Fundamental Limitations of Discrete-Time Adaptive Nonlinear Control

Transcription

1 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 9, SEPTEMBER In hs case, neher of he roos of (3) les on he agnary axs. For k > 0 and > 0, because of he second exresson n (3) a leas one of wo egenvalues les n he lef half-lane. Fro he frs exresson n (3), he su of he arguens of and has o be. Thus oher egenvalues us le n he lef half-lane. Hence he syse s sable. Thus an addon of dssave force o he syse n (9) does no desablze he syse. For k = 0(sac nsably), one of he wo egenvalues wll be a he orgn. For k < 0 (wh > 0), he su of he arguens of and s. Hence oher egenvalues us le n he rgh-hand lane n order o sasfy he exressons n (3). Thus (7) sablzed by gyroscoc forces (9) for k < 0 can be agan desablzed by he addon of dssave forces (3). For exale, he o s sablzed by a gyroscoc oen as long as he sn s suffcenly large. Evenually he sn s decreased by frcon (dssave load) o ake he o unsable. V. CONCLUSIONS In hs aer, sably ssues of arx second-order dynacal syses are dscussed. The necessary and suffcen condons of asyoc sably for e-nvaran syses n arx second-order for under varous dynac loadngs (conservave/nonconservave) are derved and a hyscal nerreaon s resened. The sably condons n he sense of Lyaunov are also derved and analyzed. As he condons are drec n ers of hyscal araeers of he syse, he effec of dfferen loadngs on he syse sably s lucd n he arx second-order for aroach. The condons are shown o be useful n he desgnng conrollers for arx second-order syses. REFERENCES [] R. E. Skelon, Adave orhogonal flers for coensaon of odel errors n arx second-order syses, J. Gudance, Conr. Dyna.,. 4, Mar [] J.-N. Juang and P. G. Magha, Robus egensyse assgnen for second order dynac syses, n Proc. AIAA Srucural Dynacs Secals Conf., 99, [3] A. M. Dwekar and R. K. Yedavall, Sar srucure conrol n arx second order for, n Proc. Norh Aercan Conf. Sar Srucures and Maerals, 995, [4] L. S. Sheh, C. D. Shh, and R. E. Yaes, Soe suffcen and soe necessary condons for he sably of ulvarable syses, Trans. ASME, vol. 00,. 4 8, Se [5] S. K. Shrvasava and S. Pradee, Sably of uldensonal lnear e-varyng syses, J. Gudance, Conr., Dyna., , Oc [6] P. Hsu and J. Wu, Sably of second-order uldensonal lnear e-varyng syses, J. Gudance, Conr. Dyna., vol. 4, , Se. 99. [7] L. S. Sheh, M. M. Meho, and H. M. Db, Sably of he second order arx olynoals, IEEE Trans. Auoa. Conr., vol. 3,. 3 33, Mar [8] F. R. Ganacher, Alcaons of Theory of Marces. New York: Wley, 959. [9] P. Lancaser, Labda Marces and Vbrang Syses. New York: Pergaon, 964. [0] A. M. Dwekar and R. K. Yedavall, Aeroelasc analyss hrough arx second order for, n Proc. AIAA Gudance, Navgaon and Conrol Conf., Aug. 995, [] H. Zegler, Prncles of Srucural Sably. London, U.K.: Blasdell, 966. [] R. Bellan, Inroducon o Marx Analyss. New York: McGraw-Hll, 970. [3] R. A. Horn and C. R. Johnson, Marx Analyss. New York: Cabrdge Unv. Press, 990. [4] E. L. Dowell and H. C. Curss, Jr., A Modern Course n Aeroelascy. London, U.K.: Kluwer, 989. [5] Z. P. Bazan and L. Cedoln, Sably of Srucures. New York: Oxford Unv., 99. Fundaenal Laons of Dscree-Te Adave Nonlnear Conrol Lang-Lang Xe and Le Guo Absrac A arcular olynoal s nroduced n hs aer whch can be used o deerne under wha condons a ycal class of dscreee nonlnear syses wh unceranes n boh araeers and noses s no sablzable by feedback, hus deonsrang he fundaenal laons of dscree-e adave nonlnear conrol. As a consequence, s shown ha for nonlnear syses wh unknown araeers and noses, he syses ay ndeed be nonsablzable, n general, whenever he usual lnear growh condon