![Preface xi
the form of a difference equation with a small parameter is represented,
where the small parameter depends on the sampling period. The design of
SISO as well as MIMO sampled-data control systems is discussed.
Chapter 13: Design of control systems with distributed parameters. The
main points of the extension of the previously presented methodology for
control system design with the highest derivative in feedback for distributed
parameter systems are highlighted, based on consideration of the parabolic-
type system.
The book aims to disseminate new results in the area of control system
design under uncertainty, and may be used as a course textbook. It con-
tains numerous examples with simulation results, as well as assignments
suitable for courses in nonlinear control system design. The core of the
book is based on a translation of an earlier book [Yurkevich (2000a)] and
lecture notes used by the author over the last ten years with students in the
Automation and Computer Engineering Department at Novosibirsk State
Technical University.
The design methodology may be useful for graduate and postgraduate
students in the field of nonlinear control systems design. It will also be of
interest to researchers, engineers, and university lecturers who are taking
aim at real-time control system design in order to solve practical problems in
the control of aircraft, robots, chemical reactors, and electrical and electro-
mechanical systems.
Any comments about the book (including any errors noticed) can be
sent to {yurkev@mail.ru) with the subject heading (book). They will be
sincerely appreciated.
It is with great pleasure that I express gratitude to many colleagues
who contributed to this book through useful discussions and helpful sug-
gestions. My students and colleagues from the Automation Department
of Novosibirsk State Technical University and, in particular, Professors
G.A. Frantsuzova, O.Ya. Shpilevaya, and A.S. Vostrikov, have provided
me with stimulating discussions of the subject. Professors A.L. Fradkov
(Institute for Problems of Mechanical Engineering, Academy of Sciences
of Russia), A.I. Rouban (Krasnoyarsk State Technical University), S.D.
Zemlyakov (Institute of Control Sciences, Academy of Sciences of Russia),
and N.D. Egupov (Kaluga Branch of Bauman Moscow State Technical Uni-
versity), offered many helpful suggestions and much moral support during
my work. I would like to thank Professors M.J. Blachuta and K.W. Wo-
jciechowski (Institute of Automatics, Silesian Technical University), with
whom I have had the pleasure of working. Reviews of the book, along with](https://image.slidesharecdn.com/30562-250312005356-837f98af/85/Design-of-Nonlinear-Control-Systems-with-the-Highest-Derivative-in-Feedback-Valery-D-Yurkevich-17-320.jpg)
Design of Nonlinear Control Systems with the Highest Derivative in Feedback Valery D. Yurkevich
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- 5. Design of Nonlinear Control Systems with the Highest Derivative in Feedback Valery D. Yurkevich Digital Instant Download Author(s): Valery D. Yurkevich ISBN(s): 9789812388995, 9812388990 Edition: Kindle File Details: PDF, 14.98 MB Year: 2004 Language: english
- 7. Design of Nonlinear Control Systems with the Highest Derivative in Feedback
- 8. SERIES ON STABILITY, VIBRATION AND CONTROL OF SYSTEMS Founder and Editor: Ardeshir Guran Co-Editors: M. Cloud & W. B. Zimmerman About the Series Rapid developments in system dynamics and control, areas related to many other topics in applied mathematics, call for comprehensive presentations of current topics. This series contains textbooks, monographs, treatises, conference proceed- ings and a collection of thematically organized research or pedagogical articles addressing dynamical systems and control. The material is ideal for a general scientific and engineering readership, and is also mathematically precise enough to be a useful reference for research specialists in mechanics and control, nonlinear dynamics, and in applied mathematics and physics. Selected Volumes in Series A Vol. 4 Asymptotic Methods in Buckling Theory of Elastic Shells Authors: P. E. Tovstik and A. L. Smirinov Vol. 5 Generalized Point Models in Structural Mechanics Author: I. V. Andronov Vol. 6 Mathematical Problems of Control Theory: An Introduction Author: G. A. Leonov Vol. 7 Analytical and Numerical Methods for Wave Propagation in Fluid Media Author: K. Murawski Vol. 8 Wave Processes in Solids with Microstructure Author: V. I. Erofeyev Vol. 9 Amplification of Nonlinear Strain Waves in Solids Author: