Decentralized Model Reference Adaptive Control Without Restriction on Subsystem Relative Degrees

Transcription

1 1464 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 44, NO 7, JULY 1999 [5] S Kosos, Fne npu/oupu represenaon of a class of Volerra polynoal syses, Auoaca, vol 33, no 2, pp , 1997 [6] S Kosos and D Lappas, A descrpon of 2-densonal dscree polynoal dynacs, IMA J Mah Conrol and Infor, vol 13, pp , 1996 [7] L Ljung, Syse Idenfcaon Theory for he User Englewood Clffs, NJ: Prence-Hall, 1987 [8] J W Rugh, Nonlnear Syse Theory Balore, MD: Johns Hopkns Unv Press, 1981 Decenralzed Model Reference Adapve Conrol Whou Resrcon on Subsyse Relave Degrees Changyun Wen and Yeng Cha Soh Absrac When he drec odel reference adapve conrol (MRAC) schee wh frs-order local esaors s eployed o desgn oally decenralzed conrollers, he sably resul can only be appled o a syse wh all of s nonal subsyse relave degrees less han or equal o wo In hs paper, hs resrcon s relaxed and s acheved by eployng he paraeer projecon ogeher wh sac noralzaon To pleen he local conrollers, no a pror knowledge of he subsyse unodeled dynacs and no nforaon exchange beween subsyses are requred Global sably s esablshed for he closed-loop syse and sall n he ean rackng error s ensured Wh hs analyss, he class of neracons and subsyse unodeled dynacs can be enlarged o nclude hose havng nfne eory Index Ters Adapve conrol, decenralzed conrol, robusness, sably I INTRODUCTION Decenralzed adapve conrol s an poran conrol schee for large scale syses, and has connued o receve a lo of aenon fro conrol researchers over he las few decades However, only a led nuber of sably resuls n hs area are avalable due o he dffcules n he analyss of gnored neracons The frs bach of resuls were obaned based on he drec odel reference adapve conrol (MRAC) approach [1] [3] A srong ason for hese resuls s ha relave degrees of all he nonal subsyse odels should be less han or equal o wo The sably resuls usng he ndrec pole assgnen desgn schee were repored laer n [4] [6] here s no resrcon on he relave degrees of he nonal subsyse Recenly, effors on relaxng he subsyse relave degrees n he case of eployng he drec adapve conrol schee have been ade by usng soe advanced adapve sraeges The concep of hgh-order uners n [7] was appled o acheve hs n [8] and [9] In hs case, a local dynac esaor wh he subsyse relave degree as s order s desgned o denfy he unknown paraeers of each subsyse The negraor backseppng echnque of [10] was also successfully ulzed o reach a slar goal n [11] [13] To oban he fnal conrol for each subsyse, a nuber of erave desgn seps should be nvolved o calculae Manuscrp receved Aprl 24, 1996; revsed March 16, 1998 Recoended by Assocae Edor, M Krsc Ths work was pored n par by NTU under he Appled Research Projec Gran RP 23/92 The auhors are wh he School of Elecrcal and Elecronc Engneerng, Nanyang Technologcal Unversy, Sngapore (e-al: ecywen@nuedusg) Publsher Ie Idenfer S (99) soe neredae vrual conrol sgnals As coened n [9], he unodeled neracons us sasfy ceran srucural condons when hese advanced schees are used However, for he convenonal MRAC schee, he proble of he relaxaon of he subsyse relave degrees s sll unsolved Due o he splcy of he convenonal MRAC schee, he soluon o such