Dedicated to: Alexander David who is the Future (D.K.A.) My father who led me into Engineering; my teachers who showed me the way in Control Engineering; and to my children who, in using this book, will lead us to the promised land (R.B.Z.) Butterworth-Heinemann Ltd Linacre House, Jordan Hill, Oxford OX2 8DP -& A member of the Reed Elsevier pic group OXFORD LONDON MUNICH NEW DELHI TOKYO TORONTO Contents BOSTON SINGAPORE SYDNEY WELLINGTON First published by Pergamon Press 1974 Second edition 1984 Third edition published by Butterworth-Heinemann © Butterworth-Heinemann Ltd 1995 Ltd 1995 All rights reserved No part of this publication may be reproduced in any material form (including photocopying or storing in any medium by electronic means and whether or not transiently or incidentally to some other use of this publication) without the written permission of the copyright holder except in accordance with the provisions of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenharn Court Road, London, England WIP 9HE Applications for the copyright holder's written permission to reproduce any part of this publication should be addressed to the publishers British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 7506 2298 Library of Congress Cataloguing in Publication Data A catalogue record for this book is available from the Library of Congress Printed in Great Britain by Hartnolls Limited, Bodmin, Cornwall Introduction 1.1 Historical Perspective 1.2 Basic Concepts 1.3 Systems Description 1.4 Design, Modeling, and Analysis 1.5 Text Outline Modeling of Physical Systems Introduction 2.1 Mechanical Systems 2.2 Electrical Systems 2.3 2.4 Electromechanical Systems 2.5 Thermal Systems 2.6 Hydraulic Systems 2.7 System Components 2.8 Summary 2.9 References 2.10 Problems 11 Models for Control Systems 3.1 Introduction System Impulse and Step Responses 3.2 The Transfer Function 3.3 Differential Equation Representation 3.4 3.5 Block Diagram Analysis 3.6 State Equation Representation 3.7 Relationship Between System Representations 3.8 Small Disturbance of Nonlinear Systems 3.9 Summary 3.10 References 71 11 14 24 31 39 42 49 61 61 62 v 71 72 76 77 80 95 104 115 119 119 VI CONTENTS 3.11 Problems • 120 Time 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 Response - Classical Method Introduction Transient Response Steady State Response , Response to Periodic Input Approximate Transient Respon.e Summary References Problems Time 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 Response - State Equation Method Introduction Solution of the State Equation • Eigenvalues of Matrix A and Stability Two Examples Controllability and Obsorvablllty Summary References Problems 133 133 136 154 165 179 187 188 189 •••••••• I I I I •• • , I I ••• I •• , •• I I I I I • , • I I I ••••• , • t Performance Criteria 6.1 Introduction , 6.2 Control System Specification • • 6.3 Dynamic Performance Indict 6.4 Steady State Performance •• 6.5 Sensitivity Functions and RobUitn •• 6.6 Summary 6.7 References 6.8 Problems I , I •• I I • ••• I I • , ••• , Assessing Stability and PertoM~ 7.1 Introduction I •• ,I., I, • 7.2 Stability via Routh-Hurwltl 0•••• 7.3 Frequency Response Method •••• 7.4 Root Locus Method ••••• 7.5 Dynamic Response PerformaDat M.uures 7.6 Summary • I'" 7.7 References 7.8 Problems • I t I •••• I I I •••• 203 203 204 221 224 228 238 239 239 243 243 243 248 252 261 271 271 272 277 277 279 288 324 342 356 358 359 CONTENTS Control Strategies and Plant Sizing 8.1 Introduction 8.2 Goals for Control System 8.3 Control and Controlled Variables 8.4 Reducing Goals to Control Strategies 8.5 Examples from Applications 8.6 Sizing of Components and Subsystems 8.7 A Design Example 8.8 Summary 8.9 References 8.10 Problems VII 365 365 366 373 379 391 396 405 408 409 410 System Compensation 9.1 Introduction 9.2 Stabilization of Unstable Systems 9.3 Types of Compensation 9.4 Cascade Comp ensation 9.5 Feedback Compensation 9.6 Feedforward Compensation 9.7 A Practical Example 9.8 Pole-Placement Design 9.9 State Observers 9.10 Pole-Placement Design with State Observer 9.11 Summary 9.12 References 9.13 Problems 419 419 420 425 426 449 455 460 465 478 484 486 486 488 10 Discrete Time Control Systems 10.1 Introduction 10.2 The Sampling Process 10.3 Data Reconstruction 10.4 The Z- Transform 10.5 The Inverse Z- Transform 10.6 Digital Transfer Functions 10.7 System Response 10.8 Stability Tests 10.9 Graphical Analysis Methods 10.10 Digital Compensators 10.11 Summary 10.12 References 10.13 Problems 495 495 497 503 508 514 517 525 532 536 545 552 553 554 CONTENTS Vlll 11 Non Linear Control Systems 11.1 Introduction 11.2 The Phase Plane-Method 11.3 Limit Cycles in Non-Linear Control Systems 11.4 Describing Function Technique 11.5 Stability Criteria 11.6 Stability Region for Non-Linear Systems 11.7 Summary 11.8 References 11.9 Problems 565 565 566 592 599 614 627 629 630 632 12 Systems with Stochastic Inputs 12.1 Introduction 12.2 Signal Properties 12.3 Input-Output Relationships 12.4 Linear Correlation 12.5 Summary 12.6 References 12.7 Problems 643 643 645 650 655 656 656 657 13 Adaptive Control Systems 13.1 Introduction 13.2 Adaptive Control Methods 13.3 Controller Design Methods 13.4 System Parameter Estimation 13.5 Adaptive Control Algorithms 13.6 Stability of Adaptive Controllers 13.7 An Application Example 13.8 Summary 