An Information-Theoretic Analysis of Discrete-Time Control and Filtering Limitations by the I-MMSE Relationships

Fundamental limitations or performance tradeoffs/limits are important properties and constraints of both control and filtering systems. Among various tradeoff metrics, total information rate that characterizes the sensitivity tradeoffs and time-averaged performance of control and filtering systems was conventionally studied by using the differential entropy rate and Kolmogorov–Bode formula. In this article, by extending the famous mutual information-minimum mean-square error (I-MMSE) relationships to the discrete-time additive white Gaussian channels with and without feedback, a new paradigm is introduced to estimate and analyze total information rate as a control and filtering tradeoff metric. Under this framework, we explore the tradeoff properties of total information rate for a variety of the discrete-time control and filtering systems, e.g., linear time-invariant, linear time-varying, and nonlinear, and propose an alternative approach to investigate total information rate via optimal estimation.