Research Article  Open Access
Teerawat Sangpet, Suwat Kuntanapreeda, Rüdiger Schmidt, "Adaptive Vibration Control of Piezoactuated EulerBernoulli Beams Using InfiniteDimensional Lyapunov Method and HighOrder SlidingMode Differentiation", Journal of Engineering, vol. 2014, Article ID 839128, 9 pages, 2014. https://doi.org/10.1155/2014/839128
Adaptive Vibration Control of Piezoactuated EulerBernoulli Beams Using InfiniteDimensional Lyapunov Method and HighOrder SlidingMode Differentiation
Abstract
This paper presents an adaptive control scheme to suppress vibration of flexible beams using a collocated piezoelectric actuatorsensor configuration. A governing equation of the beams is modelled by a partial differential equation based on EulerBernoulli theory. Thus, the beams are infinitedimensional systems. Whereas conventional control design techniques for infinitedimensional systems make use of approximated finitedimensional models, the present adaptive control law is derived based on the infinitedimensional Lyapunov method, without using any approximated finitedimension 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 highorder sliding mode differentiation technique is used to estimate the derivative. The technique features robust exact differentiation with finitetime 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 selflearning vibration control strategy for flexible plate structures. A control algorithm is based on a Ptype 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 backpropagation algorithm was utilized for adapting the controller parameters. In [12], Sangpet et al. utilized a fractionalorder 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 timedelay acceleration feedback controller for vibration suppression of cantilever beams. Stability boundaries of the closedloop system were determined by a Hurwitz stability criterion. In [14], Takács et al. presented an adaptivepredictive vibration control system using extended Kalman filter. The viability of the control method was experimentally evaluated. In [15], Sangpet et al. proposed an observerbased hysteresis compensation scheme for tracking control of a piezoactuated flexible beam. The observer based on a PI observer and a Kalmanfilter 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 distributedparameter systems and have an infinite number of vibration modes. Most of control design techniques exploit an approximated finitedimensional 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].
Infinitedimensional control for the systems modeled by PDEs has been recently investigated. Extensions of finitedimensional techniques to infinitedimensional systems as well as innovative infinitedimensional specific control design approaches have been proposed; see [18] and references therein. The Lyapunovbased control method has played a major role in nonlinear control of finitedimensional complex systems [19–22]. Infinitedimensional Lyapunovbased control is its extension to infinitedimensional systems, which has been successfully applied to many engineering problems [23–28].
In [23], Dadfarnia et al. proposed infinitedimensional Lyapunovbased 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 slidingmode boundary control of a onedimensional unstable heat conduction system modeled by parabolic PDE systems with parameter variations and boundary uncertainties. An infinitedimensional 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 onedimensional cantilever EulerBernoulli 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 EulerBernoulli 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 infinitedimensional 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, highorder slidingmode differentiation [30, 31] has been attracting interest. It is an alternative differentiation that features robust exact differentiation with finitetime 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 EulerBernoulli 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 infinitedimensional Lyapunov method, which does not involve any approximated finitedimensional models. Thus, the stability of the control system including the filter dynamic is guaranteed for all vibration modes. The highorder 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 EulerBernoulli 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.
The equation relating the applied voltage and the axial strain produced by the piezoelectric patch actuator is written as [8, 11]: where and are the dielectric constant and the thickness of the actuator, respectively. The normal stress and the bending moment on the beam produced by the actuator are where is Young’s modulus of the actuator, is the width of the actuator, , and , are the thicknesses of the beam and the actuator.
By considering the piezoelectric patch sensor as a parallel plate capacitor, the voltage across the two layers produced by the sensor is given as [23]: where is the piezoelectric constant, is the width of the sensor, is Young’s modulus of the sensor, is the dielectric constant of the sensor, is the thickness of the sensor, and is the piezoelectric capacitance.
2.2. Finite Element Equation
Consider a finite element (FE) model of a cantilever beam shown in Figure 2. It is assumed that the beam is divided into elements. Each element has two nodes and each node has two degrees of freedom: transverse direction and slope .
The transverse displacement at any position on the element can be expressed as [8, 30]: where is the nodal displacement vector and is the modeshape vector function. By using the Lagrangian method, the differential equation of motion to be solved can be expressed as [34, 35]: where is the mass matrix, is the stiffness matrix, is the damping matrix, and is the force matrix. For Rayleigh proportional damping, , where are the damping constants. The force matrix for the element without the piezoelectric actuator is and is, for the element bonded with the actuator, patch , which is directly derived from (5). Note that the piezoelectric sensor is not included in the FE model since the bonded sensor in the experimental system is very thin and light compared to those of the beam and the actuator.
