Understanding the Capability of PD Control for Uncertain Stochastic Systems

In this article, we focus on the global stabilizability problem for a class of 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 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 differs significantly from the existing results for PD controlled stochastic systems where the diffusion term does not depend on the control variables. 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.