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The authors wish to thank the large number of colleagues and friends who have worked with us in the area of control over the years. This book is really a synthesis of ideas that they helped us to formulate. All three authors spent time together in the Centre for Industrial Control Science at the University of Newcastle, Australia. This was a fertile breeding ground for many discussions on the principles of control. Financial support from the Australian Government for this centre under the Commonwealth Special Centres program is gratefully acknowledged. Also, financial and other support was provided by the Universidad T´ecnica Federico Santa Mar´ıa covering, amongst other things, several visits to Chile by the first author during the writing of this book. Many students and colleagues read drafts of the book ranging over a five year period. The authors accept full responsibility for the views expressed in the book (and all remaining errors)
CONTROL SYSTEM DESIGN Graham C Goodwin1 Stefan F Graebe2 Mario E Salgado3 Valpara´ıso, January 2000 Centre for Integrated Dynamics and Control University of Newcastle, NSW 2308 AUSTRALIA OMV Aktiengesellschaft Department of Optimization/Automation Schwechat, AUSTRIA Departamento de Electr´ onica Universidad T´ecnica Federico Santa Mar´ıa Valpara´ıso, CHILE Dedicated, in thankful appreciation for support and understanding, to Rosslyn Alice Mariv´ı CONTENTS OVERVIEW I THE ELEMENTS The Excitement of Control Engineering Introduction to the Principles of Feedback Modeling Continuous Time Signals and Systems II 21 41 65 SISO CONTROL ESSENTIALS Analysis of SISO Control Loops Classical PID Control Synthesis of SISO Controllers 117 121 157 177 SISO CONTROL DESIGN Fundamental Limitations in SISO Control Frequency Domain Design Limitations Architectural Issues in SISO Control Dealing with Constraints 195 199 239 263 291 III 10 11 IV DIGITAL COMPUTER CONTROL 12 Models for Sampled Data Systems 13 Digital Control 14 Hybrid Control 313 317 351 385 V ADVANCED SISO CONTROL 15 SISO controller Parameterizations 16 Control Design Based on Optimization 17 Linear State Space Models 18 Synthesis via State Space Methods 19 Introduction to Nonlinear Control 401 405 455 483 515 547 VI MIMO CONTROL ESSENTIALS 20 Analysis of MIMO Control Loops 21 Exploiting SISO Techniques in MIMO Control 583 587 627 VII MIMO CONTROL DESIGN 22 Design via Optimal Control Techniques 23 Model Predictive Control 24 Fundamental Limitations in MIMO Control 649 653 715 743 VIII ADVANCED MIMO CONTROL 25 MIMO Controller Parameterizations 26 Decoupling 779 783 823 vii CONTENTS CONTENTS OVERVIEW vii ACKNOWLEDGEMENTS xxi PREFACE xxiii APPENDICES xxix I THE ELEMENTS PREVIEW THE EXCITEMENT OF CONTROL ENGINEERING 1.1 Preview 1.2 Motivation for Control Engineering 1.3 Historical Periods of Control Theory 1.4 Types of Control System Design 1.5 System Integration 1.6 Summary 1.7 Further Reading 5 10 11 18 19 INTRODUCTION TO THE PRINCIPLES OF FEEDBACK 2.1 Preview 2.2 The Principal Goal of Control 2.3 A Motivating Industrial Example 2.4 Definition of the Problem 2.5 Prototype Solution to the Control Problem via Inversion 21 21 21 22 27 29 ix x Contents Overview 2.6 2.7 2.8 2.9 2.10 2.11 High Gain Feedback and Inversion From Open to Closed Loop Architectures Trade-offs Involved in Choosing the Feedback Gain Measurements Summary Further Reading MODELING 3.1 Preview 3.2 The Raison d’ˆ etre for Models 3.3 Model Complexity 3.4 Building