On Parameter Estimation and Control of Time-Varying Stochastic Systems

Bengt Lindoff

Department of Mathematical Statistics,
Lund Institute of Technology,
Lund University,

ISBN 91-628-2619-0


This thesis is about parameter estimation and control of time-varying stochastic systems. It can be divided into two parts.

The first part deals with an estimation algorithm commonly used when estimating parameters in time-varying stochastic systems, the Recursive Least Squares (RLS) algorithm with forgetting factor. The exact statistical properties for the RLS-estimator with forgetting factor are in most cases difficult to find, due to the complex dependency of the time-varying characteristics and on the forgetting factor. In the first part of this thesis, the RLS-estimator with forgetting factor is applied to different time-varying as well as stationary FIR-, AR- and ARX-structures and some distribution properties for the parameter estimates are derived.

A method to compute the exact distribution and moments of the RLS-estimator in a time-varying Gaussian AR(1)-processes is presented. For stationary vector autoregressions and stationary ARX-models the asymptotic bias and covariance function of the RLS estimates are derived. The estimated covariance matrix of the parameter estimates is important when analyzing RLS with forgetting factor. The first moment of this estimate is calculated showing that the asymptotic bias is nonzero. Furthermore, the MSE for the parameter estimate is derived for time-varying FIR-models, giving a possibility to find an optimal forgetting factor in the RLS algorithm.

The second part concerns the problem on controlling time-varying stochastic systems. Optimal control of such systems is generally a very difficult task, which simultaneously must take the character of the unknown time-varying parameters and the fulfilment of the control action into account. The optimal controller action thus must have dual features. However, the optimal dual controller is in most cases impossible to derive, so suboptimal dual controllers must be used.

In the thesis a new optimal adaptive predictive controller (APC) for time-varying stochastic systems is presented that can be explicitely computed for arbitrary prediction horizons. Also a large simulation study of different suboptimal dual controllers is made. The study shows that the APC can successfully be used as a suboptimal dual controller.

Key words:
Time-Varying Stochastic Systems, Linear Systems, Recursive Estimation, Recursive Least Squares, Forgetting Factor, Quadratic Forms, Convergence Analysis, Dual Control, Adaptive Stochastic Control, Adaptive Predictive Control