摘要
自1983年Mandelbort指出自然界及许多科学技术领域中存在大量的分数维以来,分数阶微积分在各个领域得到了广泛研究,分数阶微积分较整数阶微积分具有诸多突出优势迅速成为当前的研究热点;在实际系统中,不可避免存在非线性特征,是非线性系统。与此同时,一般控制系统由于反馈延时、信号传输延时等都具有时滞。而时滞的存在往往是系统的稳定性和性能变差的原因,时滞的存在可能导致通信网拥塞、数据丢包,可能导致交通系统中车辆流动不畅、交通拥堵。如何判定分数阶时滞非线性系统的稳定性与控制至今未得到有效解决,该课题的科学意义不仅在于发展分数阶时滞非线性系统稳定性理论实现分数阶时滞系统控制,推动分数阶控制理论和控制方法的应用,更有助于丰富非线性系统控制理论,推动控制理论和控制工程的发展。
本论文提出了分数阶无时滞与含时滞非线性系统稳定性理论;提出了矩阵配置的控制器设计方法。实现了分数阶无时滞与含时滞非线性系统的控制与同步及系统的未知参数辨识。
本论文的主要研究内容和创新点包括以下内容:
1.分数阶非线性系统稳定性与控制研究。基于整数阶非线性系统稳定性理论与分数阶线性系统稳定性理论,从系统Jacobian矩阵特征值的角度出发提出了分数阶非线性系统稳定性理论。与此同时,将整数阶Lyapunov稳定性理论拓展到分数阶系统中,提出构造正定函数V并对函数V求整数阶导数的稳定性分析理论。该稳定性理论克服了分数阶Lyapunov稳定性理论中对函数V的复杂的分数阶求导、稳定性分析困难,且系统参数发生摄动时稳定性分析失效这一难题。在此基础上,研究无时滞分数阶系统的脉冲控制理论,建立了脉冲强度与脉冲间隔的一般关系,实现了分数阶系统的脉冲控制。最后研究了分数阶系统的间歇控制稳定性理论,建立了间歇控制一般稳定性条件,实现了分数阶系统间歇控制。数值仿真结果证实所提理论的正确有效性。
2.分数阶时滞系统稳定性理论与控制研究。基于本文提出的分数阶非线性系统稳定性理论和整数阶时滞系统稳定性理论,研究了如何建立正定函数V的扩展函数并对该扩展函数求整数阶导数而不是分数阶导数的分数阶系统Lyapunov-krasovskii理论,建立了扩展函数的一般构造方法。利用该理论,整数阶时滞系统的大部分控制方法可拓展到分数阶时滞系统。且从定理的形式可以看出,当系统系数和微分阶次在一定范围内摄动时该理论仍然适用。基于特征值稳定性分析理论,结合不等式利用反证法等建立了分数阶时滞系统极限发散和收敛速率计算方法。根据分数阶时滞非线性系统的极限发散和极限收敛速率,建立了分数阶离散时滞非线性系统间歇控制稳定性理论,实现了离散时滞分数阶系统间歇控制方法。最后,仿真实例显示其结果的有效性。
3.基于矩阵配置法的分数阶非线性系统控制器设计。尽管非线性系统的控制与同步有大量报道,但如何简单有效设计控制器实现系统控制鲜见介绍。针对这一情况提出矩阵配置控制器设计法。针对无时滞参数己知、参数未知的分数阶系统提出了如何利用矩阵配置控制器设计方法设计控制器实现系统控制和未知参数辨识。基于本文提出的分数阶时滞系统稳定性理论,利用时滞分离方法,提出了含离散时滞的分数阶非线性系统矩阵配置控制器设计方法,实现了离散时滞分数阶非线性系统控制。进一步将这一方法拓展到了整数阶分布式时滞系统,实现了整数阶分布式时滞系统控制。这些工作表明该控制器设计方法具有较好的通用性。
最后,总结全文并提出了进一步研究的方向。
In view of the fractional calculus's outstanding advantage than integer or-der's, since Mandelbort point out that there are vast fractional phenomena in the nature and the domain of science and technology, the fractional calculus has been widely studied involving all fields, and became the research hotspot. Nonlinear always exists in real system, nonlinear is an essential attribute of real system. Meanwhile, owing to feedback delay and siginal transmission delay, time-delay is commonly encountered. Time-delay is frequently a source of instability poor performance, such as network congestion, data packet dropout, poor flow the ve-hicles, traffic jams, etc. However, how to predicate the stability on the fractional nonlinear system with delay have not been solved effectively, let alone realize control. So the scientific significance of the subject lies not only in developing the stability theorem and realizing control about the fractional nonlinear system, promoting the application of the control theorem and control method, but also in enriching the nonlinear system control theorem, pushing the development of control theory and control engineer.
