LAPLACE TRANSFORM AND TRANSFER FUNCTION

Transcription

1 CHBE320 LECTURE V LAPLACE TRANSFORM AND TRANSFER FUNCTION Professor Dae Ryook Yang Spring 2018 Dep. of Chemical and Biological Engineering 5-1 Road Map of he Lecure V Laplace Transform and Transfer funcions Definiion of Laplace ransform Properies of Laplace ransform Inverse Laplace ransform Definiion of ransfer funcion How o ge he ransfer funcions Properies of ransfer funcion + - Conroller Acuaor PROCESS Sensor Lecures IV o VII 5-2 1

2 SOLUTION OF LINEAR ODE 1 s -order linear ODE Inegraing facor: High-order linear ODE wih consan coeffs. Modes: roos of characerisic equaion Depending on he roos, modes are Disinc roos: Double roos: Imaginary roos: Many oher echniques for differen cases Soluion is a linear combinaion of modes and he coefficiens are decided by he iniial condiions. 5-3 LAPLACE TRANSFORM FOR LINEAR ODE AND PDE Laplace Transform No in ime domain, raher in frequency domain Derivaives and inegral become some operaors. ODE is convered ino algebraic equaion PDE is convered ino ODE in spaial coordinae Need inverse ransform o recover ime-domain soluion (D.E. calculaion) u() ODE or PDE y() U(s) Transfer Funcion Y(s) (Algebraic calculaion) 5-4 2

3 DEFINITION OF LAPLACE TRANSFORM Definiion F(s) is called Laplace ransform of f(). f() mus be piecewise coninuous. F(s) conains no informaion on f() for < 0. The pas informaion on f() (for < 0) is irrelevan. The s is a complex variable called Laplace ransform variable Inverse Laplace ransform and are linear. 5-5 LAPLACE TRANSFORM OF FUNCTIONS Consan funcion, a f() a Sep funcion, S() f() 1 Exponenial funcion, e -b f() 1 b<0 b>

4 Trigonomeric funcions Euler s Ideniy: sin() 1 Recangular pulse, P() f() h w 5-7 Impulse funcion, f() 1/ w w Ramp funcion, f() 1 1 Refer he Table 3.1 (Seborg e al.) for oher funcions 5-8 4

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6 PROPERTIES OF LAPLACE TRANSFORM Differeniaion 5-11 If f (0) = f (0) = f (0) = = f (n-1) (0) = 0, Iniial condiion effecs are vanished. I is very convenien o use deviaion variables so ha all he effecs of iniial condiion vanish. Transforms of linear differenial equaions

7 Inegraion 0 Time delay (Translaion in ime) f() Derivaive of Laplace ransform 5-13 Final value heorem From he LT of differeniaion, as s approaches o zero Limiaion: has o exis. If i diverges or oscillaes, his heorem is no valid. Iniial value heorem From he LT of differeniaion, as s approaches o infiniy

8 EXAMPLE ON LAPLACE TRANSFORM (1) f() Using he iniial and final value heorems Bu he final value heorem is no valid because 5-15 EXAMPLE ON LAPLACE TRANSFORM (2) Wha is he final value of he following sysem? Acually, canno be defined due o sin erm. Find he Laplace ransform for?

9 INVERSE LAPLACE TRANSFORM Used o recover he soluion in ime domain From he able By parial fracion expansion By inversion using conour inegral Parial fracion expansion Afer he parial fracion expansion, i requires o know some simple formula of inverse Laplace ransform such as 5-17 PARTIAL FRACTION EXPANSION Case I: All p i s are disinc and real By a roo-finding echnique, find all roos (ime-consuming) Find he coefficiens for each fracion Comparison of he coefficiens afer muliplying he denominaor Replace some values for s and solve linear algebraic equaion Use of Heaviside expansion Muliply boh side by a facor, (s+p i ), and replace s wih p i. Inverse LT:

10 Case II: Some roos are repeaed Each repeaed facors have o be separaed firs. Same mehods as Case I can be applied. Heaviside expansion for repeaed facors Inverse LT 5-19 Case III: Some roos are complex Each repeaed facors have o be separaed firs. Then, Inverse LT

11 EXAMPLES ON INVERSE LAPLACE TRANSFORM Muliply each facor and inser he zero value 5-21 Use of Heaviside expansion

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13 SOLVING ODE BY LAPLACE TRANSFORM Procedure 1. Given linear ODE wih iniial condiion, 2. Take Laplace ransform and solve for oupu 3. Inverse Laplace ransform Example: 5-25 TRANSFER FUNCTION (1) Definiion An algebraic expression for he dynamic relaion beween he inpu and oupu of he process model Transfer Funcion, G(s) How o find ransfer funcion 1. Find he equilibrium poin 2. If he sysem is nonlinear, hen linearize around equil. poin 3. Inroduce deviaion variables 4. Take Laplace ransform and solve for oupu 5. Do he Inverse Laplace ransform and recover he original variables from deviaion variables

14 TRANSFER FUNCTION (2) Benefis Once TF is known, he oupu response o various given inpus can be obained easily. Inerconneced sysem can be analyzed easily. By block diagram algebra X + - G1 G3 G2 Y Easy o analyze he qualiaive behavior of a process, such as sabiliy, speed of response, oscillaion, ec. By inspecing Poles and Zeros Poles: all s s saisfying D(s)=0 Zeros: all s s saisfying N(s)= TRANSFER FUNCTION (3) Seady-sae Gain: The raio beween ulimae changes in inpu and oupu For a uni sep change in inpu, he gain is he change in oupu Gain may no be definable: for example, inegraing processes and processes wih susaining oscillaion in oupu From he final value heorem, uni sep change in inpu wih zero iniial condiion gives The ransfer funcion iself is an impulse response of he sysem

15 EXAMPLE Horizonal cylindrical sorage ank (Ex4.7) q i L q R w i h Equilibrium poin: (if, can be any value in.) Linearizaion: 5-29 Le his erm be k Transfer funcion beween (inegraing) Transfer funcion beween (inegraing) If is near 0 or D, k becomes very large and is around D/2, k becomes minimum. The model could be quie differen depending on he operaing condiion used for he linearizaion. The bes suiable range for he linearizaion in his case is around D/2. (less change in gain) Linearized model would be valid in very narrow range near

16 PROPERTIES OF TRANSFER FUNCTION Addiive propery X 1 (s) X 2 (s) G 1 (s) G 2 (s) Y 1 (s) Y 2 (s) + + Y(s) Muliplicaive propery X 1 (s) X 2 (s) X 3 (s) G 1 (s) G 2 (s) Physical realizabiliy In a ransfer funcion, he order of numeraor(m) is greaer han ha of denominaor(n): called physically unrealizable The order of derivaive for he inpu is higher han ha of oupu. (requires fuure inpu values for curren oupu) 5-31 EXAMPLES ON TWO TANK SYSTEM Two anks in series (Ex3.7) No reacion C i C 1 C 2 V 1 V 2 q Iniial condiion: c 1 (0)= c 2 (0)=1 kg mol/m 3 (Use deviaion var.) Parameers: V 1 /q=2 min., V 2 /q=1.5 min. Transfer funcions

17 Pulse inpu 5 Equivalen impulse inpu 0.25 Pulse response vs. Impulse response