Linear stochastic optimal control under information rate constraints

The discrete-time, linear, stochastic optimal control problem is considered under information rate constraints on the feedback loop. The feedback loop, including sensor, is modelled as a communication channel which provides a specified amount of information (in the Shannon sense) about the state of the linear plant at each discrete-time instant given the current and past observations and past controls. No further specific structure for the sensor is assumed. The expected value of a positive definite quadratic loss function is used as the performance criterion to be minimized. This leads to a double minimization problem in which the performance criterion is minimized over the set of admissible controls and the set of conditional probability densities for the state given the observations and controls which achieve the specified information. A set of recursion relationships for the solution of this problem is derived using the techniques of calculus of variations and dynamic programming. The solution indicates that the optimum sensor is linear in the state with additive Gaussian noise with parameters which achieve the specified information rate; and, with the appropriate parameters in the measurement equation, the solution reduces to the well-known solution for the linear plant-quadratic cost problem with linear measurements and additive Gaussian noise.