Delay-Dependent Control for Time-Delayed T-S Fuzzy Systems Using Descriptor Representation

Transcription

1 82 Inenaonal Jounal of Conol Auomaon and Sysems Vol 2 No 2 June 2004 Delay-Dependen Conol fo me-delayed -S Fuzzy Sysems Usng Descpo Repesenaon Eun ae Jeung Do Chang Oh and Hong Bae ak Absac: hs pape pesens a desgn mehod of delay-dependen conol fo -S fuzzy sysems wh me delays Based on paallel dsbued compensaon (DC) and a descpo model ansfomaon of he sysem a delay-dependen conol s ulzed An appopae Lyapunov-Kasovsk funconal s chosen fo delay-dependen sably analyss A suffcen condon fo delay-dependen conol s epesened n ems of lnea max nequales (LMIs) Keywods: Delay-dependen conol descpo epesenaon LMI -S fuzzy model INRODUCION Snce mos physcal sysems and conol sysems ae epesened by nonlnea dffeenal equaons he sably analyss and sablzng conolle desgn of nonlnea sysems ae vey mpoan A ypcal appoach fo he analyss and synhess of nonlnea conol sysems s o ulze local lneazaon he possbly of analyss and synhess of nonlnea conol sysems began o be ealzed wh he publcaon of he -S fuzzy model noduced by akag and Sugeno [] Recenly Wang e al [23] poposed an effcen mehod fo desgnng fuzzy conolle fo -S fuzzy sysems usng he concep of DC he dea s o desgn lnea feedback conol gan fo each local lnea model heefoe he desgn mehod of he DC-ype fuzzy conolle s smla o ha of he lnea conolle me-delay ofen occus n he sae he conol npu and he measuemen oupu of numeous sysems such as anspoaon sysems communcaon sysems chemcal pocessng sysems envonmenal sysems and powe sysems [4] Snce medelay s fequenly a souce of nsably he sably Manuscp eceved July ; evsed Febuay ; acceped Febuay Recommended by Edoal Boad membe Jn Young Cho unde he decon of Edo Jn Bae ak hs eseach was fnancally suppoed by Changwon Naonal Unvesy n 2003 Eun ae Jeung s wh he Depamen of Conol and Insumenaon Engneeng Changwon Naonal Unvesy 9 Sam-dong Changwon Kyungnam Koea (e-mal: e26@samchangwonack) Do Chang Oh s wh he Depamen of Eleconcs and Infomaon Konyang Unvesy 26 Nae-dong Nonsan Chungnam Koea (e-mal: docoh@konyangack) Hong Bae ak s wh he School of Eleconc and Eleccal Engneeng Kyungpook Naonal Unvesy Daegu Koea (e-mal: hbpak@eeknuack) poblems elaed o me-delay sysems have eceved consdeable aenon ove he las 3 decades (see [4-6] and he efeences heen) I s well known ha he choce of a Lyapunov-Kasovsk funconal s decsve fo showng sably hee ae wo sably cea One s delay-ndependen he ohe s (less consevave) delay-dependen A few papes [78] fo nonlnea sysems wh me delay usng he -S fuzzy model have been wen bu hese ae only fo delayndependen conol hee ae no publcaons ye fo delay-dependen conol fo me-delayed -S fuzzy sysems In he cuen pape we pesen a mehod of desgnng delay-dependen conol fo