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? enter image description here

  • $\begingroup$ Yes, it appears to be the right approach. You can edit the question to include the expression for the closed loop transfer function. $\endgroup$
    – AJN
    Jun 18, 2023 at 12:46
  • $\begingroup$ 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. $\endgroup$
    – LukaM
    Jun 18, 2023 at 13:41

1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.