Sampled-data Boundary Observer Design for One-Dimensional ODE-Schrödinger Cascade System

In this paper, we are concerned with the sampled-data hybrid systems, described by a linear ordinary differential equation(ODE), and the partial differential equation(PDE) represented by a Schrödinger equation. We propose a small-gain method to analyse the exponential stability of this dynamical hybrid systems. This stability result proves to be useful in designing sampled-output exponentially convergent observers for systems that are represented by an ODE-PDE cascade. An illustration example is presented to confirm the effectiveness of the proposed exponential observer.

1 Introduction

The problem of designing sampled-output observer for nonlinear finite-dimensional systems have received a great deal of interest over the past decade, see for instance [1]–[4], and the references therein. Less effort is devoted to the design of the sampled output observer for partial differential equations(PDEs). In [5], a sampled-output observer with zero-order-hold(ZOH) has been developed for semilinear systems represented by a scalar diffusion equation. In [6], an observer featuring time-varying gain sampled-output has been proposed for a class of semilinear systems described by parabolic PDEs. The time-varying gain was shown to be beneficial to achieve larger maximum allowable sampling period(MASP). In [7], a small-gain approach is presented for analyzing the exponential stability of an ODE connected in series with a heat equation. Exponential stability is established under conditions involving the maximum allowable sampling period.