Sliding Mode and Active Disturbance Rejection Control to Stabilization of One-Dimensional Anti-Stable Wave Equations Subject to Disturbance in Boundary Input

In this technical note, we are concerned with the boundary stabilization of a one-dimensional anti-stable wave equation subject to boundary disturbance. We propose two strategies, namely, sliding mode control (SMC) and the active disturbance rejection control (ADRC). The reaching condition, and the existence and uniqueness of the solution for all states in the state space in SMC are established. The continuity and monotonicity of the sliding surface are proved. Considering the SMC usually requires the large control gain and may exhibit chattering behavior, we then develop an ADRC to attenuate the disturbance. We show that this strategy works well when the derivative of the disturbance is also bounded. Simulation examples are presented for both control strategies.

I. Introduction

In the last few years, the backstepping method has been introduced to the boundary stabilization of some PDEs ([6], [13]). This powerful method can also be used to deal with the stabilization of wave equations with uncertainties in either boundary input or in observation ([4], [5]). Owing to its good performance in disturbance rejection and insensibility to uncertainties, the sliding mode control (SMC) has also been applied to some PDEs, see [1], [2], [8], [9], [11], [12]. Other approaches that are proposed to deal with the disturbance include the active disturbance rejection control (ADRC) ([3]), and the adaptive control method for systems with unknown parameters ([6], [7]). In the ADRC approach, the disturbance is first estimated in terms of the output; and then canceled by its estimates. This is the way used in [4], [5], [7]. Compared with the SMC, the ADRC has not been used in distributed parameter systems.