s relaxed and he nuber of unknown araeers s large, even hough he corresondng nose-free syses are globally sablzable. Index Ters Adave conrol, dscree-e, global sablzably, nonlnear dynacs, sochasc syses. I. INTRODUCTION Adave lnear conrol and he relaed ssues have been he an focus of adave conrol over he as several decades (see, e.g., [] [3], [6], [8], [9], and []). In recen years, aes have been ade oward a heory of adave nonlnear conrol. If he nonlneary s only nvolved n he nu ar, or f he ouu ar of a syse s nonlnear bu has a lnear growh rae, hen s farly well known ha he exsng adave conrol ehods can sll be aled as long as he unknown araeers ener he syse lnearly, wheher he syse s descrbed n connuous-e or dscree-e (see, e.g., [3], [5], and [6]). However, he suaon changes draacally when one aes o deal wh syses wh ouu nonlneares havng growh raes faser han lnear. Neher of he exsng ehods are useful, nor do he slares beween adave conrol of connuous- and dscree-e syses rean. For a large class of connuous-e nonlnear syses, nonlnear-dang and/or back-seng aroaches can be successfully used n adave conrol desgn regardless of he growh rae of he nonlneares (cf., e.g., []). Ths s so even n he case where exernal dsurbances exs (c.f. [4] and [4]). For exale, consder he followng connuouse sochasc conrol odel: dy =(y b + u )d + dw; b > 0 () where s an unknown araeer and w s a sandard Brownan oon, and y and u are he syse ouu and nu sgnals, resecvely. Then can be shown easly by usng he Io forula ha () can be a.s. globally sablzed by he nonlnear dang conrol u = 0y 0 y jy b j for any b > 0. Manuscr receved May 6, 998. Recoended by Assocae Edor, M. Krsc. Ths work was suored by he Naonal Naural Scence Foundaon of Chna and he Naonal Key Projec of Chna. The auhors are wh he Insue of Syses Scence, Chnese Acadey of Scences, Bejng, 00080, Chna (e-al: Lguo@ss03.ss.ac.cn). Publsher Ie Idenfer S (99)0656-X /99$ IEEE

2 778 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 9, SEPTEMBER 999 However, he connuous-e aroaches do no work n he dscree-e case when he nonlnear funcon has a growh rae faser han lnear, as observed by several auhors (see, e.g., [0] and [6]). Thus a queson naurally arses: Can we fnd a sablzng adave conroller n hs case? A generally negave answer was recenly gven n [7], where a crcal sablzably henoenon was found for a class of nonlnear conrol syses. To be recse, for he followng ycal conrol odel: y + = y b + u + w + ; b > 0 () where s an unknown araeer and fw g s a Gaussan whe nose sequence, has been shown n [7] ha () s no a.s. globally sablzable f and only f b 4, or he followng nequaly has a soluon: z 0 bz + b 0; z (; b): (3) The above resul clearly deonsraes he laons of adave conrol n he dscree-e case and shows ha he dscree-e robles are uch ore colcaed. Ths aer s anly concerned wh dscree-e syses. We shall sudy nonlnear odels wh ul-unknown araeers, whch are exensons of he scalar araeer case () as suded n [7]. Corresondng o (3), we shall nroduce a generalzed olynoal P (z) n he ularaeer case, whch wll be used o deerne when a nonlnear syse s no sablzable by feedback, hus deonsrang he fundaenal laons of adave nonlnear conrol n he dscree-e case. The boo lne of our an resul s soewha unexeced whch shows ha for dscree-e nonlnear syses wh rando unceranes n boh araeers and dsurbances, global adave sablzaon s ossble, n general, whou osng he lnear growh condon on he nonlneares of he syses. II. MAIN RESULTS Consder he followng dscree-e olynoal nonlnear regresson odel: y + = y b + y b + + n y b + u + w + ; 0 (4) where y and u are he syse ouu and nu sgnals, resecvely, ( n) are unknown araeers, and w