13.9 References 13.10 Problems 659 659 660 663 669 673 684 685 687 687 687 A Laplace and Z-Transforms A.1 Laplace Transforms A.2 Z- Transforms A.3 References 689 689 692 695 B Symbols, Units and Analogous Systems B.1 Systems of Units B.2 Symbols and Units B.3 Comparison of Variables in Analogous Systems B.4 References 697 697 698 700 700 CONTENTS IX C Fundamentals of Matrix Theory Introduction C.1 Matrix Algebra C.2 Types of Matrices C.3 Matrix Calculus C.4 Linear Algebraic Equations C.5 Characteristic Equations and Eigenvectors C.6 Functions of a Matrix C.7 References C.8 701 701 703 705 709 711 712 713 719 D Computer Software for Control D.1 References 721 722 Index 723 About the Authors Dr D K Anand is both a Professor and Chairman of the Department of Mechanical Engineering at the University of Maryland, College Park, Maryland, U.S.A He is a registered Professional Engineer in Maryland and has consulted widely in Systems Analysis for the U.S Government and Industry He has served as Senior Staff at the Applied Physics Laboratory of the John Hopkins University and Director of Mechanical Systems at the National Science Foundation Dr Anand has published over one hundred and fifty papers in technical journals and conference proceedings and has published two othe books on Introductory Engineering As well he has a patent on Heat Pipe Control He is a member of Tau Beta Pi, Pi Tau Sigma, Sigma Xi, and is a Fellow of ASME Dr R B Zmood is the Control Discipline Leader in the Department of Electrical Engineering at Royal Melbourne Institute of Technology, Melbourne, Victoria, Australia He has consulted widely both in Australia and in the U.S on the industrial and military applications of control systems He has served as a staff member of the Telecom Research Laboratories (formerly A.P.O Research Laboratories) and the Aeronautical Research Laboratories of the Australian Department of Defence, as well as having worked in industry for a considerable period Dr Zmood joined RMIT in 1980 and since that time his research interestll have centered on the control of magnetic bearings both from a theoretical and application viewpoint and he has published widely in this area He ill a member of IEEE x Preface Since the printing of the first two editions, the use of computer software by students has become an important adjunct to the teaching and learning of control systems analysis With this in mind the entire text has been enlarged and strengthened in the third edition In addition an attempt has been made to broaden the scope of the book so that it is suitable for mechanical and electrical engineering students as well as for other students of control systems This revision has been largely carried out by the second author The advent of the desk-top computer based computer aided design (CAD) tools has removed the need for repeated hand computations previously required in control system design While this has forced a fundamental review of the material taught in control courses, it is our contention that many of the analytical and graphical tools, developed during the early days of the discipline are still important for developing an intuitive understanding, or a "mind's eye model", of system design The computer simply removes the drudgery of applying them In reviewing the content of the earlier editions we have sought to arrive at a balance between the material which has pedagogical value and that which has proved useful to the authors in research and industrial practice This has led to the deletion of some material, and the inclusion of much new material In addition the order of the material has been altered to assist in the assimilation of important concepts Class room experience has shown, for example, that when the dominant pole concept is introduced at the same time as the root locus analysis method for feedback systems students identify this idea with the analysis method, rather than accepting it as a separate concept By presenting it divorced from the root locus method it has been found they more readily accept the generality of the idea In the early chapters considerable attention is given to introducing the many methods of mathematical modeling physical systems To this end the concept of the system S is emphasized and the mathematical models Xl ~ n~a are viewed as approximate but useful descriptions of the system Their relative utilities depend upon the application in question While very little motivation for the adoption of these models is given at this time the rapid progress in later chapters to their use in design is felt to satisfy the question of the student Why all these models? Consistent with our focus on the central role of the system S, the presentation of the various models is carefully developed so as to show their interrelationships Apart from discussing steady state and transient performance measures and the sensitivity function, we have introduced unstructured robustness concepts for investigating the effect upon system operation of large changes in its parameters As the parameters of all practical control systems vary over some non-infinitesimal but defined range the robustness approach has been assuming an ever more important role in system design Although there is a rich collection of research results on system robustness our treatment of this field is necessarily