Models 3.5 Model Structures 3.6 State Space Models 3.7 Solution of Continuous Time State Space Models 3.8 High Order Differential and Difference Equation Models 3.9 Modeling Errors 3.10 Linearization 3.11 Case Studies 3.12 Summary 3.13 Further Reading 3.14 Problems for the Reader CONTINUOUS TIME SIGNALS AND SYSTEMS 4.1 Preview 4.2 Linear Continuous Time Models 4.3 Laplace Transforms 4.4 Laplace Transform Properties and Examples 4.5 Transfer Functions 4.6 Stability of Transfer Functions 4.7 Impulse and Step Responses of Continuous Time Linear Systems 4.8 Poles, Zeros and Time Responses 4.9 Frequency Response 4.10 Fourier Transform 4.11 Frequently Encountered Models 4.12 Modeling Errors for Linear Systems 4.13 Bounds for Modeling Errors 4.14 Summary 32 34 36 36 37 39 41 41 41 42 44 45 45 49 50 50 52 57 58 60 61 65 65 65 66 67 70 74 74 76 85 92 97 99 103 104 Contents Overview 4.15 Further Reading 4.16 Problems for the Reader II SISO CONTROL ESSENTIALS xi 108 109 115 PREVIEW 117 ANALYSIS OF SISO CONTROL LOOPS 5.1 Preview 5.2 Feedback Structures 5.3 Nominal Sensitivity Functions 5.4 Closed Loop Stability Based on the Characteristic Polynomial 5.5 Stability and Polynomial Analysis 5.6 Root Locus (RL) 5.7 Nominal Stability using Frequency Response 5.8 Relative Stability: Stability Margins and Sensitivity Peaks 5.9 Robustness 5.10 Summary 5.11 Further Reading 5.12 Problems for the Reader 119 119 119 123 125 126 132 136 141 143 148 150 152 CLASSICAL PID CONTROL 6.1 Preview 6.2 PID Structure 6.3 Empirical Tuning 6.4 Ziegler-Nichols (Z-N) Oscillation Method 6.5 Reaction Curve Based Methods 6.6 Lead-lag Compensators 6.7 Distillation Column 6.8 Summary 6.9 Further Reading 6.10 Problems for the Reader 157 157 157 160 160 164 167 169 172 172 174 SYNTHESIS OF SISO CONTROLLERS 7.1 Preview 7.2 Polynomial Approach 7.3 PI and PID Synthesis Revisited using Pole Assignment 7.4 Smith Predictor 177 177 177 185 187 xii Contents Overview 7.5 7.6 7.7 III Summary Further Reading Problems for the Reader SISO CONTROL DESIGN 188 190 191 195 PREVIEW 197 FUNDAMENTAL LIMITATIONS IN SISO CONTROL 8.1 Preview 8.2 Sensors 8.3 Actuators 8.4 Disturbances 8.5 Model Error Limitations 8.6 Structural Limitations 8.7 Remedies 8.8 An Industrial Application (Reversing Mill) 8.9 Design Homogeneity Revisited 8.10 Summary 8.11 Further Reading 8.12 Problems for the Reader 199 199 200 201 203 204 205 220 225 229 230 233 235 FREQUENCY DOMAIN DESIGN LIMITATIONS 9.1 Preview 9.2 Bode’s Integral Constraints on Sensitivity 9.3 Integral Constraints on Complementary Sensitivity 9.4 Poisson Integral Constraint on Sensitivity 9.5 Poisson Integral Constraint on Complementary Sensitivity 9.6 Example of Design Trade-offs 9.7 Summary 9.8 Further Reading 9.9 Problems for the Reader 239 239 240 244 246 252 254 257 258 261 10 ARCHITECTURAL ISSUES IN SISO CONTROL 10.1 Preview 10.2 Models for Deterministic Disturbances and Reference Signals 10.3 Internal Model Principle for Disturbances 10.4 Internal Model Principle for Reference Tracking 263 263 263 265 269 Section 26.11 Further Reading 869 is due to the saturated control signal not being able to annihilate the control errors sufficiently fast compared to the controller dynamics; therefore the control states continue to grow in response to the non-decreasing control These wound up states produce the transients when the loop emerges from saturation can be compensated by a direct generalization of the SISO anti wind-up implementation • The second phenomenon is specific