Robust Control for Uncertain Takagi Sugeno Fuzzy Systems with Time-Varying Input Delay

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1 Robus Conrol for Unceran akag Sugeno Fuzzy Sysems wh me-varyng Inpu Delay Ho Jae Lee e-mal: Jn Bae Park e-mal: Deparmen of Elecrcal and Elecronc Engneerng, Yonse Unversy, Seodaemun-gu, Seoul, , Korea Young Hoon Joo e-mal: School of Elecronc and Informaon Engneerng, Kunsan Naonal Unversy, Kunsan, Chonbuk, , Korea A conrol problem of akag Sugeno fuzzy sysems wh a mevaryng npu delay and norm-bounded unceranes s addressed. he npu delay s well-known n makng he closed-loop sablzaon dffcul. A suffcen condon for he robus fuzzy-modelbased sablzaon s derved based on he Lyapunov Razumkhn sably heorem, whou he assumpon of he varaon rae on he delay. A consrucve desgn scheme s presened n he form of he erave convex opmzaon problem. he effecveness of he proposed mehod s demonsraed by a numercal smulaon of a nonlnear mass-sprng-damper sysem. DOI: 0.5/ Inroducon In he conrol engneerng area, s common o encouner some praccal applcaons conanng me delays n her model confguraons. Examples nclude chemcal processes, bologcal sysems 2, and vrual laboraores 3. I s wdely beleved ha he me delay s one of he major sources of he nsably of conrol sysems 4 6. o resolve he conrol problems assocaed wh he me delay, wo generalzed approaches have araced grea aenon: One s he Lyapunov Krasovsk sably heorem and he oher s he Lyapunov Razumkhn approach 7. Alhough he former has been wdely adoped, reures a supplemenary propery n erms of he me-dervave of he me delay: he upper bound of he me-dervave mus be less han one. hs reuremen may no be sasfed n some specfc applcaons. On he conrary, he laer can be successfully appled because any properes of he me-dervave of he gven me delay are no necessary, alhough he obaned resuls may be conservave 4. urnng our aenon o anoher mporan echncal ssue, mos plans n he ndusry have severe nonlneares and unceranes. hey hus pos addonal dffcules o he sably analyss and conroller desgn. Unl now, varous conrol echnues have been developed. Among hem, he akag Sugeno --S fuzzy-modelbased conrol s popular oday snce s regarded as a powerful Conrbued by he Dynamc Sysems, Measuremen, and Conrol Dvson of HE AMERICAN SOCIEY OF MECHANICAL ENGINEERS for publcaon n he ASME JOURNAL OF DYNAMIC SYSEMS, MEASUREMEN, AND CONROL. Manuscrp receved February 2, Fnal manuscrp receved July 2, Revew conduced by: Prabhakar R. Paglla. resoluon o brdge he gap beween he fruful lnear conrol heores and he fuzzy logc conrol argeng plans ha are mahemacally ll-defned, unceran, and nonlnear. Plenful works relaed can be found n 8 4 and references heren. Despe he exensve sudes n he fuzzy-model-based conrol leraure o dae, here are relavely few research resuls acklng he --S fuzzy sysems wh me delay 3,4. Moreover, alhough he npu delay s a echncally mporan ssue of freuen occurrence, few relaed conrol sraeges seem o be avalable. I remans ye o be a heorecally challengng ssue, and hereby mus be carefully handled. Movaed by he above observaons, hs paper ams a solvng he robus conrol problem for a connuous-me unceran --S fuzzy sysem wh a