Stochastic control over finite capacity channels: Causality, feedback and uncertainty

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 to develop a mathematical framework for analysis and design of optimal communication/control system subject to uncertainty, causality and feedback. The theory is developed by generalizing the classical information theoretic theorems of Shannon to their robust analogs. Necessary conditions for stability and reconstruction of uncertain control signals over uncertain channels are presented via the causal rate distortion function for a family of sources and channel capacity for a family of channels. Further, in view of the complexity of the causal rate distortion function, a tight lower bound is introduced which applies to families of sources. Finally, using a generalized version of the information transmission theorem and the lower bound on the rate distortion function, the connection between stabilizabilty and reconstruction is related to capacity for a class of channels.