A. V. Porubov Vol. 10 Spatial Control of Vibration: Theory and Experiments Authors: S. O. Reza Moheimani, D. Halim, and A. J. Fleming Vol. 11 Selected Topics in Vibrational Mechanics Editor: I. Blekhman Vol. 12 The Calculus of Variations and Functional Analysis: With Optimal Control and Applications in Mechanics Authors: L P. Lebedev and M. J. Cloud Vol. 13 Multiparameter Stability Theory with Mechanical Applications Authors: A. P. Seyranian and A. A. Mailybaev Vol. 14 Stability of Stationary Sets in Control Systems with Discontinuous Nonlinearities Authors: V. A. Yakubovich, G. A. Leonov and A. Kh. Gelig Vol. 15 Process Modelling and Simulation with Finite Element Methods Author: W. B. J. Zimmerman
- 9. SERIES ON STABILITY, VIBRATION AND CONTROL OF SYSTEMS ^ M B L ^ Series A Volume 16 Founder & Editor: Ardeshir Guran Co-Editors: M. Cloud & W. B. Zimmerman Design of Nonlinear Control Systems with the Highest Derivative in Feedback Valery D. Yurkevich Concordia University, Canada YJ5 World Scientific N E W J E R S E Y • L O N D O N • S I N G A P O R E • B E I J I N G - S H A N G H A I • H O N G K O N G • T A I P E I • C H E N N A I
- 10. Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. DESIGN OF NONLINEAR CONTROL SYSTEMS WITH THE HIGHEST DERIVATIVE IN FEEDBACK Copyright © 2004 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in anyform or by any means, electronic or mechanical, includingphotocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 981-238-899-0 Printed by Fulsland Offset Printing (S) Pte Ltd, Singapore
- 11. To my wife, Lyudmila and to the memory of my sister and brother, Natalya and Peter
- 13. Preface Methods for the analysis and design of nonlinear control systems are grow- ing rapidly. These developments are motivated by extensive applications, in particular, to such areas as mechatronic systems, robotics, and aircraft flight control systems. A number of new ideas, results, and approaches has appeared in this area during the past few decades. This text was developed as a systematic explanation of one such new approach to control system design, which can provide effective control of nonlinear systems on the assumption of uncertainty. The approach is based on an application of a dynamical control law with the highest derivative of the output signal in the feedback loop. A distinctive feature of the control systems thus designed is that two-time-scale motions are forced in the closed-loop system. Stability conditions imposed on the fast and slow modes, and a sufficiently large mode separation rate, can ensure that the full-order closed-loop system achieves desired properties: the output transient performances are as desired, and they are insensitive to parameter variations and external disturbances. A general design methodology for control systems with the highest derivative in feedback for continuous-time single-input single-output (SISO) or multi-input multi-output (MIMO) systems, as well as their discrete-time counterparts, is presented in this book. The method of singular perturba- tion is used to analyze the closed-loop system properties throughout. The material is structured into thirteen chapters, the contents of which could be outlined as follows. Chapter 1: Regularly and singularly perturbed systems. The main pur- pose of this chapter is a short explanation of some preliminary mathemat- ical results concerning the properties and analysis of perturbed differential equations. The results constitute a background for an approximate analysis vii
- 14. viii Design of nonlinear control systems with the highest derivative in feedback and design of nonlinear control systems under uncertainty. Chapter 2: Design goal and reference model. The problem statement of output regulation for nonlinear time-varying control systems and the basic step response parameters are discussed. The model of the desired output behavior in the form of a desired differential equation is introduced; its parameters are selected based on the required output step response parameters (overshoot, settling time). Particularities of the reference model construction, in order to obtain the required system type, are also discussed. Chapter 3: Methods of control system design under uncertainty. In this chapter a short overview of robust control synthesis techniques on the as- sumption of uncertainty is given. Main attention is devoted to discussion of nonadaptive approaches, in particular, to control systems with the highest derivative of the output signal and high gain in the feedback loop, control systems with state vector