基于模糊变换的模糊系统及HX方程
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摘要
本文研究基于模糊变换的模糊系统的构造方法和模糊推理建模法问题。首先,给出了利用单入-单出模糊系统和模糊变换构造双输入-单输出模糊系统的方法,指出这种模糊系统具有泛逼近性,并给出了该模糊系统具有泛逼近性的充分条件。其次,将该模糊系统应用到模糊推理建模法中,得到了一种新的HX方程,泛逼近性定理说明:该HX方程对原系统具有很好的泛逼近性。最后,将得到的新的HX方程应用到自治Lienard系统中,得到了不含一阶导数项的简化HX方程。简化的HX方程将原先逐片求解(m-1)(n-1)个方程,简化为逐片求解m-1个方程,从而降低了计算复杂度。仿真实验说明了新HX方程的有效性。
In this paper, the method to construct a fuzzy system and fuzzy inference modeling method are investigated. Firstly, a method to construct the dual input-single output fuzzy system by using a single input-single output fuzzy system and the fuzzy transformation is estabilished. It is pointed that the contructed fuzzy system is an universal approximator and a sufficient condition of the fuzzy system as the universal approximator is proposed. Secondly, we have obtained a new HX differential equation by applying the proposed fuzzy system to fuzzy inference modeling method. The universal approximation theorem of the proposed fuzzy system has shown that the new HX differential equation is of universal approximation to the original system. Finally, by applying the new HX differential equation to automomous Lienard system, we have obtained a simplified HX differential equation which has no first derivation part and change solving(m-1)(n-1) differential equations into solving m-1 differential equations. The simulation has shown the efficiency of the simplified HX differential equation.
In this paper, the method to construct a fuzzy system and fuzzy inference modeling method are investigated. Firstly, a method to construct the dual input-single output fuzzy system by using a single input-single output fuzzy system and the fuzzy transformation is estabilished. It is pointed that the contructed fuzzy system is an universal approximator and a sufficient condition of the fuzzy system as the universal approximator is proposed. Secondly, we have obtained a new HX differential equation by applying the proposed fuzzy system to fuzzy inference modeling method. The universal approximation theorem of the proposed fuzzy system has shown that the new HX differential equation is of universal approximation to the original system. Finally, by applying the new HX differential equation to automomous Lienard system, we have obtained a simplified HX differential equation which has no first derivation part and change solving(m-1)(n-1) differential equations into solving m-1 differential equations. The simulation has shown the efficiency of the simplified HX differential equation.
引文
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[16] Yuan X H,Liu Z L,Lee E S.The center-of-gravity fuzzy systems based on the normal fuzzy implications[J].Computer and Mathematics with Applications,2011,61(9):2879～2898.
[17] 袁学海,李洪兴,孙凯彪.基于参数单点模糊化方法的模糊系统及其逼近能力[J].电子学报,2011,39(10):2372～2377.
[18] 袁学海,李洪兴,杨雪.基于模糊变换的模糊系统和模糊推理建模法[J].电子学报,2013,41(4):674～680.
[2] Li H Y,et al.Adaptive sliding mode control for Takagi-Sugeno fuzzy systems and its applications[J].IEEE Transactions on Fuzzy Systems,2018.26(2):531～542.
[3] Feng Z H,Zheng W X,Wu L G.Reachable set estimation of T-S fuzzy systems with time-varying delay[J].IEEE Transactions on Fuzzy Systems,2017,25(4):878～891.
[4] Wang L X.A course in fuzzy systems and control[M].Prentice-Hall,Inc.,1997.
[5] Li Y M,Shi Z K,Li Z H.Approximation theory of fuzzy systems based upon genuine many-valued implication-MISO case[J].Fuzzy Sets and Systems,2002,130:159～174.
[6] Wang L X,Mendel J M.Fuzzy basis functions,universal approximation,and orthogonal least sequares learning[J].IEEE Trans.On Neural Networks,1992,3(5):807～814.
[7] Ying H.Sufficient conditions on general fuzzy systems as function approximators[J].Automatica,1994,30(3):521～525.
[8] Zeng K,Zhang N Y,Xu W L.A comparative study on sufficient conditions for Takagi-Sugeno fuzzy systems as universal approximators[J].IEEE Transactions on Fuzzy Systems,2000,8(6):773～780.
[9] Zeng X J,Singh M G.Approximation theory of fuzzy systems-MIMO case[J].IEEE Transactions on Fuzzy Systems,1995,3(2):219～235.
[10] 李洪兴,王加银,苗志宏.模糊控制系统的建模[J].中国科学,A辑,2002;32(9):772～781.
[11] 李洪兴,宋文彦,袁学海,李宇成.基于Fuzzy推理的时变系统建模[J].系统科学与数学,2009,29(8):1109～1128.
[12] 赵纬经,李洪兴.自治Lienard系统的简化HX方法[J].模糊系统与数学,2013,27(1):96～103.
[13] Zadeh L A.The concept of a linguistic variable and its application to approximate reasoning I[J].Information Sciences,1975,8(3):199～251.
[14] 王国俊.模糊推理的全蕴涵三I算法[J].中国科学,E辑,1999,29(1):32～42.
[15] Zhang C L,Yuan X H,Wang Y T.The reductivity of fuzzy inference.Cao B Y,Xie X J.Fuzzy engineering and operators Research.AISC 147:155～163(Berlin/Heidelberg:Springer-Verlag,2012).
[16] Yuan X H,Liu Z L,Lee E S.The center-of-gravity fuzzy systems based on the normal fuzzy implications[J].Computer and Mathematics with Applications,2011,61(9):2879～2898.
[17] 袁学海,李洪兴,孙凯彪.基于参数单点模糊化方法的模糊系统及其逼近能力[J].电子学报,2011,39(10):2372～2377.
[18] 袁学海,李洪兴,杨雪.基于模糊变换的模糊系统和模糊推理建模法[J].电子学报,2013,41(4):674～680.