# 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 %

(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.

• "I think I understand how to find majority of these parameters" - what parameters did you find? Steady state 0 is a pretty simple constraint- at t=inf you will be where you want, exactly. The rest you should be able to model or calculate given a step input- how long does it take to settle and what is its overshoot when where you want to be suddenly shoots from one value to another?
– Abel
Commented Oct 25, 2023 at 11:08
• correct, you need the "I" term to drive steady state error to 0. You could solve this by factoring the denominator of G to get the plant's poles, then apply a design technique. The Bode graphical method is the easiest, and should work well here. Commented Jan 14 at 21:44