Optimum design of measurement channels and control policies for linear-quadratic stochastic systems

In the design of optimal controllers for linear-quadratic stochastic systems, a standard assumption is that the measurement channels are fixed and linear, and the measurement noise is Gaussian. In this paper the authors relax the first part of this restriction and raise the issue of the derivation of optimum measurement structures as a part of the overall design. Toward this end, they take the measurement process as one given by a Wiener integral, and modify the cost function so that it now places some soft constraints on the measurement strategy. Using some results from information theory, the authors show that the scalar version (for both finite and infinite horizons) of this joint design problem admits an optimum, dictating linear designs for both the controller and the measurement strategy. For the vector version, however, it is possible for a nonlinear design to improve over the best linear one. In both cases, best linear designs involve the solutions of nonlinear (deterministic) optimal control problems.