The document discusses root locus analysis, a technique for analyzing the stability and transient response of control systems. It provides rules for sketching root loci, including that branches represent closed-loop poles and the locus is symmetric about the real axis. The document also describes refining the root locus sketch by finding the imaginary axis crossing, angles of departure and arrival, and approximating higher-order systems as second-order. An example problem is given to apply these techniques.

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. Refining Sketch: Root Locus
Rules:
1. Branches equal to closed-loop poles
2. Symentrical about real axis.
3. Left of odd number of real-axis finite open-loop poles and/or zeros.
4. Begins on finite/infinite poles, ends on finite/infinite zeros of G(s)H(s).
5. Approaches asymptotes as the
locus approaches infinity:
Refinements:
1. Break-away and break-in:
2. jw-Axis crossing using Routh-Hourwitz criterion.
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4. Refining Sketch: Root Locus
3. Angles of Departure and Arrival:
Root locus departs at complex open-loop
poles and arrives at open-loop zeros at
angels given by: assuming points close to
such poles and zeros, summing all angles
drawn from all poles and zeros will equal
(2k+1)180.
Zeros - postivive
Poles - negative
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5. Refining Sketch: Root Locus
4. Plotting and Calibrating the Root Locus:
Evaluate the root locus at a point on the s-
plane by first solving if that point yields a
summation of angles (zero angles - pole
angles) equal to an odd multiple of 180.
Then calculate the gain by multiplying the
pole lengths drawn to that point divided by
product of zero lengths.
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6. Refining Sketch: Root Locus
Approximation to 2nd order systems
1. Higher-order poles are much farther into the left half of the s-plane than the
dominant second order pair of poles. The response from a higher order pole
does not appreciably change the transient response expected from the
dominant second order pole.
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7. Refining Sketch: Root Locus
Approximation to 2nd order systems
2. Closed-loop zeros nea the closed loop second-order pole pair are nearly
canceled by the close proximity of higher-order closed-loop poles.
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8. Refining Sketch: Root Locus
4. Plotting and Calibrating the Root Locus:
Evaluate the root locus at a point on the s-
plane by first solving if that point yields a
summation of angles (zero angles - pole
angles) equal to an odd multiple of 180.
Then calculate the gain by multiplying the
pole lengths drawn to that point divided by
product of zero lengths.
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Mechanical Engineering

9. Refining Sketch: Root Locus
Example: Do the following:
1. Sketch the root locus .
2. Find the imaginary-axis crossing.
3. Find the gain, K, at the jw-crossing.
4. Find the angles of departure.
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10. Refining Sketch: Root Locus
Example: Do the following:
1. Sketch the root locus .
2. Find the imaginary-axis crossing.
3. Find the gain, K, at the jw-crossing.
4. Find the angles of departure.
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Mechanical Engineering

11. Refining Sketch: Root Locus
Example: Do the following:
1. Sketch the root locus .
2. Find the imaginary-axis crossing.
3. Find the gain, K, at the jw-crossing.
4. Find the angles of departure.
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12. Refining Sketch: Root Locus Lab
Find F(s) at point s= -7+j9 Find G(s) at point s=-3+j0
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13. Refining Sketch: Root Locus Lab
a. Sketch the root locus.
b. Find Imaginary-axis crossing.
c. Find K at jw-axis crossing.
d. Find the break-in point.
e. Find the point where the locus crosses the 0.5
damping ration line.
f. Find the gain at the point where the locus crosses
the 0.5 damping ration line.
g. Find the range of gain, K, for which the system is
stable.
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14. Refining Sketch: Root Locus Lab
Sketching Rules:
1. Branches: 2
2. Symmetric about real axis
3. Real-axis segment: Left of -2 pole
4. Start at poles, ends at zeros
5. Behavior at infinity:
Sigma = 0
Theta = infinity
Use Matlab:
6. Real-axis breakaway points
7. jw - crossing
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15. Refining Sketch: Root Locus Lab
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