high performance control

High Performance Control T T Tay1 I M Y Mareels2 J B Moore3 1997 Department of Electrical Engineering, National University of Singapore, Singapore Department of Electrical and Electronic Engineering, University of Melbourne, Australia Department of Systems Engineering, Research School of Information Sciences and Engineering, Australian National University, Australia Preface The engineering objective of high performance control using the tools of optimal control theory, robust control theory, and adaptive control theory is more achievable now than ever before, and the need has never been greater Of course, when we use the term high performance control we are thinking of achieving this in the real world with all its complexity, uncertainty and variability Since we not expect to always achieve our desires, a more complete title for this book could be “Towards High Performance Control” To illustrate our task, consider as an example a disk drive tracking system for a portable computer The better the controller performance in the presence of eccentricity uncertainties and external disturbances, such as vibrations when operated in a moving vehicle, the more tracks can be used on the disk and the more memory it has Many systems today are control system limited and the quest is for high performance in the real world In our other texts Anderson and Moore (1989), Anderson and Moore (1979), Elliott, Aggoun and Moore (1994), Helmke and Moore (1994) and Mareels and Polderman (1996), the emphasis has been on optimization techniques, optimal estimation and control, and adaptive control as separate tools Of course, robustness issues are addressed in these separate approaches to system design, but the task of blending optimal control and adaptive control in such a way that the strengths of each is exploited to cover the weakness of the other seems to us the only way to achieve high performance control in uncertain and noisy environments The concepts upon which we build were first tested by one of us, John Moore, on high order NASA flexible wing aircraft models with flutter mode uncertainties This was at Boeing Commercial Airplane Company in the late 1970s, working with Dagfinn Gangsaas The engineering intuition seemed to work surprisingly well and indeed 180◦ phase margins at high gains was achieved, but there was a shortfall in supporting theory The first global convergence results of the late 1970s for adaptive control schemes were based on least squares identification These were harnessed to design adaptive loops and were used in conjunction with vi Preface linear quadratic optimal control with frequency shaping to achieve robustness to flutter phase uncertainty However, the blending of those methodologies in itself lacked theoretical support at the time, and it was not clear how to proceed to systematic designs with guaranteed stability and performance properties A study leave at Cambridge University working with Keith Glover allowed time for contemplation and reading the current literature An interpretation of the Youla-Kuˇ era result on the class of all stabilizing controllers by John Doyle gave c a clue Doyle had characterized the class of stabilizing controllers in terms of a stable filter appended to a standard linear quadratic Gaussian LQG controller design But this was exactly where our adaptive filters were placed in the designs we developed at Boeing Could we improve our designs and build a complete theory now? A graduate student Teng Tiow Tay set to work Just as the first simulation studies were highly successful, so the first new theories and new algorithms seemed very powerful Tay had also initiated studies for nonlinear plants, conveniently characterizing the class of all stabilizing controllers for such plants At this time we had to contain ourselves not to start writing a book right away We decided to wait until others could flesh out our