Root Locus

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Root Locus
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Closed-loop control system with a variable
parameter K.
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Unity feedback control system. The gain K is a
variable parameter.
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Root locus for a second-order system when
KeltK1ltK2. The locus is shown in color.
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Evaluation of the angle and gain at s1,for gain
KK1.
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(a) Single-loop system. (b) Root locus as a
function of the parameter a.
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(a) The zero and poles of a second-order
system,(b) the root locus segments,and (c) the
magnitude of each vector at s1.
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A fourth-order system with (a) a zero and (b)
root locus.
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Illustration of the breakaway point (a) for a
simple second-order system and (b) for a
fourth-order system.
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A graphical evaluation of the breakaway point.
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Closed-loop system.
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Evaluation of the (a) asymptotes and (b)
breakaway point.
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Illustration of the angle of departure(a) Test
point infinitesimal distance from p1(b) actual
departure vector at p1
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Evaluation of the angle of departure.
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The root locus for Example 7.4locating (a) the
poles and (b) the asymptotes.
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• 1.Write the characteristic equation in pole-zero
  form so that the parameter of interest k appears
  as 1kF(s)O.
• 2.Locate the open-loop poles and zeros of F(s) in
  the s-plane.
• 3.Locate the segments of the real axis that are
  root loci.
• 4.Determinethenumberofseparateloci.
• 5.Locate the angles of the asymptotes and the
  center of the asymptotes.
• 6.Determine the break away point on the real
  axis(if any).
• 7.By utilizing the Routh-Hurwitz criterion,
  determine the point at which the locus crosses
  the imaginary axis(if it does so).
• 8.Estimate the angle of locus departure from
  complex poles and the angle locus arrival at
  complex zeros.

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Root loci as a function of a and ß(a) loci as a
varies (b) loci as ß varies for one value of a
a1
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A region in the s-plane for desired root location.
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A feedback control system.
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Closed-loop system with a controller.
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Root locus for plant with a PID controller with
complex zeros.
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Laser manipulator control system.
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Root locus for a laser control system.
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The response to a ramp input for a laser control
system.
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Proposed configuration for control of the
lightweight robot arm.
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Root locus of the system if K20,and K1 is
variedfrom K10 to K18,and Gc(s)K1.
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Root locus for the robot controller with a zero
inserted at s-0.2 with Gc(s) K1.
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Using the rlocus function.
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Disk drive control system with a PD controller.
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Sketch of the root locus.
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Root Locus Plots for Typical Transfer Functions
(continued).
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Root Locus Plots for Typical Transfer Functions
(continued).
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Root Locus Plots for Typical Transfer Functions
(continued).
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Root Locus Plots for Typical Transfer Functions
(continued).
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Root Locus Plots for Typical Transfer Functions
(continued).
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Root Locus Plots for Typical Transfer Functions
(continued).
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Root Locus Plots for Typical Transfer Functions
(continued).
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Root Locus Plots for Typical Transfer Functions
(continued).
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Root Locus Plots for Typical Transfer Functions
(continued).
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Root Locus Plots for Typical Transfer Functions
(continued).
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Root Locus Plots for Typical Transfer Functions
(continued).
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Root Locus Plots for Typical Transfer Functions
(continued).
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Root Locus Plots for Typical Transfer Functions
(continued).
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Root Locus Plots for Typical Transfer Functions
(continued).
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Root Locus Plots for Typical Transfer Functions
(continued).
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