Proportional Integral Derivative Control for Continuous-time Stochastic Systems

Abstract

This chapter presents a pseudo-proportional integral derivative (PID) tracking control strategy for general non-Gaussian stochastic systems based on a linear B-spline model for the output PDFs. The objective is to control the conditional PDFs of the system output to follow a given target function. Different from existing methods, the control structure (i.e., the PID) is imposed before the output PDF controller design. Following the linear B-spline approximation on the measured output PDFs, the problem is transferred into the tracking of given weights which correspond to the desired PDF. In this case, since the output is the shape of the PDFs of the random output variables of systems, the classical PID controllers cannot be applied directly. Instead, a pseudo-PID control strategy will be presented for the weight tracking control problem, for which a dynamical model is established to describe the relationship between the weight and the output PDFs. For systems with or without model uncertainties, it is shown that the solvability can be cast into a group of matrix inequalities. Furthermore, an improved controller design procedure based on the convex optimization is proposed, which can guarantee the required tracking convergence with enhanced robustness. Simulations are given to demonstrate the efficiency of the proposed approach and encouraging results have been obtained.