approach Iven Mareels and his PhD student Zhi Wang set to work using averaging theory, and Roberto Horowitz and his PhD student James McCormick worked applications to disk drives Meanwhile, work on Boeing aircraft models proceeded with more conservative objectives than those of a decade earlier No aircraft engineer will trust an adaptive scheme that can take over where off-line designs are working well Weiyong Yan worked on more aircraft models and developed nested-loop or iterated designs based on a sequence of identification and control exercises Also Andrew Paice and Laurence Irlicht worked on nonlinear factorization theory and functional learning versions of the results Other colleagues Brian Anderson and Robert Bitmead and their coworkers Michel Gevers and Robert Kosut and their PhD students have been extending and refining such design approaches Also, back in Singapore, Tay has been applying the various techniques to problems arising in the context of the disk drive and process control industries Now is the time for this book to come together Our objective is to present the practice and theory of high performance control for real world environments We proceed through the door of our research and applications Our approach specializes to standard techniques, yet gives confidence to go beyond these The idea is to use prior information as much as possible, and on-line information where this is helpful The aim is to achieve the performance objectives in the presence of variations, uncertainties and disturbances Together the off-line and on-line approach allows high performance to be achieved in realistic environments This work is written for graduate students with some undergraduate background in linear algebra, probability theory, linear dynamical systems, and preferably some background in control theory However, the book is complete in itself, including appropriate appendices in the background areas It should appeal to those wanting to take only one or two graduate level semester courses in control and wishing to be exposed to key ideas in optimal and adaptive control Yet students having done some traditional graduate courses in control theory should find Preface vii that the work complements and extends their capabilities Likewise control engineers in industry may find that this text goes beyond their background knowledge and that it will help them to be successful in their real world controller designs Acknowledgements This work was partially supported by grants from Boeing Commercial Airplane Company, and the Cooperative Research Centre for Robust and Adaptive Systems We wish to acknowledge the typesetting and typing support of James Ashton and Marita Rendina, and proof reading support of PhD students Andrew Lim and Jason Ford Contents Preface v Contents ix List of Figures xiii List of Tables xvii Performance Enhancement 1.1 Introduction 1.2 Beyond Classical Control 1.3 Robustness and Performance 1.4 Implementation Aspects and Case Studies 1.5 Book Outline 1.6 Study Guide 1.7 Main Points of Chapter 1.8 Notes and References 1 14 14 16 16 17 Stabilizing Controllers 2.1 Introduction 2.2 The Nominal Plant Model 2.3 The Stabilizing Controller 2.4 Coprime Factorization 2.5 All Stabilizing Feedback Controllers 2.6 All Stabilizing Regulators 2.7 Notes and References 19 19 20 28 34 41 51 52 Design Environment 59 3.1 Introduction 59 x Contents 3.2 3.3 3.4 3.5 3.6 Signals and Disturbances Plant Uncertainties Plants Stabilized by a Controller State Space Representation Notes and References 59 64 68 81 89 Off-line Controller Design 4.1 Introduction 4.2 Selection of Performance Index 4.3 An LQG/LTR Design 4.4 H∞ Optimal Design 4.5 An Design Approach 4.6 Notes and References 91 91 92 100 111 115 126 Iterated and Nested (Q, S) Design 5.1 Introduction 5.2 Iterated (Q, S) Design 5.3 Nested (Q, S) Design 5.4 Notes and References 127 127 129 145 155 157 157 158 160 162 166 169 Direct