Parameter Design By The Root Locus Method

Parameter design by the root locus method

In this particular case, the parameters appear as the coefficients of the characteristic equation. The effect of varying (3 from zero to infinity is determined from the root locus equation

We note that the denominator of the above equation is the characteristic equation of the system with β = 0. Therefore, we must first evaluate the effect of varying a from zero to infinity by using the equation

where β has been set equal to zero in the third order characteristic equation. Then, upon evaluating the effect of α, a value of α is selected and used with the root locus equation to evaluate the effect of β. This two-step method of evaluating the effect of α and then β may be carried out as two root locus procedures. First, we obtain a locus of roots as α varies, and we select a suitable value of α; the results are satisfactory root locations.

Then, we obtain the root locus for β by noting that the poles of the root locus equation are the roots evaluated by the root locus of equation above. A limitation of this approach is that we will not always be able to obtain a characteristic equation that is linear in the parameter under consideration (for example, α).

Fig 1: Root loci as a function of αand β(a) Loci as αvaries (b) Loci as βvaries for one value of α= α₁

To illustrate this approach effectively, let us obtain the root locus for α and then β for the third order characteristics equation. A sketch of the root locus as α varies for the equation is shown in Figure 1(a), where the roots for two values of gain α are shown. If the gain α is selected as α₁, then the resultant roots of the equation become the poles of the root locus equation.

The root locus of Equation as β varies is shown in Figure 1(b), and a suitable β can be selected on the basis of the desired root locations. Using the root locus method, we will further illustrate this parameter design approach by a specific design example.