The randomized linear control policy method and some bounds for the adaptive multiple model problem

Using admissibility in the analysis of the multiple model, finite horizon, discrete time LQG problem, a family of lower and upper bounds of the minimum expected cost is obtained. Some of the earlier works are also shown to be related through admissibility and bounds. The well-known OLFO control is generalized to a closed-loop, Optimum Feedback Linear Policy (OFLP) control. Then, a new method of approximating the optimal solution is developed, called the Randomized Linear Control Policy (RLCP) method. Here, the learning nature of the controller about the true but unknown model is embedded in the computations of the expected cost by enlarging the class of control policies, from deterministic to stochastic. The randomized controls for the future stages are used to obtain the deterministic control for the current stage. Because of the linearity of the control policies in RLCP, it is possible to obtain analytically an approximate value of the expected cost as a function of control, for each fixed RLCP. This analytically computed cost is also bounded below and above by some appropriate bounds.