FUZZY CONTROL VIA IMPERFECT PREMISE MATCHING APPROACH FOR DISCRETE TAKAGI-SUGENO FUZZY SYSTEMS WITH MULTIPLICATIVE NOISES

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1 Jounal of Mane Scence echnology Vol. 4 No.5 pp (6) 949 DOI:.69/JMS FUZZY CONROL VIA IMPERFEC PREMISE MACHING APPROACH FOR DISCREE AKAGI-SUGENO FUZZY SYSEMS WIH MULIPLICAIVE NOISES Wen-Je Chang Che-Pn Kuo Cheung-Cheh Ku Key wods: akag-sugeno fuzzy models multplcatve noses Dscete Jensen Inequalty Lnea Matx Inequalty. ABSRAC In ths pape a fuzzy contolle desgn poblem s consdeed fo the dscete-tme stochastc akag-sugeno fuzzy systems. he man pupose of ths pape s to povde a less consevatve fuzzy contolle desgn method fo the dscetetme nonlnea stochastc systems whch ae modeled by the -S fuzzy models. Based on the concept of mpefect pemse matchng the fuzzy contolle s desgned wthout the lmtaton mposed fom shang the same membeshp functons of fuzzy model. In othe wods the mpefect pemse matchng technque povdes a genealzaton appoach n desgnng fuzzy contolle. Moeove the flexblty obustness of poposed fuzzy contolle desgn appoach can be futhe enhanced than the well-known paallel dstbuted compensaton method. Moeove based on the Lyapunov functon the stablty condtons ae efeed to lnea matx nequalty poblems fo applyng the convex optmal algothm. Fnally the nonlnea stochastc tuck-tale system s povded to show the utlty of the poposed fuzzy contolle desgn method. I. INRODUCION In ecent yeas thee s an nceasng nteest n the fuzzy contol systems that have many ssues to be efeed (Guea Vemeen ; anaka Wang ; Chang Sun 3; Chang Shng 4; Chang Chang 6; Jeon Jeong 6; Chang et al. 3; Deng u 5; L et al. 5a 5b; Zhang et al. 5; anaka et al. 6). hese woks wee focused on the stablty stablzaton of closed-loop fuzzy systems. Especally these appoaches ae Pape submtted /3/5; evsed /8/6; accepted 4//6. Autho fo coespondence: Wen-Je Chang (wchang@mal.ntou.edu.tw) Depatment of Mane Engneeng Natonal awan Ocean Unvesty Keelung awan R.O.C. developed based on the -S fuzzy models. he akag-sugeno (-S) fuzzy models ae descbed by a set of fuzzy IFHEN ules wth fuzzy sets n the antecedents dynamc systems n the consequent whch can locally epesent lnea nputoutput elatons of nonlnea systems. he Paallel Dstbuted Compensaton (PDC) concept (Guea Vemeen ) povded a useful famewok to desgn the fuzzy contolle fo the -S fuzzy models. he goal of PDC concept s to desgn lnea feedback gans fo each local lnea systems let the oveall contol nput be blended by these lnea feedback gans. hese local lnea systems ae consdeed as subsystems the aggegaton of subsystems epesents the oveall nonlnea systems. In the lteatue (anaka et al. 998; Km Lee ; exea et al. 3; E et al. 6; Fang et al. 6; Zhou 7; L et al. 8; Chang et al. 9; Lam 9; Chang et al. a b; Lu Lam 5; Sun 6) the fuzzy contolle desgn has been developed based on the concept of PDC fo -S fuzzy models. he PDC technque s ndeed a useful tool fo fuzzy contolle desgn of -S fuzzy model. Howeve unde the PDC desgn technque the fuzzy contolle shaes the same membeshp functons of the -S fuzzy model. heefoe when the pemse membeshp functons of the -S fuzzy model ae complcated t nceases the stuctual complexty of the fuzzy contolle leadng to a hghe mplementaton cost. Some appoaches to mpoved fuzzy contolle desgn va elaxed stablty condtons wee epoted n (anaka et al. 998; exea et al. 3; Fang et al. 6; Chang et al. 9). hese stablty condtons can be cast n tems of Lnea