CHARALAMBOUS Charalambos D., LOUKA Stelios
In this paper the authors consider the operational problem of optimal signalling and control, called control-coding capacity (with feedback), $C_{FB}$ in bits/second, of discrete-time nonlinear partially observable stochastic systems in state space form, subject to an average cost constraint. $C_{FB}$ is the maximum rate of encoding signals or messages into randomized controller-encoder strategies with feedback, which control the state of the system, and reproducing the messages at the output of the system using a decoder or estimator with arbitrary small asymptotic error probability. In the first part of the paper, the authors characterize $C_{FB}$ by an information theoretic optimization problem of maximizing directed information from the inputs to the outputs of the system, over randomized strategies (controller-encoders). The authors derive equivalent characterizations of $C_{FB}$, using randomized strategies generated by either uniform or arbitrary distributed random variables (RVs), sufficient statistics, and a posteriori distributions of nonlinear filtering theory. In the second part of the paper, the authors analyze $C_{FB}$ for linear-quadratic Gaussian partially observable stochastic systems (LQG-POSSs). The authors show that randomized strategies consist of control, estimation and signalling parts, and the sufficient statistics are, two Kalman-filters and an orthogonal innovations process. The authors prove a semi-separation principle which states, the optimal control strategy is determined explicitly from the solution of a control matrix difference Riccati equation (DRE), independently of the estimation and signalling strategies. Finally, the authors express the optimization problem of $C_{FB}$ in terms of two filtering matrix DREs, a control matrix DRE, and the covariance of the innovations process. Throughout the paper, the authors illustrate that the expression of $C_{FB}$ includes as degenerate cases, problems of stochastic optimal control and channel capacity of information transmission.