a proble s of praccal neres In [14], Daa and Ioannou appled he noralzaon echnque used n he sngle-loop robus adapve conroller desgn o acheve he requred relaxaon Bu he proposed local noralzng sgnals requre nforaon fro he oher subsyses o bound he effecs of neracons fro hese subsyses Thus, only parally decenralzed adapve conrollers can be desgned In hs paper, he proble wll be solved wh oally decenralzed conrollers by eployng he paraeer projecon ogeher wh a sac noralzaon echnque Global sably s esablshed for he closed-loop syse and sall n he ean rackng error s ensured Wh our analyss, he class of neracons and subsyse unodeled dynacs can be enlarged o nclude hose havng nfne eory The reanng par of he paper s organzed as follows Secon II gves he class of syses o be conrolled and Secon III presens he decenralzed conrollers The analyss of he closed-loop syse and he an resul are gven n Secon IV Fnally, he paper s concluded n Secon V II SYSTEM MODELS AND ASSUMPTIONS In hs paper, he class of nerconneced syses consdered consss of sngle-npu/sngle-oupu subsyses The h subsyse s odeled as y () =H (D)u () +H (D) j H j (D)[u j () +y j ()] + j H j (D)[u j () +y j ()] + d () (1) for ; j =1; 111;, y ;u ; and d are, respecvely, he oupu, B npu, and dsurbance of he h subsyse In (1), H (D) = (D) A (D) and s he reduced-order ransfer funcon of subsyse wh n n A (D) =D + a 01 D n a 0 B (D) =b D + b 01 D b 0 D denoes he dfferenaon operaor, <n ; j ; j are consans, and H j (D) and H j (D) denoe he subsyse neracons f 6= j and unodeled dynacs f = j Now, a reference odel gven below s chosen for he h subsyse y() =W(D)r () (2) W(D) =k 1 D (D) and r s an exernal reference npu sgnal Here, k s a consan and D(D) s a onc Hurwz polynoal of degree n 3 = n 0, e, D(D) = D n + dn 01D n d1d + d0 The conrol proble s o desgn oally decenralzed conrollers for plan (1) such ha he closed-loop syse s sable n he sense ha all sgnals n he syse are bounded for arbrary bounded r and nal condons, and he oupu y () follows he oupu y() of he odel (2) as closely as possble To solve he conrol proble, he followng asons are ade for he plan gven n (1) /99$ IEEE

2 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 44, NO 7, JULY Ason 21: A1) B (D) s Hurwz A2) An upper bound for n, he nonal relave degree n 3 = n 0 of subsyse, and he sgn of he hgh frequency gan sgn(b ) are known Furherore, he coeffcens of A (D) and B (D) are nsde a known copac convex regon C A3) H j(d) and H j(d) are sable, and H (D) H j(d) and H j (D) are srcly proper A4) d () s bounded Rearks 21: 1) Noe ha here s no resrcon on he nonal subsyse relave degrees n 3 ;=1; 2; 111; 2) Whle odelng errors are assued o sasfy A3) and A4), no a pror knowledge s requred fro he for he pleenaon of he adapve conrollers gven n he laer secons Ason A2) also ples a known lower bound for jb j III DESIGN OF ROBUST DECENTRALIZED ADAPTIVE CONTROLLERS For each subsyse, we defne he followng flered varables: _! ;1 =3! ;1 + q y ; _! ;2 =3! ;2 + q u (3) (3 ;q ) s a conrollable par sasfyng (DI 0 3 ) 01 q = 1 02 F (D) [Dn ; 111; 1] T (4) wh F (D) as an arbrary Hurwz polynoal of order n 0 1 Boh 3 and q are chosen by users Le! T =! T ;1;! T ;2;y : (5) Then he conrol s gven as u =! T + c ;0r (6) T () =[;1(); T ;2(); T ;3()] s a (2n 0 