The Root Locus Method

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The Root Locus Method

• Modern Control Systems
• Lecture 16

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Outline

• Why do we need root locus
• Magnitude and angle conditions
• The root locus procedure

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Why do we need root locus

• The location of closed-loop roots of the
  characteristic equation determines stability
  (including relative stability), transient
  response and steady-state response. It is often
  necessary to adjust one or more system parameters
  to relocate closed-loop roots such that desired
  performance is achieved. Therefore, it is
  useful to investigate how the roots of the
  characteristic equation move around the s-plane
  as we change a parameter.

The root locus technique is a graphical method
for sketching the locus of roots in the s-plane
as a parameter is varied. A root locus does not
exist for a system with fixed parameters!
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Consider the following feedback system.

• The closed-loop TF is

The characteristic equation of the system is and
the roots of the characteristic equation are
As we can see, characteristic roots move as K
changes. Plot of root position as K is increased
from zero to positive infinity is the root locus.
K cannot be negative in this example, why?
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• The closed-loop TF is

When rewriting this in standard form we have

• As K increases from zero to 8,
• closed-loop roots move toward -1j8
• ? decreases, overshoot increases.
• ?n increases
• the system always remains stable

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Magnitude and angle conditions

• For the closed-loop system shown in the figure,
  the characteristic equation is

Because s is a complex variable, the above
equation is rewritten as
Therefore, the magnitude condition is and the
angle condition is
The root locus exists at points in the s-plane
for which both the magnitude and angle conditions
are satisfied.
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Magnitude and angle conditions (contd)

• For a multiloop closed-loop system, we can use
  Masons gain formula to obtain the characteristic
  polynomial q(s).

q(s) may be written as
The characteristic equation is therefore
1F(s)0, or F(s)-1j0.
In general, F(s) is expressed by
Then the magnitude and angle conditions for the
root locus are
Normally K is taken to be positive.
where k is an integer.
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The root locus procedure

• An orderly procedure for quickly sketching the
  root locus.
• Step 1. Prepare the root locus sketch.
• Rearrange the characteristic equation so that the
  parameter of interest, K, appears as the
  multiplying factor in the form

Factor P(s) and write it in the form of poles and
zeros.
When K0, the roots of the characteristic
equation are the poles of P(s). As K?8, the roots
of the characteristic equation are the zeros of
P(s). In other words, the root locus begins at
the poles of P(s) and ends at the zeros of P(s)
as K goes from zero to 8.
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The root locus procedure (contd)

• Step 2. Locate the segments of the real axis that
  are root loci.
• The root locus on the real axis always lies in a
  section of the real axis to the left of an odd
  number of poles and zeros.
• The number of separate loci is equal to the
  number of poles. With n poles and M zeros and
  ngtM, we have n-M branches of the root locus
  approaching the n-M zeros at 8.
• Root loci must be symmetrical with respect to the
  real axis.

As an example, we have a single loop feedback
system whose characteristic equation is
We rewrite it in the required form as
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• Step 3. The root loci proceed to the zeros at 8
  along asymptotes centered at sA and with angles
  fA.
• When the number of finite zeros of P(s), M, is
  less than the number of poles, n, then n-M
  sections of loci must end at zeros at 8. These
  sections of loci proceed to the zeros at 8 along
  asymptotes as K?8.

For example, the characteristic polynomial of a
4th-order system is
The asymptotes are determined by
root locus
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• Step 4. Determine the point at which the root
  locus crosses the imaginary axis (j?-axis) if it
  does so.
• The actual point at which the root locus crosses
  j?-axis is evaluated by using Routh-Hurwitz
  criterion.

• Step 5. Determine the breakaway point on the real
  axis (if any).
• The tangents to the loci at the breakaway point
  are equally spaced over 3600.
• The breakaway point on the real axis can be
  evaluated analytically, as shown below.

To find the breakaway point on the real axis,
rearrange the characteristic equation to isolate
K so that the characteristic equation is written
as
The breakaway point is obtained by
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For example, consider a unity feedback system
with an open-loop TF G(s)
The characteristic equation is then We rewrite it
into
The breakaway point is obtained by
which is
So the breakaway point on the real axis occurs at
s-3.
root locus
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• Step 6. Determine the angle of departure of the
  locus from a pole and the angle of arrival of the
  locus at a zero.
• The angle of departure (or arrival) is of
  interest for complex poles (and zeros).
• The departure or arrival angle is obtained from
  the angle condition.

For example, consider a 3rd-order open-loop TF
To find the departure angle from the pole p1, we
know that the angles at a test point s1, an
infinitesimal distance from p1, must meet the
angle condition.
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• Step 7. Complete the root locus sketch.
• This entails sketching in all sections of the
  locus not covered in the previous six steps.

In some situation, we may want to determine a
root location sx and the parameter Kx at that
root location. The root location sx has to
satisfy the angle condition, i.e.,
where k is an integer.
The parameter value Kx at root sx is evaluated by
using the magnitude condition, and Kx is
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Consider again the unity feedback system shown in
the figure.

• A test point s1 is verified as a root location if
  the angle condition is satisfied. The angle at s1
  is

So s1 is on the root locus. Now we want to find
parameter KK1 at s1.
We use the magnitude condition to find K1 at s1
root locus
evaluation of KK1 at s1