Plotting and Reshaping the Root Locus

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This example looks at the root locus plot for a particular open-loop transfer function,
G_{p}(s). This transfer function would represent some system which is to be controlled.
We will also look at how the root locus of G_{p}(s) can be reshaped so that the final root
locus goes through a particular point in the s-plane. The reshaping is done by placing
another transfer function, G_{c}(s), in series with G_{p}(s). The poles and zeros of
G_{c}(s) are
chosen to make the total phase shift of G_{c}(s)G_{p}(s) at the chosen point equal to 180 degrees
(K > 0) or 0 degrees (K < 0). The gain of Gc(s) would be chosen to make the total magnitude
of G_{c}(s)G_{p}(s) equal to 1 at the chosen point. The transfer function
G_{p}(s) is

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A particular point s = s_{1} will be specified as a desired closed-loop pole. If the root locus
of G_{p}(s) passes through s_{1}, then s = s_{1} can be made a closed-loop pole merely by placing a
gain K_{c} in series with G_{p}(s) and adjusting the value of K_{c} to satisfy the magnitude criterion.
The phase criterion will already be satisfied if the root locus of G_{p}(s) passes through
s_{1}. If
the root locus of G_{p}(s) does not pass through s_{1}, then additional positive or negative phase
shift will be needed to satisfy the phase criterion at s_{1}.

In general, the point s = s_{1} would be chosen as a desired closed-loop pole under the
assumption that time-domain performance requirements, such as overshoot and settling time, will
be satisfied if s_{1} is a closed-loop pole. Often, 2nd-order approximations are used to decide
an appropriate s_{1}. For this example, the value for s_{1} is chosen rather arbitrarily for
illustration, without worrying about time-domain performance. The point s_{1} is

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First, we will compute the magnitude and phase of the transfer function G_{p}(s), evaluated
at the point s = s_{1}. Graphically, we can draw vectors from each of the open-loop poles and
zeros to the point s_{1}. The magnitude of G_{p}(s) at s_{1} will be the gain (8) times the lengths of
each of the vectors drawn from zeros to s_{1} divided by the lengths of each of the vectors
drawn from poles to s_{1}. In MATLAB, the "polyval" function can be used to evaluate the
numerator and denominator polynomials at s_{1}, and the "abs" and "angle" functions can be used
to determine the magnitude and phase.

Evaluating G_{p}(s) at s = s_{1}

At the point s = s_{1}, G_{p}(s) has a **magnitude of 1.0346** and a **phase of -150.6201
degrees**. Since the phase shift is not an integer multiple of 180 degrees (either even or
odd), then the root locus of G_{p}(s) does not pass through s_{1} for either K > 0 or K < 0.

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Since G_{p}(s) has 5 poles and 3 zeros, there will be 2 branches of the root locus which go
to infinity as the magnitude of K goes to infinity. The directions that these branches
take depends on the sign of K. The root locus plots assume that there is a variable gain K
in series with G_{p}(s). The root locus plots for K > 0 and K < 0 are provided. On each plot,
the point s_{1} is also shown. The plots obviously show that neither of the plots pass
through s1.

Root Locus for G_{p}(s), K > 0

Root Locus for G_{p}(s), K < 0

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Since the phase shift of G_{p}(s) at s = s_{1} is approximately -150 degrees, the easiest way
to satisfy the phase angle criterion is to add a -30 degrees of phase shift, measured at
s_{1}.
Since poles provide negative phase shift (since they are in the denominator), the minimum
amount of adjustment to G_{p}(s) would be one pole placed so as to give approximately -30
degrees of phase shift at s_{1}, and a gain which could be adjusted to satisfy the magnitude
criterion. A simple bit of trig shows that one pole located at s = -7.3285 provides the
-29.3799 degrees necessary to make the root locus of G_{c1}(s)G_{p}(s) pass through
s_{1}. The
magnitude of G_{c1}(s)G_{p}(s) without any gain is 0.1692. Therefore, the compensator gain must
be 5.9103 so that the magnitude criterion is satisfied at s_{1}. The transfer function for
the compensator G_{c1}(s) is shown below. The root locus of G_{c1}(s)G_{p}(s) is plotted, and it
is seen that the root locus of the compensated system does pass through s_{1}. Since the gain
has been chosen to satisfy the magnitude criterion at s_{1}, then s_{1} is an actual closed-loop
pole for the compensated system.

Root Locus for G_{c1}(s)G_{p}(s), K > 0

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The transfer function G_{c1}(s) is the minimum configuration required to work with
G_{p}(s) to
make the specified s_{1} a closed-loop pole. Notice that the total number of poles in
G_{c1}(s)G_{p}(s) is now 6, while the number of zeros is still 3. Since there are now 3 more
poles than zeros, there will be 2 branches of the root locus which move into the right-half
of the s-plane with increasing gain. This might not be desired. An alternate way of
achieving the necessary phase shift is to use a compensator with 1 zero and 1 pole, with
their locations chosen so as to satisfy the phase angle criterion at s_{1}. With this approach,
the excess number of poles over zeros is not increased. These locations are not unique,
although there are constraints on the locations. The gain of the compensator would again
be chosen to satisfy the magnitude criterion at s_{1}. Details of the design procedure will be
covered later. For a more detailed root locus design example, see Root Locus Design Example #1.

Root Locus for G_{c2(}s)G_{p}(s), K > 0

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