Chapter 8 Root Locus Techniques

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Chapter 8 Root Locus Techniques

• 1. The definition of a root locus
• 2. How to sketch a root locus
• 3. How to refine your sketch of a root locus
• 4. How to use the root locus to find the poles of
  a
• closed-loop system
• 5. How to use the root locus to describe
  qualitatively
• the changes in transient response and
  stability of
• a system as a system parameter is varied
• 6. How to use the root locus to design a
  parameter
• value to meet a transient response
  specification or
• system of order 2 and higher

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Root locus, a graphical presentation of the
closed-loop poles as a system parameter is
varied, is a powerful method of analysis and
design for stability and transient response
(Evans, 1948 1950)
open-loop transfer function
closed-loop transfer function
where and are factored polynomials
and signify numerator and denominator terms,
respectively.
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Vector Representation of Complex Numbers
Any complex number, , described in
Cartesian coordinates can be graphically
represented by a vector. The complex number also
can be described in polar form with magnitude
and angle , as .
Figure 8.2Vector representation of complex
numbersa. s ? j? b. (s a)c.
alternate representation of (s a)d. (s
7)s?5 j2
We conclude that is a complex number
and can be represented by a vector drawn from the
zero of the function to the point .
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Assume a function
The magnitude, , of at any point,
, is
The angle, , of at any point, , is
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Example Given
Find at the point .
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Defining the Root Locus
It is this representation of the paths of the
closed-loop poles as the gain is varied that we
call a root locus.
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Properties of the Root Locus
The closed-loop transfer function for the system
is
From the above equation, a pole, , exists when
the characteristic polynomial in the denominator
becomes zero, or
Alternately, a value of is a closed-loop pole
if and
A closed-loop pole must satisfy the magnitude
condition and angle condition.
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Problem Are points and
closed-loop poles?
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Sketching the Root Locus

• The following five rules allow us to sketch the
  root locus using minimal calculations.
• Number of branches. A branch is the path that on
  pole traverses. The number of branches of the
  root locus equal the number of closed-loop poles.
• Symmetry.
  
• The root locus is symmetrical about the real
  axis.
• 3. Real-axis segments.
• On the real axis, for the root
  locus exists to the left of an odd number of
  real-axis, finite open-loop poles and/or finite
  open-loop zeros.

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4. Starting and ending points. Where does the
root locus begin (zero gain) and end
(infinite gain)?
As approaches zero (small gain),
As approaches infinity (large gain),
The root locus begins at the finite and infinite
poles of and ends at the finite
and infinite zeros of .
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5. Behavior at infinity. A function can also have
infinite poles and zeros. If the function
approaches infinity as approaches infinity,
then the function has a pole at infinity. If the
function approaches zero as approaches infinity,
then the function has a zero at infinity.
The root locus approaches straight lines as
asymptotes as the locus approaches infinity.
Further, the equation of the asymptotes is given
by the real-axis intercept, , and angle,
, as follows
where , and the
angle is given in radians with respect to the
positive extension of the real axis.
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Problem Sketch the root locus for the system
shown in the following figure.
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Refining the Sketch
1. Real-axis breakaway and break-in points.
The point where the locus leaves the real axis is
called the breakaway point, and the point where
the locus returns to the real axis is called the
break-in point. One of methods for finding
the breakaway or break-in point is to maximize
and minimize the gain, , using differential
calculus. For all points on the root locus yields
For points along the real-axis segment of the
root locus where the breakaway and break-in
points could exist, . Hence, along the
real axis, the above equation becomes
Differentiating the gain with respect
to and setting the derivative equal to zero,
we can find the points of maximum and minimum
gain and hence the breakaway and break-in points.
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Figure 8.13Root locus example showingreal- axis
breakaway (-?1) and break-in points (?2)
Problem Find the breakaway and break-in points
for the root locus of the following figure using
differential calculus.
From the open-loop poles and zeros, it yields
Then, we find
Solving for , we find and
, which are the breakaway and
break-in points.
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2. The -Axis Crossings. The -axis
crossing is a point on the root locus that
separates the stable operation of system from the
unstable operation. To find the -axis
crossing, we can use the Routh-Hurwitz criterion
as follows forcing a row of zeros in the Routh
table will yield the gain going back one row to
the even polynomial equation and solving for the
roots yields the frequency at the imaginary axis
crossing.
Problem For the system shown in the following
figure, find the frequency and gain, , for
which the root locus crosses the imaginary axis.
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3. Angles of Departure and Arrival.
Calculating the root locus departure angle from
the complex poles and the arrival angle to the
complex zeros. If we assume a point on the
root locus close to a complex pole (or zero),
the sum of angles drawn from all finite poles and
zeros to his point is an odd multiple of
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Problem Given the unity feedback system shown in
the following figure, find the angle of departure
from the complex poles and sketch the root locus.
Complex poles For the complex pole
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4. Plotting and calibrating the root locus.
In summary, we search a given line for the point
yielding a summation of angles (zero angles -
pole angles) equal to an odd multiple of .
The gain at that point is then found by
multiplying the pole lengths drawn to that point
and dividing by the product of the zero lengths
drawn to that point.
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Problem Sketch the root locus for the system
shown in the following figure and find the
following 1. The exact point and gain where the
locus crosses the o.45 damping ratio
line. 2. The exact point and gain where the locus
crosses the -axis. 3. The breakaway point on
the real axis. 4. The range of within which
the system is stable.
Figure 8.19a. System forb. root locus sketch
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Transient Response Design via Gain Adjustment
The conditions justifying a second-order
approximation are restated 1. Higher-order poles
are much farther into the left half of the -plane
than the dominant second-order pair
of poles. 2. Closed-loop zeros near the
closed-loop second-order pole pair are
nearly canceled by the close proximity of
higher-order closed-loop poles. 3. Closed-loop
zeros not canceled by the close proximity of
higher-order closed-loop poles are far
removed from the closed-loop second-order
pole pair.
Figure 8.20Making second-order approximations
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Summarizing the design procedure for higher-order
systems, we arrive at the following

