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g s Reseach Potal DOI:.9/FSD.5.738967 Documet Veso Pee evewe veso Lk to ublcato eco g's Reseach Potal Ctato fo ublshe veso (APA): Cao. Gao X. Z. Wag X. Lam H.. & Ma J. (6).

738967 Ctg ths ae Please ote that whee the full-text ove o g's Reseach Potal s the Autho Accete Mausct o Post-Pt veso ths may ffe fom the fal Publshe veso.

1 g s Reseach Potal DOI:.9/FSD Documet Veso Pee evewe veso Lk to ublcato eco g's Reseach Potal Ctato fo ublshe veso (APA): Cao. Gao X. Z. Wag X. Lam H.. & Ma J. (6). Stablty aalyss of T-S fuzzy PD PI a PID cotol systems. DOI:.9/FSD Ctg ths ae Please ote that whee the full-text ove o g's Reseach Potal s the Autho Accete Mausct o Post-Pt veso ths may ffe fom the fal Publshe veso. If ctg t s avse that you check a use the ublshe's eftve veso fo agato volume/ssue a ate of ublcato etals. A whee the fal ublshe veso s ove o the Reseach Potal f ctg you ae aga avse to check the ublshe's webste fo ay subsequet coectos. Geeal ghts Coyght a moal ghts fo the ublcatos mae accessble the Reseach Potal ae etae by the authos a/o othe coyght owes a t s a coto of accessg ublcatos that uses ecogze a abe by the legal equemets assocate wth these ghts. Uses may owloa a t oe coy of ay ublcato fom the Reseach Potal fo the uose of vate stuy o eseach. You may ot futhe stbute the mateal o use t fo ay oft-makg actvty o commecal ga You may feely stbute the URL etfyg the ublcato the Reseach Potal Take ow olcy If you beleve that ths ocumet beaches coyght lease cotact lbayue@kcl.ac.uk ovg etals a we wll emove access to the wok mmeately a vestgate you clam. Dowloa ate: 5. Oct. 8

