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

2020/12/22 by Song Fang, Quanyan Zhu, Fang, Song +1
Computer Science · Engineering · Mathematics · #Advanced Control Systems Optimization #Applied mathematics #Computer science #Conditional entropy #Conditional expectation #Control (management) #Control Systems and Identification #Control theory (sociology) #Dynamical systems theory #Entropy (arrow of time) #FOS: Computer and information sciences #FOS: Electrical engineering #FOS: Mathematics #Information Theory (cs.IT) #Machine Learning (cs.LG) #Mathematical analysis #Mathematics #Optimization and Control (math.OC) #Physics #Principle of maximum entropy #Robotics (cs.RO) #Stability and Control of Uncertain Systems #Statistics #Systems and Control (eess.SY) #Upper and lower bounds #cs.IT #cs.LG #cs.RO #cs.SY #eess.SY #electronic engineering #information engineering #math.IT #math.OC

paper · pdf · doi:10.48550/arxiv.2012.12174

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:1912.05541

openalex publication_date 2020/12/22 · openalex created_date 2021/05/24 · arxiv created 2021/06/03 · arxiv updated 2021/06/07 · openalex updated_date 2026/07/28

Abstract

In this paper, we examine the fundamental performance limitations in the control of stochastic dynamical systems; more specifically, we derive generic 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 = ∞, 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.

Citations

• Information-Theoretic Performance Limitations of Feedback Control: Underlying Entropic Laws and Generic L_p Bounds
• A Mathematical Theory of Communication
• Approximation by superpositions of a sigmoidal function
• Human-level control through deep reinforcement learning
• Generic Variance Bounds on Estimation and Prediction Errors in Time Series Analysis: An Entropy Perspective
• Probability, Random Variables, and Stochastic Processes
• Patterns, predictions, and actions: A story about machine learning

Related

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