LQG Control with minimum directed information

We consider a discrete-time Linear-QuadraticGaussian(LQG) control problem in which Massey’s directedinformation from the observed output of the plant to the controlinput is minimized while required control performance is attainable.This problem arises in several different contexts, includingjoint encoder and controller design for data-rate minimization innetworked control systems. We show that the optimal control lawis a Linear-Gaussian randomized policy. We also identify the statespace realization of the optimal policy, which can be synthesizedby an efficient algorithm based on semidefinite programming.Our structural result indicates that the filter-controller separationprinciple from the LQG control theory, and the sensor-filterseparation principle from the zero-delay rate-distortion theoryfor Gauss-Markov sources hold simultaneously in the consideredproblem. A connection to the data-rate theorem for mean-squarestability by Nair & Evans is also established.