2 Stablzato of Takag-Sugeo Fuzzy Systems Usg Fuzzy PID PI a PD cotolles au Cao Natoal ey Laboatoy of Tuable Lase Techology Hab Isttute of Techology Hab Cha E-mal: kcaoht@gmal.co m X. Z. Gao Deatmet of Electcal Egeeg a Automato Aalto Uvesty School of Electcal Egeeg Aalto Fla E-mal: xaozh.gao@aalto.f X Wag Beg Isttute of Astoomcal Systems Egeeg Beg Cha E-mal: wx_cv@6.com H.. Lam Deatmet of Ifomatcs g s College Loo Loo Ute gom E-mal: hakkeug.lam@kcl.ac.uk Jg Ma Natoal ey Laboatoy of Tuable Lase Techology Hab Isttute of Techology Hab Cha E-mal: mag@ht.eu.c Abstact I ths ae smle a systematc ways of esgg stablzato fuzzy PID (Pootoal-Itegal-Dffeetal) PI (Pootoal-Itegal) a PD (Pootoal-Dffeetal) cotolles fo the Takag-Sugeo fuzzy moel ae oose. The state sace eesetatos of the fuzzy PID PI a PD cotolles ae fstly esete. The we equvaletly tasfom the fuzzy PID (PI o PD) cotol system to the fuzzy statc outut feeback cotol system. The fuzzy statc outut feeback cotolle esg fo the latte system ca be effcetly solve va the exstg umecal otmzato methos wth some cosevatsm. Cosequetly the fuzzy PID (PI o PD) cotolle ca be eve ue to the oe-to-oe coesoece betwee the fuzzy PID cotolle a the fuzzy statc outut feeback cotolle. A smulato examle s also gve to emostate the effectveess of ou oose methos. eywos-takag-sugeo (T-S) fuzzy moels fuzzy PID cotolles statc outut feeback cotol lea matx equaltes (LMIs) I. INTRODUCTION (HEADING ) It s well kow that the PID (Pootoal-Itegal- Devatve) cotolles have bee wely ale moe tha 9% of the ustal cotol loos because they ca offe emakable cotol efomaces at the accetable costs []. I oe to acheve goo efomaces fo olea systems the tatoal PID cotolles have bee geealze to the olea oes fo examle fuzzy PID cotolles [-4]. Thee ae two ma tyes of the fuzzy PID cotolles.e. Mama tye a Takag-Sugeo (T-S) tye. A Mama fuzzy cotolle s chaacteze by a set of fuzzy ules whch ae costucte by lgustc tems both the ateceet a cosequet ats of the fuzzy ules. I cotast the cosequet at of a T-S fuzzy ule becomes a aalytcal fucto of the emse vaables. Fakly seakg thee s o essetal ffeece betwee these two tyes of fuzzy cotolles fom the cotol esectve. They tue the coeffcets of the PID cotolle accog to ffeet mechasms. The esg aoaches of the fuzzy PID cotolles have bee extesvely eveloe a exloe the exstg lteatue. I [5] a tye of the Mama fuzzy PID cotolle s esge base o the covetoal lea PID cotolle a the Boue-Iut-Boue-Outut (BIBO) stablty aalyss of the close-loo system s gve by the small ga theoem. The escbg fucto metho s also use to aalyze the stablty of Mama fuzzy PD (Pootoal-Dffeetal) a PI (Pootoal-Itegal) cotol systems [6]. I [7] a selftug mechasm s oose fo the Mama fuzzy PI a PD cotolles to esst sue loa stubace. Comasos betwee the oose fuzzy PI cotolles a Zegle Nchols-tue PID cotolles have show that the fome outefoms the latte wth ega to vaous efomace exes such as oveshoot settlg tme a tegal absolute eo. The Geetc Algothms (GAs)-base Mama fuzzy PI+PD cotolle s use to elmate the oveshoot of a ommum hase system [8] whee the fuzzy PI cotolle s mlemete to cacel the effect of ustable zeos a the fuzzy PD cotolle s ale to efom the taset esose. It s eote [9] that a Mama fuzzy PI-lke (fuzzy PD cotolle lus a tegato) cotolle ca be effectvely use the mllg ocess a the stablty s guaatee by the ccle cteo. Paallel to the Mama fuzzy PID cotol systems the stablty aalyss a cotolle esg of the T-S fuzzy PID cotol systems have bee vestgate as well. Base o the ccle cteo the stablty cotos fo the T-S fuzzy ootoal a fuzzy PI cotol systems ae oose a exloe [] a [] esectvely. The aalytcal stuctues of the T-S fuzzy PI a fuzzy PD cotolles ae exame [] a the BIBO stablty coto s gve o the bass of the small ga theoem. I [3] the covetoal PI cotolle s use to stablze the T-S fuzzy moel. The coeffcets of the cotolle ca be eve va LMIs (Lea Matx Iequaltes). I [4] the escbg fucto metho s use to aalyze the stablty of the fuzzy PI a PD cotol systems. Both the Mama a T-S fuzzy PID cotolles ae eloye the emaet maget sychoous moto ve cotol systems oe to move the taset esose a ehace the stubace eecto ablty [5-8]. Although sgfcat effots have bee evote to the esg of the fuzzy PID PI a PD cotolles thee ae stll ot eough goous

