Robust Adaptive Asymptotic Tracking of Nonlinear Systems With Additive Disturbance

Transcription

1 54 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 5, NO. 3, MARCH 6 Robust Adaptve Asymptotc Tackg of Nolea Systems Wth Addtve Dstubace Z. Ca, M. S. de Queoz, D. M. Dawso Abstact Ths ote deals wth the tackg cotol of multpleput multple-output (MIMO) olea paametc stct-feedback systems the pesece of addtve dstubaces paametc ucetates. Fo such systems, obust adaptve cotolles usually caot esue asymptotc tackg o eve egulato. I ths wok, ude the assumpto the dstubaces ae wth bouded tme devatves, we peset a obust adaptve cotol costucto that guaatees the tackg eo s asymptotcally dve to zeo. Idex Tems Backsteppg, dstubace, Lyapuov theoy, olea cotol, obust adaptve cotol. I. INTRODUCTION Cotuous obust adaptve cotol laws the pesece of addtve dstubaces ca geeally esue closed-loop sgal boudedess covegece of the tackg eo (state) to a esdual bouded set wth sze of the ode of the dstubace magtude, but ot asymptotc tackg (o egulato). Fo example, [6] poposed a poecto-based adaptve backsteppg cotolle fo sgle-put sgle-output (SISO) mmum phase lea systems of elatve degee two the pesece of put output dstubaces, whch guaatees the tackg eo s ultmately bouded small the mea squae sese. A obust adaptve backsteppg cotolle wth a leakage-based adaptato law was desged [3] fo SISO olea paametc stct-feedback systems wth fucto ucetates (cludg exteal dstubaces) satsfyg a tagula boud. The poposed desg esues global ufom ultmate boudedess of the system state. A smla class of systems was cosdeed [5] the developmet of two obust adaptve cotol methods usg the tug fucto desg the modula desg of [9]. Both methods gve L =L estmates o the effect of the ucetates/dstubaces o the tackg eo. I [7], a adaptve backsteppg cotolle wth tug fuctos fo lea systems wth output multplcatve dstubaces was desged wth a swtchg -modfcato. The cotolle gves a tackg eo popotoal to the sze of the petubatos. I [], a class of SISO olea systems affected by ukow, tme-vayg bouded paametes addtve dstubaces was cosdeed, a obust adaptve tackg cotolle was peseted that acheves boudedess of all sgals abtay dstubace atteuato. The wok [], whch studed a class of systems smla to the oe [], poposed a obust adaptve cotolle that esues the L om of the tackg eo s bouded. The tackg cotol poblem fo SISO olea systems wth ukow cotol coeffcets tme-vayg dstubaces was ecetly studed [6]. The obust Mauscpt eceved Mach, 5; evsed July, 5 Septembe 3, 5. Recommeded by Assocate Edto A. Astolf. Ths wok was suppoted pat by the Natoal Scece Foudato ude Gats DMS-44 CMS , pat by the Lousaa Boad of Regets ude Gat LEQSF(-5)-RD-A-3. Z. Ca s wth the Depatmet of Electcal Compute Egeeg, Uvesty of Iowa, Iowa Cty, IA USA (e-mal zhu-ca@ uowa.edu). M. S. de Queoz s wth the Depatmet of Mechacal Egeeg, Lousaa State Uvesty, Bato Rouge, LA USA (e-mal dequeoz@me.lsu.edu). D. M. Dawso s wth the Depatmet of Electcal Compute Egeeg, Clemso Uvesty, Clemso, SC USA (e-mal ddawso@ces.clemso.edu). Dgtal Obect Idetfe.9/TAC adaptve cotolle poposed [6] was show to guaatee the global ufom boudedess of the