第1期 张香燕，等：π型隶属函数的典型模糊控制器的解析结构 ·37 因此，0≤ne°-p≤，-≤r-g-1≤0，得 shaped membership functions[J].Science in China(Se- ≤2，·2 ries E),2000,43(3):263274 之+2ep2.21m92≤所以， [9]FAN X Z,ZHANG N Y,Li N.Structure analysis of typical fuzzy controllers with unequally spaced fuzzy sets h(p,=lim4(p,=0. for input and output variables [J ]Chinese Journal of E- 第3种情况： lectronics,2001,10(3):326-331. 同理，·分≤e'·p10.0≤r:g≤寸 [10]DU X Y,ZHANGN Y,YU N.Structure analysis and function evaluation of a kind of fuzzy PID controllers 因此.支92ep-)+21arg2≤受 [J].Chinese Journal of Electronics,2004,13(4):654- 659. 所以，(p,g=lim(p,g=0. [11]DU X Y,ZHANG N Y.Structure analysis of MISO 第4种情况， typical fuzzy controllers [J].Advances in System Sci- 同理支≤ep-1①，之≤r.g ence and Applications,2004,4(2):169-174. [12]李少远，田永青.模糊控制器的结构化分析及系统化设 1.因此12-2(ne-p-1)2-2(nr-g 计方法[U1.控制与决策，2001,16(4)：435438. 1)2.所以，(p,=lim攻(p,=0. LI Shaoyuan,TIAN Yongqing.Structure analysis and 综合以上4种情况可得(p,g=Iim(p, systematic design of typical fuzzy controllers [J ]Con- trol and Decision,2001.16(4):435-438. y=0. [13]HU B G,MANN G K I,GOSINE R G.New method- 参考文献： ology for analytical and optimal design of fuzzy PID con trollers [J ]IEEE Transactions on Fuzzy Systems, [1]王立新.模糊系统与模糊控制教程[M].王迎军，译.北 1999,7(5):521-539. 京：清华大学出版社，2003. [2]SIL ER W,YING H.Fuzzy control theory:the linear [14]MITA IM S,KOSKO B.The shape of fuzzy sets in a- case [J ]Fuzzy Sets and Systems,1989,33(2):275- daptive function approximation [J].IEEE Transactions 290. on Fuzzy Systems,2004,9(4):637-656. [3]YING H,SIL ER W,BUCKL EYJ J.Fuzzy control the- [15]李洪兴.模糊控制的插值机理[U].中国科学(E辑)， ory:a nonlinear case [J].Automatica,1990,26(3): 1998,28(3):259267 513-520. LI Hongxing.Interpolation mechanism of fuzzy control [4]YIN G H.The simplest fuzzy controllers using different [J ]Science in China(Series E),1998,28(3):259- methods are different nonlinear proportional-integral con 267 trollers with variable gains [J].Automatica,1993,29: [16]HASSINE R,KARRA Y F.Approximation properties 1579-1589. of piecewise parabolic functions fuzzy logic systems [J]. [5]YIN G H.A nonlinear fuzzy controller with linear control Fuzzy Sets and Systems,2006,157:501-515. rules is the sum of a global two-dimensional multilevel 作者简介： relay and a local nonlinear proportional-integral control- 张香燕，女，1982年生，博士研究生，主 ler[J].Automatica,1993,29(2):499505. 要研究方向为模糊系统与分层模糊系统，在 [6]YING H.Fuzzy control and modeling:analytical founda- 国内外期刊和会议上共发表文章5篇」 tions applications M].New York:IEEE Press, 2000. [7]LI N,ZHANG N Y,JIN K Y.Structure analysis of typical fuzzy controllers with unevenly distributed input 张乃尧，男，1946年生，教授，博士生导 membership functions[J].Journal of Tsinghua Universi- 师，主要研究方向为模糊控制理论及应用、 ty,2000,40(1):120123. 机器人足球、音乐情感识别等.已发表论文 [8]曾珂，张乃尧，徐文立.采用伪梯形隶属函数的一类 100余篇，其中被SCI检索10余篇，EI检索 模糊控制器的结构分析卫].中国科学(E),2000,30(4): 30余篇. 320-330. ZEN G Ke,ZHAN G Naiyao,XU Wenli.Structure anal- ysis of a class of fuzzy controllers using pseudo trapezoid 1994-2009 China Academic Journal Electronic Publishing House.All rights reserved http://www.cnki.net因此 ,0 ≤ne 3 - p ≤ 1 2 , - 1 2 ≤nr 3 - q - 1 ≤0 , 得 1 2 ≤1 + 2 ( ne 3 - p) 2 - 2 ( nr 3 - q - 1) 2 ≤ 3 2 . 所以 , Φ∞ L ( p , q) = limn→∞ Φ∞ L ( p , q) = 0. 第 3 种情况 : 同理 , - 1 2 ≤ne 3 - p - 1 ≤0 , 0 ≤nr 3 - q ≤ 1 2 , 因此 , 1 2 ≤1 - 2 ( ne 3 - p - 1) 2 + 2 ( nr 3 - q) 2 ≤ 3 2 . 所以 ,Φ∞ L ( p , q) = limn →∞ Φ∞ L ( p , q) = 0. 