**Subject :**Advance Control System

## Response Speed from Open-Loop Frequency Response

**Response Speed from Open-Loop Frequency Response**

Equations (1) and (2) relate the closed-loop bandwidth to the desired settling or peak time and the damping ratio. We now show that the closed-loop bandwidth can be estimated from the open-loop frequency response. From the Nichols chart in Figure 1, we see the relationship between the open-loop gain and the closed-loop gain. The M = 0.707(-3 dB) curve, replotted in Figure 2 for clarity, shows the open-loop gain when the closed-loop gain is —3 dB, which typically occurs at ω_{BM} if the low-frequency closed-loop gain is 0 dB. We can approximate Figure 2 by saying that the closed-loop bandwidth,ω_{BM} (the frequency at which the closed-loop magnitude response is -3dB), equals the frequency at which the open-loop magnitude response is between —6 and —7.5 dB if the open-loop phase response is between —135° and —225°. Then, using a second-order system approximation, Eq. (1) and (2) can be used, along with the desired damping ratio, ζ, to find settling time and peak time, respectively.