1)-densonal conrol paraeer vecor and c ;0() s a feedforward paraeer scalar Fro [16], can be shown ha a desred paraeer vecor 3 of and a desred paraeer c;0 3 of c ;0 exs, and hey can be obaned when he nonal ransfer funcon H (D) of he h subsyse s known When H (D) s unknown, an adapve law s requred o updae and c ;0 To acheve hs and o ensure he robusness of he adapve conroller n he presence of odelng errors ncludng neracons, subsyse unodeled dynacs, and exernal dsurbances, we nroduce paraeer projecon operaon o he adapve law The adapve law o une and c ;0 s dvded no he followng wo cases Case 1: b = 1 In hs case, c ;0 = 1f k s chosen o be one and _ = P 0 = 0 T > 0 ; 0 0 e ;1 1+! T! e ;1 = y 0 y + T 0 v ; = W(D)I! v = W(D) T! (8)! T =! T ; T ; (1) T (n ) T ; 111; (9) and P denoes he projecon operaon defned n [17] or [18] Case 2: b s unknown In hs case, c ;0() s unknown and needs o be updaed The local adapve law n hs case s a odfed verson of ha n [15] by changng he - odfcaon and he noralzng sgnal appropraely as n Case 1 (7) Rearks 31: 1) As can be noed fro (6) (9), he noralzaon s sac Also he pleenaon of local adapve conrollers does no requre any nforaon exchange beween subsyses and he a pror knowledge on subsyse unodeled dynacs 2) The resuls for he adapve conroller n Case 2 can be obaned by followng he slar analyses as n Case 1 and [15] Thus we jus focus our aenon on Case 1 whou any furher elaboraon on Case 2 IV STABILITY OF THE DECENTRALIZED ADAPTIVE CONTROL SYSTEMS We need o esablsh he robusness of he local adapve conrollers n he presence of gnored neracons, unodeled dynacs, and exernal dsurbances Before dong hs, soe prelnary analyss s requred Fro (1) (6), can be shown ha he h subsyse can be expressed as y = W(D)! T ~ + r + () (10) 3 ~ = 0 (11) () = ()+ 1+W (D) 3 ;1 (DI 0 3 ) 01 q d (12) () = 1 j (D)[u j ()+y j ()] (13) 1 j (D) =W (D) j H j (D) ;2 (DI 0 3 ) 01 q + j H j (D) 1+W (D)( 3 ;3 + 3 ;1 (DI 0 3 ) 01 q : (14) Clearly, 1 j (D) s srcly proper and sable fro Ason 21 Fro (12), we have he followng resul Lea 41: The odelng error () n (12) sasfes j ()j j k! j( )k + d0 (15) 0 j and d0 are soe nonnegave consans Proof: Le V (D) be an arbrary Hurwz polynoal defned as Then V (D) =D n 02 + v ;n 03D n v ;0 v T =[1;v ;n 03; 111;v ;0]: () = = F j V j 1 j (u j + y j) V j F j F j 1 j vj T (! j;1 +! j;2) : (16) V j Thus fro he sably and properness of 1 j F V, he resul can be esablshed fro (16) Rearks 41: 1) In he proof of Lea 41, he effecs of soe exponenally decayng ers due o nonzero nal condons have been absorbed by d0 2) The consan j ndcaes he srengh of he neracons beween subsyses and j when 6= j, and he unodeled dynacs of he h subsyse are coupled o he nonal odel when = j 3) In ers of he boundng sgnals, he bound for he odelng error n (15) allows he effecs of he unodeled dynacs and neracons o have nfne eory, hus s looser han hose gven n exsng leraure, such as [1] and [2] The class of odelng errors consdered can also be enlarged o nclude any nonlnear unodeled dynacs sasfyng (15)

