Understanding the Capability of PD Control for Uncertain Stochastic Systems

TL;DR

This paper investigates the effectiveness of PD control in stabilizing uncertain second-order stochastic systems, revealing conditions for global stability and fundamental limitations of PD control through algebraic inequalities.

Contribution

It demonstrates that PD control can globally stabilize certain uncertain stochastic systems under specific bounds, and identifies the limitations of PD control via a polynomial condition.

Findings

PD control achieves global stabilization under algebraic bounds.

Stabilizing PD parameters form a two-dimensional convex set.

A polynomial condition indicates when PD control fails to stabilize.

Abstract

In this article, we focus on the global stabilizability problem for a class of second order uncertain stochastic control systems, where both the drift term and the diffusion term are nonlinear functions of the state variables and the control variables. We will show that the widely applied proportional-derivative(PD) control in engineering practice has the ability to globally stabilize such systems in the mean square sense, provided that the upper bounds of the partial derivatives of the nonlinear functions satisfy a certain algebraic inequality. It will also be proved that the stabilizing PD parameters can only be selected from a two dimensional bounded convex set, which is a significant difference from the existing literature on PD controlled uncertain stochastic systems. Moreover, a particular polynomial on these bounds is introduced, which can be used to determine under what conditions the system is not stabilizable by the PD control, and thus demonstrating the fundamental limitations of PD control.