and high gain in the feedback loop, and control systems with sliding motions. Chapter 4-' Design of SISO continuous-time control systems. The prob- lem of output regulation of SISO nonlinear time-varying control systems is discussed. The control system is designed to provide robust zero steady- state error of the reference input realization. Moreover, the controlled out- put transients should have some desired behavior. These transients should not depend on the external disturbances and varying parameters of the plant model. The insensitivity condition of the output transient behav- ior with respect to external disturbances and varying parameters of the system is introduced. The highest derivative in the feedback loop is used in proposed control law structures. The limit behavior of control systems with the highest derivative of the output signal in the feedback loop is dis- cussed. Closed-loop system properties are investigated on the basis of the two-time-scale technique and, as a result, slow and fast motion subsystems are considered separately. Chapter 5: Advanced design of SISO continuous-time control systems. Problems related to implementation of continuous-time control systems with the highest derivative in feedback are discussed. In particular, control accuracy and robustness of the control system, various design techniques for choosing controller parameters, the influence of high-frequency noisy measurements, and noise attenuation are considered. Chapter 6: Influence of unmodeled dynamics. The peculiarities of SISO continuous-time control system design with the highest derivative in feed- back are discussed on the assumption of uncertainty in the model descrip- tion caused by unmodeled dynamics. These dynamics reflect errors on the
- 15. Preface ix system degree (or relative degree). Their influences, such as a pure time delay in the feedback loop and the unstructured uncertainties, lead to a plant model in the form of perturbed and/or singularly perturbed systems of differential equations. Some particulars of control design in the presence of a nonsmooth nonlinearity in the control loop are discussed as well. Chapter 7: Realizability of desired output behavior. The conditions of readability of the desired output behavior are discussed in this chapter. These are connected with invertibility conditions, nonlinear inverse dynam- ics solutions, and the problem of internal behavior analysis. Concepts such as invertibility index (relative degree), normal form of a nonlinear system, internal stability analysis, degenerated system on condition of output sta- bilization, and zero-dynamics are discussed. The design methodology for SISO control systems with the relative highest derivative in feedback is considered in the presence of internal dynamics. Finally, the problem of switching controller design is discussed. Chapter 8: Design of MIMO continuous-time systems. The problem of output regulation of MIMO nonlinear time-varying control systems is discussed. Here the goals of control system design are to provide output decoupling and disturbance rejection, i.e., each output should be indepen- dently controlled by a single input, and to provide desired output transient performance indices on the assumption of incomplete information about varying parameters of the plant model and unknown external disturbances. The design methodology for SISO control systems with the highest deriva- tive in feedback are extended to cover MIMO nonlinear time-varying control systems. The control law structure with the relative highest derivative in feedback is used in order to provide desired dynamical properties and de- coupling of the output transients in a specified region of the system state space. The systematic design procedure for the control laws with the rela- tive highest output derivatives is presented. The output regulation problem is discussed on the assumption that the previously presented realizability of the desired output behavior is satisfied. Chapter 9: Stabilization of internal dynamics. This chapter is devoted to consideration of control system design where the dimension of the control vector is large, as the dimension of the output vector and redundant control variables are used in order to obtain internal dynamics stabilization. By this, the presented design methodology may be extended to more general system types. The discussed problem of internal dynamics stabilization for linear time-invariant systems corresponds to the displacement of zeroes of the transfer function in the left half of the complex plane.
