RootLocus Analysis

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Root-Locus Analysis

• Instructor Chia-Ming Tsai
• Dept. of Electronics Engineering
• National Chiao Tung University

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Contents

• Introduction
• Root-Locus Plots
• Summary of General Rules for Constructing Root
  Loci
• Positive-Feedback Systems
• Conditionally Stable Systems
• Root Loci for Systems with Transport Lag

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Introduction

• The transient response is closely related to the
  location of the closed-loop poles.
• The location of the closed-loop poles depends on
  the value of the loop gain chosen.
• It is important to know how the closed-loop poles
  move in the s plane as the loop gain is varied.
• Gain adjustment Compensator Desired result
  ??? (Chapter 7)

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Root-Locus Method

• Developed by W. R. Evans.
• A simple method for finding the roots of the
  characteristic equation.
• The basic idea is that the value of s that make
  the transfer function around the loop equal-1
  must satisfy the characteristic equation of the
  system.
• It is the locus of roots of the characteristic
  equation of the closed-loop system as a specific
  parameter (usually, gain K) is varied from 0 to
  ?.
• It shows the contribution of each open-loop pole
  or zero to the locations of the closed-loop poles.

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Root-Locus Method (Contd)

• It shows the manner in which the open-loop pole
  or zero should be modified so as the response
  meets system performance specifications.
• To determine the value of the loop gain K that
  makes ? of the dominant closed-loop poles as
  prescribed.
• To suggest the way to choose the location of an
  open-loop pole or zero.

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Root-Locus Plots
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Contd
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Contd
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Drawing a Root-Locus Plot

• Seek out the loci of possible roots using the
  angle condition.
• The loci can be scaled, or graduated, in gain
  using the magnitude condition.
• Abscissa-based analysis.

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Example 1

• Determine the root loci on the real axis.
• Determine the asymptotes of the root loci.
• Determine the breakaway point.
• Determine the points where the root loci cross
  the imaginary axis.
• Choose a test point in the broad neighborhood of
  the j? axis and the origin and apply the angle
  condition.
• Draw the root loci.
• Determine a pair of dominant complex-conjugate
  closed-loop poles such that the damping ratio ?
  meets the target.

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Contd
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Contd
-2 -1 0
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Contd
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Contd
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Contd
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Contd