modern control systems 2-1 lecture 02 modeling (i) –transfer function 2.1 circuit systems 2.2...
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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 2-2
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
Modern Control Systems 2-3
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
Modern Control Systems 2-4
)( 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)
Modern Control Systems 2-5
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:
2.2 Mechanical Systems (Translational Motion)
Modern Control Systems 2-6
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
2.2 Mechanical Systems (Translational Motion)
Fig. 2.3
Modern Control Systems 2-7
Example 2.4:懸吊系統 (Suspension system)
2.2 Mechanical Systems (Translational Motion)
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
Modern Control Systems 2-8
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
Modern Control Systems 2-9
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
Modern Control Systems 2-10
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 )
2.2 Mechanical Systems (Rotational Motion)
Modern Control Systems 2-11
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
Form Newton’s Second Law
Under ZIC, take Laplace transform both sides
2.2 Mechanical Systems (Rotational Motion)
Fig. 2.5
Modern Control Systems 2-12
)]([)( 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
2.3 Transfer Function
ZIC)(
)()(
sR
sYsG
Modern Control Systems 2-13
) ()( 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:
2.3 Transfer Function
(2.1)
Conclusion: Under ZIC, for sinusoidal input, the steady state Output is also a sinusoidal wave.
Modern Control Systems 2-14
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!
2.3 Transfer Function
With reference to (2.1), we know
and
Find the output y(t) ?
Fig. 2.6
Modern Control Systems 2-15
)()(
)()(
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
2.3 Transfer Function
(Transfer Function from r to y)
Modern Control Systems 2-16
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
2.3 Transfer Function
From Kirchihoff Voltage Law, we obtain
(Transfer Function from )oi vv to
Fig. 2.7
Modern Control Systems 2-17
)(
)(1
)(
)()(
2 sq
sp
kbsMssR
sYsG
ymkyybr
rkydt
dyb
dt
ydm
2
2
)()()()(2 sRskYsbsYsYms
Form Newton’s Second Law
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
2.3 Transfer Function
Fig. 2.8
Modern Control Systems 2-18
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)
◆
◆
◆
◆
◆
2.3 Transfer Function
)(
)()(
sq
spsG
Transfer Function
Modern Control Systems 2-19
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:
2.3 Transfer Function
Example 2.8
Fig. 2.9
Modern Control Systems 2-20
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
2.3 Transfer Function
Time Constant of a first-order system
Consider
Modern Control Systems 2-21
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
2.3 Transfer Function
Modern Control Systems 2-22
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
2.3 Transfer Function