2. Recap
Recap
„
„ For the above system the closed loop transfer function is
For the above system the closed loop transfer function is
given by:
given by:
„
„ We can see that the closed loop poles location in s
We can see that the closed loop poles location in s-
-plane
plane
depends on the open loop system gain, poles (
depends on the open loop system gain, poles (i.e
i.e a),
a),
zeros (if exist), controller gains (
zeros (if exist), controller gains (kp
kp,
, ki
ki and
and kd
kd).
).
+
_
C(t)
R(t)
1
( )
s s a
+
p d
k k s
+
2
( )
d p
d p
k s k
s k a s k
+
+ + +
3. Why Root locus?
ƒ
ƒ Varying any one or all of these parameters
Varying any one or all of these parameters
changes the closed loop poles and hence the
changes the closed loop poles and hence the
impulse response and stability of the system
impulse response and stability of the system
changes.
changes.
ƒ
ƒ And it is laborious each time to find the closed
And it is laborious each time to find the closed
loop poles by varying these parameter values.
loop poles by varying these parameter values.
ƒ
ƒ Root locus is a graphical method which can
Root locus is a graphical method which can
easily find the closed loop poles by changing
easily find the closed loop poles by changing
any one of these parameters (if others kept
any one of these parameters (if others kept
constant).
constant).
4. Root locus
Root locus
„
„ This method may not give the exact
This method may not give the exact
results but a fast approximate result
results but a fast approximate result
can be achieved.
can be achieved.
„
„ Root locus is defined as the locus of
Root locus is defined as the locus of
the roots of the characteristic equation
the roots of the characteristic equation
of the closed loop system as a specific
of the closed loop system as a specific
parameter is varied from zero to
parameter is varied from zero to
infinity.
infinity.
5. Root locus approach
Root locus approach
„
„ Consider a system shown above.
Consider a system shown above.
„
„ The closed loop transfer function of this system
The closed loop transfer function of this system
is given by
is given by
„
„ So the characteristic equation is
So the characteristic equation is
KG(s)
H(s)
R(s) +
-
C(s)
C(s) KG(s)
=
R(s) 1+ KG(s)H(s)
1+ KG(s)H(s) = 0
6. Root locus approach
Root locus approach
contd..
contd..
„
„ Which is same as
Which is same as
„
„ Now this can be written as:
Now this can be written as:
„
„ Angle condition:
Angle condition:
(k=0,1..)
(k=0,1..)
„
„ Magnitude condition:
Magnitude condition:
„
„ A locus of all the points in the complex plane
A locus of all the points in the complex plane
satisfying the angle condition alone is called root
satisfying the angle condition alone is called root
locus.
locus.
„
„ The closed loop poles for a give parameter value
The closed loop poles for a give parameter value
are determined from the magnitude condition.
are determined from the magnitude condition.
KG(s)H(s) = -1
0
G(s)H(s) = ±180 (2k +1)
∠
| KG(s)H(s) |=1
7. Root locus approach
Root locus approach
contd..
contd..
„
„ Suppose
Suppose KG(s)H(s
KG(s)H(s) is of the form:
) is of the form:
„
„ Now to sketch the root locus with K as the parameter
Now to sketch the root locus with K as the parameter
first we must know the location of poles and zeros of
first we must know the location of poles and zeros of
G(s)H(s).
G(s)H(s).
„
„ Later we test a point in complex plane for the angle
Later we test a point in complex plane for the angle
condition. If condition is satisfied the test point lies
condition. If condition is satisfied the test point lies
on the root locus.
on the root locus.
1 2 m
1 2
K(s + z )(s + z ).....(s + z )
KG(s)H(s) =
( )( ).....( )
n
s p s p s p
+ + +
8. For a system with 4
For a system with 4
poles and 1 zero
poles and 1 zero
„
„ Assuming two of these
Assuming two of these
poles are complex
poles are complex
conjugates.
conjugates.
„
„ The angle of G(s)H(s) at
The angle of G(s)H(s) at
this test point is:
this test point is:
„
„ Is this value is equal to
Is this value is equal to
then the
then the
test point lies on the
test point lies on the
root locus.
root locus.
1
G(s)H(s) = 1 2 3 4
φ θ θ θ θ
∠ − − − −
0
±180 (2k +1)
9. Rules to plot Root locus
Rules to plot Root locus
„
„ It is difficult to test each and every point on
It is difficult to test each and every point on
the complex plane for angle condition.
the complex plane for angle condition.
