A robustness property of the separation principle

We consider the stochastic linear regulator problem when the observation and driving noises are random processes with large, but finite, bandwidths. We show that as the bandwidth of the noise tends to infinity there is a natural limiting stochastic regulator problem which involves Gaussian white noise disturbances. The optimal control law of this problem, for which the Separation Principle holds, is suboptimal for the original problem. We obtain a power series expansion of the suboptimal cost of this control law in terms of the correlation time (inverse of the bandwidth) of the noise. From this expansion we conclude that as the bandwidth of the disturbances approaches infinity the suboptimal cost approaches the optimal cost of the limiting regulator problem, and so, that the Separation Principle is robust. Both finite and infinite time (steady state) problems are considered.