Fundamental Limits of Controlled Stochastic Dynamical Systems: An Information-Theoretic Approach
This work provides foundational theoretical limits for control engineers designing stochastic dynamical systems, particularly for understanding the inherent performance boundaries imposed by system characteristics and disturbances.
This paper investigates the fundamental performance limits in controlling stochastic dynamical systems, deriving generic L_p bounds for causal controllers and stochastic disturbances. The bounds are characterized by unstable poles/nonminimum-phase zeros of the plant and the conditional entropy of the disturbance for linear time-invariant systems, and solely by the conditional entropy for strictly causal plants.
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-Szegö formula.