Adaptive-Q Control 6.1 Introduction 6.2 Q-Augmented Controller Structure: Ideal Model Case 6.3 Adaptive-Q Algorithm 6.4 Analysis of the Adaptive-Q Algorithm: Ideal Case 6.5 Q-augmented Controller Structure: Plant-model Mismatch 6.6 Adaptive Algorithm 6.7 Analysis of the Adaptive-Q Algorithm: Unmodeled Dynamics Situation 6.8 Notes and References 171 176 Indirect (Q, S) Adaptive Control 7.1 Introduction 7.2 System Description and Control Problem Formulation 7.3 Adaptive Algorithms 7.4 Adaptive Algorithm Analysis: Ideal case 7.5 Adaptive Algorithm Analysis: Nonideal Case 7.6 Notes and References 179 179 180 185 187 195 204 Adaptive-Q Application to Nonlinear Systems 8.1 Introduction 8.2 Adaptive-Q Method for Nonlinear Control 8.3 Stability Properties 8.4 Learning-Q Schemes 8.5 Notes and References 207 207 208 219 231 242 Contents Real-time Implementation 9.1 Introduction 9.2 Algorithms for Continuous-time Plant 9.3 Hardware Platform 9.4 Software Platform 9.5 Other Issues 9.6 Notes and References xi 243 243 245 246 264 268 270 271 271 271 279 289 296 A Linear Algebra A.1 Matrices and Vectors A.2 Addition and Multiplication of Matrices A.3 Determinant and Rank of a Matrix A.4 Range Space, Kernel and Inverses A.5 Eigenvalues, Eigenvectors and Trace A.6 Similar Matrices A.7 Positive Definite Matrices and Matrix Decompositions A.8 Norms of Vectors and Matrices A.9 Differentiation and Integration A.10 Lemma of Lyapunov A.11 Vector Spaces and Subspaces A.12 Basis and Dimension A.13 Mappings and Linear Mappings 297 297 298 298 299 299 300 300 301 302 302 303 303 304 B Dynamical Systems B.1 Linear Dynamical Systems B.2 Norms, Spaces and Stability Concepts B.3 Nonlinear Systems Stability 305 305 309 310 C Averaging Analysis For Adaptive Systems C.1 Introduction C.2 Averaging C.3 Transforming an adaptive system into standard form C.4 Averaging Approximation 313 313 313 320 323 10 Laboratory Case Studies 10.1 Introduction 10.2 Control of Hard-disk Drives 10.3 Control of a Heat Exchanger 10.4 Aerospace Resonance Suppression 10.5 Notes and References References 325 Author Index 333 Subject Index 337 332 References Vidyasagar, M (1986) Optimal rejection of persistent bounded disturbances, IEEE Trans on Automatic Control 31(6): 527–34 Vidyasagar, M (1991) Further results on the optimal rejection of persistent bounded disturbances, IEEE Trans on Automatic Control 36(6): 642–52 Wang, L and Mareels, I M Y (1991) Adaptive disturbance rejection, Proc IEEE Conf on Decision and Control, Brighton Wang, Z (1991) Performance Issues in Adaptive Control, PhD thesis, University of Newcastle Williamson, D (1991) Digital Control and Implementation, Finite Wordlength Consideration, Prentice-Hall, Englewood Cliffs, N.J Wolovich, W A (1977) Linear Multivariable Systems, Springer-Verlag, Berlin Wonham, W M (1985) Linear Multivariable Control: A Geometric Approach, Springer-Verlag, Berlin Yan, W Y and Moore, J B (1992) A multiple controller structure and design strategy with stability analysis, Automatica 28: 1239–44 Yan, W Y and Moore, J B (1994) Stable linear matrix fractional transformations with applications to stabilization and multistage H∞ control design, International J Robust and Nonlinear Control 65 Youla, D C., Bongiorno, Jr., J J and Jabr, H A (1976a) A modern WienerHopf design of optimal controllers Part I, IEEE Trans on Automatic Control 21(1): 3–14 Youla, D C., Bongiorno, Jr., J J and Jabr, H A (1976b) A modern Wiener-Hopf design of optimal controllers Part II, IEEE Trans on Automatic Control 21(6): 319–30 Zang, Z., Bitmead, R R and Gevers, M R (1991) H2 iterative model refinement and control rebustness enhancement, Proc IEEE Conf on Decision and Control, Brighton, pp 279–84 Zhang, Z and Freudenberg, J S (1987) Loop transfer recovery with nonminimum phase zeros, Proc IEEE Conf on Decision and Control, Los Angeles, pp 956–7 Author Index Aggoun, L., v, 207, 327 Anderson, B D O., v, vi, 4, 10, 12, 13, 