最优控制课件第三章-Recovered精编版

1、 2 3 )()()()()(tutBtXtAtX )()()(tXtCty )(tX)(tu)(ty )(tu )()()(tytzte )(tz * ( )u t 4 )(tQ)(tR )(tQ )(tR )()( 2 1 ff T tPete f t t TT ff T dttutRtutetQtetPeteJ 0 )()()()()()( 2 1 )()( 2 1 （4-1） dttetQte f t t T )()()( 2 1 0 dttutRtu f t t T )()()( 2 1 0 )()()(tytzte 5 f t f t x )(tz ( ) d z tz ( )z

2、t 6 ( )( ) ( )( ) ( ) ( )( ) ( ) x tA t x tB t u t y tC t x t ( )( )( )e tz ty t 1)( )( )0( )( )( )C tIz ty tx te t 2)( )0( )( )z ty te t 3)( )0( )( )( )z te tz ty t 7 )(tX )()()()()(tutBtXtAtX )()()(tetXty ItC)(0)(tz f t t TT ff T dttutRtutXtQtXtPXtXJ 0 )()()()()()( 2 1 )()( 2 1 f t t TT ff T dttut

3、RtutetQtetPeteJ 0 )()()()()()( 2 1 )()( 2 1 8 )()()()()()()()()()()( 2 1 tutBtXtAttutRtutXtQtXH TTT )()()( 2 1 )()( )( fff T ff f tPXtPXtX tXtX t )()()()(ttAtXtQ X H T 0)()()()( ttBtutR u H T )()()()( 1 ttBtRtu T )(tx)(t 9 )(t )(tX )()()(tXtKt )(tK )()()()()(tXtKtXtKt )()()()()(tXtKtAtXtQ T )()()()(

4、)()()()( 1 tXtKtBtRtBtXtAtX T 10 0)()()()()()()()()()()()( 1 tXtQtKtAtKtBtRtBtKtAtKtK TT )(tX )()()()()()()()()()()( 1 tQtKtBtRtBtKtKtAtAtKtK TT )(tK )(tK )(tK PtK f )( 11 )( f tK f t )(tK *1 ( )( )( )( )( ) T u tRt Bt K t X t )(tK)(tX )(tK 0 , f t t 12 )(tK )(tK )(tK )()(tKtK T )()()()()()()()()()(

5、 )( 1 tQtKtBtRtBtKtKtAtAtK dt tdK TTTTT T ( )( ) , T dKtdK t dtdt )()(tKtK T 13 dt tdK dt tdK T T )()( )(tK )()()()()()()()()()()( 1 tQtKtBtRtBtKtKtAtAtKtK TT PtKtK ff T )()( )()()()()()()()()()( )( 1 tQtKtBtRtBtKtKtAtAtK dt tdK TTTTT T 14 )()()()()( 1 tXtKtBtRtu T )()()( 2 1 min tXtKtXJ T 0)(tu f t

6、tt 0 )(tK PtK f )( )()()( 2 1 )()( 2 1 fff T ff T tXtKtXtPXtXJ f tt 1 ( )( )( ) 2 f t T t d Xt K t X t dt dt f t t TT ff T dttutRtutXtQtXtPXtXJ 0 )()()()()()( 2 1 )()( 2 1 15 111 ( )( )( )()()()( )( )( ) 222 f t TTT fff t d Xt K t X t dtXtK tX tXt K t X t dt 1 ( )( )( ) 2 f t T t d Xt K t X t dt dt

7、1 ( )( )( )( )( )( )( )( )( ) 2 f t TTT t Xt K t X tXt K t X tXt K t X t dt 1 ( )( )( ) 2 f t T t d Xt K t X t dt dt 1 ( ) ( )( )( ) ( ) ( ) 2 f t TT t Xt Q t X tut R t u t dt 16 11 ()()()( )( )( ) 22 TT fff XtK tX tXt K t X t 1 ( ) ( )( )( ) ( ) ( ) 2 f t TT t Xt Q t X tut R t u t dt 11 ()()()( ) (

