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 $\mathcal{L}_p$ 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.
Code (0)
등록된 구현이 없습니다.
Similar Papers 제목 키워드 기반
Fundamental Limits of Obfuscation for Linear Gaussian Dynamical Systems: An Information-Theoretic Approach
In this paper, we study the fundamental limits of obfuscation in terms of privacy-distortion tradeoffs for linear Gaussian dynamical systems via an information-theoretic approach. Particularly, we obtain analytical formu…
Neural Markov Controlled SDE: Stochastic Optimization for Continuous-Time Data
We propose a novel probabilistic framework for modeling stochastic dynamics with the rigorous use of stochastic optimal control theory. The proposed model called the neural Markov controlled stochastic differential equat…
Stochastic OptimizationTime SeriesTime Series AnalysisA Dynamical Systems Perspective on the Analysis of Neural Networks
In this chapter, we utilize dynamical systems to analyze several aspects of machine learning algorithms. As an expository contribution we demonstrate how to re-formulate a wide variety of challenges from deep neural netw…
Sampling via Controlled Stochastic Dynamical Systems
We present a general framework for constructing controlled stochastic dynamical systems that exactly sample from a class of probability distributions with Gaussian tails. Given a target distribution and a reference stoch…
On the Structural Limits of Machine Learning Decision Systems: An Information-Theoretic, Interaction-Based, and Stochastic-Dynamical Perspective
Machine learning procedures are commonly evaluated in terms of predictive accuracy and computational efficiency. However, their achievable performance is fundamentally constrained by structural properties of the underlyi…
Computational Efficiency