2.3. HighOrder SlidingMode Differentiation
In this subsection, a highorder slidingmode differentiation method is shortly described. To explain the method, let be a function from which we want to determine its derivative. Consider the following auxiliary system: The main idea is to find to keep , resulting in . Based on a secondorder slidingmode algorithm, one obtains [30] where and is a signum function. A general form of thorder differentiators can be written as where is an approximated value of the th derivative of the function , is a Lipschitz constant of the function , and are constant parameters. The reader is referred to Levant [30, 31] for full details of the method. In this paper, this method will be used for estimating the first derivative of the signal from the piezoelectric sensor.
3. Experimental System
A schematic and a photograph of the experimental system used in this paper are shown in Figure 3. The system consists of a flexible cantilever beam, a piezoelectric patch actuator, a piezoelectric patch sensor, a highvoltage power amplifier, a 16bit ADC/DAC interface board, and a PC. The beam is made of aluminum. The dimensions and properties of the beam are summarized in Table 1. The piezoelectric actuator and sensor are lead zirconate titanate (PZT). They are bonded on the beam at the distance mm from the clamped point. The dimensions and properties of the actuator and sensor are summarized in Table 2. The sensor measures the axial strain of the beam. The system is also equipped with a laser displacement sensor for measuring the beam’s tip deflection; however, the displacement sensor is not used in the control loop. The signals from the sensors are fed back to the PC through the interface board. The control signal from the computer is converted to an analogue signal by the interface board and is transmitted through the highvoltage power amplifier to the actuator to close the control loop. The gain of the amplifier is 150. Figure 4 displays the pulse response and the frequency response functions of the beam where the signal from the piezoelectric sensor is considered as the output. The frequency response shows that the first two natural frequencies of the beam are 39.28 rad/s and 224.7 rad/s.


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4. Finite Element Model
A finite element (FE) model of the experimental beam was developed to be used in the control design process. For simplicity, was chosen. The pulse response and the frequency response functions obtained from the developed FE model are shown in Figure 5. The responses agree with those of the experimental system.
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5. Controller Design
The proposed control design is given in this section. The control objective is to suppress the vibration of the beam by using only the signal from the piezoelectric sensor. The schematic for the control loop is shown in Figure 6.
Instead of directly using and for feedback, let and pass through a lowpass filter to obtain a new feedback signal as follows: where is the time constant of the filter and are weighting parameters. Note that including a lowpass filter in the control loop is usually done in practice to attenuate the effect of noise in the feedback signal. In the following, the dynamic of the filter is also included in the controller design.
Define a Lyapunov function candidate as where is considered as a controller parameter. By taking the time derivative of (13) and substituting (1), (2), (6), and (12) into the derivative result, it yields By choosing the following control law: it yields By Barbalat’s lemma [36, 37], is achieved. Note that can be obtained directly from the piezoelectric sensor (6); that is, , and can be obtained from . In our experiment, we used the highorder slidingmode differentiation method described in Section 2.3 to determine .
Note that control law (15) has a singularity at . Thus, implementations of the control law must empirically avoid this singularity. For simplicity, we chose and in the experiment to avoid the singularity.
Moreover, the parameter is usually uncertain or even unknown. Thus, control law (15) with and is modified as where is an adaptive variable replacing . Define . Take as a new Lyapunov function candidate. Taking its derivative with respect to time and substituting (17) into the result yields By choosing the following adaptive law: it results in Applying Barbalat’s lemma [36, 37] results in .
6. Results
6.1. Simulation Study
A simulation study was conducted using the FE model described in Section 2.2 to tune the control parameter . It was observed that the convergence rate of the controlled response increases with . Here, the control gain of was chosen. Simulation results are shown in Figure 7. The controller was activated at sec. The results show that the controller is able to suppress the vibration effectively. Note that converged to a finite value.
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6.2. Experimental Results
Controller (17) was implemented digitally on the PC with the sampling period of the control loop of 1 msec. The highorder sliding differentiation technique with , , was used. The experimental results are shown in Figure 8. The controller provides satisfactory responses similar to those obtained by the simulation. Note that the controller was activated at sec.
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Note that we set the initial conditions of the FEM simulation and experiment roughly the same for a comparison purpose. It was found that there are slight differences in the FEM simulation and experimental results. This is mainly due to the measurement noise and the mismatch in the initial conditions. The difference between the maximum control inputs is about 12.08%. The difference of the steady state adaptive gains is about 3.35%. In summary, both results are practically comparable.
7. Conclusions
In this paper we proposed an adaptive controller for vibration suppression of flexible structures. The structure under consideration is a cantilever beam with a collocated piezoelectric actuator and sensor pair. The governing equation of the beam is a partial differential equation. Since the beam is an infinitedimensional system, we derived the control law using the infinitedimensional Lyapunov method, which does not require any approximated finitedimension model. Thus, the stability of the control system is guaranteed for all vibration modes. We also employed a highorder sliding mode differentiation technique to obtain the derivative of the sensor’s signal for being used in the control law. The technique features robust exact differentiation with finitetime convergence. Effectiveness of the controller is illustrated both in finiteelement model and experiment, with results that show good agreement.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the Royal Golden Jubilee Ph.D. program of the Thailand Research Fund. The authors also would like to thank anonymous referees for their valuable comments.