s he nose sgnal. Assue he followng. A) b ( n) are nonnegave real nubers akng (4) eanngful and sasfyng b >b > >b n > 0: A) fw g s a Gaussan whe nose sequence wh dsrbuon N (0; ). A3) The unknown araeer vecor =[ ; ; n] s ndeenden of fw g and has a Gaussan dsrbuon N (; I n). Our objecve s o sudy he global sablzably of (4) under he above condons. Frs, we gve a recse defnon of sablzably. Defnon : Le fy ; 0 g be he feld generaed by he observaons y ; 0. Syse (4) s sad o be a.s. globally sablzable, f here exss a feedback conrol u F y = fy ; 0 g; =0; ; ; such ha for any nal condon y 0 R, l su T! =T T = y < ; a.s. Reark : We reark ha he global sablzaon of (4) s a rval ask n eher he case where s known or he case where he nose s free (.e., w 0). To be recse, f were known, we can u u 0( y b + y b + + ny b ); whch obvously globally sablzes he syse. In he case where s unknown bu he nose s free (w 0), we can oban he rue value of he araeer by solvng n ndeenden lnear equaons. For exale, f n he frs (n +)ses we choose fu ; 0 ng o be ndeendenly dencally dsrbued rando varables wh robably densy funcon (x), hen he rue value of he araeer can be obaned easly by solvng he followng lnear equaon: A =[y 0 u ;y 3 0 u ; ;y n+ 0 u n ] where A =(y b ) nn s a nonsngular arx (cf. [5]). Hence, agan, we can ake he conrol as u = 0( y b + y b + + n y b ) for > n, whch globally sablzes he nose-free syse. For ore general araerc-src-feedback odels wh no nose, a relaed bu ore colcaed wo-hase aroach can also be aled o desgn a globally sablzng adave conroller regardless of he growh rae of he nonlneares (cf., [7]). Unforunaely, he an drawback of he above wo-hase aroach s ha he resulng adave conroller s no robus wh resec o nose. In fac, he resence of nose wll even change he sablzably of dscree-e nonlnear syses draacally f he growh rae of he nonlneares s faser han lnear, as wll be shown by he followng heore ogeher wh s corollares. Theore : Under Assuons A) A3), syse (4) s no a.s. globally sablzable whenever he followng nequaly: P (z) < 0; z (; b ) (5) has a soluon, where P (z) s a olynoal defned by P (z) =z n+ 0b z n +(b 0b )z n0 ++(b n00b n )z+b n : (6) The roof s gven n he nex secon. Reark : Obvously, for n =, P (z) concdes wh he quadrac olynoal n (3). Noe ha a rval necessary condon for (5) o have a soluon s b >, and when b (4) s always a.s. globally sablzable (see [5]). To undersand he lcaons of Theore, we now gve soe dealed dscussons on (5). Corollary : If b ( n) sasfes b > and 0 < b 0b + b =( b 0) ; n0; hen (5) has a soluon whenever n log(( b +)=( b 0 ))= log b. Consequenly, whenever b > and he nuber of unknown araeers n s suably large, here always exs 0 <b n <b n0 < <b such ha (4) s no a.s. globally sablzable. The roof of hs corollary s gven n he nex secon. Reark 3: By Corollary we know ha he usual lnear growh condon osed on he nonlnear funcon f () of he general conrol odel y + = f (y ; ;y 0) +u + w + ; R n (7) canno be essenally relaxed n general for global adave sablzaon, unless addonal condons on he nuber n and he srucure of f () are osed. Corollary : Le b >, hen for n > + log(=(b 0 ))= log(b =), (5) has a soluon for any fb g sasfyng b n < b n0 < <b <b. On he oher hand, f b, hen for any n, here always exs b n <b n0 < <b <b such ha (5) has no soluon. Corollary 3: For any n and any b >b > >b n > 0, we have he followng. n ) A necessary condon for (5) o have a soluon s = b > 4: ) A suffcen condon for (5) o have a soluon s eher b > 4, n or = b > (n + )( + (=n)) n : The roofs of Corollares and 3 are n [5], due o sace laons. III. PROOF OF THE MAIN RESULTS We frs resen he roof of Theore, whch s refaced wh wo leas.