tackg eo. I ths ote, we cosde multple-put multple-output (MIMO) olea paametc stct-feedback systems subected to bouded addtve dstubaces that ae twce cotuously dffeetable have bouded tme devatves. Fo these systems, we peset a cotuous obust adaptve cotol costucto that guaatees asymptotc tackg. The poposed costucto s based o the cotuous olea obust cotol techque of [5], whch was ogally used to compesate fo ustuctued ucetates. Hee, we use t as a obustfyg mechasm fo adaptve cotolles. That s, adaptato s used to compesate fo stuctued (paametc) ucetates whle the obust mechasm compesates fo dstubaces, hece ecoveg the dstubace-fee, asymptotc tackg popety of the adaptve cotolle. The stad adaptve backsteppg desg [9] s udcously modfed to allow the use of the obust cotol techque of [5]. Also stumetal to ou ew costucto s the use of the suffcetly smooth poecto-based adaptato law ecetly toduced [3]. Ths allows the adaptve stablzg fuctos of the backsteppg desg tobe dffeetable as may tmes as ecessay. A Lyapuov-type stablty aalyss s used to pove the poposed obust adaptve cotolle yelds semglobal asymptotc tackg. II. PROBLEM STATEMENT We cosde a class of paametc stct-feedback systems of the fom _x = ' (x ) x. _x = ' (x ;...;x) x. _x = '(x ;x ;...;x) d u (a) (b) (c) m whee x(t) ; = ;...; ae the system states, ' pm ; =;...;ae kow oleates, p s a uceta costat paamete vecto, d(t) m s a uceta addtve dstubace, u(t) m s the cotol put, y = x s the system output. We make the followg assumptos egadg the system. A) ' C ; =;...;. A) d C kd(t)k L d ; k d(t)k _ L d, kd(t)k L d whee d ; d; d ae kow postve costats. A3) The paamete vecto belogs to a compact covex set = f kk g whee s a kow postve costat. Let the output tackg eo be defed as whee the C efeece taectoy y(t) e = y y () m s such that y () (t) L ; =;...; (3) ( ) () (t) deotes the th devatve wth espect totme. Ou goal s tocostuct a C state feedback cotol u(x ;x;...;x) that esues e(t)! as t! the boudedess of all closed-loop sg- It may be possble to solve ths poblem wth dscotuous cotol; howeve, ou goal hee s to use cotuous cotol /$. 6 IEEE

2 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 5, NO. 3, MARCH 6 55 als. The followg otato wll be used thoughout the ote ^ (t) p ; =;...; ae paamete estmates ~ (t) = ^ (t); =;...; (4) deote the coespodg paamete estmato eos; Po( ; ^ ) p ; =;...; ; 8 (t) p deote C poecto opeatos used to esue ^ (t) L depedet of the stablty aalyss (see Appedx I fo detals); pp ; = ;...; ae costat, dagoal, postve-defte matces; c ; =;...;ae postve costats. To educe the otatoal complexty facltate the eadablty of the ote, the cotol costucto that follows s peseted fo the case whee m =. Note, howeve, that the ma esult s eadly applcable to the MIMO case. Remak Solutos to specal cases of the above poblem wee peseted pelmay vesos of ths ote. I [], we cosdeed the case whee = (). I [], we addessed the egulato poblem,.e., y (), fo the geeal th-ode system (). III. CONSTRUCTION OF ROBUST ADAPTIVE CONTROL LAW Step We beg by dffeetatg () substtutg fom (a) toobta _e = ' (y) x _y (5) Afte addg subtactg the tem ' (y ) to(5), the eo system becomes whee _e = ' (y ) x _y ~w (6) ~w = ' (y) ' (y ) (7) Remak Due toassumpto A) (86), we ca use the Mea Value Theoem to show s some globally vetble, odeceasg fuc- whee ( ) to. Let k ~w k (kek)k tah(e)k (8) = c (x ) (9) whee s a stablzg fucto yet to be desged. To facltate the otato, let defe Dffeetatg () alog (6) gves = e () V =l(cosh( )) ~ ~ () _V = tah( ) ' (y ) _y ~w c ~ ^ () Based o (), we desg the stablzg fucto paamete update law as follows = c tah ( ) ' (y )^ _y (3) ^ = Po( ; ^ ) = ' (y ) tah( ) (4) Substtutg (3) (4) to() gves _V c tah ( ) tah( )(c ~w ) (5) whee popety P) of the poecto opeato was used (see Appedx I). Remak 3 The fal step of ou desg wll eque that ^ (t) L; = ;...; depedet of the stablty aalyss. Ths motvates the use of the tem l(cosh( )) the Lyapuov fucto of the fst steps of the backsteppg pocedue. I patcula, due to popety P3) of the poecto opeato (see Appedx I), we kow ^ (t) L f (t) L. The boudedess of s facltated by the fact l(cosh( ))=@ = tah( ). Remak 4 Usg (9) (3), the state x ca be decomposed to x = c c tah( ) ' (y )^ _y (6) whee the tem s a fucto of, the tem b s bouded. The usefuless of ths decomposto wll become appaet the ext step. Step ( ) Let = c (x ) (7) whee s a stablzg fucto, dffeetate = =(c )(x ) toobta _ = ' c c _ (8) whee (b) was used. The devatve of ca be wtte as _ = (x ;...;x ; ^ ;...; ^ ;y ;...;y () @ ^ ^ 9 (x ;...;x ;^ ;...;^ ;y () whee 9 s kow. Usg (9), we ca ewte (8) as _ = c ' T w (x ;...;x ;^ ;...;^ ) y () (9) 9 c Addg subtactg the tem w b =c to() yelds () _ = (w b 9 c ~w ) () c

3 56 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 5, NO. 3, MARCH 6 whee w b = w (y ; b ;...; b(); ^ ;...; ^ ) ~w =(w w b ) () Remak 5 Usg assumpto A), (85), (86), we ca show k ~w kc (k k)k tah( )k (3) whee ( ) ; =;...;ae some globally vetble, odeceasg fuctos = ( ;...; ). Note that the calculato of (3) s facltated by the fact that x b() = () (see Remak 4). Defe whose devatve s _V tah( ) V = V l(cosh( )) ~ ~ (4) c tah ( ) tah( )(c ~w ) c c c (w b 9 c ~w ) ~ ^ (5) whee c =. Based o (5), we desg = c tah( ) w b ^ 9 (6) ^ = Po( ; ^ ) = w b tah( ) (7) Substtutg (6) (7) to(5) gves _V c c tah ( ) tah( )(c ~w ) c c Remak 6 Usg (7) (6), the state x ca be wtte as x = c c tah()9b() 9 9 b() w b ^ (8) (9) whee 9 b() = 9 ( b ;...; b(); ^ ;...; ^ ;y ;...;y () ); s a fucto of ;...;, b s bouded. Step I the last step, we modfy the stad backsteppg pocedue ode to use the ew obust cotol mechasm of [5] to deal wth the addtve dstubace (c). Let the vaable (t) be defed as whee Afte dffeetatg (3), we obta = _ (3) c = (x ) (3) c _ = c c c (3) Dffeetatg twce poduces = c _' _ d _u (33) whee (c) was used. We fst cocetate o the calculato of the tem (33). To that ed, we have _ x () 8 whee 8 s kow. Dffeetatg (34) gves = _ 8g whee g = f = y ' (x ;...;x ) (34) x ;...;x ; ^ ;...; ^ ;y ;...;y () f x ;...;x ; ^ ;...; ^ ;y ;...;y () (35) k= x k ^ k= y ' k (x k ' k ) ' T ^ k k ' (37) popety P3) of the poecto opeato was used. We ow tu ou atteto to the calculato of the tem _' (33). Thus, we have _' T = g whee x ;...;x ; ^ ;...; ^ ;y ;...;y () f x ;...;x ; ^ ;...; ^ ;y ;...;y @ x y () (' x ) (39) f c c ^ Afte substtutg (35), (38), (33) to(3), we obta _ = c c (f f _ d _u _ 8g g )c c (4)