me-delayed - S fuzzy conol sysems usng he concep of DC We also use he equvalen descpo model ansfomaon poposed by Fdman [9] An appopae Lyapunov-Kasovsk funconal s chosen fo delaydependen sably analyss A suffcen condon fo he exsence of a DC-ype conolle s epesened n ems of LMIs 2 A DESCRIOR MODEL RANSFOR- MAION OF -S FUZZY SYSEMS he -S fuzzy model s an effecve way o epesen a nonlnea dynamc sysem I uses a lnea sae-vaable descpon as s consequence of ndvdual plan ules A me-delayed -S fuzzy model s composed of plan ules ha can be epesened as follows: lan Rule : IF z () s M and and z p () s M p HEN x () = Ax() + D x( τ ) + Bu() () x () = φ() [ τ 0]

2 Inenaonal Jounal of Conol Auomaon and Sysems Vol 2 No 2 June whee = 2 M s he fuzzy se and s he numbe of ules x() R n s he sae u () R m s he conol npu τ > 0 s he me delay of he sysem φ() s a veco-valued nal connuous funcon A D and B ae consan maces wh appopae dmensons and z() = [ z() z2() zp()] ae known pemse vaables whch may be funcons of he saes exenal dsubances and/o me Gven a pa of [ x( ) u( ) z( )] by usng he cene of gavy fo defuzzfcaon he fnal sae of he -S fuzzy sysem s nfeed as follows: = x () = h ( z())[ Ax() + D x( τ ) + Bu()] (2) whee h ( z( )) = w ( z( )) w ( z( )) = p w ( z( )) = M ( z ( )) = M ( z ( )) s he gade of membeshp of z ( ) n M and s assumed ha w ( z( )) 0 = 2 = w ( z( )) > 0 fo all heefoe h ( z( )) 0 = 2 = h ( z( )) = fo all Fom [9] we epesen (3) n he equvalen descpo fom: x () = y () = 0 = y () + h( z ())[ Ax () + Dx ( τ ) + Bu ()](3) o 0 = η() = h ( z())[ Aη() + D E η( τ) + Bu()] (4) whee x() I 0 η() = y () E = 0 0 E0 = [ I 0] 0 I 0 0 A = A I D = D B = B In he nex secon we wll pesen a suffcen condon of delay-dependen sably and an exsence condon of sablzng sae feedback gan fo -S fuzzy me-delayed sysems Befoe closng hs secon we ecall ak s nequaly whch wll be used o pove he man esuls n Lemma [0]: Assume ha a( α) R and m b( α) R ae gven fo α Ω hen fo any maces X > 0 and M he followng holds: 2 b ( α) a( α) dα Ω a( α) X XM a( α) dα Ω b( α) M X (22) b( α) whee (22) denoes ( M X + I) X ( XM + I) 3 MAIN RESULS (5) In he sequel we pesen ou man esuls on delaydependen sably analyss and sablzaon fo medelayed -S fuzzy sysems based on he LMI mehod 3 Sably analyss Fs we deve he sably condon of unfoced me-delayed sysems as follows: η() = h( z())[ Aη() + DE0η( τ)] (6) = Snce η( τ) = η( ) η( s) ds (6) can be τ ewen as = η( ) = h ( z( ))[( A + D E ) η( ) D 0 τ ysds ( ) ] (7) Consde a Lyapunov-Kasovsk funconal fo he sysem (6) as V( η( )) = V( η( )) + V2( η( )) + V3( η( )) V ( η ( )) = η ( ) Eη ( ) 2 ( ( )) V η τ = x ( θ) Rx ( θ) d θ (8) whee V ( η()) y () s Ry() s ds 3 τ 0 τ + θ = 0 = > R > 0 (9) heoem : Fo a gven τ > 0 he equlbum of he sysem (6) s asympocally sable n he lage f

