Output feedback stabilization for 1-D unstable wave equations with boundary control matched disturbance and van der Pol nonlinear boundary

Abstract

In this paper we study the output feedback stabilization of a plant described by a one-dimensional wave equation with a van der Pol type nonlinear boundary condition, an unknown internal nonlinear uncertainty and an external disturbance acting at the control end (boundary). The simultaneous occurrence of nonlinear internal uncertainty, external disturbance, and van der Pol type nonlinear boundary term in the plant leads to the system is more complicated. First, we show that the open-loop system is well-posed and we design an infinite-dimensional estimator to estimate the disturbance. It is shown that the disturbance estimator can successfully estimate the total disturbance, in the sense that the estimation error signal is in L2(0,∞). Using the estimated disturbance, we propose an asymptotic state observer, and then we design an observer-based output feedback stabilizing controller. The closed-loop system is shown to be asymptotically stable. Finally, a numerical simulation is carried out to illustrate the theoretical results and effectiveness of the proposed control law.