Matx Inequalty (LMI) that can be solved numecally usng some convex pogammng technques (Boyd et al. 994). Howeve the elaxed stablty condtons n (anaka et al. 998; exea et al. 3; Fang et al. 6; Chang et al. 9) ae also developed based on the PDC concept. Wthout loss of genealty when a lage numbe of subsystems ae nvolved the PDC-based fuzzy contolle s consevatve. In ode to solve ths poblem the Non-PDC fuzzy contolle desgn (Zhou et al. 7; L et al. 8; Lam 9; Chang et al. a) has focused on developng the elaxed stablty cond-

2 95 Jounal of Mane Scence echnology Vol. 4 No.5 (6) tons based on the nonquadatc Lyapunov functons. In (Zhou et al. 7; L et al. 8; Lam 9; Chang et al. a) the applcaton of the nonquadatc Lyapunov functons nceases the complexty fo the fuzzy contolle devatons. Dffeent fom the appoaches of (Zhou et al. 7; L et al. 8; Lam 9; Chang et al. a) ths pape ntends to utlze the concept of Impefect Pemse Matchng (IPM) to fnd elaxed fuzzy contolles whose membeshp functons ae dffeent fom that of the -S fuzzy models. he goal of IPM technque (Lam Naman 9) s to desgn dffeent membeshp functons of fuzzy contolles fo the -S fuzzy models. Unde the IPM concept the -S fuzzy model can be uncetan n value. As a esult the authos of (Lam Naman 9) poposed fuzzy contolle dsplays the obustness popety to hle paamete uncetantes. In ode to keep the desgn flexblty obustness popety of the fuzzy contolle the fuzzy contolle does not eque shang the same pemses as those of the fuzzy model. heefoe usng the IPM concept one can desgn a fuzzy contolle wth smple membeshp functons to hle the fuzzy model wth complex membeshp functons. Howeve the dffeent pemses of nfluence can be unstable. In such a case abtay matces ae onng to elax the stablty condtons. Consequently the less consevatve stablty condtons fo detemnstc -S fuzzy systems wee poposed vefed n (Lam Naman 9). Howeve the stochastc systems often appea n pactcal ndustes. Moeove the stablty stablzaton poblems of stochastc systems ae moe dffcult than that of detemnstc systems. hus t s wothy to deve the elaxed stablty condtons va the concept of IPM to fnd fuzzy contolles fo stochastc -S fuzzy systems. In ou pevous wok (Chang et al. 5) the stablty analyss contolle synthess of contnuous-tme stochastc nonlnea systems wee dscussed nvestgated va applyng IPM concept. Based on ou expeences n (Chang et al. 5) a elaxed fuzzy contolle desgn method s developed fo dscete-tme stochastc nonlnea systems n ths pape. Recently the fuzzy contol poblem fo the nonlnea stochastc systems has been nvestgated va the -S fuzzy models n (Chang et al. 9; Chang et al. a; Ku et al. ; Chang et al. ). In these papes the consequent pat of -S fuzzy model s descbed as stochastc dffeental equatons that ae stuctued by detemnstc dffeental equatons wth multplcatve nose tems (El et al. 5). he PDC concept was employed to desgn the fuzzy contolles n (Chang et al. 9; Chang et al. a; Ku et al. ; Chang et al. ). he pupose of ths pape s to study the fuzzy contolle desgn fo the dscete-tme stochastc nonlnea systems. Based on the stochastc dffeental equatons (El et al. 5) the -S fuzzy model wth multplcatve nose (Chang et al. 9; Chang et al. a; Ku et al. ; Chang et al. ) s bult to epesent the dscete-tme stochastc nonlnea systems. Besdes the poposed fuzzy contolle desgn may not shae the same pemse membeshp functon wth the -S fuzzy model by usng the IPM concept. Fo the fuzzy contolle desgn va IPM concept abtay slack matces