一类分数阶超混沌系统的自适应有限时间控制
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摘要
为实现带有不确定参数的分数阶超混沌Lorenz系统的自适应有限时间控制,采用分数阶微积分的相关引理及有限时间Lyapunov原理,设计了一个自适应有限时间控制器。该方法将整数阶混沌系统的有限时间控制方法拓展到阶次小于1的分数阶混沌系统,数值仿真验证了该控制器的准确性及有效性。该方法简单有效,可使系统的状态变量在有限时间内收敛到平衡点,收敛速度较快,具有良好的鲁棒性能。
To solve the adaptive finite-time control problem for fractional order hyperchaotic Lorenz systems with uncertain parameters,based on the related lemmas of fractional calculus and the finite-time Lyapunov principle,an adaptive finite time controller is designed to guarantee the system's adaptive finite-time stability.The accuracy and effectiveness of the proposed controller are verified by numerical simulations.The method is simple and effective,which can make the state variables of the system converge to the equilibrium point in a finite time.The convergence speed is fast and the robust performance is good.
To solve the adaptive finite-time control problem for fractional order hyperchaotic Lorenz systems with uncertain parameters,based on the related lemmas of fractional calculus and the finite-time Lyapunov principle,an adaptive finite time controller is designed to guarantee the system's adaptive finite-time stability.The accuracy and effectiveness of the proposed controller are verified by numerical simulations.The method is simple and effective,which can make the state variables of the system converge to the equilibrium point in a finite time.The convergence speed is fast and the robust performance is good.
引文
[1]MOHAMMAD POURMAHMOOD AGHABABA.Fractional Modeling and Control of a Complex Nonlinear Energy SupplyDemand System[J].Complexity,2015,20(6):74-86.
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[2]PODLUBNY I.Fractional Differential Equations[M].New York:Academic Press,1999.
[3]BEHINFARAZ R,BADAMCHIZADEH M A,GHIASI A R.An Approach to Achieve Modified Projective Synchronization between Different Types of Fractional-Order Chaotic Systems with Time-Varying Delays[DB/OL].(2015-07-08).[2017-03-08].https:∥doi.org/10.1016/j.chaos.2015.07.008.
[4]RAYNAUD H F.State-Space Representation for Fractional Order Controllers[J].Automatica,2000,36(7):1017-1021.
[5]BEHINFARAZ R,BADAMCHIZADEH M A.Synchronization of Different Fractional-Ordered Chaotic Systems Using Optimized Active Control[C]∥IEEE International Conference on Modeling,Simulation,and Applied Optimization.[S.l.]:IEEE,2015:1-6.
[6]SONG X.Output Feedback Control for Fractional-Order Takagi-Sugeno Fuzzy Systems with Unmeasurable Premise Variables[J].Transactions of the Institute of Measurement&Control,2015,38(10):1201-1211.
[7]YIN C,DADRAS S,ZHONG S M,et al.Control of a Novel Class of Fractional-Order Chaotic Systems via Adaptive Sliding Mode Control Approach[J].Applied Mathematical Modelling,2013,37(4):2469-2483.
[8]CHEN D Y,LIU Y X,MA X Y,et al.Control of a Class of Fractional-Order Chaotic Systems via Sliding Mode[J].Nonlinear Dynamics,2011,67(1):893-901.
[9]AGHABABA M P.A Novel Terminal Sliding Mode Controller for a Class of Non-Autonomous Fractional-Order Systems[J].Nonlinear Dynamics,2013,73(1/2):679-688.
[10]KAMENKOV G V.On Stability of Motion over a Finite Interval of Time[J].Akad Nauk Sssr Prikl Mat Meh,1953(4):529-540.
[11]BHAT S P.Continuous Finite-Time Stabilization of the Translational and Rotational Double Integrators[J].IEEE Transactions on Automatic Control,1998,43(5):678-682.
[12]AMATO FRANCESCO.Finite-Time Stability of Linear Time-Varying Systems:Analysis and Controller Design[J].IEEETransactions on Automatic Control,2010,55(4):1003-1008.
[13]GARCIA.Finite-Time Stabilization of Linear Time-Varying Continuous Systems[J].Automatic Control,2009,54(2):364-369.
[14]LAZAREVIC'M P.Finite Time Stability Analysis of Linear Autonomous Fractional Order Systems with Delayed State[J].Asian Journal of Control,2010,7(4):440-447.
[15]LAZAREVIC'M P.Finite Time Stability Analysis of PDA Fractional Control of Robotic Time-Delay Systems[J].Mechanics Research Communications,2006,33(2):269-279.
[16]WANG J,GAO T,ZHANG G.Adaptive Finite-Time Control for Hyperchaotic Lorenz-Stenflo Systems[J].Physica Scripta,2015,90(2):25204-25208.
[17]KILBAS A A,SRIVASTAVA H M,TRUJILLO J J.Theory and Applications of Fractional Differential Equations[M].Amsterdam:Elsevier,2006.
[18]ZHAO Y,WANG Y,LIU Z.Finite Time Stability Analysis for Nonlinear Fractional Order Differential Systems[C]∥IEEEControl Conference.[S.l.]:IEEE,2013:487-492.
[19]李雄.分数阶非线性系统稳定性分析及同步控制[D].西安:陕西师范大学数学与信息科学学院,2016.LI Xiong.Stability Analysis and Synchronization Control of Fractional Order Nonlinear Systems[D].Xi'an:School of Mathematics and Information Science,Shanxi Normal University,2016.
[20]LI Y,CHEN Y Q,PODLUBNY I.Stability of Fractional-Order Nonlinear Dynamic Systems:Lyapunov Direct Method and Generalized Mittag-Leffler Stability[J].Computers&Mathematics with Applications,2010,59(5):1810-1821.
[21]NORELYS AGUILA-CAMACHO.Lyapunov Functions for Fractional Order Systems[J].Communications in Nonlinear Science&Numerical Simulation,2014,19(9):2951-2957.
[22]BECKENBACH E F,BELLMAN R.Inequalities[J].Ergebnisse der Mathematik und ihrer Grenzgebiete,1962,42(9):421-422.