„
„ There are 8 rules which states the procedure
There are 8 rules which states the procedure
to plot the root locus of any system.
to plot the root locus of any system.
„
„ To draw a root locus first we must locate the
To draw a root locus first we must locate the
open loop poles and zeros on the s
open loop poles and zeros on the s-
-plane.
plane.
„
„ One fact is that the root locus plots are
One fact is that the root locus plots are
symmetric about the real axis as the complex
symmetric about the real axis as the complex
roots occur in conjugate form.
roots occur in conjugate form.
10. 1. Locate the poles and zeros of on the S
1. Locate the poles and zeros of on the S-
-plane.
plane.
The root locus branches start at open loop poles and
The root locus branches start at open loop poles and
terminate at zeros.
terminate at zeros.
If no. of (poles
If no. of (poles-
-zeros) > 0, those (n
zeros) > 0, those (n-
-m) branches will
m) branches will
end at infinite.
end at infinite.
2. Determining the root loci on real axis. Taking any test
2. Determining the root loci on real axis. Taking any test
point, if the sum of no. of (poles+zeros) right to it, is
point, if the sum of no. of (poles+zeros) right to it, is
an odd number, then that point will be on Root
an odd number, then that point will be on Root-
-Loci.
Loci.
3. Determining the asymptotes of the Root
3. Determining the asymptotes of the Root-
-Loci. These
Loci. These
asymptotes show the way through which the (n
asymptotes show the way through which the (n-
-m)
m)
branches should end at infinite.
branches should end at infinite.
( ) ( )
G s H s
Set of Rules to plot Root-Locus
11. Rules contd
Rules contd………
………
4.
4. Finding the break
Finding the break-
-away and break
away and break-
-in points. These are
in points. These are
the points at which the root locus branch divides or
the points at which the root locus branch divides or
combines.
combines.
5. If there are any complex poles or zeros, we have to find
5. If there are any complex poles or zeros, we have to find
the angle of departure or arrival of root loci at that point.
the angle of departure or arrival of root loci at that point.
6. Finding the points where the root loci crosses the
6. Finding the points where the root loci crosses the
imaginary axis. This can be found from Routh
imaginary axis. This can be found from Routh’
’s stability
s stability
criterion.
criterion.
7. Taking a series of test points in the neighborhood of
7. Taking a series of test points in the neighborhood of
origin and
origin and ‘
‘jw
jw’
’ axis at the intersection points.
axis at the intersection points.
8. Determining the closed loop poles at desired
8. Determining the closed loop poles at desired ‘
‘k
k’
’ value
value
12. Rule1
Rule1
„
„ As K increases from 0 to infinity, each branch of
As K increases from 0 to infinity, each branch of
the root locus originates from an open
the root locus originates from an open-
-loop pole
loop pole
with K=0 and terminates either on open loop zero
with K=0 and terminates either on open loop zero
or on infinity with K=
or on infinity with K=
„
„ The number of branches terminating on infinity
The number of branches terminating on infinity
equals the number of open
equals the number of open-
-loop poles minus
loop poles minus
zeros.
zeros.
„
„ The proof of this statement is as follows:
The proof of this statement is as follows:
∞
13. Rule1 contd..
Rule1 contd..
„
„ The general characteristic equation can be rewritten as:
The general characteristic equation can be rewritten as:
„
„ When K=0, this equation has roots at
When K=0, this equation has roots at -
- which are the
which are the
open loop poles.
open loop poles.
„
„ The equation can also be written as:
The equation can also be written as:
„
„ When , this equation has roots at
When , this equation has roots at -
- which are
which are
open loop zeros of the system.
open loop zeros of the system.
j
p
1 1
1
( ) ( ) 0
n m
j i
j i
s p s z
K = =
+ + + =
∏ ∏
1 1
( ) ( ) 0
n m
j i
j i
s p K s z
= =
+ + + =
∏ ∏
i
z
K → ∞
14. Rule1 contd..
Rule1 contd..
„
„ Therefore m branches terminate at open
Therefore m branches terminate at open-
-loop zeros,
loop zeros,
the other (n
the other (n-
-m) branches terminate at infinity.
m) branches terminate at infinity.
