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System Control
Professor Kyongsu Yi ©2014 VDCL
Vehicle Dynamics and Control Laboratory Seoul National University
6. Root Locus Analysis
2
Characteristic Equation
2
CK )(sGR C+−
)()()(
sAsBsG =
)()()(
)(1)(
sBKsAsBK
sGKsGK
RC
C
C
C
C
+=
+=
Characteristic Equation : 0)()( =+ sBKsA C
Transient response
Characteristic roots, poles
3
)(1)(
bmsssG
+=
Characteristic Equation : 2 0Cms bs K+ + =
mmKbb
s C
242 −±−
=
0=CK ,0bsm−
=
mbKC 4
2
−= mbs
2−
=
mbKC 4
2
−⟩ jm
bmKmbs C
24
2
2−±
−=
double roots
Im
x x x Re mbKC 4
2−=
0=CK
∞=CK
3
Characteristic roots Example: Position control
4
02 =++ CKbsms
011 2 =+
+s
Kbsm
C
4
Characteristic roots
Example: Effect of Mass Variation
m1
2sKbs C+
RC
+−
5
m1
2sKbs C+
RC
+
1 1 ( )0 1 0( )
B sm m A s≈ + =
0)(1)( =+ sBm
sA
0)( =sA
0=s ; double roots
∞≈m1 0)(1)( =+ sB
msA
0)( =sB
bKs b−=
Characteristic roots
−
6
0)()(1 =+
sAsBKC
0)()( =+ sBKsA C
1
1
( ) ( )( ) 1 ( ) ( )
( )( )( ) ( ) 0 ( ) ( )
m
ij
C C n
ii
C s G sR s G s H s
s zB sG s H s K K n mA s s p
=
=
=+
−= = = ≥
−
∏
∏
1 1
( ) ( ) 0n m
i C ii j
s p K s z= =
− + − =∏ ∏
Root-Locus Plots
Closed-loop transfer function
Characteristic equation
7
1 1
( ) ( ) 0n m
i C ii j
s p K s z= =
− + − =∏ ∏
Openloop poles openloop zeros
1
1
0 ( ) 0 closed loop poles=open loop poles
1 ( ) 0 closed loop poles=open loop zeros
n
C ii
m
C ij
K s p
K s z
=
=
= − =
⟩⟩ − =
∏
∏
Characteristic Roots
Closed-loop poles
Characteristic equation
8
1 1
( ) ( ) 0n m
i C ii j
s p K s z= =
− + − =∏ ∏
Openloop poles openloop zeros
∏
∏
=
=
−
−=−= n
ii
m
ji
C ps
zs
KsAsB
1
1
)(
)(1
)()(
real, negative
s : complex number
Characteristic Roots Characteristic equation
9
1
2
1
2
j j x
j j y
x r e x e
y r e y e
θ
θ
∠
∠
= =
= =2θ1θ
y
xIm
Re
1 2( )1 2
( )
jx y r r ex y x y
x y x y
θ θ⋅⋅ = ⋅
⋅ = ⋅
∠ ⋅ = ∠ ⋅∠
1 2( )1
2
jrx ey r
xxy y
x x yy
θ θ−=
=
∠ = ∠ −∠
1
1 1
1
( )( ) ( ) 180 360
( )
m
i m nj
i inj i
ii
s zs z s p n
s p
=
= =
=
−∠ = ∠ − − ∠ − = ± ±
−
∏∑ ∑
∏
Characteristic roots
10
( ) ( ) 0
( ) ( ) 0( )
C
c
A s K B s
A s K B sB s
+ =
+ =
1⟩⟩CK
( ) 0( ) 0( ) c
B s m rootsA s K n m rootsB s
= ⇒ −
+ = ⇒ −
0)(
)(
1
1 =+−
−
∏
∏
=
=cn
ii
m
ji
Kps
zs
Characteristic roots
11
( ) 0( ) 0( ) c
B s m rootsA s K n m rootsB s
= ⇒ −
+ = ⇒ −
0)(
)(
1
1 =+−
−
∏
∏
=
=cn
ii
m
ji
Kps
zs
0
1
1
1
1
=+
+−
+−
∑
∑
=
−
=
−
cm
j
mi
m
n
i
ni
n
Kszs
sps
01
1 1=++
−− −−
= =
− ∑ ∑ cmn
n
i
m
jii
mn Kszps
01 1 =+
−
−−≈
−
= =∑ ∑
c
mnn
i
m
jii
Kmn
zps
Characteristic roots
n-m roots
12
1 1 0
n mn m
i ii j
c
p zs K
n m
−
= =
− − + = −
∑ ∑
mn
zpKs
n
i
m
jii
mncmn
−
−+⋅−=⇒∑ ∑= =
−− 1 111
)1(
( )
±
−− ⋅=−ππ n
mnj
mn e211
1)1(
complex real
n-m=2 n-m=3
Characteristic roots
n-m roots
13
Root – Locus Method
• Graphically obtains closed-loop poles as function of a parameter in feedback system.
