Decentrazed Adaptve Contro for a Cass of Large-Scae onnear Systems wth Unknown Interactons Bahram Karm 1, Fatemeh Jahangr, Mohammad B. Menhaj 3, Iman Saboor 4 1. Center of Advanced Computatona Integence, Amrkabr Unversty of echnoogy, ehran, Iran E-ma: bahram-karm@aut.ac.r. Center of Advanced Computatona Integence, Amrkabr Unversty of echnoogy, ehran, Iran E-ma: jahangr_ftm@aut.ac.r 3. Amrkabr Unversty of echnoogy, ehran, Iran E-ma: menhaj@aut.ac.r 4. Center of Advanced Computatona Integence, Amrkabr Unversty of echnoogy, ehran, Iran E-ma: saboor@aut.ac.r Abstract: hs paper presents a decentrazed adaptve controer for a cass of arge-scae nonnear systems wth unknown subsystems and nteractons. A drect adaptve controer s devsed based on Lyapunov stabty anayss so that the stabty of the cosed oop system s guaranteed by ntroducng a sutaby drven adaptve rue. he adaptve controer proposed n ths paper can guarantee the stabty of the cosed-oop system wthout knowng the sgn of the controer coeffcent. o show the effectveness of the proposed decentrazed adaptve controer, a nonnear system s chosen as a case study. Smuaton resuts are very promsng. Key Words: Adaptve contro and Decentrazed nonnear system 1 IRODUCIO In the past decades, there has been an ncreased attenton n the deveopment theores for arge-scae systems. For arge-scae systems, decentrazed contro can often provde better performance over centrazed contro []. For ths reason, an ncreasng amount of attenton s beng drected towards decentrazed contro to extend the cass of systems to whch t s appcabe [], [3]. he dffcuty and uncertanty n measurng parameter vaues wthn a arge-scae system may ca for adaptve technques. Snce these restrctons ncude a arge group of appcatons, a varety of decentrazed adaptve technques has been deveoped. Mode reference adaptve contro (MRAC based desgns for decentrazed systems have been studed n [4] [6] for the contnuous tme case. Decentrazed adaptve controers for robotc manpuators were presented n [7] and [8], whe a scheme for nonnear subsystems wth a speca cass of nterconnectons was formuated n [9]. Knowng that most of physca arge-scae systems are nonneary couped to the dynamcs of the processes, the researchers are st tryng to contro these systems [1], [13]. Mosty, they ether nvestgate subsystems whch are near n a set of unknown parameters.e. [13], or they consder soated subsystems to be known,.e. [1, [14]. For most practca appcatons, the near contro synthess s appcabe to nearzed modes of arge-scae systems. However, ths ony guarantees stabty n a regon about the operatng pont and possby degradaton n performance and nstabty over a arge doman of operaton. In [1] - [13], some decentrazed nonnear controers for arge-scae systems have been deveoped. For exampe, decentrazed adaptve controers are desgned under the assumpton that the soated subsystems s known n [11], [1] and n [13], the subsystems wth unknown parameter were consdered near n a set of unknown parameters. It has been shown that the adaptve controer proposed n [13] can guarantee the stabty of the cosed-oop system when the sgn of the controer coeffcent s known. Our objectve s to present adaptve controers for a cass of decentrazed systems wth unknown nonnear subsystems and unknown nteractons. he stabty of the cosed-oop system s guaranteed by Lyapunov s stabty theory and the proposed decentrazed adaptve contro scheme ensures that a sgnas n the cosed-oop system be bounded and the trackng error goes asymptotcay to zero wthout the requrement that the sgn of the controer coeffcent shoud be known n advance. In [15] we presented an adaptve controer for a cass of affne nonnear decentrazed arge-scae systems. hs paper consders a more genera cass of these systems n whch constrants of the subsystems nteractons have been reaxed. hs paper s organzed as foows: In Secton, the detas of the probem statement and dervaton of the error dynamcs for the decentrazed system are descrbed. Secton 3 presents the man resuts of decentrazed adaptve contro for each subsystem usng ony oca nformaton and stabty anayss for composte system. An ustratve exampe s then used n Secton 4 to demonstrate the effectveness of the decentrazed adaptve technque and fnay Secton 5 concudes the paper. 978-1-444-518-1/1/$6. c 1 IEEE 57