brief It has long been felt by the authors that, while most introductory control system texts dwell on various design techniques such as root locus and other methods at length, they gloss over two of the most important aspects of control system design These being control strategies and component sizing While in some instances these are only of minor concern, in many cases they are of utmost importance Wrong decisions on these matters during the early stages of a project can lead to poor system operation or even failure In both cases it can be very costly to correct the situation at a later stage after an expensive plant or machine has been constructed This cost can be measured both in time and money The classical design techniques of the root locus and the frequency response methods involve sequentially adjusting the parameters of the assumed controller structures to determine if the performance specification is satisfied These approaches involve a considerable amount of trial and error, as well as relying on designer inspiration for the selection of the appropriate controller structures As an alternative approach we present here a state space pole placement design method where the performance specification leads systematically and directly to the controller design by a welI defined numerical algorithm State observers, which are needed to implement these designs, are also introduced, and it is shown how these designs are integrated te complete a total control system design The design methodologies discussed in earlier chapters of the book lead to controllers with fixed parameter settings Adaptive control was developed for systems having large plant parameter changes where the controller settings are adjusted to accommodate these changes and so as to always give the desired performance In the discussion only the basics of adaptive control are presented Such important concepts as the certainty equiva- PREFACE xiii lence principle, model reference adaptive systems (MRAS), and self tuning regulators (STM) are introduced and applied to a number of examples of adaptive control systems The material in this book has been used in a variety of courses over the last twenty years by the authors, both at the University of Maryland, and the Royal Melbourne Institute of Technology (RMIT) At RMIT the material presented has been used as the basis for junior level and senior level courses in electrical engineering, each running over two semesters for ~ hours per week At the University of Maryland both authors have covered the equivalent of Chapters to in a one semester course to mechanical engineering students taking their senior year Other combinations of chapters could be easily be used as a basis for other courses For example Chapters to 4, 6, 7, and 10 could be used as an introductory course on digital control systems Apart from its use as an undergraduate text the book is well structured to be read by practicing engineers and applied scientists who need to utilize control techniques in their work A hallmark of earlier editions was the use of copious examples to illustrate the various concepts and techniques This feature has been retained, with the range of problems in each chapter being greatly expanded, both in number and in spread of difficulty To this end the teacher will find some problems are elementary exercises, some are challenging even to good students, some are open-ended, and some are design-oriented These latter problems are intended to encourage the student to approach control design problems from a holistic or integrated point of view As well they illustrate the power of computer analysis for control system design Cautious selection of problems, suited to the audience who are using the book, will need to be exercised In carrying out a task of this magnitude many people, some of them unknowingly, have contributed to its success First of all there are the many students who have suffered through our trying to get the presentation right Then there are our colIeagues with whom we have discussed the finer points of presentation Dr G Feng of the University of New South Wales deserves special mention for it was he who wrote the first draft of Chapter 13 Also Dr T Vinayagalingam of RMIT criticaIly read the complete manuscript and offered many suggestions for improvement of presentation Mr T Bergin has read and critiqued some ofthe key chapters, while Daniel Zmood, the son of the second author, read many of the sections from a student perspective and made useful suggestions for clarifying the text Ms R Luxa painstakingly typed the entire manuscript from the handwritten notes and Mr R Wang drew many of the figures To all we express our thanks Finally to our wives Asha and Devorah, and to our families, who at various times saw us disappear for long hours to write the manuscript PREFACE xiv we express our gratitude D K ANAND Bethesda, Maryland R B ZMOOD Melbourne, Australia Their forbearance is much appreciated 1994 1994 Chapter Introduction 1.1 Historical Perspective The desire to control the forces of nature has been with man since early civilizations Although many examples of control systems existed in early times, it was not until the mid-eighteenth century that several steam operated control devices appeared This was the time of the steam engine, and perhaps the most noteworthy