17, 37, 100, 102, 126, 172, 175, 176, 205, 207, 211, 246, 290, 325, 328 Annaswamy, A., 204, 330 Åstrom, K J., 20, 52, 245, 325 Athans, M., 126, 172, 328, 330, 331 Davison, E J., 85, 326 DeSilva, C., 326 Desoer, C A., 326 Dewilde, P., 85, 86, 89, 326 Dooren, P V., 85, 86, 89, 325, 326 Doyle, J C., vi, 17, 52, 100, 126, 227, 326 Elliott, R E., v, 207, 327 Balas, M J., 52, 330 Barnett, S., 297, 325 Barratt, C H., 10, 17, 47, 52, 64, 66, 89, 126, 325 Bart, H., 85, 86, 325 Bellman, R E., 297, 325 Benveniste, A., 176, 325 Bitmead, R R., vi, 12, 13, 17, 155, 172, 175, 176, 205, 325, 332 Blackmore, P., 245, 246, 325 Blight, J D., 126, 329 Bodson, M., 204, 331 Bongiorno, Jr., J J., 10, 52, 332 Boyd, S P., 10, 17, 47, 52, 64, 66, 89, 126, 325 Chakravarty, A., 326 Chen, C T., 23, 52, 54, 326 Chew, K K., 272, 326 Cybenko, G., 231, 326 Dahleh, M A., 126, 326 Feuer, A., 246, 327 Francis, B A., 10, 17, 52, 150, 326, 327 Franklin, G F., 52, 327 Freudenberg, J S., 100, 126, 332 Gangsaas, D., v, 126, 294, 296, 329 Gevers, M R., vi, 13, 155, 200, 246, 270, 327, 328, 332 Glover, K., vi, 37, 52, 53, 328, 329 Goh, C J., 207, 209, 210, 331 Gohberg, I., 85, 86, 325 Goodwin, G., 6, 17, 119, 204, 246, 270, 327, 329 Green, M., 5, 10, 17, 66, 89, 111, 327 Hakomori, K., 226, 327 Hansen, F R., 133, 155, 327 Hara, S., 53, 272, 327 Helmke, U., v, 327 Hirsch, M W., 164, 327 334 Author Index Horowitz, R., vi, 90, 155, 242, 277, 327, 330, 331 Hotz, A F., 294, 296, 329 Imae, J., 207, 212, 226, 242, 327 Irlicht, L S., vi, 207, 212, 242, 296, 327–329 Irwin, M C., 305, 328 Isidori, A., 236, 305, 328 Jabr, H A., 10, 52, 332 Jacobson, C A., 52, 330 Johnson, C R., 12, 17, 172, 175, 176, 205, 325 Kaashoek, M A., 85, 86, 325 Kailath, T., 23, 52, 54, 305, 328 Keller, J P., 246, 328 Kokotovic, P V., 12, 17, 172, 175, 176, 205, 325 Kong, X., 12, 176, 205, 331 Kosut, R L., vi, 12, 13, 17, 172, 175, 176, 205, 325, 328 Kuˇ era, V., 10, 52, 328 c Kwakernaak, 4, 17, 37, 100, 126, 328 Moore, J B., v, v, 4, 10, 17, 37, 51–53, 90, 100, 102, 109, 126, 155, 176, 207, 211, 212, 214, 223, 231, 233, 234, 242, 290, 294, 296, 325–332 Morari, M., 5, 17, 329 Murray, J., 326 Nakano, M., 272, 327 Narendra, K., 204, 330 Nemhauser, G L., 124, 330 Nett, C N., 52, 330 Obinata, G., 53, 207, 212, 242, 327, 330 Ogata, K., 4, 17, 20, 37, 52, 330 Omata, T., 272, 327 Paice, A D B., vi, 223, 242, 330 Partanen, A., 200, 330 Pearson, J B., 126, 326 Perkins, J E., 231, 233, 234, 242, 330 Polderman, J W., v, 6, 12, 17, 175, 176, 188–190, 194, 199, 204, 205, 313, 329, 330 Powell, J D., 52, 327 Praly, L., 12, 17, 172, 175, 176, 205, 325 Priouret, P., 176, 325 Lee, W S., 13, 133, 200, 328 Lehtomaki, N A., 126, 328 Li, B., 277, 327, 328 Li, G., 246, 270, 327 Limebeer, D J N., 5, 10, 17, 66, 89, 111, 327 Liu, R W., 326 Ljung, L., 20, 61, 133, 328 Riedle, B D., 12, 17, 172, 175, 176, 205, 325 Rohrs, R., 172, 330 McCormick, J., vi McFarlane, D C., 37, 52, 328 Madievski, A., 246, 328 Mareels, I M Y., v, vi, 6, 12, 13, 17, 172, 175, 176, 188, 190, 194, 199, 204, 205, 231, 233, 234, 242, 296, 313, 325, 328–330, 332 Metivier, M., 176, 325 Middleton, R H., 270, 329 Saeks, R., 326 Sage, A P., 207, 330 Sandell, Jr., N R., 126, 328 Sanders, J A., 12, 176, 314, 317, 331 Sastry, S., 204, 331 Schrama, R J P., 13, 133, 331 Sin, K., 6, 17, 119, 204, 327 Sivan, 4, 17, 37, 100, 126, 328 Smale, S., 164, 327 Author Index Söderström, T., 61, 328 Solo, V., 12, 176, 205, 331 Sontag, E D., 64, 305, 331 Stein, G., 126, 172, 330, 331 Stein, J G., 100, 126, 227, 326 Sugie, T., 53, 327 Tannenbaum, A., 17, 52, 326 Tay, T T., vi, 52, 53, 90, 100, 109, 126, 155, 176, 205, 212, 214, 234, 329, 331 Telford, A J., 52, 53, 296, 329, 331 Teo, K L., 207, 209, 210, 331 Teo, Y T., 126, 331 Tomizuka, M., 51, 53, 272, 326, 329 Valavani, L S., 172, 330 Verhulst, F., 12, 176, 314, 317, 331 Vidyasagar, M., 10, 17, 33, 52, 53, 97, 98, 126, 149, 331, 332 Wang, L., 176, 332 Wang, S H., 85, 326 