Fundamental Limits on the Maximum Deviations in Control Systems: How Short Can Distribution Tails be Made by Feedback?

In this paper, we adopt an information-theoretic approach to investigate the fundamental lower bounds on the maximum deviations in feedback control systems, where the plant is linear time-invariant while the controller can generically be any causal functions as long as it stabilizes the plant. It is seen in general that the lower bounds are characterized by the unstable poles (or nonminimum-phase zeros) of the plant as well as the level of randomness (as quantified by the conditional entropy) contained in the disturbance. Such bounds provide fundamental limits on how short the distribution tails in control systems can be made by feedback.