Observer design for 2×2 linear hyperbolic PDEs that are bidirectionally coupled with nonlinear ODEs

Abstract

A prediction-based observer is designed for systems consisting of 2×2 linear heterodirectional hyperbolic partial differential equations (PDEs) with collocated actuation and measurement at one boundary and bidirectional coupling with nonlinear ordinary differential equations (ODEs) at the other one. Following a preliminary backstepping transformation, a solution-based analysis of the plant allows to draw upon well-known results on the observer design for nonlinear ODEs. For that, the design is split into several steps, with a retarded observer for the ODE state and a prediction thereupon, thus significantly extending existing results for PDE-ODE systems. It is shown that the resulting error system admits a cascaded structure, from which (local) asymptotic convergence follows. Numerical studies underline the performance of the observer.