3 theoetcal methos fo the fuzzy cotolle esg.e. how to esg fuzzy PID PI a PD cotolles fo olea lats wth assue stablty. I ths ae we focus o how to systematcally esg the fuzzy PID PI a PD cotolles fo the T-S fuzzy moel base olea systems wth aamete ucetaty. The state sace eesetatos of the fuzzy PID PI a PD cotolles ae fstly esete. Next the fuzzy PID (PI PD) cotol system s equvaletly tasfome the fuzzy statc outut feeback cotol system. Sce the elatosh betwee the fuzzy PID (PI PD) cotolle a the fuzzy statc outut feeback cotolle s oe-to-oe coesoece the esg of fuzzy PID cotolle equals to the oblem of esgg fuzzy statc outut feeback cotolle. The latte oblem ca be effcetly hale usg some exstg umecal otmzato methos [9-]. The est of ths ae s ogaze as follows. Secto touces the T-S fuzzy moel the fuzzy PID PI a PD cotolles a the state sace eesetatos. The esg of the fuzzy PID PI a PD cotolles fo the T-S fuzzy cotol systems s eveloe Secto 3. I Secto 4 a umecal smulato examle s ove to emostate the effcecy of the oose methos. Fally some emaks a coclusos ae aw Secto 5. Notatos: s use to eote the -mesoal m Euclea sace. eotes the set of m matces. I the symmetc matces we use to eeset the tems uce by symmety. He{ M } s the shotha otato fo T M + M. II. PRELIMINARIES A. T-S fuzzy moel Assume that the olea lat to be stablze ca be aoxmate by the followg T-S fuzzy moel () whch cossts of fuzzy logc ules. x( t) = A x( t) + B u( t) y( t) = Cx ( t). () whee x() t s the state vecto; u a y ae the ut a outut esectvely; y = = A A = = B B u a C ; A a B ae the system matces of the -th local moel; : ξ ( t) [] s the membesh fucto a ξ s the emse vaable vecto. Note that a = thoughout ths ae that C s of full ow ak. =. It s assume B. State sace eesetatos of fuzzy PID PI a PD cotolles I ths ae the fuzzy PID cotolle fstly use to stablze the above T-S fuzzy moel. The PDC (Paallel Dstbute Comesato) techque [] s use to costuct the fuzzy cotolle. Theefoe the membesh fuctos of the fuzzy cotolle ae exactly the same as that of the T-S fuzzy moel. The outut of the fuzzy PID cotolle s as follows: u = u () t = whee u s the outut of each local cotolle. Each local lea cotolle ca be escbe the fequecy oma as follows: s Qs + Q s+ Q u() s = + + = () s τs+ τs + s whee s s the comlex umbe fequecy; τ s a kow u y ostve scala; ae coeffcets of the ootoal tegal a ffeetal tems esectvely; matces Q = +τ Q = +τ a Q =. Wthout ffcult comutato t ca be see that a state sace eesetato of () s as follows: xc A B xc() = τ τ τ τ u Q Q Q T Q y A B xc = C D y whee x s the state of the local cotolle a c τ τ τ A = ag y B = ag y [ ] [ ] T = [ ] [ ] I othe wos T s a emutato matx such that a a b b T = a b a b y y whee a y b. Futhemoe we have y

