root locus plot of dynamic systems
DESCRIPTION
Root Locus Plot of Dynamic Systems. [email protected]. We will cover. Root Locus of LTI models Root locus of a Transfer Function (TF) model Root Locus of 1 st order system Root Locus of 2 nd order system Root Locus of higher order systems - PowerPoint PPT PresentationTRANSCRIPT
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
KSG )(
)(SR )(SC
Consider the following unity feedback system
num=[1 0];
den=[1 1];
G=tf(num,den);
rlocus(G)
sgrid
Continued…..
1SKS)(SR )(SC
Root Locus of 1st Order System
Plot the root locus of following first order systems.
1SKS)(SR )(SC
1
)2(
S
SK)(SR )(SC
Exercise#1
Root Locus 2nd order systems
Consider the following unity feedback system
num=1;
den1=[1 0];
den2=[1 3];
den=conv(den1,den2);
G=tf(num,den);
rlocus(G)
sgrid
)3( SSK)(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
Consider the following unity feedback system
)(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
0 where ,01)(
)(0
1
13
15
2
1
2
1
2
1
DDx
xty
tux
x
x
x
Exercise#4: Plot the Root Locus for following LTI Models
)2()( )4)(3(
1)( SSHand
SSS
SSG
1)( )30)(1(
)10(3)(
SHandSSS
SSG
0 where ,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
SSG
num=[2 3];
den=[3 4 1];
G=tf(num,den);
[kd,poles]=rlocfind(G)
sgrid
Exercise#5
))()(()(
)(641
34
SSS
SSG
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
KSG
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
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