me-varyng npu delay. he man conrbuon s o propose a consrucve desgn ool n erms of marx neuales for --S fuzzy sysem of neres. he sablzng conroller s desgned so ha he Lyapunov Razumkhn sably s esablshed. I should be noed ha, under he sably condon obaned, any addonal resrcon on he me-dervave of he me delay s no necessary. Furhermore, he longer he me delay guaraneeng he sably of he closed-loop sysem s, he more desrable s. Hence, an erave convex opmzaon algorhm s presened o search he maxmal bound of he admssble me delay. he organzaon of hs paper s as follows: Secon 2 brefly revews a connuous-me unceran --S fuzzy sysem wh he npu delay. he man resuls are presened n Sec. 3. In Sec. 4, an example s ncluded o vsualze he feasbly of he proposed mehod conrol of a nonlnear mass-sprng-damper sysem. Lasly, Sec. 5 concludes hs paper. 2 Prelmnares and Problem Saemen Consder he me-varyng npu-delayed connuous-me unceran --S fuzzy sysem descrbed by he followng fuzzy rules: R : IF z s and and z n s n HEN ẋ = A + A x + B + B u d where R, I Q =,2,...,, denoes he h fuzzy nference rule; z h, hi N =,2,...,n, he premse varable; h he fuzzy se of z h n he h rule; xr n he sae; u dr m he delayed conrol npu, n whch d s he me-varyng delay represened by any admssbly bounded funcon sasfyng 0 d ; A,B he model of he h rule; and A,B real marx funcons represenng unceranes. Usng he cener-average defuzzfcaon, produc nference, and sngleon fuzzfer, he global dynamcs of of he rearded ype s gven by ẋ = za + A x + B + B u d = 2 x =,,0 where s a smooh vecor-valued funcon defned n Banach n space C,0, and z= h= h z h, z = z/ = z. In hs sudy, he followng fuzzy rule for he fuzzy-modelbased conroller s employed: R : IF z s and and z n s n HEN u = K x where K s conrol gan marx o be deermned. Is defuzzfed oupu s descrbed by u = zk x. = / Vol. 27, JUNE 2005 Copyrgh 2005 by ASME ransacons of he ASME

2 he closed-loop sysem wh 2 and 3 s descrbed by ẋ = = z j z da + A x + B + B K j x d. 4 Snce 4 has me-varyng unceran marces, s no easy o deermne K. Hence, he unceran marx funcons should be manageable under some reasonable assumpons. Assumpon. he unceranes consdered here are normbounded of he form: A B = D F E a E b where F s an unknown marx funcon wh Lebesguemeasurable elemens and sasfes F F I, n whch D, E a, and E b are known real consan marces of compable dmensons. Our goal s summarzed as follows: Problem. Fnd K for 3 such ha 4 s robusly globally asympocally sable n he sense of Lyapunov agans he admssbly norm-bounded and srucured unceranes and any mevaryng delay d less han or eual o he prescrbed. Furhermore, f possble, fnd he maxmal upper bound of whn whch he sably of he whole sysem s sll preserved. 3 Man Resuls Before proceedng, recall he followng lemmas whch wll be used for he proofs of our resuls. Lemma. For gven vecors a, b, and any symmerc posve defne marx P of approprae dmensons, and any posve scalar, he followng neualy holds: ±2a b a Pa + b P b. Lemma 2. Gven consan marces D and E, and a symmerc consan marx S of approprae dmensons, he followng neualy holds: S + DFE + E F D 0 where F sasfes F FI f and only f for some 0 S + D E D E 0. Remark. In case of 2a b0 n Lemma, s