Identification and Control of Multivariable Stochastic Feedback Systems

Abstract

This paper considers the problem of simultaneous identification and control of stochastic processes characterized by linear dynamic models with unknown systems parameter coefficients. Stochastic approximation is used to derive consistent identification algorithms for the case in which arbitrary feedback controls are present. These identification methods can also be used for determining the order of the system, if the latter is unknown, as well as the exact canonical structure for the multivariable case.An approximation to the optimal control solution is obtained by explicitly separating the functions of identification and control, and asymptotic convergence to a stochastic optimal controller is attained without on-line structural modification. IntroductionThere is currently much interest in identification and control of stochastic systems where the mathematical model of the system is assumed unknown-the so-called "black box" problem, or where only partial information about the model is given. Such problems are typical of the real world, where most physical systems are perturbed by unwanted external disturbances, and accurate mathematical models for present-day complex systems are rarely attainable.Within this general framework, a frequently recurring problem is the simultaneous identification and feedback control of a stochastic system described by a linear, dynamic, discrete-time mathematical model with unknown system parameters. The identification part of the problem is in itself a formidable task. The difficulty arises from the fact that the presence of the feedback control signal produces additional correlations in existing identification algorithms that were intended for open-loop systems only.Recently, however we have developed [1] a stochastic approximation algorithm for identification of the unknown parameters of a single-input/single-output linear system with an arbitrary feedback controller. In this algorithm, the controller is capable of utilizing current information from the identifier and asymptotically approaches the "stochastic optimal controller" as the identifier estimates converge. However, the above identification scheme had not been generalized to handle other than single-input/single-output systems. But since models of most physical systems are of the multi-input/multi-output type, the purpose of this paper is to extend the scheme to the multivariable case. To achieve this goal, the matrices of the model must be presented in a special canonical form, similar to the phase-variable canonical form for single-input/single-output systems. It will be shown that