直動送り機構を有するメカトロニクス機器のPID制御をベースとした高精度位置制御系設計
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(12) iv. List of Figure Captions Fig. 1.1. Situation of advanced control system for industry [6].................................................. 5. Fig. 2.1. Ball screw drive table .................................................................................................11. Fig. 2.2. Machine model of ball screw drive table (Kyoto University method)[25] ..................12. Fig. 2.3. Machine model of ball screw drive table. (Tokyo University of Agriculture and Technology method)[26]......................................13 Fig. 2.4. Frequency response of Kyoto University method .......................................................16. Fig. 2.5. Frequency response of Tokyo University of Agriculture and Technology method ......16. Fig. 2.6. Experimental machine.................................................................................................17. Fig. 2.7. Frequency response of ball screw drive table (measurement) .....................................19. Fig. 2.8. Frequency response of ball screw drive table (simulation) .........................................19. Fig. 2.9. Block diagram of P-PI+LPF control system................................................................20. Fig. 2.10. Fig. 2.11 Fig. 2.12. Fig. 2.13. Fig. 2.14. Root locus of P-PI+LPF control system. Kv=286.5Hz . . 1/Ti=300 1/s. fc=2000Hz. Block diagram of P-PI+LPF. . Root locus of P-PI+LPF. Kv=286.5Hz . . . ...............................................................22. NF control system.....................................................23. 1/Ti=300 1/s. . NF control system. . fc=2000Hz. . fn=1950Hz. ..........................................24. Root locus of P-PI+LPF+NF control system by cut off frequency. Kp=300 1/s . Kv=286.5Hz. Root locus of P-PI+LPF. Kv=286.5Hz . . . 1/Ti=300 1/s. . 1/Ti=300 1/s. . fn=1950Hz. . .........................................26. NF control system fc=4000Hz. . fn=1950Hz. . ..........................................27. Fig. 2.15. Block diagram of proposed control system ...............................................................28. Fig. 2.16. Composition of experimental system ........................................................................29. Fig. 2.17. Position command.....................................................................................................30. Fig. 2.18. Frequency analysis result of torque by conventional method..........................................31.
(13) v. Fig. 2.19. Frequency analysis result of torque with notch filter......................................................32. Fig. 2.20. Frequency analysis result of torque after adjustment for filter ........................................33. Fig. 2.21. Frequency analysis result of torque by proposed method ...............................................34. Fig. 2.22. Time response of position .........................................................................................35. Fig. 2.23. Fig. 2.24. Root locus of Proposed control system. Kv=286.5Hz . . 1/Ti=300 1/s. fc=4000Hz. . fn=1950Hz. . f1=360Hz. . f2=1000Hz. . .38. Root locus of P-PI+NF control system. Kv=286.5Hz . 1/Ti=300 1/s. . fn=1950Hz. . ...............................................................40. Fig. 2.25. Frequency analysis result of torque without low pass filter.............................................41. Fig. 3.1. Block diagram of control system ................................................................................47. Fig. 3.2. Block diagram of band pass filter ...............................................................................49. Fig. 3.3. Block diagram of P-P control......................................................................................50. Fig. 3.4. Block diagram of P-PI control ....................................................................................50. Fig. 3.5. Experimental machine.................................................................................................52. Fig. 3.6. Position error by thrust ripple .....................................................................................54. Fig. 3.7. Position error by thrust ripple .....................................................................................58. Fig. 4.1. Base vibration model of linear motor driving system .................................................65. Fig. 4.2. Block diagram of base vibration system .....................................................................65. Fig. 4.3. Block diagram of control system ................................................................................67. Fig. 4.4. Time response of motion command and output...........................................................72. Fig. 4.5. Experimental machine.................................................................................................75. Fig. 4.6. Base vibration mode ...................................................................................................76. Fig. 4.7. Time response for command pattern A........................................................................79. Fig. 4.8. Time response for command pattern B........................................................................80. Fig. 4.9. Time response for command pattern C........................................................................82. Fig. 4.10. Time response for command pattern D .....................................................................83.
(14) vi. List of Table Captions Table 1.1. Performance comparison of a feed mechanism device[1].......................................... 2. Table 2.1. Status of experimental machine ................................................................................18. Table 2.2. Specifications of the experimental system................................................................30. Table 2.3. Control gains ............................................................................................................35. Table 3.1. Specifications of the linear motor drive system........................................................53. Table 3.2. Control gains ............................................................................................................55. Table 3.3. Control gains ............................................................................................................56. Table 4.1. Specifications of the linear motor drive system........................................................75. Table 4.2. Control gains ............................................................................................................78.