Wang, Z., vi, 172, 176, 180, 205, 332 White, C C., 207, 330 Williamson, D., 246, 270, 332 Wittenmark, B., 20, 52, 245, 325 Wolovich, W A., 23, 332 Wolsey, L A., 124, 330 Wong, K H., 207, 209, 210, 331 Wonham, W M., 85, 89, 332 Xia, L., 53, 126, 296, 329 Xia, Y., 296, 329 Yamamoto, Y., 272, 327 Yan, W., vi Yan, W Y., 90, 155, 332 Youla, D C., 10, 52, 332 Zafiriou, E., 5, 17, 329 Zang, Z., 13, 155, 332 Zhang, Z., 100, 126, 332 335 Subject Index actuator, adaptive control, indirect, 179, 184 controller, 277 LQ control, 184 pole assignment, 184 -Q algorithm, 160, 169 application to nonlinear systems, 207 control, 214 design, 228 ADC, 245, 247, 260 addition of matrices, 298 affine, 47 aircraft model, 289 algebra, linear, 16, 297 algebraic loop, 117 algorithms for continuous-time plant, 245 all stabilizing controllers feedback, 41 feedforward/feedback, 49 all-pass, 108 almost period, 317 periodic function, 317 analog model, analog-to-digital converter, see ADC anti-aliasing, 245 approximation, nearest neighbor, 240 ARMAX model, 71, 233 asymptotic stability, 23, 30, 309, 311 attraction, domain of, 319 auto-regressive, moving average, exogenous input model, 71 autocorrelation, 62 auxiliary input, 21 average notation, 317 uniform, 319 well defined, 317 averaging analysis, 12 standard form, 316 theory, 16, 163 B-splines, 231 balanced, 308 realization, 308 truncation, 308 basis, 303 vector, 303 Bezout equation, double, 36, 49 identity, 20, 34 identity, double, 212 BIBO, 30, 31, 213 stability, 309 in an sense, 30 bijective, 304 338 Subject Index bisigmoid, 231 block partition notation, 20, 23 bounded, locally, 316 bounded-input bounded-output, see BIBO C language, 264 case studies, 14 Cayley-Hamilton theorem, 299 certainty equivalence, 185 principle, Cesáro mean, 162 Cholesky decomposition, 300 factor, 300 class of all plants stabilizable by a controller, 69 class of all stabilizing controllers, 212 class of stabilizing controllers, 10 classic adaptation algorithm, 186 classical control, closed-loop interpretation of S, 78 codomain, 304 cofactor, 298 commuting matrix, 298 completely detectable, 303 complex zero, 299 condition number, 302 constant disturbance, 51 rate drift, 60 constraints, 2, continuous, Lipschitz, 316 continuous-time plant, algorithms for, 245 control input, 21 controllability, 1, 23, 303, 306 Gramian, 307 uniform complete, 307 controllable, 22 controller, feedback, coordinate basis transformation, 23, 26 coprime factorization, 14, 20, 34 fraction description, 20 normalized factor, 36, 39, 103 critical point, 234 current-to-voltage converter, 247 DAC, 245, 260 data logging, 247 DC offset, 60 decomposition Cholesky, 300 polar, 300 singular value, 301 delay time, 93 derived variable, 8, design environment, 15, 59 detectability, 22, 37, 102, 211, 308 determinant, 298 deterministic, disturbance, 239 model, 60 diagonal matrix, 298 diagonalizable, 300 diagonally balanced realization, 308 differentiation, 302 digital model, digital signal processor, see DSP digital-to-analog converter, see DAC dimension, 303 Diophantine equations, 35 direct adaptive control, 12 -Q control, 157 feedthrough, 268 sum, 297 direct memory access, see DMA discrete frequency spectrum, 61 discrete-time, 305 linear model, 22 model, disk drive control system, 16 disk operating system, see DOS disturbance, 21 constant, 51 deterministic, 239 Subject Index input, 51 periodic, 272 response, 29, 291 signal, 2, 59 sinusoidal, 60 stochastic, 239 unmodeled, 210 white noise, 10 worst case, 10 DMA, 259 domain of attraction, 319 DOS, 255 double Bezout equation, 36, 49 identity, 212 drift, constant rate, 60 DSP, 243, 272 chip, 16 module, 260 dual control, dual vector space, 304 dual-processor solution, 259 dynamical system, 2, 305 dynamics unmodeled, 2, 8, 167, 210, 239 identification of, 129 eigenspace, 299 eigenvalue, 299 eigenvector, 299 EISPACK, 265 elementary operation, 20 elliptic filter, 72 EPROM, 247 equation