3 84 Inenaonal Jounal of Conol Auomaon and Sysems Vol 2 No 2 June 2004 hee exs maces Q > 0 Q 2 Q 3 U > 0 ha sasfy he followng LMIs Q 0 τq 2 Φ τdm DM τq τ 3 Q τmd τu 0 0 < 0 [ τq2 τq3] 0 τu 0 0 τmd 0 0 τu = 2 (0) whee + Φ = ( A + D ) Q Q + Q Q2 Q2 2 3 Q( A + D) Q2 + Q3 Q3 Q3 τ D( M M U) D () oof: he poof wll be compleed by showng V ( η ( )) < 0 he me devave of V ( η ( )) s ( η( )) = 2 η ( ) η( ) = 2 η ( ) h ( z ( )) = whee ( A DE 0) η( ) D y( s) ds + τ (2) = 2 η ( ) h( z ( ))( A + DE 0) η( ) = 2 ξ( ) = τ (3) = ξ() h ( z ()) η () D y ( s ) ds Fom lemma we have τ 2 ξ( ) y ( θ) Ry( θ) + 2 h( z( )) η ( ) DM R y( θ) τ (4) = = = h ( z( )) h ( z( )) η ( ) D ( M R+ I) R ( RM + I) D η ( ) Snce he las em n he gh-hand sde of (4) s less han o equal o h( z( )) η ( ) D[ RM + I] R [ RM + I] D η ( ) = he lef-hand sde of (4) has an uppe bound as shown below: τ 2 ξ( ) y ( θ) Ry( θ) o + 2 h( z( )) η ( ) DM R y( θ) τ (5) = = h ( z( )) η ( ) D ( M R+ I) R ( RM + I) D η ( ) he me devaves of V 2 ( η ( )) and V 3 ( η ( )) ae 2 ( η( )) = τ [ x ( ) Rx( ) x ( τ) Rx( τ)] (6) ( η( )) 3 0 τ 0 = y () Ry() y ( + θ ) Ry( + θ) = τ y () Ry() y ( + θ) Ry( + θ) τ Fom (2) (6) and (7) ( η( )) h ( z( )) η ( ) ( A + D E ) = = + ( A + DE0) } η( ) = { + 2 h ( z( )) η ( ) η () E RE η() τ x ( τ) Rx( τ) y () Ry() 0 [ η( ) ( τ )] DM R E x h ( z( )) η ( ) D[ RM + I] R [ RM + I] D η ( ) () η ( η( )) h ( z( )) x ( τ ) = Ψ DM R η() RMD x ( ) τ R τ whee (7) (8) (9)

4 Inenaonal Jounal of Conol Auomaon and Sysems Vol 2 No 2 June ( A DE0) ( A DE0) + DM RE0 + E0 RMD E 0 RE 0 E 0 RE = [ 0 I] Ψ = D ( RM I) R ( RM I) D E Fo smplcy defne Q 0 Q= = Q2 Q U = R 3 hen (0) and () ae epesened by Φ * * * τmd + E0Q τr 0 0 < τe0q 0 τr 0 τmd 0 0 τr = 2 Φ = ( A + D E ) Q+ Q ( A + D E ) 0 0 D( M + M + R ) D 0 (20) especvely whee * denoes he ansposed ems fo symmec posons Fom Schu complemens [] (20) s equvalen o o τ 0 τ τ 0 D( M M R ) D Q E0RE0Q τdm RMD 0 ( A D E ) Q Q ( A D E ) + ( MD + E Q) R( MD + E Q) < Q E0RE0Q D( RM I) R ( RM I) D Q E0RE0Q τdm RMD ( A D E ) Q Q ( A D E ) + DM RE Q+ Q E RMD + + (2) (22) + < 0 because he sum of he hd and foh ems n he lef-hand sde of (2) ae equal o D M RE Q+ Q E RMD Q E RE Q D( RM + I) R ( RM + I) D By he pe- and pos-mulplyng of and especvely o (22) we ge Ψ DM RMD < 0 (23) Fom (9) (23) mples V ( η ( )) < 0 32 Delay-dependen sablzaon of me-delayed -S fuzzy sysems Ou goal n hs subsecon s o sablze me delayed -S fuzzy sysems usng he DC appoach [2 3] ha s we pesen a suffcen condon of exsence of he DC-ype delay-dependen conolle he DC-ype conolle shaes he same fuzzy ses wh he me-delayed -S fuzzy sysem () he DCype conolle afe defuzzfcaon s = = u () = h( z ()) Kx () = h ( z( )) K E η( ) 0 (24) Subsung hs conolle no (4) we oban he closed-loop sysem = = η( ) = h ( z( )) h ( z( )) { A η D E0η d } () + ( ) (25) whee A = A + B KE0 heoem 2: Assume ha he numbe of ules ha fe fo all s less han o equal o s whee < s Fo a gven τ > 0 he equlbum of he closed-loop sysem (25) s asympocally sable n he lage f hee exs maces Q > 0 Q 2 Q 3 U > 0 Z 0 M N ha sasfy he followng LMIs ( G + G ) S + ( v ) Z * * * 2 τmd + E0Q τu 0 0 < 0 τe0q 0 τu 0 τmd 0 0 τu = 2 (26) G + G * * * * ( S + S ) 2Z τmd + E0Q τu τmd 0 E0Q 0 τu 0 0 < + τm( D + D) 0 0 2τU 0 τe0q τu 2 < (27) whee G = ( A + A ) Q+ B N E 0 + BNE + ( D+ D) QE 2Q2 2Q3 = G2 2Q 3 0 0

5 86 Inenaonal Jounal of Conol Auomaon and Sysems Vol 2 No 2 June 2004 G2 = ( A + A ) Q 2Q2 + BN + B N + ( D + D ) Q = ( + + ) S D M M U D Fuhemoe sae feedback gans fo each ule of he me-delayed -S fuzzy sysem (4) ae K = NQ = 2 (28) oof: Fo he closed-loop sysem (25) he me devave of Lyapunov-Kasovsk funconal (8) sasfes he followng nequaly: () η ( η( )) h( z( )) h( z( )) x ( τ ) = = DM R (29) Ψ η() RMD x ( ) τ R τ whee Ψ = ( A + D E ) + ( A + D E ) ( τmd + E0) τ R( τmd + E0) E RE D ( M M R ) D Fom coollay 4 and heoem 5 n [2] he ghhand sde of (29) s less han zeo f hee exs 0 > and 0 Z such ha Ψ + ( v ) Z DM R < 0 RMD τ R = 2 (30) Ψ +Ψ 2 Z ( D + D ) M R 0 RM ( D + D ) 2τ R < (3) Now he poof wll be compleed f (30) and (3) ae equvalen especvely o (26) and (27) By he Q 0 pe- and pos-mulplyng of 0 I and Q 0 0 I especvely o (30) and (3) manpulang he max nequales usng Schu complemens and leng N = KQ (30) and (3) can be ewen as (26) and (27) especvely 4 AN EXAMLE We wll desgn a DC-ype conolle fo he followng me-delayed nonlnea sysem: Fg he smulaon esuls fo τ = 2 x () = x () 02 x () x ( τ ) x () = x () x () + 4 x ( τ ) + u() (32) I s assumed ha x () s measuable and x () [ 2 2] An equvalen -S fuzzy model o (32) s epesened by Rule : IF x () s M HEN x () = x() + x( τ ) + u() Rule 2: IF x () s M 2 HEN x () = x() + x( τ ) + u() x whee x() = [ x() x2() ] M and = 4 M 2 = M Fo τ = 2 all paamees Q U Z M N N 2 sasfyng (26) and (27) ae Q 0 Q = Q2 Q = Z = U = M =

6 Inenaonal Jounal of Conol Auomaon and Sysems Vol 2 No 2 June N = [ ] N 2 = [ ] he conol gans of each ule ae K = [ ] K 2 = [ ] Fg shows he smulaon esuls fo τ = 2 wh he nal value condons of x () = ( 0) and x 2 (0) = We oban a maxmum me delay τ of 2705 wh 6 coespondng sae-feedback gans of K = 0 [ ] and K 2 = 0 6 [ ] fo each ule 5 CONCLUSIONS By choosng an appopae Lyapunov-Kasovsk funconal we have desgned a delay-dependen DCype conolle fo me-delayed -S fuzzy conol sysems We have also used equvalen descpo model ansfomaon of he sysem A suffcen condon fo exsence of he DC-ype conolle has been epesened n ems of LMIs and sae feedback gans fo each ule have been obaned fom soluons of he LMIs An llusaed example has been gven