Fixed configuration control of a linear system with random coefficients

In this paper, the optimal fixed configuration control of a linear stochastic system, having random parameters and an additive white noise input disturbance, is investigated. It is assumed that a noisy observation of the state is available. The observation is to be filtered, and the state of the filter used to control the stochastic system. The gains in the filter and control, the initial condition of the filter, and the pamameter specifying the filter dynamics, are to be selected so that optimal performance results subject to the constraint that the filter and control be a linear configuration. Optimal performance requires the minimization of the expected value of a quadratic performance criterion. A two point boundary value problem (TPBVP) is derived using the minimum principle, giving a set of necessary conditions for optimality. An important feature of the result is that when the filter and control gains are selected optimally, the selection of the filter dynamics is arbitrary. The optimization problem is singular with respect to one of the parameters of the filter which indicates a certain amount of freedom in the fixed configuration design. It should be noted that the control scheme proposed in this study must be considered suboptimal rather than optimal, since the linear configuration of the controller is specified prior to optimization. An example is included to illustrate application of the fixed configuration technique to the class of problems considered.