Signalling and Control of Nonlinear Partially Observable Stochastic Systems: Information States, Sufficient Statistics, and Applications
摘要
In this paper the authors consider the operational problem of optimal signalling and control, called control-coding capacity (with feedback), CFB in bits/second, of discrete-time nonlinear partially observable stochastic systems in state space form, subject to an average cost constraint. CFB 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 CFB 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 CFB, 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 CFB 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 CFB 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 CFB includes as degenerate cases, problems of stochastic optimal control and channel capacity of information transmission.