The plant Bode plot is shown in the postscript file: leadplant.pdf. In our example, we specify a desired phase margin of 30 degrees. Therefore, we must place the gain crossover frequency such that the phase angle is -150 degrees or more at the Gain=1 crossing. As we can quickly see in leadplant.pdf, a P-controller would have a gain of about K = 1. We can do a lot better with lead compensation.

All lead compensators in this example have a ratio pole/zero = 10, and a unity steady-state gain. The lead compensator must be FAST in order to be effective. The design command file of Lab 13 prompts you for the 'desired gain crossover frequency' that is, the frequency at which we wish to cross the Gain = 1 or 0 dB axis in the Magnitude plot. The command file places this frequency at the geometrical CENTER = Sqrt(p*z) of the Lead compensator! The three following files demonstrate the effect of designing a lead compensator for gain crossover frequencies of 1 rad/s leadcomp1.pdf. 2 rad/s, leadcomp2.pdf. , and 3 rad/s, leadcomp3.pdf. Clearly, the FASTER the compensator, the better the performance as measured by the ability to raise the loop gain K. The fastest and best compensator in this example raises the loop gain by 22 dB or a factor of 12.6. Compare this with a simple P-controller that would give us a gain of K = 1 only!

We cannot make the lead compensator too fast, though, lest we lose some of its benefits. We'll settle for a center frequency of 3 rad/s here. Lastly, we must raise the compensator gain so that the gain crossover is at the desired phase margin of 150 degrees. The amount of gain increase possible is already indicated in leadcomp3.pdf as K = 12.5. After increasing the gain K (using the Lab 13 command file), we can view the final design in leadcompfin.pdf.