3 1466 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 44, NO 7, JULY 1999 Now, f 0 k! ( )k = k! ()k and 0 k! j ( )k 0 k! ( )k for all j 6= and > 0, hen (15) becoes j ()j k! ()k + d 0 ; for all 0 (17) s a nonnegave consan dependng on j Fro (17), soe useful properes of he local esaors can be obaned In he reanng par of hs secon, we wll use k ;k=1; 111; 5 o denoe M k (D) M k (D) s a vecor conanng proper sable ransfer funcons Thus, k (); k = 1; 111; 5 also sasfy he sae bounds as n (17) under he sae condons Noe ha he consans and d 0 n (17) have been ade unfor for all = 1; 2; 111; and k = 1; 111; 5 Also n hs secon, all c j; j =1; 2; 111 sand for generc consans We now derve an equaon o descrbe he closed-loop syse of he h loop I can be shown ha he plan n he h loop has he followng sae represenaon: _x = A x + b u + 1 (); y = h T x + 2 () (18) (A ;b ;h T ) s a nal sae represenaon of H (D) Noe ha D 2 also sasfes (17) fro he src properness of H j (D) Now he closed-loop syse of he h loop can be descrbed as _x c = A c x c + b c! T ~ + b c r + 1 () y = h c T (19) x c + 2 () A c s a sable arx sasfyng (h c ) T (DI 0 A c ) 01 b c = W(D) and x c = [x T ;! ;1;! T ;2] T T Le T = [ T ; (1) (n 01) ; 111; ] Then (k) = D k W(D) I! ;k=0; 1; 111;n can have he followng realzaon: _ = A + B! = A + B ;1! ;1 + B ;2! ;2 + B ;3 h T x + B ;3 2 () (20) (k) = C k (21) C k =[0; 0; 111;I;0 1110] wh zero beng (2n 01)2(2n 01) block arces and I an deny arx a he (k +1)h poson A s a sable arx sasfyng Ck T (DI 0 A ) 01 B = D k W(D)I and B = [B ;1 ;B ;2 ;B ;3 ] Le x c = [x c ; T ] T Then we have _x c = A c x c + B c! T ~ + B c r + 3 () (22) A c A c 0 = B ;3 h T : (23) ;B ;1 ;B ;2 A B c s suably augened fro b c Clearly, A c s a sable arx We now esablsh he relaonshp beween! and x c I can be shown, by akng he odelng error no accoun and followng slar seps n he proof of [16, Th C1], ha kx ()k c 1 k! ()k + 4 () : (24) Then x c () = x T ();! ;1();! T ;2(); T T T c 2 k! ()k + 4 () : (25) Also k! ()k =! T ;1 ();! ;2(); T h c T x c () + 2 (); T ;! T 0 d0i;111;d n 01I T T c 3 x c () + 5 () : (26) Before esablshng he sably of he syse, we now explore soe properes of he esaor (7) (9) Lea 42: The esaor (7) (9), when appled o he plan gven n (1), has he followng properes 1) T j! () ~ ()j 1+! T ()!() 1=2 c 4; for 0: (27) 2) Suppose M 0 s a posve consan s d 0 =M 0 Ifk! ()k >M 0 ; 0 k! ( )k = k! ()k and 0 k! j ( )k 0 k! ( )k 8j 6= and for all >0 wh soe 0, hen 0 j! ( ) T ~ ( )j 1+! T ( )! d c5=0 +(1 + 2)( 0 0 ); 1=2 () for 0 (28) 1 =(c 6 = 0 + ); 2 =(c 6 = 0 + ) + c 7 0 (29) and > 0; 0 2 (0; 1] Proof: 1)! () T ~ () 1+! T ()! () 1=2 2) Fro (8) and (10), we have c 1 : k! ()k 1+! T ()! () 1=2 j~ ()j e ;1 = T ~ + (): (30) Then consder he followng posve defne funcon: V = 1 ~ T 0 01 ~ : (31) 2 Usng (7) and (30) gves 1 ~ _V ! T! + 1 ( ) ! T! : (32) Fro he ason of he lea and (17), we have () 1+! T! + ; 1=2 for 0 : (33) Then for 0, (32) becoes Then _V ~ T 2 1+! T! ( + )2 : (34) T ( ) ~ 2 ( ) d c 8 +( + ) 2 ( 0 0 ): (35) 1+! T! Fro (9), we can noe ha ;k=0; 1; 111;n 3 (1+!! ) are bounded Fro (3), (6), (18), and (24), we can have _! 1+! T ()!() 1=2 c 9 ( + ) +c 10 : (36) Now (n +1) D = Dn 0 D(D) D(D) I! + _! k = 0d0 (1) 0 d1 (2) 01110dn 01 (n ) + _! : (37) Therefore, slar bounds o (36) can be obaned for (n +1) d (1 +! T ; d (1 +!T 1=2! ) ()!())1=2 (1 +! T!)1=2 k

