Global Practical Output Tracking of Uncertain Nonlinear Systems By Smooth Output Feedback

Transcription

1 Poceedgs of the 9th WSEAS Iteatoal Cofeece o Automatc Cotol, odelg & Smulato, Istabul, ukey, ay 7-9, 7 7 Global Pactcal Outut ackg of Uceta Nolea Systems By Smooth Outut Feedback EYLAN ALIHAN, HIROSHI INABA Deatmet of Ifomato Sceces okyo Dek Uvesty Hatoyama-mach, Hk-gu, Satama 5-94 JAPAN keyla@j.deda.ac.j;aba@cck.deda.ac.j Abstact: hs ae cosdes a global actcal outut tackg oblem fo a famly of uceta olea systems whose Jacoba leazato s ethe cotollable o obsevable. It s show that ude some mld codtos o such a system thee s a smooth outut feedback achevg global actcal outut tackg ad such a smooth outut cotolle s exlctly costucted by a ew desg method oosed. he usefuless of ou esult s llustated by a umecal examle. ey-wods: Global actcal outut tackg, smooth outut feedback, uceta olea systems, escalg tasfomato. Itoducto ad Poblem Statemet I ths ae, we cosde the outut tackg oblem of a class of olea systems of the fom x& = x φ(, t x, u, =,,, x& = u φ(, t x, u, ( y = x whee x= [ x, x, L, x ] R, u R ad y R ae the system state, ut ad measuemet outut, esectvely, ad s a odd tege. Fo =,,, φ : R R, ae ukow olea C fuctos that volve ucetaty ad may ot be ecsely kow ad y s a efeece sgal. he goal s to egulate the outut y to zeo. he olea the oblem of olea outut egulato s to obta a feedback law makg the cotolled outut of a system asymtotcally tack a escbed smooth efeece sgal. he majo aoach to solvg of the oblem ogally toduced by Davso, Facs, ad Woham [-] fo lea systems. he study of the coesodg oblem a olea settg has bee extesvely vestgated by a umbe of eseaches ove the ast twety yeas;fo stace, see [-6]. Howeve, these aes cosdeed oly a smle case whee the efeece sgals oduced by the exosystem ae costats. Late o, the oblem fo olea systems wth tme-vayg efeece sgals was addessed by Isdo ad Byes [7]. ost of the afoemetoed cotbutos to the olea egulato theoy eque that the Jacoba leazato of a cotolled olea systems be stablzable ad detectable(at least atally []. Stablzablty ad detectablty of the the leazed system ae two key assumtos fo solvg the oblem of olea outut egulato by ethe state o eo feedback [7]. Howeve, the case whe the system ude cosdeato s geuely olea, the oblem becomes moe comlcated ad dffcult to solve. Fo olea systems wth ucotollable/uobsevable leazato, thee ae vey few esults the lteatue, hee we evew oly [8-9] ad [] that ae closely elated to ths ae. I [8], studed the local outut egulato oblem fo a class of tagula systems whose leazed system may be ot stablzable o detectable ad a local cotuous cotolle was desged, focg the tackg eo wth a o gve boud ad [] cosdeed the oblem of global outut tackg fo a secal case of a cosdeably geeal class of sgle ut sgle outut olea systems ( wth dffeets. It was oved [] that the oblem of global asymtotc outut tackg of a costat sgal s solvable by smooth state feedback. Howeve, outut asymtotc tackg s usually ot be ossble (eve locally fo such olea systems ( (eve by smooth state feedback [9]. I [9], Qa ad L eve gave a coute-examle to show that eve f the stablzato oblem s solvable, the coesodg asymtotc outut tackg oblem s ot ossble. o deal wth such a oblem, the lteatue, a moe sutable cocet was toduced, that s actcal outut tackg [8], [9].

