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KEH Process Dynamics and Control 5-1
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KEH Process Dynamics and Control 5-1
KEH Process Dynamics and Control 5-2
V
F1
F2
V
F1 F2
KEH Process Dynamics and Control 5-3
KEH Process Dynamics and Control 5-4
1 2
d
d
VF F
t
1 2
d
d
VF F
t
1 2( ) ( ) ( )s V s F s F s 1 2
1 1( ) ( ) ( )V s F s F s
s s
1
( ) 1
( )
V s
F s s
2
( ) 1
( )
V s
F s s
u y
ΔV(t F1
KEH Process Dynamics and Control 5-5
Kut
y
d
du
t
yT
d
d
Tss
K
sU
sYsG
1
)(
)()(
K T
y(t) u(t)
KEH Process Dynamics and Control 5-6
Kuyt
yT
d
d
1
1
)(
)()(
TssU
sYsG
I,
u(t) = I δ(t)
δ(t)
U(s) = I
KEH Process Dynamics and Control 5-7
1)()()(
Ts
KIsUsGsY
t/T
T
KIty e)(
ustep ,
u(t) = ustepσ(t)
KEH Process Dynamics and Control 5-8
step( )
uU s
s
step( ) ( ) ( )
( 1)
KuY s G s U s
Ts s
/step( ) (1 e )t Ty t Ku
KEH Process Dynamics and Control 5-9
t = T
t = T
1/e = 0.368
2% t ≈ 4T
∞
K = y∞ / ustep
• ustep
• y∞ t → ∞.
KEH Process Dynamics and Control 5-10
y∞ = Kustep)
KEH Process Dynamics and Control 5-11
KEH Process Dynamics and Control 5-12
L
t
L
L + T
y∞
1
e
)(
)()(
Ts
K
sU
sYsG
Ls
)e1()( /step
TtKuLty
KEH Process Dynamics and Control 5-13
ti ,yi
L
L + T
KEH Process Dynamics and Control 5-14
•
•
KEH Process Dynamics and Control 5-15
KEH Process Dynamics and Control 5-16
t35 t85
T = 0,682(t85 - t35)
L = t35 – 0,431T
K
y∞ = Kustep
− t
-1/T
t = L.
•
KEH Process Dynamics and Control 5-17
T
Lt
y
tyy
)(ln
•t
•
KEH Process Dynamics and Control 5-18
•
•
KEH Process Dynamics and Control 5-19
b2 ≠ 0
b1 = 0,
b1 ≠ 0.
KEH Process Dynamics and Control 5-20
ubt
ubya
t
ya
t
y21212
2
d
d
d
d
d
d
21
2
21
)(
)()(
asas
bsb
sU
sYsG
K ζ ωn
Tn = 1/ ωn . Tn
0 ≤ ζ < 1 , ζ = 1
ζ > 1. ζ < 0
KEH Process Dynamics and Control 5–21
2
nn
2
2
n
2)(
ss
KsG
12)(
n
22
n
sTsT
KsG
•
•
•
• (0 ≤ ζ < 1)
• (ζ > 1)
• (ζ = 1)
KEH Process Dynamics and Control 5-22
Tn = 1/ωn .
Y(s) = G(s)U(s).
I , U(s) = I
ustep U(s) ustep / s
y(t) = Kustep (1 - (1 + t / Tn) e-t / Tn)
(ζ = 1) .
KEH Process Dynamics and Control 5-23
2
n )1()(
sT
KsG
n/
2
n
)(
Tte
T
tIKty
T1 T2 ωn ζ T1 T2
(ζ > 1) .
KEH Process Dynamics and Control 5-24
)1)(1()(
21
sTsT
KsG
1
, 1
n
2
2
n
2
1
TT
1 2n
1 2 1 2
1 ,
2
T T
T T T T
ee
)( 21 --
21
t / Tt / T
TT
IKty
ee1
1)( 21 -
2
-
1
21
step
t / Tt / TTT
TTKuty
y(t) = K I ωn β-1 e-ζ ωnt sin(β ωnt)
y(t) = Kustep (1 - β-1 e-ζ ωnt sin(β ωnt + ϕ)
sin(βωnt) cos(βωnt)
(ζ < 1) .
KEH Process Dynamics and Control 5-25
10 , 1 2
ζ.
KEH Process Dynamics and Control 5-26
K
T1
T2
T1 T2 .
T1 T2 ),
T2 = 0).
KEH Process Dynamics and Control 5-27
)1)(1(
e )(
21
sTsT
KsG
Ls
t = L + 1,25(T1 + T2).
L
z = T1/(T1 + T2) t = L + 0,5(T1 + T2).
• z = 1
• z = 0,5
• 0,5 < z < 1
KEH Process Dynamics and Control 5-28
z = 0,8
KEH Process Dynamics and Control 5-29
y
y∞
τ = (t – L)/(T1 + T2) .
τ = τ72 ≈ 1,25
τ =
τz ≈ 0,5 .
