Self-Tuning Controllers Via the State Space
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SELF-TUNING CONTROLLERS VIA THE STATE SPACE by KEVIN WARWICK January 1982 A thesis submitted for the degree of Doctor of Philosophy o the University of London and for the Diploma of Membership of the Imperial College. Department of Electrical Engineering Imperial College London SW7 2BT - 2 - ABSTRACT The field of self-tuning has developed due to the need for the control of systems, with unknown parameters, that are affected by stochastic disturbances. Most of the theory has been centred around polynomial controller descriptions based on a system considered as an autoregressive moving average process. In this thesis the system model is retained to enable an estimation of the system parameters to be carried out. However, once these estimations are obtained, a state space approach is used to calculate the required control action. By use of this scheme a controller has been achieved which has different properties,some advantageous, to those previously encountered. The resultant recursive control can be cal- culated simply and is such that it deals with a variable system time delay by employing a pole placement technique. With regard to a simple ex- tension from its primary form, the controller is able to deal with non- zero external inputs by means of steady state following. The self-tuning property of the state space method is proven, and from the proof it is shown that both this approach, and certain schemes previously employed, are special cases of a more generalised format. Subsequently, by taking account of this format, a tuner is suggested which incorporates a control input dependent on state feedback used in combination with linear output feedback, the state feedback providing pole placement and the linear output feedback allowing the variance of the system output to be optimized. The addition to the controller is shown to affect neither the use of an external input nor the overall self- tuning property. - 3 - Observer theory connected with each of the control schemes is discussed with a view to the formation of optimal and non-optimal observers , and where possible simulations are employed to show the nature of the various conditions obtainable. The systems considered are, for the most part, single-input-single-output, although the extension to the multivariable case is looked at. The use of the state space in self-tuning has given rise not only to alternative tuner operating techniques, but also to a deeper theoretical understanding of self-tuning in general, and this widens its field of applications to encompass areas in process control where state space theory predominates. - 4 - PREFACE In the past decade two major factors have influenced the theory connected with control systems. The ever increasing use of computers and more especially minicomputers has pointed the way towards an ex- pansion in the practical application of control algorithms which take account of this method of implementation. More complex theory has developed because of the wider handling capabilities thus achieved and from this much control system design is now based on a state space representation rather than the more classical frequency analysis or root locus techniques. The field of self-tuning controllers,which has arisen due to the improvement in computing power, has nevertheless remained in a polynomial system description framework. This thes is is intended to extend the field of self-tuning to the state space, where it is hoped further developments and increased awareness will result from its new. found cohesion with existing modern control procedures. The first chapter serves as an explanation of how self-tuning has evolved in its own right from previous adaptive control schemes. Basic definitions, used throughout the rest of the text are included, and previous, polynomial based, controllers are reviewed. The state space formulation is discussed in Chapter 2, and the resulting control action compared with that obtained from previous methods. Hence it is shown how, as well as arriving at an equivalent state space controller, by using other means of reconstructing the state, certain advantages over the polynomial forms can be achieved. - 5 - The properties of the state space controller are discussed in Chapter 3, where the proof of the general self-tuning property of pole placement type controllers is given as a starting point. Included in this chapter is the application of an external reference input signal, where it is explained how a state space description can be of benefit. Chapter 4 is used to provide extensions to the basic state space scheme and to show how various plant conditions, e.g. non- linearities, multivariable design; affect the fundamental assumptions. Of significance in this chapter is the section in which linear output feedback is employed in combination with