# Root Sensitivity Of A Control System

**Root sensitivity of a control system**

*Fig: 1 A feedback control system.*

The characteristic equation of the feedback control system shown in Figure 1 is

The gain K will be considered to be the parameter α. Then the effect of a change in each parameter can be determined by utilizing the relations

where α_{0} and β_{o} are the nominal or desired values for the parameters α and β, respectively. We shall consider the case when the nominal pole value is β_{0} = 1 and the desired gain is α_{0} = K = 0.5. Then the root locus can be obtained as a function of α = K by utilizing the root locus equation

*Fig: 2 The root locus for K*

The nominal value of gain K = α_{0} = 0.5 results in two complex roots, —r_{i} = -0.5 *j*0.5 and —r_{2} = , as shown in Figure 2. To evaluate the effect of unavoidable changes in the gain, the characteristic equation with α = α_{0} ± Δα becomes