Firstly, the stability theories of the fractional nonlinear system with or with-out delay are proposed. Then, the Matrix configuration controller design methods are presented. Finally, control and synchronization of fractional nonlinear sys-tems with and without delay, as well as identification of unknown parameters of the systems are realized.
The main achievements and innovations of this dissertation are as follows:
1. Stability and control study of fractional nonlinear system. Based on stability theorems of integer order nonlinear systems and fractional linear systems, the stability theorem and control of fractional nonlinear system with-out delay are studied from Jacobian matrix eigenvalues point of view. In the meantime, by expanding integer order Lyapunov stability theory to fractional systems, a new stability theorem on fractional nonlinear system is proposed via constructing positive definite function V and taking integer derivatives of V. The superiority of the proposed theorem is reflected in not only avoiding difficultly taking fractional derivatives of V, but also avoiding losing stable efficacy while parameters perturbing. And on this basis we study impulsive control theorem on fractional nonlinear system, establish the general relationship between pulse strength and pulse period, and realize fractional impulsive control. Then we conduct intermittent control theorem study, create general intermittent stability condition, realize fractional intermittent control. Finally, simulation examples are given to illustrate the effectiveness of the developed theoretical results.
2. stability and control study of fractional nonlinear system with time-delay. Based on the proposed stability theorem of the fractional nonlinear system and the stability theorem of integer time-delay nonlinear system, we s-tudy fractional Lyapunov-krasovskii theory by building positive definite function V and taking integer derivatives of V instead of taking fractional derivatives, and propose general constructing method of expanding function. Using this theory, most control method of integer delayed system can be expanded to fractional delayed system. Based on the eigenvalue stability theory, combined inequali-ty and reduction to absurdum, the computing methods of limit divergence and convergence are presented. Subsequently, intermittent control theorem on frac-tional nonlinear system with discrete time-delay is set up and the intermittent control on delayed fractional nonlinear system is realized. Finally, numerical ex-amples are provided to demonstrate the effectiveness and the applicability of the proposed method.
3. Controller design of fractional nonlinear system via matrix con-figuration method. Although there are a large number of reports on nonlinear control and synchronization, but how to simply design controller is seldom report-ed so far. Stimulated by this, the matrix configuration controller design method is put forward. For non-delay fractional nonlinear system with or without unknown parameters, controller can be designed by matrix configuration method to realize system control and parameter identification. Based on the proposed stability theorem on the delayed fractional nonlinear system, by separating time-delay terms, the matrix configuration controller design method of fractional nonlinear system with discrete time-delay is presented also, and for delayed fractional non-linear system with or without unknown parameters, controller can be designed by matrix configuration method to realize system control and parameter identi-fication. We as well as expand this method to the integer order nonlinear system with distributed delay and achieve control. All the works show that the matrix configuration controller design method has good versatility and effectiveness.
Finally, the main results of the dissertation are concluded, and the issues of future investigation are proposed.
引文
[1]Yu M, Wang L, Chu T G. Stabilization of networked control systems with data packet dropout and transmission delays:Continuous-time case. Eu-ropean Journal of Control,2005, 11(1):40-49.
[2]Liu Y, Liu L G, Wang H. Study on congestion and bursting in small-world networks with time delay from the viewpoint of nonlinear dynamics. Chinese Physics Letters,2012,29(6):258-264.
[3]Orosz G, Krauskopf B, Wilson R E. Bifurcations and multiple traffic jams in a car-following model with reaction-time delay. Physica D:Nonlinear Phenomena,2005,211(3-4):277-293.
[4]Dong J W, Kim W J. Markov-chain-based output feedback control for stabilization of networked control systems with random time delays and packet losses. International Journal of Control, Automation, and Systems, 2012,10(5):1013-1022.
[5]Xu H, Jagannathan S, Lewis F L. Stochastic optimal control of unknown linear networked control system in the presence of random delays and pack-et losses. Automatica,2012,48(6):1017-1030.
[6]Sun Y G, Gao Q Z. Stability and stabilization of networked control system with forward and backward random time delays. Mathematical Problems in Engineering,2012.
[7]Balachandran K, Zhou Y, Kokila J. Relative controllability of fractional dynamical systems with distributed delays in control. Computers & Math-ematics with Applications,2012,64(10):3201-3209.
