Fundamental Limits of Controlled Stochastic Dynamical Systems: An Information-Theoretic Approach

TL;DR

This paper establishes fundamental information-theoretic limits on the performance of controlling stochastic dynamical systems, linking system properties and disturbances to control bounds across various scenarios.

Contribution

It derives generic $\\mathcal{L}_p$ bounds for causal controllers in stochastic systems, connecting system poles, zeros, and disturbance entropy to control limitations.

Findings

Lower bounds depend on unstable poles and disturbance entropy for LTI plants.

For causal plants, bounds are determined solely by disturbance entropy.

Special cases include minimum-variance and maximum deviation control bounds.

Abstract

In this paper, we examine the fundamental performance limitations in the control of stochastic dynamical systems; more specifically, we derive generic $\mathcal{L}_p$ bounds that hold for any causal (stabilizing) controllers and any stochastic disturbances, by an information-theoretic analysis. We first consider the scenario where the plant (i.e., the dynamical system to be controlled) is linear time-invariant, and 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 conditional entropy of the disturbance. We then analyze the setting where the plant is assumed to be (strictly) causal, for which case the lower bounds are determined by the conditional entropy of the disturbance. We also discuss the special cases of $p = 2$ and $p = \infty$, which correspond to minimum-variance control and controlling the maximum deviations, respectively. In addition, we investigate the power-spectral characterization of the lower bounds as well as its relation to the Kolmogorov-Szeg\"o formula.