4 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 44, NO 7, JULY and d ~ d ( T (k) (1 +! T!)1=2 ) k =0; 1; 111;n 3 Noe ha! = n k=0 d k (k) Thus f T () ~ 2 () 1+! T ()! () [ j ; j +1 j ]; and for soe j 0 (38) we can esablsh j! () T ~ ()j 1+! T ()! () 1=2 c ( + ) c 12 0 ; 8 2 [ j ; j +1 j ]: (39) Then (28) can be esablshed fro (35) and (39) Reark 42: 1 can be ade arbrary sall by reducng and 2 by akng M0 a suffcenly large nuber M0 s used here for he purpose of analyss only I s no a desgn paraeer Fro Lea 42, (25), and (26), he sably of he syse can be esablshed under a specal case Ths s presened n he followng lea Lea 43: Consder he decenralzed adapve syse conssng of plan (1) and local adapve conrollers (6) (9) Suppose ha k! ( 0 )k = M0 and for all > 0, 0 k! ( )k = k! ()k and 0 k!j( )k k!()k for all j 6= Then under Ason 21, here exss a posve consan 3 1 such ha for all 3 1 he closed-loop syse ensures ha k! ()k M; 8 =1; 2; 111; (40) 0 M = c12m0 + c13 wh c12 1 Proof: The soluon of (22) s x c () =e A (0 ) x c (0 )+ + B c r ( )+ 3 ( ) d: e A (0) B c! T ( ) ~ ( ) As A c s a sable arx, here exs posve consans c and such ha e A ce 0 : (41) Suppose ha he neredae nuber M0 s also such ha kr k1 M0; 8 = 1; 2; 111; Then usng (25), (26), (17), (27), and (41) gves k! ()k c3ce 0(0 ) c2! c14 ce 0(0)! T ( ) ~ ( ) (1 + k! ()k 2 ) 1=2 k!()k +! T ( ) ~ ( ) (1 + k! ()k 2 ) 1=2 c15m0 + c16 k! ()k + c17 0 +M0 + 3 ( ) d + 5 () ce 0(0)! T ( ) ~ ( ) 2 (1 + k! ()k 2 ) 1=2 k! ()k d + c18: (42) Afer soe rearrangeen of (42), he Bellan Grownwall lea can be appled Then fro Lea 42, we can oban ha k! ()k c19m0 + c20 + c21 k! ()k (43) 0 for 3 1 and and 3 are suffcenly sall consans sasfyng c22( ) < (44) wh 3 1 dependng on 3 1 and 3 2 on 3 Noe ha he rgh sde of (43) s nondecreasng Thus can be rewren as k! ()k c19m0 + c20 + c21 k! ()k: (45) 0 0 Then fro (45), we ge k! c19 ()k c21 M 0 + c20 (46) 1 0 c21 f 3 1 wh he posve consan 3 1 sasfyng c < 1 Fnally, he resul s proved by leng 3 1 = nf 3 1; 3 1; g and c c12 =axf 10c ; 1g; c 13 = c 10c To esablsh he sably resul for he general case, we explore he paraeer esaor furher, and hs gves Lea 44 as follows Lea 44: If k! ()k > M0 for all 0, and 0 k!j( )k c12m0 + c13 for all 2 [0;1] and j = 1; 2 111;, and 0 k! ()k = k! ()k; 0 k! j( )k 0 k! ()k; 8j 6= for all 1 wh soe 1 0, hen j! ( ) T ~ ( )j 1+! T ( )! () 1=2 d c 5=0 +(1 + 2) 0 0 ; for 0 (47) 1 =[c6=0 +(c12 + c13)](c12 + c13): (48) Proof: By nong he condon of he lea and usng (17), we have j ()j (1 + k! ()k 2 ) 1=2 (c 12 + c13) +; 8 2 [0 ;1]: (49) For 1, (33) becoes vald Nong ha c12 1, we can have (49) for all 0 Then replacng (33) by (49) and by (c12 +c13) n he proof of (28), we can esablsh (47) Reark 43: Noe ha he propery n he above lea s que slar o (28) n Lea 42 excep ha s changed o (c12 +c13) Fro Leas 42 44, we can esablsh our an sably resul as follows Theore 41: Consder he decenralzed adapve syse conssng of plan (1) and local adapve conrollers (6) (9) Under Ason 21, here exss a consan 3 such ha for all 3, we have he followng 1) The closed-loop syse s globally sable n he sense ha all sgnals rean bounded 8 and for all fne nal saes, any bounded r, and arbrarly bounded exernal dsurbances 2) The rackng error e ;1() =y 0 y sasfes e 2 ;1( ) d 1 + 2( + d0 + 0) 0 0 ; for all 0 0 (50) 1; 2 are posve consans Proof: 1) To show he boundedness of all he rajecores k! k; = 1; 2; 111;, we consder a funcon k!()k defned as k!()k = axfk!1()k; k!2()k; 111; k! ()kg: (51) Clearly, he resul s proved f k!()k s bounded I can be noed ha k!()k s connuous and hus, sarng wh 0 =0

