: 1
U sG s
C s
A
u t t 1u t u t t 212
u t t
1U s 1U ss 2
1U ss 3
1U ss
9 (2020/6/19-1) : 2
( )(t)
1t dt t 0 t 0,t01) t = 0 t 0 (t) = 0
2) 1( 1
1tL G(s)1 1y t G s g tL
1
: 3
( )
us t1 t 0
0 t 0t0
1
Inditial Responce
1su t
sL G(s)
1 1y t G s g ds
Lt
0
2: 4
ur tt t 0
0 t 0 t0
1
1
12
1ru t
sL
21 020 0
t tu t
t
13
1u ts
L t0
3
: 5
y t L 1 G s L 1 1s 1
e t
y t L 1 G s 1s
L 1 1s s 1
L 1 1s
1s 1
1 e t
G s 1s 1
: 6
y t L 1 G s 1s2 L 1 1
s2 s 1L 1 k1
s2
k2
sk3
s 1k1 s 1 k2s s 1 k3s
2
s2 s 1k2 k3 s
2 k1 k2 s k1
s2 s 1
L 1 1s2
1s
1s 1
t 1 e t
G s 1s 1
: 7
MATLAB/Simulink
m1) 2) 3)
Simulink ExcelG s 1
s 1
: 8
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
1.2
y t e t
t
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
1.2
y t 1 e t
t0 2 4 6 8 100
1
2
3
4
5
6
7
8
9
10
y t t 1 e t
t
G s 1s 1
: 9
U sG s
C s
u t t 1u t u t t 212
u t t
1U s 1U ss 2
1U ss 3
1U ss
c t g t c t g dt
0
1) tf d
1 F ss0
t
: 10
U sG s
C s
A
uy dux = Ax + b
= cx + ( ) (0) ( )tt e u dt AAx = e x + b
1 1( ) (0) ( )ss s UsX = I A X + I A b
11 1
dett ad s
e ss
jA II A
AI AL L
0
t
: 11
A
11 1
dett ad s
e ss
jA II A
AI AL L
11 1 0
11 1 0
nn
n nn n
b s b s ba sa s a s a s a
eAt
1 2
1
nn
pp
N s N ss s s s
: 12
11 1 0
11 1 0
nn
n nn n
b s b s ba sa s a s a s a
eAt
1 21 2
1 1 1 1 11 2
ir nn n n ri ll l rl
l l l ll l l i lr i
kk k ks s s s
0
1lim!
ii
n ln
i l ixi
dk s a sn l ds
111 2
1 1 !i i i
ri n n t
i ii i
ka t a s k k t t e
nL
A
: 13
11 1
dett ad s
e ss
jA II A
AI AL L
11
21) 22) 3) 2
31) 32) 2 13) 34) 2
1
11 0
1
1
1 0n
nn
nns a s
b s sabs
ba
A
: 14
d x t ax budt
1KG sTsy t cx
u(t) y(t)
1
1tTy t K e
T : time constantK : gain constant
a
1 bcT Ka a
(0)s a X x bUcbY Us a
1
1 111 1
y t G s U s
K K KTTs s s Ts
L
L L
(first order time lag element) [RC ]
: 15
9-2
MATLAB / SimulinkExcel
d x t x udt
11
G ss
y t x
3 1
15
d x t x udt
15 1
G ss
15
y t x
0.1d x t x udt
0.10.1
G ss
0.1y t x
: 16
9-2
11
G ss
15 1
G ss
0.10.1
G ss
111 1t
y t e
151 1t
y t e
1101 1t
y t e
: 17
9-2
11
G ss
15 1
G ss0.1
0.1G s
s
: 18
1
1tTy t K eG s K
Ts 1T : time constant K : (gain)
1)
2) 1
63.2%
T3) K
: 19
121 1t
y t e
12 1
G ss
: 20
11
G ss
T (t/T)T
t = 0
11 0.632K e K
2) 1
63.2%
T
1)
3) K
: 21
0 10 20 30 40 500
0.2
0.4
0.6
0.8
1
1.2
T=1
T=5
T=10
y(t)
t
K=1 T
T1 K
1
1tTy t K eG s K
Ts 1
: 22
2
1 0 02
d dx t a x t a x t b u tdt dt
0 1 0
0 1 0x xd u ta a bx xdt
02
1 0
bG ss a s a
21 0 1 00 0 0s a s a X s + sx x a x b U s
11 0 1 0
20 1 1 0
1 01 0
1s
s a s a bG s s
a s a s a s ac I A b c b
1 0y t x
(second order time lag element)[RLC , ]
u(t) y(t)
: 23
: damping factor
n : ( ) (natural angular frequency)
G s K n2
s2 2 ns n2
2 21 2
2
,
1
n n n
n
p p
z
Re
Im
n
21n
21n
n
1sin
: 24
G s K n2
s2 2 ns n2
02
1 0
bG ss a s a0 1 0
0 1 0x xd u ta a bx xdt
1 0y t x2
0 0n na a
2
0 1 02n n n
x xd u tKx xdt
1 0y t x
21 1 0
