基于五I算法的模糊系统的响应能力
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摘要
为了在实际中更好地应用基于五I算法的模糊系统,研究此类模糊系统的响应能力和泛逼近性意义非凡.本文首先给出基于S蕴涵和Zadeh蕴涵的五I算法的模糊推理表达式.本文指出对于S蕴涵和R蕴涵,采用单点模糊化、重心解模糊法、基于五I算法的单输入单输出(SISO)模糊系统只有阶跃输出条件而无函数逼近的泛性;而对Zadeh蕴涵,此类模糊系统不一定具有此性质.此外,在基于五I算法的多规则SISO模糊系统中,一定条件下模糊推理与聚合的次序无关.最后提供一个具体的例子来展示基于五I算法的模糊系统的响应能力和泛逼性.
In order to apply the fuzzy system based on the QIP algorithm well in practice,research on responsiveness and general approximation of such fuzzy systems is of great significance. In this paper, the fuzzy inference expression of 5-I algorithm based on S implication and Zadeh implication is proposed. Furthermore, by using single-point fuzzification and gravity center defuzzification method, we find that for S implication and R implication, a single-input single-output(SISO) fuzzy system based on QIP algorithm has only step output conditions and can not approximates a function well. The preceding property does not necessarily exist in such fuzzy systems with Zadeh implication. In addition, in the multi-rule SISO fuzzy system based on QIP algorithm, fuzzy reasoning is independent of the order of aggregation under certain conditions. Finally, a concrete example is provided to demonstrate responsiveness and universal approximation of fuzzy systems based on the five-I algorithm.
In order to apply the fuzzy system based on the QIP algorithm well in practice,research on responsiveness and general approximation of such fuzzy systems is of great significance. In this paper, the fuzzy inference expression of 5-I algorithm based on S implication and Zadeh implication is proposed. Furthermore, by using single-point fuzzification and gravity center defuzzification method, we find that for S implication and R implication, a single-input single-output(SISO) fuzzy system based on QIP algorithm has only step output conditions and can not approximates a function well. The preceding property does not necessarily exist in such fuzzy systems with Zadeh implication. In addition, in the multi-rule SISO fuzzy system based on QIP algorithm, fuzzy reasoning is independent of the order of aggregation under certain conditions. Finally, a concrete example is provided to demonstrate responsiveness and universal approximation of fuzzy systems based on the five-I algorithm.
引文
[1]Zadeh L A.Fuzzy sets[J].Information and Control,1965,8(3):338-353
[2]Zadeh L A.Outline of a new approach to the analysis of complex systems and decision processes[J].IEEETransaction on Systems,Man and Cybernetics,1973,3(1):28-44
[3]Lee C C.Fuzzy logic in control system:fuzzy logic control(I,II)[J].IEEE Transaction on Systems,Man and Cybernetics,1990,20(2):404-418
[4]Ross S M.Stochastic Processes[M].New York:John Wily&Sons,1983
[5]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):43-53Wang G J.Fully implication triple I method for fuzzy reasoning[J].Science in China(Series E),1999,29(1):43-53
[6]江欢.模糊推理中CRI算法与全蕴涵三I算法的等价性研究[D].重庆:西南大学,2009Jiang H.The equivalence between CRI and the full implication triple I method in fuzzy reasoning[D].Chongqing:Southwestern University,2009
[7]Dubios D,Prade H.Fuzzy sets in approximate reasoning(I)(II)[J].Fuzzy Sets and Systems,1991,40(1):143-244
[8]Zhou B K,Xu C Q,Li S J.The quintuple implication principle of fuzzy reasoning[J].Information Sciences,2015,297(3):202-215
[9]李洪兴.模糊控制的插值机理[J].中国科学(E辑),1998,28(3):259-267Li H X.Interpolation mechanism of fuzzy control[J].Science in China(Series E),1998,28(3):259-267
[10]李洪兴.常见模糊蕴涵算子的模糊系统及其响应函数[J].控制理论与应用,2005,22(3):341-347Li H X.Fuzzy systems and their response functions based on commonly used fuzzy implication operators[J].Control Theory and Applications,2005,22(3):341-347
[11]李洪兴,尤飞,彭家寅.基于某些模糊蕴涵算子的模糊控制器及其响应函数[J].自然科学进展,2003,13(10):1073-1077Li H X,You F,Peng J Y.Fuzzy controllers based on some fuzzy implication operators and their response functions[J].Progress in Natural Science,2003,13(10):1073-1077
[12]尤飞.模糊蕴涵算子及其构造的模糊系统的响应性能分析[D].北京:北京师范大学,2005You F.Fuzzy implication operators and applicability on fuzzy control system[D].Beijing:Beijing Normal University,2005
[13]侯健,李洪兴,王加银.两类模糊系统具有插值性的充要条件[J].控制理论与应用,2006,23(2):287-291Hou J,Li H X,Wang J Y.Sufficient and necessary conditions for fuzzy systems possessing interpolation property[J].Control Theory and Applications,2006,23(2):287-291