[8]Mandelbort B B. The Fractal Geometry of Nature New York:Freeman, 1983.
[9]张为民.一种采用分数阶导数的新流变模型理论.湘潭大学自然科学学报,2001,23(1):30-36.
[10]Gemant A. A method of analyzing experimental obtained from elasto-viscous bodies, Physics,1936,7:311-317.
[11]Bagley R L, Torvik P J. On the fractional calculus model of viscoelastic behavior. Journal of Rheology,1986,30(1):133-155.
[12]Shestopal V O, Goss P C J. The estimation of column creep buckling durability from the initial stages of creep. Acta Mechanica,1984,52:269-275.
[13]Hilfer R. Applications of fractional calculus in physics. New Jersey:World Scientific,2001.
[14]Matignon D. Stability results of fractional differential equations with appli-cations to control processing. Computational Engineering in Systems and Applications Multiconference,1996,2:963-968.
[15]Song L, Xu S Y, Yang J Y. Dynamical models of happiness with fractional order. Communications in Nonlinear Science and Numerical Simulations, 2010,15(3):616-628.
[16]Choudhury M D, Chandra S, Nag S. Forced spreading and rheology of starch gel:viscoelastic modeling with fractional calculus. Colloids and Sur-faces A:Physicochemical and Engineering Aspects,2012,407:64-70.
[17]Euler L. De progressionibus transcentibus, sev quarum termini algebraice dari nequeunt. Comment. Acad. Sci. Imperiales Petropolitanae,1738, 1(5):38-57.
[18]Laplace P S. Theorie analytique des probabilities. Paris:Courcier,1812.
[19]Lacroix S F. Traite du calcul differential et du calcul integral. Paris:Courci-er,1820.
[20]Fourier J. The analytical theory of heat. New York:Dover publ.,1812.
[21]Oldham K B and spanier J. The fractional calculus. New York-London: Academic Press,1974.
[22]Kilbas A A, Samko S G and Marichev O I. Fractional intergeral and deriva-tives:theory and applications. USA:Gordon and Breach Science Publish-ers,1993.
[23]Miller K S and Ross B. An introduction to the fractional calculus and fractional differential equations. New York:John Wiley and Sons Inc,1993.
[24]Podlubny I. Fractional differential equations. New York-London:Academic Press,1979.
[25]Bagley R L, Calico R A. Fractional order state equations for the control of viscoelastically damped structures. Journal of Guidance, Control, and Dynamics,1991,14:304-311.
[26]Kulish V V. Application of fractional caleulus to fluid mechanics. Journal of Fluids Engineering,2002,124(3):803-806.
[27]Rossikhin Y A, Shitikova M V. Application of fractional caleulus for dy-namic problems of solid mechanics:novel trends and recent results. Applied Mechanics Reviews,2010,63:010801-1-010801-52.
[28]Rossikhin Y A, Shitikova M V. Application of fractional caleulus for dynam-ic problems of linear and nonlinear fractional mechanics of solids. Applied Mechanics Reviews,1997,50(1):15-67.
[29]Sun H, Abdelwahab A, Onaral B. Linear approximation of transfer function with a pole of fractional Power. IEEE Transactions on Automatic Control, 1984,29(5):441-444.
[30]Kusnezov D, Bulgac A, Dand G D. Quantum Ley Processes and fractional kinetics. Physical Review Letters,1999,82(6):1136-1139.
[31]Metzler R, Klafter J. The random walk's guide to anomalous diffusion:a fractional dynamics approach. Physics Reports,2000,339(1):1-77.
[32]Li C P, Peng G J. Chaos in Chen's system with a fractional order. Chaos, Solitons & Fractals,2004,22(2):443-450.
[33]Li C P, Deng W H, Xu D. Chaos synchronization of the Chua system with a fractional order. Physica A:Statistical Mechanics and its Applications, 2006,360(2):171-185.
[34]Laskin N. Fractional market dynamics. Physica A:Statistical Mechanics and its Applications,2000,287(3-4):482-492.
[35]Hartley T T, Lorenzo C F. Dynamics and control of initialized fractional-order systems. Nonlinear Dynamics,2002,29(1):201-233.
[36]高为炳.非线性控制理论导论.北京:科学出版社,1988.
[37]廖晓听.稳定性的数学理论及应用.湖北:华中师范大学出版社,2001.
