Reserch Journl of Applied Sciences, Engineering nd echnology 6(: 46-4, 3 ISSN: 4-7459; e-issn: 4-7467 Mxwell Scienific Orgniion, 3 Submied: Jnury 5, 3 Acceped: Mrch 7, 3 Published: November, 3 Sysems Vribles nd Srucurl Conrollbiliy: An Invered Pendulum Cse Qing M nd Xio-Qun Li School of Mechnicl nd Auomoive Engineering, Hubei Universiy of Ars nd Science, XingYng, Chin Deprmen of Auomobile Engineering, ChengDe Peroleum College, Cheng De, Chin Absrc: In order o explore he essence of srucurl conrollbiliy, srucurl conrollbiliy of invered pendulum is nlyed in his presened sudy nd sufficien conrollbiliy condiion of clss of perurbed liner sysem is obined, which is essenil o prove he srucurl conrollbiliy for he perurbed invered pendulum. wo differen srucured models of invered pendulum re consruced. Srucurl conrollbiliy of boh cses re discussed nd compred, which shows h he usul model used in conroller design for invered pendulum is jus specil cse of norml model for invered pendulum. Keywords: Perurbed liner sysem, Rionl Funcions Mrix wih muli-prmeer (RFM, srucurl conrollbiliy, sysem vrible INRODUCION Here consider liner ime-invrin sysem denoed by differenil equion: x Ax Bu ( where, we denoe by x R n, u R m, A R n n, B R n m he sysem ses, sysem inpus, coefficien mrix nd inpu mrix, respecively. Define is conrollbiliy mrix by Q c = [B, AB,, A n- B] nd composie mrix by Q (s = [si - A, B]. I is clerly h sysem ( is compleely conrollble if nd only if rnk (Q c = n, which is equivlen o h for ll s C (C is he complex number field, rnk (Q (s = n. Generlly, conrollbiliy is generic propery (Lee nd Mrkus, 967, h is, if we denoe ll he conrollble sysem se by Y, hen Y is open nd dense se. In rdiionl sysemic heory, he mhemic model ( is described excly, h is, ll he elemens of coefficien mrix nd inpu mrix re ccurely numericl numbers. However, from he view poin of engineering, i is difficul o ge such ccure model, which is o sy h here my exis unceriny in elemens of A nd B. mos of hese uncerinies re resuled from he uncerinies of sysem vribles. If we include hese sysem vribles in physicl model, hen i urns ou o be sysem model wih vribles, which we cn sy srucurl model. he invered pendulum is muli-prmeer sysem nd mos reserch on i is conrol sregy. Vrible srucure mehod is used o conrol he invered pendulum, in which he sliding mold conrol lw ws obined by Lypunov mehod (M nd Lu, 9. An nher Lypunov bsed conrol mehod ws proposed o conrol he invered pendulum (Ibne e l., 5. Oher mehod such h he pssive ful olern mehod ws used o conrol double invered pendulum (Niemnn nd Sousrup, 5. On he oher hnd, he rod opiml selecion problem in muli-link invered pendulum ws reserched (Bei-Chen nd Shung, 6. By using he conrollbiliy mrix Q c which is obined by differen rods combinion, he smlles Eigen vlues of differen conrollbiliy mrices Q c re clculed. hen he sysem which is corresponding o he smlles Eigen vlue is considered o be he opiml choice. In h sudy, he problem h rod is considered o be sysem vrible is concerned, neverheless, he clculion is sill process over he rel number field. From he view poin of srucurl model, i is worh reserching wheher he sysem vribles ffec he sysem propery (such s conrollbiliy. In his sudy srucurl conrollbiliy of invered pendulum is nlyed. wo srucurl models of invered pendulum re consruced. For clss of perurbed liner sysem sufficien condiion of conrollbiliy is presened, which is bsemen for nlying srucurl conrollbiliy of he second ype of srucurl model. Also his sudy gives reserch h wheher sysem vribles ffec he conrollbiliy of invered pendulum from he view of srucurl model poin, h is, srucurl conrollbiliy. In order o explore he essence of srucurl conrollbiliy, in his sudy we jus consider he lineried invered pendulum model. Corresponding Auhor: Qing M, School of Mechnicl nd Auomoive Engineering, Hubei Universiy of Ars nd Science, Xingyng, 4453, Chin, el.: 557636 46
