New Stability Condition of T-S Fuzzy Systems and Design of Robust Flight Control Principle

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1 96 JOURNAL O ELECRONIC SCIENCE AND ECHNOLOGY, VOL., NO., MARCH 3 New Sably Conon of -S uzzy Sysems an Desgn of Robus lgh Conol Pncple Chun-Nng Yang, Ya-Zhou Yue, an Hu L Absac Unlke he pevous eseach woks analyzng he sably of he -S (akag Sugeno) fuzzy moel, an exenson on he sably conon of -S fuzzy sysems wh a ffeen saegy s pove. In he saegy a new vaable, whch s elave o he gae of fuzzy membeshp funcon, s nouce o he sably analyss an a new sably concluson s euce. he efnon of sably conon n hs pape s ffeen fom pevous woks, hough hey ae smla n fom. Wh he popose meho, he smulaon n flgh conol law shows a bee effecveness. Inex ems lgh conol, lnea max nequales, sably, akgag-sugeno (-S) fuzzy conol.. Inoucon Compang wh aonal conol mehos n nonlnea sysems, he fuzzy conolle has shown bee pefomance ha coul moel unceany o nonlnea chaacesc []. Bu s ffcul o escbe he sably of hs nonlnea fuzzy conol sysem. akag an Sugeno pesene an analycal moel of fuzzy sysem n 985, calle -S fuzzy moel []. hey analyze he sably conon an esgne he fuzzy conolle usng he Lyapunov ec meho. hs -S moel becomes he mosly aope meho n sably analyss of fuzzy conol sysem. he focus of sably analyss abou -S fuzzy sysem s how o elax he sably conons an how o Manuscp eceve May, ; evse June,. hs wok was suppoe by he Avaon Scence ounaon une Gan No. 776, Zheang Povncal Naual Scence ounaon une Gans No. Y696 an No. R95, he unamenal Reseach uns fo he Cenal Unveses une Gan No. QNA4, an Naonal Naual Scence ounaon of Chna une Gan No. 673 an No C.-N. Yang s wh he School of Aeonaucs an Asonaucs, Zheang Unvesy, Hangzhou, 37, Chna (Coesponng auho e-mal: chnyang@zu.eu.cn). H. L s wh he School of Aeonaucs an Asonaucs, Zheang Unvesy, Hangzhou, 37, Chna (e-mal: zzlhu@zu.eu.cn). Y.-Z. Yue s wh he Naonal Key Laboaoy of Scence an echnology on Acaf Conol, X An lgh Auomac Conol Reseach Insue, X an 765, Chna (e-mal: gnss68@63.com). Dgal Obec Ienfe:.3969/.ssn X.3..7 fn a bee conolle n an enlage soluon ange fo he close-loop sysem. anaka pove a suffcen conon of fuzzy sysem by seekng a common posve efne max P, an he sysem was asympocally sable f hee exse such max [3]. anaka also popose seveal new elaxe sably conons an conolle esgn poceues n [3] [6]. Cao foun was ffcul o ge a common posve efne max an e o fn a se of posve efne max p (=,,,) nsea of seekng he sngle common posve efne max P [7]. Yang e al. eve he elaxe sably of he fuzzy conol sysem wh uncean gaes of membeshp whch epesene he paamee unceanes of he fuzzy sysems [8]. Xe e al. scusse he elaxe sably conons of connuous -S fuzzy sysems va non-quaac Lyapunov funcon echnque [9],[]. hey solve he poblems (equal o confgue he egenvalues of he close-loop sysem) by he sably conon menone befoe, whch ae all he sngly feasble soluon. Howeve, somemes s no suffcen o all he subsysems, so Chal e al. sue he sably fahe bu he concluson s sll no pefec fo all subsysems []. hs pape sues moe eeply on he sably eseach woks of [], [], an []; a ffeen sably conon s pove hough Lyapunov sably heoy, an a smulaon esul of flgh conol sysem s gven compang wh he basc sably conons n []. he pape s oganze as follows. Secon oulnes he -S fuzzy sysem moel an he basc sably conon by a ffeen analyss meho an poves a new sably conon. Secon 3 manly pesens he smulaon effecveness of flgh conol law wh he popose meho. Conclusons ae gven n Secon 4.. -S uzzy Moel an he Dffeen Relaxe Sably Resuls. Descpon of uzzy Moel he onay -S connuous fuzzy sysem s escbe as If z () s M, z () s M,, an z g () s M g, hen x() = Ax() Bu () y() = C x () u() = x (), =,,, ()

