Adaptive Vibration Control of Piezoactuated Euler-Bernoulli Beams Using Infinite-Dimensional Lyapunov Method and High-Order Sliding-Mode Differentiation
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
References
[1]
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.
[2]
M. Azadi, S. A. Fazelzadeh, M. Eghtesad, and E. Azadi, “Vibration suppression and adaptive-robust control of a smart flexible satellite with three axes maneuvering,” Acta Astronautica, vol. 69, no. 5-6, pp. 307–322, 2011.
[3]
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.
[4]
C. C. Fuller, Active Control of Vibration, Academic Press, 1996.
[5]
A. Preumont, Vibration Control of Active Structures, Kluwer Academic, 1997.
[6]
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.
[7]
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.
[8]
B. Bandyopadhyay, T. C. Manjunath, and M. Umapathy, Modeling, Control and Implementation of Smart Structures: A FEM-State Space Approach, Springer, Berlin, Germany, 2007.
[9]
A. R. Tavakolpour, M. Mailah, I. Z. Mat Darus, and O. Tokhi, “Self-learning active vibration control of a flexible plate structure with piezoelectric actuator,” Simulation Modelling Practice and Theory, vol. 18, no. 5, pp. 516–532, 2010.
[10]
H. Wang, X.-H. Shen, L.-X. Zhang, and X.-J. Zhu, “Active vibration suppression of flexible structure using a LMI-based control patch,” Journal of Vibration and Control, vol. 18, no. 9, pp. 1375–1379, 2012.
[11]
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.
[12]
T. Sangpet, S. Kuntanapreeda, and R. Schmidt, “Improving delay-margin of noncollocated vibration control of piezo-actuated flexible beams via a fractional-order controller,” Shock and Vibration, vol. 2014, Article ID 809173, 8 pages, 2014.
[13]
F. An, W. Chen, and M. Shao, “Dynamic behavior of time-delayed acceleration feedback controller for active vibration control of flexible structures,” Journal of Sound and Vibration, vol. 333, no. 20, pp. 4789–4809, 2014.
[14]
G. Takács, T. Polóni, and B. Rohal'-Ilkiv, “Adaptive model predictive vibration control of a cantilever beam with real-time parameter estimation,” Shock and Vibration, vol. 2014, Article ID 741765, 15 pages, 2014.
[15]
T. Sangpet, S. Kuntanapreeda, and R. Schmidt, “Hysteretic nonlinearity observer design based on Kalman filter for piezo-actuated flexible beams,” International Journal of Automation and Computing, vol. 11, no. 6, pp. 627–634, 2014.
[16]
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.
[17]
A. Montazeri, J. Poshtan, and A. Yousefi-Koma, “Design and analysis of robust minimax LQG controller for an experimental beam considering spill-over effect,” IEEE Transactions on Control Systems Technology, vol. 19, no. 5, pp. 1251–1259, 2011.
[18]
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.
[19]
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.
[20]
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.
[21]
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.
[22]
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.
[23]
M. Dadfarnia, N. Jalili, B. Xian, and D. M. Dawson, “A lyapunov-based piezoelectric controller for flexible cartesian robot manipulators,” Journal of Dynamic Systems, Measurement and Control, vol. 126, no. 2, pp. 347–358, 2004.
[24]
J.-M. Coron, B. d'Andréa-Novel, 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.
[25]
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.
[26]
Y. F. Shang and G. Q. Xu, “Stabilization of an Euler-Bernoulli beam with input delay in the boundary control,” Systems and Control Letters, vol. 61, no. 11, pp. 1069–1078, 2012.
[27]
B.-Z. Guo and F.-F. Jin, “The active disturbance rejection and sliding mode control approach to the stabilization of the Euler-Bernoulli beam equation with boundary input disturbance,” Automatica, vol. 49, no. 9, pp. 2911–2918, 2013.
[28]
A. Luemchamloey and S. Kuntanapreeda, “Active vibration control of flexible beam based on infinite-dimensional lyapunov stability theory: an experimental study,” Journal of Control, Automation and Electrical Systems, vol. 25, no. 6, pp. 649–656, 2014.
[29]
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.
[30]
A. Levant, “Robust exact differentiation via sliding mode technique,” Automatica, vol. 34, no. 3, pp. 379–384, 1998.
[31]
A. Levant, “Higher-order sliding modes, differentiation and output-feedback control,” International Journal of Control, vol. 76, no. 9-10, pp. 924–941, 2003.
[32]
X. Wang and B. Shirinzadeh, “High-order nonlinear differentiator and application to aircraft control,” Mechanical Systems and Signal Processing, vol. 46, no. 2, pp. 227–252, 2014.
[33]
I. Salgado, I. Chairez, O. Camacho, and C. Ya？ez, “Super-twisting sliding mode differentiation for improving PD controllers performance of second order system,” ISA Transactions, vol. 53, no. 4, pp. 1096–1106, 2014.
[34]
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.
[35]
D. L. Logan, A First Course in the Finite Element Method, Wadsworth Group, 2002.
[36]
J. E. Slotine, Applied Nonlinear Control, Prentice-Hall, 1991.
[37]
H. K. Khalil, Nonlinear Systems, Prentice-Hall, 1996.