[38]Aeyels D. Asymptotic stability of nonautonomous systems by Lyapunov' s direct method. Systems and Control Letters,1995,25:273-280.
[39]Lasalle J P. Some extensions of Liapunov's second method. IRE Transac-tions on Circuit Theory,1960,7:520-527.
[40]Loria A, Panteley E, Teel A. UGAS of nonlinear time-varying systems:A o-persistency of excitation approach. Proceedings of the 39th IEEE Con-ference on Decision and Control, Sydney, Australia,2000,4:3489-3494.
[41]Lee T, Chen B. A general stability criterion for time-vary systems using a modified detectability condition.IEEE Transactions on Automatic Control, 2002,47:797-802.
[42]廖晓昕.动力系统的稳定性理论和应用.北京:国防工业出版社,2000.
[43]Hale J K and Verduyn Lunel S M. Introduction to functional differential equations. Academic Press, New York:Springer-Verlag,1993.
[44]吴敏,何勇.时滞系统鲁棒控制:自由权矩阵方法.北京:科学出版社,2008.
[45]苏宏业,褚健,鲁仁全,嵇小辅.不确定时滞系统的鲁棒控制理论.北京:科学出版社,2007.
[46]Fridman E, Shaked U. Delay-dependent stability and H∞ control:constant and time-varying delays. International Journal of Control,2003,76(1):48-60.
[47]Park P, Ko J. Stability and robust stability for systems with a time-varying delay. Automatica,2007,43(10):1855-1858.
[48]Moon Y S, Park P G, Kwon W H, Lee Y S. Delay-dependent robust stabi-lization of uncertain state-delayed systems. International Journal of Con-trol,2001,74(14):1447-1455.
[49]Gu K, Kharitonov V L, Chen J. Stability of time-delay systems. Boston: Birkhuser,2003.
[50]Gu K. Discretized LMI set in the stability problem for linear uncertain time-delay systems. International Journal of Control,1997,68(4):923-934.
[51]Han Q L. A discrete delay decomposition approach to stability of linear retarded and neutral systems. Automatica,2009,45(2):517-524.
[52]He Y, Wang Q G, Lin C, Wu M. Augmented Lyapunov functional and delay-dependent stability criterion for neutral systems. International Jour-nal of Robust and Nonlinear Control,2005,15(8):923-933.
[53]Pascal G, Arkadi N, Alan J. LMI control toolbox for use with Matlab. The MathWorks, Inc.,1995.
[54]Boyd S, Ghaoui L E, Feron E, et al. Linear matrix inequalities in system and control theory. Philadelphia:SIAM,1994.
[55]Gahinet P, Nemirovskii A, Laub A J, et al. LMI control toolbox. Mas-sachusetts:MathWorks, Natick,1995.
[56]Momani S, Hadid s. Lyapunov stability solutions of fractional integro-differential equations. Int. J. Math. Math. Sci.,2004,47:2503-2507.
[57]Sabatier J. On stability of fractional order systems, In Plenary Lecture VIII on 3r d IFAC Workshop on Fractional Differentiation and its Applications, Ankara, Turkey,2008.
[58]Lakshmikantham V, Leela S, Devi J V. Theory of fractional dynamic sys-tems. Cambridge:Cambridge Scientific Publishers,2009.
[59]Li Y, Chen Y Q, Podlubny I. Stability of fractional-order nonlinear dynamic systems:Lyapunov direct method and generalized Mittag-Leffler stability. Omputers & Mathematics with Applications,2010,59(5):1810-1821.
[60]Fahd J, Thabet A, Dumitru B. Stability of q-fractional non-autonomous systems. Nonlinear Analysis:Real World Applications,2013,14(1):780-784.
[61]Delavari H, Baleanu D, Sadati J. Stability analysis of Caputo fractional-order nonlinear systems revisited. Nonlinear Dynamics,2012,67(4):2433-2439.
[62]El-Sayed A, Gaafar F. Stability of a nonlinear non-autonomous fractional order systems with different delays and non-local conditions. Advances in Difference Equations,2011, DOI:10.1186/1687-1847-2011-47.
[63]El-Sayed A M A, Gaafar F M, Hamadalla E M A. Stability for a non-local non-autonomous system of fractional order differential equations with delays. Electronic Journal of Qualitative Theory of Differential Equations, 2010,
[64]Jiao Z, Zhong Y S. Robust stability for fractional-order systems with struc-tured and unstructured uncertainties. Computers & Mathematics with Ap-plications,2012,64(10):3258-3266.
