Nonlinear Control of Chaotic Forced Duffing and Van der Pol Oscillators
- 1 College of Engineering, Shaqra University, Riyadh, Saudi Arabia
- 2 Electrical and Computer Engineering Department, Oakland University, Rochester, MI, USA
- 3 Electrical and Computer Engineering Department, Oakland University, Rochester, MI, USA
Abstract
This paper discusses a novel technique and implementation to perform nonlinear control for two different forced model state oscillators and actuators. The paper starts by discussing the Duffing oscillator which features a second order non-linear differential equation describing complex motion whereas the second model is the Van der Pol oscillator with non-linear damping. A first order actuator is added to both models to expand on the chaotic behavior of the oscillators. In order to control the system without comprising linearization, Lyapunov non-linear control was used. A control Lyapunov function was tailored to the system. This led to improved maneuverability of the controller and the performance of the overall system. The controller was found to be highly efficient in system tracking and had swift response time. Simulations were performed on both the uncontrolled and controlled cases. Both simulation results ultimately confirmed the effectiveness of the proposed controller.
- Thompson, J.M.T. and Stewart, H.B. (2002) Nonlinear Dynamics and Chaos. John Wiley & Sons, New York.
- Zaher, A.A., Harb, A.M. and Zohdy, M.A. (2004) Recursive Backstepping Control of Chaotic Duffing Oscillators. Proceedings of the 2004 American Control Conference, 5, 4302-4306.
- Harb, A., Zaher, A. and Zohdy, M. (2002) Nonlinear Recursive Chaos Control. Proceedings of the 2002 American Control Conference, 8-10 May 2002. https://doi.org/10.1109/acc.2002.1023974
- Bermúdez-Gómez, C.R., Enriquez-Caldera, R. and Martínez-Carballido, J. (2012) Chirp Signal Detection Using the Duffing Oscillator. CONIELECOMP 2012, 22nd International Conference on Electrical Communications and Computers, Cholula, 344-349. https://doi.org/10.1109/conielecomp.2012.6189936
- Zaher, A.A. (2011) Secure Communication Using Duffing Oscillators. IEEE International Conference on Signal and Image Processing Applications (ICSIPA), Kuala Lumpur, 563-568. https://doi.org/10.1109/ICSIPA.2011.6144150
- Li, J. and Shen, Y. (2009) The Study of Weak Signal Detection Using Duffing Oscillators Array. IEEE Circuits and Systems International Conference on Testing and Diagnosis, Chengdu, 28-29 April 2009, 1-4. https://doi.org/10.1109/cas-ictd.2009.4960901
- Wang, G.Y. and He, S.L. (2003) A Quantitative Study on Detection and Estimation of Weak Signals by Using Chaotic Duffing Oscillators. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 50, 945-953. https://doi.org/10.1109/TCSI.2003.812606
- Wang, Y., Li, H. and Dai, W. (2016) Application of Duffing Oscillator in Ship Propeller Blade Number Recognition. IEEE/OES China Ocean Acoustics (COA), Harbin, 1-5.
- Peluffo-Ordónez, D.H., Rodríguez-Sótelo, J.L., Revelo-Fuelagán, E.J., Ospina-Aguirre, C. and Olivard-Tost, G. (2015) Generalized Bonhoeffer-Van der Pol Oscillator for Modelling Cardiac Pulse: Preliminary Results. IEEE 2nd Colombian Conference on Automatic Control (CCAC), Manizales, 1-6.
- Menzel, K.O., Bockwoldt, T., Arp, O. and Piel, A. (2013) Modeling Dust-Density Wave Fields as a System of Coupled van der Pol Oscillators. IEEE Transactions on Plasma Science, 41, 735-739. https://doi.org/10.1109/TPS.2012.2227497
- Kuo, C.L., Pai, N.S., Liang, S.M. and Hu, S.H. (2008) Fuzzy Sliding-Model Control for Synchronization of an Uncertain Duffing-Holmes System. 2008 IEEE International Symposium on Knowledge Acquisition and Modeling Workshop, Wuhan, 21-22 December 2008, 104-107. https://doi.org/10.1109/KAMW.2008.4810435