Root locus of_dynamic_systems

1
Root Locus Plot of Dynamic Systems
imtiaz.hussain@faculty.muet.edu.pk

We will cover........
Root Locus of LTI models
Root locus of a Transfer Function (TF) model
Root Locus of 1st order system
Root Locus of 2nd order system
Root Locus of higher order systems
Root locus of a Zero-pole-gain model (ZPK)
Root locus of a State-Space model (SS)
Gain Adjustment

Root Locus of 1st Order System
Consider the following unity feedback system
Matlab Code
num=1;
den=[1 0];
G=tf(num,den);
rlocus(G)
sgrid
S
K
SG )(
)(SR )(SC



num=[1 0];
den=[1 1];
G=tf(num,den);
rlocus(G)
sgrid
Continued…..
1S
KS)(SR )(SC


Root Locus of 1st Order System

Plot the root locus of following first order systems.
1S
KS)(SR )(SC


1
)2(


S
SK)(SR )(SC


Exercise#1

Root Locus 2nd order systems
num=1;
den1=[1 0];
den2=[1 3];
den=conv(den1,den2);
G=tf(num,den);
rlocus(G)
sgrid
)3( SS
K)(SR )(SC


Determine the location of closed
loop poles that will modify the
damping ratio to 0.8 and natural
undapmed frequency to 1.7
r/sec. Also determine the gain K
at that point.

Exercise#2
)1(
)3(


SS
SK)(SR )(SC


)(SR )(SC


)1(
)3)(2(


SS
SSK
Plot the root locus of following 2nd order systems.
(1)
(2)

Root Locus of Higher Order Systems
)(SR )(SC


)2)(1(  SSS
K
Determine the closed loop gain
that would make the system
marginally stable.
num=1;
den1=[1 0];
den2=[1 1];
den3=[1 2];
den12=conv(den1,den2);
den=conv(den12,den3);
G=tf(num,den);
rlocus(G)
sgrid

Exercise#3
)(SR )(SC


)2)(1(
)3(


SSS
SK
)(SR )(SC


)2)(1(
)5)(3(


SSS
SSK
Plot the root locus of following systems.
(1)
(2)

Root Locus of a Zero-Pole-Gain Model
k=2;
z=-5;
p=[0 -1 -2];
G=zpk(z,p,k);
rlocus(G)
sgrid
)(SR )(SC


)2)(1(
)5(3


SSS
S

Root Locus of a State-Space Model
A=[-5 -1;3 -1];
B=[1;0];
C=[1 0];
D=0;
sys=ss(A,B,C,D);
rlocus(sys)
sgrid
  0where,01)(
)(
0
1
13
15
2
1
2
1
2
1

































DD
x
x
ty
tu
x
x
x
x



Exercise#4: Plot the Root Locus for following LTI Models
)2()(
)4)(3(
1
)( SSHand
SSS
S
SG 



1)(
)30)(1(
)10(3
)( 


 SHand
SSS
S
SG
  0where,100)(
)(
1
0
0
243
100
010
3
2
1
3
2
1
3
2
1























































DD
x
x
x
ty
tu
x
x
x
x
x
x



(1)
(2)
(3)

Choosing Desired Gain
143
32
)( 2



SS
S
SG
num=[2 3];
den=[3 4 1];
G=tf(num,den);
[kd,poles]=rlocfind(G)
sgrid

Exercise#5
))()((
)(
)(
641
34



SSS
S
SG
Plot the root Loci for the above ZPK model and find out the
location of closed loop poles for =0.505 and n=8.04 r/sec.
b=0.505;
wn=8.04;
sgrid(b, wn)
axis equal
(1)

Exercise#5: (contd…)
))((
)(
41 

SS
K
SG
i) Plot the root Loci for the above transfer function
ii) Find the gain when both the roots are equal
iii) Also find the roots at that point
iv) Determine the settling of the system when two roots are
equal.
Consider the following unity feedback system(2)

Exercise#5: (contd…)
i) Plot the root Loci for the above system
ii) Determine the gain K at which the the system produces
sustained oscillations with frequency 8 rad/sec.
Consider the following velocity feedback system
)(SR )(SC


))()(( 753  SSSS
K
S53 
(3)

End of tutorial
You can download this tutorial from:
http://imtiazhussainkalwar.weebly.com/

Root locus of_dynamic_systems

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