- 16. x Design of nonlinear control systems with the highest derivative in feedback Chapter 10: Digital controller design based on pseudo-continuous ap- proach. The design of digital controllers for continuous nonlinear time- varying systems is discussed. The control task is formulated as a track- ing problem for the output variables, where the desired decoupled output transients are attained on the assumption of incomplete information about varying parameters of the system and external disturbances. A distinguish- ing feature of the approach is that a pseudo-continuous-time model of the control loop with a pure time delay is used, where the delay is the result of a zero-order-hold transfer function approximation. The linear continuous- time controller with the relative highest output derivatives in feedback is designed, where the control law parameters are selected in accordance with the requirements placed on output control accuracy and damping of fast- motion transients. In particular, the selection of the sampling period is provided based on the requirement placed on the phase margin of the fast- motion subsystem. Then the Tustin transformation is applied to calculate the parameters of a digital controller. In order to increase the sampling pe- riod, a control law with compensation of the pure time delay is introduced. Chapter 11: Design of discrete-time control systems. The method of discrete-time control systems design to provide the desired output tran- sients is introduced, and is related with the purely discrete-time systems. In the case of continuous-time plants, the first step to be performed is dis- cretization of the plant model. As a result, the discrete-time model of the plant in the form of a difference equation is used. A procedure to analyze the fast and slow motions in the discrete-time control system is given. It has been shown that if a sufficient time-scale separation between the fast and slow modes in the closed loop system and stability of the fast motions are provided, then after damping of the fast motions the output behavior in the closed loop system corresponds to the reference model and is insen- sitive to parameter variations of the plant and external disturbances. The design methodology is the discrete-time counterpart of the previously dis- cussed approach to continuous-time control system design with the highest derivative in feedback. Chapter 12: Design of sampled-data control systems. In this chapter, a design methodology for the discrete-time control system with two-time-scale motions is extended for the purpose of sampled-data control system design, by taking into account the particulars of the model of a series connection between a zero-order hold and a continuous-time system with high sampling rate. As a result, an approach to derive an approximate discrete-time model for nonlinear time-varying systems preceded by zero-order hold (ZOH) in
- 17. Preface xi the form of a difference equation with a small parameter is represented, where the small parameter depends on the sampling period. The design of SISO as well as MIMO sampled-data control systems is discussed. Chapter 13: Design of control systems with distributed parameters. The main points of the extension of the previously presented methodology for control system design with the highest derivative in feedback for distributed parameter systems are highlighted, based on consideration of the parabolic- type system. The book aims to disseminate new results in the area of control system design under uncertainty, and may be used as a course textbook. It con- tains numerous examples with simulation results, as well as assignments suitable for courses in nonlinear control system design. The core of the book is based on a translation of an earlier book [Yurkevich (2000a)] and lecture notes used by the author over the last ten years with students in the Automation and Computer Engineering Department at Novosibirsk State Technical University. The design methodology may be useful for graduate and postgraduate students in the field of nonlinear control systems design. It will also be of interest to researchers, engineers, and university lecturers who are taking aim at real-time control system design in order to solve practical problems in the control of aircraft, robots, chemical reactors, and electrical and electro- mechanical systems. Any comments