We will assume that you want to control a system with a transfer function:

That is the transfer function for G(s) in the block diagram below.  You will probably need to design a compensator for the controller - and determine the value for the gain.

We also assume that you have the following specifications:

• Phase Margin must be at least 40^o.
• The SSE should be less than 5%.
• Need to know the best response time possible.

In this plot we have drawn lines at -140^o and -180^o.  We can see that the phase plot intersects -140^o at a frequency of about 0.25 Hz.  If we set the zero db crossing there, the phase margin will be 40^o.  To get the magnitude plot to zero db at f = -.25 requires a gain of about 12 db.  Since we already have a DC gain of 6 db that will move the overall magnitude plot up so that the DC gain will be 18db - or a gain of 8.  That gives a SSE of 1/[1 + 8] or 11%.  Here's the plot with a vertical line drawn at the frequency where the phase is -140^o so you can see how it fits together.

Now, for this system, we find the following.

• If we set the zero db crossing at .25 Hz, we have to add 12 db.  Since there is a DC gain of 6db, that will raise the DC gain to 18 db, or a gain of 8.  The means SSE = 11%.
• The zero db crossing at .25 Hz means that the rise time is .35/.25 = 1.4 sec.

        Here is what we will do.

• We assume that we want the zero db crossing at the same frequency, i.e. 0.25 Hz.
• We will add a compensator that adjusts the phase to increase the phase margin at that frequency, and examine how that affects SSE.
• If we want maximum effect from the compensator at .25 Hz, we need the phase peak at that frequency.  Let's put the pole at 0.75 Hz, (i.e. 3x.25 Hz) and the zero at .083 Hz (i.e. .25/3 Hz)
• NOTE:  Choosing the pole and zero that way puts the geometric mean of the two frequencies right at 0.25 Hz, and that is where the maximum lead will be obtained.  That frequency is shown on the plot below with a vertical black line.

• Notice the following about this system.
• At f = 0.25, the phase is now at -90^o.
• At f = 0.25, the magnitude plot is at -3 db.
• If we keep the open loop zero-db crossing at f = 0.25, we can get the following.
• A 90^o phase margin - a substantial improvement
• 9 db of DC gain (since we only need to raise the magnitude plot by 3db to get to 0 db at f = 0.25 Hz.  That's a DC gain of 2.8, and we will have substantially worse performance for SSE.
• We could raise the frequency of the open loop zero-db crossing to f = 0.5 Hz, with the following effects.
• We would be back to a 40^o phase margin - the same relative stability as we had originally.
• We have the magnitude plot at -9db at f = 0.5, so we can raise the magnitude plot by 9db, giving a DC gain of 15 db for a DC gain of about 5.6 - not as good as originally.
• We would have a system twice as fast as the original system because the open loop zero-db crossing is twice as high.