4 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 5, NO. 3, MARCH 6 57 We ow add subtact the tems g b = g ( b ;...; b(); ^ ;...; ^ ;y ;...;y () ) g b = g ( b ;...; b(); ^ ;...; ^ ;y ;...;y () ) tothe ght-h sde of (4) toobta _ = c c f f _ d _u _ 8 g g b g b g P g b g b f h c c (4) whee h( b ;...; b(); ^ ;...; ^ ;y ;...;y () ) has the specal popety that h(t); _ h(t) L (43) due to (3) popetes P) P3) of the poecto opeato. Fally, we ewte (4) as _ = (8h c _ d u) f c c (44) c whee f = (f f f 3 )=c c. Remak 7 Usg (83), (85), (86), we ca show f has the specal popety that s some globally vetble, odeceasg fuc- whee f ( ) to, kfk f (kxk)kzk (45) x =( ; ;...; ;) z = (tah( ); tah( );...; tah( ); ;) (46) Based o (44), we desg _u as [5] _u = _ 8 sg( ) c c (c c ) (47) whee >. The actual C cotol put ca be wtte fom (47) as follows u(t) =8(t) 8() c c (c c )( (t) ()) t [c c (c c ) ( )sg( ())] d (48) whee u() =. Afte substtutg (47) to (44), we obta the closedloop system If the cotol ga s selected to satsfy the followg suffcet codto >kh(t)k L the k d(t)k _ L (k h(t)k c _ L kd(t)k L ) (5) whee the postve costat b s defed as t L( ) d b (5) b = [ () c c ()(h() d())] _ (53) Lemma (See the poof of [8, Th. 8.4].) Cosde that a soluto exsts fo the system _ = f (; t); f q! q (54) Let the set D be defed as D = f kk <" s g whee " s >, let V D! be a C fucto satsfyg W () V (; t) W () 8t ; 8 D (55) whose devatve alog the taectoes of (54) satsfy _V (; t) W () 8t ; 8 D (56) whee W ();W () ae C postve defte fuctos W () s a dffeetable, postve sem-defte fucto. If () S = f D W () g; <<mkk=" W (), the (t) s bouded. Futhemoe, f W () s ufomly cotuous, the W ((t))! as t! (57) Theoem The cotol law (48) esues that all system sgals ae bouded e(t)! as t!, povded s adusted accodg to (5), c > (c =c ), the cotol gas c ; =;...; ae selected suffcetly lage elatve to the system tal codtos. Poof Let the fucto P (t) be defed as follows P (t) = b t L( ) d (58) whee b L(t) wee defed Lemma. If s selected toaccodg to (5), t follows fom Lemma that P (t). We ow defe the followg fucto V V = V P (59) Usg (84), we ca boud (59) as follows _ = c f c c c (h _ d sg( )) (49) whee l(cosh(ksk)) V ksk (6) IV. MAIN RESULT Befoe pesetg the ma esult Theoem, we state two techcal lemmas. Lemma (See [5] fo poof.) Let the fucto L(t) be defed as follows L = c c (h _ d sg( )) (5) s = x ; ~ ;...; ~ ; p P = m ;m =max max ; (6) x was defed (46). Explct codtos o these cotol gas ae povded the poof.

5 58 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 5, NO. 3, MARCH 6 Takg the tme devatve of (59) substtutg fom (3), (49), (58), we obta 3 _V = _ V c f c (6) upo use of (5). Substtutg ow fom (8) fo = (45) gves _V c tah ( ) tah( )(c ~w ) c c c [ f (kxk)kzkkkc ] (63) We ca uppe boud (63) by usg (3) as follows _V c c tah ( ) c f (kxk)kzkkkc k= c c tah( ) k tah( )k [ k (k k)k tah( k )k] (64) Usg (85) (86), we ca show that fo k =;...; c k c k tah( k ) k k tah( k )k k;k (k k k)k tah( k )k c k c k (k k k ) k;k (k k k) (k k) ktah( k )kktah( k )k (65) Now, let c = c (k ); = ;...; c = c (k )whee k > ; =;...;, ewte (64) as _V (k ) tah ( ) c k= (k k) tah ( ) [ k (k k)k tah( k )kktah( )ktah ( )] c tah( ) tah ( ) c [ f (kxk)kzkkkc ] (66) whee ; = ; =. Completg squaes o the above backeted tems yelds _V tah ( ) c k (k k) 4 k= c c f (kxk) 4c kzk k(k k) tah ( ) (67) 3 If f (s; t) deotes the ght-h sde of the closed-loop system o whch the stablty aalyss s beg pefomed, otce fom (49) (5) that f (s; t) has a dscotuty o the set of Lebesgue measue zeo f(s; t) =g. Sce Lemma eques that a soluto exst fo _s = f (s; t), see [5] fo a dscusso o the exstece of Flppov s geealzed soluto. Let (k k)= (k k) 4 k= k(k k); =;...; (68) c > (c =c ) (69) = mf;c ((c )=(c )) g, the (67) ca be wtte as _V (k (k k)) tah ( ) f (kxk) kzk (7) 4c It follows fom (7) that _V kzk fo k > (k k); =;...; c > (kxk) (7) 4 whee the costat satsfes <<. We ow apply Lemma by fst detemg fom (6) (7) that W (s) = l(cosh(ksk)) W (s) = ksk W (s) =kzk (7) Fom the ga codtos (7), we defe the sets D S as follows D = s ksk < m (k ); (p f c ) ; =;...; (73) S = s D W (s) < l cosh m (k ); (p f c ) ; =;...; (74) We ca ow voke Lemma to state that s(t) L. Fom (3), we the kow x (t) L. Fom (3), (4), popety P3) of the poecto opeato, we kow (t); ^ (t) L. We ca ow use (9) to show x (t) L. Fom (a), we kow _x (t) L. Cotug wth ths pocedue, we ca show (t);x (t); L ;=;...;. We ca the state _ (t); _ (t) L ; = ;...; by usg (). Usg (49) assumpto A, we ca show _(t) L. Fom (3) (34), we kow _ (t); _ (t) L ; hece, fom the tme devatve of (3), we have _x (t) L. Fally, we ca use (c) to show that u(t) L. Now, t s clea fom (7) that _W (s(t)) L, whch s a suffcet codto fo W (s) beg ufomly cotuous. It the follows fom Lemma that kz(t)k! as t!8s() S, whch mples fom (46) that e(t)! as t!8s() S. Note that the ego of stablty (74) ca be made abtaly lage to clude ay tal codtos by ceasg the cotol gas c ; = ;...; (.e., a semglobal stablty esult). Specfcally, we ca use the secod equato (7) (74) to calculate the ego of stablty as follows k > cosh exp ks()k ; =;...; c > 4 f cosh exp ks()k (75) whee ks()k = = () () = ~ () ~ () P () (76)

6 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 5, NO. 3, MARCH 6 59 Note that the equaltes (69) (75) ca be satsfed fo lage eough gas c ; =;...;because ) ( ) s ot a fucto of c, ) ();(), P () ae oly a fucto of =c, ) () s ot a fucto of c sce u() =. k k k k " k b k " ; (83) V. CONCLUSION Ths ote cosdeed the tackg poblem fo MIMO olea paametc stct-feedback systems the pesece of addtve dstubaces paametc ucetates. The cotuous obust adaptve cotolle, whose costucto s fouded o the fuso of a adaptato law a dyamc obust cotol mechasm, explots the two tmes cotuous dffeetablty of the dstubaces. The esultg obust adaptve cotol law guaatees the sem-global asymptotc covegece of the tackg eo to zeo boudedess of all closed-loop sgals. The poposed obust adaptve asymptotc tackg cotol also ca be appled to a class of flat [] MIMO olea systems wth matched umatched dstubaces of the fom x () = f x; _x;...;x () ; G x; _x;...;x () ; (u d )d (77) See [4] fo the detals of ths costucto. APPENDIX I PROJECTION OPERATOR We use a smootheed