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(116) Ñ. 2. Ò. ÓÔUÕÖ$×"ØMÙ'ÚÔÛcÕ3ÜÝ ÞßHàMáâã-äåHÕ-æç=èé. 10. à -3Æ §Ç ¦:n:7®·=¹)%¨ ² i/°6'ò0©ª$Æ ÇR ¹ à -%;S«$Ç ÀÁ )² ¬: %7;Ç [20] Ð ê-ë-ì~íÊîcï$ð ;ñS i ° '² * ¹ ® T ¯*ë ²(· ;ñ' ð ò°- ¹ ë
(117) ³ ½ ±² ² (~´· F -$: » U ^ò V W$Æ ÇÉ H [21] %7Ç'Ю ¬ U¹ ê3ë3ìí îï ð ñ/ (c² ¹ ªSpÆ ,Ç ' ð µ%¶ ;%7Ç [22]4 5¹µ:· ì? ò¸S¹*Æ Ç G%H ì ? ,ò ¹ 6 qº»%¹ ¬ ; ¹SA;7;Ç'Ð ¼ 0½ Ï Q £¹- ÷ ¾ ¿6SÀ)Á°'> À ;« Ç ¹ÿ K Q ² (~Ä· Ã/$ ¾%¿6*ÅÆ¥O§QÇ [23] 45%7 ·=¹ ê ë ìí îï*ð ì ? à - I ÉlÊ £ ë ? E6F$¦ Â Ç $ ñ/ (^² ~ ° '0ò #:! ȹ ^ /: nÊMþÇ'Ð ì ? 8 ;S-¹ ê-ë-ì~í îcï$ð ;ñ ²3¶$Ï;*¹ % ° ' $ ò > À Æ Ç R % l ì ? § ë É ¾ ì ? 2 ÍÎ ò Ã$Ä ÆÇ'Ð q º6Ë/² ÌPCÄxQ;3Ï~ Ç ¼ ½ ì ? 0ò Ï Ðk0 R 465/²¶Ï*- S * ¹ ¼ 0½ÑÒ ¾¿6¹ ¤ ¥ ®0¾¿) ^ò ÉlÊ £ ë ? 0 RM% i%' ð > À r\² !Ï$ Ó*ÔU ì ? ¹ E6F®ò¶ ¯ P þ Ð O ² ' ð ì ? $ ^ò > À ÆÇ ò Õ´µ0 R ÀÁ H 0ò MNU¹ ¯ Q É H 0ò ȹ 0 R$ « $ r6\ò Ö~² (~· %¨ Z Æ Ç'Ð-º$ ' ð > À r\²-Å J×6Ø ^ò ¹ Ï$ ÙÚ-Æ Ç'Ð ì ÜÝB ö ¨ Z [\² !*Ï$ 8:9²¶Ï$-¹ MN$Æ Ç À Á É H Û D ¶ (,Þ$¹º$ Ö % ]_Ç'Ð 2.2 ß; ¹ êë ìí®îïð *ñ/ &ó ' ð 0ò ~*6Æ %à á ì â ÆÐ 2.3 ß; ì Ü*ÝB ö ò â ÆÐ 2.4 ß;% Ö~ $MN$Æ®Ç À Á ¿ Ã$Ä G:H À3Á Û D ² (-Ç M N H ¨ Z Æ Ç') ' ó r.ò % Ð ã ¬ $ [\: ò ;M² 2.5 ß; Ù Ú ¹ 2.6 ß;8%9; ä ò ]_Ç'Ð.
(118) Ñ 2.2. 2. Ò. ÓÔUÕÖ$×"ØMÙ'ÚÔÛcÕ3ÜÝ ÞßHàMáâã-äåHÕ-æç=èé. 11. !"#$%&. S8 ßl; ê$ë*ìí îï;ð ñ À$Á/ ÃÄ ²¦*nþÇ ê*ë$ìíkîUïð õ ë-T ì áì ò^å~Ï ¹ æ áì òñç*Æ ÇÐ Sà áì -¹ ê-ë-ì~í îcï$ð õ ëTì è ðGé à ò$¹ Ï;*¹3 1 ê:ë áìÝ í [24] ò¶ ¯ þî=Ð Sï áìÝ í$-¹6 ð ò ñ:ò ÆÊÇ ï ê:ë (óæ H ô ) ²*~õ;Ï %áìÝ %í ò¶ ¯ þî GH ;7~Ç Ð)è ð%G%é à* ¹)ö:÷*øS/ æ )ùú [25] f%û:öSügøæ~6ý [26] ²*(-Î*6Mþ¥O0Q*Rl ~ò ÿ ë ¾ ²ñ* ç Æ ÇÐ *à S$á ì ( ·B Ê Q ÇÅ ¹3& óS' ð Ê ² ! Ï;¹8*9; ~ ¹ Ï Ç $T ë*ìí î ï$ð )Ö e /² ¨SZ Ö ò¶ ¯ þî= Ð 2.2.1. . ² ê-ë-ì~íÊîcï*ð ;ñ'ñ:ò â ÆÐ ê-ë-ì~íÊîcï$ð ;ñ-¹%8A² ï$ð ë ê* ë ? (Servo motor) ò%¹ÊÏ®¹ ¤ Ý í (Coupling) ²P(· ê/ë/ì í î ²l Ê ë ê ** ë ? [%5îOQÏÇÐ $ë*? *¢)²(· ê-ë-ì~íÊî ï$ð O (Ball screw) 0 Q~¹ ¥
(119) (Nut) ²$;¢Sè ð %t ð è ð ² ìO,Q~¹
(120) Më*ì ; O,Q*R õ ë Tì (Table) ò ï$ð Æ ÇÐ ê-ë-ì~íÊî ¹ ë*? ²%&)- /¹ $ë*? ² ñ #S . . 2.1. (Support bearing). ; ñ #îOQÏǹÊÏ ñ:®òÏÇÐ. Coupling. Servo motor. Ball screw. Table. Anchor bracket Fig. 2.1. Nut. Ball screw drive table. Support bearing. Base.
