Stochastic control over finite capacity channels: Causality and feedback

Optimal communication/control analysis and design of dynamical controlled systems, when there are finite capacity communication constraints often involve information and control theoretic analysis. This paper, employees information theoretic concepts, which are subject to causality, and thus applicable to control/communication system analysis and design in which encoders, decoders and controllers are causal. The basic mathematical concepts are constructive; they are based on a modified version of Shannon's self-mutual information, known as directed information, which describes how much information a random process conveys to another random process, to account for the causality of the probabilistic channel connecting these random processes, when feedback is used. Following this construction data processing inequalities are derived while the solution to the casual information rate distortion is obtained in which the optimal reconstruction kernel is causally dependent on the source and control. Further, in view of the complexity of the causal rate distortion function, a tight lower bound on the casual rate distortion function is derived which is easy to apply in many control/communication systems. Finally, using the information transmission theorem, the connection between stabilizabilty and observability of the control system is related to the causal rate distortion function and its lower bound.