Local Model Architectures for Nonlinear Modelling and Control ...

1 downloads 0 Views 602KB Size Report
Roderick Murray-Smith and Kenneth Hunt. Daimler-Benz AG, Alt-Moabit 91 b, D-10559 Berlin, Germany. E-mail: murray or hunt @DBresearch-berlin.de.
      "!#%$ &&$ '  ( '&&&$*)+$ ,-$- 0. /214357698;:*5;?A@CBED2FG6IH>J?&KL1M3KLK43H>JONP=LKQH RTSVUXWYXZ\[^]`_aZ;bCcdefQdgYXh^]jik&SlmUXhonmp0l4f4Rq]>prAstsAn'_uZ;[^YvUXb4fweZ;[`W'Sxb&ytz {|]^W}SVUvY~2€& ‚&‚ ƒA„PkA[q…&&† ‡ˆ‰&Š&‚ ‹&Œ&‹&ƒA‚ ;…QŽx ‹A‚‘C’;†a“^”&‹ •–˜—\™;š›Cœ&™A}žLk&ŸS;YQik& CZ>Y4¡Z;h^¢qkt[`£&¤aSx[`ZuYXZVSx[`b&UXbC¥0¤`y&¤`h`Z;W¤u¢¦&UXŸ7¦Sx[`ZSlmYXZ§h`k Wk& CZ>YaSxb  Ÿ\kAb&h`[`kxYw¨Cb £&b kx¢b-b kAb&YvUXbCZVSx[0  y&b2SWUXŸ© [`k&Ÿ;Z;¤`¤`Z\¤gª[`ktW«h`¦CZ7UX[ktlm] ¤`Z;[`¬AZ; UXb ©C¨ h^]­kA¨ h`©C¨ hlQZ;¦2S;¬AUXkt¨C[;z¯®AUXW©&YXZAfCYXk&ŸS;YvYXy-SŸ;Ÿ;¨ [>Sxh`Z0Wk&  Z7YX¤TSx[`ZT¨C¤`Z;  h`kG[`Z;© [`Z;¤`Z;b&h°SG¥xYXktl2S;YvYXy±Ÿ;kAW©mYXZ;²³©C[`k&Ÿ\Z;¤`¤;z´¦CZ'ª[>SxWZ\¢qkA[`£ ¤`¨ ©C©2kt[`h`¤h`¦ Z Wk& CZ>YvYvUXbC¥³©C[`k&Ÿ;Z\¤`¤}UXb±[`ZVSVYoSx© ©mYvUXŸVSxh^UXkAb ¤l2Z;h`h`Z;[h`¦QSxb³WkA¤`hµS[`h^Uv¶2Ÿ7U9S;YPbCZ\¨m] [>SVYPb Z;h^¢qkt[`£±S[`Ÿ7¦&UXh`Z;Ÿ;h`¨ [`Z;¤;z0´¦&UX¤-©2S©2Z;[-¤`¦ kx¢¤µ¦Ck¢·h`¦ Z7UX[-¤`h`[`¨ Ÿ;h`¨C[`Z¸SVYX¤`k SVYvYXk¢¤h`¦CZ\W¹h`k-WkA[`Z'ZSx¤^UvYXyUXb&h`Z;¥A[>Sh`Z£&b kx¢qYXZ;  ¥AZtfuWZ;h`¦Ck&  ¤'SbC ±º»¯¼>½I¾&¼7½ Wk& CZ>YX¤ª[`ktW·kAh`¦ Z;[0©QSx[>S mUX¥tW¤¤`¨ Ÿ7¦ S¤ª¨ c;c;yYXkA¥UXŸAf˜¤`y&¤`h`Z;W·UX  Z;b&h^Uv¶QŸSxh^UXktb Sxb  ¤`h>Sh^UX¤`h^UXŸ;¤;z&dgYX¥tkA[^UXh`¦ W¤˜ªkA[wSx¨ h`kAW'Sh^UXŸg©QSx[>SWZ;h`Z;[˜Z\¤`h^UXW'Sxh^UXktb'Sxb  Wk& CZ7Y ¤`h`[`¨CŸ;h`¨ [`ZUX  Z;b&h^Uv¶2ŸVSxh^U9kAb S[`Z0¥xUX¬AZ\bLz žLk&ŸS;Y¯ik& CZ>YX¤gUXb&h`¨&UXh^UX¬AZ7YXyYXZ;b  'h`¦CZ\W¤`Z7YX¬AZ\¤h`k0h`¦ Z¨ ¤`Zkxªwž4k&ŸSVYw¿qkAb&h`[`kYvYXZ;[`¤;f ¢g¦ Z;[`Zah`¦CZ§¥xYXkAlQSVYQŸ;ktb&h`[`kxYvYXZ;[uUX¤˜Ÿ\kAW©2kt¤`Z; kª4SqŸ;ktWl&UXb2Sh^UXkAb'kxªÀ¤^UXW©mYXZqYXk&ŸS;YvYXy SxŸ;Ÿ;¨ [>Sxh`ZTŸ;kAb&h`[`kY2Y9S;¢¤;zCdž4k&ŸSVYL¿Áktb&h`[`kxYvYXZ;[T¡Z;h^¢qkt[`£­ž¯¿¡ÃLªkt[uŸ;ktb&h`[`kxYvYvUXb ¥ h`¦CZY9Sh`Z;[>SVYw CZ;¬AU9Sh^UXkAb*kª§SŸS[gktb-S¤`h`[>S;UX¥A¦&hT[`k&Sx UX¤  Z;WktbC¤`h`[>Sh`Z; Lz Ä Å ÆgÇ4È4É0ÊPËÌ|ÇÀÍ7ÉÆ 0Î J46IÏЯ?&Ð|350Ð4573Ï>3xKQH7ÏPFG3H>J4/214ÏÁÑ­/ 5H>J43'?m=4H>/CF?&H76I8T8/CK4Ï>H75>=48xH>6I/ K /mÑaÒI/Q8A?&Ò|F/2143Ò|K43HEB ÓH73/ F57:QÏuÏAÕQԘ@Î0Ï>J4F3Ï7/23/CH>?mJ45>ÒI3@P?&6IK257H>8;3xJ4Ö 6IH>5\3?m8H>H76IK4=4Ö5>3x?0Ï-K28t=4?&F-Я?mÕ|Õ43ÒI5a3-/m/&ÑwÑ0Ï7695>FG3ÐLÐ45>ÒI33Ï75˜3ÒIK2/2H>8t6IK4?&ÒIÖ±ÒI@'K4?&/C8K48ÒI=L69KL5\?&3tH7?&305F14@2/2K¯14?m3FÒIÏAÔt6I8'ט/2Ï>@28t?mÏEB Ò FG3H>J4/214Ïq/mѯ8/ K2H75>/ ÒIÒI6IK4ÖPÏ7@2Ï>H>3xFÏtØCÏ>=L8\J?mÏqÖQ?&6IKÏ>8;J431L=4ÒI69KLÖ¯Ø ?m5>30Ù 3x5>@'=4Ï73;Ñ­=4Ò|F3xH>J4/214Ï Ñ­/C5'K4/CK4ÒI69KL3t?&5'8x/ K2H>57/ Ò`ؘÕ4=4H143H735>FG6IK46IK4Ö*H>J43GÖQ?&6IK±Ï78\JL314=4ÒI3Õ|38/CF3Ï6IK48573t?&Ï769KLÖ ÒI@ 146ÛÚ8=LÒ9H?mÏH>J43Ð45>/283xÏ>Ï08/CFÐ4ÒI3Ü26IH`@69KL85>3t?mÏ>3ÏAÔ2Î0J43'ÝaÞAß\àmáaâTÞmã2ä`å\Þ&áæá9çxåè}çVäjéTÞmå;êAëaÐ45>3VB Ï73K2H>31 JL35>38t?&K*Õ|3'Ù26I3 Ó 3x1³?&ÏP?ÖC3K435;?&ÒuÑ­/ 57F-=4Ò?&H769/CKµ/mÑH>J43ÖQ?m6IKÏ78;J4314=4ÒI6IK4Ö-6I143t?LØ ÓH7=4J45>3x3G5>38/ H>K4J4Ï73H>57ìvÏ>=48;8J4H>33x141ñ=LH>Ò93C/Oí25>8t3?&ÐLKG5>3Õ¯Ï73'3K2?&H=LH>H>/ JLF*3?&ÐLH75>69/28A?&8ÒI3ÒIÏ7@ÏtÔ1L3Î0H>JL3573F3V6IòwK4338x1GH-Õ2/&@µÑPH7H>J4JL33Ýî8;ÞAJ4ß;/Cà&6IáL83ï(/mÞAÑð çxÒI/2áCÏ>8tH>?&57Ò=4/ 8;B5 ÖCÒ9/CÕ¯?&ÒgЯ?&5;?&F3xH>353Ï>H76IF?&H76I/ KOF3H7J4/214ϵ/CK±H>J43*573Ï>=4ÒIH769KLÖ³FG/Q1L3ÒTÏ7H>57=48H>=L5>3 6IÏ146IÏ`B 8=LÏ>Ï>3x1aÔ |ó ôó õ÷öaøuù2úûwüAý&þÿ&üAþqý&ù ±38/ K4Ï76I1435P1L69Ï785>3xH>3;B`H>6IF3K4/CK4ÒI6IK43t?&5Ï>@2Ï7H>3FGÏ0J¯?AÙ269KLÖµH7J43'Ö 3xK435\?mҘѭ/C5>F  

