仿射非线系统状态方程的任意阶近似解.doc

30 6 2008 6 Journal of Un iversity of Science and Technology BeijingVol . 30 No. 6Jun . 2008/dy999/article_396_4440_1.shtml1 ,2) 2) 2)1) , 102600 2) , 100083,. ,Volterra . ,.; ; ; Volterra ; ; TP 273Any order a pproximate sol ut ion of the state equat ion f or an aff ine nonl inear systemCA O S haoz hong1 , 2) , L I U Hepi ng2) , T U X uy an2)1) College of Info r matio n & Mechanical Engineering , Beijing Instit ute of Grap hic Co mmunicatio n , Beijing 102600 , China2) School of Info r matio n Engineering , U niversit y of Science and Technology Beijing , Beijing 100083 , ChinaABSTRACT The state equatio n of an typical affine no nlinear system was solved wit h t he ordinary differential equatio n t heory. Byutilizing t he expansio n exp ressio n of equilibrium point of t he system , t he ho mogeneous equatio ns solutio n was obtained , and t hen t heno nlinear differential equatio n was equivalent to it s no nlinear Volterras integral equatio n of t he seco nd kind by t he co nstant variatio n met hod . Any order app ro ximate solutio n of t he equatio n was p resented , and it s co nvergence was mat hematically p roved by t he successive app ro ximatio n met hod.KEY WO RDS no nlinear system ; state equatio n ; ordinary differential equatio n ; Volterras integral equatio n ; successive app ro ximatio n met hod ; app ro ximate solutio n, , , , 1- 4 . , . , , ( : 2007 04 12 : 2007 07 25: ( No . 60374032 , No .60673101) ; ( No . KM200810015003) ;( No . TXM2007 - 014223 - 044661) ;( No . 09170107019): ( 1965 ) , E mail : cszh6502 163 . co m) , ;, . 5- 7 , , . ,. , , .,. x = 0 , ; Volterra . , , .6 : 691 1 n :x = f x , u()x M , u U( 1)y = h ( x , u), M n , U .( 1) , x u . , 1- 2 := ( ) +( )xf xg x u( 3) , ( 3) :n1f i ( x) = aij ( t ) x j + i ( x ( t ) , t )( 5)j =, gij ( x) :gij ( x) = bij ( t ) + Gi ( x ( t ) , t )( 6), bij ( t ) x , G ( x ( t ) , t ) .( 5) ( 6) ( 2) , :( ) =( )( ) +( )( ) +x tA t x tB t u ty = h ( x),x M , u U( 2)T( x ( t ) , t ) + G ( x ( t ) , t ) u ( t )( 7),a11 ( t )a12 ( t )a1 n ( t )f ( x) = ( f 1 ( x) , f 2 ( x) , , f n ( x) ) ,a21 ( t )a22 ( t )a2 n ( t )g ( x) =g11 ( x)g12 ( x)g1 m ( x)g21 ( x)g22 ( x)g2 m ( x),g n1 ( x)g n2 ( x)g n m ( x)TA ( t ) =,a n1 ( t )a n2 ( t )a n n ( t )b11 ( t )b12 ( t )b1 m ( t )b21 ( t )b22 ( t )b2 m ( t )x = ( x 1 , x 2 , , x n )T, u = ( u1 , u2 , ,TB ( t ) =,u m ), y = ( y 1 , y 2 , , y p )Tbn1 ( t )bn2 ( t )bnm ( t ), h ( x ) = ( h1 ( x ) , h2 ( x ) , , h p ( x ) ).f , f ( 0) = 0 h ( x) =T( x ( t ) , t ) =( ( x ( t ) , t ) , ( x ( t ) , t ) , ( x ( t ) , t ) ) T ,12nG ( x ( t) , t) =( h1 ( x ) , h2 ( x ) , , h p ( x ) )hi ( x ) hi ( 0) = 0 ( i = 1 , 2 , , p ) , , ( 2) gi j ( x) ( i = 1 , 2 , , n ; j = 1 , 2 , , m )M . , x = 0 ( u = 0) .f ( x) g ( x) x = 0 ( u = 0) :nnnG11 ( x ( t) , t)G12 ( x ( t) , t)G1 m ( x ( t) , t)G21 ( x ( t) , t)G22 ( x ( t) , t)G2 m ( x ( t) , t).Gn1 ( x ( t) , t)Gn2 ( x ( t) , t)Gnm ( x ( t) , t)( 7) ,x ( t ) = A ( t ) x ( t )( 8)x ( t = 0) = x ( 0) .f i ( x) = aij ( t ) x j + aij 1 j 2 ( t ) x j1 x j2 +j = 1nnnj 1 = 1 j 2 = 1Picad :ai j1 j2 j3 ( t ) x j1 x j2 x j3 + +j 1 = 1 j 2 = 1 j 3 = 1x ( t ) = R ( t ) x ( 0)( 9)nnntk123k0aij1 j 2 j ( t ) x jx j x jx j + ( 3)R ( t ) = I +A ( t 1 ) d t 1 +j 1 = 1 j 2 = 1j k = 1t t 1A ( t 1 ) A ( t 2 ) d t 2 d t 1 + +0 0nni ( x ( t ) , t ) = aij1 j 2 ( t ) x j 1 x j 2 +t t 1t n - 11 )( ) nnnj 1 = 1 j 2 = 10 00A ( tA t 2ai j1 j2 j3 ( t ) x j1 x j2 x j3 + +j 1 = 1 j 2 = 1 j 3 = 1A ( t n ) d t n d t n - 1 d t 2 d t 1 + ( 10)nnntaij1 j2 j ( t ) x j x jx j + ( 4)j 1 = 1 j 2 = 1j k = 1k12kR ( t ) = exp0 A () d( 11)692 30 , ( 7) . ( 7). , , :x ( t ) = R ( t ) C ( t )( 12)t0x (0) ( t) = R ( t) x (0) + R ( t)R - 1 () B () u () d, C ( t ) , R ( t ) x ( m ) ( t )t0= R ( t ) x ( 0) + R ( t )R - 1 () R ( 0) = I ( n n ) , C ( t ) C ( 0) = x ( 0) .