3 Youla Parameterization for Descriptor Systems

Abstract

For control engineers, the stability of the given control systems has always been the main concern. We often encounter controller synthesis problems of the following form: Given a plant $\vec{G}$, design a controller $\vec{K}$ such that the closed-loop system is stable and satisfies certain given performance criteria. It is thus convenient to have a parameterization of the class of controllers which stabilize the plant and to optimize the performance criteria within this class. This approach has been quite successful in the modern control systems theory, the result being the Youla-Kucera parameterization. The objective of this chapter is to extend the results which gave the Youla parameterization for nonlinear systems, as obtained via a state-space approach, of [46] and [76] to the nonlinear descriptor systems case. By providing such a parameterization for descriptor systems, it is hoped that controller synthesis problems for descriptor systems may also become tractable. In particular, one would like to tackle the $\mathcal{H}_\infty$ control problems in this way.