Adaptive output tracking for a hybrid wave-ODE system subject to unknown sinusoidal disturbance

In this paper, we are concerned with the output tracking problem for a hybrid wave-ODE system coupling an elastic string with a rigid body at one end. The control force acts on the rigid body and the output is at the other end where is subject to an unknown sinusoidal disturbance. By constituting auxiliary system and internal model dynamic, an adaptive method is developed to identify the unknown frequency of disturbance. An error-based adaptive dynamic compensator is proposed to achieve exponential output tracking. The well-posedness and stability of the closed-loop system are proved by employing the semigroup theory. The numerical simulations are presented to confirm the effectiveness of the proposed control strategy.

1 Introduction

Output tracking is one of the central problems in control theory. The main objective of output tracking is to find a control such that the performance output of the control plant can track the given reference signal in the presence of disturbance. One of the main difficulties in output tracking is the disturbance treatment. There are many approaches to cope with disturbance in the problem of partial differential equation control. These include adaptive control [1], sliding mode control [2], active disturbance rejection control [3] and internal model principle (IMP) [4], [5]. In [6], performance output tracking for a wave equation with measurement/disturbance collocated configuration is considered by the method of active disturbance rejection control. The regulation problem for 1-d wave equation is considered in [7] where disturbance is dealt with by the adaptive control. However, the reference signal to be tracked and the disturbance to be rejected usually are might not be available in real-world applications.