Dynamics of a Switched by Hysteresis Linear System in R2
Main Article Content
Keywords
Hysteresis, periodic orbit, piece-wise dynamic systems, basins of attraction.
Abstract
This paper examines the local and global behavior of a two-variable piece-wise smooth dynamic system by studying the flow of two nonhomogeneous linear dynamic systems that switch on the boundaries of a hysteresis band. Both equilibrium solutions and parametric domains that guarantee the existence of periodic orbits within the band, were determined analytically. While, phase portraits and basins of attraction of the system for different values of the parameters were performed through numerical simulations. Results show the coexistence of multiple stationary states of different types when different parameter values are used.
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References
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K. J. Å ström, G. C. Goodwin, and P. R. Kumar, Adaptive Control, Filtering, and Signal Processing, ser. The IMA Volumes in Mathematics and its Applications. New York, NY: Springer New York, The IMA Volumes in Mathematics and Its Applications Series., 1995, vol. 74. [Online]. Available: http://link.springer.com/10.1007/978-1-4419-8568-2
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M. Dawande, V. Kanetkar, and G. Dubey, “Three-phase switch mode rectifier with hysteresis current control,” IEEE Transactions on Power Electronics, vol. 11, no. 3, pp. 466–471, May 1996. [Online]. Available: http://ieeexplore.ieee.org/articleDetails.jsp?arnumber=491640
H. Lee and V. I. Utkin, “Chattering suppression methods in sliding mode control systems,” Annual Reviews in Control, vol. 31, no. 2, pp. 179–188, jan 2007. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S1367578807000363
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J. N. Hincapié, A. Trejos, M. E. Moncada, and A. Escobar, “Electrónica de potencia para el calentamiento por inducción doméstico: revisión del estado del arte,” Ingeniería y ciencia, vol. 9, no. 18, pp. 237–262, 2013.
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L. Perko, Differential Equations and Dynamical Systems, ser. Texts in Applied Mathematics. New York, NY: Springer-Verlag New York, 2001, vol. 7.
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htm?arnumber=1219573
Z. Zhusubaliyev and E. Soukhoterin, “Oscillations in a relay control system with hysteresis and time dead zone,” Mathematics and Computers in Simulation, vol. 58, no. 4-6, pp. 329–350, mar 2002. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0378475401003767
M. Spivak, Calculus on Manifolds: a Modern Approach to Classical Theorems of Advanced Calculus. Benjamin-Cummings, 1965. [Online]. Available: http://en.wikipedia.org/wiki/Calculus_on_Manifolds_(book)
Stephen Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, ser. Texts in Applied Mathematics. New York: Springer-Verlag, Texts in Applied Mathematics Volume 2., 2003, vol. 2. [Online]. Available: http://link.springer.com/10.1007/b97481
C. S. Hsu, Cell-to-Cell Mapping, A Method of Global Analysis for Nonlinear Systems, ser. Applied Mathematical Sciences. New York, NY: Springer New York, Applied Mathematical Sciences., 1987, vol. 64. [Online]. Available: http://www.springerlink.com/index/10.1007/978-1-4757-3892-6