Youla–Kuc̆era Parameterization in Contraction Framework

Abstract

In this article, we study incrementally exponentially stable (IES) image and kernel representations for nonlinear systems with the aim of generalizing the Youla–Kuc̆era parameterization in the contraction framework. We first construct these representations and their stable inverses in the contraction framework and then provide a parameterization of stabilizing controllers by additionally assuming incremental input-to-state stability for the image representation. Focusing on constant metrics results in a parameterization of all stabilizing controllers rendering the closed-loop systems IES with respect to constant metrics if an observer having the same dimension as a system can be designed. After that, we revisit the presented image and kernel representations from the variational viewpoint and show that their variational systems are, respectively, image and kernel representations for the variational systems of the original nonlinear systems. Then, we interpret the proposed controller parameterization in terms of the Youla–Kuc̆era parameterization for variational systems.