(121) Ñ. 2. Ò. ÓÔUÕÖ$×"ØMÙ'ÚÔÛcÕ3ÜÝ ÞßHàMáâã-äåHÕ-æç=èé. 12. ê-ë-ì~íÊîcï$ð õ ëTì è ðGé à/²)! Ï *¹*ÏC! ¬ %Û) á ÛMN OQ¶ ·¹)}%~A;* þ $ ² ö ÷ ø:æ G àkIû%ö%ü*g6ø:æ G à S7ÇÐ 2.2 ²$ö÷ ø:æ G à~ áì ò ¹ 2.3 $² ûö%ü*gø:æ G à~ á-ì ò â Æ~Ð ö÷ ø:æ G à ;S ² ./¹!). ò" < ÏÇÏ Ä¤* # S7Ê·¹ûö%ü*gø:æ G à ;S ê3ë-ì~íÊî *¢)%è ð ê-ë-ì~íÊî $: % è ð " & OQÏÇ~Ï (' ² ¤#*S7ÇÐ xn. ). Kb. Jm. Table. Nut. ). m Tm Kc. xt Kn. Mt. Ct. b. Kg. Jc/2. Jc/2. Coupling. Jb. Ball screw. *. Mbs p R p/2. +. Ks. Motor. ,. Tm Motor torque. ,. xt. Displacement of the table. ,. b Mt. ,. Jb. Angle of rotation of ball screw. ,. Kg Ct p. ,. ,. ,. ,. m Angle of rotation of motor Mbs. Ball screw inertia. Jc. , ,. Elastic displacement of nut part. Mass of the table. Jm Rotor inertia Kb. ,. xn. ,. Ks. Torsional rigidity of ball screw. Kn. Mass of the ball screw. Coupling inertia. Kc. Rigidity of ball screw. ,. ,. ,. Torsional rigidity of coupling. ,. Rigidity of support bearing Rigidity of nut. Table sliding side viscous damping coefficient Lead Fig. 2.2. Machine model of ball screw drive table (Kyoto University method)[25].
(122) Ñ. 2. Ò. ÓÔUÕÖ$×"ØMÙ'ÚÔÛcÕ3ÜÝ ÞßHàMáâã-äåHÕ-æç=èé. xn. xt. Table. Nut. -. Ks m. Kb. Kc. Jm. -. .m. Kn. 13. Mt. Ct. b. Kg. Jc/2. Jc/2 Jb. Coupling. /. Mbs p R p/2. 0 Ball screw. Motor Fig. 2.3. Machine model of ball screw drive table. (Tokyo University of Agriculture and Technology method)[26]. ð G:é à®òR;;¹6*à Sá$ì òñ*ç Æ ¯ Q Sá$ì ¬ è S à áì )!¡ 1¹ ¨$© ò¶ ¯ þ Î R$Ð à (2.1) $² ö÷ ø:æ G à¹à (2.2) ²$û%ö%ü*gøSæ G à~è ð%Gé 3 2 Ó34 ² (Ä· à áì $ë*? ¥ ì ½ Ð 5² $Æ Ç $ë*? ò78Æ ÇÐ 2.4 ²ö÷ ø:æ G à~ ê-ë-ìíÊîcï$ð ;ñ áì $² û%ö%ü*gø:æ G à~ ê-ë-ì~íÊîcï$ð ;ñ áì ¾:¿ ¤. ò. Ç3Ð ¯;¯ ; ¯ Q . àò â Æ~Ð)è ðG%é à ø m »6 ¾:¿ ¤. ¾:¿ ¤./¹ 2.5 â Æ~Ð.
