Control Systems. Mathematical Modeling of Control Systems.

Transcription

1 Conrol Syem Mahemacal Modelng of Conrol Syem

2 Oulne Mahemacal model and model ype. Tranfer funcon model Syem pole and zero Chbum Lee -Seoulech Conrol Syem

3 Mahemacal Model Model are key elemen n he degn and analy of conrol yem qualave mahemacal model??? u y We mu make a comprome b/w he mplcy of he model v. he accuracy of he reul of analy Chbum Lee -Seoulech Conrol Syem

4 Lnear v. nonlnear yem Lnear yem: he prncple of uperpoon hold Lneary n mahemac Le V and W be vecor pace over he ame feld K. A funcon f: V W ad o be a lnear map f for any 2 vecor x and y n V and any calar α n K, he followng condon are afed: Lneary n yem addvy homogeney A general yem can be decrbed by operaor H, ha map an npu x a a funcon of o an oupu y a ype of black box decrpon. Lnear yem afy he propere of uperpoon and homogeney. Chbum Lee -Seoulech Conrol Syem

5 Lnear Tme Invaran Syem Lnear Tme Invaran = Lnear & me nvaran A me-nvaran TIV yem one whoe oupu doe no depend explcly on me. If he npu gnal x produce an oupu y, hen any me hfed npu, x+, reul n a me-hfed oupu y+ Tme nvaran mean ha he coeffcen n he dfferenal equaon are conan and don change wh repec o me. We can apply mpule repone & Laplace ranform n LTI yem Chbum Lee -Seoulech Conrol Syem

6 Tme-Varyng Model A me-varyng yem a yem ha no me nvaran oupu depend explcly upon me Eg. a pacecraf conrol yem. The ma of fuel conumpon change due o fuel conumpon Dynamc yem Lnear Nonlnear Chbum Lee -Seoulech Lnear Tme Invaran Our focu Lnear Tme Varyng Conrol Syem

7 Tranfer Funcon Example F ma my kyby F. e my by ky F my y = dplacemen from prng equlbrum Chbum Lee -Seoulech Conrol Syem

8 Tranfer Funcon Aumng zero nal condon, ake he Laplace Tranform of boh de m 2 Y by ky F 2 m b k Y F oupu Y F G 2 m b k Tranfer Funcon npu npu G oupu Chbum Lee -Seoulech Conrol Syem

9 Tranfer Funcon Tranfer Funcon: he rao of he Laplace ranform of he npu and oupu of a lnear me-nvaran yem wh zero nal condon and zero-pon equlbrum. Raonal funcon n he complex varable Le x : npu, y : oupu a n n y a n n y b m m a x b y m a 0 x y m b x b 0 x n > m Chbum Lee -Seoulech Conrol Syem

10 Conrol Syem Chbum Lee -Seoulech Tranfer Funcon n-h order yem nce he hghe power n he denomnaor n. Noe: lmed o me-nvaran, dfferenal equaon ndependen of he npu magnude. homogeney no nformaon on phycally rucure. MKS and RLC 0 0 IC zero ] [ ] [ TF: a a a a b b b b X Y npu L oupu L G n n n n m m m m

11 Conrol Syem Chbum Lee -Seoulech Syem Pole and Zero Roo of N=0 : he yem zero z, z 2,, z m Roo of D=0 : he yem pole p, p 2,, p n Noe Syem Pole and zero: real or eher complex conjugae par n n m m n n n n m m m m p p p p z z z z K D N a a a a b b b b G numeraor denomnaor

12 Syem Pole and Zero Ex. G K 3 2 Chbum Lee -Seoulech Conrol Syem

13 Oulne Convoluon Impule Repone Chbum Lee -Seoulech Conrol Syem

14 Conrol Syem Chbum Lee -Seoulech Convoluon Inegral If a ranfer funcon The oupu can be wren a Y = GU The nvere Laplace ranform gven by he convoluon negral G U Y oupu npu d u g d g u y for 0 where u g

15 Impule Repone Funcon Impule repone of a dynamc yem oupu when npu a un mpule Un mpule G Impule repone g Y G U when U y L [ G U ] g L[ ] Impule repone funcon Laplace ranform of he un mpule L[ ] Chbum Lee -Seoulech Conrol Syem

