# Parameter Design By The Root Locus Method

**Parameter design by the root locus method**

Initially, the root locus method was developed to determine the locus of roots of the characteristic equation as the system gain, K, is varied from zero to infinity. However, as we have seen, the effect of other system parameters may be readily investigated by using the root locus method. Fundamentally, the root locus method is concerned with a characteristic equation, which may be written as

Then the standard root locus method may be applied. The question arises: How do we investigate the effect of two parameters, α and β? It appears that the root locus method is a single-parameter method; fortunately, it can be readily extended to the investigation of two or more parameters. This method of parameter design uses the root locus approach to select the values of the parameters.

The characteristic equation of a dynamic system may be written as

Hence, the effect of the coefficient ax may be ascertained from the root locus equation

If the parameter of interest, α, does not appear solely as a coefficient, the parameter may be isolated as

For example, a third-order equation of interest might be

To ascertain the effect of the parameter α, ωc isolate the parameter and rewrite the equation in root locus form, as shown in the following steps:

Then, to determine the effect of two parameters, we must repeat the root locus approach twice. Thus, for a characteristic equation with two variable parameters, α and β, we have

The two variable parameters have been isolated, and the effect of α will be determined. Then, the effect of β will be determined. For example, for a certain third order characteristic equation with α and β as parameters, we obtain