SKEDSOFT

Control Systems - 1

Root-contour plots

In the above equation, the parameter a is written as a multiplying factor. For a given value of K, the effect of a on the closed-loop poles can be investigated from the equation. The root contours for this system can be constructed by following the usual procedure for constructing root loci.

We shall now construct the root contours as K and a vary, respectively, from zero to infinity. The root contours start from the poles (at s = ±j√K) and terminate at the zeros (at s = 0 and infinity).

 

Fig: 2

 

We shall first construct the locus of roots when a = O. This can be done easily as follows:

Substitute a = 0 into the characteristics equation, Then

 

 

The open-loop poles are thus a double pole at the origin. The root-locus plot of the above equation is shown in Figure 2(a).

To construct the root contours, let us assume that K is a constant; for example, K = 4. Then the characteristics equation becomes

 

 

The open-loop poles are s = ±j2.The finite open-loop zero is at the origin. The root locus plot corresponding to the above equation is shown in Figure 2(b). For different values of K, the equation yields similar root loci. The root contour, the diagram showing the root loci corresponding to 0 ≤ K≤ ∞, 0≤ a ≤ ∞, can be plotted as in Figure 2(c). Clearly, the root contours start at the poles of and end at the zeros of the transfer function as/(s2 K). The arrowheads on the root contours indicate the direction of increase in the value of a.

The root contours show the effects of the variations of system parameters on the closed loop poles. From the root-contour plot shown in Figure 2(c),we see that, for 0 < K < ∞, o< a < ∞, the closed-loop poles lie in the left-half s plane and the system is stable. Note that if the value of K is fixed, say K = 4, then the root contours become simply the root loci, as shown in Figure 2(b).

We have illustrated a method for constructing root contours when the gain K and parameter a are varied, respectively, from zero to infinity. Basically, we assign one parameter a constant value at a time and vary the other parameter from 0 to ∞ and sketch the root loci. Then we change the value of the first parameter and repeat sketching the root loci. By repeating this process we can sketch the root contour.