o demonsae ou esuls REFERENCES [] akag and M Sugeno Fuzzy denfcaon of sysems and s applcaons o modelng and conol IEEE ans on Sys Man Cyben vol 5 no pp [2] H Wang K anaka and M Gffn aallel dsbued compensaon of nonlnea sysems by akag and Sugeno s fuzzy model oc FUZZ-IEEE Yokohama Japan pp [3] H O Wang K anaka and M F Gffn An appoach o fuzzy conol of nonlnea sysems: Sably and desgn ssues IEEE ans on Fuzzy Sys vol 4 no pp 4-23 Feb 996 [4] M Malek-Zavae and M Jamshd me-delay Sysems Analyss Opmzaon and Applcaons Sysems and Conol Sees vol 9 Noh- Holland Amsedam 987 [5] V Kolmanovsk and A Myshks Appled heoy of Funconal Dffeenal Equaons Mahemacs and Is Applcaons (Sove Sees) Boson Kluwe 999 [6] M Mahmoud Robus Conol and Fleng fo me-delay Sysems New Yok Macel Decke 2000 [7] Y-Y Cao and M Fank Analyss and synhess of nonlnea me-delay sysems va fuzzy conol appoach IEEE ans on Fuzzy Sys vol 8 no 2 pp Apl 2000 [8] K R Lee J H Km E Jeung and H B ak Oupu feedback obus H conol of uncean fuzzy dynamc sysems wh me-vayng delay IEEE ans on Fuzzy Sys vol 8 no 6 pp Dec 2000 [9] E Fdman New Lyapunov-Kasovsk funconals fo sably of lnea eaded and neual ype sysems Sys Conol Le vol 43 no 4 pp [0] ak A delay-dependen sably ceon fo sysems uncean me-nvaan delays IEEE ans on Auomac Conol vol 44 no 4 pp Apl 999 [] S Boyd L E Ghaou E Feon and V Balakshnan Lnea Max Inequales n Sysem and Conol heoy SIAM 994 [2] K anaka Ikeda and H O Wang Fuzzy egulaos and fuzzy obseves: Relaxed sably condons and LMI-based desgns IEEE ans on Fuzzy Sys vol 6 no 2 pp May 998

7 88 Inenaonal Jounal of Conol Auomaon and Sysems Vol 2 No 2 June 2004 conol H Eun ae Jeung eceved he BS MS and hd degees n Eleconc Engneeng fom Kyungpook Naonal Unvesy n and 996 especvely He s cuenly an Assocae ofesso n he Depamen of Conol and Insumenaon Engneeng a Changwon Naonal Unvesy Hs eseach neess nclude obus conol me delay sysems and fuzzy conol Do Chang Oh eceved he BS MS and hd degees n Eleconc Engneeng fom Kyungpook Naonal Unvesy n and 997 especvely He s cuenly an Asssan ofesso n he Depamen of Eleconcs and Infomaon Engneeng a Konyang Unvesy Hs eseach neess nclude model and conolle educon obus conol H conol and me delay sysems Hong Bae ak was bon n Koea on Mach 6 95 He eceved he BS and MS degees n Eleconc Engneeng fom Kyungpook Naonal Unvesy Daegu Koea n 977 and 977 especvely and hd degee n Eleccal and Compue Engneeng fom he Unvesy of New Mexco Albuqueque New Mexco USA n 988 He s cuenly a ofesso a he School of Eleccal Engneeng and Compue Scence Kyungpook Naonal Unvesy He eceved he "Hae-Dong ape Awad" fom IEEK n 988 Hs cuen eseach neess nclude obus conol opmal conol o ndusal applcaons gudance conol undewae acousc sgnal pocessng and bomedcal conol He s a membe of IEEE IEEK ICASE and ASK