[65]Li C, Wang J C. Robust stability and stabilization of fractional order in-terval systems with coupling relationships:The 0<α<1 case. Journal of the Franklin Institute:Engineering and Applied Mathematics,2012, 349(7):2406-2419.
[66]Burton T A. Fractional differential equations and Lyapunov functional-s. Nonlinear Analysis-Theory Methods & Applications,2011,74(6):5648-5662.
[67]Zhao L D, Hu J B, Fang J A, Zhang W B. Studying on the stability of fractional-order nonlinear system. Nonlinear Dynamics,2012,70(1):475-479.
[68]胡建兵,韩焱,赵灵冬.基于Lyapunov方程的分数阶混沌系统同步.物理学报,2008,57(12):7522-7526.
[69]胡建兵,韩焱,赵灵冬.分数阶系统的一种稳定性判定定理及在分数阶统一混沌系统同步中的应用.物理学报,2009,58(7):4402-4407.
[70]胡建兵,韩焱,赵灵冬.一种新的分数阶系统稳定理论及在Backstepping方法同步分数阶混沌系统中的应用.物理学报,2009,58(4):223-2239.
[71]Fioravanti A R, Bonnet C, Ozbay H. A numerical method for stability windows and unstable root-locus calculation for linear fractional time-delay systems. Automatica,2012,48(11):2824-2830.
[72]Kaslik E, Sivasundaram S. Analytical and numerical methods for the sta-bility analysis of linear fractional delay differential equations. Journal of Computational and Applied Mathematics,2012,236(16):4027-4041.
[73]Li C P, Zhang F R. A survey on the stability of fractional differential equations. European Physical Journal Special Topics,2011,193(1):27-47.
[74]Balasubramaniam P, Lakshmanan S, Rakkiyappan R. LMI optimization problem of delay-dependent robust stability criteria for stochastic systems with polytopic and linear fractional uncertainties. International Journal of Applied Mathematics and Computer Science,2012,22(2):339-351.
[75]Liang T N, Chen J J, Lei C. Algorithm of robust stability region for interval plant with time delay using fractional order (PID mu)-D-lambda controller. Communications in Nonlinear Science and Numerical Simulation,2011, 17(2):979-991.
[76]Lazarevic M P. Finite time stability analysis of PD a fractional control of robotic time-delay systems, Mech. Res. Comm.,2006,33:269-279.
[77]Zhang X. Some results of linear fractional order time-delay system. Appl. Math. Comp.,2008,197:407-411.
[78]Baleanu D, Ranjbar N A, Sadati R S J. Lyapunov-krasovskii stability the-orem for fractional systems with delay. Romanian Journal of Physics,2011, 56(5-6):636-643.
[79]Matignon D. Stability properties for generalized fractional differential sys-tems. Proc. Essaim,1998,5:145-158.
[80]Yakar C. Fractional differential equations in terms of comparison results and lyapunov stability with initial time difference. Abstract and Applied Analysis,2010, DOI:10.1155/2010/762857.
[81]Li Z B, Zhao X S. The parametric synchronization scheme of chaotic sys-tem. Commun Nonlinear Sci Numer Simulat,2011,16(7):2936-29440.
[82]Li Y, Chen Y Q, Podlubny I. Mittag-Lefffler stability of fractional order nonlinear dynamic systems. Automatica,2009,45(8):1965-1969.
[83]Krishna B T. Studies on fractional order differentiators and integrators:A survey. Signal Processing 2011,91(3):386-426.
[84]Jiang X, Xu M, Qi H. The fractional diffusion model with an absorption term and modified Fick's law for non-local transport processes. Nonlinear Anal-Real,2009,11(1):262-270.
[85]Tang Y, Wang Z D, Fang J A. Pinning control of fractional-order weighted complex networks. Chaos,2009,19(1).
[86]Shahiri M, Ghaderi R, Ranjbar N, Hosseinnia S, Momani S. Chaotic fractional-order Coullet system:synchronization and control approach. Commun Nonlinear Sci Numer Simul,2010,15(3):665-674.
[87]Meral F, Royston T, Magin R. Fractional calculus in viscoelasticity: an experimental study. Commun Nonlinear Sci Numer Simul,2010, 15(4):939-945.
[88]Ahmed E, Elgazzar A S. On fractional order differential equations model for nonlocal epidemics. Physica A,2007,379(2):607-14.
[89]Ahmad W, El-Khazali R. Fractional-order dynamical models of love. Chaos Soliton Fract,2007,33(4):1367-75.
