Youla-Kučera Parametrization in the Contraction Framework

In this paper, we study incrementally exponentially stable (IES) image and kernel representations for nonlinear systems with the aim of generalizing the Youla-Kučera parametrization in the contraction framework. We first construct these representations and their stable inverses in the contraction framework and then provide a parametrization of stabilizing controllers by additionally assuming incremental input-to-state stability for the image representation. Focusing on constant metrics results in a parametrization 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. Finally, we interpret the proposed controller parameterization in terms of the Youla-Kučera parametrization for variational systems.