Observer Design for ODE-PDE Cascade with Lipschitz ODE Nonlinear Dynamics and Arbitrary Large PDE Domain

Abstract

This paper deals with observer design for systems constituted of an Ordinary Differential Equation (ODE), containing a Lipschitz function of the state, and a linear Partial Differential Equation (PDE) of diffusion-reaction heat type. In addition to nonlinearity, system complexity also lies in the fact that no sensor can be implemented at the junction point between the ODE and the PDE. Boundary observers providing state estimates of the ODE and PDE do exist but their exponential convergence is only ensured under the condition that either the Lipschitz coefficient is sufficiently small or the PDE domain length is sufficiently small. Presently, we show that limitation is removed by using two connected observers and implementing one extra sensor providing the PDE state at an inner position close to the ODE-PDE junction point. The measurements provided by both sensors are used by the first observer which estimates, not only the PDE state, but also its spatial derivative along the PDE subdomain located between both sensor positions. Using these estimates, the second observer determines estimates of the ODE state and the remaining PDE states. All state estimates are shown to be exponentially convergent to their true values.