ae ntoduced to compensate unstable elements. Hence the IMP-based fuzzy contolle can be enhanced to ncease the flexblty obustness popety. Fnally the IMP-based fuzzy contolle desgn poblem can be effcently solved by usng convex pogammng technques to solve the elaxed LMI stablty condtons. o valdate the effectveness of the poposed fuzzy contolle desgn method a numecal example fo the contol of dscete-tme nonlnea tuck-tale system s ncluded n ths pape. II. SYSEM DESCRIPIONS AND PROBLEM SAEMENS In ths pape a dscete-tme stochastc -S fuzzy model s used to epesent a nonlnea stochastc system. he stochastc -S fuzzy model s descbed by fuzzy IF-HEN ules whch epesent local lnea nput-output elatons of the nonlnea systems. Let be the numbe of fuzzy ules descbng the dscete-tme stochastc -S fuzzy model. he -th ule of the stochastc -S fuzzy model has the followng fom. Rule : IF z k s s M HEN M z k s M z k A B A B x k x k u k x k u k w k fo () whee z k z k ae the pemse nn vaables s the numbe of pemse vaables A nm B M s the fuzzy set nn A nm B ae known constant m u k s the contol nput vecto (Kaatzas Sheve 99) whch satsfes E w k n matces xk s the state vecto wk s a scala Bownan moton E w k whee E denote the stad expected opeato fo. he fnal plant of the fuzzy model () can be efeed as follows. x k h z k A x k B u k A x k B u k w k whee zk z k z k f zk f zk h z k f z k M z k h z k (3) ()

3 W.-J. Chang et al.: Fuzzy Contol va Impefect Pemse Matchng Appoach 95 whee M ( z ( k)) s the gade of membeshp functon of the z (k) n M. It s assumed that f zk fo all h zk fo all f z k = z k. heefoe h z k z k. Based on the IPM concept one can desgn a fuzzy contolle that the membeshp functon s dffeent fom that of the fuzzy plant. hus the fuzzy contolle can be descbed as follows. Rule : IF z (k) s N z (k) s N HEN u k N z k s G x k fo (4) whee N s the fuzzy set z (k) z (k) ae the pemse vaables s the numbe of pemse vaables G mn s the feedback gans of ule. he nfeed output of the fuzzy contolle (4) s gven by. whee zk z k z k u k m z k G x k (5) zk zk g m z k g z k N z k m z k g whee N ( z ( k)) s the gade of membeshp functon of the z (k) n N. It s assumed that g zk g zk fo all z(k). heefoe m zk fo all (6) = m z k z k. Substtutng (5) nto () one can obtan coespondng closed-loop system as follows. A BG x k h z k m z k x k xkwk In the followng statements denoted as h A B G (7) h zk m z k ae m fo bevty. In (7) the fuzzy model fuzzy contolle do not shae the same membeshp functons that lead to IPM. he IPM concept can be efeed to (Lam Naman 9). Based on the IPM concept the pupose of ths pape s to deve elaxed stablty condtons fo fndng the fuzzy contolle (5) that can stablze the closed-loop system (7). In next secton the Lyapunov theoy s used to deve the elaxed stablty condtons. Besdes the LMI technque s employed to fnd the solutons of these elaxed stablty condtons such that the fuzzy contolle (5) can be obtaned. III. IPM-BASED FUZZY CONROLLER DESIGN FOR DISCREE SOCHASIC -S FUZZY SYSEMS Accodng to the closed-loop fuzzy system (7) the IPMbased fuzzy contolle s developed n ths secton by devng the elaxed stablty condtons. One can fnd that these stablty condtons can be solved by usng the convex pogammng technque because they can be tansfeed nto the LMI poblems. In the followng theoem the stablty condtons fo the exstence of IPM-based fuzzy contolle ae fst deved. By tansfeng the stablty condtons of heoem the LMI stablty condtons wll be ntoduced n the heoem. heoem he closed-loop system (7) s asymptotcally stable f the membeshp functons of the fuzzy model fuzzy contolle satsfy m h fo all zk whee thee exst negatve defnte matces R R S S postve defnte matx P abtay matces Λ V feedback gans G such that the followng condtons ae satsfed. Ω (8) Ψ R (9) K R R () Θ S Y S S R R R R R R R R R R S S S S S S S S S S () () (3)