4 whee = τc + D τ C = τc 3 = τ C τ C. u y C C C T = C C. Remak : It ca be see fom (3) that the coeffcets a a ae lealy eeet o D ae eteme the coeffcets ae kow cosequetly. C a D. If C a (3) Remak : Although x c() t s the state of the local cotolle t s also the state of the fuzzy PID cotolle. To see ths ot ote that the system matces A a B ae costat matces the the states of the local cotolles ae always same f they have same tal states. Base o the emaks the state sace eesetato of the fuzzy PID cotolle ca be wtte as follows: () () xc xc A B xc() = = = = u u C D y A B xc = C D y = = = whee C = C a D D. The state sace eesetato of the fuzzy PID cotolle (4) ca be geealze fo the fuzzy PI a PD cotolles the same way. The state sace eesetato of the fuzzy PI cotolle ca be exesse by (4) xc A B xc() = (5) u C D y whee A = B = I a the aametes of the fuzzy PI cotolles ae eve by = D = C. Smlaly the state sace eesetato of the fuzzy PD cotolle ca be escbe by (6) xc A B xc() = (7) u C D y whee A = ag{ τ τ } B = I a the coeffcets of the fuzzy PD cotolles ae eve by = D τc = τ C. (8) III. MAIN THERETICAL ANALYSIS RESULTS I ths secto the obust fuzzy PID PI a PD cotolles esg methos wll be oose to stablze the T-S fuzzy moel (). To acheve ou obectve the stablzato of the fuzzy PID (PI o PD) cotol systems s equvaletly tasfome to the oblem of stablzg the fuzzy statc outut feeback cotol systems. I the followg aalyss the tme t s oe the tme assocate vaables e.g. x a u() t s eote as x a u esectvely. A. System tasfomato T T T T T T Defg y= [ xc y ] a u = [ xc u ] as the oututs of the ew T-S fuzzy moel a fuzzy cotolle esectvely we ca ewte () a (4) as follows: c x = A x+ B u y = Cx (9) = = T T T whee x = [ x x ] A A B A whee A = = = B a A B B = I I C = C. = a = u= y () A B =. C D Remak 3: Obvously the yamcal chaactestcs ae ot chage wth the tasfomato. Theefoe the ogal system a the tasfome system have the same stablty cotos. B. Stablty aalyss Substtutg () to (9) the close-loo system s obtae as follows: x= ( A + B C) x. T The quaatc Lyauov caate fucto V ( x) = x P x P > s use hee fo the aalyss. The tme evatve of V ( x ) s Aaetly f { } T V ( ) = He ( + ) x x P A B C P P x. { } He AP+ BCP < () the close-loo system s asymtotcally stable. Note that equalty () s o-covex sce thee ae ouct tems betwee vaables a P. To ccumvet

5 the oblem the metho touce [] s emloye. Assume that RC = CP a let L = L whee = [ A B ]R () L = R = Lc a Lc = [C D ]R a R s a matx wth aoate mesos the we ca ewte () as follows: Θ < (3) PD) cotolle (4) ((5) o (7)) whee τ > s evously kow s asymtotcally stable wth eefe ecay ate υ f thee exst matces P = P T > R a Lc such that (4)(6) ae satsfe whee Θ = He{A P + B LC}+υ P. Futhemoe let [C D ] = Lc R ( C D C D ca be efe the same way) the the coeffcets of the fuzzy PID (PI o PD) cotolle ca be eve by (3) ((6) o (8)). = = IV. whee Θ = He{A P + B L C}. Summazg the above aalyss a usg the techques of [3] to hale the fuzzy summato equalty (3) we have the followg Theoem fo the stablty aalyss. It s ote that the above aalyss s also sutable fo the esg of fuzzy PI a PD cotolles. Theoem : The fuzzy PID (PI o PD) cotol system cosstg of the T-S fuzzy moel () a the fuzzy PID (PI o PD) cotolle (4) ((5) o (7)) whee τ > s evously kow s asymtotcally stable f thee exst matces P = P T > R a Lc such that RC = CP Θ < = Θ + (Θ + Θ ) < = Plat ule : If ξ s M x = A x + B u the y = Cx. whee x = [ x x ]T ξ = y a A = A = B = B = (5) C = [3.5 ].. (6) Futhemoe let [C D ] = Lc R ( C D C D ca be efe the same way) the the coeffcets of the fuzzy PID (PI o PD) cotolle ca be eve by (3) ((6) o (8)). Remak 4: The matx R s always vetble because both of the matces C a CP ae of full ow ak. Remak 5: Note that thee exst ouct tems of a R (3) (see ()). Because A a B ae kow costat matces a C a D ae vaable matces the ew vaable matx Lc = [C I ths secto a smulato examle [5] s gve so as to show the effectveess of the oose esg methos. Cose the followg two-ule T-S fuzzy moel: (4). A SIMULATION EXAMPLE D ]R ca be efe whch makes L to be lea eeet o vaable matces The membesh fuctos ae efe as follows: = ( ( + e 7(ξ π /4) ) )( + e 7(ξ +π /4) ) =. I ou examle the fuzzy PID cotolle (4) s fst use to stablze the T-S fuzzy moel we ca f the followg matces satsfy Theoem wth τ = P= R = R a Lc. Covesely R a Lc ca be use to eve the matces C c a D c. It s woth metog that solvg cotos (4)-(6) s actually a SDP (Sem-Defte Pogammg) oblem. The MATLAB Toolbox YALMIP [4] s use hee whch ca ecogze a solve SDP oblems effcetly Coollay : The fuzzy PID (PI o PD) cotol system cosstg of the T-S fuzzy moel () a the fuzzy PID (PI o Lc = [ ] Lc = [ ]. Futhemoe we have [C D ] = Lc R = [ ]

