How to design a Controller that would adjust the steady-state error of a system?

Question

I have a simple model (input-plant-output) with a transfer function : $$ T(s)=\frac{4}{s^{2}+2s+4} $$ The goal of this problem is to design a system such that it has the following specification for a step input:

1. $\%\;\text{overshoot}\leq6\%$
2. $t_{s}<1\;\text{sec}$

Moreover, when the system is given a ramp input, it must have a steady-state error of at most $0.1$

My strategy :

First try : I first thought of adding a Lag-Lead compensator and tune the poles but this appears to force the step-response to be prone to heavy oscillations. Moreover, the settling time and the maximum overshoot were far off from the desired values.

Second try : I decided to add a PI controller to the lag-lead compensator and this helped in decreasing the maximum overshoot and the oscillations in general but it did not resolve the issue of the settling time and in fact the settling time was much far.

I would hope for the best optimal method to use that would be able to satisfy both requirements if possible because I am not aware if it might not be possible to satisfy both requirements (i.e., step input requirement and the ramp input requirement)

Answer 1

I'm not gonna give you a complete answer but I can help you.

Your open-loop transfer function has 2 poles at $ -1 ±j \sqrt(3) $

Good news, they are stable. However, they are not damped and they are slow.

Your strategy is this

1 - Damp the poles, this will reduce the overshoot.
2 - Try to increase the poles frequency. This will reduce the settling time.
3 - Your allowed steady-state error to a ramp input is less than 0.1. It means that the steady-state error to a step input must be 0. You need a PI controller since your open-loop transfer function has no integrator.
4 - Another strategy cancel the poles in your controller and add a simple PI controller. This will allow you to shape the time response as needed. It should be good enough for a homework. In real-life, you should check for robustness.

Thanks to Tim Wescott for pointing out my original mistake about the steady-state error.