Stochastic stability of nonlinear dynamical systems under information constraints

Abstract

We study the following problem: Given a stochastic nonlinear system controlled over a possibly noisy communication channel, what is the largest class of such channels for which there exist coding and control policies so that the closed-loop system is stochastically stable? The stability criterion considered is asymptotic mean stationarity (AMS). We first develop a general method (based on ergodic theory) to derive fundamental lower bounds on information transmission requirements leading to stabilization. Through this method we develop a new notion of entropy which is tailored to derive lower bounds for asymptotic mean stationarity for both noise-free and noisy channels. Our fundamental bounds are consistent, and more refined in comparison with the bounds obtained earlier via information theoretic methods. Moreover, our approach is more versatile in view of the models considered and allows for finer lower bounds when the AMS measure is known to admit further properties such as moment bounds.

A Novelty in Blahut-Arimoto T ype Algorithms: Optimal Control over Noisy Commu nication Channels

“…Overview C Ontrol-theory with an integration of information theory has received brilliant attention in the recent decade. There arises a totally important question, in this context, how to optimally control the system over a noisy communication channel with feedback [1], [2], [3], [4], [5], [6], [7], [8]. Undoubtedly, the principle of distortion-function as the criterion of estimation-error 1 plays an entirely central role.…”

“…In relation to the information-constraint based issues in control theory, [4], [5], [6], [7], [8] are brilliant. Some probabilitytheoretic procedures as well as information-theoretic ones were applied in [4], [5].…”

“…Some probabilitytheoretic procedures as well as information-theoretic ones were applied in [4], [5]. In [6], invariance properties of nonlinear stochasticities for dynamical systems were taken into account with regard to information constraints. In [7], the problem of optimal zero-delay coding as well as the estimation of a stochastic dynamical system over a memoryless noisy communication-channel were explored.…”

“…The exact estimation of the objective function is indeed impossible in reality due to parameter uncertainties. Simultaneous perturbation stochastic approximation (SPSA) [22] is a pseudo-gradient descent stochastic optimisation algorithm 4 which is of a purely useful nature for the cases in which the objective function 5 and its gradients are unknown 6 . Specifically, SPSA is totally important when the function is not differentiable at the optimising point.…”

“…5 Call loss-function. 6 Which is a fundamental beneficiary compared to traditional gradient decent algorithms. 7 It achieves the optimal value via the local optima of the objective function successfully moving towards the global optimum.…”