Dynamic Behavior of Closed-Loop Control Systems
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Next, we develop a transfer function for each of the five elements in the feedback control loop. For the sake of simplicity, flow rate w1 is assumed to be constant, and the system is initially operating at the nominal steady rate.
ProcessIn section 4.3 the approximate dynamic model of a stirred-tank blending system was developed:
1 21 2 (11-1)
τ 1 τ 1K KX s X s W ss s
where
11 2
ρ 1, , and (11-2)wV xK Kw w w
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(11-5)sp me t x t x t
or after taking Laplace transforms,
(11-6)sp mE s X s X s
The symbol denotes the internal set-point composition expressed as an equivalent electrical current signal. This signal is used internally by the controller. is related to the actual composition set point by the composition sensor-transmitter gain Km:
spx t
spx t spx t
(11-7)sp m spx t K x t
Thus
(11-8)spm
sp
X sK
X s
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Current-to-Pressure (I/P) TransducerBecause transducers are usually designed to have linear characteristics and negligible (fast) dynamics, we assume that the transducer transfer function merely consists of a steady-state gain KIP:
(11-9)tIP
P sK
P s
Control ValveAs discussed in Section 9.2, control valves are usually designed so that the flow rate through the valve is a nearly linear function of the signal to the valve actuator. Therefore, a first-order transfer function usually provides an adequate model for operation of an installed valve in the vicinity of a nominal steady state. Thus, we assume that the control valve can be modeled as
2 (11-10)τ 1
v
t v
W s KP s s
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Composition Sensor-Transmitter (Analyzer)We assume that the dynamic behavior of the composition sensor-transmitter can be approximated by a first-order transfer function:
(11-3)τ 1
m m
m
X s KX s s
ControllerSuppose that an electronic proportional plus integral controller is used. From Chapter 8, the controller transfer function is
11 (11-4)τc
I
P sK
E s s
where and E(s) are the Laplace transforms of the controller output and the error signal e(t). Note that and e are electrical signals that have units of mA, while Kc is dimensionless. The error signal is expressed as
P s p t p
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Transfer Functions of Control Systems
1
Set-Point Change: ?
Load Change: ?
sp
X sX s
X sX s
1. Summer
2. Comparator
3. Block
• Blocks in Series
are equivalent to...
G(s)X(s)Y(s) Cha
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Closed-Loop Transfer Function for Set-Point Change (Servo
Problem)
2
0
1
p v p
v p c
v p c sp
v p c m sp m
m c v p
sp c v p m
D s
Y s X s G s U s G s G s P s
G s G s G s E s
G s G s G s Y s B s
G s G s G s K Y s G Y s
K G G GY sY s G G G G
Closed-Loop Transfer Function for Load Change (Regulator
Problem)
1 2
0
1
sp
d p
d v p c m
d v p c
m s
m
d
c v m
p
p
Y s
Y s X s X s
G s D s G s U s
G s D s G s G s G s GK Y Y s
G s D s G s G s G s G Y s
Y s GD s G G
s
G G
General Expression of Closed-Loop Transfer Function
1where
, : any two variables in block diagram: product of the transfer functions in the
path from to : product of the transfer functions in the
feedback loop.
f
e
f
e
YX
X Y
X Y
Servo Problem
open-loop transfer function
sp
f m c v p
e c v p m OL
X Y
Y YK G G G
G G G G G
Regulator Problem
open-loop transfer function
f d
e c v p m OL
X DY Y
G
G G G G G
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I O
Example2 1
42 1 2
1 1 2 3
1 2 3 1
1 1 2 3
1 2 3 1
1 1 2 1 2 3
2 1 2 1 2 1
4
4
2 3 1
2 1
2 1 2
2 1
2 1 2
1
1
1
11
1
c
c m
m c
sp c m
m c
c m
c
c m
c
c
m c c
c m c c m
m
GG
G GG G
G GO GI G G G
K G G GCY G
G
G G G
K G G G
G G G G
K G G G G GG G G G G G
GG G
G
G
G
G
G
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Closed-Loop Response with P Control (Set-Point Change)
1
1
1
1
111
1where
and open-loop gain = 01
1
pm c v
pspc v m
OLOL m c v p
OL
OL
KK K KH s Ks
KH s sK K Ks
KK K K K K KK
K
Closed-Loop Response with P Control (Set-Point Change)
1/1
1
1
1
offset1
offset and
sp sp
t
spOL
OL
Mh t M S t H ss
h t K M e
Mh h M K MK
K
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Closed-Loop Response with P Control (Load Change)
1
2
1 1
2
/1 1 2
2
1
11
11
where 1
1
offset 01
offset and
p
pc v m
p
OL
t
psp
OL
OL
KH s Ks
KQ s sK K K
sK
KK
Mq t M S t Q s h t K M es
K Mh h K M
K
K
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Closed-Loop Response with PI Control (Set-Point Change)
11
11
1 1
1
111
1
1
c cI
ppm vm v
p pspv mc m
ccI
cI
v
G
K
s Ks
KK K KK KH s ssK KH s
G s
K K K Ks
Ks s
G
Closed-Loop Response with PI Control (Load Change)
2 23
1
3
2 3 3
3 3 3
1 1
1 11 1
11
where
1
11
2 1
1, , 2
p p
p pv m v m
I
c v m
OLII
OL OL
OLI I I
c v m O
c
O
I
L
c
L
K KH s s s
K KQ sK K K K
s s
sK K K K s
Ks s
K K
KK
K
G
K K K K
Ks
s s
Closed-Loop Response with PI Control (Load Change)
3
3
1 1
32 23 3 3
233
233 3
1
2 1
1sin
1
Note that offset is eliminated!
t
q t S t Q ss
KH ss s
Kh t e t
Closed-Loop Response with PI Control (Load Change)
3 3
3 3
faster response , less oscillatory
faster response , more oscillatory
c
I
K
EXAMPLE 1: P.I. control of liquid level
Block Diagram:
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Assumptions
1. q1, varies with time; q2 is constant.
2. Constant density and x-sectional area of tank, A.
3. (for uncontrolled process)
4. The transmitter and control valve have negligible dynamics (compared with dynamics of tank).
5. Ideal PI controller is used (direct-acting).
)h(fq3
0KAs1)s(G
As1)s(G
K)s(G
K)s(Gs
11K)s(G
CL
P
VV
MMI
CC
For these assumptions, the transfer functions are:
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MPVC
L
GGGGG
QH
DY
11
MVI
C KAs
Ks
K
AsDY
1111
1
MPCIMVCI
I
KKKsKKKsAs
DY
2
02 MPCIMVCI KKKsKKKsA
1s2sK)s(G 22
The closed-loop transfer function is:
Substitute,
Simplify,
Characteristic Equation:
Recall the standard 2nd Order Transfer Function:
(11-68)
(2)
(3)
(4)
(5)
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For 0 < < 1 , closed-loop response is oscillatory. Thus decreased degree of oscillation by increasing Kc or I (for constantKv, KM, and A).
To place Eqn. (4) in the same form as the denominator of the T.F. in Eqn. (5), divide by Kc, KV, KM :
01ssKKK
AI
2
MVC
I
AKKK
21 IMVC
10
Comparing coefficients (5) and (6) gives:
Substitute,
22
KKKA
KKKA
II
MVC
I
MVC
I2
• unusual property of PI control of integrating system• better to use P only
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