Determining Gain Value For Stability From A Bode' Plot.

Assume that you have the feedback control system shown below. In this system you are using proportional control, and you have a gain, Kp, that you can adjust. (If you are using integral control, or something else, you can still put it into this framework.)

If you have a Bode' plot for the system system being controlled (G(s)), then you can determine the gain at which the system becomes unstable.

Here is a Bode' plot for a simple system.

The magnitude starts off at 20db (a gain of 10) at low frequencies. The phase starts at zero degrees and goes to -270o. The form of the plot is one that could encircle -1. (It's probably a rosette.)

o When all else fails, check what the Nyquist plot looks like. Try to sketch it.

To calculate the gain, do the following.

Determine the frequency at which the phase crosses -180o. o That frequency is about 0.28 on this plot.

Determine the gain at that frequency. o On this plot, the gain is about 1 db.

Modify the gain so that the plot goes through 0 db at this frequenc. o If the gain is larger than 1 (0 db), you need to decrease the gain. o If the gain is less than 1, you need to increase the gain. o In this case, you need to decrease the gain by 1 db, so you need a gain

of -1 db, i.e Kp = 0.89.