Optimal adaptive control of uncertain stochastic linear systems

The problem of optimal control of stochastic linear time-invariant uncertain systems on finite time interval is formulated and 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 new canonical representation of linear time-invariant systems enables direct application of the existing 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 the a linear time varying optimal estimator of the augmented state composed of the state of the system and the parameters, and of the optimal LQR controller. The estimator is causal. The controller is non-causal. As causal approximation control schemes based on the SPOF and the certainty equivalence principle and separation theorems are presented.