state feedback, providing, in an output optimization sense, an improved control action under the pole placement criterion. The underlying filtering and observer theory is considered, as a slight diversification, in Chapter 5, where the techniques behind the numerous control schemes are discussed. This comparison is, of course, only possible by means of the state space representation. The thesis is set out in a way such that Chapter 5 may be read after Chapter 2 if so desired, with hopefully no loss of continuity, although in its present position it also serves as an explanation of certain results contained in Chapters 3 and 4. In conclusion I would like to express my gratitude and sincere appreciation to all those who have helped me in the completion of this work. Firstly, and most importantly, I wish to thank my supervisor, Professor J.H. Westcott, but I must extend my thanks to include Professor o P. Antsaklis, Professor K.J. Astrom and Dr. D.W. Clarke, who all took time - 6 - for useful discussion with me in the earlier stages. The helpful comments and suggestions from my fellow research students and departmental lecturers will never be forgotten, and to this end Dr, A. Voreadis must be singled out for a special mention. Without the financial support of the U.K. Science and Engineering Research Council I would be heavily in debt, and therefore I acknowledge, gratefully, their assistance. Finally I wish to thank Mrs. D. Abeysekera for her careful typing of the manuscript, which included the correction of my numerous spelling mistakes. - 7 - Glossary of Terms Used Sec. i.j. Represents Chapter i, Section j Sec. i. j(k) Represents Chapter i, Section j, subsection k. Ci.j.k) Denotes equation number k in Chapter i, Section j. Fig. i.j. Denotes figure number j in Chapter i. E{ •} The expected value of the function contained within the parentheses. -1 z Discrete time backward shift operator. y(s) ,u(.s) Output and Input signals respectively, in the frequency domain. G(s) Frequency domain transfer function. T1 Transport delay (total) T1 Sampling period y(t),u(t),e(t) Output, Input and disturbance signals respectively, at time t in the discrete time domain. k Integer part of the system time delay. T Fractional part of the system time delay. ACz'^^Cz'Setc. Polynomials in the backward shift operator. 6. Kronecker delta; i = j 6. = 1 ij ij i 1 j + 6.ij. =0 Hz) Spectral Density function. ft Covariance function. ARMA Autoregressive Moving Average. CARMA Controlled ARMA. k , k Maximum value of k. m max 0(t) Vector of regressors in parameter estimation (at time t), or otherwise a general cost function. 8 - 0(t) Vector of parameter estimates. (at time t) A(.t) Variable forgetting factor. n,m,p Denote dimension or number of parameters, n often has a subscript, e.g. n^. VC0) Maximum Likelihood function. e(.t) Prediction error estimate. /N -1 /N -1 ^ A(z ),B(z ),etc. Polynomials in the backward shift operator containing estimated parameters. x(t) State vector at time t. x(t) Estimate of state vector at time t. x(.t) State vector at time t formed using estimated parameters. A x(t) Estimate of x(t). Sl,S2,S3,S^ Weighting values in L.Q.G. design T [ ] Transpose of a matrix, vector or in the limiting case a scalar value. T Time domain or matrix containing specified pole polynomial parameters. t Present time instant. F or F(t) State feedback vector. x1(t) Modified state estimate. W Matrix used in pole placement design. 0 Null (zero) matrix A(t) Reconstruction error at time t. - 9 - CONTENTS Page TITLE PAGE 1 ABSTRACT ' 2 PREFACE 4 GLOSSARY OF TERMS USED 7 CONTENTS 9 1: Introduction to Self-Tuning Controllers 11 1.1 The Necessity for Adaptive Controllers 12 1.2 Adaptive Control Systems 15 1.3 System Models 20 1.4 Recursive Estimation for Discrete Time Processes 29 1.5 Self-Tuning 38 1.6 Concluding Remarks 51 2: State Space Controllers 53 2.1 Linear Quadratic Gaussian Control 54 2.2 State Space Formulations 55 2.3 State Estimation 60 2.4 Single Stage Cost Function Design 64 2.5 Pole Placement Design 74 2.6 Concluding Remarks ^ 3: Properties of State Space Self-Tuners 100 3.1 Self-Tuning Property 101 3.2 Self-Assigning Poles 114 3.3 Response to an External Input Signal 124 3.4 Concluding Remarks 147 - 10 - 4: Extensions to the Formulation 149 4.1 State Space Self-Tuning with Linear Output Feedback 150 4.2 Multivariable Controller Design 169 4.3 Nonlinear Systems Control 182 4.4 Ill-Defined System Models 186 4.5 Concluding Remarks 188 5: Filtering and Observer Theory 191 5.1 State Reconstruction for the Primary Model 192 5.2 Modified State Reconstruction Part I: Optimal Observer 198 5.3 Modified State Reconstruction Part II: Non-Optimal Observer 207 5.4 Concluding Remarks 213 6: Conclusions 214 APPENDICES 222 REFERENCES 240 CHAPTER 1 INTRODUCTION TO SELF TUNING CONTROLLERS Adaptive control is a general technique of which self-tuning is a special, recently initiated, classification.