I need to design a controller for a system, need to choose between P, PI, PD, and PID, given only the desired ss, settling, and overshoot %

Question

(Apologies for terrible formatting, first post and don't know how to make equations look nice)

The system is

$$G(s)=\frac{10}{4s^2+7s+23}$$

and the requirements are steady state error of zero, $10ms$ settling time, and at most $5\%$ overshoot. I think I understand how to find majority of these parameters, I'm just struggling with actually finding the $K_p$, $K_i$, and $K_d$ values for the controller.

Looking at steady state it seems I need either a PI or PID, as the others do not achieve $ss=0$.

The settling time and overshoot were used to find $ξ$ and $ω$, using $t=4/ξω$ and $$ξ=-\frac{\ln M}{\sqrt{\pi^2+\ln^2 M}}$$ with $M=0.05$, to give $ξ=0.69$ and $ω=579.6$.

My question now is how do I translate that into the variables used in the PID controller. Using some information previously given and posts here it looks like I need to find the characteristic equation using something like $(s+p)(s^2+2ωξs+ω^2)$, as the closed loop TF is third order (I think?), but everytime I try I end up with wild results that definitely do not give the correct step response.

Unless I'm wrong, the controller TF is either $(K_ps+K_i)/s$ or $(K_p s+K_i+K_d s^2)/s$, depending on whether using PI or PID, which would make the closed loop TF:

$$\frac{10 K_p s+K_i}{4s^3+7s^2+23s+10K_p s+10K_i}$$

OR

$$\frac{10K_p s+K_i+K_d s^2}{4s^3+7s^2+23s+10K_p s+10K_i+K_d s^2}$$

Again, depending on what controller. Either way I'm stumped. TIA.

Answer 1

After deriving for closed-loop transfer function including controller, you need to find dominant poles of your system and aproximate your system as second-order system. The you can use root lous or pole placement to fit your specification.