Stochastic Weak Passivity Based Stabilization of Stochastic Systems with Nonvanishing Noise

TL;DR

This paper introduces stochastic weak passivity and stability concepts for stochastic systems with persistent noise, providing new stabilization methods that ensure weak stability despite nonvanishing noise at the desired state.

Contribution

It proposes stochastic weak passivity and asymptotic weak stability, offering a novel framework for stabilizing stochastic systems with nonvanishing noise using feedback control.

Findings

Stochastic weak passivity ensures stabilization outside a certain region.

Theorems guarantee global or local stabilization under feedback.

Applications demonstrated on linear and nonlinear stochastic systems.

Abstract

For stochastic systems with nonvanishing noise, i.e., at the desired state the noise port does not vanish, it is impossible to achieve the global stability of the desired state in the sense of probability. This bad property also leads to the loss of stochastic passivity at the desired state if a radially unbounded Lyapunov function is expected as the storage function. To characterize a certain (globally) stable behavior for such a class of systems, the stochastic asymptotic weak stability is proposed in this paper which suggests the transition measure of the state to be convergent and the ergodicity. By defining stochastic weak passivity that admits stochastic passivity only outside a ball centered around the desired state but not in the whole state space, we develop stochastic weak passivity theorems to ensure that the stochastic systems with nonvanishing noise can be globally\locally stabilized in weak sense through negative feedback law. Applications are shown to stochastic linear systems and a nonlinear process system, and some simulation are made on the latter further.