Res. J. Appl. Sci. Eng. echnol., 6(: 46-4, 3 SOME DEFINIIONS AND WO INERMEDIAE CONCLUSION For liner ime-invrin sysem (, n imporn concepion is conrollbiliy, h is, wheher we cn find some suible inpus u such h for ny given iniil se x i cn be seered o ny desired desinion se. Wih he liner ime-invrin sysems, here exis some definiions bou conrollbiliy s follow: Definiion : For liner ime-invrin sysem ( if here exis some dmissible conrol u ( (wih he consrin: u B x A ( e W C(, ( P where, (x - P is he iniil se ime. hen he soluion x ( of liner ime-invrin sysem ( ime cn be denoed o: A A( x( e ( x P e Bu( A A( A WC (, ( e ( x P e BB e x P A A A A e W (, C e ( x P e e BB ( x P u( A A e ( x P e W (, W (, ( x P C C during he ime inervl (, nd rnsfer he iniil se x ( o x ( =, hen se x ( is sid o be conrollble over he inervl (,, or conrollble ime. Definiion : If ny se of liner ime-invrin sysem ( is conrollble over he ime inervl (,, hen he sysem ( is sid o be compleely conrollble. Here we consider clss of liner ime-invrin sysem ( wih perurbion denoed by: x Ax Bu P ( where, we denoe by x R n, u R m, A R n n, B R n m nd P R n he sysem ses, sysem inpus, coefficien mrix, inpu mrix nd consn perurbion respecively. Now define symmeric mrix: A e ( x P e ( x P A By he Definiion, liner ime-invrin sysem ( is conrollble. Lemm : For liner ime-invrin sysem (, he non-singulriy of conrollbiliy Grmin mrix W C (, is equivlen o h rnk (Q c = n. hen for sysem (, we define mrix denoed by c, which is lso sid o be he conrollbiliy mrix. Anoher conclusion bou conrollbiliy of sysem ( is immedie. heorem : Liner ime-invrin sysem ( is conrollble if is conrollbiliy mrix c is of full rnk, h is, rnk ( c = n. Proof: By he heorem nd Lemm, he proof is immedie. A ( A ( (3 W (, e B( B ( e d C SOME NOAIONS OF SRUCURAL CONROLLABILIY For sysem ( symmeric mrix W C (, is sid o be he conrollbiliy Grmin mrix. Similrly, for sysem ( we cn lso define is symmeric conrollbiliy Grmin mrix denoed by C (, which differ from h of sysem (. For sysem (, here gives conclusion bou is conrollbiliy. heorem : Liner ime-invrin sysem ( is conrollble if is conrollbiliy Grmin mrix C (, is nonsingulr. Proof: Becuse C (, is nonsingulr, hen (, exiss. Suppose he sysem inpu is wih he form s follow: 47 In clssicl conrol heory, he elemens of mrices A nd B re considered o be ccure excly, bu for some physicl resons some of hese elemens re pproxime numericl numbers. So someimes he vribles re preserved in sysem mhemic model. he model wih vribles is considered o be more resonble nd only he known eros re decided o be ccure. he concepion of srucurl conrollbiliy is presened firs (Lin, 974. During he nlysis of srucurl conrollbiliy only he srucure of mrices A nd B is concerned, which mens h he objecive reserched is no he numericlly known mrices A nd B, bu he corresponding sme dimension srucured mrices denoed by nd, respecively.
Res. J. Appl. Sci. Eng. echnol., 6(: 46-4, 3 Definiion 3: he elemen of srucured mrix is eher ero or undeermined vlue differing from ny oher undeermined elemen. If ll he elemens of srucured mrix re concree numericl numbers, hen i is considered o be dmissible numericl reliion. If he numericl mrices M nd N re boh he dmissible numericl reliion of srucured mrix, hen hey re srucurlly equivlen. Oher srucured mrices re proposed by reserchers (Anderson nd Clemens, 98; Ymd nd Luenberger, 985; Muro, 999 fer Lin s srucurl mrix. However, here exiss problem h he inverse mrix of hese srucured mrices is no longer o be heir corresponding srucured mrix. Clculing inverse mrix is frequen during exploring he properies of sysem ( nd (, so ll hese srucured mrices re no fied. hen clss of more generl srucured mrix ws proposed, which is cll rionl funcions mrix wih Muli-Prmeer (RFM (Lu nd Wei, 99, 994. Le,, be he prmeer vecor in sysem ( or (. R q is clled he definiion domin of r or prmeer spce. Le F (r denoe he field of ll rionl funcions wih rel coefficiens in q prmeers r,, r q. F (r [λ] denoe he polynomil ring in λ wih coefficiens in members over F (r. Definiion 4: If ech member of mrix M is member over F (r, hen he mrix M is