veso of the poecto opeato toduced [4] that was ecetly poposed [3]. The poecto opeato of [3] eplaces the Lpschtz cotuty popety of the poecto [4] wth the stoge popety of abtaly may tmes cotuous dffeetablty. Ths ew poecto opeato s useful fo backsteppg, obust adaptve cotolles whch eque multple dffeetatos of the adaptato law. The poecto s gve by [3] Po( ; ^ )= p(^ ) 4(" " ) ; =;...; (78) whee p(^ )=^ T ^ (79) = p (^ ) f p(^ ) > othewse = p(^ ) p(^ ) (8) (8) was defed (7), s the gadet opeato, "; ae abtay postve costats, was defed assumpto A3). It ca be pove that the above poecto opeato has the followg popetes [3] If ^ (), the P) k^ (t)k " 8t ; P) ~ T Po( ; ^ ) ~ T ; P3) Po( ; ^ )= b kpo( ; ^ )kk k[ (( ")=( )) ]( ")=(), whee = p(^ ) p(^ ) b = 4(" " ) p(^ ) p(^ ) 4(" " ) (8) P4) Po( ; ^ ) C. APPENDIX II HYPERBOLIC FUNCTION INEQUALITIES It ca be show that the followg equaltes hold 8 q tah (kk) l(cosh(kk)) q = l(cosh( )) kk (84) tah(kk) ktah()k (85) kk kk (86) tah(kk) whee Tah() = (tah( ); tah( );...; tah( q)). REFERENCES [] Z. Ca, M. S. de Queoz, B. Xa, D. M. Dawso, Adaptve asymptotc tackg of paametc stct-feedback systems the pesece of addtve dstubace, Poc. IEEE Cof. Decso Cotol, Paadse Isl, Bahamas, 4, pp [] Z. Ca, M. S. de Queoz, D. M. Dawso, Asymptotc adaptve egulato of paametc stct-feedback systems wth addtve dstubace, Poc. Ame. Cotol Cof., Potl, OR, 5, pp [3] Z. Ca, M. S. de Queoz, D. M. Dawso, A suffcetly smooth poecto opeato, IEEE Tas. Autom. Cotol, vol. 5, o., pp , Ja. 6. [4] Z. Ca, M. S. de Queoz, D. M. Dawso, Robust adaptve asymptotc tackg of olea systems the pesece of addtve dstubace, Lousaa State Uv., Bato Rouge, LA, Tech. Rep. ME-MS-5, Ju. 5. [5] R. A. Feema, M. Kstc, P. V. Kokotovc, Robustess of adaptve olea cotol to bouded ucetates, Automatca, vol. 34, o., pp. 7 3, 998. [6] S. S. Ge J. Wag, Robust adaptve tackg fo tme-vayg uceta olea systems wth ukow cotol coeffcets, IEEE Tas. Autom. Cotol, vol. 48, o. 8, pp , Aug. 3. [7] F. Ikhouae M. Kstc, Robustess of the tug fuctos adaptve backsteppg desgs fo lea systems, IEEE Tas. Autom. Cotol, vol. 43, o. 3, pp , Ma [8] H. Khall, Nolea Systems. New Yok Petce-Hall,. [9] M. Kstc, I. Kaellakopoulos, P. Kokotovc, Nolea Adaptve Cotol Desg. New Yok Wley, 995. [] R. Mao P. Tome, Robust adaptve state-feedback tackg fo olea systems, IEEE Tas. Autom. Cotol, vol. 43, o., pp , Ja [] R. Otega, A. Loa, P. J. Ncklasso, H. Sa-Ramez, Passvty- Based Cotol of Eule-Lagage Systems. Lodo, U.K. Spge- Velag, 998. [] Z. Pa T. Başa, Adaptve cotolle desg fo tackg dstubace atteuato paametc stct-feedback olea systems, IEEE Tas. Autom. Cotol, vol. 43, o. 8, pp , Aug [3] M. M. Polycapou P. A. Ioaou, A obust adaptve olea cotol desg, Automatca, vol. 33, o. 3, pp , 996. [4] J.-B. Pomet L. Paly, Adaptve olea egulato Estmato fom Lyapuov equato, IEEE Tas. Autom. Cotol, vol. 37, o. 6, pp , Ju. 99. [5] B. Xa, D. M. Dawso, M. S. de Queoz, J. Che, A cotuous asymptotc tackg cotol stategy fo uceta olea systems, IEEE Tas. Autom. Cotol, vol. 49, o. 7, pp. 6, Jul. 4. [6] Y. Zhag P. A. Ioaou, A ew class of olea obust adaptve cotolles, It. J. Cotol, vol. 65, o. 5, pp , 996.