I have a block function like this and I need to calculate parameters a and K so the system will be overcritically damped. I calculated the closed-loop transfer function and I believe that by comparing the denominator of the closed-loop transfer function
$$ \frac{0.5K(s+a)}{s^2+s(3+0.5k)+0.5ka} = \frac{1}{s^2+2\xi ws + w^2}$$
, I should be able to calculate the a and K parameters, when $\xi$ is overcritically damped but I can't solve the system of equations. Is this the right approach, or should I try something else?
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$\begingroup$ Yes, it appears to be the right approach. You can edit the question to include the expression for the closed loop transfer function. $\endgroup$– AJNJun 18 at 12:46
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$\begingroup$ I added the denominator of the closed loop function and compared it to the general equation for calculating the damping ratio and w. But I am still struggling with how to resolve the system of equations that I get from the second and third part of the equation when comparing the with the right equation. $\endgroup$– LukaMJun 18 at 13:41
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1 Answer
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Ignore the numerator for the time being. The criticality of damping is dependent on the denominator. The coefficient of $s^2$ already matches on both sides.
Assuming that the expressions given in the question are correct, and that $K, a>0$, $$ \omega^2 = K\cdot a/2 \implies \omega = \sqrt{(K\cdot a/2)} $$ and $$ 2\xi\omega = 3+K/2 \\ \implies \xi = \frac{(3+K/2)}{2\cdot\sqrt{(K\cdot a/2)}} \ge 1 \\ \implies (3+K/2)^2 \ge 2 K\cdot a\\ \implies K^2/4 + 3K + 9 - 2K\cdot a \ge 0 $$
Solve the equality to get critical values of $K$. It will divide the values of $K$ into 3 ranges. Check each range manually for super criticality.
I have not checked the above for correctness. Be particularly careful with inequalities and square roots and squarings.