(123) Ñ. 2. Ò. ÓÔUÕÖ$×"ØMÙ'ÚÔÛcÕ3ÜÝ ÞßHàMáâã-äåHÕ-æç=èé. 14. Mt ⋅ xt + Ct ⋅ xt + Kn( xt − xn) − Kn ⋅ R ⋅ θb = 0 Mbs ⋅ xn + Kn( xn − xt ) + Kl ⋅ xn + Kn ⋅ R ⋅ θb = 0 Jb +. Jc θb + Kg1(θb − θm) + R ⋅ Kn( xn − xt ) + R 2 ⋅ Kn ⋅ θb = 0 2. Jm +. Jc θm + Kg1(θm − θb) = Tm 2. (2.1). Mt ⋅ xt + Ct ⋅ xt + Kn( xt − xn) = 0 Mbs ⋅ xn + Kn( xn − xt ) + Kl ( R ⋅ θb − xn) = 0 Jb +. Jc θb + R ⋅ Kl ( R ⋅ θb − xn) + Kg1(θb − θm) = 0 2. Jm +. Jc θm + Kg1(θm − θb) = Tm 2. ¯ ¯ ; R b êë-ì/íÊî $¢)²($Ç ¥ 3ÿ ò â Æ~Ð Kg1. ë*ìí¸î í î ·96. !.;:;7 ·U¹à (2.3) ² â Æ/ÐSä:R ¹ Kl íÊî ¾ Ê ¾ ¥ G=< ).:;S7Ê·¹à (2.4) ² â Æ~Ð Kg1 =. Kl =. Kc ⋅ Kg Kc + Kg. Kb ⋅ Ks Kb + Ks. (2.2). ¤ $ Ý % í ê ñ # ! *ê ëì. (2.3). (2.4). ( · ö%÷*øSæ G à % á*ì ¾¿ ¤ .l;¹ &ó:' ð 315Hz ¹ 1140Hz ¹ û ö%ü*gø:æ G à áì ¾:¿ ¤.;S-¹&ó%' ð 2250Hz ;S7ÇÐ3 2.5 (·Ä% 7 ÇÐ 3 !:&ó%' ðS$ë
(124) -¹ 1 > Kn ²?:·$Æ Ç õ ë 317Hz ¹ 1140Hz ¹ 2240Hz ;S T ì ¾ Ê ¾ ¥= G < ² A Æ @ Æ Ç $ ë
(125) ¹ 2 > Kn ¹ Kc ¹ Kg ²?:·$Æ Ç ê-ë-ì~íÊî í î · = G < ²A Æ @ Æ Ç * ë
(126) ¹ 3 > Kn ¹ Kb ¹ Ks ²?:$· Æ Ç ê-ë-ì~í î ¾ Ê ¾ ¥ G=< : & ó%' ðS$ ë
(127) ;S7 ÇÐ ¯ Q 2 ! à á ì :& ó%' ð ²ø-« þCBÏS w . 2.4.
(128) D. 2. E. FHG9IKJ L
(129) MONPQGRSITVUWXZYO[Q\]^_ZI`Hacbd. 15. Qþ*ÏR ¹ 89*; *¹ö:÷øæ :á$ì ò *À ÁÂ$Ã~Ä )à Sá$ì ¹~Ï Ç3Ðà ö÷ ø:æ áì ±%² Gé àò â ~ Æ Ð (2.5) ²à (2.1) − M −1C x= I 4×4 Mt 0. 0 Mbs. 0. 0. Jb +. 0. 0. 0. M=. C=. − M −1 K x + B ⋅ Tm 04×4 0 0. Ct. 0 0 0. 0 0 0. 0 0 0 0 0 0 0 0 0. Jc 2. 0 Jm +. Kn − Kn Kn + Kl − Kn K= − R ⋅ Kn R ⋅ Kn 0. 0 0 Jc 2. − R ⋅ Kn 0 R ⋅ Kn 0 2 R ⋅ Kn + Kg1 − Kg1 − Kg1. 0. Kg1. R = p (2π ). [. x = xt. xn θb θm. xt. xn θb θm. ]. T. T. B= 0 0 0. 1 Jm +. Jc 2. 0 0 0 0. (2.5).
(130) D. 2. E. 315Hz. Fig. 2.4. 1140Hz. 16. 2250Hz. Frequency response of Kyoto University method. 317Hz. Fig. 2.5. FHG9IKJ L
(131) MONPQGRSITVUWXZYO[Q\]^_ZI`Hacbd. 1140Hz. 2240Hz. Frequency response of Tokyo University of Agriculture and Technology method.