linear, 302 state, 305 equivalence certainty, 185 relation, 300 equivalent order functions, 315 erasable programmable read-only memory, see EPROM error signal specifications, 94 Euclidean norm, 301 339 excitation, 12 stochastic, 12 exciting signals, 162 exogenous input, 21 auto-regressive, moving average model, 71 exponential asymptotic stability, 311 forgetting, 165 factor Cholesky, 300 normalized coprime, 36 factor, normalized coprime, 39, 103 factorization coprime, 14, 20, 34 minimal, 87 spectral, 37 fast universal controller, see FUC Fatou’s theorem, 310 feedback controller, stabilizing, 30 state estimate, 104 stabilizing, 37 feedforward/feedback controller, 32 feedthrough, direct, 268 fiber, 304 fictitious noise disturbance, filter elliptic, 72 frequency shaping, 4, 5, filtered excitation algorithm, 185 finite word length, 269 finite-dimensional system, flight control, 16 flutter, wing, 289 forgetting factor, 231 frequency -shaped modeling error, 85 weights, 96 shaping, 291 filter, 4, 5, 340 Subject Index spectrum, 62 discrete, 61 Frobenius norm, 302 frozen system, 321, 322 FUC, 259 full loop recovery, 107 full rank, 298 function, 304 representation, 232 transfer, 305 functional learning, 231 gain ∞, 65 65 , 65 rms, 65 Gaussian truncated, 231 global asymptotic stability, 311 learning, 238 gradient search, Gramian controllability, 307 observability, 308 grid points, 237 grid size, 240 p, H∞ control, 10 design strategy, 140 minimization, 97 norm, 65 optimal control, 5, 97 design, 111 optimization, 68 H2 norm, 67 hard-disk drive, 271 Hardy 2-space, 309 heat exchanger, 16, 251, 279 Hermitian, 297 skew, 297 hierarchical design, 3, 13 high performance, control, Hilbert space, 309 identification of unmodeled dynamics, 129 technique, 207 identity matrix, 298 image, 304 space, 299 implementation aspects, 14 impulse response, 65 indirect adaptive control, 12, 179, 184 induced ∞-norm, 65 p norm, 65 inequality, Schwartz, 301 infinite dimensional controller, 207 ∞-norm, induced, 65 ∞-norm, specification in, 98 information state, injective, 304 input disturbance, 51 input sensitivity recovery, 104 input/output representation, integer representation, 269 integration, 302 interior subset, 320 internal model, 51, 53 internally stable, 31, 150, 224 interpolating function spread of, 237 interpolation function, 233 inverse, 25, 299 isomorphism, 304 iterated control-identification principle, 133 design, (Q, S), 129 Kalman’s realization theorem, 308 KBM function, 317 kernel, 299, 304 Subject Index Krylov Boguliobov Mitropolski function, see KBM function ∞ gain, 65 control, 10 norm, 65 optimal control, p gain, 65 norm, 63 induced, 65 gain, 65 sense, BIBO stability in an, 30 leakage, 161 learning functional, 231 global, 238 least squares, 233 -Q, 16, 218, 231 least squares, 214 algorithm, 186 learning, 233 Lebesque 2-space, 309 limit-of-performance, 115 curve, 116, 121 linear algebra, 16, 297 dynamical system, 7, 305 equation, 302 mapping, 304 model, discrete-time, 22 operator, 304 programming, 121 quadratic, see LQ Gaussian, see LQG system operator, 212 transformation, 304 linearization, 208 techniques, 16 LINPACK, 265 Lipschitz, continuous, 316 locally bounded, 316 locally Lipschitz continuous, 316 loop recovery, 5, 106 loop transfer recovery, see LTR LQ control, adaptive, 184 design, 95, 137 regulation, 95 tracking, 95 LQG control, 4, 10, 100 design, 101, 227 method, 4, 37 LQG/LTR, 208, 211 design, 100, 227 LTR, 100 Luenberger observers, 52 Lyapunov function, 311 strict, 311 lemma of, 302 McMillan degree, 84, 108, 308 map, 304 mapping, 304 linear, 304 MATLAB -to-C converter, 268 environment, 265 M-file compiler, 268 matrices addition of, 298 multiplication of, 298 similar, 300 matrix, 297 commuting, 298 determinant, 298 diagonal, 298 identity, 298 inversion lemma, 299 nonsingular, 298 norm, 301 orthogonal, 298 permutation, 298 positive definite, 300 rank, 298 sign, 298 341 342 Subject