CONTROL SYSTEMS Chaper Mahemaical Modelling of Phyical Syem-Laplace Tranform Prof.Dr. Faih Mehme Boalı
Definiion Tranform -- a mahemaical converion from one way of hinking o anoher o make a problem eaier o olve problem in original way of hinking ranform oluion in ranform way of hinking invere ranform oluion in original way of hinking
The Logarihm Tranform Algebric operaion Logarihmic operaion Algebric operaion Muliplicaion Diviion Taking power Logarihmic operaion Addiion Subracion Muliplicaion
Laplace Tranform problem in ime domain Laplace Tranform oluion in Complex frequency domain Invere Laplace Tranform oluion in ime domain Oher ranform Fourier z-ranform wavele 4
Laplace Tranform Moon Earh 5
Laplace ranformaion ime domain linear differenial equaion Inegral or Differenial operaion ime domain oluion Laplace ranform invere Laplace ranform Laplace ranformed equaion Algebraic operaion Laplace oluion complex frequency domain 6
The Laplace Tranform Time domain Frequency domain Time domain operaion Differeniaion Inegraion Differenial / Inegral operaion Frequency domain operaion Muliplicaion by complex frequency Diviion by complex frequency Algebric operaion 7
Laplace Tranform L [ f ] F f e d 0 Conver ime-domain funcion and operaion ino frequency-domain : complex frequency j 8
Complex frequency : j Re Im an co jin 9
Rericion Laplace Tranform i applicable o only linear yem. A linear yem aifie he properie of uperpoiion and homogeneiy. If Fx+Fy=Fx+y Superpoiion propery and If Fx=A and Fkx=kFx Homogeneiy propery hen funcion F i a linear funcion. 0
Rericion Coninuiy : Laplace Tranform i applicable required ha f i a lea piecewie coninuou for 0
Rericion Boundedne: Laplace Tranform i applicable required ha f Me γ where M and γ are conan. Where >- f Me
Laplace Tranform of Common Funcion Name f F Impule Sep 0 h 0 0 Ramp r Exponenial f a e a Sine f in Coine f co
Laplace Tranformaion of ome funcion Pule funcion Impule funcion 4
Laplace Tranformaion of ome funcion Exponenial funcion Sep funcion Uni Sep funcion 5
Laplace Tranformaion of ome funcion Ramp funcion Sinuoidal funcion 6
Laplace ranform of ome funcion 7
Laplace ranform of ome funcion 8
Laplace ranform of ome funcion 9
Laplace Tranform Properie.Lineariy L f F F f.conan muliplicaion L a f a F.Complex hif L e a f F-a 4.Real ime hif L T f - T e F 5.Scaling L f/a a Fa 0
Laplace Tranform Properie 6.Differeniaion 7.Inegraion F d f L 0 ''0... '0 0 n n n n n n n f f f f F f d d L 0 f F f d d L n fold n F d f L
Laplace Tranform Properie 8.Iniial Value Theorem 9.Final Value Theorem 0.Convoluion. * 0 G F d τg τ f g f L L lim lim 0 F f lim lim 0 F f
Laplace Tranform e e f Take Laplace ranform of he following funcion Uing LT able and LT properie F
Laplace Tranform >> ym F=laplace+**exp-*+^*exp-* implifyf preyan F = / + ^ + / + ^ + / an = / + ^ + / + ^ + / -------- + -------- + - + + 4
Laplace Tranform >> ym f=+**exp-*+^*exp-*; F=laplacef,, implifyf preyan F = / + ^ + / + ^ + / an = / + ^ + / + ^ + / -------- + -------- + - ---- + + >> 5
Example of Soluion of an ODE d y d y0 dy 6 d y'0 8y 0 ODE w/iniial condiion Y 6Y 8Y / Y 4 Apply Laplace ranform o each erm Solve for Y Y y 4 4 4 Apply parial fracion expanion 4 e e Apply invere Laplace ranform o each 6 4 4 erm
Example of Soluion of an ODE '0 0; 0 8 6 y y y d dy d y d ODE w/iniial condiion Apply Laplace ranform o each erm Solve for Y Y Y Y Y Y Y Y y Y y y Y / 8 0 6 / 8 0 6 0 / 8 0 6 0 0 8 6 8 6 / Y 7
Example of Soluion of an ODE Apply parial fracion expanion Apply invere Laplace ranform o each erm 4 4 4 Y 4 4 4 e e y 4 4 4 8 6 Y Y 4 / ; 4; / 8