PROBLEM FORMULAIO AD DERIVAIO OF HE ERROR DYAMICS A arge-scae nonnear system comprsed of nterconnected subsystems, s consdered. he th subsystem (S, whch s assumed to be snge-nput snge-output, s gven as x = ( x1,..., x + g ( x u, y = h( x1,..., x, = [ x, x,..., x ], 1 = [,,..., ] where x,1,, n s state vector, x x1 x x s fu state of the overa n system, (., (. g, h (., are unknown smooth functons, u and y are respectvey the nput and output of the th subsystem. If each subsystem has strong reatve degree m, then the output dynamcs may be rewrtten as [16] ( m = α, k, k + +Δ 1 ( m y s the y f ( x u ( x,..., x ( where m th tme dervatve of y and α, k, are unknown parameters. o desgn the controer, n Eq. ( we do not need to know the sgn of n a subsystems. Assumpton 1. he nterconnecton Δ ( x1,..., x s bounded by Δ ( x1,..., x ρ, where ρ s unknown and ρ >. ow defne the trackng error e = r y for S, where r s the desred output trajectory and y s the output of th subsystem. Our objectve s to desgn an adaptve contro system for each subsystem whch w cause the output y to track the desred output trajectory r n the presence of nterconnectons and, usng ony oca measurements. hs requrement eads to the foowng assumpton. Assumpton. he desred output trajectory and ts ( m dervatves r,..., r for the th subsystem S are measurabe and bounded. Let the output error vector for the th subsystem be defned ( m 1 by [,,..., e = e e e ] and wrte the tme dervatve of e as ( ( [,,..., e = e e e ] (3 he error dynamcs may be expressed as ( m ( m ( m e = r y (4 It s desred that the output error of the th subsystem ( ( 1 foows m m m, 1 +... + + a, =, where m the coeffcents are chosen so that each L ( s = s + (1 m 1 a, m 1... s + + a, has ts roots n the open eft-haf pane. ow use ( and (4 to obtan ( m ( m e = r α, kf, k( x u Δ (5 3 DECERALIZED ADAPIVE COROLLER AD SABILIY AALYSIS hs secton presents an adaptve controer for ( wth unknown nterconnecton functons. An adaptve agorthm s defned to estmate α, k wth α, k, wth, and ρ wth ρ and compensate the unknown nteractons. Defne a controer whch compensates for the dynamcs of each subsystem. For the th subsystem, the controer s defned by v u = (6 where the sgna v s defned as ( v a e + a e + a e +... + a e ( m 1 (,,1,, 1 m ( m α, kf, k( x + r + ρsgn( epb, where b and P are gven n (13 and (14 respectvey. Substtute (6 nto (5 to obtan ( m ( m = α, k, k( Δ (7 e r f x v (8 he thrd term,, n (8 s equa to: = 1 = 1 (9 where the s defned as =. Use (8 and (9 to obtan: ( m ( m e = r α, kf, k( x v + v Δ(1 From (7 and (1, equaton (1 becomes: ( m e = α f ( x,, ( m 1 ( a, e + a,1 e +... + a, m 1e k k ρ sgn( e Pb + v Δ, (11 where the α, k s defned as α, k = α, k α, k. Substtutng (11 n (3, the error dynamcs (3 can be wrtten n a matrx form as: 1 Chnese Contro and Decson Conference 571
e = A e + b ( α f ( x, k, k ρ sgn( e Pb + v Δ (1 where the Hurwtz matrx A and vector b are 1... 1........ A..... =,........ 1 a, a,1 a,... a, m 1 b = [... 1]. (13 Snce A s Hurwtz, a unque postve defnte souton P to the foowng Lyapunov equaton exsts A P + PA = Q (14 where the matrx Q s postve defnte. Consder the foowng update aws: α, k = γ α f, ( k x epb (15 / = γ e Pbv (16 ρ = γ ρ e Pb (17 where γα, γ, γ ρ > are constant desgn parameters. he man resuts of ths paper are now summarzed n the foowng theorem. heorem: Gven the error dynamca system (1 for the decentrazed system (1 wth a reference mode satsfyng Assumpton and the nteracton between subsystems satsfyng Assumpton 1, then the contro aw (6 wth adaptaton aws (15 (17 makes the trackng error asymptotcay converge to zero and a sgnas n the cosed oop system bounded. Proof: Consder the foowng Lyapunov functon V = ( V,1 + V, (18 = 1 Wth V = e Pe (19,1 α k 1, k = ρ V, = + + γα γ γ ρ ( where α, k = α, k α, k, = and ρ = ρ. ρ Use the defnton of the error dynamc (1 to wrte the tme dervatve of V as = 1 ( V = e A P + PA P α, kf, k + e b ( x + e Pb ρ sgn( e Pb + v Δ + V, (1 From (14, we have V = eqe = 1 P α, kf, k + e b ( x + e Pb ρ sgn( e Pb + v Δ + V, ( Furthermore, knowng that Δ ( x1,..., x ρ, we can obtan the foowng upper bound for the tme dervatve of V : V eqe+ epb α, kf, k( x = 1 e Pb ρ + ( e Pb v + e Pb ρ + V,, eqe+ ( epb α, kf, k( x = 1 (3 (, e Pb ρ + e Pb v + V, From (18, ( and (3, we can wrte, k V α eqe + α, k ( e Pbf, k ( x + = 1 γ α e Pbv ρ + ( + ρ( e Pb γ γ ρ (4 Use the parameters adaptve rues (15 (17, to obtan V = 1 e V eqe (5 Usng Barbaat s emma, convergence of the trackng error to zero s guaranteed. hs w make the convergence of the update aws, equatons (15 (17, possbe. hs competes the proof. Q.E.D. Remark 1: he proposed method gven n secton 3 s apparenty more genera than the subsystems as consdered n [15]. Resuts of the proposed controer to the nverted 57 1 Chnese Contro and Decson Conference