about the book (including any errors noticed) can be sent to {yurkev@mail.ru) with the subject heading (book). They will be sincerely appreciated. It is with great pleasure that I express gratitude to many colleagues who contributed to this book through useful discussions and helpful sug- gestions. My students and colleagues from the Automation Department of Novosibirsk State Technical University and, in particular, Professors G.A. Frantsuzova, O.Ya. Shpilevaya, and A.S. Vostrikov, have provided me with stimulating discussions of the subject. Professors A.L. Fradkov (Institute for Problems of Mechanical Engineering, Academy of Sciences of Russia), A.I. Rouban (Krasnoyarsk State Technical University), S.D. Zemlyakov (Institute of Control Sciences, Academy of Sciences of Russia), and N.D. Egupov (Kaluga Branch of Bauman Moscow State Technical Uni- versity), offered many helpful suggestions and much moral support during my work. I would like to thank Professors M.J. Blachuta and K.W. Wo- jciechowski (Institute of Automatics, Silesian Technical University), with whom I have had the pleasure of working. Reviews of the book, along with
- 18. xii Design of nonlinear control systems with the highest derivative in feedback many useful comments and pieces of advice were kindly provided by Pro- fessors J-P. Barbot (Equipe Commande des Systemes) and L.M. Fridman (Universidad Nacional Avtonoma de Mexico) and were very much appreci- ated. I am grateful to Professor A. Guran (series Editor-in-Chief, Institute of Structronics of Canada) for the opportunity to publish this book, Pro- fessor M. Cloud (Lawrence Technological University) for editing the entire manuscript, and Mr. Yeow-Hwa Quek (World Scientific editorial staff) for assistance in the production of this book. Finally, I am grateful to Profes- sors N. Esmail and K. Khorasani for many-sided and considerate support, and for accommodating me with the possibility of creative and fruitful work on the Faculty of Engineering & Computer Science at Concordia University. Most of all, I would like to thank my wife, Lyudmila, for her love and moral support in my life and work. Montreal, 2003 Valery D. Yurkevich
- 19. Contents Preface vii 1. Regularly and singularly perturbed systems 1 1.1 Regularly perturbed systems 1 1.1.1 Nonlinear nominal system 1 1.1.2 Linear nominal system 3 1.1.3 Vanishing perturbation 5 1.1.4 Nonvanishing perturbation 6 1.2 Singularly perturbed systems 7 1.2.1 Singular perturbation 7 1.2.2 Two-time-scale motions 8 1.2.3 Boundary-layer system 10 1.2.4 Stability analysis 10 1.2.5 Fast and slow-motion subsystems 14 1.2.6 Degree of time-scale separation 15 1.3 Discrete-time singularly perturbed systems 18 1.3.1 Fast and slow-motion subsystems 18 1.3.2 Degree of time-scale separation 19 1.4 Notes 21 1.5 Exercises 21 2. Design goal and reference model 23 2.1 Design goal 23 2.2 Basic step response parameters 24 2.3 Reference model 25 2.4 Notes 30 xiii
- 20. xiv Design of nonlinear control systems with the highest derivative in feedback 2.5 Exercises 31 3. Methods of control system design under uncertainty 33 3.1 Desired vector field in the state space of plant model . . . 33 3.2 Solution of nonlinear inverse dynamics 36 3.3 The highest derivative and high gain in feedback loop . . . 37 3.4 Differentiating filter and high-gain observer 40 3.5 Influence of noise in control system with the highest derivative 43 3.6 Desired manifold in the state space of plant model 45 3.7 State vector and high gain in feedback loop 46 3.8 Control systems with sliding motions 49 3.9 Example 52 3.10 Notes 54 3.11 Exercises 55 4. Design of SISO continuous-time control systems 57 4.1 Controller design for plant model of the 1st order 57 4.1.1 Control problem 57 4.1.2 Insensitivity condition 58 4.1.3 Control law with the 1st derivative in feedback loop 59 4.1.4 Closed-loop system properties 61 4.2 Controller design for an nth-order plant model 64 4.2.1 Control problem 64 4.2.2 Insensitivity condition 65 4.2.3 Control law with the nth derivative in the feedback loop 66 4.2.4 Fast-motion subsystem 68 4.2.5 Slow-motion subsystem 72 4.2.6 Influence of small parameter 73 4.2.7 Geometric interpretation of control problem solution 74 4.3 Example 75 4.4 Notes 76 4.5 Exercises 77 5. Advanced design of SISO continuous-time control systems 79 5.1 Control accuracy 79 5.1.1 Steady state of fast-motion subsystem 79 5.1.2 Steady state of slow-motion subsystem 80
