LQG control with minimum directed information

We consider a framework of LQG optimal control in which Massey's directed information from the state sequence to the control sequence is minimized. The use of directed information in this study is motivated by the data-rate minimization in quantized LQG control. We show that an optimal control policy in our framework can be realized by a simple three-stage architecture comprising (i) a linear sensor with additive Gaussian noise, (ii) a Kalman filter, and (iii) a certainty equivalence controller. This result can be viewed as an integration of two previously known separation theorems: the filter-controller separation theorem in the standard LQG control theory, and the sensor-filter separation theorem that arises in zero-delay rate-distortion theory for Gauss-Markov sources. A tractable computational algorithm based on semidefinite programming is proposed to synthesize an optimal policy.