Dear PF Mentor, this is NOT homework assignment! This my own personal research intended to use a theoretical approach to develop a transfer function for an overdamped system with a low settling time. This could be used for creating faster robots. Step 1: Initializing the transfer function G(s) = 1 / ((s+A) (s+B)) G(s) = 1 / (s^2 + (A+B)s + AB) if A = B G(s) = 1 / (s^2 + 2As + A^2) G (s) = (1/(A^2)) / (s^2 + (2A/A^2)s + 1 ) Step 2: G = 1 / ( (1 / (natural frequency)^2)s^2 + (2* (damping ratio) / (natural frequency))s + 1 Step 3: Finding the values of the quadratic equation a = 1 b = 2 * (damping ratio) c = (natural frequency)^2 Step 4: The equation for the settling time of a second order system is Ts = 4 / (damping ratio * settling time). May I remind you that in this case, the system is not critically damped, because the damping ratio exceeds 1. I'm choosing Ts = 1. Ts = 1 = 4/(damping ratio * natural frequency) => damping ratio = 4/ natural frequency Step 5: Criteria: b^2 > 4ac = (2*damping ratio)^2 > 4*(natural frequency)^2 i.e. (2*damping ratio)^2 = 2*(4*(natural frequency)^2) This way, I avoid having complex poles in my system. Step 6: Finding the damping ratio and the natural frequency I replace the damping ratio with (4 / natural frequency) to find the actual value of the natural frequency. This is called substitution in mathematics: (2*(4 / natural frequency))^2 > 4*(natural frequency)^2 natural frequency = 8^(1/4), in other words; the fourth root of 8. Thus the damping ratio = 4 / natural frequency = 4 / 8^(1/4) Makes sense? Step 7: Now I can insert the values for the damping ratio and natural frequency into step 3 to find b and c. G(s) = 1 / (s + 8^(1/4)) + (s + 8^(1/4)) Thus, I've found the second order transfer function of an overdamped system with a settling time of 1 second, where A = B and b^2 > 4ac. Do you agree?