INC341 Root Locus

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INC341Root Locus

• Lecture 7

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Rectangular vs. polar
s 4 j3
Rectangular form 4 j3 Polar
form magnitude5, angle 37
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Rectangular form
Add, Subtraction
Polar form
Multiplication
Division
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b
r
?
a
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Vector representation of a transfer function
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Vector s
(sa)
(s7) s 5j2
(sa)
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Example
Find F(s) at s -3j4
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What is root locus and why is it needed?

• Fact I poles of closed-loop system are an
  important key to describe a performance of the
  system (transient response, i.e. peak time,
  overshoot, rise time), and stability of the
  system.
• Fact II closed-loop poles are changed when
  varying gain.
• Implication Root locus paths of closed-loop
  poles as gain is varied.

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Cameraman Object Tracking using infrared
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Varying gain (K)
Varying K, closed-loop poles are moving!!!
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• Transient
• Klt25 ? overdamped
• K25 ? critically damped
• Kgt25 ? underdamped
• Settling time remains the same under underdamped
  responses.
• Stability
• Root locus never crosses over into the RHP,
  system is always stable.

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Concept of Root Locus
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Closed-loop transfer function
Characteristic equation
magnitude
phase
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• If there is any point on the root locus, its
  magnitude and phase will be consistant with the
  follows

magnitude
phase

• Note that phase is an odd multiple of 180

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Example
Is the point -23j a closed-loop pole for some
value of gain? Or is the point on the root locus?
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-23j is not on the root locus!!! What about
?
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The angles do add up to 180!!!
is a point on the root locus
What is the corresponding K?
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Sketching Root Locus

• Number of branches
• Symmetry
• Real-axis segment
• Starting and ending points
• Behavior at infinity

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1. Number of branches
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2. Symmetry
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3. Real-axis segment
On the real axis, the root locus exists to the
left of an odd number of real-axis
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• Sum of angles on the real axis is either 0 or 180
  (complex poles and zeroes give a zero
  contribution).
• Left hand side of even number of poles/zeros on
  the real axis give 180 (path of root locus)

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Example
root locus on the real axis
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4. Starting and ending points
closed-loop transfer function
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K0 (beginning) poles of T(s) are
K8 (ending) poles of T(s) are
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Example
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5. Behavior at infinity
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Rule of thumb

• of poles of zeroes
• has 3 finite poles at 0 -1 -2, and 3 infinite
  zeroes at infinity

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Example
Sketch root locus
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