CONTROL THEORY a guided tour

J. R. Leigh

Peter Peregrinus Ltd. on behalf of the Institution of Electrical Engineers

Contents

Page The s tructure , p u r p o s e and content of the book xi

1 Control concepts : a n o n - m a t h e m a t i c a l in troduct ion 1 1.1 General Systems ideas 1 1.2 Wha t is control theory? - an initial discussion 3 1.3 Wha t is automatic control? 6 1.4 Some examples of control Systems 7

2 Control d e s i g n ideas: a n o n - m a t h e m a t i c a l t rea tment? 10 2.1 Initial discussion 10 2.2 Question: Can the best possible control actions be synthesised by

some mechanism? 11 2.3 Requirements for an automatic control System 12 2.4 Automatic feedback control 13 2.5 Diagrams illustrating and amplifying some of the concepts described

so far and showing relationships to a Software engineering context 13

3 Synthes i s of a u t o m a t i c feedback control loops: a m o r e quant i tat ive v i ew 18 3.1 Feedback loops: further discussion 18 3.2 What sorts of control laws are there? 20 3.3 How feedback control works - a practical view 21 3.4 General conditions for the success of feedback control strategies 26 3.5 Alternatives to feedback control 26

4 H o w the Laplace t r a n s f o r m great ly s impl i f i e s S y s t e m representa t ion and m a n i p u l a t i o n 28 4.1 Laplace transform techniques 28 4.2 Definition of the Laplace transform 28 4A Convergence of the integral that defines the Laplace transform 29 4B Problems with 0* and 0~ 29 4.3 Use of the Laplace transform in control theory 30 4.4 The concept of transfer function 30 4.5 System simplification through block diagram manipulation 31 4.6 How the transfer function can be obtained from a differential

equation 32 4.7 Poles and zeros of a transfer function 32

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4.8 Understanding System behaviour from a knowledge of pole and zero locations in the complex plane 32

4.9 Pole placement: synthesis of a Controller to place the closed loop poles in desirable positions 36

4.10 Moving the poles of a closed loop System to desirable locations - the root locus technique 37

4.11 Obtaining the transfer function of a process from either a frequency response curve or a transient response curve 38

4C Convolution - what it is 39 4.12 Determination of transfer functions by cross-correlation 41 4D Calculation of resonant frequency from the pole—zero diagram 41 4E Derivation of a formula for damped natural frequency 43 4F The root locus of a System with open-loop poles and zeros located as shown in

Fig. 4.19 will include a circle centred on the zero 44

5 Frequency r e s p o n s e m e t h o d s 45 5.1 Introduction 45 5.2 Design using frequency response methods - initial explanation 45 5.3 Frequency response of a linear System 45 5.4 The Bode diagram 46 5.5 Frequency response and stability: an important idea 46 5.6 Simple example of the use of the foregoing idea in feedback loop

design 48 5.7 Practical point - the need for stability margins 49 5.8 General idea of control design using frequency response methods 49 5.9 Obtaining the frequency response of a System experimentally - a list

of difficulties 50 5.10 Design based on knowledge of the response of a System to a unit

step input 50 5.11 How frequency response is obtained by calculation from a

differential equation 51 5.12 Frequency response testing can give a good estimate of a system's

transfer function 53 5.13 Frequency response of a second order System 54 5A The frequency response of a System with poles andjor zeros near to the

imaginary axis 56 5B Some interesting and useful ideas that were originated by Bode 56 5.14 Nyquist diagram and Nichols chart 57

6 M a t h e m a t i c a l m o d e l l i n g 58 6.1 Fundamental approaches to mathematical modelling 58 6.2 Methods for the development of mathematical modeis 59 6.3 Modelling a System that exists, based on da ta obtained by

experimentation 59 6.4 Construction of modeis from theoretical considerations 60 6.5 Methods/approaches/techniques for parameter estimation 61 6.6 Why modelling is difficult - an important discussion 62 6.7 Fixing of parameters 63 6.8 Parameter estimation 63 6.9 Regression analysis 63 6.10 Analysis of residuals 64 6A Doubl and certainty 65 6B Anticipatory Systems 65 6C Chaos 68

Contents vii

L i m i t s to P e r f o r m a n c e 70 7.1 Stability - initial discussion 70 7.2 Stability for control Systems - how it is quantified 71 7A The ingenious methodof Hurwitz 73 7.3 Linear System stability tests 73 7.4 Stability margin 73 7.5 Stability tests for non-linear Systems 73 7.6 Local and global stability 74 7.7 Lyapunov's second (direct) method for stability determination 76 7B Geometrie Interpretation of Lyapunov's second method 11 7.8 Wha t sets the limits on control Performance? 78 7.9 Limits and constraints on synthesis: summary of points 80 7.10 Systems that are difficult to control: unstable Systems 80 7C Cancellation of an unstable pole by a matching zero in the Controller 81 7D Shifting an unstable pole by feedback 82 7.11 Systems that are difficult to control - non-minimum phase Systems 82 7E Motivation for the name: non-minimum phase System 83 7F Mapping of complex funetions - a few points that underlie classical control

theory 85 7G Derivatives of a complex funetion G(sj 87 7H Singularities a complex funetion G(s) 87

