Modern Control Systems 2-1 Lecture 02 Modeling (i) –Transfer function 2.1 Circuit Systems 2.2 Mechanical Systems 2.3 Transfer Function.

Modern Control Systems 2-1

Lecture 02 Modeling (i) –Transfer function

2.1 Circuit Systems

2.2 Mechanical Systems

2.3 Transfer Function


dt

dvCi 21

dt

diLv 21

model

iRv 21

21vL

i

21vC

i

21vR

Resistor

應用的定律．克希荷夫電流定律 (Kirchhoff Current Law)．克希荷夫電壓定律 (Kirchhoff Voltage Law)．歐姆定律 (Ohm’s Law)

Inductor Capacitor

2.1 Circuit Systems

i


R

vv oi idt

dvC o

ioo v

RCv

RCdt

dv 11

)()(1)()(

0

trdttvLdt

tdvC

R

tv too

ovdt

diL

dt

tdrtv

Ldt

tdv

Rdt

tvdC o

oo )()(

1)(1)(2

2

iv

i

R

C

ov

L)(tr R C

Example 2.1 ： RC Series Circuit

Example 2.2： By Kirchhoff Current Law (Node Analysis)

By Kirchhoff and Ohm’s Law

2.1 Circuit Systems

Fig. 2.1

Fig. 2.2


)( on translati:, 21 位移yy

force :F

tension:21yk

k

21 ,vv

ydt

dyv

b

b

1v2v

F

21bv

121221 vvyyv

21bvF

F

2y

21ky1221 yyy 1y

21kyF

modelk

Spring(彈簧 )

(張力 )

: spring constant ( 彈簧常數 )

: Velocity (速率 )

21vb : Viscous Friction Force(黏滯摩擦力 )

: viscous friction Constant (黏滯摩擦係數 )

Damper( 阻尼器 )

2.2 Mechanical Systems (Translational Motion)


vM

Mv

F

Mdt

dvM

Madt

dvMF

Mass(質量 )

: inertia force (慣性力 )

: Mass ( 質量 )

maF

a

2

2

dt

yd

dt

dva

F：所有外力之和

： acceleration (加速度 )

(外力的方向與位移相同為正 )

應用的定律

牛頓定律：



ymkyybr

rkydt

dyb

dt

ydm

2

2

)()()()(2 sRskYsbsYsYms

kbsmssR

sY

2

1

)(

)(

Form Newton’s Second Law

Under ZIC, take Laplace transform both sides

Example 2.3: Mass-Spring-Damper

(Transfer Function)

彈簧的彈性係數:k

外力:r

黏滯磨擦係數:b

)(tr)( ),( tyty

k

mb

Note: ZIC=Zero Initial Condition


Fig. 2.3


Example 2.4：懸吊系統 (Suspension system)


Mathematical Model:

M

Mg

)( 12 yyK )( 12 yyb

)( 21 yyb )( 21 yyK

m

2yKωxKω

Fig. 2.4

1y

2y

x

b K

K

caraofbody

m

M

12121 )()( yMyyKyybMg

221212 )()( ymyKyyKyybxK ωω

K Tire spring constant


21 ,1221

1 2 T

model

21KT

1 2 T

21bT

: angle 角度T： torque 轉矩

21 ,

1221

11

： angular velocity (角速度 )

2

2

dt

d

dt

d

JJT

T

： angular acceleration (角加速度 )

：對轉動軸的慣量

2.2 Mechanical Systems (Rotational Motion)

Spring(彈簧 )k

b

Damper( 阻尼器 )

Inertia(慣量 )

J


Gear train11 , rN

22 , rN

111 ,,

222 ,,

1

2

1

2

2

1

2

1

2

1

r

r

N

N

2

1

2

1

N

N

r

r(1)

2211 rr (2) 2211 (3)

2211 rr (4) 21 SS no energy loss

Nr

21 SS


JT

J

應用的定律

= 物體的加速度

= 轉動慣量

T =所有外加轉矩之和(與角位移方向相同者為正 )



b

J

k

T

Example2.5：Mass－ Spring－ Damper (Rotational System)

JkbT

Tkdt

db

dt

dJ

2

2

)()()()(2 sTsksbssJs

Function)(Transfer 1

)(

)(2 kbsJssT

s


Under ZIC, take Laplace transform both sides


Fig. 2.5


)]([)( trsR L

Operator Laplace:L

)]([)( tysY L

0)0()0()0()(

)(

yyy

sR

sY

)(tr)(sR

)0( , )0( , )0( yyy

)(ty

)(sYSystem

Definition

ZIC: Zero Initial Condition


ZIC)(

)()(

sR

sYsG


) ()( tsinRtr o

)()( tsinARty o

jssGjGjGjGA )()( ),( ,)(

Transfer Function: Gain that depends on the frequency of input signal

Under ZIC, the steady state output

where

is also called the DC gain.