to MIMO systems is due to uncompensated interactions arising from the input vector loosing its original design direction • In analogy to the SISO case, there can be regions in state space, from which an open loop unstable MIMO system with input saturation cannot be stabilized by any control • More severely than in the SISO case, MIMO systems are difficult to control in the presence of input saturation, even if the linear loop is stable and the controller is implemented with anti wind-up This is due to saturation changing the directionality of the input vector • This problem of preserving decoupling in the presence of input saturation can be addressed by anti wind-up schemes that scale the control error rather than the control signal 26.11 Further Reading Dynamic decoupling Desoer, C and Gă undes, A (1986) Decoupling linear multi-input multi-output plants by dynamic output feedback An algebraic theory IEEE Transactions on Automatic Control, 31(8):744–750 Falb, P and Wolowich, W (1967) Decoupling in the design and synthesis of multivariable control system Automatica, 12:651–669 Gilbert, E (1969) The decoupling of multivariable systems by state feedback J SIAM Control, 7(1):50–63 Goodwin, G., Feuer, A., and G´ omez, G (1997) A state-space technique for the evaluation of diagonalizing compensator Systems and Control Letters, 32(3):173– 177 Hammer, J and Khargonekar, P (1984) Decoupling of linear delay equations Journal Mathematical System Theory, pages 135–137 870 Decoupling Chapter 26 Hautus, M and Heymann, M (1983) Linear feedback decoupling–transfer function analysis IEEE Transactions on Automatic Control, 28(8):823–832 Lin, C-A and Hsie, T.F (1991) Decoupling controller design for linear multivariable plants Automatica, 36:485–489 Morse, A and Wonham, W (1973) Status of noninteracting control Transactions on Automatic Control, 16:568–581 IEEE Williams, T and Antsaklis, P (1986) A unifying approach to the decoupling of linear multivariable systems International Journal of Control, 44(1):181–201 Wonham, W (1985) Linear Multivariable Control: A Geometric Approach Springer– Verlag, third edition Decoupling invariants Commault, C., Descusse, J., Dion, J., Lafay, J., and Malabre, M (1986) New decoupling invariants: the essential orders International Journal of Control, 44(3):689–700 Dion, J M and Commault, C (1988) The minimal delay decoupling problem: feedback implementation with stability SIAM Journal of Control and Optimization, 26(1):6681 Gă undes, A (1990) Parameterization of all decoupling compensators and all achievable diagonal maps for the unity-feedback system In Proceedings of the 29th CDC, Hawai, pages 2492–2493 Lin, C-A (1995) Necessary and sufficient conditions for existence of decoupling controlles IEEE Transactions on Automatic Control, 42(8):1157–1161 Cost of decoupling G´ omez, G and Goodwin, G (1996) Integral constraints on sensitivity vectors for multivariable linear systems Automatica, 32(4):499–518 Section 26.12 26.12 871 Problems for the Reader Problems for the Reader Problem 26.1 A discrete time MIMO system has a nominal model with transfer function Gq (z), where Gq (z) = z−a (z − 0.7)(z − 0.9) 0.5 −0.5 z (26.12.1) 26.1.1 For a = 1, build a decoupled loop such that the closed loop poles are located inside the circle with radius 0.4 26.1.2 Repeat for a = Problem 26.2 Consider a stable MIMO plant having nominal transfer function Go (s) 26.2.1 If Go (s) is minimum phase, discuss the feasibility to design a controller such that the input sensitivity Sio (s) is diagonal 26.2.2 Repeat your analysis for the case when Go (s) is non-minimum phase Problem 26.3 Consider a MIMO system having a model given by 2(−s + 3) 0.5(−s + 3) (−s + α) 0.5(s + 1) −β −(s + 2) Go (s) = (s + 1)2 (s + 2) −1 0.5 2.5 (26.12.2) with α = −2 and β = It is desired to achieve dynamic decoupling 26.3.1 Is it possible without spreading the non minimum phase zeros in to the three channels? 