)()()(
)()(1
)()(
sBKsAsBK
sAsBK
sAsBK
CR
⋅+⋅
=⋅+
⋅=
closed-loop poles : (characteristic roots)
0)()( =⋅+ sBKsA
0)( =sA ; open-loop poles ip
0)( =sB ; open-loop zeros iz
0)()(1
=−∏==
i
n
ipssA
0)()(1
=−∏==
i
m
izssB Where, mn ≥
0)()(11
=−∏+−∏==
i
m
ii
n
izsKps
14
Root – Locus Method
1) K=0 : closed-loop poles = open-loop poles
closed-loop characteristic eq. ; ⇒ closed-loop poles are asymptotic to open-loop poles for small K 2) K>>1 0)()( =⋅+ sBKsA
( ) ( ) 0( )
A s K B sB s
+ ⋅ =
(n-m)-poles m-poles
⇒ m closed-loop poles are asymptotic to open-loop zeros n-m poles
0)()(11
=−∏+−∏==
i
m
ii
n
izsKps
15
Root – Locus Method n-m Poles
0)(
)(
1
1 =+−∏
−∏
=
= Kzs
ps
i
m
i
i
n
i 01
1
1
1=+
⋅⋅⋅+
∑−
⋅⋅⋅+
∑−
−
=
−
= Ksps
sps
mi
m
i
m
ni
n
i
n
01
11=+⋅⋅⋅+
∑−∑− −−
==
− Kszps mni
m
ii
n
i
mn
(approximation) 011=+
−
∑−∑
−≅
−
== Kmn
zps
mn
i
m
ii
n
i
11 1
1 11 1
( )
( 1)
n m
i ii i n m
n m
i ii i n m n m
p zs K
n m
p zK
n m
= = −
= = − −
∑ −∑ = + −
− ∑ −∑ = + − ⋅
−real image
π)21(1 Nie +±=− ⋅⋅⋅= 3,2,1N
mnNimn e −+±− =− /)21(1
)1( π
16
Root – Locus Method
16
1=− mn 1)1( 11
−=−
17
Root – Locus Method
17
iNiNi eee 222/)21(2
1
)1(πππ
π ±
+±
+± ===−2=− mn
18
Root – Locus Method
18
3=− mn ππ
π iiNi eee ,)1( 33/)21(31
±+± ==−
19
Root – Locus Method
19
4=− mnii
ee 43
441
,)1(ππ
±±=−
20
Root – Locus Method
20
Example) 3=− mn n m poles−
21
Root – Locus Method
21
Angle condition
1 1
1
1
( ) ( ) 0
( ) ( ) 0
( )1 0( )
( ) 1( )
( ) 1
( )
n m
i ii i
m
iin
ii
A s K B s
s p K s z
B sKA s
B sA s K
s z
Ks p
= =
=
=
+ ⋅ =
∏ − + ∏ − =
+ ⋅ =
= −
∏ −= −
∏ −
real negative
1
1 1
1
1 1
( )( ) ( )
( )
( ) ( )
180 360 360180(1 2 ) 1,2,3
m
m nii
i in i ii
im n
i ii i
s zs z s p
s p
s z s p
N N
=
= =
=
= =
∏ −= ∏ − − ∏ −
∏ −
= ∑ − −∑ −
= ± ± ⋅⋅⋅= ± + = ⋅⋅⋅
22
Root – Locus Method
3) Real axis: angle condition
( ) 1 ,( )
B sA s K
= −1 1
( ) ( ) ( ) 100(1 2 )( )
n n
i ii i
B s s Z s P NA s = =
= − − − = ± +∑ ∑
23
Root – Locus Method
23
Example)
4 open-loop poles, 1 open loop zero
1
1 2
3
4
4
1
180
360
180
0
( ) ( ) 360i ii
s z
s p s p
s p
s p
s z s p=
∠ − =
∠ − +∠ − =
∠ − =
∠ − =
= ∠ − −∑ − = −
Root Roci in the real axis
XX
X
X
O
; Segments which have an odd number of open loop poles & zeros lying to the right on the real axis become portions of the root locus
24
Root – Locus Method
24
Portions of the root locus
n-m=3
25
Root – Locus Method
25
4) Angle of departure
2 1 2 1 2 2 3 2 4( ) ( ) ( ) ( ) ( ) 180(1 2 )p z p p s p p p p p N∠ − −∠ − −∠ − −∠ − −∠ − = ± +
)()()()()21(180)( 423212122 ppppppzpNps −∠+−∠+−∠+−∠−+±=−∠