2 YANG e al.: New Sably Conon of -S uzzy Sysems an Desgn of Robus lgh Conol Pncple 97 whee M, (=,,, g) s he fuzzy ses, x() s he sae veco, u() s he npu veco, an y() s oupu veco, x() R n, y() R k, u() R m, A R n n, B R n m, C R k n, an R m n, s he numbe of I-HEN ules, n s he numbe of sae vaables, m s he numbe of npu vaables, k s he numbe of oupu vaables, an z (), z (),, z g () ae he pemse vaables. Gven a pa of (x(), u()), f we esgn fuzzy local sae feeback conolle base on each fuzzy sub-sysem (such as paallel sbue compensaon [6] ) an use fuzzy easonng meho an fuzzfcaon pocess, he fnally oupu of he fuzzy sysem () s gven by x() = = ω ( z( )){ A x( ) B u( )} = ω ( z( )) = h( z( ))( Ax( ) Bu ( )) = = = = h( z( )) h( z( )){ A B} x ( ) ω ( z( )) C x( ) y z C x = () = = ( ()) h = ω ( z( )) () =. () Accong o he esgn meho of lnea sysem heoy, f (A, B ), =,,, s a pa of conollable maces, he fuzzy sysem s local conollable, hen we can assgn he egenvalues of (A, B ) a wll. he open-loop sysem (u()=) of () s: ω ( z( )) A x( ) x z A x = () = = ( ()) h = ω ( z( )) () =. (3) Evey funcon escbe by A x() s calle a sub-sysem, whee g ω ( z( )) = M ( z ( )),, = h( z( )) = ω ( z( )) = ω ( z( )) an z()=[z (), z (),, z g ()], M, (z ()) s he gae of he membeshp funcon of pemse vaable z () n se M,. Suppose o all, ω (z()), =,,,, hen h (z()), =,,,, an = ω (()) z >, = h ( z ( )) = o all.. Sably Conon of -S uzzy Sysem akak an Sugeno have analyze he sably va Lyapunov s ec meho, pove he sably conon of a fuzzy sysem, an gave a conolle esgn meho []. sly, he sysem moel s escbe by a -S fuzzy moel, hen a esgne conolle seeks a common symmec max P o sasfy he Lyapunov funcon; f hee exss such max P, he fuzzy sysem s asympocally sable. heoem []. If hee exss a common posve efne max P o all he fuzzy subsysems ha sasfes A P PA < =,,, (4) hen he fuzzy sysem (3) s asympocally sable n he lage o all he subsysems. o smplcy, we ewe A, = A B, by Lyapunov s ec meho, he suffcen conon of he local sae feeback ha ensues he asympocal sably of he close-loop fuzzy sysem s A P PA. (5),, < he manly elaxe sably esul of connuous fuzzy sysem was gven by anaka [7]. heoem [],[3]. Assume ha he numbe of ules ha fe fo all s less han o equal o s, whee s. he equlbum of he connuous fuzzy conol sysem () s asympocally sable n he lage f hee exss a common posve efne max P an a common posve semefne max Q such ha A, P PA, ( ) Q< (6) s A, A, A, A, P P Q, < (7) fo all an excepng he pas (, ). he sably conon of heoem eles on each subsysem A, B an, =,,,, bu he gaes of he membeshp funcon s no aken no accoun. In he wok below, we wll popose an exenson of [8], wh a ffeen saegy fom [7] an eve a ffeen sably esul. Due o whee h (z()). hen h ( z( )) h ( z( )) h ( z( )) = < = { h ( ( )) ( ( ))} h z z (8) < h ( z ( )) =, =,,, (9) = h ( z( )) h ( z( )) h ( z ( )) (a) = < when m, m s an nege vaable. hen h ( z( )) h ( z( )) h ( z ( )). (b) = m <

