Fuzzy Control of Inverted Robot Arm with Perturbed Time-Delay Affine Takagi-Sugeno Fuzzy Model

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1 7 IEEE Inenaonal Confeence on Robocs an Auomaon Roma Ialy -4 Al 7 FD5. Fuzzy Conol of Invee Robo Am wh Peube me-delay Affne akag-sugeno Fuzzy Moel Wen-Je Chang We-Han Huang an We Chang Absac A sably analyss an conolle synhess mehoology fo an nvee obo am sysem s oose n hs ae. hs uncean sysem s moele by a sae sace akag-sugeno (-S) fuzzy moel wh lnea nomnal a an sucue boune aamee unceanes n he sae equaons of each fuzzy ule. Fs a suffcen conon on obus sably of he Connuous Peube me-delay Affne -S (CPDAS) fuzzy moels of nvee obo am s oose. hen H -subance aenuaon efomance of he fuzzy moels s analyze. A las a numecal examle shows he use of he oose aoach on he sablzaon an H -subance aenuaon fo he nvee obo am sysems. I. INRODUCION he nvee obo am s an ubquous examle of nonlnea conol sysems analyss an esgn. he sana oblem elae o nvee obo am s sablzaon of ehe ownwa o ugh equlbums a a escbe oson. he uose of hs ae ams o make he nvee obo am balance; afewas he nvee obo am coul ge he ynamc balance. he ack of he nvee obo am can be hozonal o gaen. In hs ae we oose he sae feeback fuzzy conol base on he -S moel fo he nvee obo am. In ecen yeas he -S fuzzy conol [-8] has become one of he useful conol aoaches fo comlex nonlnea conol sysems. I s well known ha me elay ofen occus n many ynamcal sysems. heefoe me elay coul be consee as an moan ssue n -S fuzzy conol sysems. In he me-elay -S fuzzy moel [8] local ynamcs n ffeen sae sace egons ae eesene by lnea moels. he oveall -S fuzzy moel of he sysem s acheve by fuzzy blenng of hese me-elay lnea subsysems. he conol esgn s cae ou base on he so-calle Paallel Dsbue Comensaon (PDC) [-8] scheme. he ea s ha fo each local lnea moel a lnea feeback conol s esgne. he esulng oveall conolle whch s n geneal nonlnea s also a fuzzy blenng of each nvual lnea conolle. he conol aoach s o esgn lnea feeback gans fo each local lnea moel an o le he oveall conol nu can be blene by hese lnea feeback gans. One of he mos moan equemens fo a conol sysem s he so-calle obusness. hose can solve he subance an eubaons oblems fom nheen unceanes n he eal sysem. In hs ae he H conol scheme [9] s use o eal wh he obus efomance esgn oblems n CPDAS fuzzy moels. I can ove he guaanee H efomance fo he aenuaon γ whch can coe wh he wos subances n sysems. We also oose he ssue of obus sably n he esence of nom-boune unceany. he uncean moels ae escbe by a sae-sace moel an me-vayng nom-boune aamee unceany n he sysem maces. In geneal base on Lnea Max Inequales (LMI) [] mehos one can fn suable lnea feeback gans fo each fuzzy ule fo close-loo homogeneous -S fuzzy sysems. Howeve he synhess of he CPDAS fuzzy moels s a ffcul oblem fo he esgnes because he close-loo sably conons ae no LMI fomulaons bu Blnea Max Inequales (BMI) ones. he BMI conons canno be easly solve va a convex omzaon algohm. Fo hs eason an Ieave LMI (ILMI) [6-8] algohm s ale o solve he oose BMI oblem n hs ae. II. DYNAMIC MODEL OF INVERED ROBO ARM In hs secon we fs nouce he mahemacal moel of nvee obo am sysem. Refeng o Fg. we oose a smlfe ynamc moel o escbe oen-loo nvee obo am sysem as follows []. g k θ() = sn θ() θ l m () () + ev () whee l s he lengh of he o m s he mass of he bob g s he acceleaon ue o gavy k s he fcon θ( ) s he angle by he o an he vecal axs v ( ) enoes he subances. he oque s he conol nu an s assume ha he θ =β. In conol objec s o manan a consan angle oe o manan θ ( ) =β he oque mus have a seay-sae comonen ss ha sasfes sn ss o ss = ( β) a β + c = asn c () /7/$. 7 IEEE. 48

