The bode plot given is corresponding to a proportional contr

The bode plot given is corresponding to a proportional control system with K - 20. Estimate the stability margins and the associated crossover frequencies. omega_cg = GM = omega_phi = PM = Is the closed-loop system with K - 20 stable or unstable? What is the range of K for which the closed-loop sy stem is stable? Give necessary explanation.

Solution

GAIN MARGIN
- Find the frequency where the PHASE becomes -180 degrees = 5 rad/s approx.

Find the GAIN, G (in dB), at this SAME FREQUENCY (from the upper plot) = -8 dB

Then, we define the GAIN MARGIN as:

Gain Margin = 0 - G dB   = 0 - -8   = 8 dB

PHASE MARGIN
- Find the frequency where the GAIN is 0 dB = 0.8 rad/s

Find the PHASE, P (in degrees), at this SAME FREQUENCY = -94 deg

we define the PHASE MARGIN as:

Phase Margin = +P + 180 degrees = -94 + 180 = 86 deg

Yes the system is stable because both GM and PM are positive.