5 1468 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 44, NO 7, JULY 1999 and k = 1; 2; 111; we can dvde he e axs [0; 1) no he followng wo subsequences: < 0 k =[ k01;s k ] < + k =(s k; k ) < 0 k = f jk!()km0g and <+ k e, [0; 1) = 1 < 0 k [ k=1 = f jk!()k >M0g (52) 1 < + : (53) =1 Obvously, suffces o prove ha k!()k s bounded n < + ; k 8k 1 Ths can be shown hrough nducon Thus we consder 2< + 1 frs Fro he connuy of k!()k; < + 1 and an 2f1; 2; 111;g such ha 0 k!( )k = k!()k and k!()k = k! ()k for all 1 and 2< + 1 Thus 0 k! ( )k = k! ()k and 0 k! j ( )k 0 k! ( ) 8j 6=, for all 1 and 2 < + 1 Therefore, he condons of Lea 43 are sasfed for all 1 and 2< + 1 Then fro hs lea and nong ha k! ( 0 )k = k!( 0 )k = M 0, we can have, for all 1 and all 3 1, ha e, 0 0 k!( )k M (54) k! ( )k M; 8 =1; 2; 111;: (55) If he condons of Lea 422 or 43 are volaed for 1 and 2 < + 1, he followng wo possbles ay occur o k!()k Case 1: 0 k!( )k = k!()k bu k!()k = k! j()k; j2f1; 2; 111;gn for all > 1 In hs case, he condon ha 0 k! j( )k 0 k! ( )k; 8j 6= canno be sasfed Thus Lea 43 canno be appled for > 1 However, we now consder k! j()k Clearly, here exss a 1 j such ha k! j ( 1 j )k = M 0 and k! j ()k >M 0 for all 2 [ 1 j ; 1 ] < + 1 Also n hs case, we have, for > 1 0 and 0 k! j( )k = k! j()k k! ( )k 0 k! j ( )k 2f1; 2; 111;gnj: (56) Thus fro (55) and (56), Lea 44 can be appled o k! j()k for 1 j Then followng he sae seps as n he proof of Lea 43 and applyng Lea 44 wh nal condon k! j( 1 j )k = M 0, we shall oban (54) or (55) for all 1 and all 3 3 = 3 1 c 12 + c 13 : (57) Case 2: 0 k!( )k 6= k!()k for 2 [ 1 ; 2 ] < + 1 and 0 k!( )k = k!()k for > 2 In hs case, he condon ha 0 k! ( )k = k! ( )k canno be sasfed for 1 However, (54) or (55) auoacally holds for 2 [ 1 ; 2 ]If 2 s nfne, he resul s proved For a fne 2 and when > 2, (54) or (55) can be shown under he condon (57) by followng he sae arguen as n Case 1 In hs way, he boundedness of k!()k s esablshed over < + 1 Now assung (54) or (55) holds 8 2< +, can be shown ha, k by followng he proof of Lea 43 and he above arguen, (54) s also rue 8 2< + k+1 fro Lea 44 wh he nal condon k! ( k+1 )k = M 0 for 2f1; 2; 111; g and k+1 2< + k+1 Afer esablshng he boundedness of k! ()k; 8 =1; 2; 111;, we can have y () and u () bounded 2) Once he boundedness of all he sgnals s esablshed, hen he rackng properes can be obaned by followng slar analyses used n [15] Rearks 44: 1) Noe ha rajecory k!()k only has hree possbles Tha s, sasfes he condon of Leas 422 and 43, or Case 1, or Case 2 2) In he sably analyss, we only need o ake care of he subsyse n whch he sac noralzng sgnal has axu agnude aong all he subsyses over a ceran e nerval and consder he suaon ha he agnude exceeds a ceran level, e, k!()k M 0 In hs case, he locally noralzed paraeer esaon predcon error n he subsyse concerned becoes sall and sasfes ceran condons specfed n Leas 42 and 44 Also he nducve echnque used and he dvson of e nerval no wo subsequences are crucal n he esablshen of a unfor bound for k!