Lecture 15 ME 176 7 Root Locus Technique

ME 176
Control Systems Engineering
Root Locus Technique

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Mechanical Engineering

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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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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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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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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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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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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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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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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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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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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Lecture 15 ME 176 7 Root Locus Technique

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Lecture 15 ME 176 7 Root Locus Technique