invention was the speed control ftyball governor invented by James Watt Around the beginning of the twentieth century much of the work in control systems was being done in the power generation and the chemical processing industry Also by this time, the concept of the autopilot for airplanes was being developed The period beginning about twenty-five years before World War Two saw rapid advances in electronics and especially in circuit theory, aided by the now classical work of Nyquist in the area of stability theory The requirements of sophisticated weapon systems, submarines, aircraft and the like gave new impetus to the work in control systems before and after the war The advent of the analog computer coupled with advances in electronics saw the beginning of the establishment of control systems as a science By the mid-fifties, the progress in digital computers had given engineers a new tool that greatly enhanced their capability to study large and complex systems The availability of computers also opened the era of data-logging, computer control, and the state space or modern method of analysis The Russian sputnik ushered in the space race which led to large governmental expenditures on the U.S space program as well as on the devel1 opment of advanced military hardware During this time, electronic circuits became miniaturized and large sophisticated systems could be put together very compactly thereby allowing a computational and control advantage coupled with systems of small physical dimensions We were now capable of designing and flying minicomputers and landing men on the moon The post sputnik age saw much effort in system optimization and adaptive systems Finally, the refinement of the micro chip and related computer developments has created an explosion in computational capability and computercontrolled devices This has led to many innovative techniques in manufacturing methods, such as computer-aided design and manufacturing, and the possibility of unprecedented increases in industrial productivity via the use of computer-controlled machinery, manipulators and robotics Today control systems is a science; but with the art still playing an important role Much mathematical sophistication has been achieved with considerable interest in the application of advanced mathematical methods to the solution of ever more demanding control system problems The modern approach, having been established as a science, is being applied not only to traditional engineering control systems, but to newer fields like urban studies, economics, transportation, medicine, energy systems, and a host of fields which are generating similar problems that affect modern man 1.2 Basic Concepts Control system analysis is concerned with the study of the behavior of dynamic systems The analysis relies upon the fundamentals of system theory where the governing differential equations assume a cause-effect (causal) relationship A physical system may be represented as shown in Fig 1-1, where the excitation or input is x(t) and the response or output is y(t) A simple control system is shown in Fig 1-2 Here the output is compared to the input signal, and the difference of these two signals becomes the excitation to the physical system, and we speak of the control system as having feedback The analysis of a control system, such as described in Fig 1-2, involves the determination of y(t) given the input and the characteristics of the system On the other hand, if the input and output are specified and we wish to design the system characteristics, then this is known as synthesis A generalized control system is shown in Fig 1-3 The reference or input variables rl, r2, , rm are applied to the comparator or controller The output variables are Cl, C2, • , Cn The signals fl, f2, , fp are actuating or control variables which are applied by the controller to the system or plant The plant is also subjected to disturbance inputs Ul, U2, , uq If the output variable is not measured and fed back to the controller, then the total system consisting of the controller and plant is an open loop system If the output is fed back, then the system is a closed loop system 1.3 Systems Description Because control systems occur so frequently in our lives, their study is quite important Generally, a control system is composed of several subsystems connected in such a way as to yield the proper cause-effect relationship Since the various subsystems can be electrical, mechanical, pneumatic, biological, etc., the complete description of the entire system requires the understanding of fundamental relationships in many different disciplines Fortunately, the similarity in the dynamic behavior of different physical systems makes this task easier and more interesting As an example of a control system consider the simplified version of the attitude control of a spacecraft illustrated in Fig 1-4 We wish the satellite to have some specific attitude relative to an inertial coordinate system The actual attitude is measured by an attitude sensor on board the satellite If the desired and actual attitudes are not the same, then shows an illustration of the conceptual design of a proposed Sun Tracker Briefly, it consists of an astronomical telescope mount, two silicon solar cells, an amplifier, a motor, and gears The solar cells are attached to the polar axis