Zagazig University
Faculty of Engineering
Course: CSE 425 – Industrial Process Control
4th Year Computers & Systems Engineering
Sheet 6
Q1 The data below the process reaction curve for two processes.
Determine the tuning for a PID controller using the Ziegler-Nichols
method for each process.
(a)
1
(b)
Q2 In a unity feedback system, a PID controller with
transfer function


1
Gc ( s)  K p 1 
 Td s ,
 Ti s

is used to control a process with transfer function
.


1

G ( s )  
 (2s  1)( s  1) 
Using the given Ziegler Nichols tuning rules,
determine the values of Kp, Ti and Td.
2
Answer
The given process has two real poles (with no poles at s = 0), and hence
the process step response takes the following s-shaped reaction curve and
the ZN first method is applicable.
In order to use the table above, we need to calculate
 The process gain K
 The process time constant T
 The process time delay L
The process unit-step response Y(s) is given by
Y ( s) 
1
1
(2s  1)( s  1) s
Using partial fraction
1
4
1
Y (s)  

s 2s  1 s  1
y (t )  1  2 exp( 0.5t )  exp( t )
The step response has a steady state value of 1.0. Therefore, we need to
find the time at which the response becomes approximately 0.28 and 0.63
(28% and 63%, respectively).
y(t1 )  0.28  1  2e 0.5t1  e t1
y(t 2 )  0.63  1  2e 0.5t2  e t2
As given in the lecture, these two equations are quadratic equations in
exp(-0.5t1) and exp(-0.5t2). After some manipulations, we obtain
t28% = 1.5
t63% = 3.15
This gives
K=1
T = 1.5(3.15-1.51) = 2.46
L = t63% - T =3.15-2.46 = 0.69
3
From the ZN PID rules in the table:
Kp = 1.2*T/L=4.28
Ti = 2L = 1.38
Td = 0.5L=0.345
Q3 In a unity feedback system, a process transfer function is given by
G p ( s) 
( s  12)
s( s  1)( s  5)
Use Ziegler Nichols tuning rules to determine the PID controller
parameters Kp, Ti and Td, to control this process.
Hint: Kp = 0.6 Kcr,
Ti = 0.5 Pcr,
Td = 0.125 Pcr
4
Answer
The given process has a poles at s = 0 (which means an integrating
action), and hence the process step response will not take the s-shaped
reaction curve and thus the ZN first method is not applicable. Hence we
use the second method.
In the second method we put the process under feedback proportional
control with gain K and we increase the gain till sustained oscillation is
obtained. The value corresponding to this situation is called Kcr and the
observed oscillation period is called Pcr.
Using a proportional controller Gc(s) = K, the closed loop characteristic
equation will be given by
1  Gc ( s)G p ( s)  0
s( s  1)( s  5)  K ( s  12)  0
s 3  6s 2  (5  K ) s  12 K  0
Using Routh criterion:
s3
1
5 K
s2
6
12 K
s1
5 K
s0
12 K
For stability, K < 5. Hence, Kcr = 5.
When Kc = 5, Routh array becomes
s3
1
5 K
s2
6
60
s1
0
s0
60
The third row is zero. So, the auxiliary equation obtained from the second
row is
5
6 s 2  60  0
 by solving the auxiliary equation
s   j 10 

  3.16 rad/sec 
2
Tu

Tu  Pcr  2.
Using the given ZN 2nd method settings, the controller
parameters can be calculated as
Kp = 0.6 Kcr = 3.0
Ti = 0.5 Pcr = 1.0
Td = 0.125 Pcr = 0.25
6