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September  2017, 7(3): 251-271. doi: 10.3934/naco.2017017

## Adaptive Neuro-Fuzzy vibration control of a smart plate

 School of Production Engineering and Management, Technical University of Crete, GR-73100, Chania, Greece

* Corresponding author: gestavr@dpem.tuc.gr

Received  November 2016 Revised  June 2017 Published  July 2017

In the present paper, the vibration supression of a smart plate with the use of ANFIS (Adaptive Neuro-Fuzzy Inference System) is investigated. The whole system consists of a nonlinear mechanical model, which is an extension of the von Kármán plate model with control. The structure is subjected to external disturbances and generalized control forces. Initial and boundary conditions are set up. The initial boundary value problem is spatially-discretized by a time spectral method. The obtained discretized model is a system of nonlinear ordinary differential equations (ODEs) with respect to time. A neuro-fuzzy inference system is built and tested in order to create a nonlinear controller for the vibration supression of the plate. More specifically, a Sugeno-type fuzzy inference system is employed and trained through ANFIS. The inputs of the controller are the displacement and the velocity and the output is the control force. An effective optimization procedure is proposed and numerical results are presented.

Citation: Aliki D. Muradova, Georgios K. Tairidis, Georgios E. Stavroulakis. Adaptive Neuro-Fuzzy vibration control of a smart plate. Numerical Algebra, Control and Optimization, 2017, 7 (3) : 251-271. doi: 10.3934/naco.2017017
##### References:
 [1] Ph. G. Ciarlet, Mathematical Elasticity, Ⅴ. Ⅱ: Theory of Plates, Elsevier, Amsterdam, 1997. [2] P. Ciarlet and P. Rabier, Les Equations de von Kármán, Springer-Verlag, Berlin, Heidelberg, New York, 1980. [3] Ph. Destuynder and M. Salaun, Mathematical Analysis of Thin Plate Models, Mathématiques & Applications (Berlin) [Mathematics & Applications], Springer, 1996. doi: 10.1007/978-3-642-51761-7. [4] D. Driankov, H. Hellendoorn and M. Reinfrank, An Introduction to Fuzzy Control, 2nd edition, Springer-Verlag, Berlin, Heidelberg, New York, 1996. [5] G. Duvaut and J. L. Lions, Les Inequations en Mecaniques et en Physiques, Dunod, 1972. [6] N. R. Fisco and H. Adeli, Smart structures: Part Ⅱ: Hybrid control systems and control strategies, Scientia Iranica, 18 (2011), 285-295. [7] A. Isidori, Nonlinear Control Systems, 3rd edition, Springer Verlag, London, 1995. doi: 10.1007/978-1-84628-615-5. [8] S. Korkmaz, A review of active structural control: challenges for engineering informatics, Comput. and Struct., 89 (2011), 2113-2132. [9] P. Koutsianitis, G. K. Tairidis, G. A. Drosopoulos, G. A. Foutsitzi and G. E. Stavroulakis, Effectiveness of optimized fuzzy controllers on partially delaminated piezocomposites, Acta Mechanica, 228 (2017), 1373-1392.  doi: 10.1007/s00707-016-1771-6. [10] A. D. Muradova, A time spectral method for solving the nonlinear dynamic equations of a rectangular elastic plate, J. Eng. Math., 92 (2015), 83-101.  doi: 10.1007/s10665-014-9752-z. [11] A. D. Muradova and G. E. Stavroulakis, Fuzzy vibration control of a smart plate, Int. J. Comput. Meth. Eng. Sci. Mech., 14 (2013), 212-220.  doi: 10.1080/15502287.2012.711427. [12] A. D. Muradova and G. E. Stavroulakis, Hybrid control of vibrations of smart von Kármán, Acta Mechanica, 226 (2015), 3463-3475.  doi: 10.1007/s00707-015-1387-2. [13] R. E. Precup and H. Hellendoorn, A survey on industrial applications of fuzzy control, Computers in Industry, 62 (2011), 213-226. [14] A. Preumont, Vibration Control of Active Structures, Springer, 2002. doi: 10.1007/978-94-007-2033-6. [15] J. N. Reddy, Theory and Analysis of Elastic Plates and Shells, CRC Press, Taylor & Francis, 2007. [16] A. H. N. Shirazi, H. R. Owji and M. Rafeeyan, Active vibration control of an FGM rectangular plate using fuzzy logic controllers, Procedia Engineering, 14 (2011), 3019-3026. [17] G. K. Tairidis, G. E. Stavroulakis, D. G. Marinova and E. C. Zacharenakis, Classical and soft robust active control of smart beams, Computat. Struct. Dynamics and Earthquake Engineer. (eds. Papadrakis, M. , Charmpis, D. C. Lagaros and N. D. , Tsompanakis), CRC Press/Balkema and Taylor & Francis Group, London, UK. , Ch. 11 (2009), 165–178. [18] A. R. Tavakolpour, M. Mailah, I. Z. M. Darus and O. Tokhi, Self-learning active vibration control of a flexible plate structure with piezoelectric actuator, Simul. Model. Prac. and Theory, 18 (2010), 516-532. [19] Q. Wenzhonga, S. Jincaib and Q. Yangc, Active control of vibration using a fuzzy control method, J. of Sound and Vibration, 275 (2004), 917-930.  doi: 10.1016/S0022-460X(03)00795-8. [20] I. J. Zeinoun and F. Khorrami, An adaptive control scheme based on fuzzy logic and its application to smart structures, Smart Mater. Struct., 3 (1994), 266-276.