of the telescope so that if the pointing direction is in error, more of the sun's image falls on one cell than the other This pair of cells, when connected in parallel opposition, appear as a current source and act as a positional error sensing device A simple differential input transistor amplifier can provide sufficient gain so that the small error signal produces an amplifier output sufficient for running the motor This motor sets the rotation rate of the polar axis of the telescope mount to match the apparent motion of the sun This system is depicted in block diagram form in Fig 1-7 The use of this device is not limited to an astronomical telescope, but can be used for any system where the Sun must be tracked For example, the output of a photovoltaic array or solar collector can be maximized using a Sun Tracker the comparator sends a signal to the valves which open and cause gas jet firings These jet firings give the necessary corrective signal to the satellite dynamics thereby bringing it under control A control system represented this way is said to be represented by block diagrams Such a representation helps in the partitioning of a large system into subsystems This allows each subsystem to be studied individually, and the interactions between the various subsystems to be studied at a later time If we have many inputs and outputs that are monitored and controlled, the block diagram appears as illustrated in Fig 1-5 Systems where several variables are monitored and controlled are called muItivariable systems Examples of multi variable systems are found in chemical processing, guidance and control of space vehicles, the national economy, urban housing growth patterns, the postal service, and a host of other social and urban problems As another example consider the system shown in Fig 1-6 The figure In recent years, the concepts and techniques developed in control system theory have found increasing application in areas such as economic analysis, forecasting and management An interesting example of a multivariable system applied to a corporation is shown in Fig 1-8 The inputs of Finance, Engineering and Management when compared to the output which include products, services, profits, etc., yield the excitation variables of available capital, labor, raw materials and technology to the plant There are two feedback paths, one provided by the company and the other by the marketplace The number of control systems that surround us is indeed very large The essential feature of all these systems is the same They all have input, control, output, and disturbance variables They all describe a controller and a plant They all have some type of a comparator Finally, in all cases we want to drive the control system to follow a set of preconceived commands 1.4 Design, Modeling, and Analysis Prior to the building of a piece of hardware, a system must be designed, modeled and analyzed Actually the analysis is an important and essential feature of the design process In general, when we design a control system we so conceptually Then we generate a mathematical model which is analyzed The results of this analysis are compared to the performance specifications that are desired for the proposed system The accuracy of the results depends upon the quality of the original model of the proposed design The Sun Tracker proposed in Fig 1-6 is a conceptual design We shall show, in Chapter 9, how it is analyzed and then modified so that its performance satisfies the system specifications The objective then may be considered to be the prediction, prior to construction, of the dynamic behavior that a physical system exhibits, i.e its natural motion when disturbed from an equilibrium position and its response when excited by external stimuli Specifically we are concerned with the speed of response or transient response, the accuracy or steady state response, and the stability By stability we mean that the output remains within certain reasonable limiting values The relative weight given to any special requirement is dependent upon the specific application For example, the air conditioning of the interior of a building may be maintained to ±1°C and satisfy the occupants However, the temperature control in certain cryogenic systems requires that the temperature be controlled to within a fraction of a degree The requirements of speed, accuracy and stability are quite often contradictory and some compromises must be made For example, increasing the accuracy generally makes for poor transient response If the damping is decreased, the system oscillations increase and it may take a long time to reach some steady state value It is important to remember that all real control systems are nonlinear; however, many can be approximated within a useful, though limited, range as linear systems Generally, this is an acceptable first approximation A very important benefit to be derived by assuming linearity is that the superposition theorem applies If we obtain the response due to two different inputs, then the response due to the combined input is equal to the sum of Table D-l Computer Program ACET ACSL CC Ctrl-C Software Vendor Comments Information and Control Systems, 28 Research Drive, Hampton, VA 23666 Mitchell and Gauthier Assoc., 73 Junction Square Drive, Concord, MA 01742 System Technology Inc., 13766 S Hawthorne