References
 Z. Mohamed, J. M. Martins, M. O. Tokhi, J. Sá da Costa, and M. A. Botto, “Vibration control of a very flexible manipulator system,” Control Engineering Practice, vol. 13, no. 3, pp. 267–277, 2005. View at: Publisher Site  Google Scholar
 M. Azadi, S. A. Fazelzadeh, M. Eghtesad, and E. Azadi, “Vibration suppression and adaptiverobust control of a smart flexible satellite with three axes maneuvering,” Acta Astronautica, vol. 69, no. 56, pp. 307–322, 2011. View at: Publisher Site  Google Scholar
 T. P. Sales, D. A. Rade, and L. C. G. De Souza, “Passive vibration control of flexible spacecraft using shunted piezoelectric transducers,” Aerospace Science and Technology, vol. 29, no. 1, pp. 403–412, 2013. View at: Publisher Site  Google Scholar
 C. C. Fuller, Active Control of Vibration, Academic Press, 1996.
 A. Preumont, Vibration Control of Active Structures, Kluwer Academic, 1997. View at: Publisher Site  MathSciNet
 A. M. Aly, “Vibration control of buildings using magnetorheological damper: a new control algorithm,” Journal of Engineering, vol. 2013, Article ID 596078, 10 pages, 2013. View at: Publisher Site  Google Scholar
 A. Moutsopoulou, G. E. Stavroulakis, and A. Pouliezos, “Innovation in active vibration control strategy of intelligent structures,” Journal of Applied Mathematics, vol. 2014, Article ID 430731, 14 pages, 2014. View at: Publisher Site  Google Scholar
 B. Bandyopadhyay, T. C. Manjunath, and M. Umapathy, Modeling, Control and Implementation of Smart Structures: A FEMState Space Approach, Springer, Berlin, Germany, 2007. View at: MathSciNet
 A. R. Tavakolpour, M. Mailah, I. Z. Mat Darus, and O. Tokhi, “Selflearning active vibration control of a flexible plate structure with piezoelectric actuator,” Simulation Modelling Practice and Theory, vol. 18, no. 5, pp. 516–532, 2010. View at: Publisher Site  Google Scholar
 H. Wang, X.H. Shen, L.X. Zhang, and X.J. Zhu, “Active vibration suppression of flexible structure using a LMIbased control patch,” Journal of Vibration and Control, vol. 18, no. 9, pp. 1375–1379, 2012. View at: Publisher Site  Google Scholar  MathSciNet
 Z. Qiu, X. Zhang, and C. Ye, “Vibration suppression of a flexible piezoelectric beam using BP neural network controller,” Acta Mechanica Solida Sinica, vol. 25, no. 4, pp. 417–428, 2012. View at: Publisher Site  Google Scholar
 T. Sangpet, S. Kuntanapreeda, and R. Schmidt, “Improving delaymargin of noncollocated vibration control of piezoactuated flexible beams via a fractionalorder controller,” Shock and Vibration, vol. 2014, Article ID 809173, 8 pages, 2014. View at: Publisher Site  Google Scholar
 F. An, W. Chen, and M. Shao, “Dynamic behavior of timedelayed acceleration feedback controller for active vibration control of flexible structures,” Journal of Sound and Vibration, vol. 333, no. 20, pp. 4789–4809, 2014. View at: Google Scholar
 G. Takács, T. Polóni, and B. Rohal'Ilkiv, “Adaptive model predictive vibration control of a cantilever beam with realtime parameter estimation,” Shock and Vibration, vol. 2014, Article ID 741765, 15 pages, 2014. View at: Publisher Site  Google Scholar
 T. Sangpet, S. Kuntanapreeda, and R. Schmidt, “Hysteretic nonlinearity observer design based on Kalman filter for piezoactuated flexible beams,” International Journal of Automation and Computing, vol. 11, no. 6, pp. 627–634, 2014. View at: Google Scholar
 X. Dong, Z. Peng, W. Zhang, H. Hua, and G. Meng, “Research on spillover effects for vibration control of piezoelectric smart structures by ANSYS,” Mathematical Problems in Engineering, vol. 2014, Article ID 870940, 8 pages, 2014. View at: Publisher Site  Google Scholar
 A. Montazeri, J. Poshtan, and A. YousefiKoma, “Design and analysis of robust minimax LQG controller for an experimental beam considering spillover effect,” IEEE Transactions on Control Systems Technology, vol. 19, no. 5, pp. 1251–1259, 2011. View at: Publisher Site  Google Scholar
 R. Padhi and S. F. Ali, “An account of chronological developments in control of distributed parameter systems,” Annual Reviews in Control, vol. 33, no. 1, pp. 59–68, 2009. View at: Publisher Site  Google Scholar