2 FD5. θ= θ= π/ θ=π/ l Fg. θ=π θ mg Invee obo am sysem Choose he sae vaables as () = θ( ) β x () =θ () an he conol vaable as () = ( ) ss x = = x u. hen he new equlbum on s () x an u () =. he nvee obo am equaon () can be hus eesene as () = x () () () x (a) { } () () x = a sn x +β sn β + cu + ev (b) hen we wll conse hs class nclung emse nomnal aamee unceany: () =ϕ() () () cu() ev() () (. sn() ) x (). sn x (4a) { sn. cos () () sn} x = a x +β + x β ϕ = ρ+ + (4b) ( ) x ( () ) + ρ + τ (4c) n whee ϕ() R s a me-elay weghng funcon an ρ [ ] s he weghng coeffcen. III. HE AFFINE -S FUZZY MODEL OF INVERED ROBO ARM Geneally he man feaue of he -S fuzzy sysem can be exesse by jonng ynamcs of each fuzzy ule of lnea subsysems. Gven a a of ( x () u () ) he eube me-elay affne -S fuzzy moel of nvee obo am sysem () can be nfee as follows [8]: () = h( z() ) ( +Δ ) x() + ( +Δ ) x τ() x = { } A A A A ( ()) { } () h z B B u a a E v (5) = + +Δ + +Δ + whee ( ) = ( ) ( ) ( ) z() z z z... z h ( z () ) = ω ( z ( ) ) () ω ( z () ) = ( z ()) ω = M j j j= h z an h z () = (6) he quanes A A B a an E ae consan maces. Beses ΔA ΔA ΔB an Δa ae me-vayng maces wh aoae mensons an hey ae sucue n he followng nom-boune fom: = [ Δ Δ Δ Δ ] = ( )[ ] A A B a DΔ Q Q Q Q (7) 4 whee D Q Q Q an Q 4 ae known eal consan Δ s an unknown max funcon wh Lebesgue-measuable Δ Δ I. maces of aoae mensons an ( ) elemens an sasfes Fo a nonlnea -S fuzzy sysem eesene by (5) a fuzzy conolle s esgne o shae he same fuzzy ses wh he lan. I s base on he PDC conce []. he ouu of he PDC-base fuzzy conolle s eemne by he summaon such as () () { () } F μ (8) = u = h z x + Subsung (8) no (5) one can oban he coesonng close-loo sysem x () = h ( ˆ x() ) hj( ˆx() ) = j= {( Aj + DΔ Aj ) + ( Aj + DjΔ Aj )} x () + Ev() whee () = () τ() x x x A = j G A g j j Gj = A BF j gj = a Bμ j an Δ A j = Δ Q Q F Q Q j 4 Q μ j. Base on he PDC ye fuzzy conolle (8) a suffcen conon fo ensung elay-neenen sably of conolle me-elay affne -S fuzzy moel (9) s nouce n hs ae. Moeove a H conol efomance wh γ > s also consee n hs ae. hs consan s of he followng fom. (9) 48

3 FD5. f () () <γ () () f x S x v v () wh zeo nal conon fo all v() L[ f] whee γ s a escbe value whch enoes he wos case effec of v () on x (). Beses S= S > s a osve-efne n n weghng max an S R. he uose of hs ae s o fn saable fuzzy conolles (8) such ha he close-loo sysem (9) s obusly sable wh sasfyng he H consan (). IV. SUFFICIEN CONDIIONS OF ROBUS FUZZY CONROLLER DESIGN A fuzzy conolle s esgne o shae he same fuzzy ses wh he affne -S fuzzy moel (5) base on he PDC scheme. In hs secon he elay-neenen sably conons fo he CPDAS fuzzy moel (9) ae escbe n he followng heoem. heoem Gven a H aenuaon aamee γ >. he CPDAS fuzzy sysem escbe by (9) s quaacally sable n he lage an he H conol efomance () s guaanee fo an aenuaon γ f hee exs common osve efne maces P > S > N > conol gans F μ an scalas ξjq such ha c c R ϒ j Γj + Θj < fo ˆ I P QQ + QjQj + N (a) c c R ϒ j Γj + Θj < P QQ + QjQj + N fo ˆ I (b) whee c Gj + Gj Gj + Gj Γj = P+ P () c Γj ξjqjq q c = Γj () gj + gj P ξjq jq ξjqv jq n q= q= R Θ = U + P+ PE I γ E P+ S (4) j j R Θ R j Θj U j U = PA N A P+ PA N A P j j j + P D D j { I } D Dj P (5) ( Q Q F ) ( Q Q F ) ( ) + j j + Q Q F Q Q F (6) j j j j U Q Q μ Q Q μ j = 4 j 4 j + Q Q μ Q Q μ (7) 4j j 4j j Fom heoem can be noe ha he max nequales n P F an μ j belong o he class of BMIs an he conolle synhess canno be solve wh ease by a convex omzaon algohm. In oe o solve he esen obus fuzzy conolle esgn oblem s necessay o ewe he conons of heoem l. In nex secon a new heoem s ove o nouce new sably conons whch can be solve by an ILMI algohm. V. ROBUS FUZZY CONROLLER DESIGN VIA ILMI ALGORIHM In hs secon an ILMI algohm s ove o ge a suable soluon fo he sably conons of heoem. he ecay ae α s consee n he sably conons n oe o elax he LMI seach oceue an make feasble. heoem he sably conons () escbe n heoem ae hel an he CPDAS s quaacally sable n he lage f hee exss a ecay ae α < osve efne maces P > S > N > conol gans F μ an scalas ξjq such ha c Γj < P QQ + QjQj + N fo Î (8a) c Γj < P QQ + QjQj + N fo Î (8b) whee R αp L4j L5jP c L6jP N Γj L7jP L8j L9j EP 48

