Post on 14-Jan-2016
description
The Root Locus Method
Introduction
The performance of a feedback system can be described in terms of the location of the roots of the characteristic equation in the s-plane.
It is very useful to determine how the roots of the characteristic equation move around the s-plane as we change one parameter.
The root locus method was introduced by Walter R. Evans in 1948.
Introduction
Root-locus is a graphical technique for sketching
the locus of roots in the s-plane as a parameter is
varied in a given range.
Root-locus is based on the poles and zeros of
the open-loop transfer function for determining the
poles of the closed-loop transfer function of a
system.
Root Locus Concept
The characteristic equation
022 Kss
roots Ks 112,1
em.f the systfunction o transfer open-looples of theare the possK 22
01
0
. 1 1 21 otsme real roare the sassK
jsK
jsK
1
11 2
2,1
2,1
K)2(
1
ss
The open-loop transfer function:
)2()(
ss
ksG
0)2(
1)(1
ss
ksG
. 10 2 1 al rootsfferent reare two disandsK
reasing.with k incoots grow n of the rary sectioThe imagin
t.is constanthe roots ection of The real s
x roots.ate comple of conjugare a pairsands 21
Root Locus Concept
According to above discussion: we can sketch the locus of the roots varying with k from 0 to + on the S-plane.
Definition of Root locus:
The root loci is the path of the roots of the
characteristic equation (or the poles of the
systems closed-loop transfer function), traced out in the s-plane as a system parameter is
changed.
Root Locus Concept
Root - root of s polynomial equation(the characteristic equation)
Locus - Set of points (roots)
system.order -second theof 1 case with therespected
becan This poles. real thebeing poles theof because dampedlly exponentia is
response system theofportion transientthe
,10 range in the changes When (2)
k
What information can be got from
the root-loci of the system?
(1) when k changes in the range 0 ~ +, the system is always stable, because the loci of the closed-loop
poles are always in the left-half of
the s-plane.
Root Locus Concept
system.order -
second theof 10 case with therespected
becan This poles.complex conjugate the
being poles theof because sinusoid damped
lly exponentia a is response system theof
portion transient the,1 When (3)
k
Root Locus Concept
constant. being poles theofsection real theof because not varied be willtime
setting theAnd axis. real thefromaway poles theof beause augmented being
with increased be willresponse system theofovershoot the1 when (4)
k
k
... etc.
).0( 0 pole loop-open a of because system typea is system The (5) ks
system. a of eperformanc theanalyze tolocus-root use why weisIt system.
a of loci-root thefromn informatio oflot aget can that weobvious isIt
Root Locus Concept
We usually interested in determining the locus of the roots
as K* varies as K0
)(sG
)(sH
n
j
j
m
i
i
ps
zs
KsHsG
1
1
)(
)(
)()(
The characteristic equation is
0)()(1 sHsG
rearrange the equation, if necessary, so that the parameter of
interest K, appears as the multiplying factor in the form, and
write the polynomial in the form of poles and zeros as follows:
Two Basic Criterion
K*: The root-locus gain, which is proportional to
open-loop gain K. 121
121
22
22
sTsTsTs
sssKsHsG
kkkiv
lllj
(For reasons that will become clear later, this is the definition of the positive or 180
degree locus. Will later define the negative, or 0 degree locus.)
)(
)(
)()(
1
1 eex variabls: a compl
ps
zs
KsHsGn
j
j
m
i
i
criterionMagnitude
ps
zs
Kn
j
j
m
i
i
1
||
||
1
1
criterionAngle kkpszsn
j
j
m
i
i ,2 ,1 ,0 )12()()(11
Two Basic Criterion
0)()(1 sHsG 01)()( jsHsG
the magnitude and angle requirement
for the root locus are:
Two Basic Criterion
1. The number of separate loci is equal to the order of the characteristic equation. Number of loci branches = n
2. The loci are symmetrical about the real axis. The root locus is symmetrical about the real axis since
the roots of 1+G(s)H(s)=0 must either be real or appear
as complex conjugates.
Root Loci Construction Rules
Basic approach: The complete root loci can be constructed
point-by-point finding all points in the s-plane that satisfy the
angle criterion.(usually dont use this approach directly.)
The rapid sketching procedure of the root locus
shown as follows:
0)(
)(
1
1
1
n
j
j
m
i
i
ps
zs
K 0)()(11
m
i
i
n
j
j zsKps
0K njps j ,2,1 , 0)(1
n
j
jps
3. The root loci begin at the open loop poles and end at open zeros. (1) The root loci begin at the open loop poles.
Root Loci Construction Rules
(2) The root loci end at the open loop zeros.
0
)(
)(
1
1
1
n
j
j
m
i
i
ps
zs
K
Kps
zs
n
j
j
m
i
i1
)(
)(
1
1
mizs i ,2,1 , 0)(1
m
i
izsK
Example Second-order system
)4(
)2(
)125.0(
)15.0()()(
*
ss
sK
ss
sKsHsG
zero
poles
-4
-2
0
j
Root Loci Construction Rules
. ,
:
zeropole
symbolsthe withhe s-planezeros in tpoles and open-loop locate the
system physical actual
:zeros loop-open and poles loop-open of loci-root For the
mn
mn
4. Root Loci on the Real Axis
Root Loci Construction Rules
The locus covers the section of the real axis to the left of an odd number of poles and zeros of G(s)H(s).