3 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 9, SEPTEMBER Lea : Le fc ; g and fd ; qg be wo sequences of osve nubers sasfyng c > c > > c ; d > d > > d q > 0; q and c c + + ; 0 ; for soe >0: Also, le f ; ; g and fj ; ;j g be wo sequences of negers arbrarly aken fro f; ; ; qg, wh k 6= l and j k 6= j l, for k 6= l. If here exss an neger 0( 0 ) such ha eher 6= 0 or j 6= 0, hen c d +d c d +d c d +d c c d c d n(;) n (d 0d ) c d where by defnon d q+ = 0. Proof: By he assuons d >d > >d q > 0 and k 6= l ;j k 6= j l for k 6= l, s easy o see ha k= d k 0 (d + d j ) 0; 8 [; ]: (8) k= Moreover, whou loss of generaly assue ha 0 =nff: 6= or j 6= ; g: Then k= d k 0 (d + d j ) k= =d 0 (d + d j ) d 0 d + > 0: (9) Now, by he assuon c c + + ;c and (8) and (9), we have c d c d c d c d +d c d +d c d +d d 0(d +d ) d 0(d +d ) d 0(d = c c c +d ) (d 0d 0d ) (d 0d 0d ) d 0d c c c 0d 0 (d 0d 0d ) c + c = 0 c (d 0d 0d ) c = n(;) c (d 0d 0d ) = n(;) (d 0d 0d ) c n(;)(d 0d ) c n(;) n (d 0d ) c : (d 0d 0d ) (d 0d 0d ) Hence Lea s rue. The followng lea lays a key role n he roof of Theore. Lea : Assue ha for soe > 0 and, jy j jy 0j + ;= ; ; ;and ha he nal condon jy 0j s suffcenly large, hen he deernan of he arx P 0 + sasfes 0 0n+ jp+j n+ (0) where by defnon y =for <0, and P+ s defned recursvely by wh P + = P 0 P'' P +' P ' ; P 0 = I () ' =[y b ;y b ; ;y b ] ; b >b > >b n > 0: () Snce he roof of hs lea s raher nvolved, we gve n he Aendx. Proof of Theore : We only need o rove ha f (5) has a soluon, hen for any feedback conrol u F y, here always exss an nal condon y 0 and a se D 0 wh osve robably such ha he ouu sgnal y of he closed-loo conrol syse ends o nfny a a rae faser han exonenal on D 0. Now consder he followng sae sace equaon ( 0): + = ; 0 = ; y + = ' + u + w + (3) where s he unknown araeer vecor defned n Assuon A3) and ' s defned by (). By our Assuons A) and A3) and he fac ha u F y, we know ha (3) s a condonal Gaussan odel, and hence he condonal execaon ^ = E[jF y ] can be generaed by he Kalan fler and he condonal covarance arx of he esaon error ( 0 ^ ) can be generaed by he Rcca equaon () or where ~ = 0 ^ (see, e.g., [3, Sec. 3.]). Nex, by (3) we know ha E[ ~ ~ jf y ]=P (4) y + = ' ~ +(' ^ + u )+w + : (5) Consequenly, by he fac ha E[ ~ jf y ]=0and E[w +jf y ]=0 follows fro (4) and (5) ha for any u F y E[y +jf y ]=' P ' +(' ^ + u ) + ' P ' +: (6) Furherore, by he arx nverson forula follows fro () ha P 0 + = P 0 + ' ' : Then jp+j 0 = jp 0 + ' ' j = jp 0 (I + P ' ' )j = jp 0 j( + ' P ' ): Hence, follows fro hs and (6) ha Le us defne D 0 = E[y+jF y 0 jp+ ] j jp 0 ; 8 0: (7) j =0 f!: E[y +jf y ] ( +)5= y +g: Then by he condonal Gaussan roery of he sequence fy g,a coleely slar arguen as ha used n [7, Aendx B] shows ha Prob(D 0 ) > 0. Hence, by (7) we have y+ jp +j 0 ( +) 5= jp 0 j ; 0; on D 0: (8) Now, le z 0 (; b ) be a soluon of (5). We roceed o rove ha on D 0 jy jjy 0j z ; =; ; : (9)

