分数阶统一混沌系统的非脆弱模糊控制
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摘要
针对分数阶统一混沌系统,研究了非脆弱模糊控制。首先,基于T-S模糊建模理论,将分数阶统一混沌系统进行T-S模型重构;然后利用李雅普诺夫稳定性理论,研究分数阶统一混沌系统的稳定性分析问题,给出以设定衰减率全局渐近稳定的稳定性分析条件;最后,采用线性矩阵不等式技术,设计了能够镇定闭环系统的分数阶非脆弱模糊控制器。给出的3个仿真结果证明了设计方法的有效性。
This paper considers the design problem of nonfragile fuzzy control for fractional-order(FO) unified chaotic systems. Firstly, in terms of the T-S fuzzy modelling theory, the FO unified chaotic system is modeled into the T-S fuzzy system; then with the application of Lyapunov stability theory, sufficient conditions are given for the stability analysis; finally, the existence conditions of the FO nonfragile fuzzy controller are discussed in this paper. The FO nonfragile fuzzy controller designed here guarantees the stability of the closed-loop system. Three numerical simulation examples are given to illustrate the effectiveness of the designed approach.
This paper considers the design problem of nonfragile fuzzy control for fractional-order(FO) unified chaotic systems. Firstly, in terms of the T-S fuzzy modelling theory, the FO unified chaotic system is modeled into the T-S fuzzy system; then with the application of Lyapunov stability theory, sufficient conditions are given for the stability analysis; finally, the existence conditions of the FO nonfragile fuzzy controller are discussed in this paper. The FO nonfragile fuzzy controller designed here guarantees the stability of the closed-loop system. Three numerical simulation examples are given to illustrate the effectiveness of the designed approach.
引文
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[20]Liu X,Zhang Q.New approaches to H￥controller designs based on fuzzy observers for T-S fuzzy systems via LMI[J].Automatica,2003,39(9):1571-1582.
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[23]Zheng Y,Nian Y,Wang D.Controlling fractional order chaotic systems based on Takagi-Sugeno fuzzy model and adaptive adjustment mechanism[J].Physics Letters A,2010,375(2):125-129.
[24]Lin TC,Kuo CH.H￥synchronization of uncertain fractional order chaotic systems:adaptive fuzzy approach[J].ISA Transactions,2011,50(4):548-556.
[25]Chen D,Zhang R,Sprott JC,et al.Synchronization between integer-order chaotic systems and a class of fractional-order chaotic system based on fuzzy sliding mode control[J].Nonlinear Dynamics,2012,70(2):1549-1561.
[26]Wang B,Xue J,Chen D.Takagi-Sugeno fuzzy control for a wide class of fractional-order chaotic systems with uncertain parameters via linear matrix inequality[J].Journal of Vibration and Control.26August,2014,22(10):414-416.
[27]Kuntanapreeda S.Robust synchronization of fractional-order unified chaotic systems via linear control[J].Computers and Mathematics with Applications,2012,63(1):183-190.
[28]Xie L.Output feedback H￥control of systems with parameter uncertainty[J].International Journal of Control,1996,63(4):741-750.
[29]Shahri ESA,Balochian S.Analysis of fractional-order linear systems with saturation using Lyapunov’s second method and convex optimization[J].International Journal of Automation and Computing,2015,12(4):440-447.
[2]薛定宇,陈阳泉.高等应用数学问题的MATLAB求解[M].北京:清华大学出版社,2004.Xue D Y,Chen Y Q.Solution of Higher Applied Mathematics Problems by MATLAB[M].Beijing:Tsinghua University Press,2004.
[3]Bagley RL,Torvik P.On the appearance of the fractional derivative in the behavior of real materials[J].Journal of Applied Mechanics,1984,51(2):294-298.
[4]Bagley RL,Calico RA.Fractional-order state equations for the control of viscoelastic damped structures[J].Journal of Guidance Control and Dynamics,1991,14(2):304-311.
[5]Jenson VG,Jeffreys GV.Mathematical methods in chemical engineering[M].New York:Academic Press,1977.
[6]Weber E.Linear transient analysis[M].New York:Wiley,1956.
[7]卢俊国,魏荣,汪小帆,等.连续混沌系统控制与同步的自适应方法[J].控制与决策,2002,17(1):114-116.Lu J G,Wei R,Wang X F,et al.Adaptive approach to controlling and synchronizing continuous-time chaotic systems[J].Control and Decision.2002,17(1):114-116.