sid o be rionl mrix or mrix over F (r. If ll he mrices of sysems re consider o be RFM, hen he sysems is sid o be rionl funcion sysems wih muliprmeers r or sysems over F (r. he mos dvnge of RFM over oher srucured mrices is h he inverse mrix of RFM is sill RFM, which is more convenien in nlying sysem properies. he srucured mrix used in his presened sudy is RFM defined wih Definiion 4. Now we cn define he mhemic model of liner ime-invrin sysem over F (r: x Arx ( Bru ( (4 or liner ime-invrin sysem wih consn perurbion over F (r: x Ar ( xbru ( P (5 Fig. : One sge invered pendulum sysem Proof: By heorem nd Definiion 4, he proof is immedie. SRUCURAL CONROLLABILIY ANALYSES FOR INVERED PENDULUM he invered pendulum now is considered o be norml sysem, so here we jus give he simple illusrion figure shown by Fig., if ny doub my refer o Shen e l. (3. In order o exploring he srucurl properies here he one sge lineried invered pendulum is concerned. In his secion wo differen lineried cses re considered s follow: Lineriion up-vericl equilibrium poin: his is he usul cse sudied. he dynmic behvior of one sge invered pendulum is described by 4 ses, h is, posiion of cr, velociy of cr, ngulr of rod nd velociy of rod ngulr, denoed by,, nd respecively. In order o deermine he mhemic model governing his sysem, we firs concern he horionl posiion of he cr, menwhile he posiion of he pendulum is +lsin. hen by Newon s second lw, h is, mss ccelerion = force, in he horionl direcion yields: d M d ( l sin m u (6 Agin pplying he Newon s second lw o pendulum mss, in he up-vericl direcion o he rod, we cn ge: m cos ml mg sin (7 where, r denoes he prmeer vecor in sysems (4 nd (5. heorem 3: Liner ime-invrin sysem wih consn perurbion (5 is srucurl conrollble if is conrollbiliy mrix c (r is of full rnk, h is, rnk c (r = n. 48 where, noion g is he ccelerion of grviy in he downwrd direcion. he differenil Eq. (6 nd (7 re nonliner, we need furher simplificion. Now le, where is he equilibrium poin, which we cll i he sic opering poin. is consn, is he deviion h
Res. J. Appl. Sci. Eng. echnol., 6(: 46-4, 3 deprs from. When = nd is no lrge, we cn ge is lineried mhemic model described by se spce form: where, X A( r X B( r U (8 mg M Ar ( M Br ( ( M m g Ml Ml X Le prmeer r = (M, m, l, g R 4, hen he conrollbiliy mrix Q C (r is: mg M M l mg M M l ( M mg Ml M l ( M m g Ml M l 3 Qc ( r B AB A B A B Since deerminn of Q C (r is: g M l M m l m nd when (M + m l = m, Q c (r hs full rnk nd he sysem is compleely conrollble. I is known h relion (M + m l = m is n lgebric vriey. Le se S be S = {r Q c (r = }, hen he Lebesgue mesuremen of invered pendulum (8 being nonsrucurl conrollble is ero, which mens h invered pendulum (8 is srucurl conrollble. sin cos (9 cos cos( cos cos sin sin cos sin cos ( he effeciveness of Eq. ( depends on he ssumpion h 6. becuse of sin. According o Eq. (9, we hve: d sin cos d sin cos ( ( Subsiuing he Equliies ( nd ( ino Eq. (6, we cn hve n pproxime differenil equion for he invered pendulum sysem: ( M m ml cos u (3 Agin pplying he sme mehod o Eq. (7, we hve: cos l g g (4 Becuse of: d cos sin ( Equion (4 kes he form: cos l g g (5 cos sin According o Eq. (3 nd (4, we hve: Lineriion ny up opering poin: Wh we hve discussed in he bove subsecion is he usul lineried cse, under which condiion h he sic opering poin is =. Now we will discuss lrge rnge h he ngulr kes vlue from he inervl [-6, 6 ]. If we do his, hen we re sure h dynmic model described by differenil Eq. (6 nd (7 is conrollble he opering poin = is excly relible when 6. Becuse he equilibrium poin, now we pply, where is smll perurbion, which is close o ero. According o he fc h cos nd sin, hen pplying hese o he rigonomeric expnsions, we hve: sin sin( sin cos cos sin 49 ( M m g cos cos ( M m mcos l ( M m mcos ( M m g sin ( M m m cos l u l M m ( g ( M m mcos l cos cos g sin u (6 M m mgcos mgcos sin u M m mcos M m mcos M m mcos mgcos mgcos sin u M m mcos M m mcos M m mcos (7