Invere Laplace Tranform L [ F ] f j j j F e d 9
Invere Tranform by parial fracion We generally do no ue invere Laplace inegral for aking invere ranform. Forunaely, we can ake invere Laplace ranform by uing parial fracion expanion of complex domain funcion. Parial fracion are everal fracion whoe um equal a given fracion Purpoe -- Working wih ranform require breaking complex fracion ino impler fracion o allow ue of able of ranform 0
Invere Tranform by parial fracion Conider ha F i a raio of polynomial expreion F N D The n roo of he denominaor, D are called he pole. Pole deermine abiliy and repone characeriic of he yem The m roo of he numeraor, N, are called he zero
Invere Tranform by parial fracion Three poible cae need proper raional, i.e., n>m. imple pole real and unequal. imple complex roo conjugae pair. repeaed roo of ame value
Cae : imple pole real and unequal Simple pole are placed in a parial fracion expanion F N D K0 z zm n p p pn p p pn The conan, i can be found by uing he mehod of reidue i p F i Invere Laplace Tranform of each erm i calculaed o a o find f p i L - p n f e e e p n p
Cae : imple pole real and unequal F p p Expand ino a erm for each facor in he denominaor. Solve he coefficien and. 4
Example imple pole F F e e F L f Take invere ranform of each erm 5
MATLAB Soluion imple pole >> ym F F=+/+*+ f=ilaplacef implifyf preyan F = + / + * + f = /exp* - /exp* an = -exp - /exp* exp - - ---------------- exp >> 6
7 Cae : complex-conjugae pole Complex funcion F i wrien in he form of parial fracion expanion The conan, i can be found by uing he mehod of reidue Invere Laplace ranform i deermined by uing he formula m m p p p p p D N F 4 4 i i p i p i i D N p F p p k m k a k e b b a b e f in co jb a p jb a p
Example complex-conjugae pole 8 Deermine Invere Laplace Tranform of he complex funcion F given below Complex funcion F i wrien in he form of parial fracion expanion b a j p j p F F
Example complex-conjugae pole 9.. 0 0 0 F F 0 0 /
Example complex-conjugae roo 40 0.5 0.5 F 0.5 in co 0.5 0.5 0.5in 0.5co 0.5 in 0.5 0.5co e f e f e f p k m k a k e b b a b e f in co
MATLAB Soluion complexconjugae pole >> ym F F=/*^+*+ f=ilaplacef implifyf preyan F = /*^ + * + f = / - co + in/*exp an = / - co + in/*exp co + in / - --------------- exp 4
Conider a complex funcion F having repeaed pole: Parial fracion expanion of F can be wrien a: 4 Cae : muliple repeaed pole n p N D N F... p p p p N F n n n n n
Coefficien can be calculaed by uing he formula 4 Cae : muliple repeaed pole.! p n k n k n k D N p d d k n
Example repeaed pole? - L F 44
Example repeaed pole 0!! 0 0 p p p D N D N d d D N d d 45
Example repeaed pole 0!! d d D N d d D N d d 46
Example repeaed pole!! d d d d D N d d D N d d 47
Example repeaed pole 48 F F L L L e f e e f f F L
MATLAB Soluion repeaed pole MATLAB Soluion >> ym F=^+*+/+^; f=ilaplacef f = /exp + ^/exp >> 49
Uing Malab for Laplace : Tranform Ue MATLAB o find he ranform of F e 4 MATLAB Code: >>ym, laplace*exp-4*,, an = /+4^ 50
Uing Malab for Laplace Tranform Ue Malab o find he invere ranform of F 6 6 8 >>ym >>ilaplace*+6/+*^+6*+8 an = -exp-*+*exp-**co* 5
Uing Malab for Laplace Tranform Ue Malab o find he invere ranform of F.5.5e.5e f=-.5+.5**exp-*+.5*exp-* 5
>> ym f=-.5+.5**exp-*+.5*exp-*; F=laplacef,, implifyf preyan F = 5/4* + + 7/* + ^ - 5/4* an = - 5/* + ^ - 5 ---------- + >> 5
Solving differenial equaion by LT 54
Solving differenial equaion by LT 55
Solving differenial equaion by LT 56