Introduction to Root Locus.
Root Locus analysis seeks to find the behaviour of the closed loop poles
of a control system as some parameter is adjusted. This parameter is
usually the gain but can also be controller variables.
Here we will be concerned with systems with unity feedback H(s) = 1
and forward transfer function KG(s). The open loop poles and zeros are
therefore the open loop poles and zeros of G(s) and the closed
loop poles are determined by the zeros of the characteristic equation
1 + KG(s) = 0.
Since s is complex this condition can be represented as 2 conditions
one |KG(s)| = 1 on the magnitude of KG(s) and one arg[KG(s)] =
± (2n + 1)pi on the phase angle of G(s).
One of the quickest ways of finding the root locus is to find the
points where the angle condition is satisfied. Having found such a point
then the value of K corresponding to that point can be found by
substituting the s coordinates
of the point into the magnitude condition.
In the following pages applets are used to
illustrate the general behaviour of the root locus
of some simple pole-zero systems using the angle and magnitude
In finding the root locus one is interested in (a) whether or not the
root locus crosses the imaginary axis (unstable system) and (b) to
what extent the root locus can be moved away from the imaginary axis
by the addition/movement of poles and zeros to G(s).
Some simple rules to help locate the root
1. Loci start at the open loop poles and end at the open loop
zeros or at inf.
2. Number of loci must be equal to the number of poles.
3. A point on the real axis is on the loci if the number of poles and zeros
on the real axis to its right is odd.
4. Loci ending at inf do so at angles to the +ve real axis ginen by
(2k + 1)pi/(n(p) - n(z)), k goes from 0 to (n(p) - n(z)). n(p) is the
number of poles and n(z) is the number of zeros.
5. Asymptotes intersect the real axis at the c.g. of the pole-zero
configuration, ie [Sum(numerical value of poles) - Sum(numerical
value of zeros)]/([n(p) - n(z)].
6. If coefficients of G(s) are real, then locus is symmetric about the
7. Loci leave the real axis where dK/ds = 0 and angle condition
8. Intersection of loci with imaginary axis can be found by the
Designing a system by the root locus approach means setting the system parameters so that the closed loop poles are at locations which produce acceptable dynamical behaviour for the system.
The first step in designing a system by the root locus approach means finding the roots as K is varied. If adjusting K is not enough to
set the poles at at the desired locations, then open loop poles and or zeros
may have to be added by controllers or compensators to achieve the
desired closed loop pole locations.
Gif images of the applets shown below indicate how the root locus depends on the number of poles and zeros and their locations. They
give a good starting point to decide how poles and zeros may be
added to achieve desired closed loop pole locations. The applets
also show the transient and closed loop frequency response to assist
in locating desirable pole locations.
When activated the appearance of some of the applets are shown below.
In the applet below the closed loop poles for any K can be found by
setting K to that value and noting the starting point of the loci.
Applets above and below not only show the root locus but the step
response is shown in white and the frequency response in light
magenta(vertically oriented). The frequency scale is linear and the range is the same as the range of the V axis.
The applet below locates the closed loop poles by locating the zeros
of the characteristic equation.
The applet below shows another way in which the closed loop can be found. It is by locating the point on the angle condition where the
magnitude condition is satisfied.
The applet below shows another way in which the closed loop can be found. It is by locating the point on the magnitude condition where the
angle condition is satisfied.
In the applet below poles and zeros can be entered in the s plane
by clicking the mouse in the appropriate location.
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COPYRIGHT © 1999 Cuthbert A. Nyack.