4 780 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 9, SEPTEMBER 999 We ado he nducon arguen. Frs, we consder he case where =. Snce P 0 0 = I, we have by () jp 0 j = jp ' 0 ' 0j =+k' 0 k > 0 : Therefore, by (8) we have for jy 0 j, jy j jp 0 j=jp 0 0 j > jy 0j b > jy 0j z,ond 0. Hence (9) s rue for =. Now le us assue ha for soe, jy jjy 0j z ; = ; ; ;; on D 0 ; hen by Lea, follows ha and 0n+ jp 0 +j 3 0 y0n b jp 0 Consequenly, by (8) we have on D 0 jy +j = = j 3 0 jp +j ( +) 5= jp 0 j 3( +) 5= 0n+ 0 y0n: b = 0n+ 0 0n = 3( jyjb jy 0n+j b +) 5=4 jy 0j b jy 0nj b 3( +) 5=4 jy j b jy 0j b 0b jy 0n+j b 0b jy 0nj 0b : (0) However, by he nducon assuon we have jy 0j jy j z ; ; and so by b + 0 b < 0( n) jy 0j b 0b jy j (b 0b )z ; n: () Noe ha hs nequaly also holds for >, snce y j =for j<0 by defnon. Hence, follows fro (0) and () ha on D 0 jy +j 3( +) 5=4 b +(b 0b )z ++(b 0b )z 0b z jy j 3( +) 5=4 jyj0z P (z ) jy j z [jy 0j 0z P (z ) ] z jy j z jy j z 3( +) 5=4 where he las nequaly holds for suffcenly large jy 0 j because 0z 0n 0 P (z 0 ) > 0. Hence, by nducon, (9) s rue. Thus for all large nal condons jy 0 j, he ouu rocess jy j dverges o nfny a a rae faser han exonenal on D 0. Ths colees he roof of Theore. Proof of Corollary : Take z 0 = b (; b ). We need only verfy ha P (z 0 ) < 0. By he assuon for n, we have z n 0 = ( b ) n ( b + )=( b 0 ): Hence by (6) and he condons on b, we have P (z 0 ) <z n 0 (z 0 0 b )+ b b 0 (z n )+b = 0 z n 0 b b 0 + b z0 n 0 b 0 z b = 0 z n b 0 b b 0 + b 0 z n 0 0 b b 0 + b b = 0 b 0 z n b 0 + b + b b + 0 b 0 b 0 0 b + =0: Hence, Corollary holds. IV. CONCLUDING REMARKS I s farly well known ha for nonlnear sochasc syses descrbed by nonlnear regresson odels wh lnear unknown araeers, a globally sablzng adave conroller can be desgned whenever he nonlnear funcon [say f (x)] nvolved has a lnear growh rae,.e., jf (x)j = O(jxj); as jxj!. However, n conras o he connuous-e case, essenal dffcules eerge n he dscree-e case when he nonlnear funcon has a growh rae faser han lnear. In fac, he nonlnear growh rae has been he crux n dscree-e adave nonlnear conrol for years. Naurally, one would ask he followng quesons: ) Can we reove he usual lnear growh condon n he dscree-e case? ) How far can we go fro lnear growh o nonlnear growh for global sablzaon? A frs se n hs drecon was recenly ade n [7], where was shown ha n he unknown scalar araeer case (n =), he nonlnear conrol syse n queson s globally sablzable f and only f jf (x)j = O(jxj b ) wh b<4. In he resen aer, we have deal wh he general ularaeer case (n ) by consderng he olynoal regresson odel descrbed by (4). By nroducng a new and ore general olynoal (6), we have found a creron abou suaons where (4) s no globally sablzable (Theore ). Based on ha, varous exlc cases are dscussed n Corollares 3. Perhas he os rearkable consequence of our an resul s he followng lcaon for general nonlnear regresson odels. I s ossble n general o essenally relax he usual lnear growh condon for global sablzaon, unless addonal condons are osed (see Reark 3). APPENDIX Proof of Lea : By he arx nverson forula follows fro () ha P 0 + = P 0 + ' ' ; hence, we have (a) shown a he boo of he nex age. b +b Now, le us denoe () = [y ; ; ; ], n; 0, and le (0) = e,.e., he h colun of he deny arx I, hen jp 0 +j = de =0 (); =0 (); ; =0 By he eleenary roeres of deernans, we have jp 0 +j = ; ; ; =0 n() : de( ( ); ( ); ; n(n)): ()

5 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 9, SEPTEMBER I s clear ha f n he grou ( ; ; ;n) here are a leas wo negers havng he sae value (bu dfferen fro 0), hen de( ( ); ;n(n)) = 0. So n he dscussons below we wll exclude hs knd of zero-valued deernan. We roceed o rove (0) by consderng wo cases searaely. Case I) <n0: In hs case, n order ha de( ( ); ( ); ;n(n)) 6= 0, he nuber of (0) s n ( ; ; ;n) us a leas be (n 0 0 ) and he oher negers us be dsnc. Then each er n he exanson of he nonzero deernan de( ( ); ( ); ;n(n)) conans a os [n0(n00)] = ( +)dfferen facors, whose general for s ; + (3) where 6= l ;j 6= jl, and k 6= kl for 6= l. Noe ha one such er s y b 0 0 (fro he roducs of he an dagonal eleens of he arx [ (); ; +(0); +(0); ; n(0)] ), and s dfferen fro oher ers. Now, we roceed o rove ha he absolue value of any oher er s no greaer han =[jy 0 j n(;)n (b 0b ) ] 0 0 : We dvde hs roof no wo subcases. Subcase ): If n he general for (3) = +, hen we can rewre (3) as 0 0. By our assuons, s easy o see ha Lea s alcable, hence we have 0 0 n(;)n (b 0b ) jy 0j 0 0 : Subcase ): If n he general for (3) < +, hen we can add soe ( +; ; +) o ( ; ; ) so ha ( ; ; ; + ; ; + ) T (; 0 ; ; 0): Here and hereafer, we use T (; 0 ; ; 0) o denoe he class of all eruaons of he neger sequence (; 0 ; ; 0). Then by + <n, we can choose suable j + ; ;j + ;k + ; ;k + f; ; ;ngwh j 6= jl, k 6= kl for 6= l; ; l + such ha he er s dfferen fro y b Subcase ), we have 0 0. Thus by he concluson of n(;)n (b 0b ) jy 0j Hence, he desred nequaly s rue. 0 : Now, rewre () as jp 0 + j = R + y b 0 0 ; where R denoes he suaon of all he ers dfferen fro 0 0. I s obvous ha R has a os [( +) n n! 0 ] (n n n! 0 ) ers. Hence, by he resuls roved above, we oban n n n! 0 jr j n(;) n (b 0b ) jy 0 j 0 : Therefore, by choosng he nal value jy 0j large enough, we can ake jr j less han b (=)y 0 0. Consequenly (0) follows. Case II) n 0 : Frs of all, any nonzero deernan de( ( ); ( ); ; n(n)); 0 ; ; n can be exanded as he suaon of n! ers whose general for s ; where (j ; ;j n) T (; ;n); k f0; 0; ;g; k n, and as noed before, any wo k s canno have he sae value dfferen fro (0). Obvously one such er s y b 0 0n+ [one er n de( (); ; n ( 0 n + ))]. We now show ha for any oher ers ( k 6= k or j k 6= k; for soe k n), he followng nequaly holds: j j (=(+)) n (b 0b ) jy 0n+j 0n+: (4) We roceed o rove hs by consderng hree subcases. Subcase ): If ( ; ; ; n ) T (; 0; ;0n +), hen we can drecly aly Lea o ge j j n(;) n (b 0b ) jy 0n+ j =(+) n (b 0b ) jy 0n+ j 0n+ 0n+: Subcase ): If n he grou ( ; ; ; n) here are a leas wo negers less han ( 0 n +), hen we ay fnd ( 0 ; 0 0 ; ; n ) 0 T (; 0 ; ;0n +) such ha l l ( l n) and s dfferen fro y b Consequenly, by he concluson of Subcase ), we have (=(+)) n (b 0b ) jy 0n+ j 0 0n+ : 0n+: jp 0 + j = P = ' ' = + =0 =0 =0 =0 +. =0 =0. =0 = =0 (a)

6 78 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 9, SEPTEMBER 999 Subcase 3): If n he grou ( ; ; ;n) here s jus one neger less han ( 0 n +),.e., 0 n; for soe n, hen by jy 0nj jy 0n+j =(+) we have j j 0n +b 0n+ yb (=(+))(b +b ) jy 0n+j 0 (=(+))(b +b ) 0n+ jy 0n+j 0 0n+ jy 0n+j (=(+))b (=(+)) n (b 0b ) jy 0n+j 0n+ : Hence (4) s rue. Thus, slar o he arguens n Case I), we rewre () as jp 0 + j = R + 0 0n+ ; where R denoes he suaon of [( +) n n! 0 ] ers n he deernan exansons, whch are dfferen fro y0n+. b Then by (4) we know ha jr j ( +) n n! 0 (=(+)) n (b 0b ) jy 0n+j ( +) n n! 0 (+) (=(+))n (b 0b ) jy 0 j 0 0n+ 0 0n+ 0 0n+ where he las nequaly holds for suffcenly large jy 0 j. Hence, he roof of Lea s coleed. ACKNOWLEDGMENT The auhors would lke o hank Prof. P. R. Kuar for hs helful suggesons. REFERENCES [] K. J. Åsrö and B. Wenark, Adave Conrol, nd ed. New York: Addson-Wesley, 995. [] P. E. Canes, Lnear Sochasc Syses. New York: Wley, 988. [3] H. F. Chen and L. Guo, Idenfcaon and Sochasc Adave Conrol. Boson, MA: Brkhäuser, 99. [4] H. Deng and M. Krsć, Sochasc nonlnear sablzaon I: A backseng desgn, Sys. Conr. Le., vol. 3, no. 3, , 997. [5] L. L. Xe and L. Guo, Fundaenal laons of adave conrol, Ins. Syses Scence, Chnese Acadey of Scences, Tech. Re., 998. [6] L. Guo, Self-convergence of weghed leas-squares wh alcaons o sochasc adave conrol, IEEE Trans. Auoa. Conr., vol. 4, , Jan [7], On crcal sably of dscree-e adave nonlnear conrol, IEEE Trans. Auoa. Conr., vol. 4, , Nov [8] L. Guo and H. F. Chen, The Åsrö Wenark self-unng regulaor revsed and ELS-based adave rackers, IEEE Trans. Auoa. Conr., vol. 36,. 80 8, July 99. [9] P. A. Ioannou and J. Sun, Robus Adave Conrol. Englewood Clffs, NJ: Prence-Hall, 996. [0] I. Kanellakooulos, A dscree-e adave nonlnear syse, IEEE Trans. Auoa. Conr., vol. 39,. 36 3, Nov [] P. R. Kuar and P. Varaya, Sochasc Syses: Esaon, Idenfcaon and Adave Conrol. Englewood Clffs, NJ: Prence-Hall, 986. [] M. Krsć, I. Kanellakooulos, and P. V. Kokoovć, Nonlnear and Adave Conrol Desgn. New York: Wley, 995. [3] W. Ln and J.