The active disturbance rejection and sliding mode control approach to the stabilization of the Euler–Bernoulli beam equation with boundary input disturbance

Abstract

In this paper, we are concerned with the boundary feedback stabilization of a one-dimensional Euler–Bernoulli beam equation with the external disturbance flowing to the control end. The active disturbance rejection control (ADRC) and sliding mode control (SMC) are adopted in investigation. By the ADRC approach, the disturbance is estimated through an extended state observer and canceled online by the approximated one in the closed-loop. It is shown that the external disturbance can be attenuated in the sense that the resulting closed-loop system under the extended state feedback tends to any arbitrary given vicinity of zero as the time goes to infinity. In the second part, we use the SMC to reject the disturbance by removing the condition in ADRC that the derivative of the disturbance is supposed to be bounded. The existence and uniqueness of the solution for the closed-loop via SMC are proved, and the monotonicity of the “reaching condition” is presented without the differentiation of the sliding mode function, for which it may not always exist for the weak solution of the closed-loop system. The numerical simulations validate the effectiveness of both methods.

Introduction

In the past three decades, the Euler–Bernoulli beam equation has been a representative model for the control of systems governed by partial differential equations (PDEs). This is spurred by the outer space applications, such as flexible link manipulators and antennas, for which the suppression of the vibration by the boundary feedback is a central issue. We refer Han, Benaroya, and Wei (1999) for engineering interpretation of the beam equations.

There are many works contributed to the stabilization of the beam equation. The examples can be found in Chen, Delfour, Krall, and Payre (1987), Guo and Yu (2001), He, Ge, How, Choo, and Hong (2011), He, Zhang, and Ge (2012), Luo, Guo, and Morgul (1999), Luo, Kitamura, and Guo (1995), Nguyen and Hong (2012) and the references therein. However, most of the control designs for the beam equation are collocated control based on the passive principle and do not take the disturbance into account. The earlier non-collocated control design for the beam equation is Luo and Guo (1997). Recently, a powerful backstepping method is introduced to stabilize the Euler–Bernoulli beam equation via completely non-collocated control (Smyshlyaev, Guo, & Krstic, 2009). Once again, the external disturbance is not considered in these works.

There are several different approaches to deal with the uncertainties in system control. The sliding mode control (SMC) that is inherently robust is the most popular one that has been studied widely for both finite-dimensional systems and infinite-dimensional counterparts. For the latter, many works require the input and output operators are to be bounded (Pisano, Orlov, & Usai, 2011). Recently, a boundary SMC controller for a one-dimensional heat equation with boundary input disturbance is designed in Cheng, Radisavljevic, and Su (2011). In Guo, Guo, and Shao (2011), Krstic (2010), the adaptive controls are designed for one-dimensional wave equations in which the uncertainties are the unknown parameters in disturbance. Another powerful method in dealing with uncertainties is based on Lyapunov functional approach. In Ge, Zhang, and He (2001), a boundary control is designed by the Lyapunov method for an Euler–Bernoulli beam equation with spatial and boundary disturbance. Generally speaking, there are not so many works, to the best of our knowledge, to the stabilization of the beam equation with disturbance.

The active disturbance rejection control (ADRC), as an unconventional design strategy, was first proposed by Han in 1990s (Han, 2009). It has been now acknowledged to be an effective control strategy for lumped parameter systems in the absence of proper models and in the presence of model uncertainty. Its power has been demonstrated by many engineering practices such as motion control, tension control in web transport and strip precessing systems, DC–DC power converts in power electronics, continuous stirred tank reactor in chemical and process control, micro-electro-mechanical systems gyroscope (Gao, 2006, Guo and Zhao, 2011, Han, 2009). For more details on practical perspectives, we refer to a nice recent review paper Zheng and Gao (2010). The main idea of the ADRC is using the estimation/cancellation strategy in dealing with the uncertainties. Its convergence has been proved for finite-dimensional systems in Guo and Zhao (2011). Very recently, this approach is successfully applied to the attenuation of disturbance for a one-dimensional anti-stable wave equation in Guo and Jin (2013).

In this paper, we are concerned with the stabilization of a one-dimensional Euler–Bernoulli beam equation with uncertainty at the input boundary via both SMC and ADRC approaches. The system is governed by the following PDEs:

$$\{\begin{array}{c}{u}_{tt}(x,t)+u_{xxxx}(x,t)=0\text{,}x\in (0,1),t>0\text{,}\hfill \\ u(0,t)=u_x(0,t)=0\text{,}t⩾0\text{,}\hfill \\ u_{xx}(1,t)=0\text{,}t⩾0\text{,}\hfill \\ u_{xxx}(1,t)=U\left(t\right)+d\left(t\right)\text{,}t⩾0\text{,}\hfill \end{array}$$

where $u(x,t)$ is the transverse displacement of the beam at time $t$ and position $x,U$ is the control input through shear force, $d$ is the external disturbance at the control end. This one end fixed and another end free beam equation (1) models typically the vibration control of a single link flexible robot arm with the external disturbance in the free (working) end (see e.g., Han et al., 1999, Luo & Guo, 1997).

It is well-known that when there is no disturbance, the collocated feedback control $U\left(t\right)=k{u}_t(1,t),k>0$ will stabilize exponentially the system (1) (Chen et al., 1987). However, this stabilizer is not robust to the external disturbance. For instance, when $d\left(t\right)=d$ is a constant, the system (1) under the feedback $U\left(t\right)=k{u}_t(1,t)$ has a solution $(u,u_t)=(-\frac{d}{2}{x}^2+\frac{d}{6}{x}^3,0)$. Therefore, in the presence of the disturbance, the control must be re-designed.

We proceed as follows. In Section  2, we use the ADRC approach to attenuate the disturbance by designing an estimator to estimate the disturbance. After canceling the disturbance by the approximated one, we design the collocated like feedback controller. The closed-loop system is shown to tend any arbitrary given vicinity of zero as the time goes to infinity. Section  3 is devoted to the disturbance rejection by the SMC approach, in which the boundedness of the disturbance required in ADRC is removed. The existence and uniqueness of the solution are proved, and the monotonicity of the “reaching condition” is presented without the differentiation of the sliding mode function, for which it does not always exist for the weak solution of the closed-loop system. The numerical simulations are presented in Section  4 for illustration of the effectiveness of both methods.