∑Ti τ72 ,
τz
z
∑Ti = T1 + T2 z =Ti / ∑Ti T1 T2
τ72
τz .
t
t72
∑Ti
∑Ti = 0,8(t72 – L)
L
L = 0
KEH Process Dynamics and Control 5-30
tz τz = 0,4τ72
KEH Process Dynamics and Control 5-31
tz = 0,4t72 + 0,6L
−
tz yz
− yz / y∞
y∞ yz
− z
yz / y∞
− yz / y∞ < 0,27
L
yz z
− yz / y∞ > 0,4 L
− L tz
T1 = z ∑Ti , T2 = ∑Ti - T1
T1 T2 t72 = 1,25 t72
z
KEH Process Dynamics and Control 5–32
t72
𝜏72 z
∑Ti
∑Ti = (t72 – L) / τ72
τ72 z
z
yz
KEH Process Dynamics and Control 5-33
)12)(14)(16(
1)(
ssssG
u U(s) = 1 /
T1 = 6 T2 = 4 T3 = 2
KEH Process Dynamics and Control 5-34
sssssUsGsY
)12)(14)(16(
1)()()(
ssTsTsTsF
)1)(1)(1(
1)(
321
332211 /
2313
2
/
3212
2
/
3121
2
e))((
e))((
e))((
1)(TtTtTt
TTTT
T
TTTT
T
TTTT
Ttf
2/4/6/ e2
1e4e
2
91)( tttty
K
ustep = 1 , y∞ = 1.
K = 1
KEH Process Dynamics and Control 5-35
G(s).
t ≈ 2,5 L ≈ 2,5
y∞ y = y63 = 0,63 y∞ = 0,63.
t ≈ 12,5 L + T ≈ 12,5.
K =1 T = 10 L = 2,5
U(s) = 1 / s Y1(s) = G1(s)U(s)
KEH Process Dynamics and Control 5-36
s
ssG 5,2
1 e110
1)(
10/
1 e1)5,2( tty
K =1,02 T = 9,05
L = 3,83.
KEH Process Dynamics and Control 5-37
G(s)
G1(s
G(s)
t72 ≈ 15.
L ≈ 1
L = 1,5 ∑Ti ≈ 10,8 .
tz
tz ≈ 6,9 .
y = yz ≈ 0,275 z ≈ 0,6 y∞ = 1.
τ72 ≈ 1,264
∑Ti ≈ 10,68.
T1 = z ∑Ti ≈ 6,41 T2 = ∑Ti - T1 ≈ 4,27.
KEH Process Dynamics and Control 5-38
K = 1,002 T1 = T2 = 5,35 L = 1,38.
KEH Process Dynamics and Control 5–39
s
sssG 5,1
2 e)127,4)(141,6(
1)(
27,4/41,6/
2 27,441,614,2
11)5,1( tt eety
KEH Process Dynamics and Control 5-40
G(s)
G2(s
G(s)
(0 ≤ ζ < 1)
•
•
KEH Process Dynamics and Control 5-41
y∞ >0
ymax
M
M = (ymax − y∞) / y∞
P
tr
y∞ .
y∞
KEH Process Dynamics and Control 5-42
tδ
(1 − δ)y∞ (1 + δ)y∞ (1 − δ)y∞ ≤ y(t) ≤ (1 + δ)y∞
δ = 0,05 = 5% δ = 0,02 = 2%
KEH Process Dynamics and Control 5-43
10 , 2
)(2
nn
2
2
n2
ss
KsG
21
/max e
y
yyM
n
2
P
n
)/arctan(
rt
Mt ,
)ln(
n
M P
ζ
ωn
K
•
•
n ymax,n n
ymin,n
n+k n
KEH Process Dynamics and Control 5-44
/2
R
min,
min,
max,
max,e kk
n
kn
n
knM
yy
yy
yy
yy
kM R
0
20 0 2,5
ζ
ζ = 0,2485
ωn = 0,998.
ζ = 0,25 ωn = 0,1.
KEH Process Dynamics and Control 5-45
5,625,375,9447,05,1
5,117,2max
Py
yyM
)(ln
)ln(
22 M
M
L
L
y(t) u(t)
y(t + L) = u(t)
L
G(s) = e-Ls
KEH Process Dynamics and Control 5–46
e-Ls
e-Ls ≈ 1 − Ls
KEH Process Dynamics and Control 5-47
...!3
)(
!2
)(1e
32
LsLsLsLs
L
L
KEH Process Dynamics and Control 5-48
...!3
)(
!2
)(1
1
e
1e
32
LsLsLs
Ls
Ls
Ls
Ls
1
1e
KEH Process Dynamics and Control 5-49
Ls
Ls
Ls
2
1
2
1
e
ee
LsLs
LsLs
2
11
21
2
11
2
11
e
K = 1
KEH Process Dynamics and Control 5-50
2
2
)(12
1
2
11
)(12
1
2
11
e
LsLs
LsLsLs
e−Ls = 1 / eLs
n
n = 1
KEH Process Dynamics and Control 5-51
n
n
x
n
x
1lime
n
Ls
Lsn
11
1e
KEH Process Dynamics and Control 5-52
KEH Process Dynamics and Control 5-53
13
12
11
15,2
15,1
13
12
11
15,2
15,1
13
12
11
15,2
15,1
2
1
0
sT
sT
sT
sT
sT
G
sT
sT
sT
sT
sT
G
sT
sT
sT
sT
sT
G
G = G1 + G2
T1 > T2 > 0 G
KEH Process Dynamics and Control 5-54
11 2
22
1
11
sT
KG
sT
KG
12
1
2
1 K
K
T
T
•
•
KEH Process Dynamics and Control 5–55
KEH Process Dynamics and Control 5–56
