Optimal stochastic control of discrete linear systems with unknown gain

The optimal, adaptive control of linear discrete systems with constant but imprecisely known gain is considered. It is assumed that a priori, Gaussian distribution functions are available for the unknown gain and for any random disturbances or observation error. The identification and control aspects are separated to the extent that the identification of the system gain is formulated as an optimal estimation problem; while selection of the optimal control policy is accomplished by minimization of the expected value of a quadratic cost functional, conditioned up on the information available at the time of control application and subject to the constraints imposed by the identification techniques. The interaction between identification and control is explicitly shown by the form of these constraints. The effects of this interaction are examined by considering a simple, first order system and the average performance of the optimal adaptive control law is compared to that of the optimal deterministic control.