Index singular, 298 transfer function, 19 unitary, 298 maximum overshoot, 93 singular value, 310 undershoot, 93 microcontroller, 14, 244 module, 261 minimal factorization, 87 realization, 87, 308 representation, 23 stable linear fractional, 85 minimality, 308 minimization, H∞ , 97 minimum phase, 108 plants, 107 model internal, 51 matching controllers, 53 plant, reduction, 290 moving average, auto-regressive, exogenous input model, 71 multiplication of matrices, 298 multirate sampling, 14 nearest neighbor approximation, 240 nested controller, 14 design, (Q, S) design, 145 neural network, 242 no adaptation approximation, 322 noise, white, 62 nominal controller, 19 plant model, nonlinear control, 207 plant, 207 system adaptive-Q application to, 207 fractional map, 219 stability, 310 nonminimum phase plant, 108 nonrepeatable run-out, see NRRO nonsingular matrix, 298 norm, 62, 63 Euclidean, 301 Frobenius, 302 H∞ , 65 H2 , 67 induced ∞-, 65 induced p , 65 ∞-, 98 , 65 p , 63 matrix, 301 vector, 301 normalized coprime factor, 36, 39, 103 NRRO, 272 Nyquist sampling theorem, 243 O, order symbol, 314 o, order symbol, 315 observability, 1, 23, 308 Gramian, 308 observable, 22 offset, DC, 60 on-line identification, one-degree-of-freedom controller, 29 one-to-one, 304 onto, 304 operating system, 255 operator, linear, 304 optical encoder, 247 optimal control, 4, 187 controller disk, 274 H∞ , 97 design, H∞ , 111 optimization, H∞ , 68 optimizing a stabilizing controller, 15 order, 314 function, 314 Subject Index 343 equivalent, 315 small, 315 symbol O, 314 o, 315 orthogonal, 298 matrix, 298 output, 21 injection, 37 sensitivity recovery, 105 overhead crane, 250 overshoot, maximum, 93 positive definite matrix, 300 precompensator, 50 prefilter, 214, 291 preimage, 304 probability theory, 16 projection, 161 proper transfer function, 22 proper, strictly, 308 proportional integral differential, see PID pseudo-inverse, 299 pulse-width-modulated, 247 parallel connection, 24 parameter adjustment, Parseval’s theorem, 67, 171 PC, 262 peak spectral response, 15 time, 93 performance, enhancement, 218 index, selection, 92 measure, 2, period, almost, 317 periodic disturbance, 272 permutation matrix, 298 Pernebo and Silverman theorem, 309 persistence of excitation, 6, 238 personal computer-based solution, 262 PID, 95 controller, 247 plant, -model mismatch, 166 controller stabilized, 68 model, 2, 20 parameter variation, uncertainty, 64 plug-in controller, 13 design, polar decomposition, 300 pole assignment, adaptive, 184 pole-placement strategy, 135 pole/zero cancellation, 31, 100 polynomial, 299 (Q, S) design, iterated, 129 Q-parameterization, 15, 41 QR-decomposition, 301 quadrature phase pulse, 247 R p , 22 Rsp , 22 radical basis function, 231 RAM, 247 random access memory, see RAM range, 304 space, 299 rank, 298 full, 298 row, 303 rational proper transfer function, 22 real zero, 299 realization balanced, 308 diagonally, 308 minimal, 87, 308 theorem, Kalman’s, 308 reconstructible, 234 recovery full loop, 107 loop, 5, 106 loop transfer, see LTR sensitivity, 103, 106 input, 104 output, 105 recursive controller, 14 344 Subject Index design, 13 regulation LQ, 95 robust, 90 regulators, stabilizing, 15, 51, 53 relation, equivalence, 300 repeatable run-out, see RRO representation minimal stable linear fractional, 85 theorem, 231 residuals, 291 resonance suppression, 289 response disturbance, 29 impulse, 65 peak spectral, 15 transient, 93 Riccati equation, 3, 103, 113, 211 rise time, 93 rms, 101 gain, 65 robust control, regulation, 90 stability, 19 stabilization, 75, 90 robustness, 2, 6, 64 root mean square, 101 row rank, 303 space, 303 RRO, 272 run-out nonrepeatable, see NRRO repeatable, see RRO S-parameterization, 15 sampling, multirate, 14 scalar, 297 scaling, 218 Schwartz inequality, 301 selection of performance index, 92 self-tuning, sensitivity recovery, 103, 106 sensor, separation principle, 115 