)4(
)2(
)125.0(
)15.0()()(
*
ss
sK
ss
sKsHsG
Example Second-order system
zero
poles
-4
-2
0
j
s
criterionAngle
kkpszsn
j
j
m
i
i
,2 ,1 ,0 )12()()(11
5. The loci proceed to the zeros at infinity along asymptotes.
These linear asymptotes are centered at a point on the real
axis given by
mn
zpm
i
i
n
j
j
a
11
The angle of the asymptotes with respect to the real axis is
)1,,1,0( )12(
mnk
mn
ka
Root Loci Construction Rules
j
0)3)(1(
)2(1)(1
sss
sKsG
Example: Third-order system
A single-loop feedback control
system has a characteristic
equation follows: -1 -2 -3 0
113
)2()310(
a
)1( 270
)0( 90180
13
120
0
0
k
kka
Root Loci Construction Rules
Sequential Example: A single-loop feedback control system
has a characteristic equation as follows:
0)2)(1(
1)()(1
sss
KsHsG
The intersection of the asymptotes is
13
210
a
The angles of the asymptotes are
)2( 300
)1( 180
)0( 60
1803
)12(
0
0
0
0
k
k
kk
a
Root Loci Construction Rules
-2 -1
j
6. Breakaway points on the root loci.
Breakaway points on the root loci correspond to multiple-
order roots of the equation 1+G(s)H(s)=0. The breakaway
point d can be computed by solving
It is important to point out that the condition for the breakaway
point given in Eq.(above) is necessary but not sufficient. In
other words, all breakaway points on the root loci must satisfy
Eq.(above), but not all solutions of Eq. are breakaway points.
m
i
n
j ji pdzd1 1
11
Root Loci Construction Rules
Necessary conditions only
7. The tangents to the loci at the breakaway points.
In general, due to the phase criterion, the tangents to the
loci at the breakaway point are equally spaced over 3600
Breakaway
point
090
045
045
Root Loci Construction Rules
The two loci at the breakaway
point are spaced 180o apart. The four loci at the breakaway
point are spaced 90o apart.
0)2)(1(
1)()(1
sss
KsHsG
02
1
1
11
ddd
58.1 ,42.0 21 dd
Root Loci Construction Rules
Sequential Example: A single-loop feedback control system
has a characteristic equation as follows:
-2
j
-1
m
i
n
j ji pdzd1 1
11
here no zeros
8. Intersection(the crossing points)of the root loci with the imaginary axis and corresponding values of K*.
0)2)(1(
1)()(1
sss
KsHsG
023 23 Ksss
023 23 Kjj
portion real 03
portionimaginary 022
3
K
6 2
0 0
K
K
The characteristic equation is
Root Loci Construction Rules
To substitute into the characteristic
equation then to solve the value for
js
Sequential Example:
With a little practice, you should be able to sketch root loci very rapidly.
-2
j
-1
Root Loci Construction Rules
Sequential Example: 0)2)(1(
1)()(1
sss
KsHsG
The intersection of the asymptotes is
13
210
a
The angles of the asymptotes are
)2( 300
)1( 180
)0( 60
1803
)12(
0
0
0
0
k
k
kk
a
6 2
0 0
K
K
58.1 ,42.0 21 dd
Breakaway points on the root loci
Intersection of the root loci with the
imaginary axis.
kp
s
)( kps )(lim kps
p psk
k
9. The angles of arrival and departure.
m
i
n
kjj
jkikp ppzpKk1 1
)()()12(
Root Loci Construction Rules
Departure angle
kz
s
)( kzs )(lim kzs
z zsk
k
9. The angles of arrival and departure.
n
j
jk
m
kii
ikz pzzzkk11
)()()12(
Root Loci Construction Rules
Arrival angle
Example: ))(1(
)()(jisjss
KsHsG
jpjpp 1 1 0 321
1
-1
-1 0
j 2p
3p
1p
)()()12( 32122 ppppkp
000 90135180)12(2
kp
0452
p
0453p
Root Loci Construction Rules
m
i
n
kjj
jkikp ppzpKk1 1
)()()12(
Example: )5.15.0)(5.2(
)2)(5.1()(
*
jSss
jssKsG
Root Loci Construction Rules
10. The sum of closed loop poles
0
)(
)(
1
1
1
n
j
j
m
i
i
ps
zs
K
0)()(11
m
i
i
n
j
j zsKps
0)(1
n
i
iss
012
2
1
1
nn
nnn asasasas
constpsan
j
j
n
i
i
11
1
If the order of the denominator of the open loop transfer
function is greater than the numerator by at least 2 ,
then the sum of the closed loop poles is a constant.
Root Loci Construction Rules
)2( mn
constpsn
j
j
n
i
i
11
. 0)((s)1 ofroot theis pole, loop-open theis where sHGsp ij
)2( mn
Root Loci Construction Rules
Example:
Root Loci Construction Rules
In terms of above rules we can rapidly sketch the
root-loci of a control system.
11. Determine the parameter value Kx at any point Sx on the root loci using the magnitude requirement.
xss
zs
ps
K
ps
zs
KSm
i
i
n
j
j
n
j
j
m
i
i
x
1
1
1
1*
||
||
1
||
||
ocriterionMagnitude
ps
zs
Kn
j
j
m
i
i
1
||
||
1
1
xss
psK
ps
KSn
j
jn
j
j
x
1
1
* || 1
||
1 ocriterionMagnitude
ps
Kn
j
j
1
||
1
1
If there is no zeros
Sketch the root-loci for the following open-loop transfer functions:
)4(
)6)(2()( )1(
ss
ssKsG
)2(
)3()( )3(
ss
sKsG
)6(
)4)(2()( )2(
ss
ssKsG
)1(
)3)(2()( )4(
ss
ssKsG
)2(
)136()( )5(
2
ss
ssKsG
)136(
)2)(1()( )6(
2
ss
ssKsG
Re
Im
Re
Im
Re
Im
Re
Im
Re
Im
Re
Im
Examples
(1) (2) (3)
(4) (5) (6)