I need to design a prefilter and PID controller for a plant whose transfer function is given by
$$G(s)=\frac{3}{s(s^2+4s+5)}$$
subject to the following performance requirements:
An acceleration error constant of $K_a=2$.
Phase margin equal to $45$ degrees.
A bandwidth greater than $2.8\ \text{rad}/s$.
I only have an idea about how to use the acceleration error constant to calculate the $K_i$ to be $3.3333$ based on the theory of steady-state error, but I do not know how to use it for the other two requirements.
The issue of compensation is quite broad, there are several techniques to address it. Here we will stick to the basic concepts of closed loop stability.
Calling
$$ G(s) = \frac{3}{s(s^2+4s+5)}\\ PID(s) = \frac{K_i}{s}+s K_d + K_p $$
we have the stability conditions
$$ \left|G(j\omega_c)PID(j\omega_c)\right| = 1\\ \angle G(j\omega_c)PID(j\omega_c) = -\pi+\phi $$
where $\omega_c$ defines practically the bandwidth and $\phi$ the phase margin
Considering also the steady state acceleration error as
$$ \lim_{s\to 0}s \left(1-\frac{G\cdot PID}{1+G\cdot PID}\right)\frac{1}{s^3} = \frac{5}{3K_i} $$
we have three conditions and three parameters to determine or
$$ \frac{9 \left(K_d^2 \omega _c^4+\omega _c^2 \left(K_p^2-2 K_d K_i\right)+K_i^2\right)}{\omega _c^4 \left(\omega _c^4+6 \omega _c^2+25\right)}=1\\ \frac{\omega _c \left(\omega _c^2 (K_p-4 K_d)+4 K_i-5 K_p\right)}{K_d \omega _c^4+5 K_i-\omega _c^2 (5 K_d+K_i-4 K_p)}=-\tan\phi\\ \frac{5}{3K_i}= \frac{1}{K_a} $$
now with $\phi = \frac{\pi}{4}, \omega_c = 3, K_a = 2$ solving for $K_p,K_d, K_i$ we obtain
$$ K_p = 5.66, K_d = 4.14, K_i = 3.33 $$
Follows the close loop response showing in in red $G(s)$ and in blue $PID(s)G(s)$
The overshoot issue can be handled with a proper prefilter and we leave it to the reader.