[8]GE S S,Wang C.Uncertain chaotic system control via adaptive neural design[J].International Journal of Bifurcation and Chaos,2002,12(5):1097-1109.
[9]Liu H,Li S G,Sun Y G,et al.Adaptive fuzzy synchronization for uncertain fractional-order chaotic systems with unknown nonsymmetrical control gain[J].Acta Physica Sinica,2015,64(7):331-334.
[10]Lu J.Chaotic dynamics and synchronization of fractional-order Chua's circuits with a piecewise-linear nonlinearity[J].International Journal of Modern Physics B,2005,19(20):3249-3259.
[11]Yu Y,Li H,Wang S,et al.Dynamic analysis of a fractional-order Lorenz chaotic system.Chaos,Solitons and Fractals,2009,42(2):1181-1189.
[12]Chen X R,Liu C X,Wang F Q,et al.Study on the fractional-order Liu chaotic system with circuit experiment and its control[J].Acta Physica Sinica,2008,57(3):1416-1422.
[13]Li C,Chen G.Chaos in the fractional order Chen system and its control[J].Chaos Solitons and Fractals,2004,22:549-554.
[14]Li Z,Chen D,Zhu J,et al.Nonlinear dynamics of fractional order Duffing system[J].Chaos Solitons and Fractals,2015,81:111-116.
[15]Dang H.Dynamics and synchronization of the fractional-order sprott E system[J].International Forum on Materials Analysis and Testing Technology.2014,850-851:876-879.
[16]Shao S,Gao X.Synchronization in time-delayed fractional order chaotic rossler systems[C].International Conference on Communications,Circuits and Systems,2008,6(3):652-654.
[17]Bai J,Yu Y,Wang S,et al..Modified projective synchronization of uncertain fractional order hyperchaotic systems[J].Communications in Nonlinear Science and Numerical Simulation,2012,17(4):1921-1928.
[18]Takagi T,Sugeno M.Fuzzy identification of systems and its applications to modeling and control[J].IEEE Transactions on Systems Man and Cybernetics,1985,15(1):116-132.
[19]Feng G.A survey on analysis and design of model-based fuzzy control systems[J].IEEE Transactions on Fuzzy Systems,2006,14(5):676-697.
[20]Liu X,Zhang Q.New approaches to H￥controller designs based on fuzzy observers for T-S fuzzy systems via LMI[J].Automatica,2003,39(9):1571-1582.
[21]杨志红,常凤云,姚琼荟.基于T-S模糊模型的离散混沌系统变结构控制[J].控制工程,2007,14(2):161-163.Yang Z H,Chang F Y,Yao Q H.Variable structure control of design chaotic system based on T-S fuzzy models[J].Control Engineering of China,2007,14(2):161-163.
[22]张乐,井元伟.基于非脆弱控制器设计的不确定模糊系统稳定性研究[J].控制与决策,2007,22(3):329-332.Zhang L,Jing Y W.Stability of a class of uncertain fuzzy systems based on non-fragile controller design[J].Control and Decision,2007,22(3):329-332.
[23]Zheng Y,Nian Y,Wang D.Controlling fractional order chaotic systems based on Takagi-Sugeno fuzzy model and adaptive adjustment mechanism[J].Physics Letters A,2010,375(2):125-129.
[24]Lin TC,Kuo CH.H￥synchronization of uncertain fractional order chaotic systems:adaptive fuzzy approach[J].ISA Transactions,2011,50(4):548-556.
[25]Chen D,Zhang R,Sprott JC,et al.Synchronization between integer-order chaotic systems and a class of fractional-order chaotic system based on fuzzy sliding mode control[J].Nonlinear Dynamics,2012,70(2):1549-1561.
[26]Wang B,Xue J,Chen D.Takagi-Sugeno fuzzy control for a wide class of fractional-order chaotic systems with uncertain parameters via linear matrix inequality[J].Journal of Vibration and Control.26August,2014,22(10):414-416.
[27]Kuntanapreeda S.Robust synchronization of fractional-order unified chaotic systems via linear control[J].Computers and Mathematics with Applications,2012,63(1):183-190.
[28]Xie L.Output feedback H￥control of systems with parameter uncertainty[J].International Journal of Control,1996,63(4):741-750.
[29]Shahri ESA,Balochian S.Analysis of fractional-order linear systems with saturation using Lyapunov’s second method and convex optimization[J].International Journal of Automation and Computing,2015,12(4):440-447.