( 12) ( 7) , ( x ( m - 1) () ,) + B () +G ( x ( m - 1) () ,) u () d, m 1( 16)d R ( t )d tC ( t ) + R ( t )d C ( t ) = A ( t ) R ( t ) C ( t ) +d t, u = 0 , ( 16) 8- 9 ,B ( t ) u ( t ) + ( x ( t ) , t ) + G ( x ( t ) , t ) u ( t ) ,x ( 0) ( t ) = R ( t ) x ( 0)d R ( t )( m ) ( )t( )( 0) + R ( t )d t= A ( t ) R ( t ) , :R ( t ) d C ( t ) =d txt= R t x( x ( m - 1) () ,) dR - 1 ( ) 0( 17)( x ( t ) , t ) + ( B ( t ) + G ( x ( t ) , t ) ) u ( t ) .R ( t ) R - 1 ( t ) , 0 t 2 , C ( t ) :t. C ( t ) = x ( 0) + 0 R- 1 () ( x () ,) +( 14) , ( B () + G ( x () ,) ) u ( t ) d( 13). 0t( 13) ( 12) , ( 7) :( t) = R ( t) x (0) + R ( t) R - 1 () B () u () d,F ( x , u , t ,) =- 1tx ( t) = R ( t) x (0) + R ( t) 0 R() ( x () ,) +R ( t ) R- 1 () ( x ,) + G ( x ,) u () , B () + G ( x () ,) u () d( 14)( 14) Volterra ,:t0x ( 0) ( t) = R ( t ) x ( 0) + R ( t )R - 1 () B () u () dt( 14) :tx ( t ) = ( t ) + 0 F ( x , u , t ,) d( 18), :x ( 0) ( t ) = ( t )t00x(1) ( t) = R ( t) x (0) + R ( t)R - 1 () ( x(0) () ,) +x ( m ) ( t ) = ( t ) +F ( x , u , t ,) d, m 1( 19) B () + G ( x ( 0) () ,) u () d, tx(2) ( t) = R ( t) x (0) + R ( t)R - 1 () ( x(1) () ,) +m x ( m ) (t ) , 0 B () + G ( x ( 1) () ,) u () dt0x ( m ) ( t) = R ( t ) x ( 0) + R ( t )R - 1 () ( x ( m - 1) () ,) + B () +( 14) .i ( t ) F ( x , u , t ,) :(1) i ( t ) t 0 t d , | i ( t ) | Ci ;( 2) Fi ( x , u , t ,) 0 t d , 0 t , ai x i bi , ci u i ei ,G ( x ( m - 1) () ,) u () d| Fx , u , t ,| Ma ( t ) b , i ()iiii( 15)( 15) , , ( 14) Fi ( x , u , t ,) Lip schit | Fi ( x , u , t ,) - Fi ( x , u , t ,) | 6 : 693 N( l)( l - 1)K | x j - x j | , j = 1 , 2 , , N ,j = 1, K ., 10 ,( 19) x ( m ) ( t ) , .miiiix ( m ) ( t ) = x ( 0) ( t ) + x ( l) ( t ) - x ( l - 1) ( t ) ,1 - N Kt , x i ( t ) - x i( t ) l = 1, x ( m ) ( t ) ., , . , , ; , l = 1, x ( l)( l - 1).l = 1.i ( t ) - x i( t ) 3 :| x ( 1)( 0)i ( t ) - x i ( t ) | . 0tt| Fi ( x () , u () , t ,) | M i d= M i t ,0, Volterra - 1| x (2)(1)t(1),m,m, .i ( t) - x i ( t) | 0 Fi ( x() , u () , t ,) -Fi ( xtN( 0)() , u () , t ,) d1 Alberto I. N onli nea r Cont rol S yste ms . 3rd ed. Lo ndo n :( 1)( 0) 120 K | x j () - x j () | dKM t ,Sp ringer Verlag , 1995j = 1N, M = M j .j = 1t22 Ho ng Y G , Cheng D Z. A nal ysis a n d Cont rol on N onli nea r S yste ms . Beijing : Science Press , 2005( , . . :| x (3)(2)(2), 2005)i ( t) - x i ( t) | 0( 1) Fi ( x() , u () , t ,) -3 Xie L L , Guo L . Adaptive co nt rol of a class of discrete time affineFi ( xtN() , u () , t ,) dno nlinear systems. S yst Cont rol L et t , 1998 , 35 : 2024 Popescu M . On minimum quadratic f unctio nal co nt rol of affine( 2)( 1) 1 2 3no nlinear systems. N onli nea r A nal , 2004 , 56 : 11650 K | x j () - x j () dM N K t ,j = 12 3, 5 Pei H L , Zho u Q J . App ro ximate linearizatio n of no nlinear systems a neural net wo r k app roach . Cont rol T heory A p pl , 1998 , 15| x ( l)( l - 1) 1l - 2l - 1 l( 1) : 34i ( t ) - x i( t ) | M Nl !,Kt ,6 Ver hulst F. N onli nea r Di f f erent i al Equat ions a n d Dy na m icalS yste ms . New Yo r k : Spinger Verlag , 1992| x ( l + 1)( l)7 Xu Z G , Hauser J . Higher o rder app ro ximate feedback linearizai( t ) - x i ( t ) | ltio n abo ut a manifold fo r multi inp ut systems. I E E E T ra ns A u0 Fi ( x