3 98 Assume ha he numbe of fuzzy ules ha fe fo all s,, when m, (b) s also equvalen. Because = = < h ( z( )) = h ( z( )) h( z( )) h ( z ( )) =. () Subsue () o () an so Rewe (), we ge h( ( )) h( ( )) z z < h( z()) h( z ()). () < x() = h ( z()), x() =,, h( z()) h( z()) x() (3) < an hen oban he followng heoem. heoem 3. Assume ha he numbe of ules ha fe fo all s,, an s less han o equal o m whee <m. If hee exss he common max P> an Q such ha A, P PA, ( ) Q< (4) m,,,, P P Q, < JOURNAL O ELECRONIC SCIENCE AND ECHNOLOGY, VOL., NO., MARCH 3 (5) he equlbum of he connuous fuzzy conol sysem (3) s asympocally sable n he lage. Poof. Selec he Lyapunov funcon as bellow: V( x()) = x () Px () we ge V( x()) = x () Px() x () P x () = h( z( )) h( z( )) x ( ){( A B) P = = PA ( B)} x ( ) z x, P P, x = h( z()) h( z()) x () < = h ( ()) (){ } () A,,,, P P x(). om he conons an (8) (), we ge he followng esuls: V = h = ( ( )) ( ( )) ( ){ x } ( ) z x P P x,, h( z()) h( z()) x () Qx () < z x Qx = < < h ( ( )) ( )( s ) ( ) h ( z()) h ( z()) x () Qx () = ( m ){ h( z( )) h( z( ))} x ( ) Qx( ) < h( z()) h( z()) x () Qx () < ( m )(/ ) x ( ) Qx ( ). Noe ha: n fac, he obane sably conon has a ffeen efnon fom heoem 5 n [7], hough s smla n fom. Because he solvng poceues base on lnea max nequales always ge a sngle feasble soluon, he popose esgn meho coul gve many soluon hough unng he value of m (we can efne m a posve numbe nee, as shown n he followng example). So s helpful o selec a favoable esul an avo he non-ancpan egenvalues of close-loop fuzzy sysem. 3. Robus uzzy Conolle Desgn n lgh Conol Sysem A flgh conol sysem s a ypcal nonlnea, unceany paamee moele an me vayng sysem. I s no easy o keep sable an obus when esgnng conolle n flgh. We selec he smple aplane as same as ha n [9] an esgn he fuzzy conolle wh he popose meho. he known conons ae: he hegh of flgh h=3 km, an he mach numbe M a =.5,.8, an.9. he pupose s o esgn a feeback conolle ha coul sablze an aplane n he ange of M a [.4,.]. he esgn poceue of fuzzy conol law s shown as below. We efne hee fuzzy subses M =.5, M =.8, M 3 =.9, conseng he hegh of flgh h s a consan. g. shows he gae of he fuzzy membeshp funcon of he mach numbe. M =M M =M M 3 =M 3 3(.5) 5(.8) 9(.9) M g.. Gae of membeshp funcon abou mach numbe.

4 YANG e al.: New Sably Conon of -S uzzy Sysems an Desgn of Robus lgh Conol Pncple 99 uzzy ules accong o mach numbe ae aken as he pemse vaables. he h fuzzy ule : f M a s M, hen x() = Ax() Bu (), =,, 3 y() = Cx(), =,, 3 u() = x (), =,, 3. he global conolle s [ ] u = h ( M ) h ( M ) h ( M ) x ( ) (6) a a 3 a 3 whee u s he conol npu of unng angle of elevao. An a a a 4 a 5 a a A = a3 a3 a33 a5 a54 C = c6 c63 B = [ b b b ] 3 u=[δ e ]. he followng esuls show he soluon of heoem 3 (m=.5): P = Q = = [ ] = [ ] 3 = [ ]. g. o g. 4 show he smulaon esuls une he nal conon x=[ /57.3 ]. We can see ha he esuls fom heoem 3 ae bee n pefomance evaluaon, such as he sae value, ausmen me an oveshoo, an also a bee obusness o eec he paamee peubaon. A ffeen ampng consan coul be obane hough unng he value of hs vaable, so helps o acheve bee conol effecveness. α ( ) ωz( /s) θ ( ) α ω z me (s) g.. Response of angle of aack α, pch ae ω Z, an angle of pch θ. H (m) g. 3. Response of alue. V (Ma) θ me (s) me (s) g. 4. Response of velocy. 4. Conclusons hs pape uses a ffeen saegy o scuss he elaxe sably conon va Lyapunov s ec meho. he eve esul s ffeen fom he pevous papes, whch s flexble o gan a bee conol pefomance by ausng a vaable, an can be ulze o seach a bee conol ule n lage ange vae moels. he applcaon of he new saegy n he flgh conol sysem shows ha he obane sably conon s useful o esgn a sable conolle of a nonlnea sysem an he conolle also has goo obusness o eec he paamee peubaon of he plane.