26.3.2 If α = and β = −1, are your conclusions still valid? Problem 26.4 Consider again the plant of problem 26.3 26.4.1 Repeat problem 26.3, but aiming for a triangular (lower and upper) design 26.4.2 Discuss the issue of dynamic decoupling cost when β = −1 and α has an uncertain value around α = (This is a hard problem.) 872 Decoupling Chapter 26 Problem 26.5 Consider a process with a nominal model given by Go (s) = (s + 1)(s + 2) 0.5 −1 s+2 (26.12.3) Design a Q − controller to achieve dynamic decoupling and zero steady state error for constant references and step disturbances Problem 26.6 Discuss the difficulties of achieving dynamic decoupling for input disturbances for stable although not necessarily minimum phase plants Problem 26.7 Consider a plant with the same nominal model as in problem 26.5 Design a digital controller assuming that the sampling rate is ∆ = 0.1[s] Name Index Ahlen, A., 473 Anderson, B., 295, 579 Antsaklis, P., 687 Araki, M, 294 Astră om, K., 24, 99, 188, 227, 294, 295, 348 Athani, V., 613 Athans, M., 579 Barratt, C., 614 Bellman, R., 579 Bensoussan, D., 579 Bernstein, D., 472 Black, H., 24 Bode, H., 24, 403 Bohlin, T., 38, 427 Bongiorno, J., 348, 427 Boyd, S., 614 Braslavsky, J., 227, 403, 404, 428 Bristol, E., 579 Brockett, R., 135, 171, 172 Brown, J., 403 Bryant, G., 257 Bryson, A., 579 Bucy, R., 579, 580 Cadzow, J., 99 Callier, F., 531 Campbell, D., 38 Campo, P., 472 Cannon, R., 38 Chan, S., 580 Chen, C., 134, 226 Chen, J., 613 Chen, T., 294 Churchill, R., 403 Clark, M., 257 Commault, C., 687 Crisafulli, S., 188, 189 D’Azzo, J., 24 Dahleh, M., 614 Daoutidis, P., 473 Davison, E., 643 De Don´ a, J., 473 De Souza, C., 580 Dendle, D., 257 Descusse, J., 687 Desoer, C., 348, 531, 642, 686 Diaz-Bobillo, I., 614 Dion, J., 687 Distefano, J., 78 Doeblin, E., 24 Doetsch, G., 77 Dorato, P., 427 Dorf, R., 24 Doyle, J., 227, 256, 347, 348, 403, 426, 427, 531, 532, 579, 580, 614, 642 Edwards, J., 614 Edwards, W., 257 Elsley, G., 24 Emami–Naeini, A., 24 Evans, W., 171 Falb, P., 532, 579, 642, 687 873 874 Name Index Feuer, A., 99, 100, 294, 295, 473, 531, 687 Fortmann, T., 227 Francis, B., 227, 256, 294, 347, 381, 403, 426, 427 Franklin, G., 24, 294 Freudenberg, J., 403, 404, 613 Fuller, A., 24 Garcia, C., 473 Gevers, M., 428, 580 Gilbert, E., 135, 473, 687 Glover, K., 348, 426, 427, 531, 579, 580, 614, 642 G´ omez, G., 257, 531, 613, 687 Goodwin, G., 24, 38, 99, 100, 257, 294, 295, 348, 381, 404, 427, 428, 472, 473, 580, 613, 643, 687 Graebe, S., 24, 38, 78, 227, 257, 427, 472, 473 Green, M., 532 Gă ucálă u, A., 643 Gă undes, A., 686, 687 Hagander, P., 295 Hă agglund, T., 188, 227 Hagiwara, T., 294 Hamid, N., 78, 99 Hammer, J., 687 Hanus, R., 472 Hautus, M., 687 Henrotle, J., 472 Heymann, M., 687 Hitz, K., 227 Ho, Y., 135, 579 Horowitz, I., 227, 404 Houpis, C., 24 Hovd, M., 579, 643 Hsie, T., 687 Hung, Y., 532 Isaksson, A., 227, 348 Isidori, A., 472 532, 227, 403, 531, 348, Ito, Y., 294 Jabr, H., 348, 427 James, H., 24 Jankovi´c, M., 472 Joseph, P., 579 Jury, E., 294 Kailath, T., 134, 135, 531, 580 Kalman, R., 135, 