The Attempt at a Solution
For the above modeled transfer function of the plant, 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?