Making a plant stable with a compensator based on given requirements

Question

I have a transfer function of the plant modeled as the following:

I'm trying to design a compensator that satisfies some requirements.

a. The maximum control bandwidth (0 dB crossover frequency) is 100 rad/s.

b. The minimum phase margin at crossover is 30 deg.

c. The loop transmission magnitude at 2000 rad/s must be less than -6 dB.

d. The step response overshoot of the closed loop system must be less than 20%.

e. The 2% settling time must be less than 0.75 s.

f. The steady-state error to the unit step reference is less than 1%.

Open loop poles reside in -0.05 +/- i and - 5 +/- 2000i, so there is a lot of potential for instability for some variable gain K. This means that I would need a two zeros for the poles to sink in.

Designing a compensator that makes this system stable is easy, I can throw in two zeros in the LHP such as s^2 + 2 s + 8. What I'm not too sure on is the method of satisfying the requirements. Normally, I'd use a PID controller to tune the settling time, step response etc. But in this system, a PID controller cannot be applied, because the addition of pole would make this system unstable. (Edit: Well I guess I could use PID, as long as I define the bounds for gain K, but this seems like a crummy solution as the range of K would be really small)

What are some alternative methods of achieving the requirements while making the system stable?

Answer 1

Your intuition to use a compensator, not a PID, is correct in this instance as you need to meet gain and phase margin requirements, which is generally what compensators are for--you can easily get a. and b. using the right compensator. The gain crossover frequency in the Bode/Nyquist plot is roughly the reciprocal of the timeconstant for the response--e., the steady state error can be solved for using any controller by taking the DC limit--f. Overshoot always has to do with damping so you could work out what the effective damping ratio of the CLTF is and constrain it that way--d. Finally, I've never heard of c.

Another approach which is likely to work with much less effort is to select a compensator, put the poles/zeros where you get a good frequency response for a/b, then tune the poles/zeros around these points, checking the response characteristics using a simulation.