一种有效的不确定分数阶T-S模糊系统的控制器设计方法
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摘要
考虑分数阶非线性系统的稳定和镇定问题,基于线性矩阵不等式(LMI)方法,对分数阶T-S模糊系统进行研究.利用并行分布补偿法,设计分数阶T-S模糊系统的控制器.考虑阶次满足0 <α<1的分数阶系统,给出可以利用Matlab求解的LMI形式的T-S模糊控制器设计镇定判据.该判据的优点是可以处理具有正实部特征根的分数阶T-S模糊系统的稳定性和镇定问题,能够保持与Matignon分数阶系统稳定性结论的一致性,并克服其他方法只能处理特征根在负实部的方法的局限性和保守性.数值仿真结果验证了所提控制器设计方法的有效性.
Considering the stability and stabilization of a class of nonlinear fractional order systems, based on the linear matrix inequality(LMI) approach, fractional order T-S fuzzy systems are studied. Using the method of parallel distributed compensation, controllers of fractional order T-S fuzzy systems are designed. Considering the fractional order T-S fuzzy systems with the order α satisfying 0 < α < 1, stabilization criterion is given in terms of LMI, which can be solved by Matlab. This criterion can handle the problems of the stability and stabilization of fractional order T-S fuzzy systems which have positive real eigenvalues, while maintaining the consistency with the stability criterion of fractional order systems from Matignon. The limitation and conservatsm of the eigenvalues in negative real parts in the other methods are solved. Numerical simulation results verify the effectiveness of the proposed controller design method.
Considering the stability and stabilization of a class of nonlinear fractional order systems, based on the linear matrix inequality(LMI) approach, fractional order T-S fuzzy systems are studied. Using the method of parallel distributed compensation, controllers of fractional order T-S fuzzy systems are designed. Considering the fractional order T-S fuzzy systems with the order α satisfying 0 < α < 1, stabilization criterion is given in terms of LMI, which can be solved by Matlab. This criterion can handle the problems of the stability and stabilization of fractional order T-S fuzzy systems which have positive real eigenvalues, while maintaining the consistency with the stability criterion of fractional order systems from Matignon. The limitation and conservatsm of the eigenvalues in negative real parts in the other methods are solved. Numerical simulation results verify the effectiveness of the proposed controller design method.
引文
[1]赵春娜,李英顺,陆涛.分数阶系统分析与设计[M].北京:国防工业出版社,2011:1-3.(Zhao C N,Li Y S,Lu T.Analysis and design of fractional order systems[M].Beijing:National Defend Industry Press,2011:1-3.)
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[3]邱宁.时间分数阶延迟微分方程在流体力学中的应用[J].沈阳大学学报:自然科学版,2016,28(2):170-172.(Qiu N.Application of Time Fractional Delay Differential Equations in Fluid Dynamics[J].J of Shenyang University:Natural Science,2016,28(2):170-172.)
[4]杨平,董国威.互联电网AGC的分数阶PID控制[J].电力系统及其自动化学报,2013,25(3):124-129.(Yang P,Dong G W.Fractional Order PID Control for AGC of Interconnected Power System[J].Proc of the Chinese Society of Universities for Electric Power System and its Automation,2013,25(3):124-129.)
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[8]Oustaloup A,Mathieu B,Lanusse P.The CRONEcontrol of resonant plants:Application to a flexible transmission[J].European J of Control,1995,1(2):113-121.
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[10]Tanaka K,Sugeno M.Stability analysis ans design of fuzzy control systems[J].Fuzzy Sets and Systems,1992,45(2):135-156.
[11]Huang Xia,Wang Zhen,Lia Yuxia,et al.Design of fuzzy state feedback controller for robust stabilization of uncertain fractional order chaotic systems[J].J of the Franklin Institute,2014,351(12):5480-5493.
[12]Tian Engang,Yue Dong,Zhang Yijun.Delay-dependent robust H∞control for T-S fuzzy system with interval time-varying delay[J].Fuzzy Sets and Systems,2009,160(12):1708-1719.
[13]Zheng Yongai,Nian Yibei,Wang Dejin.Controlling fractional order chaotic systems based on Takagi-Sugeno fuzzy model and adaptive adjustment mechanism[J].Physics Letters A,2010,375(2):125-129.
[14]Lin Chong,Chen Bing,Wang Qingguo.Static output feedback stabilization for fractional order systems in T-Sfuzzy models[J].Neurocomputing,2016,218:354-358.
[15]Wang Bin,Chen Diyi.Takagi-Sugeno fuzzy control for a wide class of fractional-order chaotic systems with uncertain parameters via linear matrix inequality[J].J of Vibration and Control,2016,22(10):2356-2369.
[16]卫一恒,朱敏,彭程,等.不确定分数阶时滞系统的鲁棒稳定性判定准则[J].控制与决策,2014,29(3):511-516.(Wei Y H,Zhu M,Peng C,et al.Robust stability criteria for uncertain fractional order systems with time delay[J]Control and Decision,2014,29(3):511-516.)
[17]Ji Y,Su L,Qiu J.Design of fuzzy output feedback stabilization for uncertain fractional order systems[J]Neurocomputing,2016(173):1683-1693.
[18]Li Yuting,Li Junmin.Stability analysis of fractional order systems based on T-S fuzzy model with the fractional orderα:0<α<1[J].Nonlinear Dynamics,201478(4):2909-2919.