579, 580 Kanellakopoulos, I., 474 Kapoor, N., 473 Keerthi, S., 473 Keller, J., 295 Khalil, H., 472 Khargonekar, P., 426, 687 King, W., 257 Kinnaert, M., 472 Kokotovi´c, P., 472, 474 Kondo, K., 257 Kothare, M., 472 Krsti´c, M., 474 Krzyz, J., 403 Kucera, V., 580 Kuo, B., 24, 294 Kwakernaak, H., 99, 135, 579 Lafay, J., 687 Lancaster, P., 580 Landau, I., 189 Lathi, B., 77 Levine, W., 24 Levinson, N., 403 Limebeer, D., 532 Lin, C-A., 687 Liu, R., 348, 642 Ljung, L., 38, 39, 427 Lo, C-P., 381 Longman, R., 381 Looze, D., 403, 613 Luenberger, D., 135 MacFarlane, A., 532, 614 Maciejowski, J., 531, 532 MacLeod, I., 188, 189 875 Name Index Maddock, R., 172 Malabre, M., 687 Manousiouthakis, V., 579 Matsumori, Y., 257 Maxwell, T., 24 Mayne, D., 473, 532 Mayr, O., 24 McFarlane, D., 532 Mellquist, M., 188, 189 Melsa, J, 134 Michalska, H., 473 Michel, A., 472 Middleton, R., 78, 100, 257, 381, 404 Miller, D., 428 Misaka, Y., 257 Miyagi, T., 257 Moore, J., 579 Moran, W., 295 Morari, M., 189, 227, 347, 348, 427, 472, 473, 531, 642 Morse, A., 687 Mu, Z., 257 Murray, J., 348, 642 Muscato, G., 427 Muske, K., 473 Narendra, K., 135 Nett, C., 472, 613 Nichols, N., 24, 189 Ninness, B., 295, 428 Nyquist, H., 172 Ogata, K., 24, 38, 134, 294 Okamato, M., 257 Oppenheim, A, 78, 99 ¨ uler, B., 643 Ozg¨ Papoulis, A., 78 Payne, R., 38, 427 Person, P., 348 Phillips, R., 24 Pollington, A., 295 Poor, H, 100 Postlethwaite, I., 256, 404, 531, 532, 578, 613, 614 Powell, J., 24, 294 Prett, D., 473 Rawlings, J., 473 Redheffer, R., 403 Rivera, D., 189, 227, 348 Rodman, L., 580 Rodr´ıguez, J., 474 Romagnoli, J., 474 Rosenbrock, H., 134, 532 Ryu, Y., 381 Saeks, R., 348, 642 Salgado, M., 295, 404, 643 Schultz, D., 134 Sepulchre, R, 472 Seron, M., 227, 403, 428, 472, 473, 643 Shaked, U., 404 Shim, J., 257 Shinskey, F., 381 Simon, H., 580 Sin, K., 580 Sivan, R., 99, 135, 579 Skogestad, S., 189, 227, 256, 348, 404, 531, 532, 578, 579, 613, 643 Smith, O., 348 Să oderstră om, T., 39 Sourlas, D., 579 Stein, G., 614 Stephanopoulos, G., 38, 227, 381, 579 Sternby, J., 295 Stoorvgel, A., 614 Stubberud, A., 78 Sule, V., 613 Tannenbaum, A., 227, 256, 347, 403, 427 Teel, A., 473 Tempo R., 427 Thomas, P., 257 Truxal, J., 24 Vidyasagar, M., 348, 472, 531 Villanueva, H., 474 Voda, A., 189 876 Wang, L., 428 Wang, S., 643 Weller, S., 295, 643 West, M., 614 Whittle, P., 532 Wiberg, D., 134 Willems, J., 78, 171, 172, 580 Williams, I., 78 Williams, T., 687 Wilsky, A., 78, 99 Wimmer, H., 580 Wittenmark, B., 24, 99, 294 Wolowich, W., 532, 642, 687 Wonham, W., 381, 687 Woodyatt, A., 257 Workman, M., 294 Youla, D., 348, 427 Zafiriou, E., 227, 347, 348, 427, 531, 642 Zames, G., 403, 427, 579 Zhou, K., 348, 427, 531, 532, 579, 580, 614, 642 Ziegler, J., 189 Name Index Subject Index actuator, 45 actuator limitations, 201 actuator saturation, 571 affine, 61 affine parameterization, 405 NMP stable plants, 409 stable plants, 406 aliasing, 318 ammonia plant, 588 analytic function, 768 anti wind-up mechanism state saturation, 299 anti-aliasing filter, 318 anti-backlash, 222 anti-windup, 222 at-sample design, 360 at-sample response, 319 bandwidth, 90 Bezout identity, 841 Bode’s integral, 240 Bode’s theorems, 785 canonical form controllability, 495 controller, 496 observer, 502 cascade control, 222, 280 primary controller, 281 secondary controller, 281 Cauchy integral, 239, 770 Cayley-Hamilton theorem, 490 channel, 587 characteristic loci, 604, 626 closed loop bandwidth and actuator limitations, 203 and disturbances, 204 and