1
1 1
1
( )( ) ( )
( )
180(1 2 ) 1,2,3
m
m nii
i in i ii
i
s zs z s p
s p
N N
=
= =
=
∏ −= ∑ − −∑ −
∏ −
= ± + = ⋅⋅⋅
26
Root – Locus Method
26
n=2, m=1
K)(sR )(sC+
− )( 1
1
psszs
−−
)(1
1
1
psszsK
−−
+
27
Root – Locus Method
27
5) Double root s=b
Characteristic eq. 0) ()( 2 =− bs
[ ] 0 ==bsdsd
At double root
0)()( =⋅+ sBKsA ⇒ )()(
sBsAK −=
( ) ( ) 0
( ) ( ) ( ) 0( )
( ) ( )( ) ( ) 0
dA s dB sKds ds
dA s A s dB sds B s ds
dA s dB sB s A sds ds
− =
− =
− =
28
Root Locus: Summary
1. K=0 2. K>>1 3. Real axis 4. Angle of departure 5. Double roots
0)()(1 =+
sAsBKC
( ) ( )( ) 1 ( ) ( )
C s G sR s G s H s
=+
Closed-loop transfer function
Characteristic equation
29
a slight change in the pole–zero configuration may cause significant changes in the root-locus configurations
System with 4 poles and 1 zero: two possible root loci
30
Root – Locus Method
30
• Positioning system
m
x
u oiln Lubricatio : b
P control
cK)(sR )(sX+
−1
( )s ms b+
)(sU
0)(
11 =+
+bmss
Kc
)()( bmsssA += 1)( =sBn=2 m=0
Open loop poles : 0, mb
−
31
Root – Locus Method
31
I control )(sR )(sX+
−1
( )s ms b+
)(sU
sKi
2
11 0( )
11 0( )
i
i
Ks s ms b
Ks ms b
+ =+
+ =+
n=3 m=0
angle of departure
)21(180)()(11
Npszs i
n
ii
m
i+±=−∑−−∑
==
)21(180)()()( 321 Npspsps +±=−∠−−∠−−∠−
180)(2 1 ±=−∠− ps
90)( 1 =−∠ ps
Open loop poles : mb
−0,0
double loot
32
Root – Locus Method
32
PI control )(sR )(sX+
−1
( )s ms b+
)(sU
+
sTK
ic
11
0)(
111 =+
++
bmsssTsTK
i
ic n=3 m=1 n-m=2
Open loop pole : 0, 0, mb
−
Open loop zero : iT
1
33
Root – Locus Method
33
PID control - Mass change, - PI, I gain variation
)(sR )(sX+
−1
( )s ms b+
)(sU1(1 )C di
K T ST S
+ +
21 11 0( )
i dC
i
T s T sKT s s ms b
+ ++ =
+2 Open loop zeros
mb
−,0,03 Open loop poles :
34
Root – Locus Method
34
2 0ms bs K+ + =
2
11 0Cbs Km s
++ =
Example)
Effect of increasing mass, m, for fixed gain Kc 1 0; s=0, 0
1 ; c
closed loop polesm
Ks sm b
=
= ∞ = − = −∞
1 0 ( )mm= = ∞
35
Plotting Root Loci with MATLAB 1 0numK
den+ =
rlocus(num, den) r=rlocus(num,den) Plot(r, ‘o’)
36
MATLAB Program % --------- Root-locus plot --------- num = [1 3]; den = [1 5 20 16 0]; rlocus(num,den) v = [-6 6 -6 6]; axis(v); axis('square') grid; title ('Root-Locus Plot of G(s) = K(s + 3)/[s(s + 1)(s^2 + 4s + 16)]')
37
End of root locus
38
Root – Locus Method
38
Minimum phase systems: 시스템의 모든 pole과 zero가 LHP에 있는 경우 Non minimum phase systems: 적어도 시스템의 한 개의 pole이나 zero가 평면의 RHP에 있는 경우
(1 )( )( 1)
dT sG ss Ts−
=+
(1 )1 0( 1)
dT sKs Ts−
+ =+
1 1( 1)
dT ss Ts K
−=
+
1
dT−+
