Output feedback control of general linear heterodirectional hyperbolic ODE-PDE-ODE systems

This paper considers the backstepping design of observer-based compensators for general linear heterodirectional hyperbolic ODE-PDE-ODE systems, where the ODEs are coupled to the PDEs at both boundaries and the input appears in an ODE. A state feedback controller is designed by mapping the closed-loop system into a stable ODE-PDE-ODE cascade. This is achieved by representing the ODE at the actuated boundary in Byrnes-Isidori normal form. The resulting state feedback is implemented by an observer for a collocated measurement of the PDE state, for which a systematic backstepping approach is presented. The exponential stability of the closed-loop system is verified in the ∞-norm. It is shown that all design equations can be traced back to kernel equations known from the literature, to simple Volterra integral equations of the second kind and to explicitly solvable boundary value problems. This leads to a systematic approach for the boundary stabilization of the considered class of ODE-PDE-ODE systems by output feedback control. The results of the paper are illustrated by a numerical example.