Journal of Engineering

Volume 2014 (2014), Article ID 839128, 9 pages

http://dx.doi.org/10.1155/2014/839128

## Adaptive Vibration Control of Piezoactuated Euler-Bernoulli Beams Using Infinite-Dimensional Lyapunov Method and High-Order Sliding-Mode Differentiation

^{1}Department of Mechanical and Aerospace Engineering, King Mongkut’s University of Technology North Bangkok, 1518 Pracharat Sai 1, Bangkok 10800, Thailand^{2}Institute of General Mechanics, RWTH Aachen University, Templergraben 64, 52056 Aachen, Germany

Received 31 July 2014; Revised 26 November 2014; Accepted 8 December 2014; Published 22 December 2014

Academic Editor: Jyoti Sinha

Copyright © 2014 Teerawat Sangpet et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper presents an adaptive control scheme to suppress vibration of flexible beams using a collocated piezoelectric actuator-sensor configuration. A governing equation of the beams is modelled by a partial differential equation based on Euler-Bernoulli theory. Thus, the beams are infinite-dimensional systems. Whereas conventional control design techniques for infinite-dimensional systems make use of approximated finite-dimensional models, the present adaptive control law is derived based on the infinite-dimensional Lyapunov method, without using any approximated finite-dimension model. Thus, the stability of the control system is guaranteed for all vibration modes. The implementation of the control law requires a derivative of the sensor output for feedback. A high-order sliding mode differentiation technique is used to estimate the derivative. The technique features robust exact differentiation with finite-time convergence. Numerical simulation and experimental results illustrate the effectiveness of the controller.

#### 1. Introduction

Flexible structures have attracted interest because of their lighter weight compared to traditional structures. They have been widely used in aerospace applications and robotics [1–3]. However, the flexibility leads to vibration problems. Therefore, vibration control is needed. Over the past few decades, active vibration control has drawn more interests from researchers since it can effectively suppress the vibration [4–7].

Piezoelectric actuators and sensors provide an effective means of vibration suppression of flexible structures [8]. The advantages of using piezoelectric actuators and piezoelectric sensors include nanometer scale resolution, high stiffness, and fast response. Many researchers have studied the vibration suppression of flexible structures using piezoelectric actuators and piezoelectric sensors. In [9], Tavakolpour et al. proposed a self-learning vibration control strategy for flexible plate structures. A control algorithm is based on a P-type iterative learning with displacement feedback. Wang et al. [10] presented a simple control law for reducing the vibration of the flexible structure. Linear feedback control was derived using a linear matrix inequality method. Qiu et al. [11] proposed a neural network controller based on PD control with collocated piezoelectric actuator and sensor. The back-propagation algorithm was utilized for adapting the controller parameters. In [12], Sangpet et al. utilized a fractional-order control approach to improve the delay margin in the control system of a piezoactuated flexible beam. The controller parameters were tuned experimentally. An et al. [13] presented a time-delay acceleration feedback controller for vibration suppression of cantilever beams. Stability boundaries of the closed-loop system were determined by a Hurwitz stability criterion. In [14], Takács et al. presented an adaptive-predictive vibration control system using extended Kalman filter. The viability of the control method was experimentally evaluated. In [15], Sangpet et al. proposed an observer-based hysteresis compensation scheme for tracking control of a piezoactuated flexible beam. The observer based on a PI observer and a Kalman-filter algorithm was employed for estimation of the hysteresis in the beam. The scheme was successfully implemented in experimental tests.

The governing equation of vibrating flexible structures is a partial differential equation (PDE). Thus, the systems are distributed-parameter systems and have an infinite number of vibration modes. Most of control design techniques exploit an approximated finite-dimensional model by ignoring the higher frequency modes. However, this approach can cause the control system to become unstable due to a spillover effect [16, 17].

Infinite-dimensional control for the systems modeled by PDEs has been recently investigated. Extensions of finite-dimensional techniques to infinite-dimensional systems as well as innovative infinite-dimensional specific control design approaches have been proposed; see [18] and references therein. The Lyapunov-based control method has played a major role in nonlinear control of finite-dimensional complex systems [19–22]. Infinite-dimensional Lyapunov-based control is its extension to infinite-dimensional systems, which has been successfully applied to many engineering problems [23–28].

