1. The document describes the dynamic behavior of closed-loop control systems using transfer functions to model each element in a feedback control loop. 2. It develops transfer functions for a process, sensor, controller, valve and other elements to determine the overall closed-loop transfer function. 3. The closed-loop transfer function is used to analyze the response of the system to set-point changes or load disturbances, for both proportional and proportional-integral control.

1. Dynamic Behavior of Closed-
Loop Control Systems
Chapter
11
4-20 mA

2. Chapter
11

3. 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.
Process
In section 4.3 the approximate dynamic model of a stirred-tank
blending system was developed:
     
1 2
1 2 (11-1)
τ 1 τ 1
K K
X s X s W s
s s
   
  
 
   
 
   
where
1
1 2
ρ 1
, , and (11-2)
w
V x
K K
w w w


  
Chapter
11

4. Chapter
11

5.       (11-5)
sp m
e t x t x t
 
 
or after taking Laplace transforms,
      (11-6)
sp m
E 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:
 
sp
x t

 
sp
x t

 
sp
x t

    (11-7)
sp m sp
x t K x t
 

Thus
 
 
(11-8)
sp
m
sp
X s
K
X s



Chapter
11

6. Current-to-Pressure (I/P) Transducer
Because 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)
t
IP
P s
K
P s



Control Valve
As 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 K
P s s


 
Chapter
11

7. 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 K
X s s


 
Controller
Suppose that an electronic proportional plus integral controller is
used. From Chapter 8, the controller transfer function is
 
 
1
1 (11-4)
τ
c
I
P s
K
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
Chapter
11

8. Chapter
11

9. Transfer Functions of Control
Systems
 
 
 
 
1
Set-Point Change: ?
Load Change: ?
sp
X s
X s
X s
X s







10. 1. Summer
2. Comparator
3. Block
•Blocks in Series
are equivalent to...
G(s)X(s)
Y(s) 

Chapter
11

11. Chapter
11

12. 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 G
Y s
Y s G G G G

  

 
 
 
 
 
 




13. 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 G
K Y Y s
G s D s G s G s G s G Y s
Y s G
D s G G
s
G G

 
 
 
  
 
  
 
 




14. General Expression of Closed-
Loop Transfer Function
1
where
, :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
Y
X
X Y
X Y







15. Servo Problem
open-loop transfer function
sp
f m c v p
e c v p m OL
X Y
Y Y
K G G G
G G G G G





 


16. Regulator Problem
open-loop transfer function
f d
e c v p m OL
X D
Y Y
G
G G G G G





 


17. Chapter
11
I O

18. Example
2 1
4
2 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
1
1
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
G
G
G G
G G
G G
O
G
I G G G
K G G G
C
Y G
G
G G G
K G G G
G G G G
K G G G G G
G G G G G G
G
G G
G
G
G
G
G
 
 
 






 
 









19. Chapter
11

20. Chapter
11

21. Closed-Loop Response with
P Control (Set-Point Change)
 
 
1
1
1
1
1
1
1
1
where
and open-loop gain = 0
1
1
p
m c v
p
sp
c v m
OL
OL m c v p
OL
OL
K
K K K
H s K
s
K
H s s
K K K
s
K
K K K K K K
K
K




 
 
 
 


  

 


22. Closed-Loop Response with
P Control (Set-Point Change)
     
   
   
1
/
1
1
1
1
offset
1
offset and
sp sp
t
sp
OL
OL
M
h t M S t H s
s
h t K M e
M
h h M K M
K
K



 
   
  
 
      

    

23. Chapter
11

24. Chapter
11

25. Closed-Loop Response with
P Control (Load Change)
 
 
         
   
1
2
1 1
2
/
1 1 2
2
1
1
1
1
1
where
1
1
offset 0
1
offset and
p
p
c v m
p
OL
t
p
sp
OL
OL
K
H s K
s
K
Q s s
K K K
s
K
K
K
M
q t M S t Q s h t K M e
s
K M
h h K M
K
K






 
 
 




  
      
 
       

    

26. Chapter
11

27. Closed-Loop Response with
PI Control (Set-Point Change)
 
 
 
1
1
1
1
1 1
1
1
1
1
1
1
c c
I
p
p
m v
m v
p p
sp
v m
c m
c
c
I
c
I
v
G
K
s K
s
K
K K K
K K
H s s
s
K K
H s
G s
K K K K
s
K
s s
G





 
 

 
 
 

 
 
 
 
 
 
 

 

 
 

28. Closed-Loop Response with
PI Control (Load Change)
 
 
2 2
3
1
3
2 3 3
3 3 3
1 1
1 1
1 1
1
1
where
1
1
1
2 1
1
, ,
2
p p
p p
v m v m
I
c v m
OL
I
I
OL OL
OL
I I I
c v m O
c
O
I
L
c
L
K K
H s s s
K K
Q s
K K K K
s s
s
K K K K s
K
s s
K K
K
K
K
G
K K K K
K
s
s s
 
 



  
 
 



  
 

 
 
 

 

  
 

 
 
 

29. Closed-Loop Response with
PI Control (Load Change)
     
 
 
3
3
1 1
3
2 2
3 3 3
2
3
3
2
3
3 3
1
2 1
1
sin
1
Note that offset is eliminated!
t
q t S t Q s
s
K
H s
s s
K
h t e t


  


 

 
  
 
 
 

  

 
  

30. Closed-Loop Response with
PI Control (Load Change)
   
   
3 3
3 3
faster response , less oscillatory
faster response , more oscillatory
c
I
K
 

 

  

  

31. EXAMPLE 1: P.I. control of liquid level
Block Diagram:
Chapter
11

32. 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
(
f
q3 
0
K
As
1
)
s
(
G
As
1
)
s
(
G
K
)
s
(
G
K
)
s
(
G
s
1
1
K
)
s
(
G
C
L
P
V
V
M
M
I
C
C


















For these assumptions, the transfer functions are:
Chapter
11

33. M
P
V
C
L
G
G
G
G
G
Q
H
D
Y





1
1
M
V
I
C K
As
K
s
K
As
D
Y


















1
1
1
1
1

M
P
C
I
M
V
C
I
I
K
K
K
s
K
K
K
s
A
s
D
Y






2
0
2


 M
P
C
I
M
V
C
I K
K
K
s
K
K
K
s
A 

1
s
2
s
K
)
s
(
G 2
2





The closed-loop transfer function is:
Substitute,
Simplify,
Characteristic Equation:
Recall the standard 2nd Order Transfer Function:
(11-68)
(2)
(3)
(4)
(5)
Chapter
11

34. For 0 <  < 1 , closed-loop response is oscillatory. Thus
decreased degree of oscillation by increasing Kc or I (for constant
Kv, 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 :
0
1
s
s
K
K
K
A
I
2
M
V
C
I





A
K
K
K
2
1 I
M
V
C 


1
0 


Comparing coefficients (5) and (6) gives:
Substitute,















2
2
K
K
K
A
K
K
K
A
I
I
M
V
C
I
M
V
C
I
2
•unusual property of PI control of integrating system
•better to use P only
Chapter
11