Optimal Adaptive Control of Uncertain Stochastic Discrete Linear Systems

The problem of optimal control of stochastic discrete linear time-invariant uncertain systems on finite time interval is formulated and partially solved. This optimal solution shows that previously published adaptive optimal control schemes and indirect adaptive control schemes do not need heuristics for their rationalization. It is shown that these schemes are suboptimal causal approximations of the optimal solution. The solution is achieved by the introduction of the State and Parameters Observability form SPOF. This representation of linear time-invariant systems enables application of tools from the LQR-LQG theory of control and estimation of discrete linear time-varying systems. The optimal solution is exact and non causal. It is composed of a causal optimal estimator of the augmented state composed of the state of the system and the parameters and of a non-causal controller. The solution shows that certainty equivalence principle applies for the state and parameters, but the separation does not apply. A causal suboptimal controller, using certainty equivalence, is proposed as an ad-hoc solution. This controller needs only the knowledge of the order of the system. The scheme is bibo stable for sufficiently low noises. As an example, the proposed algorithm, is applied to an unstable nonminimum phase model of a dynamic vehicle.