Fundamental Limitations of Control and Filtering in Continuous-Time Systems: An Information-Theoretic Analysis

While information-theoretic methods have been introduced to investigate the fundamental control and filtering limitations for a few decades, currently, there is no direct method or trade-off metric to analyze the limitations of continuous- and discrete-time problems within a unified framework. To answer this challenge, we lift the traditional information-theoretic methods to infinite-dimensional spaces and formulate various control and filtering systems uniformly as noisy communication channels. Channel capacity and total information rate are studied from the perspective of general control and filtering trade-off, and computed from the estimation errors of channel inputs. Fundamental constraints on the trade-off metrics are derived and used to capture the limitations of continuous-time control and filtering systems. For the control and filtering systems in the linear case, the general trade-offs serve as the performance limits related to the characteristics of plant models. For the systems with nonlinear plants, we compute the general trade-offs and their lower bounds by resorting to the Stratonovich-Kushner equation.