(132) D 2.2.2. 2. E. FHG9IKJ L
(133) MONPQGRSITVUWXZYO[Q\]^_ZI`Hacbd. 17. . à Sá$ì ¨ Z :R ²$¹SÖ¸)¾ ¿%*¤. òo ¹ ëe ¤. ' ¡31Êò3¶ ¯ þ * Î R$Ðf 2.6 ²Ö3®Kgh¹i 2.1 ² ê-ë-ì~íÊîcï$ð ;ñC j k ò â ~ Æ Ð Table. Coupling. Servo motor. Ball screw Fig. 2.6. Support bearing. Guide rail. Experimental machine. :Öʾ¿% ¤.D¹: ë~ê ; ë? ¥ ì ½ Ð5;l!m ; ¹%-n ¥ ì ½ 40% ';o~÷ø p=q)¿ (10 r 3000Hz) ò$Ð35µ¹ ºÊ« ë? ø%m»A6Êò FFT Û kÊ$ ¯ ; Ä s ÆÊÇ GH ò$ÈS¹ R Ð;f 2.7 ²)%-Ö ë? ø%m / ë? ¥ ì ½ ¾S¿ ¤ .: ¹Af 2.8 ² à %*á ì ( ·c78®R ë? ø%m / ë? ¥ ì ½ ¾S¿ ¤ .Êò â Æ/Ð ¯ 2 ! ¾:¿ ¤ . :& ó%' ð ò!¡ 1$Æ®Ç~M¹3Ö ®:& ó%' ð 840 ¹ 2040Hz ;~ 7 Ç3;² *¹)à %*á ì S& ó:' ð l 814 ¹ 2070Hz t3 ï u*Ï Ç ¯ ( ·¹à á ì ó r*;S 7 Ç ¯ ^ v3w;;« ÇÐã x¹ ¯ à á ì ò À-Á ¿ Ã;Ä . ë e3Ó 4 ² ¹ ÏÇÐ.
(134) D. Table 2.1. 2. E. FHG9IKJ L
(135) MONPQGRSITVUWXZYO[Q\]^_ZI`Hacbd. 18. Status of experimental machine. Ball screw. Maximum travel Root diameter Lead. Coupling. mm. 302. mm. 9.2. mm. 4. Hole diameter(motor side). mm. Hole diameter(ball screw side) N m. Torsional rigidity. N m/rad. Inertia. y. mm. 8. y. Permissible torque. y. 8. kg m2. 3.5 2940. z. 10-4. 0.12. Support bearing. Hole diameter. mm. Servo motor. Rated output. W. 100. Rated torque. N m. y. 0.318. Peak output torque Rated speed. y. 0.955. N m. rpm. Maximum speed Rotor inertia. 10. 3000. y. rpm. kg m2. 5000 0.0364. z. 10-4.
(136) D. Fig. 2.7. 2. E. FHG9IKJ L
(137) MONPQGRSITVUWXZYO[Q\]^_ZI`Hacbd. Frequency response of ball screw drive table (measurement). 2070Hz 814Hz. Fig. 2.8. Frequency response of ball screw drive table (simulation). 19.
(138) D. 2. E. FHG9IKJ L
(139) MONPQGRSITVUWXZYO[Q\]^_ZI`Hacbd. 2.3.
(140) . 2.3.1.
(141)
(142) . 20. q º%Ë® £$¤ ¥U§3¼½-¾ ;¿¸ À$Á ¿ ²¶Ï;¹i®°:'Êò > À Æ Ç6R ² § ë É ¾ % ìl? fk¼ ^½ % ìl? ¥ ì ½ lm ²$Ï:ÐîOQ®Ç ¯ ^{ Ï;Ð § ë É ¾ %. ìl? $ °/ ' > À ¹¼ ^½ % ìl? ¤:- ¾:¿ :3"/|}/òC~kSi°'6p î þ-Ïl(k ²lx) Q -ÏÇÐ ¯ Q % ì?  Q:n þ~ÇR 7 ä ·Ï6¾:¿ ² à - Æ Ç~ À-Á ò%+S~ - ²Æ Ç ¯ ^S7ÇR ¹3øm À-Á $»36%¾:¿ ¤. é ¬ ² à - Æ Ç ¯ ^ ä ϸO Q ÏÇÐ (4 r 6 m ) ã ¯¯ ;* ¹3{!S. ô ¾ õö 6S 3& óS' ð ò#*! ê$ë*ìí îï;ð ñʲ¶ Ï ¹ P-PI À*Á ¿¸² § ë É ¾ S ìk? 6¼ ½ S ìD? òÏÐ $R À*Á/ òȹ $R « ¹  +S-6. òO !% ! ÷$Ï» 6/ . ¹3Æ~þ%;x ÷:l* ò$ ËÆ Ç% ìl? Ã;Ä ²) ! Ï;3Ó 4 ò¶ ¯ þî= Ð ä /¹ £*¤ ¥§-¼/½¾ ¿¸² Dx / Q Ϯǹ ë~ê Û* ²*¶;Ï~C~ A þ ;S 7 Ç P-PI À-Á ¥ ì ½ % ìl? $ § ë É ¾ % ìl? (Low pass filter) òÕ ´ ^R À-Á ¿ [27] ²*Å® $3 Ó 4 ò¶ ¯ þî= Ðf 2.9 ² ¯ À-Á T § ½ ò â Æ~Ð.