2 Poceedgs of the 9th WSEAS Iteatoal Cofeece o Automatc Cotol, odelg & Smulato, Istabul, ukey, ay 7-9, 7 8 Fo system (, the global actcal outut tackg va state feedback has bee solved [9]. Howeve, f the cotolle s equed to oly deed o the outut, whch s meagful the actcal mlemetato, the oblem s stll oe. Hee, we fst gve a ecse defto of the oblem. Pactcal outut tackg va Outut Feedbak: Let y ( t be a bouded C efeece sgal whose devatve y& ( t s also bouded. Fo ay gveε >, desg a outut feedback cotolle of the fom & m ζ = α( ζ, y, ζ R, u = u( ζ, y ( Evey state of the closed-loo system (-( s well-defed o [, ad globally bouded. Fo evey x( R, thee s a fte-tme > the outut of the closed-loo system (-( satsfes y = x ( t ( t < ε, t >. ( I ode to fd such a outut feedback cotol, we eed to costuct a obseve to ecove the state fomato. Recetly a ovel olea obseve desg s toduced []. I [], t has bee show that the global stablzato of the system ( ca be solved by smooth outut feedback ude sutable gowth codtos. hese gowth codtos guaatees, the exstece of smooth outut cotolles. I ths ae, we exted the esult [] to solve the global outut tackg oblem ad dcates that the oblem of global actcal outut tackg s solvable. o ths ed, the followg codtos ae toduced. Assumto : Fo the olea system (, thee exsts a eal umbe C > fo =,,,, ( φ (, txu, C x L x C (4 Assumto : hee s a kow costat D such that y( t D, y& ( t D (5 Remak. Assumto s slghtly moe geeal tha the assumto mosed [] whch cotas o costat tem. Due to the lack of ths costat tem gowth-codtos of [], the smooth outut feedback cotol schemes develoed [] caot be aled to such a class of geuely olea systems such as x& = x x, x& = u, y = x (6 Notably, (6 volve a ucotollable/uobsevable leazato. oeove, the ucotollable mode has a egevalue o the oe ght-half lae. As a esult, thee ae o smooth state/outut feedback cotol laws stablzg ethe (6 at the og, eve locally. Hece, eedless to say, the oblem of global outut tackg s, moe challegg ad dffcult tha the feedback stablzato. Due to the gowth codto (4, the famly of olea systems cosdeed ths ae s lage tha those cosdeed []. I ths sese, the esult ths ae s a geealzato of the esult [].. Global Pactcal Outut ackg by Outut Feedback I ths secto we show how to exted the outut feedback stablzato esults []to acheve actcal tackg of system ( heoem : Ude Assumtos -, the oblem of global actcal outut tackg fo system ( s solvable by a smooth dyamc outut feedback cotolle of the fom (. Poof: Defe e = y, e = x, =,. Note that, the defto of the eo sgal e, =,,, we oly chage the coodate of the fst state x. It s dffeet to the commo defto used solvg asymtotc tackg, whee the eo s defed as the dffeece betwee all the states ad steady values. he e& = e ϕ(, t e, u, =,,, (7 e= u ϕ(, t e, u, y = e whee ϕ( teu,, = φ( e y( t, e,, e, u y& ( t, ϕ( teu,, = φ( e y( t, e,, e, u, =,, By assumtos -, t s eadly to show that, fo =,,,, ϕ (, teu, C e e L e C (8 ( whee C > s a costat oly deedg o C >, D ad. Next, toduce the followg escalg tasfomato L z,, = e z = e =,,, L v= u (9 whee s a escalg facto to be assged late. I the escalg coodates z s, the uceta system (7 ca be tasfeed to z& = z ψ (, t z, v, =,,, z& = v ψ (, t z, v, y = z (

