Minimum Rate Coding for LTI Systems Over Noiseless Channels

This paper studies rate requirements for state estimation in linear time-invariant (LTI) systems where the controller and the plant are connected via a noiseless channel with limited capacity. Using information theoretic arguments, we obtain first for scalar systems, and subsequently for multidimensional systems, lower bounds on the data rates required for state estimation under three different stability criteria, namely monotonic boundedness of entropy, asymptotic stability of distortion, and support size stability. Further, the minimum data rate achievable by any source-encoder is computed under each of these criteria, and the best rate achievable with quantization is shown to be in agreement with the information-theoretic bounds in some specific cases (such as if the system coefficient is an integer or if the criterion is an asymptotic one). Existence of optimal variable-length and fixed-length quantizers are studied and optimal quantizers are constructed under each of these criteria. One observation is that, the uniform quantizer is, in addition to being simple, efficient in linear control systems

[1]  D. Delchamps Stabilizing a linear system with quantized state feedback , 1990 .

[2]  Robin J. Evans,et al.  Stabilizability of Stochastic Linear Systems with Finite Feedback Data Rates , 2004, SIAM J. Control. Optim..

[3]  David L. Neuhoff,et al.  Quantization , 2022, IEEE Trans. Inf. Theory.

[4]  T. Basar,et al.  Quantization for LTI systems with noiseless channels , 2003, Proceedings of 2003 IEEE Conference on Control Applications, 2003. CCA 2003..

[5]  Paul L. Zador,et al.  Asymptotic quantization error of continuous signals and the quantization dimension , 1982, IEEE Trans. Inf. Theory.

[6]  Gustav Herdan Principles of Information Theory , 1966 .

[7]  T. Başar,et al.  Coding and Control over Discrete Noisy Forward and Feedback Channels , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[8]  Ian R. Petersen,et al.  Set-valued state estimation via a limited capacity communication channel , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[9]  Herbert Gish,et al.  Asymptotically efficient quantizing , 1968, IEEE Trans. Inf. Theory.

[10]  Tamer Basar,et al.  Simultaneous design of measurement and control strategies for stochastic systems with feedback , 1989, Autom..

[11]  Marcel Paul Schützenberger,et al.  On the Quantization of Finite Dimensional Messages , 1958, Inf. Control..

[12]  Sekhar Tatikonda,et al.  Control under communication constraints , 2004, IEEE Transactions on Automatic Control.

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

[14]  Sekhar Tatikonda,et al.  Control over noisy channels , 2004, IEEE Transactions on Automatic Control.

[15]  Robin J. Evans,et al.  EXPONENTIAL STABILISABILITY OF MULTIDIMENSIONAL LINEAR SYSTEMS WITH FINITE DATA RATES , 2002 .

[16]  D. Luenberger Optimization by Vector Space Methods , 1968 .

[17]  Tamás Linder,et al.  Optimal entropy-constrained scalar quantization of a uniform source , 2000, IEEE Trans. Inf. Theory.

[18]  A. Matveev,et al.  Shannon Zero Error Capacity and the Problem of Almost Sure Observability over Noisy Communication Channels , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[19]  Bruce A. Francis,et al.  Limited Data Rate in Control Systems with Networks , 2002 .

[20]  Nicola Elia,et al.  Stabilization of linear systems with limited information , 2001, IEEE Trans. Autom. Control..

[21]  Serdar Yüksel,et al.  State Estimation and Control for LTI Systems Over Communication Channels , 2003 .

[22]  T. Başar,et al.  On the absence of rate loss in decentralized sensor and controller structure for asymptotic stability , 2006, 2006 American Control Conference.

[23]  Sandro Zampieri,et al.  Stability analysis and synthesis for scalar linear systems with a quantized feedback , 2003, IEEE Trans. Autom. Control..

[24]  Anant Sahai,et al.  The Necessity and Sufficiency of Anytime Capacity for Stabilization of a Linear System Over a Noisy Communication Link—Part I: Scalar Systems , 2006, IEEE Transactions on Information Theory.

[25]  Tamer Basar,et al.  Control of Multi-Dimensional Linear Systems over Noisy Forward and Reverse Channels , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.