On stochastic stabilization of sampled systems

TL;DR

This paper investigates stochastic stabilization of continuous systems under digital control, establishing conditions under which Markov policies ensure practical stability in sample-and-hold implementations, especially for systems driven by Brownian motion.

Contribution

It extends stochastic stability results to sample-and-hold control, linking ideal continuous policies to digital implementations for systems with Brownian noise.

Findings

Markov policies stabilize systems in sample-and-hold mode

Stability results apply to systems driven by Brownian motion

Generalizations include non-smooth Lyapunov functions and bounded noise

Abstract

This paper addresses stochastic stabilization in case where implementation of control policies is digital, i. e., when the dynamical system is treated continuous, whereas the control actions are held constant in predefined time steps. In such a setup, special attention should be paid to the sample-to-sample behavior of the involved Lyapunov function. This paper extends on the stochastic stability results specifically to address for the sample-and-hold mode. We show that if a Markov policy stabilizes the system in a suitable sense, then it also practically stabilizes it in the sample-and-hold sense. This establishes a bridge from an idealized continuous application of the policy to its digital implementation. The central result applies to dynamical systems described by stochastic differential equations driven by the standard Brownian motion. Generalizations are discussed, including the case of non-smooth Lyapunov functions for systems driven by bounded noise. A brief overview of bounded noise models is given.