Nonlinear Observer Design for an Undamped Wave Equation with Distributed Disturbance

This paper addresses the design of a nonlinear observer for an undamped wave equation with distributed disturbance and pointwise position measurement at the boundary. Using the sliding-mode based robust exact differentiator, the velocity at the boundary is estimated. Based on this estimate, a standard Luenberger observer is designed for the nominal, i.e., undisturbed system, and robust convergence is ensured in the presence of bounded distributed disturbances that are constant in time, by adequately combining this observer with an algebraic correction. The proposed design approach achieves exponential convergence of the observer and, for special observer gains, even ensures finite-time convergence. The approach is illustrated using numerical simulations for a representative case study.