Stochastic control through capacity limited channels

Abstract

In the present thesis, we are concerned with reliable data reconstruction and stability of dynamical systems which are controlled over limited capacity communication channels. Necessary conditions for reliable data reconstruction and stability of sequences in r-mean and probability are derived which depend on the entropy rate of the input to the encoder and the type of reconstruction and stability criteria. These conditions are given in terms of the Shannon lower bound and they are applicable to linear and nonlinear systems. Using these conditions, some of necessary conditions which are already available in the literature, are obtained as a special case. We discuss some of them throughout the thesis. Moreover, under certain conditions these necessary conditions are also sufficient. The results are applied to a linear stochastic partially observed control system subject to measurement noise when the channel is an Additive White Gaussian Noise (AWGN) channel. Here, we derive an encoder, decoder, and controller for mean square stability and reconstruction using the standard detectability and stabilizability assumptions of Linear Quadratic Gaussian (LQG) theory. From obtained conditions for reliable data reconstruction and stability, it is concluded that the Shannon capacity is still an adequate measure for describing the conditions for moment reliable data reconstruction and stability. We find the continuous version of the well known eigenvalue rate condition for continuous time systems when they are controlled over continuous time AWGN channels. This condition is described by the summation of the real parts of the unstable eigenvalues of the open loop time-invariant system. This eigenvalue rate condition is obtained by addressing the necessary condition for stability of a fully observed linear continuous time-invariant noiseless plant; and by constructing an encoding scheme and a stability scheme for reliable data reconstruction and stability of a continuous time plant driven by Brownian motion. Necessary conditions for uniform reliable data reconstruction and robust stability of the uncertain dynamical systems which are controlled over communication channels, are also derived. These conditions are given in terms of the robust entropy rate of the inputs to the encoder and an additional term which is related to the reconstruction and stability criteria. The uncertainty in the dynamical system is described by a relative entropy constraint. Such uncertainty description is a natural generalization of the sum quadratic uncertainty description. These conditions are applied to specific uncertain systems by computing the robust entropy rate. Moreover, a relation between robust entropy rate and the Algebraic Riccati equation appearing in the H infinity estimation and control problem, is established. Under certain conditions the obtained necessary conditions are also sufficient. Furthermore, We study the stability problem of a fully observed controlled Gauss Markov uncertain system which is subject to the sum quadratic uncertainty restriction. In addition, throughout it is shown that the Shannon lower bound is an adequate measure for describing the conditions for reliable data reconstruction and stability of sequences related to a dynamical system which is controlled over a limited capacity communication channel.