3 Poceedgs of the 9th WSEAS Iteatoal Cofeece o Automatc Cotol, odelg & Smulato, Istabul, ukey, ay 7-9, 7 9 whee ϕ (, teu, ψ(, tzv, = ϕ(, teu,, ψ (, tzv, =, L Usg (8 ad the fact that, t s easy to see that (,, ϕ(,, ψ tzv = teu C z C, C ( L ψ (, tzv. C z z ( fo =,,. I ths way, a ew aamete-the escalg facto s toduced fo the desg of dyamc outut comesatos. It ceates a exta feedom ad lays a motat ole dealg wth the system ucetaty,.e., ψ (,, (. State Feedback Desg: Fo the escaled system ( satsfyg the gowth codto (, global actcal outut tackg s achevable by smooth state feedback. Let ξ = z z wth z = ad choose the Lyauov fucto U( ξ = ξ. he t s easy to deduce fom ( that P ( ξ ξz aξ ξ( z z α ( whee α >, a > ae kow ad deedet of. hus the vtual cotolle z * = aξ s ( ξ ( 5 ξ ξ( z z α ( Usg a ductve agumet smla to the oe [9], oe ca fd a set of vtual cotolles, tasfomatos, ad Lyauov fuctos * * z =, z = a ξ, =,,, * ξ = z z, =,,, (4 ξ U = U, =,,, wheeu = ad the smooth state feedback cotol law * * v = z = ( aξ = [ bz L bz] (5 ξ,, ξ 6ξ ξ v z α = = (6 whee all costats a, b, =,, ad α ae kow ad deedet of. Outut Feedbak Desg: Sce ( z,, z of the escaled system ( ae umeasuable but y = z s measuable, we eed oly to desg a (--dmesoal obseve fo * ( (. he umeasuable vaables ( z,, z defed by η, = z LLz =,, (7 ae ga co- whee the aametes stats to be detemed late. L,, L Fom (7, t follows that ( η ψ ψ & η = v ψ( LLL( z ψ( & = z ( L LL z (, =,, (8 I vew of (8, oe ca costuct the ( dmesoal obseve ˆ& η = ˆ η L LL z L LL ˆ η L z ( ( ˆ& η = v L LL ˆ L z, =,,, ( η (9 whch does ot volve the uceta fuctos ψ (, =,, (. oeove, the estmates of z s ca be obtaed based o the elatoshs zˆ ˆ = η LLz, =,, ( Let ε ˆ ˆ = z z = η η, =,,, be the estmate eos. he, the eo dyamcs s gve by ( ˆ ( ( ˆ ψ L & ε = z z L L z z L LLψ (, =,, ( ˆ & ε = ψ ( L LL z z L LL ψ ( ( By the cetaty equvalece cle, the umeasuable state ( z,, z the cotolle (5 ca be elaced by ts estmate ( zˆ,, zˆ geeated by the olea obseve (9-(. I ths way, oe obtas the mlemetable feedback cotolle v= [ bz bzˆl bzˆ] ( Substtutg ( to (6, we have ξ ε z ε 6ξ = = = = = α deedet of L,,. By (4, the afoemetoed equalty ca be smlfed as whee > s eal costat elated to b s ad

4 Poceedgs of the 9th WSEAS Iteatoal Cofeece o Automatc Cotol, odelg & Smulato, Istabul, ukey, ay 7-9, 7 ξ ε α 5 = = = > whee L,,. ( s a costat deedet of o deteme the obseve gas L, =,,, cosde the chage of coodates % ε = ε, % ε = ε Lε,, % ε = ε Lε. (4 I the coodates of ξ ad ε%, ( ca be eeseted as 5 ξ c( L,, L % ε = (5 ( % ε % ε α = L c L c he eo dyamcs ( the coodate ε% ca be ewtte as &% ε = ( z zˆ ψ( L( z zˆ Lψ ( &% ε = ψ ( L z zˆ L ψ (, =,,, ( Now, cosde the Lyauov fucto W (% ε,, % ε = (% ε L % ε A dect comutato gves W& = 4ξ L ε % = = = (,, ( c% L L L ε ( % ε } % (6 α = c% L (7 Fally, choose the Lyauov fucto V( ξ, % ε,, % ε = U( ξ,, ξ W( % ε,, % ε Fo the closed-loo system the coodate ( ξ, ε%. he, t follows fom (5 ad (7 that L V& { ξ L ξ C ( L,. L = ( L L L C ( L ε % α = = α (8 Whee C ( L,. L,, C ( L, L, C ( L ae ostve costats deedet of, whle C > ad >, ae ostve costats deedet of L s ad. Fom (8, t s easy to coclude that f the ga a ametes L s ad ae assged oe by oe. We have V & ( ζ ξ % ε F( = = (9 Whee ( ( ζ = ζ,, ζ = ξ,, ξ, % ε,, % ε, F ( = = ( α α. It s easy to see F( s ostve ad mootoe deceasg to zeo as ceases. Next, we wll show that (9 mles the exstece of a ga to acheve the global actcal tackg