[19]Zhang X,Chen Y.D-stability based LMI criteria of stability and stabilization for fractional order systems[D]Proc of the ASME 2015 Int Design Engineering Technical Conf&Computers and Information in Engineering Conf Boston,2015:1-6.
[20]Podlubny I.Fractional differertial equations[M].San Diego:Acdemic Press,1999:111-116.
[21]Xie L.Output feedback protect H∞control of systems with parameter uncertainty[J].Int J of Control,199663(4):741-750.
[22]Boyd S,Ghaoui L,Freon E,et al.Linear matrix inequalities in system and control theory[M]Philadelphia:SIAM,1994.
[23]Li Bingxin,Zhang Xuefeng.Observer-based robust control of(0<α<1)fractional-order linear uncertain control systems[J].IET Control Theory&Applications2016,10(14):1724-1731.
[2]汪纪锋.分数阶系统控制性能分析[M].北京:电子工业出版社,2010:1-8.(Wang J F.Control performance analysis for fractional order systems[M].Beijing:Publishing House of Electronics Industry,2010:1-8.)
[3]邱宁.时间分数阶延迟微分方程在流体力学中的应用[J].沈阳大学学报:自然科学版,2016,28(2):170-172.(Qiu N.Application of Time Fractional Delay Differential Equations in Fluid Dynamics[J].J of Shenyang University:Natural Science,2016,28(2):170-172.)
[4]杨平,董国威.互联电网AGC的分数阶PID控制[J].电力系统及其自动化学报,2013,25(3):124-129.(Yang P,Dong G W.Fractional Order PID Control for AGC of Interconnected Power System[J].Proc of the Chinese Society of Universities for Electric Power System and its Automation,2013,25(3):124-129.)
[5]Li Y,Chen Y Q,Podlubny I.Technical communique:Mittag-Leffler stability of fractional order nonlinear dynamic systems[J].Automatica,2009,45(8):1965-1969.
[6]朱呈祥,邹云.分数阶控制研究综述[J].控制与决策2009,24(2):161-169.(Zhu C X,Zou Y.Summary of research on fractionalorder control[J].Control and Decision,2009,24(2):161-169.)
[7]Podlubny I.Fractional-order systems and PIλDμcontrollers[J].IEEE Trans on Automatic Control,1999,44(1):208-214.
[8]Oustaloup A,Mathieu B,Lanusse P.The CRONEcontrol of resonant plants:Application to a flexible transmission[J].European J of Control,1995,1(2):113-121.
[9]Matignon D.Stability results for fractional differential equations with applications to control processing[J].Computational Engineering in Systems Applications,1996,2:963-968.
[10]Tanaka K,Sugeno M.Stability analysis ans design of fuzzy control systems[J].Fuzzy Sets and Systems,1992,45(2):135-156.
[11]Huang Xia,Wang Zhen,Lia Yuxia,et al.Design of fuzzy state feedback controller for robust stabilization of uncertain fractional order chaotic systems[J].J of the Franklin Institute,2014,351(12):5480-5493.
[12]Tian Engang,Yue Dong,Zhang Yijun.Delay-dependent robust H∞control for T-S fuzzy system with interval time-varying delay[J].Fuzzy Sets and Systems,2009,160(12):1708-1719.
[13]Zheng Yongai,Nian Yibei,Wang Dejin.Controlling fractional order chaotic systems based on Takagi-Sugeno fuzzy model and adaptive adjustment mechanism[J].Physics Letters A,2010,375(2):125-129.
[14]Lin Chong,Chen Bing,Wang Qingguo.Static output feedback stabilization for fractional order systems in T-Sfuzzy models[J].Neurocomputing,2016,218:354-358.
[15]Wang Bin,Chen Diyi.Takagi-Sugeno fuzzy control for a wide class of fractional-order chaotic systems with uncertain parameters via linear matrix inequality[J].J of Vibration and Control,2016,22(10):2356-2369.
[16]卫一恒,朱敏,彭程,等.不确定分数阶时滞系统的鲁棒稳定性判定准则[J].控制与决策,2014,29(3):511-516.(Wei Y H,Zhu M,Peng C,et al.Robust stability criteria for uncertain fractional order systems with time delay[J]Control and Decision,2014,29(3):511-516.)
[17]Ji Y,Su L,Qiu J.Design of fuzzy output feedback stabilization for uncertain fractional order systems[J]Neurocomputing,2016(173):1683-1693.
[18]Li Yuting,Li Junmin.Stability analysis of fractional order systems based on T-S fuzzy model with the fractional orderα:0<α<1[J].Nonlinear Dynamics,201478(4):2909-2919.
[19]Zhang X,Chen Y.D-stability based LMI criteria of stability and stabilization for fractional order systems[D]Proc of the ASME 2015 Int Design Engineering Technical Conf&Computers and Information in Engineering Conf Boston,2015:1-6.
[20]Podlubny I.Fractional differertial equations[M].San Diego:Acdemic Press,1999:111-116.
[21]Xie L.Output feedback protect H∞control of systems with parameter uncertainty[J].Int J of Control,199663(4):741-750.
[22]Boyd S,Ghaoui L,Freon E,et al.Linear matrix inequalities in system and control theory[M]Philadelphia:SIAM,1994.
[23]Li Bingxin,Zhang Xuefeng.Observer-based robust control of(0<α<1)fractional-order linear uncertain control systems[J].IET Control Theory&Applications2016,10(14):1724-1731.