When input

0)0( sGSpecial Case:


(2.1)

Conclusion: Under ZIC, for sinusoidal input, the steady state Output is also a sinusoidal wave.


1

1)(

ssG )()( tsintr

1oR 1

)45(2

2

2

1

2

1

1

1)1()(

j

jjGjG

445)1( ,

2

2)1( jGjGA

)4

sin(2

2)(

tty

2

1

2

1

mI

eR

Example 2.6

A=0.707 <1, Attenuation!


With reference to (2.1), we know

and

Find the output y(t) ?

Fig. 2.6


)()(

)()(

01012

2

2 trbdt

tdrbtya

dt

dya

dt

tyda

)()()()( 01012

2 sRbsbsYasasa

012

2

01)()(

)(

asasa

bsbsG

sR

sY

Set I.C. =0 and Take L.T. both sides

A Second-Order Example

Derivation of T.F. from Differential Equation


(Transfer Function from r to y)


L R

C

)(tvi )(ti

)(tvo

)()(

tidt

tdvC o

)()()()(

2

2

tvtvdt

tdvRC

dt

tvdLC io

oo

)()()(

)( tvtvdt

tdiLtRi io

LCs

LR

s

LC

sV

sV

i

o

1)1/(

)(

)(

2

Example 2.7: A second-order Circuit


From Kirchihoff Voltage Law, we obtain

(Transfer Function from )oi vv to

Fig. 2.7


)(

)(1

)(

)()(

2 sq

sp

kbsMssR

sYsG

ymkyybr

rkydt

dyb

dt

ydm

2

2

)()()()(2 sRskYsbsYsYms


Transfer Function

Take Laplace Transfrom both sides

)(tr)( ),( tyty

K

mb

彈簧的彈性係數:K

外力:r

黏滯磨擦係數:b

Example 2.3: Mass-Spring-Damper


Fig. 2.8


轉移函數的相關名詞

p(s)=分子多項式，q(s)= 分母多項式 (特性多項式 , characteristic polynomial)

q(s)=之階數稱為此系統之階數 (order)

q(s) 之根稱為系統之極點 (pole)

p(s)之根稱為系統之零點 (zero)

方程式 q(s)=0 稱為特性方程式 (characteristic equation)

◆

◆

◆

◆

◆


)(

)()(

sq

spsG

Transfer Function


rdt

dry

dt

dy

dt

yd 234

2

2

34

12

)(

)()(

2

ss

s

sR

sYsG

3 , 1 , 0342 sss

2

1 , 012 ss

0342 ss

3

Im

eR1

2

1極點零點

Under ZIC, take L.T., we get the transfer function

Poles ：

Zeros：

Char. Equation:


Example 2.8

Fig. 2.9


1)(

τs

b/a

as

bsG

a

1

a

bG )0(

: 稱為系統的時間常數 (Time Constant)

:稱為穩態直流增益


Time Constant of a first-order system

Consider


RCτsRCssV

sV

sVRCssV

tvdt

tdvRCtv

i

o

oi

oo

i

,1

1

1

1

)(

)(

)()1()(

)()(

)(

0,

0,0)(

tA

ttvi is called step-function. When A=1, it is called unit step function.

)1()1()()(

)11

()(

10

t

ato

o

eAeAsVtV

assA

as

a

s

AsV

L

iv

i

R

C

ov

Example 2.9

The output voltage

1

1

s

For RC=1

Fig. 2.10



A

1 2 3 4 5

AeAtvt

AeAtvt

AeAtvt

AeAtvt

AeAtvt

o

o

o

o

o

993.0)1()( 5

981.0)1()( 4

95.0)1()( 3

864.0)1()( 2

632.0)1()( ,

5

4

3

2

1

5st

gain-time constant form

6,10 ,16

10)(

kgain

ssG

pole-zero form

6

1 ,

61

610

)(

poles

sG

A0.632

)(0 tv

0

Time Constant: Measure of response time of a first-order system

Fig. 2.11


Modern Control Systems 2-1 Lecture 02 Modeling (i) –Transfer function 2.1 Circuit Systems 2.2 Mechanical Systems 2.3 Transfer Function.

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