()k over all subnervals V CONCLUSION In hs paper, we have relaxed he subsyse relave degrees requreen posed n odel reference decenralzed adapve conrol usng frs-order local esaors These local esaors are desgned usng paraeer projecon ogeher wh sac noralzaon In he pleenaon of he local conrollers, no a pror knowledge of he subsyse unodeled dynacs and no nforaon exchanges beween subsyses are requred I has been shown ha global sably of he overall adapve feedback syse can be ensured, provded he srengh of he neracons and subsyse unodeled dynacs s suffcenly weak For each subsyse, he effec of he odelng error, ncludng neracons fro oher subsyses, can be allowed o have nfne eory Despe he odelng error, we have shown ha sall n he ean rackng error can be acheved REFERENCES [1] P Ioannou and P Kokoovc, Decenralzed adapve conrol of nerconneced syses wh reduced-order odels, Auoaca, vol 21, pp , 1985 [2] P Ioannou, Decenralzed adapve conrol of nerconneced syses, IEEE Trans Auoa Conr, vol 31, pp , 1986 [3] D T Gavel and D D Sljak, Decenralzed adapve conrol: Srucural condons for sably, IEEE Trans Auoa Conr, vol 34, pp , 1989 [4] C Wen and D J Hll, Decenralzed adapve conrol of lnear e varyng syses, n Proc 11h World Congr Auoac Conrol, Tallnn, 1990, vol 4 [5], Global boundedness of dscree-e adapve conrol jus usng esaor projecon, Auoaca, vol 28, pp , 1992 [6] C Wen, Indrec robus oally decenralzed adapve conrol of connuous-e nerconneced syses, IEEE Trans Auoa Conr, vol 40, pp , 1995

6 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 44, NO 7, JULY [7] A S Morse, A coparave sudy of noralzed and unnoralzed unng errors n paraeer-adapve conrol, n Proc 30h IEEE Conf Decson and Conrol, 1991 [8] R Orega and A Herrera, A soluon o he decenralzed sablzaon proble, Sys Conr Le, vol 20, pp , 1993 [9] R Orega, An energy aplfcaon condon for decenralzed adapve sablzaon, IEEE Trans Auoa Conr, vol 41, pp , 1996 [10] M Krsc, I Kanellakopoulos, and P Kokoovc, A new generaon of adapve conrollers for lnear syses, IEEE Trans Auoa Conr, vol 39, pp , 1994 [11] C Wen, Decenralzed adapve regulaon, IEEE Trans Auoa Conr, vol 39, pp , 1994 [12] C Wen and Y C Soh, Decenralzed adapve conrol usng negraor backseppng, Auoaca, vol 33, pp , 1997 [13] S Jan and F Khorra, Global decenralzed adapve conrol of large scale nonlnear syses whou src achng, n Proc Aercan Conrol Conf, 1995, pp [14] A Daa and P Ioannou, Decenralzed adapve conrol, n Advances n Conrol and Dynac Syses, C T Leondes, Ed New York: Acadec, 1992 [15] P A Ioannou and K S Tsakals, A robus drec adapve conroller, IEEE Trans Auoa Conr, vol 31, pp , 1986 [16] K S Narendra and A M Annasway, Sable Adapve Syses Englewood Clffs, NJ: Prence-Hall, 1989 [17] S M Nak, P R Kuar, and B E Ydse, Robus connuous-e adapve conrol by paraeer projecon, IEEE Trans Auoa Conr, vol 37, pp , 1992 [18] J-B Poe and L Praly, Adapve nonlnear regulaon: Esaon fro he Lyapunov equaon, IEEE Trans Auoa Conr, vol 37, pp , 1992 [19] R H Mddleon, G C Goodwn, D J Hll, and D Q Mayne, Desgn ssues n adapve conrol, IEEE Trans Auoa Conr, vol 33, pp 50 58, 1988 A Dagonal Recurren Neural Nework-Based Hybrd Drec Adapve SPSA Conrol Syse Xao D J and Babajde O Falon Absrac A drec adapve sulaneous perurbaon sochasc approxaon (DA SPSA) conrol syse wh a dagonal recurren neural nework (DRNN) conroller s proposed The DA SPSA conrol syse wh DRNN has