Convergence of fundamental limitations in information, estimation, and control

The emerging study of integrating information theory and control systems theory has attracted considerable attention by researchers, mainly motivated by problems involving communication channels in loops. The main goals of the present paper are to introduce a new perspective to this study, namely the perspective of integrating information, estimation, and control, and to unify fundamental limitations in information, estimation, and control. In particular, we establish, over a discrete-time Gaussian channel with memory, a general equivalence among feedback communication, estimation, and feedback stabilization. We then show that, the achievable information rate in the feedback communication problem can be alternatively given by the decay rate of the Cramer-Rao bound (CRB) in the associated estimation problem, or by the Bode sensitivity integral in the associated control problem. Therefore, we conclude that the fundamental limitations in information transmission (i.e. achievable information rate), in information processing (i.e. CRB), and in information utilization (i.e. Bode integral), seemingly different and usually separately treated, are in fact three sides of the same entity. We also briefly discuss the possibility of extending the results to channels in continuous-time and channels not in feedback loops.