penduum probem are fuy presented n [15]. As we w show n next secton, the controer exhbts faster response compared wth that of [15], whch confrms superorty of the ths controer. 4 A ILLUSRAIVE EXAMPLE In ths secton, an nverted penduums connected by a sprng [13], s used as a case study to ustrate the capabty of the proposed decentrazed adaptve contro. Each penduum may be postoned by a torque nput u apped by a servomotor at ts base. It s assumed that both θ and θ (anguar poston and rate are avaabe to the th controer for = 1,. he nonnear equatons whch descrbe the moton of the penduums are defned by (6 where x1,1 = θ1 and x,1 = θ are the anguar dspacements of the penduums from vertca. he parameters m1 = kg and m =.5kg are the penduum end masses, j1 =.5kg and j =.65kg are the moments of nerta, /m s the sprng constant of the connectng sprng, r =.5m s the penduum heght, =.5m s the natura ength of the sprng, and g = 9.81 m / s s gravtatona acceeraton. he dstance between the penduum hnges s b=.4m. A, b< ndcates that the penduums repe one another when both are n the uprght poston [17]. Here we w attempt to reguated the anguar d postons to zero, so that e = θ [.e., x =, = 1, ]. x 1,1 x1, = mgr 1 kr kr x 1, = sn ( x1,1 + ( b j1 4j1 j1 u1 kr + + sn ( x,1 j1 4 j1 y = x 1 1,1 x,1 = x, mgr kr kr x, = sn ( x,1 ( b j 4j j u kr + + sn ( x 1,1 j 4 j y = x,1 (6 o show the effectveness of the proposed method, two controers are studed for the purpose comparson. We w frst demonstrate how a smpe decentrazed proportona pus ntegra (PI controer 1 t u = e + edτ, = 1, (7 woud contro the system. We fnd that the penduums exhbt undesrabe response wth reatvey arge oscatory behavor due to the ack of dampng as shown n Fgs 1 -. x1,1 x,1.5 -.5 4 6 8 1 1 14 me(sec. Fg. 1. PI controer for the frst subsystem ( x 1,1 = θ1.4. -. -.4 -.6 4 6 8 1 1 14 me(sec. Fg.. PI controer for the second subsystem ( x,1 = θ decentrazed adaptve controer proposed n Secton 3 s then apped to ths system. he controer s taken as 1 ( m 1 u = ( a, e + a,1 e +... + a, m 1e (8 ( m α, kf, k( x + r + ρsgn( epb, where, α, k, ρ are updated by adaptve rues (15 (17. he controer parameters are taken as γα = γ 1 =, γ ρ =. Fgures 3-4 show the smuaton resuts for the desgned controer and ustrate that, after a short transent perod, the states track the gven trajectores very cosey. Comparng the resuts n Fgs. 1- and 3-4, t can be seen that the proposed decentrazed adaptve controer presents desrabe performance whch confrms fast convergence of the adaptve parameters. he parameters α 1,1, α1,, 1, α,1, α,, and ρ 1, ρ, the adaptve parameters, are depcted n Fgs. 5-7 respectvey. Fgures 8-9 are aso shown the hstory of the contro nput u, = 1,. From fgures 5-9, t s nterestng to note that (8 mantans a robust performance to a wde cass of perturbatons n the system dynamcs, as ong as the nteractons are bounded. In ths sense, the controers guarantee robustness aganst modeed dynamcs naccuraces. 1 Chnese Contro and Decson Conference 573
x 1,1 (rad.1.5 -.5 -.1 -.15 -. -.5 4 6 8 1 me(sec Fg. 3. Decentrazed adaptve controer for the frst subsystem ( x1,1 = θ1 x,1 (rad..1 Updated Interactons Upper Bound 15 1 5 4 6 8 1 me(sec. Fg. 7. Convergence of the nteractons upper bounds ( ρ, ρ 4 1 -.1 u 1 -. 5 1 me(sec. Fg. 4. Decentrazed adaptve controer for the second subsystem x = θ Frst Subsystem Updated Parameters.8.6.4. (,1 4 6 8 1 me(sec. Fg. 5. Convergence of the frst subsystem adaptve parameters ( α, α, Second Subsystem Updated Parameters 1.5 1,1 1, 1 4 6 8 1 me(sec. Fg. 6. Convergence of the second subsystem adaptve parameters ( α, α,,1, u - 4 6 8 1 mes(sec. Fg. 8. Contro nput u 1 for the frst subsystem - -4 4 6 8 1 me(sec. Fg. 9. Contro nput u for the second subsystem 5 Concuson hs paper ntroduced a decentrazed adaptve controer for a cass of arge-scae nonnear systems. hs cass conssts of a such systems wth unknown SISO subsystems and unknown nteractons. he proposed adaptve controer ensured the cosed-oop stabty and convergence of the trackng errors asymptotcay to zero. he stabty anayss was performed usng the Lyapunov s theory. he smuaton resuts approved the vadty of the proposed controer. REFERECES 574 1 Chnese Contro and Decson Conference
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