esmaed upper bound may be no good and nroduce conservasm. However, opmzng over can raher reduce he nroduced conservasm. he man resul s now presened n he followng heorem: heorem. If here exs a symmerc posve defne marx Q, marces M, and posve scalars, 2, j,, and 2j such ha he followng neuales are sasfed: j Q + j D D B 2 Q 0, E a Q + E b M j E b M j j I, j IQ IQ 5 Q + D D QA Q 0, 0 E a Q I I Q 6 2Q + D D 2j j B Q 0, 0 E b M j 2j I, j I Q I Q 7 hen 2 s robusly globally asympocally sablzable by 3 n he presence of he norm-bounded unceranes and for all d, where j =QA +A Q+M j B +B M j, Q= P, M =K P, and denoes he ransposed elemens n he symmerc posons. Proof. Choose a Lyapunov funconal canddae as Vx =x Px where P s a symmerc posve defne. Clearly, Vx s posve defne and radally unbounded. he me dervave of Vx along any rajecory of 4 s gven by Noe ha x d=x d ẋd. hen, pluggng no 8 resuls n k= V x = z j z d x A + A P = + PA + A + K j B + B P + PB + B K j x 2 d l= x PB + B K j k z l z d V x = = z j z dx A + A P + PA + A x + x d K j B + B Px + x PB + B K j x d. 8 A k + A k x + B k + B k K l x dd. Applyng Lemma o 9 mples 9 Journal of Dynamc Sysems, Measuremen, and Conrol JUNE 2005, Vol. 27 / 303

3 k= l= V x z j z d x A + A P = + PA + A + K j B + B P + PB + B K j x + d x PB + B K j A k + A k P A k + A k K j B + B Px + 2 x PB + B K j B k + B k K l P K l B k + B k K j B + B Px + x Px + 2 x d Px dd. 0 From he Razumkhn sably heorem 7, and assumng ha for any real number, we have VxVx, 2,. Suppose ha A k + A k P A k + A k P, k I Q B k + B k K l P K l B k + B k 2 P, k,l I Q I Q 2 hen, s no dffcul o undersand he rgh-hand sde of 0 s less han = z j z dx A + A P + PA + A + K j B + B P + PB + B K j x +2dx PB + B K j P K j B + B Px + d + 2 x Px. 3 From he observaon ha 3 s monooncally ncreasng wh respec o d, f he followng holds, A + A P + PA + A + K j B + B P + PB + B K j +2PB + B K j P K j B + B P P 0,, j I Q 4 hen 4 s robusly globally asympocally sable agans all mevaryng npu delays no larger han and he srucured unceranes. Moreover, from he connuy of he egenvalues of 4 wh respec o, here exss suffcenly small such ha 4 wh = sll holds. Wh some effors, we can show ha 5 7 guaranee he negave defneness of 4 whenever x s no zero. Frs, we show ha 6 and 7 are drecly derved from and 2. Ineualy can be represened as follows: A + A P PP A + A P 0. 5 Applyng he Schur complemen and Assumpon o 5 gves P P A P + D F 0 E a P P E F D a Accordng o Lemma 2, 6 holds f and only f here exss a consan /2 0 such ha P P A P + D 0 0 P E a I 0 0 I D E a P Usng he Schur complemen wce and denong P =Q resuls n 6. We can agan esablsh a smlar argumen o ha above wh 2 o oban 7. Nex, 4 can be rewren as follows: A + A P + PA + A + K j B + B P + PB + B K j P + PB + B K j P 2PP K j B + B P 0. 