c [ D ] = = [-43.355 7.5839 7.35] C L R [ ] C C = = [ 3.338 4.69] C T [ ] C C = = [ 43.355 7.5839] C T whee T s a ut matx. Fom (3) we obta = 4.4354 =.8 =.589 = 68.75 =.379 =.88.

CONCLUSIONS I ths ae smle a effectve fuzzy PID PI a PD cotolles esg methos have bee oose a stue. Fstly state sace eesetatos of the fuzzy PID PI a PD cotolles ae esete.

6 c [ D ] = = [ ] C L R [ ] C C = = [ ] C T [ ] C C = = [ ] C T whee T s a ut matx. Fom (3) we obta = =.8 =.589 = =.379 =.88. The esoses fo the tal coto x () = [ π / 4 ] T ae ecte Fg. a Fg. whee we ca see that the close loo s asymtotcally stable. = = Fo fuzzy PD cotolle we have = 4.57 =.457 = 69.4 =.83. V. CONCLUSIONS I ths ae smle a effectve fuzzy PID PI a PD cotolles esg methos have bee oose a stue. Fstly state sace eesetatos of the fuzzy PID PI a PD cotolles ae esete. The the fuzzy PID (PI o PD) cotolle esg oblem s tasfome equvaletly to the oblem of esgg the fuzzy statc outut feeback cotolle. Next the latte oblem ca be effcetly hale by the covex otmzato techques. A smulato examle has bee fally gve to emostate the effectveess of ou aoaches. ACNOWLEGMENT Ths wok was suote by Helogag Postoctoal Fu (No. LBH-Z4) a the ey Laboatoy Pogam (No. BAQQ57855 a No. BAQQ578545) Fg.. Resoses of x a x () t wth tal coto x () = [ π / 4 ] T. Fg.. The outut of the fuzzy PID cotolle. Smlaly the fuzzy PI a PD cotolles ca be use to stablze the T-S fuzzy moel. To save the sace oly the esg esults ae ove. Fo the fuzzy PI cotolle we have = = 7.67 REFERENCES []. J. Åstöm a T. Hägglu PID Cotolles Theoy: Desg a Tug. Reseach Tagle Pak NC: Istum. Soc. Ame [] G. Che. Covetoal a fuzzy PID cotolles: A ovevew. Itellget cotol a system 996 : [3] J. Cavaal G. Che H. Ogme. Fuzzy PID cotolle: Desg efomace evaluato a stablty alayss. Ifomato Sceces 3: [4] C. Y. L W. X. Jg. Fuzzy PID cotolle fo D ffeetal geometc guace a cotol oblem. IET Cotol Theoy a Alcatos 7 (3): [5] D. Ms H. A. Malk G. Che. Desg a aalyss of a fuzzy ootoal-tegal-evatve cotolle. Fuzzy Sets a systems : [6] J. Aacl F. Gollo. Descbg fucto metho fo stablty aalyss of PD a PI fuzzy cotolles. Fuzzy Sets a Systems. 4 43: [7] R.. Mu N. R. Pal. A obust self-tug scheme fo PI- a PD-tye fuzzy cotolles. IEEE Tasactos o Fuzzy Systems 999 7(): - 6. [8] T. H. S. L a M. Y. Sheh. Desg of a GA-base fuzzy PID cotolle fo o-mmum hase systems. Fuzzy Sets a Systems : [9] 9. R. E. H. Guea G. S. Baess R. H. Habe A. Alque J. R. Alque. Usg ccle ctea fo vefyg asymtotc stablty PI-lke fuzzy cotol systems: alcato to the mllg ocess. IEE Poceegs Cotol Theoy a Alcatos 3 5(6): [] X. J. Ba X. Z. Gao X. L. Huag A. V. Vaslakos. Stablty aalyss of the smlest Takag-Sugeo fuzzy cotol system usg ccle cteo. Ifomato Sceces 7 77: []. R. Cao X. Z. Gao X. L. Huag X. J. Ba. Stablty aalyss of a tye of Takag-Sugeo PI fuzzy cotol systems usg ccle cteo. Iteatoal Joual of Comutatoal Itellgece Systems. 4():96-7. [] Y. S. Dg H. Yg S. H. Shao. Tycal Takag-Sugeo PI a PD fuzzy cotolles: aalytcal stuctues a stablty aalyss. Ifomato Sceces 3 5: 45-6.