of system (. By [9], thee s a fte > ( ζ V( ζ( t F(, t, =,,. ( hs mles that all the solutos ζ = ( ζ,, ζ = ( ξ,, ξ, % ε,, % ε of the closed-loo system ae globally bouded ad well defed ove [,. hs, tu, leads to the cocluso that the state ( x,, x ae globally bouded, because of the elato (9,(4 ad boudedess of y. o acheve atcal outut tackg, we must show that by choosg aoately, the outut eo ξ ( t = y( t = x( t ( t ca be made abtaly small a fte tme. o ths ed, we choose { } ( ( ( max L, L,, L, so thee s a fte tme ( ζ (, >, ( ( ( = ξ( (, >. x t y t t F t Now, fom mootoe deceasg oety of F(, fo ay gve ε > thee s a suffcetly lage ( F( ε. hat s yt ( = x ( t ( t ε, fo all t >. hs comletes the oof of heoem.. A Illustatve Examle Examle: We wok out a smle umecal examle to llustate the esult descbed heoem. he examle we cosde s a -dmesoal system of the followg fom:

5 Poceedgs of the 9th WSEAS Iteatoal Cofeece o Automatc Cotol, odelg & Smulato, Istabul, ukey, ay 7-9, 7 ( ( & η = η ηηl η η ( & η = u, y = η he cotol objectve s to foce the state η to tack the efeece y = s( t usg the measuemet yt ( oly. It s easy to vefy that system ( satsfes Assumto ad Assumto. Accodg to (9 ad (, we ca costuct a dyamc outut feedback cotolle as ˆ& u η = L ( ˆ η L( η 4 u = 4 ˆ η L η y 4 η y ( ( ( ( ( By heoem, wth oely chose, followg the desg ocedue above, the tackg eo ca be made abtaly small. I Fg. we lot out the smulato esults y Fg.. Smulato esult fo the closed-loo system (-(. (Whee L = 4, = L, y = s( t 4. Coclusos Fo a class of olea systems havg ostablzable lea aoxmato, a method fo the actcal outut egulato usg a smooth outut feedback has bee develoed ad successfully tested by umecal smulatos. Ackowledgmet hs wok was suoted at by the Jaaese sty of Educato, Scece, Sots ad Cultue ude the st Cetuy Cete of Excellece (COE Pogam. x x y Refeeces [] E. J. Davso, he obust cotol of a sevomechasm oblem fo lea tme-vaat multvaable systems, IEEE as. Automat.Cot., vol. AC-,. 5 4, Ja [] B. A. Facs ad W.. Woham, he teal model cle of cotol theoy, Automatca, vol., , 976. [] Aathaam, V. ad Desoe, C. A. ackg ad dstubace ejecto of IO olea systems wth a PI o PS cotolle, Poc. of the 4th IEEE CDC,.67-68(985. [4] J. S. A. Hebu adw. A.Woham, Eo feedback ad teal model o dffeetable mafolds, IEEE as. Automat. Cot., vol. AC-9,. 97 4, ay 984. [5]. D. D Beedetto, Sythess of a teal model fo olea outut egulato, It. J. Cotol, vol. 45,. 4, 987. [6] J. Huag ad W. J. Rugh, O a olea multvaable sevomechasm oblem, Automatca, vol. 6, , 99. [7] A. Isdo ad C. I. Byes, Outut egulato of olea system, IEEE as. Automat. Cot., vol. 5,. 4, Feb. 99. [8] S. Celkovsky ad J. Huag, Cotuous feedback actcal outut egulato fo a class of olea systems havg ostablzable leazato, Poc.8th IEEE Cof. Decso Cotol, Phoex, AZ, 999, [9] C. Qa ad W. L, Pactcal outut tackg of olea systems wth ucotollable ustable leazato, IEEE as. Automat. Cot., vol. 47,. 6, Ja.. [] Yag, B. ad L, W. Robust outut feedback stablzato of uceta olea systems wth ucotollable ad uobsevable leazato, IEEE as. Automat. Cot., Vol. 5,. 69-6, 5. [] W. L ad C. Qa, Robust egulato of a cha of owe tegatos etubed by a lowetagula vecto feld, It. J. Robust Nolea Cotol, vol.,. 97 4,. [] C. I. Byes, F. Dell Pscol, ad A. Isdo, Outut Regulato of Uceta Nolea Systems. Bosto, A: Bkhäuse, 997.