I'm trying to understand how to obtain the Kp, Ki, Kd values after finding a combination of K and a that works for me. Do I just expand the equation and take the coefficients?
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$\begingroup$ I just expanded the equation. I found that Kp=2Ka, Ki=K(a^2) and Kd=K $\endgroup$– OzymandiasMar 7, 2015 at 21:56
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$\begingroup$ Yes, your expansion appears to be correct. But the equation form you mentioned will not be able to create every PID because you only have two variables. $\endgroup$– fibonaticMar 8, 2015 at 6:05
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$\begingroup$ Then, if your aim is to realize this PID in the SW, good luck :). The formula in the title is not causal. Why don't you rephrase the title and the question so that the question does not assume that the problem is stated only in the title? $\endgroup$– Ugo PattaciniMar 8, 2015 at 9:18
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2 Answers
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It is in frequency domain instead of time domain.
$$ G(s)=\frac{K(s+a)^2}{s} = \frac{Ks^2+2Kas+Ka^2}{s} $$
according to the Laplace form of the PID controller $$ G(s)=\frac{K_ds^2+K_ps+K_i}{s} $$
so $$ K_d = K, K_p = 2Ka, K_i = Ka^2 $$
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The general PID form in Laplace domain is: $$ G(s)=\frac{K_ds^2+K_ps+K_i}{s} $$ The equation you've mentioned has a $Ks^3$ in the numerator. Hence, it cannot be transformed into PID form unless $K=0$, but that does not do you any good I guess.
