Stochastic Modelling and Control: (Monographs on Statistics & Applied Probability)

This book aims to provide a unified treatment of input/output modelling and of control for discrete-time dynamical systems subject to random disturbances. The results presented are of wide applica- bility in control engineering, operations research, econometric modelling and many other areas. There are two distinct approaches to mathematical modelling of physical systems: a direct analysis of the physical mechanisms that comprise the process, or a 'black box' approach based on analysis of input/output data. The second approach is adopted here, although of course the properties ofthe models we study, which within the limits of linearity are very general, are also relevant to the behaviour of systems represented by such models, however they are arrived at. The type of system we are interested in is a discrete-time or sampled-data system where the relation between input and output is (at least approximately) linear and where additive random dis- turbances are also present, so that the behaviour of the system must be investigated by statistical methods. After a preliminary chapter summarizing elements of probability and linear system theory, we introduce in Chapter 2 some general linear stochastic models, both in input/output and state-space form. Chapter 3 concerns filtering theory: estimation of the state of a dynamical system from noisy observations. As well as being an important topic in its own right, filtering theory provides the link, via the so-called innovations representation, between input/output models (as identified by data analysis) and state-space models, as required for much contemporary control theory.

Table of Contents:
1 Probability and linear system theory.- 1.1 Probability and random processes.- 1.2 Linear system theory.- Notes and references.- 2 Stochastic models.- 2.1 A general output process.- 2.2 Stochastic difference equations.- 2.3 ARMA noise models.- 2.4 Stochastic dynamical models.- 2.5 Innovations representations.- 2.6 Predictor models.- Notes and references.- 3 Filtering theory.- 3.1 The geometry of linear estimation.- 3.2 Recursive estimation.- 3.3 The Kalman filter.- 3.4 Innovations representation of state-space models.- Notes and references.- 4 System identification.- 4.1 Point estimation theory.- 4.2 Models.- 4.3 Parameter estimation for static systems.- 4.4 Parameter estimation for dynamical systems.- 4.5 Off-line identification algorithms.- 4.6 Algorithms for on-line parameter estimation.- 4.7 Bias arising from correlated disturbances.- 4.8 Three-stage least squares and order determination for scalar ARMAX models.- Notes and references.- 5 Asymptotic analysis of prediction error identification methods.- 5.1 Preliminary concepts and definitions.- 5.2 Asymptotic properties of the parameter estimates.- 5.3 Consistency.- 5.4 Interpretation of identification in terms of systems approximation.- Notes and references.- 6 Optimal control for state-space models.- 6.1 The deterministic linear regulator.- 6.2 The stochastic linear regulator.- 6.3 Partial observations and the separation principle.- Notes and references.- 7 Minimum variance and self-tuning control.- 7.1 Regulation for systems with known parameters.- 7.2 Pole/zero shifting regulators.- 7.3 Self-tuning regulators.- 7.4 A self-tuning controller with guaranteed convergence.- Notes and references.- Appendix A A uniform convergence theorem and proof of Theorem 5.2.1.- Appendix B The algebraic Riccati equation.- Appendix C Proof of Theorem 7.4.2.- Appendix D Some properties of matrices.- Appendix E Some inequalities of Holder type.- Author index.