8、 )( )( ) ( ) ( ) 22 f t TTT fff t XtK tX tXt Q t X tut R t u t dt 1 ( )( )( ) 2 T Xt K t X t 17 0)(tu 0 min J )(tK f tt PtK f )( f t min 1 ( )( )( ) 2 T JXt K t X t f t t TT ff T dttutRtutXtQtXtPXtXJ 0 )()()()()()( 2 1 )()( 2 1 18 *1 ( )( ) ( ) T u tR B K t x t * 1 ( ), ( )( ) ( ) 2 TT Jx t tx tP t

9、x t )()()()()()()()()()()( 1 tQtKtBtRtBtKtKtAtAtKtK TT () f K tP 19 ux xx 2 21 0)0(,1)0( 21 xx f t f t dtuxJ 0 22 1 2 1 00 10 A 1 0 B 0P 00 01 Q 1R 20 2212 1211 )( kk kk tK 2 12112222 2 1112221222 1( )( )( )( ) ( ) ( )( )( )2( )( ) ktktkt kt K t ktkt ktktkt )(1)( 2 1211 tktk )()()()( 22121112 tktktk

10、tk )()(2)( 2 221222 tktktk 0)()()( 221211 fff tktktk 21 )()( lim)( 0 tKttK tK dt d )()(1)( 11 2 1211fff tktktk )()()()()( 1222121112fffff tktktktktk )()()(2)( 22 2 221222ffff tktktktk f tt )2( fij tk0 kt f 22 10 f t 0246810 0 0.5 1 1.5 k12 k11 k22 0246810 -0.5 0 0.5 1 x1 x2 0246810 -1 -0.8 -0.6 -0.4

11、 -0.2 0 0.2 0.4 图图 4-1 例例4-1中各变量的变化过程中各变量的变化过程 23 , 0f tt)(tK )(tK f t 常值)(tK 24 ( )( )( )x tAx tBu t Q R 00 ( )x tx f t ( )u t * ( )u t 0 1 ( )( )( )( ) 2 TT t JXt QX tut Ru tdt 25 f t lim( ) f t K tK ( )0K t f t ()0 f K t ( )K t ( )K tK ( )K t 0P 0)()(PKtK f 26 0 1 2 TT t JX QXu Ru dt )()()(tButAx

12、tX QR )()( 1 tKXBRtu T 27 0 1 QKBKBRKAKA TT )()( 1 tXKBBRAtX T 00 ( )X tX min00 1 ( )( ) 2 T JXtKX t 28 )(tX )(tu KBBRA T1 ( )V x( )V x f t 29 KXXxV T )( )(xV XKXKXXxV TT )( XKBBRAKXKXKBBRAX TT T TT11 XKBKBRKAKAX TTT1 2 XKBKBRQXxV TT1 )( )(xV 30 31 ( )( )( )x tax tu t 0 22 )()( 2 1 dttrutqxJ 0 1 qK

13、r KaKKa 0a 0,q 0r q r 32 02 2 rqraKK 2 a r q rarK 2 a r q rarK )( 1 )(tKx r tu )( 1 )(tXK r atX 0 1 2 a r q K r a 33 0q 0 2 )( 2 1 dttruJ 02 2 raKK 0)(tu 0K02 arK0K 0u 34 )()()( 21 tutaxtx )()( 22 tbxtx)0( b f t dttuthxtxJ 0 22 2 2 1 )()()( 2 1 )()()()()( 1 tXtKtBtRtu T b a A 0 0 0 1 B h Q 0 01 1R 由

