Effect of Addition of Poles and Zeros on Control System - Desklib

This article discusses the effect of addition of poles and zeros on control system stability, peak overshoot, risetime, phase margin and bandwidth. It includes solved examples and MATLAB code to demonstrate the changes in system properties with addition of poles and zeros. The gain crossover frequency and phase margin are also discussed.

1. The control system is given by the following block diagram.
Here, G ( s ) = 10
s2 +2 s+10
G(s) is the open loop transfer function and M(s) is the closed loop transfer function
given by,
M(s) = G(s)H(s)/ (1 + G(s)H(s))
Now, with no addition of poles and zero H(s) = 1 is considered here.
So, M(s) = G(s)/(1+G(s)) ¿ 10
s2+2 s+20
Now, the step response and Bode plot of M(s) are shown below.
>> num =[10];
>> den =[1,2,20];
>> M = tf(num,den)
M =
10
--------------
s^2 + 2 s + 20
Continuous-time transfer function.

>> step(M)
>> title('step response of M(s)')
0 1 2 3 4 5 6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
step response of M(s)
Time (seconds)
Amplitude
>> bode(M); title('bode plot of M(s)')

-60
-40
-20
0
Magnitude (dB)
10-1 100 101 102
-180
-135
-90
-45
0
Phase (deg)
bode plot of M(s)
Frequency (rad/s)
>> num = [10];
>> den =[1,2,10];
>> G = tf(num,den)
G =
10
--------------
s^2 + 2 s + 10
Continuous-time transfer function.
>> bode(G); title('bode plot of G(s)')

-60
-40
-20
0
20
Magnitude (dB)
10-1 100 101 102
-180
-135
-90
-45
0
Phase (deg)
bode plot of G(s)
Frequency (rad/s)
2. Now, a pole can be added to the system of the form H(s) = 1/(ps+1).
Hence, M(s) = G(s)/(1+ G(s)H(s)) =
10
s2+2 s+10
1+
10
s2+ 2 s+10 ∗1
ps +1
= 10( ps+1)
( s2 +2 s+10 )∗( ps+1 )+10 = 10 ps+ 10
s3 p+ s2 ( 1+2 p )+10 ps+20
Now, the step response of the transfer function for adding poles at different points
p = [0.2624,0.6122,1.4286,3.3333] is obtained by the following MATLAB code.
MATLAB code:
p = [0.2624,0.6122,1.4286,3.3333];
num =10;
den =[1,2,20];
M = tf(num,den);

num1 = [10*p(1),10];
den1 = [p(1),(1+2*p(1)),10*p(1),20];
M1 = tf(num1,den1);
num2 = [10*p(2),10];
den2 = [p(2),(1+2*p(2)),10*p(2),20];
M2 = tf(num2,den2);
num3 = [10*p(3),10];
den3 = [p(3),(1+2*p(3)),10*p(3),20];
M3 = tf(num3,den3);
num4 = [10*p(4),10];
den4 = [p(4),(1+2*p(4)),10*p(4),20];
M4 = tf(num4,den4);
subplot(2,3,1)
step(M); title('step response of M')
subplot(2,3,2)
step(M1); title('step response of M1')
subplot(2,3,3)
step(M2); title('step response of M2')
subplot(2,3,4)
step(M3); title('step response of M3')
subplot(2,3,[5,6])
step(M4); title('step response of M4')
Plot:

0 5
0
0.2
0.4
0.6
0.8
step response of M
Time (seconds)
Amplitude
0 100
-5
0
5 10 12step response of M1
Time (seconds)
Amplitude
0 50
-0.5
0
0.5
1
1.5
step response of M2
Time (seconds)
Amplitude
0 5 10
0
0.5
1
1.5
step response of M3
Time (seconds)
Amplitude
0 2 4 6 8
0
0.5
1
1.5
step response of M4
Time (seconds)
Amplitude
3. Now, the zeros are added in the same location as poles. The step responses of the
system are observed.
Zeros are added to z = [0.2624, 0.6122, 1.4286, 3.3333].
Hence, the transfer function becomes M(s) = G(s)/(1+ G(s)H(s))
=
10
s2 +2 s +10
1+ 10
s2 +2 s +10∗( zs +1 )
= 10
s2+2 s+10+10 zs +10 = 10
s2+ 2 s+10 zs +2 0
MATLAB function:
z = [0.2624,0.6122,1.4286,3.3333];
num =10;
den =[1,2,20];
M = tf(num,den);

num1 = 10;
den1 = [1,(10*z(1)+2),20];
M1 = tf(num1,den1);
num2 = 10;
den2 = [1,(10*z(2)+2),20];
M2 = tf(num2,den2);
num3 = 10;
den3 = [1,(10*z(3)+2),20];
M3 = tf(num3,den3);
num4 = 10;
den4 = [1,(10*z(4)+2),20];
M4 = tf(num4,den4);
subplot(2,3,1)
step(M); title('step response of M')
subplot(2,3,2)
step(M1); title('step response of M1')
subplot(2,3,3)
step(M2); title('step response of M2')
subplot(2,3,4)
step(M3); title('step response of M3')
subplot(2,3,[5,6])
step(M4); title('step response of M4')
Plot:

0 5
0
0.2
0.4
0.6
0.8
step response of M
Time (seconds)
Amplitude
0 1 2 3
0
0.2
0.4
0.6
step response of M1
Time (seconds)
Amplitude
0 1
0
0.2
0.4
0.6
step response of M2
Time (seconds)
Amplitude
0 2 4
0
0.2
0.4
0.6
step response of M3
Time (seconds)
Amplitude
0 2 4 6 8 10 12
0
0.2
0.4
0.6
step response of M4
Time (seconds)
Amplitude
4. Now, addition of poles how affects the Peak Overshoot (POS), risetime (Tr), Phase
margin (PM) and Bandwidth (BW) of the system is observed and recorded in a table.
Now, the bandwidth of the system can be obtained by using the bandwidth function in
MATLAB.
MATLAB code for bandwidth:
p = [0.2624,0.6122,1.4286,3.3333];
num =10;
den =[1,2,20];
M = tf(num,den);
num1 = [10*p(1),10];
den1 = [p(1),(1+2*p(1)),10*p(1),20];
M1 = tf(num1,den1);
num2 = [10*p(2),10];

den2 = [p(2),(1+2*p(2)),10*p(2),20];
M2 = tf(num2,den2);
num3 = [10*p(3),10];
den3 = [p(3),(1+2*p(3)),10*p(3),20];
M3 = tf(num3,den3);
num4 = [10*p(4),10];
den4 = [p(4),(1+2*p(4)),10*p(4),20];
M4 = tf(num4,den4);
b = bandwidth(M);
b1 = bandwidth(M1);
b2 = bandwidth(M2);
b3 = bandwidth(M3);
b4 = bandwidth(M4);
b
b1
b2
b3
b4
Output:
b =
6.6987
b1 =
5.9561
b2 =
5.8879

b3 =
5.9148
b4 =
5.9412
Risetime (Tr) : Rise time is the time required for the response to reach from 10% to
90% of the steady-state value.
Peak Overshoot: The maximum value attained by the response.
Phase margin: The difference between the phase angle of the transfer function and
180° obtained at gain crossover frequency. The gain and phase margin can be directly
obtained from MATLAB.
[Gm,Pm] = margin(M)
Gm =
Inf
Pm =
73.1630
>> [Gm,Pm]=margin(M1)
Gm =
0.9064
Pm =
1.1538
>> [Gm,Pm]=margin(M2)
Gm =
Inf
Pm =
23.1292

>> [Gm,Pm]=margin(M3)
Gm =
Inf
Pm =
38.8916
>> [Gm,Pm]=margin(M4)
Gm =
Inf
Pm =
46.8169
Furthermore, the stepinfo function in MATLAB all the properties of the step response
of the system.
stepinfo(M)
ans =
struct with fields:
RiseTime: 0.2767
SettlingTime: 3.7804
SettlingMin: 0.3819
SettlingMax: 0.7426
Overshoot: 48.5150
Undershoot: 0

Peak: 0.7426
PeakTime: 0.7368
>> stepinfo(M2)
ans =
struct with fields:
RiseTime: 0.2448
SettlingTime: 47.0894
SettlingMin: -0.1796
SettlingMax: 1.2519
Overshoot: 150.3718
Undershoot: 35.9181
Peak: 1.2519
PeakTime: 0.9032
Items – effect of adding
pole
Results/Comments
POS From the graphs it is
observed that POS is
increased when a pole is
added to G(s).
Tr The risetime decreases.
PM Phase margin also

decreases.
BW Bandwidth also decreases.
5. Similarly, addition of zeros has the opposite effects on the properties of the system.
This can be seen from the graphs and employing MATLAB functions like before also.
Items – effect of adding
zero
Results/Comments
POS Decreases
Tr Increases
PM Increases
BW Increases
6. Now, from the above data and graphs it is seen than with the increase of Peak
overshoot (POS) the Phase Margin (PM) of the system decreases and system stability
is decreases.
7. The gain crossover frequency is the frequency at which the gain of the system is unity
and the frequency at which Phase margin is obtained. Now, as obtained from above
addition of a pole decreased the bandwidth of the system and decreased the phase
margin. The gain crossover (Wgc) along with Phase Margin(PM) can be obtained by
MATLAB as below for addition of poles.
>> [Gm,Pm,Wcg,Wcp] = margin(M)
Gm =
Inf
Pm =
73.1630

Wcg =
Inf
Wcp =
4.7850
>> [Gm,Pm,Wcg,Wcp] = margin(M1)
Gm =
0.9064
Pm =
1.1538
Wcg =
4.3661
Wcp =
4.4533
Hence, it is seen that with the decrease of gain crossover the phase margin has also
decreased.