Introduction to Root Locus.

Cuthbert Nyack

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 condition.
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 locus.

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 real axis.
7. Loci leave the real axis where dK/ds = 0 and angle condition is satisfied.
8. Intersection of loci with imaginary axis can be found by the Routh Criterion.

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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