 S. Zhao, H. Lin, and Z. Xue, “Switching control of closed quantum systems via the Lyapunov method,” Automatica, vol. 48, no. 8, pp. 1833–1838, 2012. View at: Publisher Site  Google Scholar  MathSciNet
 S. Kuntanapreeda and P. M. Marusak, “Nonlinear extended output feedback control for CSTRs with van de Vusse reaction,” Computers & Chemical Engineering, vol. 41, pp. 10–23, 2012. View at: Publisher Site  Google Scholar
 B. Zhang, K. Liu, and J. Xiang, “A stabilized optimal nonlinear feedback control for satellite attitude tracking,” Aerospace Science and Technology, vol. 27, no. 1, pp. 17–24, 2013. View at: Publisher Site  Google Scholar
 A. Yang, W. Naeem, G. W. Irwin, and K. Li, “Stability analysis and implementation of a decentralized formation control strategy for unmanned vehicles,” IEEE Transactions on Control Systems Technology, vol. 22, no. 2, pp. 706–720, 2014. View at: Publisher Site  Google Scholar
 M. Dadfarnia, N. Jalili, B. Xian, and D. M. Dawson, “A lyapunovbased piezoelectric controller for flexible cartesian robot manipulators,” Journal of Dynamic Systems, Measurement and Control, vol. 126, no. 2, pp. 347–358, 2004. View at: Publisher Site  Google Scholar
 J.M. Coron, B. d'AndréaNovel, and G. Bastin, “A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws,” IEEE Transactions on Automatic Control, vol. 52, no. 1, pp. 2–11, 2007. View at: Publisher Site  Google Scholar  MathSciNet
 M.B. Cheng, V. Radisavljevic, and W.C. Su, “Sliding mode boundary control of a parabolic PDE system with parameter variations and boundary uncertainties,” Automatica, vol. 47, no. 2, pp. 381–387, 2011. View at: Publisher Site  Google Scholar  MathSciNet
 Y. F. Shang and G. Q. Xu, “Stabilization of an EulerBernoulli beam with input delay in the boundary control,” Systems and Control Letters, vol. 61, no. 11, pp. 1069–1078, 2012. View at: Publisher Site  Google Scholar
 B.Z. Guo and F.F. Jin, “The active disturbance rejection and sliding mode control approach to the stabilization of the EulerBernoulli beam equation with boundary input disturbance,” Automatica, vol. 49, no. 9, pp. 2911–2918, 2013. View at: Publisher Site  Google Scholar  MathSciNet
 A. Luemchamloey and S. Kuntanapreeda, “Active vibration control of flexible beam based on infinitedimensional lyapunov stability theory: an experimental study,” Journal of Control, Automation and Electrical Systems, vol. 25, no. 6, pp. 649–656, 2014. View at: Publisher Site  Google Scholar
 W. He, S. Zhang, and S. S. Ge, “Adaptive boundary control of a nonlinear flexible string system,” IEEE Transactions on Control Systems Technology, vol. 22, no. 3, pp. 1088–1093, 2014. View at: Publisher Site  Google Scholar
 A. Levant, “Robust exact differentiation via sliding mode technique,” Automatica, vol. 34, no. 3, pp. 379–384, 1998. View at: Publisher Site  Google Scholar  MathSciNet
 A. Levant, “Higherorder sliding modes, differentiation and outputfeedback control,” International Journal of Control, vol. 76, no. 910, pp. 924–941, 2003. View at: Publisher Site  Google Scholar  MathSciNet
 X. Wang and B. Shirinzadeh, “Highorder nonlinear differentiator and application to aircraft control,” Mechanical Systems and Signal Processing, vol. 46, no. 2, pp. 227–252, 2014. View at: Publisher Site  Google Scholar
 I. Salgado, I. Chairez, O. Camacho, and C. Yañez, “Supertwisting sliding mode differentiation for improving PD controllers performance of second order system,” ISA Transactions, vol. 53, no. 4, pp. 1096–1106, 2014. View at: Publisher Site  Google Scholar
 S. Y. Wang, “A finite element model for the static and dynamic analysis of a piezoelectric bimorph,” International Journal of Solids and Structures, vol. 41, no. 15, pp. 4075–4096, 2004. View at: Publisher Site  Google Scholar
 D. L. Logan, A First Course in the Finite Element Method, Wadsworth Group, 2002.
 J. E. Slotine, Applied Nonlinear Control, PrenticeHall, 1991.
 H. K. Khalil, Nonlinear Systems, PrenticeHall, 1996.
Copyright
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.