spler archecure and paraeer vecor sze ha s saller han a feedforward neural nework (FNN) conroller The sulaon resuls show ha has a faser convergence rae han FNN conroller I resuls n a seady-sae error and s sensve o SPSA coeffcens and ernaon condon For rajecory conrol purpose, a hybrd conrol syse schee wh a convenonal PID conroller s proposed Index Ters Dagonal recurren neural nework (DRNN), neural nework conroller (NNC), sulaneous perurbaon sochasc approxaon (SPSA) I INTRODUCTION Nonlnear adapve conrol syse desgn s a challenge n nonlnear conrol syse heory In general, one ay use neural neworks (NN) o denfy and/or conrol unknown and/or unceran nonlnear Manuscrp receved May 9, 1996; revsed March 6, 1998 Recoended by Assocae Edor, J C Spall The auhors are wh he Deparen of Elecrcal Engneerng, The Unversy of Mephs, Mephs, TN USA (e-al: falon@ephsedu) Publsher Ie Idenfer S (99) Fg 1 Block dagra of he DA SPSA neural nework-based conrol syse syses To accoplsh hs, one needs o ran (usually, offlne) an nverse neural nework as a conroller Ths s generally dffcul snce he syse s unknown An deal schee s a drec adapve (DA) neural nework conrol syse Spall descrbed a generalzed NN based on he sulaneous perurbaon sochasc approxaon (SPSA) approach o esae he graden of he perforance funcon of an unknown nonlnear syse [1] Such a drec adapve SPSA approach does no requre any pror knowledge of he unknown syse and does no need a separae ranng phase An SPSA drec adapve conrol syse wll converge o an opal neural nework paraeer se, f exss [3] The NN-based SPSA approach as dscussed n [2] and [3] uses a forward neural nework (FNN) as he conroller The paraeer vecor sze, n general, s large For exaple, a nework whch conans four layers, wo npus, one oupu, wh wo hdden layers conanng 20 and 10 nodes, respecvely (denoed 4 2;20;10;1), has 280 eleens n he paraeer vecor Oher hngs beng equal, s ncreased copuaonal cos resuls n a slow perforance easureen perod (e, saplng perod), and he perforance easureen perod s very poran for a real-e conrol applcaon As s well known, a recurren neural nework (RNN) has soe advanages over FNN such as faser convergence, ore accurae appng ably, ec, bu s dffcul o apply he graden-descen ehod o updae he neural nework weghs n RNN [4] Ku e al [5], [6] proposed he DRNN schee ha capures he dynac behavor of a syse and, snce s no fully conneced, ranng s expeced o be uch faser han RNN DRNN wh e delay has RNN behavor bu sple connecons and s easy o use when applyng he graden-descen ehod Therefore, n hs paper, a dagonal recurren neural nework (DRNN) s eployed n a DA SPSA conrol syse Sulaon resuls are copared wh hose of he FNN SPSA schee These resuls also show ha n general, afer he SPSA process, he fxed DA SPSA neural nework-based conrol resuls n a seady-sae error because of he fne saple consran of he SPSA approach To prove he conrol perforance, a convenonal PID conroller was eployed o for a hybrd DA SPSA schee The proposed hybrd DA SPSA conrol syse was exaned by sulaon and showed good perforance II SPSA BACKGROUND Consder he proble of fndng a roo 3 of he graden equaon (1) for soe dfferenable cos funcon L : R p! R 1 There are any ehods for fndng 3 In he case L s observed n he presence of nose, an SA algorh of he generc /99$ IEEE