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(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.

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    $\begingroup$ "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? $\endgroup$
    – Abel
    Oct 25, 2023 at 11:08
  • $\begingroup$ 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. $\endgroup$
    – Pete W
    Jan 14 at 21:44

1 Answer 1

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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.

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