4 95 Jounal of Mane Scence echnology Vol. 4 No.5 (6) whee Λ Λ V V = ae abtay matces Ω Φ Λ Ψ Φ Λ V Θ Λ V K Φ Λ V Φ Λ V Y V Λ V Φ A BG PA BG A BG P A B G P. Poof : Choose the Lyapunov functon as Vxk x kp xk. Fom (Kaatzas Sheve 99) one has the popetes as Ewk E w k. heefoe one can obtan the dffeence of chosen Lyapunov functon wth stad expectaton opeato such as P P E V x k E x k x k x k x k whee E hm x k A BG PA B G A BG PA BG P xk E x k Γ x k (4) Γ hm Φ (5) Note that (5) mples the fuzzy model fuzzy contolle have dffeent membeshp functons that lead the concept of IPM. Hee let us consde the followng equalty h m hh Λ V V h h m h m h h m h m Λ V V nn nn whee Λ Λ V V = ae abtay matces. he abtay matces Λ V ae ntoduced to compensate unstable elements. hese tems ae ntoduced to (5) to allevate the consevatveness < < = ae desgned such that m h fo all zk. o obtan m h t s necessay to on abtay matces Λ V. heefoe the Eq. (5) can be epesented as Γ h m h h Φ whee h h m h h Λ V V Φ Λ h h h m h Φ Λ V hh Λ V hh h m h Φ Λ hh Φ Λ Φ Λ V Φ Λ V hh Λ V hh Λ V Λ V 3 hh Π Π Π V Π h m h Ω Π h Ψ hh K (6) Π 3 h Θ hh Y. Fom (6) f the condton m h the condton (8) ae satsfed then one has Π (7) It s evdent that f the condtons (9) () () () ae held one can obtan Π hrh (8) Π 3 hsh (9) whee h h h hn. Accodng to the nequaltes (7) (8) (9) one can obtan the followng nequalty fom (6). Γ hrh hsh ()

5 W.-J. Chang et al.: Fuzzy Contol va Impefect Pemse Matchng Appoach 953 Snce R S one can obtan Γ fom (). Γ the nequalty Due to E V x k can be obtaned fom (4). heefoe f the condtons of heoem hold then one has EV xk the closed-loop system (7) s asymptotcally stable heoem pesents the suffcent condtons that ae of blnea matx nequaltes. In the next theoem applyng the mathematc tansfom technque the blnea matx nequaltes can be tansfomed nto the LMI poblems. hough usng the MALAB LMI-toolbox to solve the LMI poblems one can obtan the stable fuzzy contolles fo the closed-loop fuzzy systems. heoem he closed-loop system (7) s asymptotcally stable f the membeshp functons of the fuzzy model fuzzy contolle satsfy m h fo all zk whee thee exst negatve defnte matces R R S S postve defnte matx P abtay matces Λ V feedback gans G N such that the followng condtons ae satsfed. whee Η Ξ () Η D () K U (3) Θ S (4) Y S S (5) R R 3 n 3 n R 5n5n D U Ξ Λ ABN ABN * * * Λ V A B N A B N * * * # K Χ A BN A BN * * * * * * * * * A BN A BN * * denotes the tansposed elements o matces fo symmetc poston. Besdes R R S S ae defned n (3) P N G P Χ Λ V Λ V Θ Λ V Y Λ V Λ V. Moeove Λ P Λ P V P V P ae abtay matces. Poof : By usng the Schu complement (Boyd et al. 994) one can get that Ω PΞ P Ψ PΗP K PK P Θ PΘ P Y PY P espectvely. Obvously f ()-(5) ae held then (8)-() can also be satsfed. heefoe f the condtons of heoem ae held then the condtons of heoem can also be satsfed. Accodng to heoem f the condtons ()-(5) ae satsfed then the closed-loop system (7) s asymptotcally stable heoem tansfoms the condtons of heoem nto LMI foms. Based on the stablty condtons of heoem one can solve these condtons va MALAB LMI-toolbox fo obtanng the feasble solutons of fuzzy contolles (5). Howeve the guaanteeng of condton as m h s an nteestng poblem. o clafy the poblem a emak s added as follows: Remak Befoe dscussng stablty stablzaton poblem of the closed-loop model (7) the membeshp functons of fuzzy contolle fuzzy model ae known. Accodng to the value of satsfyng a small enough value can be thus chosen such that m h s held.. # #

