Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Time Response of Overdamped System

  1. Oct 31, 2012 #1
    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? :wink:

    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?
     
    Last edited: Nov 1, 2012
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Similar Discussions: Time Response of Overdamped System
  1. Control - response time (Replies: 10)

Loading...