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In this paper, a robust fractional-order proportional-derivative (PDμ) control is designed for controling in integer-order unified chaotic systems with parametric uncertainties. Equivalent plant is obtained by transforming the controlled dynamic system, and then the PDμ controller as an equivalent controller is applied to the equivalent plant. In the uncertain controlled unified chaotic systems, one equation is certain, and the other two equations are uncertain . The equivalent controller for the certain system is then designed based on a fractional-order proportional-derivative controller, in which three specifications for phase margin, gain crossover frequency, and robustness should be met. On the other hand, the robustness of uncertain systems is achieved by an improved Monje-Vinagre tuning method, however, the pre-specified frequency band should be replaced by the gain crossover frequency in order to reduce the complexity in determining the controllers for the uncertain systems. Specifications related to phase margin for the lower bound of the phase, gain crossover frequency for the upper bound of the gain, and robustness for the lower bound of the phase constraints are satisfied by the uncertain system. Parameters of the equivalent controller are determined based on a graphical method. Origins of the unstable equilibrium can be asymptotically stabilized by the proposed strategy for the integer-order unified chaotic systems with parametric uncertainties. Numerical simulation examples for Chen system, L system, and Lorenz system, are given to show the effectiveness of the proposed method.
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Keywords:
- unified chaotic systems /
- parametric uncertainties /
- robust /
- fractional-order proportional-derivative control
[1] Smaoui N, Karouma A, Zribi M 2011 Commun. Nonlinear Sci. Numer. Simulat. 16 3279
[2] Cheng C 2012 Applied Mathematics and Computation 219 2698
[3] Wang H, Han Z, Xie Q, Zhang W 2009 Nonlinear Dynamics 55 323
[4] Wei W, Li D H, Wang J 2011 Chin. Phys. B 20 040510
[5] Wang H, Chen B, Lin C 2013 ICIC Express Letters 7 423
[6] Mohan J J, Deekshitulu G V S R 2012 International Journal of Differential Equations 2012 780619
[7] Hu J B, Zhao L D 2013 Acta Phys. Sin. 62 240504 (in Chinese) [胡建兵, 赵灵东 2013 62 240504]
[8] Yang J, Qi D L 2010 Chin. Phys. B 19 020508
[9] Liu J, Li X, Zhao J 2011 Proceedings of the 2011 Chinese Control and Decision Conference Mianyang, China, May 23-25, 2011 p2093
[10] Qi D L, Wang Q, Yang J 2011 Chin. Phys. B 20 100505
[11] Zou D, Gao L, Li S 2014 ICIC Express Letters 8 2745
[12] Hung M L, Lin J S, Yan J J, Liao T L 2008 Chaos, Solitons and Fractals 35 781
[13] Niu P F, Zhang J, Guan X P 2007 Acta Phys. Sin. 56 3759 (in Chinese) [牛培峰, 张君, 关新平 2007 56 3759]
[14] Monje C A, Vinagre B M, Feliu V, Chen Y 2008 Control Engineering Practice 16 798
[15] Li H S, Luo Y, Chen Y Q 2010 IEEE Transactions on Control Systems Technology 18 516
[16] Vinopraba T, Sivakumaran N, Narayanan S, Radhakrishnan T K 2012 Journal of Control Theory and Applications 10 297
[17] Gao X, Liu X W 2007 Acta Phys. Sin. 56 84 (in Chinese) [高心, 刘兴文 2007 56 84]
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[1] Smaoui N, Karouma A, Zribi M 2011 Commun. Nonlinear Sci. Numer. Simulat. 16 3279
[2] Cheng C 2012 Applied Mathematics and Computation 219 2698
[3] Wang H, Han Z, Xie Q, Zhang W 2009 Nonlinear Dynamics 55 323
[4] Wei W, Li D H, Wang J 2011 Chin. Phys. B 20 040510
[5] Wang H, Chen B, Lin C 2013 ICIC Express Letters 7 423
[6] Mohan J J, Deekshitulu G V S R 2012 International Journal of Differential Equations 2012 780619
[7] Hu J B, Zhao L D 2013 Acta Phys. Sin. 62 240504 (in Chinese) [胡建兵, 赵灵东 2013 62 240504]
[8] Yang J, Qi D L 2010 Chin. Phys. B 19 020508
[9] Liu J, Li X, Zhao J 2011 Proceedings of the 2011 Chinese Control and Decision Conference Mianyang, China, May 23-25, 2011 p2093
[10] Qi D L, Wang Q, Yang J 2011 Chin. Phys. B 20 100505
[11] Zou D, Gao L, Li S 2014 ICIC Express Letters 8 2745
[12] Hung M L, Lin J S, Yan J J, Liao T L 2008 Chaos, Solitons and Fractals 35 781
[13] Niu P F, Zhang J, Guan X P 2007 Acta Phys. Sin. 56 3759 (in Chinese) [牛培峰, 张君, 关新平 2007 56 3759]
[14] Monje C A, Vinagre B M, Feliu V, Chen Y 2008 Control Engineering Practice 16 798
[15] Li H S, Luo Y, Chen Y Q 2010 IEEE Transactions on Control Systems Technology 18 516
[16] Vinopraba T, Sivakumaran N, Narayanan S, Radhakrishnan T K 2012 Journal of Control Theory and Applications 10 297
[17] Gao X, Liu X W 2007 Acta Phys. Sin. 56 84 (in Chinese) [高心, 刘兴文 2007 56 84]
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