On One Analytic Method of. Constructing Program Controls

Transcription

1 Appled Mahemacal Scences, Vol. 9, 05, no. 8, HIKARI Ld, hp://dx.do.org/0.988/ams On One Analyc Mehod of Consrucng Program Conrols A. N. Kvko, S. V. Chsyakov and Yu. E. Balyna San-Peersburg Sae Unversy Unverses pr., , 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

2 400 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

3 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

4 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 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

5 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:

6 404 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)

7 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

8 406 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 ), 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( ) ( ) ( ) ( ) ; x ( ) ( ) ( ) ( ) ( ) ( ) ( 3 ) 3 ( ) ( ), ( 3 )

9 On one analyc mehod Concluson 3 3 x ( ) ( ) ( ) ( ) ( ) ( ) ; 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), hp://dx.do.org/0.080/ [] 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