Fundamental Limits on the Maximum Deviations in Control Systems: How Short Can Distribution Tails be Made by Feedback?

In this paper, we adopt an information-theoretic approach to investigate the fundamental lower bounds on the maximum deviations in feedback control systems, where the plant is linear time-invariant while the controller can generically be any causal functions as long as it stabilizes the plant. 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 level of randomness (as quantified by the conditional entropy) contained in the disturbance. Such bounds provide fundamental limits on how short the distribution tails in control systems can be made by feedback.

[1]  Quanyan Zhu,et al.  Information-Theoretic Performance Limitations of Feedback Control: Underlying Entropic Laws and Generic $\mathcal{L}_{p}$ Bounds , 2019, 2021 American Control Conference (ACC).

[2]  Quanyan Zhu,et al.  Fundamental Limitations in Sequential Prediction and Recursive Algorithms: Lp Bounds via an Entropic Analysis , 2020, 2020 54th Annual Conference on Information Sciences and Systems (CISS).

[3]  Lijun Chen,et al.  An Integrative Perspective to LQ and $\ell _\infty$ Control for Delayed and Quantized Systems , 2020, IEEE Transactions on Automatic Control.

[4]  Naira Hovakimyan,et al.  A Simplified Approach to Analyze Complementary Sensitivity Tradeoffs in Continuous-Time and Discrete-Time Systems , 2018, IEEE Transactions on Automatic Control.

[5]  Lei Guo,et al.  Feedback and uncertainty: Some basic problems and results , 2020, Annu. Rev. Control..

[6]  Naira Hovakimyan,et al.  Sensitivity analysis of linear continuous-time feedback systems subject to control and measurement noise: An information-theoretic approach , 2019, Syst. Control. Lett..

[7]  Generic Bounds On The Maximum Deviations In Sequential Prediction: An Information-Theoretic Analysis , 2019, 2019 IEEE 29th International Workshop on Machine Learning for Signal Processing (MLSP).

[8]  Quanyan Zhu,et al.  Generic Variance Bounds on Estimation and Prediction Errors in Time Series Analysis: An Entropy Perspective , 2019, 2019 IEEE Information Theory Workshop (ITW).

[9]  Jie Chen,et al.  Power Gain Bounds of MIMO Networked Control Systems: An Entropy Perspective , 2019, IEEE Transactions on Automatic Control.

[10]  Yorie Nakahira Connecting the Speed-Accuracy Trade-Offs in Sensorimotor Control and Neurophysiology Reveals Diversity Sweet Spots in Layered Control Architectures , 2019 .

[11]  Jie Chen,et al.  Fundamental limitations and intrinsic limits of feedback: An overview in an information age , 2019, Annu. Rev. Control..

[12]  Karl Henrik Johansson,et al.  A Frequency-Domain Characterization of Optimal Error Covariance for the Kalman-Bucy Filter , 2018, 2018 IEEE Conference on Decision and Control (CDC).

[13]  Jie Chen,et al.  An Integral Characterization of Optimal Error Covariance by Kalman Filtering , 2018, 2018 Annual American Control Conference (ACC).

[14]  Jie Chen,et al.  Fundamental error bounds in state estimation: An information-theoretic analysis , 2017, 2017 IEEE 56th Annual Conference on Decision and Control (CDC).

[15]  Jie Chen,et al.  Design constraints and limits of networked feedback in disturbance attenuation: An information-theoretic analysis , 2017, Autom..

[16]  Jie Chen,et al.  Tradeoffs in Networked Feedback Systems: From Information-Theoretic Measures to Bode-Type Integrals , 2017, IEEE Transactions on Automatic Control.

[17]  Jie Chen,et al.  Towards Integrating Control and Information Theories , 2017 .

[18]  G. Picci,et al.  Linear Stochastic Systems: A Geometric Approach to Modeling, Estimation and Identification , 2016 .

[19]  Vijay Gupta,et al.  The effect of delayed side information on fundamental limitations of disturbance attenuation , 2015, 2015 54th IEEE Conference on Decision and Control (CDC).

[20]  Vijay Gupta,et al.  On disturbance propagation in leader-follower systems with limited leader information , 2014, Autom..

[21]  Mf Marcel Heertjes,et al.  Self-tuning in master–slave synchronization of high-precision stage systems , 2013 .

[22]  Chaouki T. Abdallah,et al.  Information theoretic conditions for tracking in leader-follower systems with communication constraints , 2013 .

[23]  Naira Hovakimyan,et al.  Bode-like Integral for Continuous-Time Closed-Loop Systems in the Presence of Limited Information , 2013, IEEE Transactions on Automatic Control.

[24]  Shinji Hara,et al.  Achievable sensitivity bounds for MIMO control systems via an information theoretic approach , 2011, Syst. Control. Lett..

[25]  Naira Hovakimyan,et al.  Bode-like integral for stochastic switched systems in the presence of limited information , 2011, Proceedings of the 2011 American Control Conference.

[26]  G. Vinnicombe,et al.  Fundamental limits on the suppression of molecular fluctuations , 2010, Nature.

[27]  Chaouki T. Abdallah,et al.  Limitations in tracking systems , 2010 .

[28]  Prashant G. Mehta,et al.  Bode-Like Fundamental Performance Limitations in Control of Nonlinear Systems , 2010, IEEE Transactions on Automatic Control.

[29]  Shinji Hara,et al.  Characterization of a complementary sensitivity property in feedback control: An information theoretic approach , 2009, Autom..

[30]  Munther A. Dahleh,et al.  Feedback Control in the Presence of Noisy Channels: “Bode-Like” Fundamental Limitations of Performance , 2008, IEEE Transactions on Automatic Control.

[31]  Sang Joon Kim,et al.  A Mathematical Theory of Communication , 2006 .

[32]  Munther A. Dahleh,et al.  Fundamental Limitations of Disturbance Attenuation in the Presence of Side Information , 2005, IEEE Transactions on Automatic Control.

[33]  Pablo A. Iglesias,et al.  Nonlinear extension of Bode's integral based on an information-theoretic interpretation , 2003, Syst. Control. Lett..

[34]  Lei Guo,et al.  How much uncertainty can be dealt with by feedback? , 2000, IEEE Trans. Autom. Control..

[35]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[36]  R. Gray Entropy and Information Theory , 1990, Springer New York.

[37]  John G. Proakis,et al.  Probability, random variables and stochastic processes , 1985, IEEE Trans. Acoust. Speech Signal Process..

[38]  H. W. Bode,et al.  Network analysis and feedback amplifier design , 1945 .