Blvd., Hawthorne, CA 90250 System Control Technology Inc, 2300 Geng Road, Palo Alto, CA 94303 Simulation and design package Incorporates linear and some nonlinear design techniques platforms PC to supercomputer Non-linear simulations using state variable models and block diagrams Linear system analysis and design methods PC platform, Command driven linear system analysis and design sMultiloop and state variable, and z-domain Workstation platform, state variable analysis tern design 721 matrix and Linear sys- APPENDIX D COMPUTER SOFTWARE FOR CONTROL 722 Table D-1 Computer Program Vendor IBM Corp., New York, NY Simulation Services, Chatsworth, CA Boeing Computer Services, PO Box 24346, Seat tie, WA CSMP CSSL Easy-5 MATLAB The Math Works Inc, 24 Prime Parkway, Natick, MA 01760 MATRIXX Integrated Systems, Inc., 2500 Mission College Blvd., Santa Clara, CA 95054 Analogy, Inc., PO Box 1669, Beaverton, OR 97075 SABER SIMNON SIMULAB D.l SSPA Systems, PO Box 24001, S-40022 Goteberg, Sweden The Math Works, Inc., 24 Prime Parkway, Natick, MA 01760 Software (contd.) Comments Mainframe platform non-linear simulation Mainframe platform non-linear simulation Linear and Linear and Mainframe and work-station platforms Block diagram structure analysis Linear and non-linear continuous and discrete simulation PC and workstation platforms Classical and state space control tools for linear systems (Used with toolboxes: Control system, signal processing, robust control, system iden tification.) Mainframe and workstation platforms Classical and state space control tools Both discrete and continuous systems Workstation platforms, graphical display on PC Simulates non-linear systems in discrete and continuous domains Can include electrical circuit models PC platform Simulation of linear and non-linear discrete and continuous systems PC platform Non-linear system simulation References D Frederick, et aI., "The Extended List of Control Software", Vol 61, No.6 (1988) p 77 Electronics, Index A.C control motor, 54 Absolute stability, 278 Acceleration error constant, 155 Accelerometer, 52 Ackermann's formula, 468, 481 Actuating variables, Adaptive control, 659 Additive uncertainty, 268 Adjoint matrix, 205, 707 Amplifier, 27, 49 Analog element, 517 Analog-to-digital conversion, 496 Angle condition, 326 Approximate transient response, 179 Asymptotes, 291 Asymptotically stable, 615 Auto regressive moving average (ARMA) model, 664 Autocorrelation function, 646 Autonomous state equation, 577 Auxiliary equation, 283 A verage value, 644 Backlash, 565 Backward difference integration, 546 Backward difference transfer function, 547 Bandwidth, 245, 503 Bendixson theorem, 594 723 Block diagram, Block diagram algebra, 71 Block diagram reduction, 86 Bode plot, 173, 288, 538, 606 Break frequency, 291 British absolute units, 697 British gravitational units, 697 Cancellation compensation, 426, 445 Canonical decomposition, 236 Cascade (series) compensation, 425 Cascade (series) connection, 80 Cascade control, 373 Cascaded elements, 81 Cauchy's theorem, 306 Causal system, Cayley-Hamilton theorem, 204, 216, 714 Center, 567, 571, 575 Certainty equivalence principle, 660, 663 Characteristic equation, 136, 147, 205, 216, 279, 286, 306, 325, 713 Characteristic polynomial, 101, 107 Characteristic value, 713 Chatter, 598 Chattering states, 589 Circle stability criteria, 624 Classical method, 8, 133 724 Closed loop frequency response, 324 Closed loop poles, 136-137 Closed loop pulsed system, 521 Closed loop response, 136, 306 Closed loop system, 3, 82, 262 Closed loop transfer function, 84, 137 Co-energy, 33 Coefficient matrix, 98, 222 Cofactor, 702· Companion matrix, 102, 107, 111 Comparator, Compensating poles, 425 Compensating zeros, 425 Compensator, 342, 419-420 Completely state controllable, 229 Completely state observable, 232 Complex frequency, 689 Complex matrix, 709 Complimentary sensitivity, 264 Component sizing, 396 Computed variable control, 388 Conformal map, 536 Constitutive equations, 25 Contactor,603 Control strategy, 365, 372 Control system goals, 379 Control tasks, 380 Control variable range, 396 Control variables, 3, 370, 373 Controllability, 228 Controllability canonical form, 112 Controllable pair, 231 Controlled variables, 370, 373 Controller, 3, 252, 342 Controller canonical form, 105 Controller tuning, 379 Convergence, 684 Convolution integral, 73 Corner frequency, 291, 293 Corner plot, 289 INDEX Coulomb friction, 400 Coulomb's law, 25 Critical damping, 147 Critical point, 312, 567 Cross-spectral density, 647 Crosscorrelation function, 646 Crossover non-linearity, 580 d'Alembert's principle, 13-14 D.C control motor, 35 Damped natural frequency, 147 Damping ratio, 147, 245 Darcy's law, 14 Data reconstruction, 503, 517 Data sequence, 496 Deadband, 565, 604 Decade, 289 Decibel, 289 Decoupling controllers, 391 Degenerate node, 571 Degree of freedom, 17 Degree of stability, 345 Delay time, 245, 320 Derivative control, 254 Describing function, 566, 599,602 Design, Determinant, 205, 702 Diagonal dominance, 391 Diagonal matrix, 705 Difference method, 514, 516 Digital compensators, 545 Digital control systems, 495 Digital convolution operator, 518 Digital element, 517 Digital transfer function, 517-518 Digital-to-analog conversion, 496 Diophantine equation, 666 Dirac 8-function, 170 Direct adaptive controller, 663 