Further s i m p l e i d e a s of control and feedback 88 8.1 How efficient is it to control an unknown process by a Controller

that consists only of a high gain of value C? 88 8.2 Another Standard approach to the control of an unknown process 89

8.2.1 The three term Controller 89 8.2.2 Illustration of the value of an integral term in removing any

constant error 89 8.2.3 Illustration of the value of a derivative term to control the

degree of damping 90 8.3 Effect of changes in the process, compared with the nominal process

model against which the Controller was designed 91 8.4 A closed-loop System reduces the effects of disturbances 92 8.5 Converting a user's requirement into a control speeification 92

Linear i sat ion 94 9.1 The motivation for linearisation 94 9.2 What is linearisation? 94

9.2.1 An initial trivial example 94 9.2.2 Comments 95

9.3 Linearisation about a nominal trajeetory: illustration 95 9.4 The derivative as best linear approximation 97 9A The inverse funetion theorem 98 9B The coneept of transversality 99

Mult ivar iable l inear p r o c e s s e s 100 10.1 Transfer funetion representations 100 10.2 State space representations 101 10.3 Design of feedback control Systems 102 10.4 Time Solution of the State space equation 102 10 A It seems remarkable that an oscillatory Solution can be generated by repeated

multiplication by a constant matrix 104 10.5 Discrete and continuous time modeis: a unified approach 104

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10B Generation of a control sequence 105 10.6 The concept of controllability for multivariable Systems 105 IOC Conservation of dimension under linear transformations 106

11 Discre te t i m e and Computer control 108 11.1 Computers as System components - devices that can change their

State only at discrete times 108 IIA A simple and informative laboratory experiment 110 11.2 Discrete time algorithms 110 11.3 Approaches to algorithm design 110 IIB A clever manipulation — how the digital to analogue convertor (zero order

hold) is transferred for calculation purposes to become part of the process to be controlled 112

HC Takahashis's algorithm 113 11.4 Overview: concluding comments, guidelines for algorithm choice

and some comments on procedure 115 HD Some difflculties in movingfrom differential equations to approximating

difference equations 115 11E Discretisation 116 HF A simple matter of quadratic behaviour 116 11G Continuous is not the limit of discrete as T -* 0 119 11H Non-uniqueness of inverse Z transforms 120 HI Stability is normally considered to be a property of a System so that for any

bounded input a stable System should produce a bounded Output 120

12 State e s t i m a t i o n : the K a i m a n filter and predic t ion 121 12.1 State estimation - what it attempts to do 121 12.2 How a state estimator works - the Kaiman filter 121 12.3 The Kaiman filter - more detail 122 12.4 Obtaining the optimal gain matrix 124 12.5 Prerequisites for successful application of the Kaiman filter in the

form shown in Fig. 12.4 125 12.6 Discussion of points arising 125 12.7 Planning, forecasting and prediction 126

12.7.1 Approaches to prediction 126 12.7.2 Physically-based modeis for prediction 127

12.8 Predictive control 128

13 Nonl inear i ty 129 13.1 Wha t is meant by nonlinearity 129 13.2 Approaches to the analysis of nonlinear Systems 131 13.3 The describing function method for analysis of control loops

containing nonlinearities 132 13.4 Linear second-order Systems in the State plane 134 13.5 Nonlinear second-order Systems in the state plane 135 13.6 Process nonlinearity - large signal problems 137 13.7 Process nonlinearity - small signal problems 138

14 O p t i m i s a t i o n 139 14.1 Initial discussion 139 14.2 Optimisation - a few ideas that can form building blocks 139 14.2.1 Discussion 145

Contents ix

14.3 Time-optimal control 146 14A Time-optimal control - a geometric view 148 14B The following geometric argument canform a basisfor the development of

algorithms orfor the proqf of the Pontryagin maximum principle 151 14C Construction of time-optimal controls 151

15 Dis tr ibuted S y s t e m s 153 15.1 The cosy idealisation that a System can be characterised by its

behaviour at a point in space 153 15.2 Alternative approaches to the modelling of distributed Systems 154

15.2.1 The representation of a spatial region as the summation of elemental regions 154

15.2.2 A 'Fourier type' approach, in which an arbitrary func t i on / on an interval [0, 1] is approximated by a summation of

15A 15B

luncuons/ j When can the behaviour in a region be well approximated at a point? Oscillating and osculating approximation of curves

16 References and further reading 16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8 16.9 16.10 16.11 16.12 16.13 16.14 16.15 16.16 16.17 16.18 16.19 16.20 16.21 16.22 16.23 16.24

Notat ion

Library books Other sources Mainstream control literature Mathematical modelling Discrete time Systems State estimation Nonlinear Systems Optimisation Distributed parameter Systems Statistical aspects of control Robust and adaptive control and other model-based control Functional analysis applied to linear control Time vector methods Anticipatory Systems and causality Pattern recognition Historical control books General mathematics references Ordinary differential equations Differential- topology/differential-geometry Theory of equations Chaos Miscellany Useful tables References and further reading

156 156

158 B 8 158 158 159 139 160 160 160 161 161 161 161 162 162 163 164 164 164 165 165 165 166 166 166

175

Index 178