LQG optimality and separation principle for general discrete time partially observed stochastic systems over finite capacity communication channels

• Published in: Automatica (Oxford) Vol. 44; no. 12; pp. 3181 - 3188
• Main Authors: Charalambous, Charalambos D., Farhadi, Alireza
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Ltd 01.12.2008; Elsevier
• Subjects: Applied sciences; Capacity; Computer science; control theory; systems; Computer systems and distributed systems. User interface; Control system analysis; Control system synthesis; Control theory. Systems; Exact sciences and technology; Modelling and identification; Networked; Rate distortion; Separation; Software; Stability; Networked; Rate distortion; Stability; Separation; Capacity; Control synthesis; Entropy; Stochastic system; Stabilizability; Uncertain system; Gaussian noise; White noise; Controllability; Additive noise; Lower bound; Channel capacity; Gaussian channel; LQG control; Rate distortion theory; Transmission channel; Detectability; Decoding circuit; Separation principle; Shannon theory; Information transmission; Gaussian process; Discrete time; Stochastic control; Capacity constraint; Optimality principle; Non deterministic system
• ISSN: 0005-1098; 1873-2836
• DOI: 10.1016/j.automatica.2008.05.021

This paper is concerned with control of stochastic systems subject to finite communication channel capacity. Necessary conditions for reconstruction and stability of system outputs are derived using the Information Transmission theorem and the Shannon lower bound. These conditions are expressed in terms of the Shannon entropy rate and the distortion measure employed to describe reconstruction and stability. The methodology is general, and hence it is applicable to a variety of systems. The results are applied to linear partially observed stochastic Gaussian controlled systems, when the channel is an Additive White Gaussian Noise (AWGN) channel. For such systems and channels, sufficient conditions are also derived by first showing that the Shannon lower bound is exactly equal to the rate distortion function, and then designing the encoder, decoder and controller which achieve the capacity of the channel. The conditions imposed are the standard detectability and stabilizability of Linear Quadratic Gaussian (LQG) theory, while a separation principle is shown between the design of the control and communication systems, without assuming knowledge of the control sequence at the encoder/decoder.