(143) . . . m. cmd. Kp.
(144)
(145) . Kv 1 Tis. s. Kp Ti. . ,. ,. Position loop gain Integral time. cmd. ,. s. Position command. Fig. 2.9. Kv. . ,. ,. Velocity loop gain. Laplacian operator m. ,. Motor position. Block diagram of P-PI+LPF control system.
(146) . 2. . H9K
(147) OQSVZOQ ¡¢£Z¤H¥c¦§. 21. ¯ ¯ ; â ^ R ¨ ² § ë É ¾ % ìl?¨ ;~Å G (s) ò â Æ~Ð $ ¹ 1> ¨ § ë É ¾ % ìl? S; 7ǹ ëê Û ² ( Î-S-¹ 2 > ¨ § ë É ¾ % ìl? © ª; « ^Ç 45)7ÇÐ. à. (2.6). GLP ( s) =. LF. 2π ⋅ fc s + 2π ⋅ fc. fc «¬¯®±°³²´Hµ!¶·. (2.6). ¸ ¹¯º!»C¼¾½Q¿ ÀÁÂÃÅÄSÆ3ÇÈ;ÉAÊ!Ë!Â=ºÌ P-PI Í 1 ÎÏÐ ÁÑÒÓ3ÔQÂÖÕ Ï×Ø ÙÅÚ É;Û!ÜÝCÞß!àáâãä¯å Kv=286.5Hz(1800 1/s) Ìß!àæ;çAãä=å 1/Ti=300 1/s Ì Ð ÁÑAÒÓ ÔQÂ¯Õ Ïèé±ê
(148) ë ÓQì3íî fc=2000Hz ïcð;ñ ½OÌóòZôõö Ï÷øùú ºKû±üQý þÿ Ú Ý á!âãäÖå Kp Ú
(149) (0 rad/s) ï ½¿ ï þ Ï Ú 2.10 º¼3ÜÞ 2.10 Kp Ú »! ï ÈÉ Ï 1 Î Ì 2 ÎÏ#"%$& Ç Ý 800Hz Ì 2000Hz '( Ï% ) *+ñ-, º/.Ç ÜÝ0 ï1)#23 ÝKÞ4 ¿!Ì ×Ø Ù Ý ü657;á Ï%8 :9#; Ú/< »=ÝCÞ00=K¼ ½Q¿ ×Ø Ù É#> =?Ì:@ Ï÷ø!ú Ú:A SÜÖÝC¿B!º;Ó ÔÁDCFE éG ãä¯å Ú »H ï ÌÈÉ Ï#"%$& Ǻ Ý & Ç )-IJ »KCÝ ïML N J ÝCÞ4 ¿!Ì Kp Ï+ñOP ? 10000rad/s ï ÿ#Q »=ÝCÞ.
(150) R 2 S T6U V%WXZY\[]1U_^MV-`ba c_dfe\g1h/i:j/kfV:l6mon_p. (a) General view. 2nd natural frequency. 1st natural frequency. (b) Enlarged view Fig. 2.10. Root locus of P-PI+LPF control system. q Kv=286.5Hz r. 1/Ti=300 1/s. r. fc=2000Hz. s. 22.
(151) R 2 S T6U V%WXZY\[]1U_^MV-`ba c_dfe\g1h/i:j/kfV:l6mon_p. 23. t è êcÐuvG ÒÈ Ù Ï"%$& Çî )w ì3íîxy z ß!à ×دÏ÷3ø ì3íî ùú=Ï 4 {}|#~ ) ò + = J Ì ~ Ï×Ø Ù ºv é/ Ó3ÔÂÖÕ (Notch filter) Ú/ ½» & Ç Ú_ × ÜÝ_ ): 2 J ÝCÞ} é1 Ó ÔO¯Õ?Ì-EÖåCZ3ä ê Ó ÔQ¯ÕÌ-E åC/} G ê Ó;ÔÂÕ ï ü :Jù;ñÏ ì;í3î>Aç Ï-8 5Öº% 2:J ÝÞK00 =v? Ì '( Ï*+ñ: Ú%% Ü¿B!º é\ Ó ÔQÂ¯Õ Ú1# ÜÝKÞ 2.11 º P-PI × 2000Hz Ø ïÐ ÁÑAÒÓ ÔQ¯Õ!º é1 Ó ÔQÂ¯Õ Ú% ½Q¿ ×Ø @ ÏÐAéG/ AÚ ¼3ÜÞ. .
(152) cmd. Kp. .