16 Impule Repone Funcon The ranfer funcon and mpule repone funcon of a LTI yem conan he ame nformaon abu he yem dynamc. Tranfer funcon n -doman Impule repone funcon n me doman Excng a yem wh an mpule npu and meaurng he repone can oban he dynamc characerc of he yem mpac hammer e Chbum Lee -Seoulech Conrol Syem

17 Impule Repone Funcon Example You can fnd he yem repone o an arbrary npu Example: y y y f Impule repone funcon HW Fnd repoe o he followng npu Chbum Lee -Seoulech Conrol Syem

18 Impule Repone Funcon Example A ample of 0.8 dcree funcon Approxmaon wh mpule Chbum Lee -Seoulech Conrol Syem

19 Conrol Syem Chbum Lee -Seoulech Impule Repone Funcon Example By uperpoon: 0, ummaon convoluon negraon. 2 2 u u g u g u g u g u g y n n n n n d u g y 0

20 Conrol Syem Chbum Lee -Seoulech Impule Repone Funcon Example Summaon of he repone Each mpule repone caled and delayed g u g u g u g u g u g y

21 Impule Repone Funcon Example Smaller me ep Summaon of he repone Chbum Lee -Seoulech Conrol Syem

22 Oulne Block dagram Chbum Lee -Seoulech Conrol Syem

23 Block Dagram Block dagram: a pcoral repreenaon of he funcon performed by each componen and of he flow of gnal U X G Y Chbum Lee -Seoulech Conrol Syem

24 Block Dagram Block n cacade R Y G G 2 R G 2 G Y Block n parallel G R + Y G 2 + R G +G 2 Y Chbum Lee -Seoulech Conrol Syem

25 Conrol Syem Chbum Lee -Seoulech Block Dagram Feedback yem Ex. G R H Y E + - H G G R Y R G Y H G Y H R G E G Y Y H R E T /S R 5 Y /S+6 + -

26 Block Dagram Chbum Lee -Seoulech Conrol Syem

27 Block Dagram Chbum Lee -Seoulech Conrol Syem

28 Block Dagram Example Chbum Lee -Seoulech Conrol Syem

29 Chbum Lee -Seoulech Conrol Syem

30 Oulne Mahemacal model of mechancal yem Mahemacal model of elecrcal yem Chbum Lee -Seoulech Conrol Syem

31 Mechancal Syem Ma-prng-damper yem F ma my kyby F. e my by ky F my y = dplacemen from prng equlbrum y F m b m y k m y Chbum Lee -Seoulech Conrol Syem

32 Mechancal Syem Block Dagram repreenaon F /m y / / b/m y y k/m Y F G 2 m b k Chbum Lee -Seoulech Conrol Syem

33 Elecrcal Syem RC Crcu R v v o C v v R vo C dv RC d o v C d v o R v C d Capacance C v C 2 v d v v 2 For ep V, RCV V v o o o V / RC RC / RC / / / RC e o / RC V v o RC RC: Tme conan 63.2% of fnal value Chbum Lee -Seoulech Conrol Syem

34 Elecrcal Syem RL crcu L v R v L R v d d L v R R v L L I I V R R I / R V L / R d d R Inducor L v v 2 d v v2 d L For ep V, L / RI I / RV / L I R / L / R/ / R / L / R e R / L / R L / R L/R: Tme conan 63.2% of fnal value Chbum Lee -Seoulech Conrol Syem

35 Conrol Syem Chbum Lee -Seoulech Elecrcal Syem LRC crcu o V d C V d C R d d L v o v 2 RC LC V V o V I C V I C RI LI o

36 Conrol Syem Chbum Lee -Seoulech Recall Impedance mehod Reor Capacor Inducor T-doman S-doman Impedance Z v R d C v d d v L v R v L v C RI V I CS V LI V R C L R L C v o v LC L R L RC LC C C L R V I I C L R V / / / / / 2 2 LC L R LC V I C V V CV I I C V o o o / / / 2 Elecrcal Syem