The open-loop transfer function for a particular system is described by the transfer
function
The open-loop poles and zeros of the system are fixed, but the gain K can be varied. One question of concern is, "How does varying K affect the closed-loop stability of the system?"
The closed-loop characteristic equation is given by the denominator of the closed-loop
transfer function. The roots of that equation are the closed-loop poles. The characteristic
equation is
From this equation, it is evident that the locations of the closed-loop poles are functions of the locations of the open-loop poles (from D(s) = 0), the locations of the open-loop zeros (from N(s) = 0), and the value of the gain K. As K is varied, the locations of the closed-loop poles also vary. For a system with n open-loop poles (6 in this example), there will be n closed-loop poles. The root locus is a plot of the locations in the s-plane of all n closed-loop poles as K varies from 0 to positive or negative infinity. (Separate plots are generally made for K > 0 and K < 0.)
The closed-loop poles are those points in the s-plane that make the closed-loop characteristic
equation equal to zero. Therefore, a point s = s1 is a closed-loop pole if and only if the
following relationships hold.
Since the phase angle of the real number K depends only on the sign of K, we will make separate plots for K > 0 and for K < 0. With this approach, the phase angle of K is fixed for a particular plot, and only the phase angles of N(s) and D(s) vary as we evaluate the phase shift of G(s)H(s) at different points in the s-plane.
A root locus plot consists of n lines (branches) in the s-plane. Each branch represents
the movement of 1 closed-loop pole as K is varied in value. A particular value of
K makes 1 specific point on each of the n branches be a
closed-loop pole. Likewise, a particular point on any
branch of the root locus corresponds to some value of K, with the appropriate sign.
If a point s =
s1 satisfies the phase angle criterion, then that point is a
potential closed-loop pole. That point will be a closed-loop pole if the gain K
is chosen to satisfy the magnitude criterion at that point, and used with the
appropriate sign. If the point s = s1 does not satisfy
the phase angle criterion, then that point is not on the root locus and will not
be a closed-loop pole for any real value of K.
Since
only the magnitude of K varies on the root locus (phase of K is fixed), an alternate
definition of the root locus is that it is a map of all points in the s-plane where the
phase angle of the open-loop system takes on the appropriate value (odd integer multiple of
180 degrees if K > 0, and even integer multiple of 180 degrees if K < 0).
The root locus plots for the transfer function shown above are made for K > 0 and for
K < 0.
For each of the plots, note what parts of the real axis are on the root locus, where the
break-away and break-in points lie on the real axis, the angles of the asymptotes, the
crossings of the imaginary axis, and the closed-loop stability of the system as the gain is
varied.
Root Locus for K > 0
Root Locus for K < 0
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Lastest revision on
Wednesday, June 7, 2006 12:15 PM