In [23], Dadfarnia et al. proposed infinite-dimensional Lyapunov-based control for a flexible Cartesian robot with a piezoelectric patch actuator attachment. The robot was modeled as a flexible cantilever beam with a translational base support and a tip mass. The controller was designed to exponentially suppress the vibration of the beam and regulate the base motion. Coron et al. [24] presented a strict Lyapunov function for hyperbolic systems of conservation laws. The time derivative of this Lyapunov function can be made strictly negative definite by an appropriate choice of the boundary conditions. The method was illustrated by a hydraulic application. Cheng et al. [25] proposed sliding-mode boundary control of a one-dimensional unstable heat conduction system modeled by parabolic PDE systems with parameter variations and boundary uncertainties. An infinite-dimensional sliding manifold was constructed via Lyapunov’s direct method. Simulation studies were conducted to verify the effectiveness of the sliding mode control law. In [26], Shang and Xu proposed a new control strategy for a one-dimensional cantilever Euler-Bernoulli beam with an input delay. The control system is either exponentially or asymptotically stable, depending on the time delay. In [27], Guo and Jin presented an active disturbance rejection and sliding mode control approach for a stabilization problem of Euler-Bernoulli beams with boundary input disturbance. The controller design is based on a PDE model. In [28], Luemchamloey and Kuntanapreeda presented an experimental study of active vibration control of flexible beams. The flexible beam was modeled by PDE. The controller was designed based on the infinite-dimensional Lyapunov method. He et al. [29] considered vibration control of a flexible string system. With Lyapunov’s direct method, adaptive boundary control was developed to suppress the string’s vibration and the adaptive law was designed to compensate for the system parametric uncertainties. Numerical simulations were carried out to verify the effectiveness of the controller.

In addition, high-order sliding-mode differentiation [30, 31] has been attracting interest. It is an alternative differentiation that features robust exact differentiation with finite-time convergence. It has been successfully used in control laws, replacing the use of derivative measurements [32, 33].

This paper presents an adaptive feedback control law to suppress vibration of Euler-Bernoulli beams with a collocated piezoelectric actuator/sensor pair. The actuator and sensor are lead zirconate titanate (PZT). The filtered signal from the sensor is used to provide the feedback signal. The governing equation of the beams is a PDE. Thus, it has an infinite number of vibration modes. The controller is designed based on an infinite-dimensional Lyapunov method, which does not involve any approximated finite-dimensional models. Thus, the stability of the control system including the filter dynamic is guaranteed for all vibration modes. The high-order differentiation is used to estimate the derivative of the sensor signal and then used in the control law. The rest of the paper is organized as follows. Section 2 provides some preliminaries. Section 3 describes the flexible beam that is used as an experimental test bench. Section 4 presents a finite element model of the experimental beam. The proposed controller design is given in Section 5. Simulation and experimental results are presented in Section 6. The last section concludes the paper.

#### 2. Preliminaries

##### 2.1. System Modeling

Consider a flexible cantilever beam with piezoelectric actuator and sensor patches. The patches are bonded on the top and bottom surfaces of the beam as shown in Figure 1. A governing equation of the beam can be derived based on an Euler-Bernoulli beam equation and Hamilton’s principle [19] as with the boundary conditions where the dot and prime notations represent derivatives with respect to time and the variable , respectively, is the transverse displacement of the beam, is the input voltage applied to the actuator, is the length of the beam, , , are the densities of the beam and the piezoelectric patch actuator, respectively, , are the widths of the beam and the actuator, , are the thicknesses of the beam and the actuator, , is the Heaviside function, are the distances from the clamped end to the leading edge and the tailing edge of the actuator, respectively, is the moment constant of the actuator, is the bending stiffness, , , are the Young’s moduli of the beam and the actuator, and , are the viscous and structural damping coefficients, respectively.