Zero dynamics and funnel control of general linear differential - algebraic systems ∗

We study linear differential-algebraic multi-input multi-output systems which are not necessarily regular and investigate the zero dynamics and tracking control. We use the concepts of autonomous zero dynamics and (E,A,B)-invariant subspaces to derive the so called zero dynamics form which decouples the zero dynamics of the system and exploit it for the characterization of system invertibility. Asymptotic stability of the zero dynamics is characterized and some implications for stabilizability in the behavioral sense are shown. A refinement of the zero dynamics form is then exploited to show that the funnel controller (that is a static nonlinear output error feedback) achieves for a special class of right-invertible systems with asymptotically stable zero dynamics tracking of a reference signal by the output signal within a pre-specified performance funnel. It is shown that the results can be applied to a class of passive electrical networks.