Asymptotic Series Solutions to a Class of Stochastic Dual Control Problems

The stochastic optimal control solution to the Linear-Quadratic-Gaussian problem with partial information is considered where the system parameters operating on the control are assumed to be unknown Gaussian random processes. This class of problems has special properties that are exploited. First, the state and the random parameters are conditionally Gaussian with respect to the measurement history. This allows the optimal stochastic control problem to be reformulated as a stochastic optimal control problem of full information where the new state space is the state and parameter estimates and the error variance of the state and parameters. Even in this somewhat simple form, there is no closed form solution. Therefore, it is assumed that there is only a first order dependence on the random fluctuation of the parameter about its mean. In this way an asymptotic series solution is obtained in terms of the new state space. For this problem a closed form solution for the controller is obtained up to second order. The controller is derived for the scalar case and then extended to the case of vector state with single input and multiple outputs. The characteristics of the new solution is investigated and a scalar numerial example is given to assess the performance of this new controller.