sensor limitations, 200 closed loop characteristic matrix, 602 closed loop poles undesirable, 428 closed loop polynomial, 178, 517, 525 minimal degree, 180 Cohen-Coon PID tuning, 165 complementary sensitivity peak, 253 complex variable, 768 control automatic, 29 feedback, 35 control law, 30 control loop nominal, 127 control trimming, 224 controllability canonical from, 495 test, 500 controller high gain, 33 biproper, 182 integral action, 182 robust, 44 strictly proper, 182 Controller canonical form , 496 coprime matrices right, 759 877 878 coprime polynomials, 180 CTARE stabilizing solution, 662 cut-off frequency, 90 damping factor, 78 dead-beat control, 366 deadzone, 571 decentralized control, 627 decoupling, 823 degrees of freedom first, 123 one, 123 MIMO design, 840 second, 123, 269 third, 276 two, 123 MIMO design, 838 Delta transform, 330 derivative time constant, 159 design homogeneity, 229 detectability, 502 diagonal dominant decoupling, 640 Diophantine equation, 189 direction canonical, 848 Dirichlet kernel, 393 discrete time systems optimal filter, 682 optimal regulator, 668 dispersion multivariable, 588 disturbance, 203 feedforward, 272 generalized, 265 generating polynomial, 264 models, 265 output, 122 disturbance compensation integral control, 210 steady state, 265 via cascade control, 280 disturbance rejection, 410 dither Subject Index input, 224 domain simply connected, 766 dual systems, 500 dual-range controllers, 224 duality, 672 dynamic decoupling, 639 cost, 848 dynamic programming, 656 error sensitivity, 149 extended Kalman filter, 569 fat systems MIMO, 769 feedback, 26 feedforward, 26, 272, 637 filters all pass, 91 band pass, 89 band reject, 89 bandwidth, 91 high pass, 89 low pass, 89 finite settling time, 326 Fourier transform, 92 Fră obenius norm, 666 function space methods, 575 fundamental frequency, 85 Fundamental Laws of Trade-Off frequency domain, 258 MIMO, 773 SISO, 231 gain margin, 143 Green theorem, 766 Hamiltonian matrix, 790 hold-up effect, 278 H2 optimization, 456 hybrid systems, 45 IMP, 185, 265 impulse response, 74 879 Subject Index impulse sampler, 336 independence of the path, 763 initial conditions, 126 discrete systems, 325 inner-outer factorization, 459 innovation process, 519 input saturation, 201 instability, 36 integral action, 410 in the plant, 268 integral effect, 208 integral operator, 559 integrating systems PID control, 434 interaction MIMO system, 587 interactor matrix left, 786 right, 786 Internal Model Principle, 267, 691 MIMO, 745 state space approach, 533 internal stability nominal, 128 interpolation constraints, 428 intersample behavior, 389 intersample response, 353, 365 inverse approximate, 32 restricted, 32 inverse response, 219 inversion, 26, 29, 405 Inverted pendulum, 771 inverted pendulum, 254 Kalman filter, 673 extended, 569 stochastic approach, 673 Laplace transform, 66 region of convergence, 67 Leibnitz rule, 49 linear quadratic filter, 673 linear quadratic regulator, 659 linearization, 52 LQR, 659 matrix fraction description, 591, 758 left, 759 right, 759 Matrix inversion lemma, 518 matrix norm induced spectral, 607 matrix transfer function, 590 McMillan degree, 757 measurement, 35 requirements, 36 responsiveness, 37 measurement noise, 37 MIMO control loop anti wind-up, 853 disturbance compensation, 610 Nyquist theory, 604 robust stability, 616 robustness, 616 saturation error scaling, 854 stability, 601 steady state, 605, 745 time domain constraints, 747 tracking, 609 triangular coupling, 