7 By applyng he Schur complemen, Assumpon, and Lemma 2, 7 s euvalen o where j + E a + E b K j P K j E b E a + E b K j PD j I j I E b K j P 0 8 D P 0 j = A P + PA + K j B P + PB K j P P K j B P P 2 f and only f here exss a consan /2 j 0. Seuenal applyng he Schur complemen wce and a congruence ransformaon wh dagp,i,i,i o 8, denong Q= P and M =K P yelds 5, whch complees he proof of he heorem. Remark 2. In order o dmnsh he conservasm nroduced by he overesmaed upper bound n 6 and 7, proper values of and 2 should be chosen such ha s maxmzed. However, he erms Q and 2 Q make 5 7 be nonlnear, whch are dffcul o solve. hus, an erave convex opmzaon algorhm based on he lnear marx neualy LMI echnue s ulzed. In order o fnd he maxmal, whou loss of generaly, we replace j wh n 5. Carryng ou smple algebrac manpulaon, he followng LMI s euvalen o 5: j + D D Q E a Q + E b M j I Q 0. j B j E b 2 9 Now, he followng convex opmzaon algorhm s proposed. Sep : Fnd Q, M, and j such ha he followng LMI consrans are sasfed: QA + A Q + M j B + B M j + j D D E a Q + E b M j j I 0,, j I Q I Q. Sep 2: For Q gven n he prevous sep, fnd, 2,, 2j, and M such ha he followng generalzed egenvalue problem GEVP has soluons 304 / Vol. 27, JUNE 2005 ransacons of he ASME

4 Fg. delay Nonlnear mass-sprng-damper sysem wh an npu Maxmze subjec o 9, 6, and 7. M,, 2,, 2j Sep 3: For, 2, and M gven n he prevous sep, fnd, 2j, and Q such ha he followng GEVP has soluons Maxmze subjec o 9, 6, and 7. Q,, 2j Sep 4: Reurn o Sep 2 unl he convergence of s aaned wh a desred accuracy. 4 An Example An example s presened o vsualze he proposed desgn echnue. Consder he followng nonlnear mass-sprng-damper mechancal sysem llusraed n Fg. : M + D + k = F d where s he relave poson of he mass; F d he delayed exernal force; M = he mass of hs sysem; k=0. he sffness of he sprng. = he npu coeffcen. he dampng coeffcen of he nonlnear damper s assumed o be D = Furhermore, we assume ha k s unknown bu bounded whn 0% of s nomnal values. Choosng he sae as x=, and he npu varable u as F yelds he followng sae-space represenaon. ẋ ẋ 2 = 0.75x 3 0.5x x 2 + u d x 20 where 2. he sysem 20 has one nonlnear erm, 0.75x 3. Assume x, and f hs nonlnear erm can be represened as a convex sum, he --S fuzzy sysem of 20 can be consruced. Consder he followng euaons: 0.75x 3 = x 0+ 2 x x Fg. 2 Behavor of he maxmal bound of he me delay va he proposed algorhm: compued from Sep sold ; obaned from Sep 2 sold-damond R : IF x s abou 2 2 HEN ẋ = A 2 + A 2 x + B 2 + B 2 u d where he assocaed marces are gven by A = A 2 = 0, A = A 2 = , 0., B = B 2 = and B =B 2 =0 2. By applyng heorem and he erave opmzaon echnue, we ge K = , K 2 = , and = Fgure 2 shows he behavor of obaned by he proposed algorhm. I means ha he desgned --S fuzzy-model-based conroller can robusly sablze 20 n he presence of any me-varyng npu delay sasfyng d= and he paramerc unceranes. Durng he smulaon process, he sysem parameer k s randomly vared whn 0% of s nomnal value. Fgure 3 shows d appled o he smulaon. Indeed, he assumed me delay does no exceed and s maxmal me-dervave s much larger han one. Noce ha, accordngly, he Lyapunov Krasovsk funconal approach canno be appled. Compared o ha, he proposed approach seems o be suable, snce allows for he delay o be me-varyng wh an arbrary fas rae of change. he nal value s x0=x 0 =0,. For comparson purpose, a convenonal fuzzy-model-based conroller desgn echnue whou consderaon of he npu-delay 9 s smulaed. he smulaon resul s shown n Fg. 4. he conrol npu s acvaed a =3 s. Afer =3 s, he convenonal mehod produces oscllaory rajecores, as