6 954 Jounal of Mane Scence echnology Vol. 4 No.5 (6) In the followng secton a numecal example s povded to demonstate the applcaton effectveness of the poposed fuzzy contol method. IV. A NUMERICAL EXAMPLE In ths secton usng a numecal example the usefulness of the appoach poposed n the pevous sectons s llustated. Hee a dscete-tme nonlnea tuck-tale system s consdeed. he dynamc equaton of the dscete-tme nonlnea tuck-tale system has been descbed n (anaka Sano w k s 994). In ths example the multplcatve nose consdeed fo descbng the pactcal stochastc behavos. Wthout loss of genealty t s assumed that x k uk ae always small the hozontal poston moton k x4 can be neglected. heefoe one can epesent smplfy the ognal model as follows. t t x k x k u k L L L (6a) t x k x k x k (6b) t x3 k t x k x k x3 k L sn (6c) In ths example the followng paamete values ae used fo smulatons L =.8m L =5.5m t =m =-.msec whee L s the length of tuck L s the length of tale t s the samplng tme s the constant speed of backng up. Fo x k 9 9 coespond to two ackknfe postons. he ackknfe phenomenon cannot be avoded f the steeng s not contolled dung the backwad movement. o succeed n the backng contol we need to avod the ackknfe phenomenon. he contollng pupose of ths pape s to back up a tuck tale along a staght lne x3 wthout fowad movements as shown n (anaka Sano 994) that s x k x k x3 k. t Assume that the pemse vaable x k x k s L opeated between. hen a -S fuzzy model whch appoxmately epesents the dynamc of the tuck-tale stochastc system (6) can be obtaned as follows -S Fuzzy Model: π Rule Rule Rule Fg.. Membeshp functons of fuzzy model. t x k x k s about HEN L Rule : IF A B A B x k x k u k x k u k w k (7a) t x k x k s about o HEN L Rule : IF A B A B x k x k u k x k u k w k (7b) whee t t L L t A t A L L t t t t L L t L B B fo a =.. A a. B. π a. Fg. shows the membeshp functons of the above -S fuzzy model. Based on the IPM technque the poposed fuzzy contolle s desgned to possess dffeent membeshp functons fom the fuzzy model (7). In ths example t s assumed that the t pemse vaable x k x k of the fuzzy contolle L s constaned wthn the coespondng mem-

7 W.-J. Chang et al.: Fuzzy Contol va Impefect Pemse Matchng Appoach 955 π Rule Rule Rule Fg.. Membeshp functons of fuzzy contolle. π Rule Rule Rule Fg. 3. Membeshp functons. beshp functons ae gven n Fg.. hen the IPM-based fuzzy contolle has the followng fom. Fuzzy Contolle: t x k x k s about HEN L Rule : IF u k Rule : IF G x k (8a) t x k x k s about o HEN L u k G x k (8b) Fo detemnng the values of to satsfy the condton m h the membeshp functons of the fuzzy model fuzzy contolle ae stated n Fg. 3. Fom Fg. 3 t s easy to detemne = =.8 to satsfy m h. By applyng the poposed fuzzy contolle desgn technque one can fnd the feasble solutons by solvng the stablty condtons ()-(5) wth the settng = =.8. hen the common postve defnte matx P = - the negatve defnte matces R S ae obtaned as follows P π π R R R S S S In the same tme the contol feedback gans fo the fuzzy contolle ae obtaned as follows G (9a) G (9b) Applyng the IPM-based