  !#"$%"&$&'() *+,*+'.-!/!01*+ ! 2   "  *+  43   65

N35>3 Ø 6IϘH7J43Ï>@2Ï7H>3F / =4H7Ð4=4HtØ H>J43g69KLÐ4=4Ha?&KL1 6IÏa? 35>/mBjFG3t?&K146IÏ7H>=457Õ¯?&K48x3 H735>FOÔ ±3573Ï>H75>6I8H'?&H7H>3K2H769/CK*H>/-Ï>6IK4ÖCÒI3;Bj6IK4ÐL=4HTÏ76IK4Ö ÒI3;B`/ =4H7Ð4=4H0Ï7@QÏ7H>3FGÏPÏ7/-H>J¯?mH

³Ø "& 45  m? K41 *+  5   ³Ô $ 573Ð4573Ï>3K2H7Ï?H>6IF3143Ò?A@2Ô&Î0J46IÏTH`@2Ð|3P/mÑ F/2143xÒ¯6IÏ0:2K4/ Ó K?mÏ0H>J43  .P<   / K4ÒI6IK43t?m5 .PH7=4146I31 Ó 6I143xÒ9@-6IK*K4/CK4ÒI69KL3t?&5Ï>@2Ï>H73FÏ06I143xKQH76¯8t?&H76I/ KaÔ ñJL3K Ó 3143|  K43'H>JL36IK2Ñ^/ 5>F*?&H769/CKµÙC38H7/ 5?mÏ        !  !#"&  $!%"& $  ( !#*   !%*+ .-!   H>JL3K H>JL3Ï7@2Ï>H>3xF 8A?&K*Õ¯3 5>6IH>H73K ?&Ï /

Ó



  &   /.0 *  !



 

Î0J43  6IK !  1L3K4/ H73ÏPH7J43'H75\?&KLÏ>Ð|/ Ï>3/ Ð|35\?mH>/ 5AÔ4Î0J43?&6IF6IK3FGÐ46I5>6I8t?mÒ¯FG/Q1L3ÒIÒ96IK4Ö6IÏ H>"/ |K41 ?Я?&5;?&F3xH>35769Ï731*Ï>H75>=48H7=45>3 Ó J46I8;J*3F-=4Ò?&H73ÏTH7J43K4/ KLÒ96IK43A?&5gÑ­=4K48H76I/ K Ô |ó ô# $&%'qù)(Lý°õ÷öaøuù2ú D2H;?&K41¯?m5>1 ÒI6IK43t?m5P?&Ð4ÐL5>/Q?m8;J43ÏPH7/-F/2143ÒIÒI6IK4Ö'8/CK4Ï>6I143x5?ÒI6IK43t?&576IÏ\?&H76I/ K/mÑ ?mÕ¯/C=4H? K4/CF6IK¯?&Òq/CЯ35;?&H769KLÖ³Ð|/ 6IK2H  3CÔ Ö|Ô'Õ2@OH\?&:26IK4Ö³*? ¯5>Ï7H`B`/ 5>1L35-Î?A@QÒI/C5Ï735>6I3Ï-/3xÜQ Ð|?&K4Ï769/CK  Ó J46I8;J*5>3Ï7=4ÒIH>ÏP6IK*H>JL3ÒI6IK43t?m+5 .< ÷F/2143Ò

-, ./   060

 

Ó J43573

, ./ !'   ?&KL1

*+ 

21

 )

3 4 E 

!'  !  !#"! $!,"! $ ( *+!'!,* ! .-   

0  576  598;: =< 8;@ A 6 BA 8DC 3 6IÏ? 8/ K4Ï7H\?mKQH*Я?m5\?mF3H735ÙC38H>/C5tÔ+ / H73 H>J¯?mH*69K F)1 G 4  ?mÒIÒPÏ>6IÖCK¯?&ÒIÏ-K4/ Ó Ï7H>5>6I8H7ÒI@ 5>3xÐ45>3Ï73K2H143Ù26?&H>6I/CK4ÏÑ^5>/ F¹H>J43'KL/ F6IK¯?mÒ2/ Ð|35\?mH>6IK4Ö-Ð|/ 6IK2HtÔ ñ6IH>J-H7J43'143H¯K46IH>6I/ KLÏ 4  ?&KL1 H7J43ÒI6IK43t?m5µÏ7@2Ï>H>3xF 2)1  8t?mK Õ¯3 Ó 5>6IH>H73K6IKH7J43-Ñ`?&F6IÒI6?&5H>5\?mK4Ï`Ñ^35`B^Ñ­=4K48xH>6I/ K Ñ­/C5>F !E+ 6 6   R    JI9KMP LON I96 K " .0QP I9K 6 *+      IK IK NP3x5>3 Ø I9K 6 6IÏ0H>JL3'=4K46IH143Ò?A@/CЯ35;?&H7/ 5P?&KL1H7J43Я/CÒI@QKL/ F6?mÒ9Ï P  N ?&KL1 Q ?&573 P  I K 66   0 576 I K 6 6 0  0 598;: I K 8;8 : @  N  I K 6   S

Suggest Documents