- 21. Contents xv 5.1.3 Velocity error due to external disturbance 82 5.1.4 Velocity error due to reference input 83 5.1.5 Control law in the form of forward compensator . . 84 5.2 Root placement of FMS characteristic polynomial 85 5.2.1 Degree of time-scale separation 85 5.2.2 Selection of controller parameters 86 5.2.3 Root placement based on normalized polynomials . 87 5.3 Bode amplitude diagram assignment of closed-loop FMS . 88 5.3.1 Block diagram of closed-loop system 88 5.3.2 Bode amplitude diagram of closed-loop FMS . . . . 89 5.3.3 Desired Bode amplitude diagram of closed-loop FMS 91 5.3.4 Selection of controller parameters 92 5.4 Influence of high-frequency sensor noise 93 5.4.1 Closed-loop system in presence of sensor noise . . . 93 5.4.2 Controller with infinite bandwidth 94 5.4.3 Controller with finite bandwidth 96 5.5 Influence of varying parameters 100 5.5.1 Influence of varying parameters on FMS and SMS . 100 5.5.2 Michailov hodograph for FMS 100 5.5.3 Variation of FMS bandwidth 102 5.5.4 Degree of control law differential equation 103 5.5.5 Root placement of FMS characteristic polynomial . 104 5.6 Bode amplitude diagram assignment of open-loop FMS . . 105 5.7 Relation with PD, PI, and PID controllers 107 5.8 Example 109 5.9 Notes Ill 5.10 Exercises 112 6. Influence of unmodeled dynamics 115 6.1 Pure time delay 116 6.1.1 Plant model with pure time delay in control . . . . 116 6.1.2 Closed-loop system with delay in feedback loop . . 117 6.1.3 Fast motions in presence of delay 118 6.1.4 Stability of FMS with delay 119 6.1.5 Phase margin of FMS with delay 121 6.1.6 Control with compensation of delay 122 6.1.7 Velocity error with respect to external disturbance . 124 6.1.8 Example 124 6.2 Regular perturbances 126
- 22. xvi Design of nonlinear control systems with the highest derivative in feedback 6.2.1 Regularly perturbed plant model 126 6.2.2 Fast motions in presence of regular perturbances . . 127 6.2.3 Selection of controller parameters 128 6.2.4 Control with compensation of regular perturbances 129 6.2.5 Example 130 6.3 Singular perturbances 132 6.3.1 Singularly perturbed plant model 132 6.3.2 Fast motions in presence of singular perturbances . 133 6.3.3 Selection of controller parameters 134 6.4 Nonsmooth nonlinearity in control loop 136 6.4.1 System preceded by nonsmooth nonlinearity . . . . 136 6.4.2 Describing function analysis of limit cycle in FMS . 138 6.4.3 Effect of chattering on control accuracy 141 6.4.4 Example 143 6.5 Notes 145 6.6 Exercises 146 7. Realizability of desired output behavior 149 7.1 Control problem statement for MIMO control system . . . 149 7.1.1 MIMO plant model 149 7.1.2 Control problem 150 7.2 Invertibility of dynamical systems 151 7.2.1 Role of invertibility of dynamical systems 151 7.2.2 Definition of invertibility of dynamic control system 152 7.2.3 Invertibility condition for nonlinear systems . . . . 154 7.3 Insensitivity condition for MIMO control system 157 7.3.1 Desired dynamics equations 157 7.3.2 Insensitivity condition 158 7.4 Internal stability 159 7.4.1 Boundedness of ATD-control function 159 7.4.2 Concept of internal stability 160 7.4.3 Normal form of the plant model 161 7.4.4 Internal stability of linear systems 164 7.4.5 Internal stability of nonlinear systems 167 7.4.6 Degenerated motions and zero-dynamics 168 7.4.7 Example 170 7.5 Output regulation of SISO systems 171 7.5.1 Realizability of desired output behavior 171 7.5.2 Closed-loop system analysis 174
- 23. Contents xvii 7.5.3 Example 175 7.6 Switching regulator for boost DC-to-DC converter 176 7.6.1 Boost DC-to-DC converter circuit model 176 7.6.2 Model with continuous control variable 177 7.6.3 Switching regulator 180 7.6.4 External disturbance attenuation 183 7.7 Notes 185 7.8 Exercises 186 8. Design of MIMO continuous-time control systems 189 8.1 MIMO system without internal dynamics 189 8.1.1 MIMO system with identical relative degrees . . . . 189 8.1.2 MIMO system with different relative degrees . . . . 191 8.2 MIMO control system design (identical relative degrees) . 192 8.2.1 Insensitivity condition 192 8.2.2 Control system with the relative highest derivatives in feedback 194 8.2.3 Fast-motion subsystem 195 8.2.4 Slow-motion subsystem 197 8.2.5 Control system design with zero steady-state error . 198 8.2.6 Example 200 8.3 MIMO control system design (different relative degrees) . 202 8.3.1 Insensitivity condition and control law structure . . 202 8.3.2 Closed-loop system analysis 203 8.3.3 Control accuracy 205 8.4 MIMO control system in presence of internal dynamics . . 207 8.4.1 Fast-motion subsystem 209 8.4.2 Slow-motion subsystem 210 8.4.3 Example 211 8.5 Decentralized output feedback controller 212 8.6 Notes 214 8.7 Exercises 215 9. Stabilization of internal dynamics 217 9.1 Zero placement by redundant control 217 9.2 Internal dynamics stabilization (particular case) 221 9.3 Internal dynamics stabilization (generalized case) 222 9.4 Stabilization of degenerated mode and zero dynamics . . . 225
