Time Response of Overdamped System

[CognitiveNet]

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?

 

[anorlunda]

Not bad. You used some approximate assumptions, but it should work fairly well.