[14]李洪兴,彭家寅,王加银.基于三I算法的模糊系统及其响应性能[J].系统科学与数学,2005,25(5):578-590Li H X,Peng J Y,Wang J Y.Fuzzy systems based on triple I algorithm and their response performance[J].Journal of Systems Science and Mathematical Sciences,2005,25(5):578-590
[15]侯健,尤飞,李洪兴.由三I算法构造的一些模糊控制器及其响应能力[J].自然科学进展,2005,15(1):29-37Hou J,You F,Li H X.Some fuzzy controllers constructed by triple I algorithm and their response ability[J].Progress in Natural Science,2005,15(1):29-37
[16]潘海玉,裴道武,陈汉香.基于三I算法的模糊系统的响应能力[J].控制理论与应用,2011,28(1):24-30Pan H Y,Pei D W,Chen H X.Response ability of fuzzy systems based on triple I method[J].Control Theory and Applications,2011,28(1):24-30
[17]Klement E P,Mesiar R,Pap E S.Triangular Norms[M].Dordrecht:Kluwer Academic Publishers,2000:13-29
[18]Baczy′nski M.Residual implication revisited.Note on the Smets-Magrez theorem[J].Fuzzy Sets and Systems,2004,145(2):267-277
[19]李得超,史忠科,李永明.布尔型模糊系统逼近的充分条件[J].控制与决策,2007,22(12):1400-1402Li D C,Shi Z K,Li Y M.Sufficient conditions for Boolean fuzzy systems as universal approximators[J].Control and Decision,2007,22(12):1400-1402
[2]Zadeh L A.Outline of a new approach to the analysis of complex systems and decision processes[J].IEEETransaction on Systems,Man and Cybernetics,1973,3(1):28-44
[3]Lee C C.Fuzzy logic in control system:fuzzy logic control(I,II)[J].IEEE Transaction on Systems,Man and Cybernetics,1990,20(2):404-418
[4]Ross S M.Stochastic Processes[M].New York:John Wily&Sons,1983
[5]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):43-53Wang G J.Fully implication triple I method for fuzzy reasoning[J].Science in China(Series E),1999,29(1):43-53
[6]江欢.模糊推理中CRI算法与全蕴涵三I算法的等价性研究[D].重庆:西南大学,2009Jiang H.The equivalence between CRI and the full implication triple I method in fuzzy reasoning[D].Chongqing:Southwestern University,2009
[7]Dubios D,Prade H.Fuzzy sets in approximate reasoning(I)(II)[J].Fuzzy Sets and Systems,1991,40(1):143-244
[8]Zhou B K,Xu C Q,Li S J.The quintuple implication principle of fuzzy reasoning[J].Information Sciences,2015,297(3):202-215
[9]李洪兴.模糊控制的插值机理[J].中国科学(E辑),1998,28(3):259-267Li H X.Interpolation mechanism of fuzzy control[J].Science in China(Series E),1998,28(3):259-267
[10]李洪兴.常见模糊蕴涵算子的模糊系统及其响应函数[J].控制理论与应用,2005,22(3):341-347Li H X.Fuzzy systems and their response functions based on commonly used fuzzy implication operators[J].Control Theory and Applications,2005,22(3):341-347
[11]李洪兴,尤飞,彭家寅.基于某些模糊蕴涵算子的模糊控制器及其响应函数[J].自然科学进展,2003,13(10):1073-1077Li H X,You F,Peng J Y.Fuzzy controllers based on some fuzzy implication operators and their response functions[J].Progress in Natural Science,2003,13(10):1073-1077
[12]尤飞.模糊蕴涵算子及其构造的模糊系统的响应性能分析[D].北京:北京师范大学,2005You F.Fuzzy implication operators and applicability on fuzzy control system[D].Beijing:Beijing Normal University,2005
[13]侯健,李洪兴,王加银.两类模糊系统具有插值性的充要条件[J].控制理论与应用,2006,23(2):287-291Hou J,Li H X,Wang J Y.Sufficient and necessary conditions for fuzzy systems possessing interpolation property[J].Control Theory and Applications,2006,23(2):287-291
[14]李洪兴,彭家寅,王加银.基于三I算法的模糊系统及其响应性能[J].系统科学与数学,2005,25(5):578-590Li H X,Peng J Y,Wang J Y.Fuzzy systems based on triple I algorithm and their response performance[J].Journal of Systems Science and Mathematical Sciences,2005,25(5):578-590
[15]侯健,尤飞,李洪兴.由三I算法构造的一些模糊控制器及其响应能力[J].自然科学进展,2005,15(1):29-37Hou J,You F,Li H X.Some fuzzy controllers constructed by triple I algorithm and their response ability[J].Progress in Natural Science,2005,15(1):29-37
[16]潘海玉,裴道武,陈汉香.基于三I算法的模糊系统的响应能力[J].控制理论与应用,2011,28(1):24-30Pan H Y,Pei D W,Chen H X.Response ability of fuzzy systems based on triple I method[J].Control Theory and Applications,2011,28(1):24-30
[17]Klement E P,Mesiar R,Pap E S.Triangular Norms[M].Dordrecht:Kluwer Academic Publishers,2000:13-29
[18]Baczy′nski M.Residual implication revisited.Note on the Smets-Magrez theorem[J].Fuzzy Sets and Systems,2004,145(2):267-277
[19]李得超,史忠科,李永明.布尔型模糊系统逼近的充分条件[J].控制与决策,2007,22(12):1400-1402Li D C,Shi Z K,Li Y M.Sufficient conditions for Boolean fuzzy systems as universal approximators[J].Control and Decision,2007,22(12):1400-1402