s expeced. On he oher hand, he rajecores conrolled by he proposed mehod are uckly guded o he orgn whou oscllaon. = x + 2 x. Solvng 2 yelds x = x x = x 2 2. Now, by adopng hese as fuzzy ses, he --S fuzzy sysem of 20 can be consruced as follows under he assumpon of =0.865: R : IF x s abou HEN ẋ = A + A x + B + B u d Fg. 3 he appled npu delay: d =0, «0,3 s ; 0.689, «3, 3.5 s ; , «3.5, 5 s ; oherwse Journal of Dynamc Sysems, Measuremen, and Conrol JUNE 2005, Vol. 27 / 305

5 Fg. 4 me responses of x by: Ref. 9 whou consderaon of he npu delay doed ; he proposed mehod wh consderaon of he npu delay sold 5 Conclusons In hs paper, we have dscussed he robus conrol of --S fuzzy sysems conanng unceranes and npu delays. he Lyapunov Razumkhn sably heorem has been ulzed as a synhess ool. he suffcen condon for he exsence of he sablzng conroller has been gven n erms of marx neuales. he maxmal bound of he npu delay s searched n an erave manner. he effecveness of he proposed desgn mehodology has been horoughly verfed n he smulaon example. hs means a grea poenal for ndusral applcaons. Acknowledgmen hs work was suppored by KOSEF R References He, J. B., Wang, Q. G., and Lee,. H., 2000, PI/PID Conroller unng Va LQR Approach, Chem. Eng. Sc., 55, pp Roh, Y. H., and Oh, J. H., 2000, Sldng Mode Conrol Wh Uncerany Adapaon for Unceran Inpu-Delay Sysems, In. J. Conrol, 733, pp Oversree, J. W., and zes, A., 999, An Inerne-Based Real-me Conrol Engneerng Laboraory, IEEE Conrol Sys. Mag., 95, pp Kolmanovsk, V. B., Nculescu, S. I., and Rchard, J. R., 999, On he Lapunov Krasovsk funconals for sably analyss of lnear delay sysems, In. J. Conrol, 724, pp Inamdar, S. R., Kumar, V. R., and Kulkarn, B. D., 99, Dynamcs of Reacng Sysems n he Presence of me-delay, Chem. Eng. Sc., 463, pp Moon, Y. S., Park, P. G., and Kwon, W. H., 200, Robus Sablzaon of Unceran Inpu-Delayed Sysems Usng Reducon Mehod, Auomaca, 37, pp Hale, J. K., 977, heory of Funconal Dfferenal Euaons, Sprnger- Verlag, New York. 8 anaka, K., Ikeda,., and Wang, H. O., 998, Fuzzy Regulaors and Fuzzy Observers: Relaxed Sably Condons and LMI-Based Desgns, IEEE rans. Fuzzy Sys., 62, pp Lee, H. J., Park, J. B., and Chen, G., 200, Robus Fuzzy Conrol of Nonlnear Sysems Wh Paramerc Unceranes, IEEE rans. Fuzzy Sys., 92, pp Joo, Y. H., Chen, G., and Sheh, L. S., 999, Hybrd Sae-Space Fuzzy Model-Based Conroller Wh Dual-Rae Samplng for Dgal Conrol of Chaoc Sysems, IEEE rans. Fuzzy Sys., 74, pp Chang, W., Park, J. B., Joo, Y. H., and Chen, G., 2002, Desgn of Robus Fuzzy-Model-Based Conroller Wh Sldng Mode Conrol for SISO Nonlnear Sysems, IEEE rans. Fuzzy Sys., 25, pp Chang, W., Park, J. B., Joo, Y. H., and Chen, G., 2002, Desgn of Sampled- Daa Fuzzy-Model-Based Conrol Sysems by Usng Inellgen Dgal Redesgn, IEEE rans. Crcus Sys., I: Fundam. heory Appl., 494, pp Lee, K. R., Km, J. H., Jeung, E.., and Park, H. B., 2000, Oupu Feedback Robus H Conrol of Unceran Fuzzy Dynamc Sysems Wh me-varyng Delay, IEEE rans. Fuzzy Sys., 86, pp Cao, Y. Y., and Frank, P. M., 2000, Analyss and Synhess of Nonlnear me-delay Sysems Va Fuzzy Conrol Approach, IEEE rans. Fuzzy Sys., 82, pp / Vol. 27, JUNE 2005 ransacons of he ASME