fuzzy contolle (8) wth the feedback gans (9) to contol the dscete-tme nonlnea tuck-tale system (6) the smulaton esults of states ae shown n Fgs he ntal condton of the smulaton n ths example s gven as x() [8835 ]. Fom these smulated esponses one can fnd that the contolled dscete-tme nonlnea tuck-tale system (6) s asymptotcally stable. he contol goal of ths pape can be acheved by desgned the IPM-based fuzzy contolle (8). Obvously the poposed desgn method s useful effectve fo desgnng IPM-based fuzzy contolle fo the dscete-tme nonlnea stochastc systems. o emphasze the mpotance of consdeng stochastc behavo a compason between ths pape (Lam Naman 9) s poposed. Refeng to (Lam Naman 9) an IPM-based fuzzy contolle desgn method was developed fo detemnstc nonlnea systems. Applyng method

8 956 Jounal of Mane Scence echnology Vol. 4 No.5 (6) 5 he Poposed Method Lam Naman 9 (deg) -5 3 (deg) he Poposed Method Lam Naman me (sec.) Fg. 4. Responses of state x (k) me (sec.) Fg. 6. Responses of state x 3 (k). (deg) he Poposed Method Lam Naman 9 me (sec.) Fg. 5. Responses of state x (k). 5 of (Lam Naman 9) to the -S fuzzy model (7) the followng gans can be obtaned wth the same = =.8. G G (3) Applyng IPM-based fuzzy contolle (8) wth gans (3) the esponses of (7) ae also stated n Fgs 4-6. Fom the fgues one can fnd that se tme settlng tme of (7) dven by contolle desgned by ths pape ae smalle than that dven by contolle desgned by (Lam Naman 9). hough the smulaton esults the poposed desgn method povdes some mpovements fo the method of (Lam Naman 9) n contollng nonlnea stochastc systems. V. CONCLUSIONS In ths pape a fuzzy contolle desgn appoach has been developed fo a class of dscete-tme stochastc nonlnea systems. he -S fuzzy model wth multplcatve nose was utlzed to epesent ths class of nonlnea systems. Usng the IPM concept one can dectly desgn a fuzzy contolle that the membeshp functon s dffeent fom that of the fuzzy plant. he elaxed stablty condtons deved based on the IPM concept can be categozed as the LMI poblems. By solvng the LMI stablty condtons va convex pogammng technque the fuzzy contolle can be obtaned to stablze the dscete-tme -S fuzzy model wth multplcatve nose. Fnally a numecal example was povded to demonstate the effectveness applcaton of the poposed fuzzy contolle desgn method. REFERENCES Boyd S. L. E. Ghaou E. Feon V. Balakshnan (994). Lnea Matx Inequaltes n Systems Contol heoy SIAM Phladelpha PA. Chang W. J. C. H. Chang C. C. Ku (a). Fuzzy contolle desgn fo akag-sugeno fuzzy models wth multplcatve noses va elaxed nonquadatc stablty analyss. Poc. of the Insttuton of Mechancal Engnees Pat I: J. Systems Contol Engneeng Chang W. J. W. Chang (6). Dscete fuzzy contol of tme-delay affne akag-sugeno fuzzy models wth H constant. IEE Poceedng Pat D Contol heoy Applcatons Chang W. J. C. C. Ku Z. G. Fu (3). Robust passve constaned fuzzy contol fo dscete fuzzy systems wth multplcatve noses nteval tme delay. Mathematcal Poblems n Engneeng 3 Atcle ID Chang W. J. C. C. Ku P. H. Huang (b). Robust fuzzy contol of uncetan stochastc tme-delay akag-sugeno fuzzy models fo achevng passvty. Fuzzy Sets Systems 6-3. Chang W. J. C. C. Ku P. H. Huang (). Robust fuzzy contol va obseve feedback fo passve stochastc fuzzy systems wth tme-delay multplcatve nose. Intenatonal Jounal of Innovatve Computng Infomaton Contol Chang W. J. C. C. Ku P. H. Huang W. Chang (9). Fuzzy contolle

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