Performance output tracking for a 1-D unstable Schrödinger equation with input delay

In this paper, we investigate the performance output tracking for a one-dimensional unstable Schrödinger equation with input delay and disturbance, in which the control input is subject to a time delay. Both the performance output and the disturbance are non-collocated to the controller. By writing the time delay as a first-order hyperbolic equation, the control system can be modeled as a cascade construction in which the transport equation can be regarded as the actuator dynamics of the unstable wave equation. With the help of the actuator dynamics compensation, the difficulties caused by input delay are addressed. The full state feedback law is constructed to achieve the performance output tracking and an error based observer is designed successfully. The exponential stability of the closed-loop system is obtained.

1 Introduction

Time delay is an inevitable and universal phenomenon in routine production and engineering practice. There are many researches on time delay in partial differential control, which can be found in [1] –[4], among many others. In [5], where the Smith predictor is developed, we can find the early study of delay compensation. This method is useful to solve ordinary differential equation (ODE) systems with time delay. In [6], the unstable heat equation with input delay is considered by partial differential equation (PDE) backstepping. The PDE backstepping method is powerful and can also be used to stabilize other unstable reaction-diffusion equation with delay [7] –[9]. The approach of predictor-feedback is another way to cope with delayed distributed actuation for unstable heat equation [8]. In [10], the full state proportional integral regulation controller of a linear reaction-diffusion PDE with time delay is constructed by the way of spectral reduction. The performance output tracking is one of the most concerning topics in control theory. For the past few years, researchers have paid attention to the performance output tracking of infinite-dimensional systems in increasing numbers, see, for instance, [11] –[13]. In [9], [14], the PDE backstepping method is devoted to dealing with the stability problem with time delay.