Appled Mahemacal Scences, Vol. 9, 05, no. 8, 409-407 HIKARI Ld, www.m-hkar.com hp://dx.do.org/0.988/ams.05.54349 On One Analyc Mehod of Consrucng Program Conrols A. N. Kvko, S. V. Chsyakov and Yu. E. Balyna San-Peersburg Sae Unversy Unverses pr., 35 98504, San-Peersburg, Russa Copyrgh 05 A. N. Kvko, S. V. Chsyakov and Yu. E. Balyna. Ths arcle s dsrbued under he Creave Commons Arbuon Lcense, whch perms unresrced use, dsrbuon, and reproducon n any medum, provded he orgnal work s properly ced. Absrac The arcle proposes an analycal mehod for consrucng conrol funcon ha ensures ransferrng lnear nhomogeneous saonary sysem from an nal sae o a gven fnal sae. Condons under whch he specfed ransfer s guaraneed are presened. Mahemacs Subec Classfcaon: 34H5, 93C5, 93B05 Keywords: conrol sysem, boundary condons, sablzaon, phase coordnaes Inroducon Among he mos mporan and dffcul aspecs of he mahemacal conrol heory are ssues relaed o he developmen of mehods for buldng conrol funcons, wheren soluons of lnear saonary sysems of ordnary dfferenal equaons connec he gven pons n phase space. There s a wealh of research papers on he subec. Mos closely hs work s conneced o he research presened n [] [3]. In [] he lnear me-nvaran homogeneous sysem s consdered. An algorhm for consrucng he desred conrol funcon and he correspondng funcons of phase coordnaes presened n [] s reduced o solvng a sysem of lnear algebrac equaons. Ths sysem mgh be of que hgh order. Therefore, an mplemenaon of hs algorhm nvolves compuaonal dffcules. Mehods of consrucon of
400 A. N. Kvko, S. V. Chsyakov and Yu. E. Balyna conrol funcons for lnear nhomogeneous sysems proposed n [, 3] do no allow, n general, o fnd he requred funcons n an analycal form. The man dfference of he presen arcle from he ohers s he smplcy of developed algorhm mplemenaon. The laer s acheved due o he fac ha he desred conrol funcon and he correspondng funcon of phase coordnaes are beng found n analycal form. The obec of he sudy s a conrolled sysem of dfferenal equaons where x x x (,..., n ) T, x n R ; n n T, (,..., ) consan vecor; x Px Qu f, (.) u u u u R (,..., r ) T, r, [0,] ; f R f f f P p,,,..., n ; Q q,,..., n,,..., r consan marces; n rank(,,..., ) B AB A B n. (.) Saemen of he problem. Fnd funcons x ( ) C [0,], u( ) C [0,], sasfyng sysem (.) and condons x( 0) x0, x( T) xt. (.3) In (.3) x ( x,..., x ) s a fxed vecor. Le us agree for he menoned par n * T T T of funcons o be called a soluon o he problem (.), (.3). Le us make a change of varables n he sysem (.) replacng he dependen and ndependen varables x and accordng o he formulas x y x,. (.4) 0 0 Then n he new varables sysem (.) and boundary condons (.3) wll be as follows: dy Py Qu Px0 f d, (.5) where x x x. T 0 T 0 y y T x, (.6) (0) 0, ( 0) T 0 Changng he ndependen varable o by he formula (.7) T 0 brngs he sysem (.5) and he boundary condons (.6) o he form dy P( T 0) y Q( T 0) u ( T 0)( Px0 f ), d (.8) y(0) 0, y() x, x x x. T 0