800 zeros and decoupling, 842 MIMO system achieved sensitivity, 785 approximate inverse, 793 basic control loop, 599 directionality, 611 NMP zeros, 795 pole polynomial, 757 poles, 757 properness in, 785 relative degree, 785 Smith-McMillan form, 753 transmission zeros, 596 zero geometric multiplicity, 596 zero left direction, 596 zero polynomial, 757 880 zero right direction, 596 zeros, 757 minimal prototype, 362 minimal realization, 504 minimum phase, 76 model, 30, 42 basic principles, 44 calibration, 43 error, 43 input-output, 50 linearized, 52 nominal, 43 small signal, 54 state space, 45 structures, 45 model matching problem, 665 model predictive control, 303 modeling black box approach, 44 phenomenological approach, 44 modeling errors, 50, 99, 204 additive, 51 in unstable systems, 101 missing pole, 101 missing resonance, 102 multiplicative, 51 time delay, 101 natural frequencies, 487 natural frequency damped, 78 undamped, 78 noise, 200 non minimum phase, 77 non-minimum phase zeros constraints due to, 247 in MIMO systems, 746 nonlinear plants observer, 569 nonlinear system inversion, 556 relative degree, 557 nonlinear systems stability, 573 Subject Index nonlinearity dynamic, 556 input, 556 non-smooth, 571 smooth, 568 Nyquist in MIMO control loops, 604 path, 140 modified, 141 plot, 140 stability theorem, 141 observability, 499 canonical form, 502 observable subspace, 501 observer, 519 disturbance estimation, 534 MIMO systems, 672 nonlinear, 569 unbiased, 528 observer states uncontrollability, 530 open loop, 34 open loop control, 34 operating point, 52 operator, 29 identity, 576 invertible, 576 minimum phase, 576 stable, 576 unstable, 576 optimal regulator discrete time, 668 optimality principle, 656 optimization quadratic, 656 output equation, 484 overshoot, 212 and slow zeros, 84 Parseval’s theorem, 96 PBH test, 505 pH neutralization, 561 phase margin, 143 881 Subject Index PID controller, 177 classic, 157 derivative mode, 158 derivative time, 158 empirical tuning, 160 integral mode, 158 model based, 416, 430 modes, 157 pole placement, 185 proportional band, 158 proportional mode, 158 reset time, 158 series form, 158 standard form, 158 tuning Ziegler-Nichols methods, 160 plant, 29 dynamics, 45 input, 29 output, 29 Poisson integral, 775 Poisson summation formula, 393 Poisson-Jensen formula, 239 pole assignment, 177 polynomial approach, 177 state feedback, 517 pole placement, 180 pole zero cancellation, 183 implementation, 429 in PID design, 420 unstable, 128, 182, 514 poles dominant, 80 fast, 79 MIMO, 757 slow, 80 polynomial nominal characteristic, 125 polynomial matrices, 753 column proper, 754 coprime, 754 equivalent, 753 rank, 754 row proper, 754 Smith form, 754 postdiagonalization, 831 prediagonalization, 831 principal directions, 609 principal gains, 606, 608 process reaction curve, 164 pulse transfer function, 337 Q synthesis, 405 H2 optimal, 456 controller properness, 410 quadruple tank, 632, 749 reachability, 489 canonical decomposition, 493 test, 491 reachable subspace, 495 reconstruction, 319 reference, 29 feedforward, 269 generating polynomial, 265 reference filter, 123 relative gain array, 631 repetitive control, 372 resonant systems PID control, 420 response forced, 86 natural, 86 RGA, 631, 632 Riccati equation continuous time dynamic, 661 discrete time algebraic, 670 discrete time dynamic, 669 dynamic solution, 789 ringing, 326, 366 rlocus, 138 rltool, 138 robustness, 44, 204 and IMC control, 452 performance, 149 roll eccentricity, 698 