„
„ Examining the magnitude condition
Examining the magnitude condition
„
„ We find that this is satisfied by as
We find that this is satisfied by as
„
„ Hence (n
Hence (n-
-m) branches terminate at infinity as
m) branches terminate at infinity as
s → ∞ K → ∞
1
1
| ( ) |
1
| ( ) |
m
i
i
n
j
j
s z
K
s p
=
=
+
=
+
∏
∏
K → ∞
15. Rule 2
Rule 2
„
„ A point on the real axis lies on the locus if the number
A point on the real axis lies on the locus if the number
of open
of open-
-loop poles plus zeros on the real axis to the
loop poles plus zeros on the real axis to the
right of this point is odd.
right of this point is odd.
„
„ This can be easily verified by checking the angle
This can be easily verified by checking the angle
criterion at that point.
criterion at that point.
„
„ Where are number of open loop zeros to the right of
Where are number of open loop zeros to the right of
the point and are the number of open
the point and are the number of open-
-loop poles to
loop poles to
the right of the point.
the right of the point.
0
r r
[G(s)H(s)] = (m - n )180
∠
r
m
r
n
0
(2 1)180 ; 0,1,2...
q q
= ± + =
16. Rule 3 (determining the
Rule 3 (determining the
asymptotes of root loci)
asymptotes of root loci)
„
„ The (n
The (n-
-m) branches of the root locus which tend
m) branches of the root locus which tend
to infinity, do so along straight line asymptotes
to infinity, do so along straight line asymptotes
whose angles are given by
whose angles are given by
„
„ Proof: Consider a point in s
Proof: Consider a point in s-
-plane which is far
plane which is far
away from the open
away from the open-
-loop poles and zeros, then
loop poles and zeros, then
the angle made by all the phasors at this point is
the angle made by all the phasors at this point is
almost same. Hence,
almost same. Hence,
0
(2 1)180
; 0,1,2...,( 1)
A
q
q n m
n m
φ
+
= = − −
−
[G(s)H(s)] = -(n - m)φ
∠
17. Rule 3 (determining the
Rule 3 (determining the
asymptotes of root loci) contd..
asymptotes of root loci) contd..
„
„ If this point has to lie on the root locus then it must
If this point has to lie on the root locus then it must
satisfy
satisfy
„
„ Hence
Hence
„
„ The asymptotes cross the real axis at a point known as
The asymptotes cross the real axis at a point known as
centroid
centroid, determined by the relationship: (sum of real
, determined by the relationship: (sum of real
parts of poles
parts of poles –
– sum of real parts of zeros)/(number of
sum of real parts of zeros)/(number of
poles
poles –
– number of zeros)
number of zeros)
0
-(n - m) (2 1)180
q
φ = ± +
0
(2 1)180
; 0,1,2...,( 1)
A
q
q n m
n m
φ
+
= = − −
−
18. Rule 3 (determining the
Rule 3 (determining the
asymptotes of root loci) contd..
asymptotes of root loci) contd..
„
„ The open loop transfer function can also be
The open loop transfer function can also be
written as
written as
„
„ This can be written as
This can be written as
„
„ The denominator is like a expansion of
The denominator is like a expansion of
which is of the form
which is of the form
n m
n-m n-m-1
j i
j=1 i=1
K
G(s)H(s) =
[s + ( - z )s +...]
p
∑ ∑
m
m
m m-1
i i
i=1 i=1
n
n
n n-1
j j
j=1 j=1
K[s + ( z )s + ...+ ( z )]
G(s)H(s) =
[s + ( )s + ...+ ( )]
p p
∑ ∏
∑ ∏
( )n m
A
s σ −
+
n-m 1
[s ( ) .....]
n m
A
n m s
σ − −
+ − +
19. Rule 3 (determining the
Rule 3 (determining the
asymptotes of root loci)
asymptotes of root loci)
contd..
contd..
„
„ Comparing both as we get
Comparing both as we get
„
„ Because all the complex poles and zeros (if
Because all the complex poles and zeros (if
exist) are conjugate pairs , is always a real
exist) are conjugate pairs , is always a real
quantity.
quantity.
„
„ Hence is (sum of real parts of poles
Hence is (sum of real parts of poles –
– sum
sum
of real parts of zeros)/(number of poles
of real parts of zeros)/(number of poles –
–
number of zeros)
number of zeros)
n m
j i
j=1 i=1
( ) - (-z )
n - m
A
p
σ
−
− =
∑ ∑
s → ∞
A
σ
−
A
σ
−