14、于 35 2 x 2 x 2 x 1)()( 2 1111 tktk )()()()()( 1211121112 tktktbktaktk htktbktaktk)()(2)(2)( 2 12121122 )()()()()( 212111 txtktxtktu 0)( PtK f 0a 0)( 12 tk)(tu 2 x 0a 0)( 12 tk 36 btktktk)()()( 111212 121211 0 ( )(0)exp( ) t ktkkb d 0)( 12 tk, 0f ttt f tt 0)( 12 f tk时 0)0( 12 k 121211 0 ( )(0)exp( )0

15、 f t f ktkkb d )(tu 2 x。 37 )()( 21 txtx utx)( 2 dttutaxtxtbxtxJ 0 22 221 2 1 )()()()(2)( 2 1 , 0 2 ba 00 10 A 1 0 B ab b Q 1 1R 2212 1211 kk kk K 38 )()()( 222112 txktxktu 1 12 k 0 221211 bkkk 2 1222 20kka 1 12 k bkkk 221211 akk 1222 2 39 0 11 k 0 2 122211 kkk 0 2 ba bak2 11 1 12 k 2 22 ak )(2)()(

16、21 txatxtu 012 2 sas 40 )(tu )(tu 0 11 ()()( ) ( ) ( )( )( ) 22 f t TTT ff t JYtPY tYt Q t Y tut Ru tdt )()()()()(tutBtXtAtX )()()(tXtCtY 41 P )(tQ)(tR f t )(tY )()()()( 2 1 fff T f T tXtPCtCtXJ f t t TTT dttutRtutXtCtQtCtX 0 )()()()()()()()( 2 1 )()()(tXtCtY f t t TT ff T dttutRtutXtQtXtPXtXJ 0 )()

17、()()()()( 2 1 )()( 2 1 42 )()( ff T tPCtC)()()(tCtQtC T P )(tQ )()( ff T tPCtC )()()(tCtQtC T 43 0)()()()()(tXtCtQtCtX TT )()()(tXtCtY )(tX0)()()(tCtQtC T )()()(tCtQtC T )()( ff T tPCtC , 0)( 0f ttttC 0)()()(tYtQtY T 0)(tQ )(tY 44 )()()()()( 1 tXtKtBtRtU T )()()()()()()( 1 tXtKtBtRtBtAtX T 00) (XtX

18、)()()()()()()()()()()()()( 1 tCtQtCtKtBtRtBtKtKtAtAtKtK TTT )()()( ff T f tPCtCtK )(tK 45 )(tY )(tY 46 f t f t )()()(tButAXtX )()(tCXtY 0 1 ( )( )( )( ) 2 TT t JYt QY tut Ru tdt 47 )(tuQ R )()( 1 tKXBRtu T 0 1 QCCKBKBRKAKA TTT )()( 1 tXKBBRAtX T KBBRA T1 48 )()()()()(tutBtXtAtX )()()(tXtCtY )(tY)(tZ

19、 )(tY )()()(tYtZte )(tu)(tY)(tZ t t TT ff T dttutRtutetQtetPeteJ 0 )()()()()()( 2 1 )()( 2 1 f t 49 )()()()()()()( 2 1 tXtCtZtQtXtCtZH T )()()()()()()()( 2 1 tutBtXtAttutRtu TT 0)()()()( ttBtutR u H T )()()()( 1 ttBtRtu T )()()()()()()()()()(tZtQtCttAtXtCtQtCt TTT 50 )()( 2 1 )( )( ff T f f tPete tX

20、 t )()()()()( ff T fff T tPZtCtXtPCtC )( f t)( f tX)( f tZ )()()()(tgtXtKt )(tK)(tg )()()()()()(tgtXtKtXtKt 51 )()()()()()()()()()( 1 tXtKtBtRtBtKtAtKtKt T )()()()()()( 1 tgtgtBtRtBtK T )()()()()()()()()()()( 11 tgtBtRtBtXtKtBtRtBtAtX TT )(tu )()()()()()()()()()()()(tZtQtCtgtAtXtKtAtCtQtCt TTTT 52 0)()()()( )()()()()()()()()()(