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##### References:
 [1] Ph. G. Ciarlet, Mathematical Elasticity, Ⅴ. Ⅱ: Theory of Plates, Elsevier, Amsterdam, 1997. [2] P. Ciarlet and P. Rabier, Les Equations de von Kármán, Springer-Verlag, Berlin, Heidelberg, New York, 1980. [3] Ph. Destuynder and M. Salaun, Mathematical Analysis of Thin Plate Models, Mathématiques & Applications (Berlin) [Mathematics & Applications], Springer, 1996. doi: 10.1007/978-3-642-51761-7. [4] D. Driankov, H. Hellendoorn and M. Reinfrank, An Introduction to Fuzzy Control, 2nd edition, Springer-Verlag, Berlin, Heidelberg, New York, 1996. [5] G. Duvaut and J. L. Lions, Les Inequations en Mecaniques et en Physiques, Dunod, 1972. [6] N. R. Fisco and H. Adeli, Smart structures: Part Ⅱ: Hybrid control systems and control strategies, Scientia Iranica, 18 (2011), 285-295. [7] A. Isidori, Nonlinear Control Systems, 3rd edition, Springer Verlag, London, 1995. doi: 10.1007/978-1-84628-615-5. [8] S. Korkmaz, A review of active structural control: challenges for engineering informatics, Comput. and Struct., 89 (2011), 2113-2132. [9] P. Koutsianitis, G. K. Tairidis, G. A. Drosopoulos, G. A. Foutsitzi and G. E. Stavroulakis, Effectiveness of optimized fuzzy controllers on partially delaminated piezocomposites, Acta Mechanica, 228 (2017), 1373-1392.  doi: 10.1007/s00707-016-1771-6. [10] A. D. Muradova, A time spectral method for solving the nonlinear dynamic equations of a rectangular elastic plate, J. Eng. Math., 92 (2015), 83-101.  doi: 10.1007/s10665-014-9752-z. [11] A. D. Muradova and G. E. Stavroulakis, Fuzzy vibration control of a smart plate, Int. J. Comput. Meth. Eng. Sci. Mech., 14 (2013), 212-220.  doi: 10.1080/15502287.2012.711427. [12] A. D. Muradova and G. E. Stavroulakis, Hybrid control of vibrations of smart von Kármán, Acta Mechanica, 226 (2015), 3463-3475.  doi: 10.1007/s00707-015-1387-2. [13] R. E. Precup and H. Hellendoorn, A survey on industrial applications of fuzzy control, Computers in Industry, 62 (2011), 213-226. [14] A. Preumont, Vibration Control of Active Structures, Springer, 2002. doi: 10.1007/978-94-007-2033-6. [15] J. N. Reddy, Theory and Analysis of Elastic Plates and Shells, CRC Press, Taylor & Francis, 2007. [16] A. H. N. Shirazi, H. R. Owji and M. Rafeeyan, Active vibration control of an FGM rectangular plate using fuzzy logic controllers, Procedia Engineering, 14 (2011), 3019-3026. [17] G. K. Tairidis, G. E. Stavroulakis, D. G. Marinova and E. C. Zacharenakis, Classical and soft robust active control of smart beams, Computat. Struct. Dynamics and Earthquake Engineer. (eds. Papadrakis, M. , Charmpis, D. C. Lagaros and N. D. , Tsompanakis), CRC Press/Balkema and Taylor & Francis Group, London, UK. , Ch. 11 (2009), 165–178. [18] A. R. Tavakolpour, M. Mailah, I. Z. M. Darus and O. Tokhi, Self-learning active vibration control of a flexible plate structure with piezoelectric actuator, Simul. Model. Prac. and Theory, 18 (2010), 516-532. [19] Q. Wenzhonga, S. Jincaib and Q. Yangc, Active control of vibration using a fuzzy control method, J. of Sound and Vibration, 275 (2004), 917-930.  doi: 10.1016/S0022-460X(03)00795-8. [20] I. J. Zeinoun and F. Khorrami, An adaptive control scheme based on fuzzy logic and its application to smart structures, Smart Mater. Struct., 3 (1994), 266-276.
The structure of a fuzzy inference system
Displacement (input 1) membership functions
Velocity (input 2) membership functions
Control force (output) membership functions
Displacement before and after control with Mamdani FIS ($\omega=10\pi$)
Velocity before and after control with Mamdani FIS ($\omega=10\pi$)
External and Control forces with Mamdani FIS ($\omega=10\pi$)
Clusters of input 1 (Displacement)
Clusters of input 2 (Velocity)
Displacement before and after control with Sugeno FIS ($\omega=10\pi$)
Velocity before and after control with Sugeno FIS ($\omega=10\pi$)
External and Control forces with Sugeno FIS ($\omega=10\pi$)
Displacement before and after control with Sugeno FIS ($\omega=5\pi$)
Velocity before and after control with Sugenoi FIS ($\omega=5\pi$)
External and Control forces with Sugenoi FIS ($\omega=5\pi$)
Displacement before and after ANFIS with $\omega=10$, $D=10$ (the linear problem)
Displacement before and after ANFIS with $\omega=10$, $D=50$ (the linear problem)
Displacement before and after using LQR with $\omega=10$, $D=10$
Displacement before and after using LQR with $\omega=10$, $D=50$
Loading and control forces with ANFIS with $\omega=10$, $D=10$ (the linear problem)
Loading and control forces with ANFIS with $\omega=10$, $D=50$ (the linear problem)
Loading and control forces with using LQR with $\omega=10$, $D=10$
Loading and control forces with using LQR with $\omega=10$, $D=50$
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