• Sketch the root locus for the given system.
• Assume the system is a second-order system
  without any zeros and then find the gain to meet
  the transient response specification.
• Justify your second-order assumption by finding
  the location of all higher-order poles and
  evaluating the fact hat they are much farther
  from the -axis than the dominant
  second-order pair. As a rule of thumb, this
  textbook assumes a factor of 5 times farther.
  Also, verify that closed-loop zeros are
  approximately canceled by higher-order poles. If
  closed-loop zeros are not canceled by
  higher-order closed-loop poles, be sure that the
  zero is far removed from the dominant
  second-order pole pair to yield approximately the
  same response obtained without the finite zero.
• If the assumptions cannot be justified, your
  solution will have to be simulated in order to be
  sure it meets the transient response
  specification.

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Problem Consider the system shown in the
following figure. Design the value of gain, ,
to yield 1.52 overshoot. Also estimate the
settling time, peak time, and steady-state error.
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Root Locus for Positive-Feedback Systems
Figure 8.26Positive-feedback system
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• Number of branches.
• The number of branches of the root locus
  equal the number of closed-
• loop poles.
• 2. Symmetry.
  
• The root locus is symmetrical about the real
  axis.
• 3. Real-axis segments.
• On the real axis, the root locus for
  positive-feedback systems exists
• to the left of an even number of real-axis,
  finite open-loop poles
• and/or finite open-loop zeros.
• 4. Starting and ending points.
• The root locus for positive-feedback systems
  begins at the finite and
• infinite poles of and ends
  at the finite and infinite zeros
• of .
• 5. Behavior at infinity.
• The root locus approaches straight lines as
  asymptotes as the locus approaches infinity.
  Further, the equation of the asymptotes is given
  by the real-axis intercept, , and angle,
  , as follows

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Problem Sketch the root locus as a function of
negative gain, , for the system shown in the
following figure.
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Pole Sensitivity
Root sensitivity is the ratio of the
fractional change in a closed-loop pole to the
fractional change in a system parameter, such as
gain. The sensitivity of a closed-loop pole,
, to gain, , can be written as where
is the current pole location, and is the
current gain. From the above the equation, the
actual change in the closed-loop poles can be
approximated as Where is the change in
pole location, and is the fractional
change in the gain, .
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Problem Find the root sensitivity of the
following system at and
. Also calculate the change in the pole
location for a 10 change in .
The systems characteristic equation
is Differentiating with respect , we
have from which The sensitivity is found to be