter control -- 492 13.2.1 Problem description -- 492 13.2.2 The helicopter model -- 493 13.2.3 ${\cal H _{\infty $ mixed-sensitivity design -- 494 13.2.4 Disturbance rejection design -- 496 13.2.5 Comparison of disturbance rejection properties of the two designs -- 499 13.2.6 Conclusions -- 499 13.3 Aero-engine control -- 500 13.3.1 Problem description -- 500 13.3.2 Control structure design: output selection -- 502 13.3.3 A two degrees-of-freedom ${\cal H _{\infty $ loop-shaping design -- 506 13.3.4 Analysis and simulation results -- 508 13.3.5 Conclusions -- 509 13.4 Distillation process -- 509 13.4.1 Idealized $LV$-model -- 510 13.4.2 Detailed $LV$-model -- 512 13.4.3 Idealized $DV$-model -- 513 13.4.4 Further distillation case studies -- 514 13.5 Conclusion -- 514 A MATRIX THEORY AND NORMS -- 515 A.1 Basics -- 515 A.1.1 Some useful matrix identities -- 516 A.1.2 Some determinant identities -- 517 A.2 Eigenvalues and eigenvectors -- 518 A.2.1 Eigenvalue properties -- 519 A.2.2 Eigenvalues of the state matrix -- 519 A.2.3 Eigenvalues of transfer functions -- 520 A.3 Singular value decomposition -- 520 A.3.1 Rank -- 521 A.3.2 Singular values of a $2\times 2$ matrix -- 521 A.3.3 SVD of a matrix inverse -- 522 A.3.4 Singular value inequalities -- 522 A.3.5 SVD as a sum of rank $1$ matrices -- 523 A.3.6 Singularity of matrix $A+E$ -- 524 A.3.7 Economy-size SVD -- 524 A.3.8 Pseudo-inverse (generalized inverse) -- 524 o Principal component regression (PCR) -- 525 A.3.9 Condition number -- 525 A.4 Relative gain array -- 526 A.4.1 Algebraic properties of the RGA -- 527 A.4.2 RGA of a non-square matrix -- 528 A.5 Norms -- 530 A.5.1 Vector norms -- 531 A.5.2 Matrix norms -- 532 o Induced matrix norms -- 533 o Implications of the multiplicative property -- 534 A.5.3 The spectral radius -- 535 A.5.4 Some matrix norm relationships -- 536 A.5.5 Matrix and vector norms in Matlab -- 537 A.5.6 Signal norms -- 537 A.5.7 Signal interpretation of various system norms -- 539 A.6 All-pass factorization of transfer function matrices -- 541 A.7 Factorization of the sensitivity function -- 542 A.7.1 Output perturbations -- 542 A.7.2 Input perturbations -- 542 A.7.3 Stability conditions -- 543 A.8 Linear fractional transformations -- 543 A.8.1 Interconnection of LFTs -- 544 A.8.2 Relationship between $F_l$ and $F_u$ -- 545 A.8.3 Inverse of LFTs -- 545 A.8.4 LFT in terms of the inverse parameter -- 545 B PROJECT WORK AND SAMPLE EXAM -- 547 B.1 Project work -- 547 B.2 Sample exam -- 548 BIBLIOGRAPHY -- 553 INDEX -- 563