rolling mill, 225, 268 882 flatness control, 702 rolling mill control, 698 root locus, 134 sampling zeros, 354, 366 satellite tracking, 692 saturation, 31, 221, 368, 852 in MIMO systems, 852 input, 571 sensitivity algebraic constraints, 207 complementary, 125 control, 125 functions, 125 input, 125 interpolation constraints, 207, 208 sensitivity peak, 143, 249, 253 sensor limitations, 200 sensor response, 201 sensors, 200 separation theory, 524 shaping filter, 410 for nonlinear systems, 556 shift operator, 320 singular values, 607 maximum, 608 properties, 608 slew rate limitation, 202, 294 slew rate limitation, 223 Smith canonical form, 754 Smith controller, 425 Smith-McMillan canonical form, 753 split control, 770 stability, 73 critical, 141 internal, 127, 434 relative, 143 robust, 145 stability boundary, 74 stabilizable plant, 495 stable transformation, 31 state Subject Index observer, 519 state equation, 484 state estimate, 519 nonlinear, 559 optimal, 539 state estimation, 519 error, 525 state feedback, 515 MIMO systems, 653 optimal, 538 state saturation, 294 state space, 483 controllability canonical form, 495 controller canonical form, 496 observer canonical form, 502 similarity transformation, 484 state variables, 45 statically decoupled systems, 639 steady state errors, 265, 410 constant inputs, 410 sine wave inputs, 411 step response, 74 stochastic system, 673 sugar mill case, 758 Summary Chapter 1, 18 Chapter 2, 37 Chapter 3, 58 Chapter 4, 103 Chapter 12, 343 Chapter 17, 508 Chapter 5, 150 Chapter 6, 172 Chapter 7, 188, 540 Chapter 8, 230 Chapter 13, 379, 394 Chapter 15, 445 Chapter 10, 284 Chapter 9, 257 Chapter 16, 476 Chapter 19, 304, 577 Chapter 20, 618 Chapter 21, 644 Chapter 22, 705 883 Subject Index Chapter 24, 773 Chapter 25, 816 Chapter 26, 861 Sylvester matrix, 180 Sylvester’s theorem, 179 system zeros, 353 time constant, 78 time delay, 31 all pass approximation, 99 constraints due to, 252 Pad´e approximation, 424 time delayed systems and digital control, 452 and PID control, 424 and Q synthesis , 425 tracking error integral control, 209 transfer function, 71 biproper, 72 delta form, 334 improper, 72 poles, 72 multiplicity, 72 relative degree, 72, 76 shift form, 325 strictly proper, 72 zeros, 72 transformation bilinear, 357 step invariant, 356 transition matrix, 50 triangular coupling, 640 undershoot, 212, 213 and NMP zeros, 82 unimodular matrix, 753 unity feedback loop, 37 unobservable subspace, 502 unreachable subspace, 495 unstable MIMO plants control design, 833 unstable poles constraints due to, 252 in MIMO systems, 746 unstable zeros, see non-minimum phase zeros vibration control, 704 wind up, 222 wind-up, 159 in MIMO systems, 853 Youla parameterization, 407 z-interactors, 795, 798 zeros, 80 fast, 80 MIMO, 757 non minimum phase, 77 slow, 80 Ziegler-Nichols tuning oscillation method, 160 reaction curve method, 165 zinc coating, 694 ... DIGITAL COMPUTER CONTROL 12 Models for Sampled Data Systems 13 Digital Control 14 Hybrid Control 313 317 351 385 V ADVANCED SISO CONTROL 15 SISO controller Parameterizations 16 Control Design Based... The Excitement of Control Engineering 1.4 Chapter Types of Control System Design Control system design in practice requires cyclic effort in which one iterates between modeling, design, simulation,... PREVIEW THE EXCITEMENT OF CONTROL ENGINEERING 1.1 Preview 1.2 Motivation for Control Engineering 1.3 Historical Periods of Control Theory 1.4 Types of Control System Design 1.5 System Integration 1.6