7 [3] F. Zheg Q. G. Wag T. H. Lee X. G. Huag. Robust PI cotolle esg fo olea systems va fuzzy moelg aoach. IEEE Tasactos o Systems Ma a Cybeetcs-Pat A: Systems a Humas 3(6): [4] J. Aacl F. Gollo. Descbg fucto metho fo stablty aalyss of PD a PI fuzzy cotolles 4 43: [5] A. V. Sat. R. Raagoal. PM Sychoous moto see cotol usg hyb fuzzy-pi wth ovel swtchg fuctos. IEEE Tasactos o Magetcs 9 45(): [6] R. E. Pecu S. Petl P. oo. Fuzzy cotolles wth maxmum sestvty fo sevosystems. IEEE Tasactos o ustal electocs 7 54(3): [7] J. W. Jug Y. S. Cho V. Q. Leu H. H. Cho. Fuzzy PI-tye cuet cotolles fo emaet maget sychoous motos. IET Electc Powe Alcatos 5(): [8] B. T. Zhag Y. P. Robust factoal oe ooto-lus-ffeetal cotolle base o fuzzy feece fo emaet maget sychoous moto. IET Cotol Theoy a Alcatos 6(6): [9] L. E. Ghaou F. Ousty M. AtRam. A Coe Comlemetay Leazato Algothm fo Statc Outut-Feeback a Relate Poblems. IEEE Tasactos o Automatc Cotol 997 4(8): [] Y. Y. Cao J. Lam Y. X. Su. Statc Outut Feeback Stablzato: A ILMI Aoach. Automatca (): [] C. A. R. Cusus A. Tofo. Suffcet LMI cotos fo outut feeback cotol oblems. IEEE Tasactos o Automatc Cotol (5): []. Taaka H. O. Wag. Fuzzy Cotol Systems Desg a Aalyss: A Lea Matx Iequalty Aoach New Yok: Wley. [3] H. D. Tua P. Akaa T. Nakyo Y. Yamamoto. Paameteze lea matx equalty techques fuzzy cotol system esg. : 9(): [4] J. Löfbeg. YALMIP: A toolbox fo moelg a otmzato MATLAB. Poceegs of 4 IEEE Iteatoal Symosum o Comute Ae Cotol Systems Desg Tae Tawa Setembe [5] J. C. Lo M. L. L. Robust olea cotol va fuzzy statc outut feeback. IEEE Tasactos o Ccuts a Systems-I: Fuametal Theoy a Alcatos 3 5():

Robust Decentralized Control - A Linear Matrix Inequality(LMI)-Based Design

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