Res. J. Appl. Sci. Eng. echnol., 6(: 46-4, 3 Now we denoe he se vecor o being wih he form: X hen we my wrie Eq. (4 nd (7 in se spce form: X mg cos M m m cos ( M m g cos ( M m m cos l mg cossin + M mmcos M mmcos (8 u cos ( M m gsin ( M mmcos l ( M mmcos l Here we consider he cse in subsecion A h nd subsiue i ino Eq. (8. hen i urns ou Eq. (6, which mens h Eq. (8 is relible. Supposing h,, i.e., his invered pendulum runs he sic poin. hen by Eq. (6 we cn ge h: u ( M m g sin gg (9 ( M m cos Subsiuing Eq. (9 ino (7 yields: ( M m g sin mgcos sin M m mcos cos gg ( I is clerly h (M + m = (M + m gg, which is in fc he Newon s second lw. Le u = u + Δu where u = (M + m gg, hen by subsiuing hem ino Eq. (7 we hve: hen by subsiuing u = u + Δu nd u = (M + m gg ino Eq. (6 we ge: ( M m g cos ( M m m cos l cos u M m m l ( cos ( According o Eq. ( nd ( we cn ge noher se spce form wih he form s follow: X mg cos M m m cos ( M m g cos ( M m m cos l M m m cos cos (3 u ( M m m cos l I is clerly h bove sysem (3 is he sme form of sysem (5. Now le us es he rnk of is conrollbiliy mrix c (r. Firs le prmeer vecor r be (M, m, l, g,. hen for clculing esily, we le N denoe (M + m - m cos, hen we ge he conrollbiliy mrix c (r wih he form: mg cos cos N N l mg cos cos N N l Qc ( r cos ( M m gcos cos Nl N l cos ( M m gcos cos Nl N l I is cler h rnk c (r = 4, which mens h by heorem 3 sysem (3 is srucurl conrollble. CONCLUSION mg cos M m m cos ( M m gg u mg cos sin M m m cos = mg cos + u ( M m mcos M m m cos where gg. 4 From he srucurl model of view poin, he srucurl conrollbiliy of lineried one-sge invered pendulum is nlyed in his presened sudy. For he wo cses h lineriion up-vericl equilibrium poin nd lineriion ny up opering poin, srucurl conrollbiliy of he wo cses re discussed, which proves h he former cse is he specil form of ler cse. he liner srucurl model obined in he second cse is no he sndrd liner srucurl model (4, bu i is sndrd liner srucurl model wih consn perurbion. So he conrollbiliy
Res. J. Appl. Sci. Eng. echnol., 6(: 46-4, 3 condiion of srucurl model (5 is explored. However, he condiion obined in his presened sudy is only sufficien condiion, so exploring he necessry condiion is he fuure work. ACKNOWLEDGMEN he firs uhor wishes o hnk professor Lu Kisheng, whose some useful suggesion helps him improving his sudy. Professor Lu s pioneering work in reserching liner sysems nd elecricl neworks over F (r mkes him ineresed in srucurl properies reserch. he uhors lso wish o give heir hnks o nonymous reviewers. his sudy of he firs uhor ws suppored in pr by Nionl Nure Science Foundion of Chin under grn: No. 597769. REFERENCES Anderson, B.D.O. nd D.J. Clemens, 98. Algebric chrceriion of fixed modes in decenrlied conrol sysems. Auomic, 7(5: 73-7. Bei-Chen, J. nd C. Shung, 6. Opiml link group selecion for riple invered pendulum. Compu. Simul., 3: 58-6, (In Chinese. Ibne, C.A., O.G. Fris nd M.S. Csnon, 5. Lypunov bsed conroller for he invered pendulum cr sysem. Nonliner Dynm., 4: 367-374. Lee, E.B. nd L. Mrkus, 967. Foundions of Opiml Conrol heory, J. Willey, New York. Lin, C.., 974. Srucurl conrollbiliy [J]. IEEE. Auom. Conr., 9(3: -8. Lu, K.S. nd J.N. Wei, 99. Rionl funcion mrices nd srucurl conrollbiliy nd observbiliy. IEE Proceedings-D Con. heory Appl., 38: 388-394. Lu, K.S. nd J.N. Wei, 994. Reducibiliy condiion of clss of rionl funcion mrices. SIAM J. Mrix. Anl. Appl., 5: 5-6. M, Q. nd K.S. Lu, 9. Design nd reliion of invered pendulum conrol bsed on vrible srucure. Proceeding of Inernionl Conference on Informion Engineering nd Compuer Science (ICIECS, pp: -3. Muro, K., 999. On he degree of mixed polynomil mrices. SIAM J. Mrix Anl. Appl., : 96-77. Niemnn, H. nd J. Sousrup, 5. Pssive ful olern conrol of double inver ed pendulum: A cse sudy. Con. Eng. Prc., 3: 47-59. Shen,., S. Mei, H. Wng e l., 3. he reference design issue of robus conrol: Conrol of n invered pendulum. Con. heory Appl., (6: 974-975, (In Chinese. Ymd,. nd D.G. Luenberger, 985. Generic properies of column-srucured mrices. Liner Algebr. Appl., 65: 86-6. 4