This document provides background information on root locus analysis and techniques for sketching root loci. It discusses rules for determining the number and symmetry of root locus branches, where the locus begins and ends, and how to find breakaway/break-in points and jw-axis crossings. The document concludes by posing a problem to sketch the root locus for a given transfer function. The key information is that root locus analysis graphically shows how closed-loop poles vary with gain and provides insights into stability and transient response.

1. ME 176
Control Systems Engineering
Root Locus Technique
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2. Introduction : Root Locus
"... a graphical representation of closed loop poles as a system
parameter is varied, is a powerful method of analysis and design
for stability and transient response."
"...real powere lies in its ability to provide for solutions for systems of
order higher than 2."
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3. Background: Control Systems
Open Loop vs. Closed Loop
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4. Background: Complex Number Vector Representation
Representation of complex Number:
Cartesian Form :
Vector Form :
Vector complex arithmetic:
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5. Background: Complex Number Vector Representation
Example
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6. Background:
Root Locus
"... is the representation of the
paths of the closed-loop poles as
the gain is varied." Department of
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7. Background: Root Locus
Given Poles and Zeros of a closed-loop
transfer function KG(s)H(s) a point in the
s-plane is on the root locus for a gain K if : angles of zeros minus
angles of poles add up to (2k+1)180 degrees. K is found by dividing
the product of pole lengths to that of zeros.
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8. Sketching: Root Locus
Rule 1 of 5 : The number of branches of the root locus equals the
number of closed-loop poles.
2 poles therefore 2 branches
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9. Sketching: Root Locus
Rule 2 of 5 : The root locus is symmetrical about the real axis
Symmetry of segments between:
Branch 1 : -5 onward
Branch 2 : -5 onward
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10. Sketching: Root Locus
Rule 3 of 5 : On the real axis, for K > 0 the root locus exists to the left of
an odd number of real-axis, finite open-loop poles and/or finite open-loop
zeros.
Root Locus Exists:
Left of -3 : zero number 1
Left of -1 : pole number 1
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11. Sketching: Root Locus
Rule 4 of 5 : The root locus begins at the finite and infinite poles of G(s)H
(s) and ends at the finite and infinite zeros of G(s)H(s).
Root Locus Exists:
Starts on poles at real axis : -1, -2
Ends on zeros at real axis : -3, -4
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12. Sketching: Root Locus
Rule 5 of 5 : The root locus approaches straight lines as asymptotes as
the locus approaches infinity. Further, the equation of the asymptotes is
given by :
Angle is radians with respect to positive
extension of the real axis.
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13. Refining Sketch: Root Locus
Real Axis Breakaway and Break-in :
Breakaway and break-in points
satisfy the relationship:
where zi and pi are vnegative
of the zero and pole values,
respectively.
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14. Refining Sketch: Root Locus
jw - Axis Crossing :
Use Routh-Hurwitz criterion,1) forcing
a row of zeros in the Routh table to
establish the gain; 2) then going back
one row to the even polynomial
equation and solving for the roots
yields the frequency at the imaginary
axis crossing.
1)
2)
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15. Problem:
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16. Problem:
Sketch the root locus and its asymptotes for a unity feedback system that has the
forward transfer function:
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