Formula Sheets: The Root Locus Technique | Control Systems - Electrical Engineering (EE) PDF Download

 Page 1


Control S ystems F ormula Sheet: Root
Locus T echnique
1. Basics of Root Locus
• Definition : Plot of the closed-loop poles as a system par ameter (usually gain
K ) varies from 0 to8 .
• Char acteristic Equation : F or a system with open-loop tr ansfer functionG(s)H(s) ,
the closed-loop char acteristic equation is:
1+KG(s)H(s) = 0
• Root Locus Equation : Satisfies magnitude and angle conditions:
|KG(s)H(s)| = 1, ?G(s)H(s) =±180
?
(2k +1), k = 0,1,2,...
• Open-Loop Tr ansfer Function:
G(s)H(s) =
K(s+z
1
)(s+z
2
)···(s+z
m
)
(s+p
1
)(s+p
2
)···(s+p
n
)
wherez
i
are z eros,p
i
are poles,m is number of zeros,n is number of poles.
2. Rules for Root Locus Construction
• Rule 1: Number of Br anches : Equal to the number of open-loop poles (n ).
• Rule 2: Starting Points : Root locus starts at open-loop poles (K = 0 ).
• Rule 3: Ending Points : Root locus ends at open-loop zeros or infinity ( K?8 ).
• Rule 4: Real Axis Segments : Locus exists on the real axis to the left of an odd
number of real poles and zeros.
• Rule 5: Asymptotes : F orn>m , (n-m) br anches approach asymptotes:
Angle of asymptotes =
±180
?
(2k +1)
n-m
, k = 0,1,...,(n-m-1)
Centroid of asymptotes =s
A
=
?
real parts of poles-
?
real parts of zeros
n-m
1
Page 2


Control S ystems F ormula Sheet: Root
Locus T echnique
1. Basics of Root Locus
• Definition : Plot of the closed-loop poles as a system par ameter (usually gain
K ) varies from 0 to8 .
• Char acteristic Equation : F or a system with open-loop tr ansfer functionG(s)H(s) ,
the closed-loop char acteristic equation is:
1+KG(s)H(s) = 0
• Root Locus Equation : Satisfies magnitude and angle conditions:
|KG(s)H(s)| = 1, ?G(s)H(s) =±180
?
(2k +1), k = 0,1,2,...
• Open-Loop Tr ansfer Function:
G(s)H(s) =
K(s+z
1
)(s+z
2
)···(s+z
m
)
(s+p
1
)(s+p
2
)···(s+p
n
)
wherez
i
are z eros,p
i
are poles,m is number of zeros,n is number of poles.
2. Rules for Root Locus Construction
• Rule 1: Number of Br anches : Equal to the number of open-loop poles (n ).
• Rule 2: Starting Points : Root locus starts at open-loop poles (K = 0 ).
• Rule 3: Ending Points : Root locus ends at open-loop zeros or infinity ( K?8 ).
• Rule 4: Real Axis Segments : Locus exists on the real axis to the left of an odd
number of real poles and zeros.
• Rule 5: Asymptotes : F orn>m , (n-m) br anches approach asymptotes:
Angle of asymptotes =
±180
?
(2k +1)
n-m
, k = 0,1,...,(n-m-1)
Centroid of asymptotes =s
A
=
?
real parts of poles-
?
real parts of zeros
n-m
1
• Rule 6: Breakawa y/Break-in Points : Points where multiple roots occur , found
b y solving:
d
ds
[G(s)H(s)] = 0 or
d
ds
(
1
G(s)H(s)
)
= 0
• Rule 7: Angle of Departure (from a pole) :
?
departure
= 180
?
+
?
?( pole to zeros)-
?
?( pole to other poles)
• Rule 8: Angle of Arrival (at a zero) :
?
arrival
= 180
?
+
?
?( zero to poles)-
?
?( zero to other zeros)
• Rule 9: Intersection with Imaginary Axis : F ound b y applying Routh-Hurwitz
criterion to the char acteristic equation to determineK ands where locus crosses
the imaginary axis.
3. Gain Calculation
• Magnitude Condition : F or a points
1
on the root locus:
K =
1
|G(s
1
)H(s
1
)|
=
?
| distances froms
1
to poles|
?
| distances froms
1
to zeros|
4. Stability Analysis
• Stability : S ystem is stable if all closed-loop poles (points on root locus) lie in
the left half-plane (LHP).
• Critical Gain (K
crit
) : V alue of K where the root locus crosses the imaginary
axis (found using Routh-Hurwitz or solving char acteristic equation fors =j? ).
• Marginal Stability : Occurs when root locus l ies on the imaginary axis (poles
ats =±j? ).
5. Time Domain Par ameters
• Damping Ratio (? ): F or a pole ats =s +j? :
? =
-s
v
s
2
+?
2
• Natur al Frequency (?
n
):
?
n
=
v
s
2
+?
2
• Damped Frequency (?
d
):
?
d
=? =?
n
v
1-?
2
2