Value of Information in Feedback Control: Global Optimality

Abstract

The rate-regulation tradeoff, defined between two objective functions, one penalizing the packet rate and one the regulation cost, can express the fundamental performance bound of networked control systems. However, the characterization of the set of globally optimal solutions in this tradeoff for multi-dimensional Gauss&#x2013;Markov processes has been an open problem. In the present article, we characterize a policy profile that belongs to this set without imposing any restrictions on the information structure or the policy structure. We prove that such a policy profile consists of a symmetric threshold triggering policy based on the value of information and a certainty-equivalent control policy based on a non-Gaussian linear estimator. These policies are deterministic and can be designed separately. Besides, we provide a global optimality analysis for the value of information <inline-formula><tex-math notation="LaTeX">$\boldsymbol{\operatorname{VoI}}_{\boldsymbol{\mathit {k}}}$</tex-math></inline-formula>, a semantic metric that emerges from the rate-regulation tradeoff as the difference between the benefit and the cost of a data packet. We prove that it is globally optimal that a data packet containing sensory information at time <inline-formula><tex-math notation="LaTeX">$\boldsymbol{\mathit {k}}$</tex-math></inline-formula> be transmitted to the controller only if <inline-formula><tex-math notation="LaTeX">$\boldsymbol{\operatorname{VoI}}_{\boldsymbol{\mathit {k}}}$</tex-math></inline-formula> becomes nonnegative. These findings have important implications in the areas of communication and control.