Boundary Exponential Stabilization of a One-Dimensional Anti-Stable Wave Equation with Control Matched Disturbance

In this paper, we are concerned with output feedback stabilization for a one-dimensional anti-stable wave equation with disturbance. First, we design a disturbance estimator for the original system. Then, we propose an output feedback controller for the original system. By calculation, the closed-loop of original system is proved to be exponentially stable and well-posed. Finally, this paper is summarized.


Introduction
The wave equation is a set of differential equations derived from Maxwell's equations, which describes the wave characteristics of electromagnetic field. It is an important partial differential equation and has important research significance in the field of control. In recent years, anti-stable one-dimensional wave equation with boundary disturbance has been researched in different ways in the field of control. On this issue, there is used the Lyapunov function approach to design controller in Guo [1] (2014). Besides, active disturbance rejection control (ADRC) has established itself as a powerful control technology in dealing with vast uncertainty in control system. A class of nonlinear systems is dealt with a modified nonlinear extended state observer (ESO) of a time-varying gain in active disturbance rejection control (ADRC) in Zhao [2] (2015). In Wu [3], they apply the active disturbance rejection control (ADRC), an emerging control technology, to output feedback stabilization for a class of uncertain multi-input Boundary stabilization is considered for a multi-dimensional wave equation with boundary control matched disturbance that depends on both time and spatial variables. The active disturbance rejection control (ADRC) approach is adopted in investigation in Guo [4]. An algorithm with the active disturbance rejection control approach is developed to reject time and spatially varying boundary disturbances from a multidimensional Kirchhoff plate via boundary control in [5] (2014). A vital step toward ADRC is to estimate the disturbance through an extended state observer (ESO). In Zhao [6] (2015) and [7] (2011), a nonlinear ESO is designed for a kind of lower triangular nonlinear systems with large uncertainty. A nonlinear extended state observer (ESO) is investigated that constructed from piece-wise smooth functions consisted of linear and fractional power functions in Zhao [8] (2015). In Wu [9], they construct a nonlinear ESO for a class of uncertain lower triangular nonlinear systems with stochastic disturbance and show its convergence, where the total disturbance includes internal uncertain nonlinear part and external stochastic disturbance. Besides, the disturbance is then compensated in feedback loop by its estimate. In Guo [10] (2015), the active disturbance rejection control (ADRC) approach is adopted in investigation. In addition, the ADRC can also efficiently reduce the control energy in practice [11]. Moreover, there are many kinds of disturbances, the "backstepping" method for the problem of stabilization of one-dimensional wave equation with input harmonic disturbance is adopted in the design of the adaptive regulator in Guo [12] (2013). And there are many design methods for a boundary controlled one-dimensional wave equation with external disturbance.
A new method is proposed to estimate the total disturbance without using high gain in Zhou [13] (2018). In particular, the derivative of disturbance isn't longer commanded to be bounded, instead, we should relax the disturbance to be The requirement is different from that in the ADRC of lumped parameter systems by high gain [14] [15] [16] [17]. Our main focus is on stabilization for the following anti-stable one-dimensional wave equation with general boundary disturbance: erally represents an unknown external disturbance. The observer design for system (1.1) can describe the measurement of knocking in the combustion process of automotive engine [18]. For the presence of external disturbance, this system became unstable. A state feedback control was designed in [16]. Next, we con- . This is enough to show in Feng [19] (2017), it is obvious that using three output signals in his paper. Besides, five equations are used in his study of stabilization and tracking problems, including three-wave equations and two transport equations in the recent work Zhou [20] (2017). There exists the approach of "backstepping" in designing controller. Compared with the method he studied, I simplify it. In this paper, I reduce the number of applying dynamic compensators, using threewave equations and a transport equation to solve the problem. In this paper, we use a new approach to disturbance estimation by directly designing an infinite-dimensional disturbance estimator with using two measurements. Specifically, the stability of closed-loop system is proved by the method of matrix change, which is in the process of innovation.
The paper is organized as follows. In Section 2, we design a disturbance estimator for original system (1.1). The new system is constructed by the known system, and then the disturbance estimator is designed. In Section 3, we are devoted to designing the output feedback control, and proving the exponential stability of the resulting closed-loop system. Finally, gives the concluding remarks.

Disturbance Estimator Design
In this section, we devote to design a disturbance estimator for system (1.1). To this end, we first introduce the following coupled system consisting of transport equation and wave equation . Then z is governed by the following wave equation.
We construct an auxiliary dynamic system as follows: The error between systems (2.2) and (2.
Define the state space where 3 0 c > . In the rest of this paper, we write norm H  without discrimination.
Define the system operator ( ) Then system (2.4) can be written as an abstract evolutionary equation in H.
The following lemma is Lemma 2.1 in [19].
, system (2.4) admits a unique bounded solution Next, we design an observer for system (2.4) which implies (2.15) holds.
When the initial value ( ) ( ) , r x t is governed by the following PDEs Notice that the initial value ( ) . Therefore, observer (2.9) can recover state and disturbance simultaneously for (2.4).

Output Feedback Controller Design
We design the output feedback of control for system (1.1) as follows where the initial value of system (3.2) is ( ) where the initial value We will consider the r-subsystems of (3.7) respectively. For any initial value

( ) Engineering
Then the γ-subsystem can be written as an abstract evolutionary equation in H It is well-known that operator 1 A generates an exponentially stable 0 C -semigroup 1 e A t on H. It is a routine exercise that 1 B is admissible for 1 e A t ([22]). Then the solution to system (3.9) can be written as

Conclusion Remarks
In this paper, the anti stable wave equation with disturbance is controlled. Firstly, the disturbance estimator is designed for the original system, and then the disturbance is estimated. Then, the controller is designed to realize the control. In addition, many references, especially the relevant data about ADRC, are used to verify the closed-loop system, the closed-loop system is simplified and the content of this paper is obtained.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.