Observer Design for n + m Linear Hyperbolic ODE-PDE-ODE Systems

Abstract

In this letter, an observer design is presented for a system of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n + m$ </tex-math></inline-formula> hetero-directional transport partial differential equations (PDEs) coupled on both boundaries of the domain to ordinary differential equations (ODEs). This class of systems can represent, for instance, actuator and load dynamics at the boundaries of a hyperbolic system. The results in this letter provide a constructive way to reconstruct the state of the ODEs and the PDEs using only available measurements on one of the two ODEs. This observer design completes existing full-state feedback designs for this class of systems and enables, together with previous results, the construction of output-feedback stabilizers. As a complementary result, this letter includes a simple (constructive) method to obtain a stable left-inverse of a specific transfer matrix required in the observer design, which can also be applied (after a transposition) to the computation of the analogous stable right-inverses required for the control design.