-M. Yong, Drec adave conrol of a class of MIMO nonlnear syses, In. J. Conr., vol. 56, no. 5,. 03 0, 99. [4] Z. Pan and T. Basar, Adave conroller desgn for rackng and dsurbance aenuaon n araerc-src-feedback nonlnear syses, IEEE Trans. Auoa. Conr., vol. 43, , Aug [5] C. We and L. Guo, Adave conrol of a class of nonlnear sochasc syses, n Proc. 996 Chnese Conrol Conf., Qngdao, Se. 5 0, 996, [6] P.-C. Yeh and P. V. Kokoovć, Adave conrol of a class of nonlnear dscree-e syses, In. J. Conr., vol. 6, no., , 995. [7] J.-X. Zhao and I. Kanellakooulos, Adave conrol of dscree-e src-feedback nonlnear syses, n Proc. IEEE Conf. Decson and Conrol, San Dego, CA, Dec Coens on he Couaon of Inerval Rouh Aroxans Chy Hwang and Shh-Feng Yang Absrac In recen aers [6], [7], he Rouh aroxaon ehod was exended o derve reduced-order nerval odels for lnear nerval syses. In hs aer, he auhors show ha: ) nerval Rouh aroxans o a hgh-order nerval ransfer funcon deend on he leenaon of nerval Rouh exanson and nverson algorhs; ) nerval Rouh exanson algorhs canno guaranee he success n generang a full nerval Rouh array; 3) soe nerval Rouh aroxans ay no be robusly sable even f he orgnal nerval syse s robusly sable; and 4) an nerval Rouh aroxan s n general no useful for robus conroller desgn because s dynac unceranes (n ers of robus frequency resonses) do no cover hose of he orgnal nerval syse. Index Ters Inerval syses, odel reducon, Rouh aroxaon. I. INTRODUCTION In he las wo decades, he Rouh aroxaon ehod oneered by Huon and Fredland [] and s varans [] [5] have been recevng uch aenon n he feld of odel reducon. The ehod s based on usng he Rouh sably array o derve reduced-order odels for hgh-order lnear syses. The an advanages of he Rouh aroxaon ehod are ha has he ably o yeld sable reduced-order odels for sable orgnal hgh-order syses, o roduce a faly of reduced-order odels of dfferen orders va a sngle se of algebrac couaons, and o oban reduced odels ha rean he frs several e-oens and/or Markov araeers of he orgnal syses. Recenly, he Rouh aroxaon ehod has been exended o derve reduced-order nerval odels for hgh-order nerval ransfer funcons [6], [7]. The exenson s based on usng nerval arhec o erfor Rouh or canoncal connued-fracon exanson and nverson. Surrsngly, s observed fro he leraure [7] Manuscr receved Deceber, 997. Recoended by Assocae Edor, A. Varga. Ths work was suored by he Naonal Scence Councl of he Reublc of Chna under Gran NSC86-4-E C. Hwang s wh he Dearen of Checal Engneerng, Naonal Chung Cheng Unversy, Cha-Y 6, Tawan (e-al: chch@ccunx.ccu.edu.w). S.-F. Yang s wh he Dearen of Checal Engneerng, Naonal Cheng Kung Unversy, Tanan 70, Tawan. Publsher Ie Idenfer S (99) /99$ IEEE