[90]Song L, Xu S, Yang J. Dynamical models of happiness with fractional order. Commun Nonlinear Sci Numer Simul.2010,15(3):616-628.
[91]Chen D Y, Liu Y X, Ma X Y. Control of a class of fractional-order chaotic systems via sliding mode. Nonlinear Dynamics,2012,67(1):893-901.
[92]Tang Y, Fang J A. Synchronization of N-coupled fractional-order chaotic systems with ring connection. Commun Nonlinear Sci Numer Simulat,2010, 15(2):401-412.
[93]Hu J B, Han Y, Zhao L D. Synchronizing fractional chaotic systems based on Lyapunov equation. Acta Phys Sin,2008,57(12):7522-7526.
[94]Sadati S J, Baleanu D, Ranjbar A, Ghaderi R. Mittag-Leffler Stability The-orem for Fractional Nonlinear Systems with Delay. Abstract and Applied Analysis,2010,108651.
[95]Lu J H, Chen G R, Yu X H. Dynamical behaviors of a unified chaotic system.2nd International Symposium on Intelligent and Complex Systems, Wuhan, China,2003:115-124.
[96]Wang X Y, Chao G B. Hyperchaos generated from the unified chaotic system and its control. International Journal of Modern Physics B,2010, 24(23):4619-4637.
[97]Caputo M. Linear models of dissipation whose Q is almost frequency inde-pendent. Part Ⅱ. J Roy Austral Soc 1967,13:529-539.
[98]Daftardar-Gejji V, Babakhani A. Analysis of a system of fractional differ-ential equations. J Math Anal Appl,2004,293(2):511-522.
[99]Peng G J, Jiang Y L. Two routes to chaos in the fractional Lorenz system with dimension continuously varying. Physica A,2010,389(19):4140-4148.
[100]Li L X, Peng H P, Luo Q. Problem and analysis of stability decidable theory for a class of fractional order nonlinear system. Acta Physica Sinica,2013, 62(2).
[101]Wang X D, Tian L X. Bifurcation analysis and linear control of the Newton-Leipnik system. Chaos, Solitons & Fractals,2006,27(1):31-38.
[102]Liu S, Li X Y, Jiang W. Mittag-Leffler stability of nonlinear fraction-al neutral singular systems, Commun Nonlinear Sci Numer Simul,2012, 17(10):3961-3966.
[103]Li Y, Chen Y Q, Podlubny I. Stability of fractional-order nonlinear dynamic systems:Lyapunov direct method and generalized Mittag-Leffler stability. Comput Math Appl,2010,59(5):1810-1821.
[104]Sabatier J, Moze M, Farges C. LMI stability conditions for fractional order systems. Comput Math Appl,2010,59(5):1594-1609.
[105]Matignon D. Stability results of fractional differential equations with ap-plications to control processing. Lille, France:IMACS, IEEE-SMC,1996.
[106]Wang J. Zhou Y, Wei W. A class of fractional delay nonlinear integrodiffer-ential controlled systems in Banach spaces. Commun Nonlinear Sci Numer Simul,2011,16(10):4049-4059.
[107]Deng W H, Li C P, Lu J H. Stability analysis of linear fractional differential system with multiple time delays, Nonlinear Dynamics,2007,48(4):409-416.
[108]Meisam Y, Abolfazl R N, Reza G. Synchronization of chaotic systems with known and unknown parameters using a modified active sliding mode con-trol. ISA Transactions,2011,50(2):262-267.
[109]Zhang H, Ma X K, Liu W Z. Synchronization of chaotic systems with parametric uncertainty using active sliding mode control. Chaos, Solitons & Fractals,2004,21 (5):1249-1257.
[110]Gan Q. Synchronization of chaotic neural networks with unknown parame-ters and random time-varying delays based on adaptive sampled-data con-trol and parameter identification. IET Control Theory and Applications, 2010,6(10):1508-1515.
[111]Tang Y G, C M Y, Li L X. Parameter identification of time-delay chaot-ic system using chaotic ant swarm. Chaos Solitions & Fractals,2012, 41(4):2097-2102.
[112]Yang X S, Zhu Q X, Huang C X. Lag stochastic synchronization of chaotic mixed time-delayed neural networks with uncertain parameters or pertur-bations. Neurocomputing,2011,74(10):1617-1625.
[113]Tang Y, Fang J A, Lu S J. Adaptive synchronization for unknown stochas-tic chaotic neural networks with mixed time-delays by output coupling. Modern Physics Letters B,2008,22(24):2391-2409.