(153)
(154)
(155) . Kv s. Fig. 2.11. m. 1 Tis. Block diagram of P-PI+LPF. . NF control system. º é1. Ó ÔQÂ¯Õ Ï }¡ î Gn(s) Ú ¼ÜÞ Q ? é1 Ó ÔQÂÖÕ Ï ì3íî#¢ £ Ú ð;ñ ÜÝQÑK¤ t ÁÕ ï ÿ Ý ) Ì¥00=?̦#§¨!Á;À#©Aå ª Ï%«¬ õ ï ½Q» ð;ñ! J »=ÝOõ Ï 0.7 º "ñ ½Q» |® _ Ï ¯#° º%§=ÝCÞ. . (2.7). s 2 + (2π ⋅ fn) 2 2π ⋅ fn s2 + s + (2π ⋅ fn) 2 Q Q = fn ( fh − fl ). G N ( s) =. fn ±»º¼ ½¾_¿%·¸ ¹. fh, fl ±³²´µ ¶:·¸ ¹. (2.7). ¸ ï_ÀÁ=Ï À ÁÂñÄSÆ3ÇÈ;ÉAÊ ËÂ=ºÌ P-PI Í 1 ÎÏÐ ÁÑAÒÓ ÔQÂÖÕ Í é/ Ó ÔÂ±Õ Ï× Ø Ù Ú ÉÛ;ÜóÝÞß;àá3â¯ã ä±å Kv=286.5Hz(1800 1/s) Ì!ßà3æç=ã ä±å ÁÑÒÓ ÔÂÖÕ Ïè±éóê
(156) ë ÓQì3í!î fc=2000Hz Ì é1 Ó3ÔÂ¯Õ Ï é 1/Ti=300 1/s Ð }Âà ì3íî fn=1950Hz ïcð;ñ ½OÌ}. áâãä¯å Kp Ú
(157) (0 rad/s) ï ½Q¿ ï þ Ï # Ú 2.12 º¼3ÜÞ.
(158) Ä 2 Å Æ6Ç È%ÉÊZË\ÌÍ1Ç_ÎMÈ-ÏbÐ Ñ_ÒfÓ\Ô1Õ/Ö:×/ØfÈ:Ù6ÚoÛ_Ü. (a) General view. 2nd natural frequency. 1st natural frequency. (b) Enlarged view. Fig. 2.12. Root locus of P-PI+LPF. q Kv=286.5Hz r. 1/Ti=300 1/s. r. . NF control system. fc=2000Hz. r. fn=1950Hz. s. 24.
(159) Ä 2 Å Æ6Ç È%ÉÊZË\ÌÍ1Ç_ÎMÈ-ÏbÐ Ñ_ÒfÓ\Ô1Õ/Ö:×/ØfÈ:Ù6ÚoÛ_Ü. 25. ÈÉ Ï 2 ÎAÏ#"%$& Ç Ý 2000Hz '( Ï% ?Ì +ñ-, º Ú/< þ Ì é1 Ó ÔQÂ¯Õ Ï_Ý Þ =ßà J ÝCÞ0 Ï ¿B 2000Hz Ï á& ? ×â J Ý ) ÌÈÉ Ï Ç Ý 800Hz '( Ï& ÇãÌ Kp Ú »! ïM*+ñ-, º_.Ç ÜÝ 1 ÎÏ#"%$& '( Ï% º Ý & Ç )ävå ÜÝ ïML N J ÝCÞ} é1 Ó ÔQÂ=Õ # º Ý/@ Ïæ 420Hz JÏ º Ì P-PI ×ØÅÍ 1 ÎÏÐ ÁÑAÒÓ ÔQÂ¯Õ ÏÖï þ ï áç;Ì Kp Ï+ñOP ? ÿ#Q »=ÝCÞ 2770rad/s ïo8 Î º × Ø Ù º} é_ ÓÔÂÕ Ú_v ½¿44=Ì Ð Á ÑÒ=Ó;ÔÂ±Õ Ï3èÅé ê ë Óì íî Ú¥è ½¿ ï þ Ï3×Ø @ Ï+ÖñAú ºé¥¯» ¯¥° Úê 0 ÿìë ÞH =áâÅãä å Ìß!àáâãä¯å Kv=286.5Hz(1800 1/s) Ìß!àæçAãä¯å 1/Ti=300 1/s Ì é Kp=300 1/s Ó3ÔÂÖÕ Ï é_ Â-à ìí!î fn=1950Hz ïVðAñ ½QÌ Ð Á!ÑÒ=Ó3ÔÂÖÕ Ï!è±é ê ë Ó ì3íî fc Ú 500 10000Hz º Ç Mí ¿ ï þ Ï%# Ú 2.13 º¼3ÜÞ 2.13 fc Ú w ½Q»!º é J »!Ì î ?Ì « B!ºïðº (ñ ëSÿ Ú < »=ÝCÞ N º w ½Q»! ï Ì:ïð ~Ú .Ç ® Ì Ý ?/òó ) ý þ ÿ 57 á )8 ÿ ÝKÌ#4 ¿?Ì 2 Î @ Ï Òô é ª ÷3ø º ê õ Ý\ö57 Ï÷øÏ ë º ÷ø!ú) 8 ÿ Ý ëÿ v Ú:< ùú )vû N J Ý Þ4¿Ì 2.13 =?;Ì¥ü:ýþ=? ÿ ) º ÝVÈÉ Ï 1 ÎÏ#"%$& Ç Ý ?Ì èéÅê
(160) ë ÓQì3íî Ú w SÜÝ- ÿ OÌ 800Hz 57;á )8 ÿ ÝCÞ00=?Ì fc=4100Hz ÏÖï þ ºÌ î ) ÿ -ï = Ì ö57= ÿ ÷ø ï ÿ#Q ¿ Þ Ú ÿ õ=º ð;ñ ÜÖÝ0 ï º Ì ×Ø @º Ý & Ç Ú_ × Ü Ý )¬
(161) = þ Ý fc ï ¯ = þ ÝCÞ . 2.12.