- 24. xviii Design of nonlinear control systems with the highest derivative in feedback 9.5 Methods of internal dynamics stabilization 225 9.6 Example 228 9.7 Notes 231 9.8 Exercises 232 10. Digital controller design based on pseudo-continuous approach 233 10.1 Continuous system preceded by zero-order hold 233 10.1.1 Control problem 233 10.1.2 Pseudo-continuous-time model with pure delay . . . 234 10.2 Digital controller design 235 10.2.1 Insensitivity condition 235 10.2.2 Pseudo-continuous closed-loop system 236 10.2.3 Influence of sampling period 237 10.2.4 Digital realization of continuous controller 239 10.2.5 Example 242 10.3 Digital controller design with compensation of delay . . . . 242 10.3.1 Control law structure 242 10.3.2 Closed-loop system analysis 244 10.3.3 Digital realization of continuous controller 245 10.3.4 Example 246 10.4 Notes 248 10.5 Exercises 250 11. Design of discrete-time control systems 253 11.1 SISO two-time-scale discrete-time control systems 253 11.1.1 Discrete-time systems 253 11.1.2 Control problem and insensitivity condition 254 11.1.3 Discrete-time control law 256 11.1.4 Two-time-scale motion analysis 257 11.1.5 Robustness of closed-loop system properties . . . . 260 11.1.6 Control accuracy 262 11.1.7 Example 265 11.2 SISO discrete-time control systems with small parameter . 266 11.2.1 System with small parameter 266 11.2.2 Two-time-scale motion analysis 268 11.2.3 Interrelationship with fixed point theorem 271 11.2.4 Root placement of FMS characteristic polynomial . 273 11.2.5 FMS design based on frequency-domain methods . 274
- 25. Contents xix 11.3 MIMO two-time-scale discrete-time control systems . . . . 278 11.3.1 MIMO discrete-time systems 278 11.3.2 Control law 278 11.3.3 Two-time-scale motion analysis 281 11.3.4 Example 283 11.4 Notes 284 11.5 Exercises 285 12. Design of sampled-data control systems 287 12.1 SISO sampled-data control systems 287 12.1.1 Reduced order pulse transfer function 287 12.1.2 Input-output approximate model of linear system . 290 12.1.3 Control law 291 12.1.4 Closed-loop system analysis 293 12.1.5 Selection of controller parameters 296 12.1.6 Nonlinear sampled-data systems 297 12.1.7 Example 300 12.2 MIMO sampled-data control systems 300 12.2.1 Control problem 300 12.2.2 MIMO continuous-time system preceded by ZOH . 302 12.2.3 Control law 303 12.2.4 Fast-motion subsystem 305 12.2.5 Selection of controller parameters 306 12.2.6 Slow-motion subsystem 307 12.2.7 Example 308 12.3 Notes 309 12.4 Exercises 311 13. Control of distributed parameter systems 313 13.1 One-dimensional heat system with distributed control . . . 313 13.2 Heat system with finite-dimensional control 317 13.3 Degenerated motions 321 13.4 Estimation of modes 322 13.5 Notes 323 13.6 Exercises 323 Appendix A Proofs 325 A.I Proof of expression (8.29) 325
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- 58. Welcome to our website – the ideal destination for book lovers and knowledge seekers. With a mission to inspire endlessly, we offer a vast collection of books, ranging from classic literary works to specialized publications, self-development books, and children's literature. Each book is a new journey of discovery, expanding knowledge and enriching the soul of the reade Our website is not just a platform for buying books, but a bridge connecting readers to the timeless values of culture and wisdom. With an elegant, user-friendly interface and an intelligent search system, we are committed to providing a quick and convenient shopping experience. Additionally, our special promotions and home delivery services ensure that you save time and fully enjoy the joy of reading. Let us accompany you on the journey of exploring knowledge and personal growth! ebookultra.com