1,2
42
a a a
11
0
22naaa
00
0n
bb K Ka
2 21,2
2 1
n n n
n
: 25
: damping factor
n : ( ) (natural angular frequency)
G s K n2
s2 2 ns n2
1 0 1 2
1 2
k k ks s p s p
L
21
2 2
12
n
n n
Ky ts s s
L
1 20 1 2
p t p tk k e k e+
2 21 2
2
,
1
n n n
n
p p
z
1
2
1 11 1 2
1 n
s p
Kk s p G ss p p p
2
01 20
1 n
s
Kk sG ss p p
2
2
2 22 2 1
1 n
s p
Kk s p G ss p p p
Re
Im
n
21n
21n
n
1sin
: 26
2
2 22n
n n
KG ss s
2 1ns
1) 1 2
1 1 n tny t K t e
22 1
2
111 sin 1 tan1
n tny t K e t
22 1
2
111 sinh 1 tan1
n tny t K e t
0 1
12)
3)
2 21 12 2
2
11 1 12 1
n nt ty t K e e
: 27
G s K n2
s2 2 ns n2
2 21 2
2
,
1
n n n
n
p p
1 2
2 2 2
1 2 1 1 2 2 2 1
p t p tn n nK K Ky t e ep p p p p p p p
+ +
2 21 1
2 2 2 2 2 21 1 2 1 1 2 1
n nt tK K Ke e+ +
2 22 2
1 1
2 2
1 1
2 1 2 1n nt tK K
K e e+ +
2 22 21 1
2 2
1 11
2 1 2 1n nt t
K e e+
(2)2
: 28
2
2 22n
n n
KG ss s
2 21 2
2
,
1
n n n
n
p p
1 1 n tnK t e
1
21
2 2
12
n
n n
Ky ts s s
L
12
n
n n
KK Ks s s
L
(3)
: 29
G s K n2
s2 2 ns n2
2 21 2
2
,
1
n n n
n
p p
1 2
2 2 2
1 2 1 1 2 2 2 1
p t p tn n nK K Ky t e ep p p p p p p p
+ + 0 12
1 2, 1np p j2
2 31 122 2
1
1
n n
n n
k s kkKs s
L1 21 1k k
21 2 32 1 0n n nk k k
3 21k
2
12 22 2 22 2
1111 1
nn
n n n n
sK
s s sL
(4)2
: 30
y t2
12 22 2 22 2
1111 1
nn
n n n n
sK
s s sL
2 2
21 cos 1 sin 1
1n nt t
n nK e t e t
1
21
2
2
11 cos 11
n tnK e t
2
11
1
2tan
1
2
2
11 sin 11
n tnK e t
21 1tan
(4)2
G s K n2
s2 2 ns n2
: 31
2
2 22n
n n
KG ss s
2 1ns
1) 1 2
1 1 n tny t K t e
22 1
2
111 sin 1 tan1
n tny t K e t
22 1
2
111 sinh 1 tan1
n tny t K e t
0 1
12)
3)
2 21 12 2
2
11 1 12 1
n nt ty t K e e
: 32
2
10.2 1
G ss s
2
12 1
G ss s
2
120 1
G ss s
9-3
MATLAB / SimulinkExcel
: 33
9-3
2
10.2 1
G ss s
2
12 1
G ss s
2
120 1
G ss s
: 34
0 5 10 15 200
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Ä=0.1Ä=1Ä=2
K=1, n=1
0 1 …
1 …
…1
n t
: 35
0 5 10 15 200
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
1) 1 2
22 1
2
111 sinh 1 tan1
n tny t K e t
n
0
2 21 12 2
2
11 1 12 1
n nt ty t K e e
2 1
: 36
Re
Im
n
21n
21n
n
1sin
0 13)
0 5 10 15 200
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
n
n
22 1
2
111 sin 1 tan1
n tny t K e t
: 37
2
10.2 1
G ss s
2
0.250.1 0.25
G ss s
2
0.06250.05 0.0625
G ss s
9-4
MATLAB / SimulinkExcel
: 38
9-4
2
10.2 1
G ss s
2
0.250.1 0.25
G ss s
2
0.06250.05 0.0625
G ss s
: 39
0 2 4 6 8 10 120
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Tr
Td
Ts
Amax
Tmax
2%
1) (rise time): Tr10%
90%2) (delay time): Td
50%
3) (maximum overshoot): Amax
%
4) (peak time): Tmax
5) (settling time): Ts2%
5%
Transient Characteristics : 40
G s K n2
s2 2 ns n2
2 1ns
0 1
21maxA Ke
21, 2, ,
1l
n
lt l n
211 1 1, 2, ,l
ll ly t K e l n
22 1
2
111 sin 1 tan1
n tny t K e t
Re
Im
n
21n
21n
n
0d y tdt
al al+2 1 ( )21 11
ll
la e2
2
12l
l
a ea
0.02la c 1 ln 0.02sn
T c4
n
5%3
n
1sin
l =1 Tmax