4 FD5. γ I < (9) R αp R R Lj Lj Lj c L4j Γj L5jP L6jP L7jP L8j L9j EP () < N γ I an Lj = μ μ j Lj = Q4 Qμ j Lj = Q4 j Qjμ L4j = F F j L5j = B B j L6j = A A j L7j = D D j L8j = Q QF j L9j = Qj QjF = ξ () R R jq jq q= R = ( a Byj) P+ ( aj Bjy) P ( y j μ j ) z ( y μ ) zj jqnjq q= + + ξ = yy+ yy yμ μ y R j j j j μjy j jqvjq q= y μ ξ () () = A P + PA + A P + PA + Y Y + Y Y R j j j j j j Yj ( B P Fj ) ( PB Fj ) Yj Yj ( Bj P F ) PB+ F Y + zz+ zz zbp PBz zbp j j j j j j PB jz j + P + S (4) n whch Y = B P+ F z = B P y = μ (5) j j Accong o he conons of heoem he soluons of fuzzy conol oblem of CPDAS fuzzy sysems can be obane by alyng ILMI algohm whch s eveloe on he LMI echnque. he flowcha of ILMI algohm whch can be use o solve he conons of heoem s nouce as follows. <ILMI Algohm> In whch ( ) Oban nal P fom he followng Rca equaon. ˆ ˆ ( ) ˆˆ ( ) AP + P A P BBP + Q= ( ) Ge nal F by sana ole lacemen echnque. ( k) ( k) ( k) Se Yj y an z fom (5). Solvng by LMI oolbox Mn ( k) α P S N F μ ξ ( k) ( k) ( k) ( k) ( k) ( k) jq Subjec o ( k) ( k) ( k) ( k) P > S > N > ξjq an (8) ( k) Yes α < Feasble soluon No ( k) Mn ace ( P ) k P Subjec o ( k) ( k) ( k) ( k) P > S > N > ξ an (8) { jq ( k) ( k) ( k) Yj B P + Fj j ( k) ( k) + z B P } k k + y μ <υ Yes Infeasble an so neaon = Fg. Aˆ = A ILMI Algohm = No k = k+ Bˆ = B an Q > υ s a eeemne small. Alyng he above ILMI algohm one can oban a fuzzy conolle (8) o sablze he CPDAS fuzzy sysems (5) wh sasfyng he H efomance consan (). In nex secon a numecal examle s ove o emonsae he usefulness an effecveness of he oose esgn aoach. VI. NUMERICAL SIMULAIONS 48