On one analyc mehod 40 We assume below ha he ransformaons (.4) and (.7) are sasfed and he boundary condons for sysem (.) afer he subsuon of varable y by x have he form of (.8). Heren we assume P P, Q Q, f f, P P( T 0), Q Q( T 0), f ( T 0)( Px0 f ),. Problem soluon Theorem. Le he condon (.) be fulflled. Then n xt R here exss a soluon o he problem (.), (.3), whch can be obaned afer solvng he sablzaon problem for lnear me-dependen sysem of a specal ype and he subsequen soluon o he Cauchy problem for he auxlary lnear sysem of ordnary dfferenal equaons. Proof. We wll look for a funcon x (), whch s he soluon o he consdered problem, n he followng form: x( ) a( ) x. (.) Afer subsung (.) n (.) we oban he sysem a Pa Qu Px f. (.) Le us seek funcons condons a( ) C [0,], u( ) C [0,], sasfyng (.) and a(0) x, a( ) 0 as. (.3) Replacng he varable o by he formula e ; [0, ), (.4) where 0 s a fxed number, convers he sysem (.) and condons (.3) no he form e Pc e Qd e Px e f, (.5) d c( ) a( ( )), d( ) u( ( )), [0, ). (.6) We wll look for funcons and condons c(0) c( ) C [0, ), d( ) C [0, ) ha sasfy (.5) x, c( ) 0 as. (.7) Le us make he change of varable c accordng o he formula c c ( Px f ) e. (.8) In ha case, sysem (.5) and condon (.7) ake on he form e Pc e Qd e P( Px f ), (.9) d
40 A. N. Kvko, S. V. Chsyakov and Yu. E. Balyna Nex, we do he ransformaon of varable c (0) x ( f Px ). (.0) c accordng o he formula c c e P( Px f ) (.) Then he sysem (.9) and condons (.8) ake he form 3 e Pc e Qd e P ( Px f ), (.) d c (0) x ( f Px ) P( f Px ). (.3) In urn, he ransformaon 3 3 c c e P ( Px f ) (.4) 3! brngs he sysem (.) and he nal condon (.3) o he form 3 3 4 3 e Pc e Qd e P ( Px f ), d 3! 3 c (0) x ( f Px ) P( f Px ) P ( f Px ). 3! Usng he laer sysem, as well as he nal condons of he sysem and he nducve approach, we oban he ransformaon ( ) c c e P ( Px f ), (.5)! ha resuls n he orgnal sysem (.5) and he nal condon ang he followng form: ( ) e Pc e Qd e P ( Px f ), (.6) d! k k c (0) x ( ) P ( f Px ). (.7) k k! Togeher wh (.6) le us consder he sysem We wll search for d ( ) ( ) e Pc e Qd. (.8) d( c, ) M ( ) c ha provde exponenal sably for he sysem (.8). Le q,,..., r be he -h column of marx Q. Le us consruc a marx
On one analyc mehod 403 S { q,..., P q,..., q,..., P q }, (.9) kr r r where k,,..., r s he maxmum number of columns of he form k k q, Pq,..., P q,,..., r, so ha he vecors,,..., kr q Pq P q,..., q,..., P q are lnearly ndependen. Condon (.) mples ha he rank of he marx (.9) equals n. Transformaon brngs he sysem (.8) o he form c Sy (.0) r r dy d S PSe y S Qe d. (.) Based on [], marces S PS and S Q have he form e (0,...,,...,0) T n S PS e ek g k e kr ek g r kr {,...,,,...,,...,, },, where s n he -h place and g ( g,..., g,..., g,..., g,0,...,0) ; 0 k 0 * k k n k k 0 0 P q g P q... g P q,,..., r. (.) In (.) g, 0,..., k,..., g, 0,..., k are coeffcens of he vecor k decomposon no vecors P q ; 0,..., k,..., P q ; 0,..., k, S Q e ek e Le us consder he sablzaon problem for he sysem r {,...,,..., }; k. dyk { e,..., ek, g } ;,...,, k e y k e e d r d y y y e T T k (,..., ) ; (0,...,,...,0), (.3) 0 k where s n he -h place, and T r T g ( g,..., g ) ; d ( d,..., d ). In scalar form, he sysem (.3) can be wren as:
404 A. N. Kvko, S. V. Chsyakov and Yu. E. Balyna dy 0 g ke y, k e d d dy e yk g, k e y d... (.4) yk e yk g, k e y d dy e yk g k e y k. d Le y. Usng he las equaon from (.4) and he nducve approach, we oban y, y e g, y e ( e e g ) g, () (.5) y e r ( )... r ( ) g. ( ) ( ) ( ) () Afer dfferenang he las equaon (.5), from he frs equaon of sysem (.4) we ge k ( )... ( ) e d ;,..., r. (.6) ( ) ( ) 0 In (.5) r ( ),..., r( ) are lnear combnaons of exponenals wh k exponens no greaer han ( ). Expressons ( ),..., 0( ) n (.6) are lnear combnaons of exponenals wh exponens no greaer han zero. Le Le us assume ha k e d,,... r. (.7) ( ) ( k ( ) ) ;,..., r, (.8) k where ;,..., k are seleced so ha he roos,..., of he equaon sasfy he followng condons:... 0;,..., r 0 k,, n,,..., k,,..., r. (.9)
On one analyc mehod 405 Usng (.0), (.5), (.7), and (.8), we oban d e k T k S k c where k ( k ( ) k,..., 0( ) 0) ; y ( ) T ; ( k T,..., ) ; ;,..., r, (.30) k T s he marx from (.5),.e. k S s he marx conssng of he correspondng k -rows of S. Le us subsue (.30) no he rgh sde of he sysem (.8). Le ( ), (0) be he fundamenal marx of he sysem (.8) wh a conrol (.30). From he condons (.9), (.5), and (.0) we oban ( ) Ke, n, [0, ). (.3) Sysem (.6) wh he conrol (.30) (for he case n) can be represened n he followng form where n d n (n) n A( ) c e P ( Px f ), (.3) n! A e P e Qe T S k ( ) k k k ; M e T S e T S e T S. k k kr T ( ) k k k ( k,..., ) k k kr kr kr The soluon o he sysem (.3) wh nal condons (.7) (for n) has he form n n n (n) n! 0 c ( ) ( ) c (0) P ( Px f ) ( ) ( ) e d, [0, ). (.33) Condons (.5) and (.3) ensure he exsence of a consan K 0 so ha ( ) ( ) K e e, n, [0, ). (.34) ( ) ( n) Formulas (.33), (.34) mply ha n n c ( ) Ke c (0) n ( ) n 0 P ( Px f ) e K e d, K 0, [0, ). n! (.35) Based on (.35) we ge c ( ) Ke c (0) K e, K 0, [0, ). (.36) n n n
406 A. N. Kvko, S. V. Chsyakov and Yu. E. Balyna Condon (.36) guaranees ha n c ( ) 0 as. (.37) Subsung he funcon (.33) n (.30) wh ( n) and movng o he orgnal varable c() usng formulas (.5) (wh n), (.4), (.), and (.8) wll provde a par of funcons c( ) C [0, ), d ( ) C [0, ) whch, accordng o (.7) (wh n,n,...,) and (.37), sasfy he sysem (.5) and he condons (.7). If n he obaned par of funcons we reurn o he nal dependen and ndependen varables usng formulas (.6), (.4), (.), (.7), (.4) and move o he lm as, hen we oban he soluon o (.), (.3 ). In urn, he ranson o he nal dependen and ndependen varables by formulas (.4) and (.7) gves he soluon o he orgnal problem (.), (.3). The heorem has been proved. Modelng example. Le vecor f, marces PQ,, and sysem (.) wh condons (.3) have he form: 0 0 T P, Q, f, x ( x, x ), 0 0 0 0 x(0), x() x, x ( x, x ), x 0 0. Usng he developed algorhm, afer ordnary calculaons we oban he desred conrol funcon and he correspondng funcons of phase coordnaes n he followng form: 3 u( ) (( )( ( ) ) a ( ) a 3 ( )) ; ( ) 3 ( ) ( ) a ( ) ( ) ( ) ( ) ( 3 ) 3 ( ) ( ), ( 3 ) 3 3 ( ) ( ) a( ) ( ) ( ) ( ) ; 3 3 3 x ( ) ( ) ( ) ( ) ( ) ( ) ( 3 ) 3 ( ) ( ), ( 3 )
On one analyc mehod 407 3 Concluson 3 3 x ( ) ( ) ( ) ( ) ( ) ( ) ; 3 3. 4 The analyss of he proof shows ha he mehod proposed n he arcle allows he possbly of fndng he requred conrol n analycal form. Ths fac sgnfcanly smplfes he mplemenaon of he developed algorhm. References [] Alon A. Langholz G. More on conrollably of lnear me-nvaran sysem, In. J. Conrol, 44, No.4 (986), 6-76. hp://dx.do.org/0.080/000778608933657 [] Kalman R., Falb P., Arbb M. Topcs n Mahemacal Sysem Theory, McGraw-Hll, New York, 969. [3] Zubov V. I. Lecs po eor upravlenya (Lecures on Conrol Theory), Moscow, 975 (n Russan). Receved: Aprl 30, 05; Publshed: May 7, 05