Root Sensitivity Of A Control System
Root sensitivity of a control system
Therefore, to obtain the effect of reducing β, we determine the zero-degree locus in contrast to the 180° locus, as shown by a dotted locus in Figure 3. To find the effect of a 20% change of the parameter β, we evaluate the new roots for Δβ = ±0.20, as shown in Figure 3. The root sensitivity is readily evaluated graphically and, for a positive change in β, is
Fig: 3 The root locus for the parameter β
As the percentage change Δβ/β decreases, the sensitivity measures Sriβ and Sriβ- will approach equality in magnitude and a difference in angle of 180°. Thus, for small changes when Δβ/β < 0.10, the sensitivity measures are related as
Often, the desired root sensitivity measure is desired for small changes in the parameter. When the relative change in the parameter is of the order Δβ/β = 0.10, we can estimate the increment in the root change by approximating the root locus with the line at the angle of departure θd. This approximation is shown in Figure 3 and is accurate for only relatively small changes in Δβ. However, the use of this approximation allows the analyst to avoid sketching the complete root locus diagram. Therefore, for Figure 3, the root sensitivity may be evaluated for Δβ/β = 0.10 along the departure line, and we obtain
The root sensitivity measure for a parameter variation is useful for comparing the sensitivity for various design parameters and at different root locations. Comparing Equation above for β with the equation of the root sensitivity for r1 for a, we find (a) that the sensitivity for β is greater in magnitude by approximately 50% and (b) that the angle for Sriβ- indicates that the approach of the root toward the jω-axis is more sensitive for changes in β. Therefore, the tolerance requirements for β would be more stringent than for α. This information provides the designer with a comparative measure of the required tolerances for each parameter.