Direct digital control, 366 Discrete signals, 495 INDEX 725 Disturbance feedforward compensation, 457 Disturbance inputs, Disturbance sensitivity, 263 Disturbance variable, 373 Dominant poles, 139, 181 Du Hamel integral, 75 Duality, 233 Dynamic decoupling, 391 Dynamic errors, 164 Dynamic feedforward control, 384 Feedforward compensation, 425, 455 Feedforward control, 382 Fictitious sampler, 519 Final value theorem, 513, 692 Finite stability, 615 First order systems, 144 First-order hold, 506 Focus, 571 Forced response, 136 Forward difference integration, 546 Forward difference transfer function, 546 Eigenvalue, 216, 568, 713 Eigenvector, 713 Electrical system, 12 Elements, 701 Energy, 33 Ensemble average, 645 Forward path, 82 Fourier series, 166, 600 Fourier transform, 169 Fourier's heat conduction law, 14 Free body diagram, 16 Frequency domain representation, Equalizer network, 419 Equations of first approximation, 576 168 Frequency response, 288 Frequency response function, Equilibrium point, 567,573,615 Ergodic hypothesis, 645 Error budgeting, 401 Error constants, 154 Error constants (coefficients), 246 Error detecting device, 53, 81 Error sensing device, 57 Error series, 160-161 Error-input transfer function, 84 Estimation error, 670 Expected value, 644 Exponential matrix, 217 Extrapolation, 504 171 Frequency spectrum, 165, 167, 169 Frequency warping, 548 Friction limited backlash, 612 Friction load, 398 Fundamantal component, 167 Fundamantal matrix, 212 Fundamental component, 601 Fundamental frequency, 166 Faraday's law, 25 Feedback, 3, 83 Feedback compensation, Feedback path, 82 425, 449 165, Gain crossover frequency, 346 Gain form, 157 Gain margin, 345 Gain scheduling, 660 Gaussian (normal) distribution, 644 Gear train backlash, 611 Gear transmission, 55 Generalized error coefficients, 161 INDEX 726 Geometric consistency 14 Global stability, 615 Gyroscope, 57 condition, Harmonic components, 167 Heaviside's partial fraction expansion, 207 Heirarchical control, 368 Hold circuit, 496 Homogeneous equation, 98, 211 Hydraulic act uator, 57 Hydraulic pump, 56 Hydraulic system, 47 Hydraulic valve, 57 Hysteresis, 565 Ideal sampler, 497 Impulse response, 72, 652 Impulse sampler, 499 Incremental value, 12-13 Indirect adaptive controller, 663 Inertance, 43 Inertially limi ted backlash, 613 Initial value theorem, 692 Input, Input variables, Input vector, 98 Input-output, 72 Integral control, 255 Integral squared error, 249 Integrating gyro, 59 Interacting control systems, 388 Internal model principle, 669 In ternational System of Uni ts (SI), 697 Interpolation, 504 Invariant sets, 575 Inverse Laplace transform, 138, 207 Inverse matrix, 705 Inverse Z-transform, 514 Inversion integral, 516 Inversion integral method, Isocline method, 578 Isolated solution, 592 514 Joint probability density function, 644 Jordan canonical form, 105 Jury's test, 532 Kinetic energy, 616 Kirchoff's laws, 14, 24 Lag network, 428 Lag-lead compensation, 426, 444 Lag-lead network, 444 Laplace transform, 12, 204, 689, 710 Lienard's method, 597 Limit cycle oscillations, 611 Limit cycles, 592 Linear, 11 Linear (Hookean) spring, 15 Linear (viscous) friction, 15 Linear correlation, 655 Linear differential equations, 12 Linear quadratic optimal regulator, 475 Linear variable differential transformer (LVDT), 51 Linearity, 11 Linearly correlated, 646 Local stability, 615 Log magnitude, 289 Logarithmic plot, 289 Loop method, 25 Loop sensitivity form, 175, 325 Loop tuning, 382 727 INDEX Luenberger state observers, Lyapunov function, 616 Lyapunov method, 615 Lyapunov stability, 616 478 M -circles, 353 Magnitude condition, 326 Magnitude response, 172 Marginally stable, 322 Marginally stable system, 318 Mathematical model, 11 Matrix, 701 Matrix function, 713 Matrix inverse, 708 Matrix polynomial, 713 Mean value, 644 Measurement noise sensitivity, 263 Mechanical nodal network, 20 Mechanical system, 13 Method of residues, 694 Micro-computers, 545 Micro-controllers, 545 Minimal state realization, 237 Minimum phase, 296 Minor, 702 Minor-loop control, 381 Model, Model reference adaptive control, 673 Model reference adaptive system (MRAS), 660 Model reference controller, 664 Modified frequency response, 626 Modified Nyquist plot, 318 Multiple-input multiple-output (MIMO) systems, 377 M ul tiplicative uncertainty, 268 Multivariable system, Mutual inductance, 28 n-dimensional space, 702 Natural frequency, 147,293 Nested feedback loops, 381 Neutral stability, 338 Neutrally stable, 322 Newton's law, 13 Nichols chart, 354 Node, 569 Node method, 25 Non-linear element, 614 Non-minimum phase, 296 Non-singular matrix, 213 Nonlinear, 11 Nonlinearity, 565 Nonsingular matrix, 708 Null matrix, 706 Nutation frequency, 93 Nyquist criterion, 306, 539 Nyquist frequency, 503 Nyquist path, 310 Nyquist plot, 288, 300, 539, 606 Nyquist stability criterion, 311 Observability, 228, 232 Observability canonical form, 108 Observer canonical form, III Ohm's law, 14 Open loop poles, 137 Open loop system, 3, 83, 261 Open loop transfer function, 311 Open loop zeros, 137 Orthogonal matrix, 707 Output, Output equation, 98 Output variable, Output vector, 98 Over damping, 147 Overshoot, 148 Parabolic input, 135 728 Parallel connection, 80 Parameter estimation, 669 Parameter vector, 670 Partial fraction method, 514 Partitioned matrix, 709 Peak overshoot, 244, 