5 Refeences [] L. A. Zaeh, uzzy ses, Infomaon an Conol, vol. 8, no. 8, pp , 965. []. akag an M. Sugeno, uzzy enfcaon of sysems an s applcaons o moelng an conol, IEEE ans. on Sysems, Man, an Cybenecs, vol. 5, no., pp. 6 3, Jan [3] K. anaka an M. Sugeno, Sably analyss an esgn of fuzzy conol sysems, uzzy Ses an Sysems, vol. 45, no., pp , 99. [4] K. anaka,. Ikea, an H. Wang, Desgn of fuzzy conol sysems base on elaxe LMI sably conons, n Poc. of he 3 IEEE Conf. on Decson an Conol, Kobe, 996, pp [5] K. anaka,. Ikea, an H. Wang, Robus sablzaon of a class of uncean nonlnea sysems va fuzzy conol: quaac sablzaon, H-nfny conol heoy, an lnea max nequales, IEEE ans. on uzzy Sysems, vol. 4, no., pp. 3, 996. [6] K. anaka,. Ikea, an H. Wang, uzzy egulaos an fuzzy obseves: elaxe sably conons an LMI-base esgns, IEEE ans. on uzzy Sysems, vol. 6, no., pp. 5 65, 998. [7] S. G. Cao, N. W. Rees, an G. eng, Quaac sably analyss an esgn of connuous me fuzzy conols sysems, In. Jounal of Sysems Scence, vol. 7, no., pp. 93 3, 996. [8] C. Yang an J. An, A Relaxe sably conon of fuzzy sysem base on lnea -S fuzzy moel, Jounal of Poecles, Rockes, Mssles an Guance, vol. 3, no. 3, pp. 3 6, 3. [9] Z. Xe, uzzy conolle esgn an sably analyss fo flgh conol sysems, lgh Dynamcs, vol. 8, no., pp. 3 33,. [] H. K. Lam an. H.. Leung, Sably analyss of fuzzy conol sysems subec o uncean gaes of membeshp, JOURNAL O ELECRONIC SCIENCE AND ECHNOLOGY, VOL., NO., MARCH 3 IEEE ans. on Sysems, Man, an Cybenecs PAR B: Cybenecs, vol. 35, no. 6, pp. 3 35, 5. [] M. Chal, D. Maqun, an J. Rago, Relaxe sably conons fo akag-sugeno fuzzy sysems, n Poc. of IEEE In. Conf. on Sysems, Man, an Cybenecs, Vanoeuve,, pp Chun-Nng Yang was bon n Shannx, Chna n 97. He eceve hs Ph.D. egee n navgaon guance an conol fom he Depamen of Elecc Engneeng an Auomacs, Nohwesen Polyechncal Unvesy n 5. He s cuenly wokng as an assocae eseach fellow wh Zheang Unvesy. Hs eseach neess nclue acaf ynamcs moelng, negae navgaon echnolog, flgh conol, ec. Ya-Zhou Yue was bon n Shannx, Chna n 97. He eceve hs Ph.D. egee n navgaon guance an conol fom Chnese Aeonaucal Esablshmen. Hs eseaches nclue navgaon an conol echnology, negae navgaon echnology, ec. Hu L was bon n Helongang, Chna n 977. He eceve hs Ph.D. egee n communcaon an nfomaon sysem fom he Depamen of Eleconcs an Infomaon Engneeng, Habn Insue of echnology n 6. He s cuenly wokng as an assocae pofesso wh Zheang Unvesy. Hs eseaches nclue space communcaon, weless newoks, oung algohms an cong, ec.