theorem, time scale, 313 serial port operating system, see SPOS series connection, 25 servo system, 273 settling time, 93 sign matrix, 298 signal disturbance, 2, 59 signature, 300 similar matrices, 300 similarity transformation, 300 simulation languages, 16 simultaneous stabilization, 53 single stepping, 256 singular matrix, 298 value, 301, 308 decomposition, 301 maximum, 310 sinusoidal disturbance, 60 skew Hermitian, 297 symmetric, 297 small gain stability, 30 small order, 315 software environment debugging, 255 development, 252 platform, 264 space Hardy 2-, 309 Hilbert, 309 image, 299 Lebesque 2-, 309 range, 299 row, 303 vector, 303 dual, 304 specification in ∞-norm, 98 using 2-norms, 95 Subject Index spectral factorization, 37 spectrum, frequency, 62 discrete, 61 SPOS, 255 spread of the interpolating function, 237 stability, 311 asymptotic, 23, 30, 309, 311 exponential, 311 global, 311 BIBO, 309 of nonlinear systems, 310 properties, 219 robust, 19 small gain, 30 stabilizability, 37, 102, 211, 306 stabilizable, 22 stabilization result, 224 robust, 75, 90 simultaneous, 53 stabilizing controller, 14, 28 feedback, 30 feedforward/feedback, 32 optimizing, 15 regulator, 15 regulators, 51, 53 state feedback, 37 stable, 305 internally, 31, 150, 224 invariant subspace, 85, 87 linear fractional representation, 82 modes uncontrollable, 26 unobservable, 26 strongly, 153 standard form, 316 state, 4, 10 equation, 305 estimate feedback, 104 controller, 37 space equation, 19 representation, 81 steepest descent, 161, 164, 169 step response specification, 93 stochastic, disturbance, 239 model, 61 excitation, 12 system, strictly proper, 308 transfer function, 22 strongly stable, 153 structural resonance mode, 289 structured uncertainties, 15 subparitioning, 23 subset, interior, 320 subspace, 303 stable invariant, 85, 87 vector, 303 sufficiently exciting, rich signals, 162 sum, direct, 297 surjective, 304 Sylvester’s Inertia Theorem, 300 symmetric, 297 skew, 297 system dynamical, 2, 305 linear, 7, 305 finite-dimensional, frozen, 321, 322 stochastic, time-invariant, time-varying, 53, 212 Taylor series expansion, 208 three-term-controller, time -invariant system, -varying system, 53, 212 domain uncertainty, 68 scale approach, 12 separation, 313 trace, 299 345 346 Subject Index tracking, 10 LQ, 95 trajectory selection, 237 transfer function, 19, 305 proper, 22 rational, 22 strictly, 22 transformation coordinate basis, 23, 26 linear, 304 similarity, 300 transient response, 93 transpose, 25, 297 truncated Gaussian, 231 truncation, balanced, 308 tunability, 188 tuning property, 188, 189 with excitation, 189 two-degree-of-freedom controller, 20, 33, 53 2-norm, 301 specification using, 95 worst case design using, 97 2-space Hardy, 309 Lebesque, 309 UIO, 265 undershoot, maximum, 93 uniform average, 319 complete controllability, 307 unit step, 93 unitary matrix, 298 universal input-output, see UIO unmodeled disturbance, 210 dynamics, 2, 8, 167, 210, 239 unstructured uncertainties, 15 up/down counter, 247 Usenet, 265 variation of constants formula, 160 vector, 297 basis, 303 norm, 301 space, 303 subspace, 303 voltage to current converter, 247 well-posed, 150, 224 white noise, 62, 101 disturbance, 10 wing flutter, 289 worst case design using 2-norms, 97 disturbance, 10 Z -transform, 305 zero complex, 299 real, 299 ... Disturbances Real world plant Control input Robust controller Sensor output Commands Performance enhancement controller Robust stabilizing controller FIGURE 3.2 Performance enhancement controller distinguish... 1.7 Main Points of Chapter High performance control in the real world is our agenda It goes beyond classical control, optimal control, robust control and adaptive control by blending the strengths...Preface The engineering objective of high performance control using the tools of optimal control theory, robust control theory, and adaptive control theory is more achievable now than ever