(162) Ä 2 Å Æ6Ç È%ÉÊZË\ÌÍ1Ç_ÎMÈ-ÏbÐ Ñ_ÒfÓ\Ô1Õ/Ö:×/ØfÈ:Ù6ÚoÛ_Ü. Fig. 2.13. 26. Root locus of P-PI+LPF+NF control system by cut off frequency. q Kp=300 1/s r. r 1/Ti=300 1/s r fn=1950Hz s 0=!Ì Ð ÁÑAÒÓ ÔOÂ¯Õ ÏèéÅê
(163) ë ÓQì3íî Ú w ð;ñ ½Q¿ P-PI Í 1 ÎÏÐ ÁÑAÒ Ó ÔQÂ¯Õ Í é1 Ó ÔÂ¯Õ ÏC×Ø ÙÅÚ É#>3ܱÝCÞß!àáâãä¯å Kv=286.5Hz(1800 1/s) Ì ß à!æAçAãä¯å 1/Ti=300 1/s Ì Ð ÁÑÒÓ ÔÂ¯Õ Ï!è±éÅê ë ÓQìíî fc=4000Hz Ì é ÓÔÂ Õ Ï é:HÂà ì=íî fn=1950Hz ïðÖñ ½ÌH áâ±ãä å Kp Ú#
(164) } ½Q¿ ï þ Ï%# Ú 2.14 º¼ ÜÞ00= fc=4100Hz ïoí º 4000Hz (0 rad/s) ï ï v= §= Ý å êcÐ Á¥¤º» ð;ñ = þ ÝQû3ýõ ) 4000Hz = Ý0 ï ½Q¿ Ï ?Ì ï º ÝCÞ Ð ÁÑAÒÓ ÔQÂ¯Õ ÏèÖéÅê
(165) ë ÓQì3íî Ú 2000Hz ï ½Q¿ ï þ ºCáç;Ì Ï 9#; ) ïð 0 = ;¿BÌ 57;á )w ÷øÏ%#¾ï ÿ#Q »=ÝKÞ4 ¿!Ì *+ñ-, 4 ~ Ú = Ï 9#; ) ÿ#Q » ê Ì Kp Ï+ñOP ? 4370rad/s 4=ú ~ ½Q»:=ÝCÞ0 Ï! Ì Ð ÁÑAÒÓ ÔQÂ¯Õ ÏèÖéÅê
(166) ë ÓQì3íî Ú ÿ õ=º ð;ñ ÜÝ0 ï º Ì ×Ø Ù º Ý ß " ) J Ý ïML N J ÝCÞÈÉ & Ç Ï%8 5#ã ×Ø ãä=å Ï:w ãä¯#å "¯º Ý ÷øÏ:w Ï 2 ÎÏ#"%$& Çî¥?Ì 2.12 ï_À_Á º é1 Ó ÔQ¯Õ:= +ñ " J »=ÝCÞ Kv=286.5Hz.
(167) Ä 2 Å Æ6Ç È%ÉÊZË\ÌÍ1Ç_ÎMÈ-ÏbÐ Ñ_ÒfÓ\Ô1Õ/Ö:×/ØfÈ:Ù6ÚoÛ_Ü. (a) General view. 2nd natural frequency. 1st natural frequency. (b) Enlarged view. Fig. 2.14. Root locus of P-PI+LPF. q Kv=286.5Hz r. 1/Ti=300 1/s. r. . NF control system. fc=4000Hz. r. fn=1950Hz. s. 27.
(168) Ä 2 Å Æ6Ç È%ÉÊZË\ÌÍ1Ç_ÎMÈ-ÏbÐ Ñ_ÒfÓ\Ô1Õ/Ö:×/ØfÈ:Ù6ÚoÛ_Ü 2.3.2.
(169) . ¸ ¹=?ÌKÈ;É Ï 2 ÎÏ"$#& ÔQÂ¯Õ ÏèéÅê
図
+7
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