第 4 种情况 , 同理 , - 1 2 ≤ne 3 - p - 1 ≤0 , - 1 2 ≤nr 3 - q - 1 ≤0. 因此 1 ≤2 - 2 ( ne 3 - p - 1) 2 - 2 ( nr 3 - q - 1) 2 ≤2. 所以 ,Φ∞ L ( p , q) = limn→∞ Ф∞ L ( p , q) = 0. 综合以上 4 种情况可得Φ∞ L ( p , q) = limn→∞ Φ∞ L ( p , q) = 0. 参考文献 : [1 ]王立新. 模糊系统与模糊控制教程[ M ]. 王迎军 ,译. 北 京 : 清华大学出版社 , 2003. [2 ] SIL ER W , YIN G H. Fuzzy control theory : the linear case [J ]. Fuzzy Sets and Systems , 1989 , 33 (2) : 2752 290. [3 ] YIN G H , SIL ER W , BUCKL EYJ J. Fuzzy control the2 ory : a nonlinear case [J ]. Automatica , 1990 , 26 ( 3) : 5132520. [4 ] YIN G H. The simplest fuzzy controllers using different methods are different nonlinear proportional2integral con2 trollers with variable gains [J ]. Automatica , 1993 , 29 : 157921589. [ 5 ] YIN G H. A nonlinear fuzzy controller with linear control rules is the sum of a global two2dimensional multilevel relay and a local nonlinear proportional 2integral control2 ler [J ]. Automatica , 1993 , 29 (2) : 4992505. [6 ] YIN G H. Fuzzy control and modeling : analytical founda2 tions & applications [ M ]. New York : IEEE Press , 2000. [7 ]L I N , ZHAN G N Y, J IN K Y. Structure analysis of typical fuzzy controllers with unevenly distributed input membership functions[J ]. Journal of Tsinghua Universi2 ty , 2000 , 40 (1) :1202123. [8 ]曾 珂 , 张乃尧 , 徐文立. 采用伪梯形隶属函数的一类 模糊控制器的结构分析[J ]. 中国科学( E) , 2000 , 30 (4) : 3202330. ZEN G Ke , ZHAN G Naiyao , XU Wenli. Structure anal2 ysis of a class of fuzzy controllers using pseudo trapezoid shaped membership functions[J ]. Science in China ( Se2 ries E) , 2000 , 43 (3) : 2632274. [9 ] FAN X Z , ZHAN G N Y , Li N. Structure analysis of typical fuzzy controllers with unequally spaced fuzzy sets for input and output variables [J ]. Chinese Journal of E2 lectronics , 2001 , 10 (3) : 3262331. [ 10 ]DU X Y , ZHAN G N Y , YU N. Structure analysis and function evaluation of a kind of fuzzy PID controllers [J ]. Chinese Journal of Electronics , 2004 , 13 (4) : 6542 659. [11 ] DU X Y , ZHAN G N Y. Structure analysis of MISO typical fuzzy controllers [J ]. Advances in System Sci2 ence and Applications , 2004 , 4 (2) : 1692174. [ 12 ]李少远 , 田永青. 模糊控制器的结构化分析及系统化设 计方法[J ]. 控制与决策 , 2001 , 16 (4) : 4352438. L I Shaoyuan , TIAN Yongqing. Structure analysis and systematic design of typical fuzzy controllers [J ]. Con2 trol and Decision , 2001 , 16 (4) :4352438. [13 ] HU B G, MANN G K I , GOSIN E R G. New method2 ology for analytical and optimal design of fuzzy PID con2 trollers [J ]. IEEE Transactions on Fuzzy Systems , 1999 , 7 (5) : 5212539. [14 ]MITAIM S , KOSKO B. The shape of fuzzy sets in a2 daptive function approximation [J ]. IEEE Transactions on Fuzzy Systems , 2004 , 9 (4) : 6372656. [15 ]李洪兴. 模糊控制的插值机理[J ]. 中国科学 ( E 辑) , 1998 , 28 (3) : 2592267. L I Hongxing. Interpolation mechanism of fuzzy control [J ]. Science in China ( Series E) , 1998 , 28 (3) : 2592 267. [16 ] HASSIN E R , KARRA Y F. Approximation properties of piecewise parabolic functions fuzzy logic systems [J ]. Fuzzy Sets and Systems , 2006 ,157 : 5012515. 作者简介 : 张香燕 ,女 ,1982 年生 ,博士研究生 ,主 要研究方向为模糊系统与分层模糊系统 ,在 国内外期刊和会议上共发表文章 5 篇. 张乃尧 ,男 ,1946 年生 ,教授 ,博士生导 师 ,主要研究方向为模糊控制理论及应用、 机器人足球、音乐情感识别等. 已发表论文 100 余篇 ,其中被 SCI 检索 10 余篇 ,EI 检索 30 余篇. 第 1 期 张香燕 ,等 :π型隶属函数的典型模糊控制器的解析结构 ·37 · © 1994-2009 China Academic Journal Electronic Publishing House. All rights reserved. http://www.cnki.net