5 FD5. o conse he me elay effec n he acualy suaon s assume ha he senso fo exlong he x ( ) = θ ( ) s eube by me elay gven as: whee () () Δ A = DΔ Q (6a) Δ A = DΔ Q (6b) (). cos D. Δ () sn() Q Q o oban he CPDAS fuzzy moel of he nvee obo am sysem s necessay o aly he lneazaon echnque []. Le us choose hee oeang ons as follows: ( x x u ) = ( 58 ) oe ( x x u ) = oe ( ) an ( x x u ) = ( 8 ) oe (7) hen hee lnea subsysems can be consuce by hese hee oeang ons. In whch ( x x u ) s he oe manan equlbum on an he ohes ae he off-equlbum ons. hough consucng he above hee lnea subsysems an efnng membesh funcons as Fg. one can oban he me-elay affne -S fuzzy moel whch s comose by hee ules as follows: Rule : IF x () s abou M HEN x () = ( A +Δ A ) x() + ( A +ΔA ) x( τ( ) ) + u() + + v() Rule : IF x () s abou M HEN x () = ( A +Δ A ) x() + ( A +ΔA ) x τ( ) + u() + + v() Rule : IF x () s abou M HEN x () = ( A +Δ A ) x() + ( A +ΔA ) x τ( ) + u() + + v() whee B a E (8a) B a E (8b) B a E (8c) A. 495 A A.9 B = B = B. sn() Δ A =Δ A = Δ A.cos (). 5 A = A = A. sn() Δ A = Δ A =Δ A a a a 7.5 E 5. Accong o he membesh funcons efne n Fg. he S-oceue s esene as follows. Fo Rule s.e. x ( ) 6 he maces of S-oceue ae gven as follows: ( π / π/ 8 ) n an v = π / 8 6π / 8 (9) Fo Rule s.e. x () he maces of S-oceue ae gven as follows: ( π/ 8 π/ 8) n an v = π / 8 π / 8 () Fo he above CPDAS fuzzy moel (8) he fuzzy conolle can be esgne by alyng heoem an he ILMI algohm. In hs examle s assume ha he H conol efomance s guaanee fo an aenuaon γ =.. Beses he subance s v ( ) =. 5sn( ). hen we can ge a feasble soluon afe fou eaons of he fuzzy conolle esgn oceue. he fnal ecay ae α s.578 an he feasble soluons ae obane as follows: P.478. S e9 N e9 ξ =.6 ξ = () 484

6 FD5. An he fuzzy conolle has he followng fom: Rule : IF x () s abou M HEN u [ ] x Rule : IF x () s abou M HEN u( ) [ ] x( ) Rule : IF x () s abou M HEN u [ ] x = (a) = (b) = (c) he smulaon esuls ae shown n Fg. 4 an Fg. 5. Fom he smulae esuls one can fn ha he conolle nonlnea me-elay nvee obo am sysem (4) s globally sable. VII. CONCLUSIONS In hs ae we have shown ha he eube nvee obo am sysem wh subance can be conolle by he -S fuzzy conolles. he oose fuzzy conolle esgn aoach was eveloe va PDC meho an ILMI algohm. Fnally he smulaon esuls showe ha he fuzzy conolle esgne n hs ae can sablze he nonlnea nvee obo am subjec o sasfyng H efomance consan. [8] W. J. Chang an W. Chang Dscee Fuzzy Conol of me-delay Affne akag-sugeno Fuzzy Moels wh H Consan IEE Poceeng Pa D Conol heoy an Alcaons Vol. 5 No [9] K. Zhou J. C. Doyle an K. Glove Robus an Omal Conol Englewoo Clffs NJ: Pence Hall 996. [] H. K. Khall Nonlnea Sysems E. Pence Hall. M M M Fg. Membesh funcons of x () x ( ) ACKNOWLEDGEMEN hs wok was suoe by he Naonal Scence Councl of he Reublc of Chna une Conac NSC95--E-9-. REFERENCES [] K. anaka an H. O. Wang Fuzzy Conol Sysems Desgn an Analyss A Lnea Max Inequaly Aoach John Wley & New Yok. [] W. J. Chang an C. C. Sun Consane Fuzzy Conolle Desgn of Dscee akag-sugeno Fuzzy Moels Fuzzy Ses an Sysems Vol. No [] W. J. Chang Fuzzy Conolle Desgn va he Invese Soluon of Lyaunov Equaons ASME J. Dynamc Sysems Measuemen an Conol Vol. 5 No [4] W. J. Chang C. C. Sun an H. Y. Chung Fuzzy Conolle Desgn fo Dscee Conollably Canoncal akag-sugeno Fuzzy Sysems IEE Poceeng Pa D Conol heoy an Alcaons Vol. 5 No [5] W. J. Chang an S. M. Wu Connuous Fuzzy Conolle Desgn Subjec o Mnmzng Conol Inu Enegy wh Ouu Vaance Consans Euoean Jounal of Conol Vol. No [6] E. Km an D. Km Sably Analyss an Synhess fo an Affne Fuzzy Sysem va LMI an ILMI: Dscee Case IEEE ans. Sysems Man an Cybenecs Pa B Vol. No [7] E. Km an S. Km Sably Analyss an Synhess fo an Affne Fuzzy Conol Sysem va LMI an ILMI: A Connuous Case IEEE ans. Fuzzy Sysems Vol. No Fg. 4 Resonses of x () Fg. 5 Resonses of x () 485

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

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