349 Performance index, 248, 475 Performance robustness, 268 Period, 166 Periodic input, 135 Periodic signal, 166 Periodic solution, 592 Permitted region, 343 Persistent excitation, 673 Phase crossover frequency, 345 Phase margin, 345 Phase plane, 566 Phase plane trajectory, 566 Phase portrait, 566 Phase response, 172 Phase-area theorem, 296 Phase-lag compensation, 426 Phase-lag element, 428 Phase-lead compensation, 426, 436 Phase-lead element, 437 Phase-plane, 566 ¢>-circles, 354 PID digital controller, 551 Piecewise linear, 580 Plant, Plant sizing, 365 Plant uncertainty, 268 Poincare, 594 Polar plot, 300 Pole placement adaptive control, 676 Pole placement controller, 666 Pole-placement design, 465 Pole-placement regulator, 475 Pole-zero diagram, 79 Poles, 78 Popov criterion, 625 INDEX Popov's method, 624 Position error constant, 155 Positive definite, 617 Potential energy, 620 Potentiometer, 50 Power series expansion method, 514 Power-spectral density, 647 Practical sampler, 497 Pre-warped trapezoidal transformation, 549 Primary feedback loop, 381 Principle of least squares, 670 Principle of the argument, 307 Probability density function, 643 Proportional control, 253 Proportional response, 11 Proportional, integral and derivative control (PID), 256 Quiescent point, 13 Ramp input, 135 Rate gyro, 58 Recursive least squares algorithm, 672 Recursive projection algorithm, 671 Reduced-order observers, 483 Reference signal, 81 Reference variables, Regressor, 670 Regulator problem, 471 Relative stability, 277, 345 Relay, 565, 603 Relay controller, 584 Resolvent matrix, 207 Resonant frequency Wm, 293 Resonant peak (gain) Mm' 293 Return difference, 263 Rise time, 245 INDEX Robustness, 268 Root locus, 325, 541 Root locus method, 278 Rotational motion, 20 Routh array, 280 Routh-Hurwitz criterion, 278-279 Routh-Hurwitz test, 532 Saddle point, 567, 569 Sampled data systems, 495 Sampler, 495,517 Sampling frequency, 503 Sampling rate, 692 Saturation, 565, 602 Second order systems, 146 Secondary feedback loop, 381 Self-tuning regulator, 660 Semi-stable limit cycle, 593 Sensitivity function, 261 Separatrices, 575 Separatrix, 593 Servo problem, 471 Settling time, 245 Shannon's sampling theorem, 503 Sifting property, 73 Signal to noise ratio (SNR), 655 Similarity transformation, 114 Single-input single-output (SISO) systems, 377 Singular matrix, 708 Singular point, 567 Skew-symmetric matrix, 706 Slew rate, 398 Sliding regimes, 589 Small disturbances, 115 Stability, 7, 136, 222, 277, 615, 684 Stability diagram, 287 Stability oflinearized systems, 622 Stability region, 627 Stabilization, 420 729 Stable in the sense of Lyapunov, 615 Stable limit cycle, 593 Stable system, 154, 222 Standard deviation, 644 State equation, 16,71,96-98,204 State equation representation, 204 State equations of first approximation, 115 State observer gain matrix, 481 State representation, 96 State space, 1, 8, 96 State trajectory, 96 State variable, 8, 96 State variable feedback, 465 State vector, 96 Static decoupling, 391 Static feedforward control, 384 Static loop sensitivity, 326 Static resolution, 401 Steady state error, 136, 154 Steady state response, 7, 136,530 Stefan-Boltzmann law, 41 Step input, 134 Stochastic input, 643 Summing junction, 81 Sun tracker, 10 Superposition, 11, 565 Superposition property, 73 Supervisory control, 366 Switching servos, 584 Symmetric matrix, 706 Symplectic, 213 Synchro, 52 System, 3, 104 System analysis, System compensation, 419 System input, 98 System response, 133, 525 System sensitivity, 263 System synthesis, INDEX 730 System Type, 157, 246, 252, 295, 531 System Type retention, 451 System viewpoint, 72 Tachometer, 53 Tachometer feedback, 452 Tachometer plus lead compensation, 455 Taylor series, 12 Thermal constraints, 403 Thermal system, 39 Time average, 645 Time constant, 144 Time domain representation, 168 Time invariant, 72 Time stationary variable, 645 Torque to friction ratio, 399 Trajectory, 566 Transducers, 59 Transfer function , 13 " 71 77-78 , 99, 601 Transformer, 28 Transient response, 7, 136,204 Transition matrix, 212 Transpose matrix, 706 Triangular matrix, 707 Tustin approximation, 546, 548 Undamped natural frequency, 246 Under damping, 147 Unit matrix, 705 Units, basic, 697 Units, derived, 697 Units, supplementary, 697 Unity feedback, 84 Unstable limit cycle, 593 Unstable system, 153, 222, 420 van der Pol equation, 597 Variable control structure, 386 Variance, 644 Variation of parameters formula , 212 Vector, 701 Velocity error constant, 155 Vortex, 567 White noise, 652 Z-transform, 496, 509, 692, 710 Zero input response, 204, 238 Zero state response, 204, 238 Zero-order-hold, 504 Zero-order-hold (ZOH), 498 Zero-order-hold equivalence , 546 , 550 Zeros, 78 ... 342 356 358 359 CONTENTS Control Strategies and Plant Sizing 8.1 Introduction 8.2 Goals for Control System 8.3 Control and Controlled Variables 8.4 Reducing Goals to Control Strategies 8.5 Examples... Control Systems 13.1 Introduction 13.2 Adaptive Control Methods 13.3 Controller Design Methods 13.4 System Parameter Estimation 13.5 Adaptive Control Algorithms 13.6 Stability of Adaptive Controllers... analyzing control systems are listed in Appendix D Up to this point a wide range of control system analysis tools have been introduced Before we can proceed to the final system design and 10 CHAPTER INTRODUCTION