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Fundamental Limits of Controlled Stochastic Dynamical Systems: An Information-Theoretic Approach
In this paper, we examine the fundamental performance limitations in the
control of stochastic dynamical systems; more specifically, we derive generic
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 and ,
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.Comment: Note that this is an extended version of the original submission
"Fundamental Limits on the Maximum Deviations in Control Systems: How Short
Can Distribution Tails be Made by Feedback?"; arXiv admin note: text overlap
with arXiv:1912.0554