I have an issue with simulink, basically it's to do with a second order system, well two first order systems in series. As I understand it as you increase the damping factor (above 1), the system should respond slower and be more sluggish. The damping factor = (tau1+tau2)/(2root(tau1*tau2)). So, looking at this system with damping factor 1(with it's response): enter image description here

and this system with damping factor 1.19 and it's response: enter image description here

What gives?! The system with the damping factor 1 hasn't even reached steady state by 250 secs while the system with damping factor 1.19 has had a faster response, why is this happening?

Thanks for reading.

  • $\begingroup$ bump can someone please help $\endgroup$ Jan 17 '16 at 17:50

Time Constant and systems

A second order LTI system in Laplace domain:

$\hspace{2.5em}$ $H(s) = \frac{{\omega_{n}}^{2}}{s^{2}+\zeta\omega s+{\omega_{n}}^{2}}$

The solution is:

$\hspace{2.5em}$ $h(t) = \frac{{\omega_{n}}}{\sqrt{1-\zeta^{2}}}e^{-\zeta {\omega_{n}} t}sin({\omega_{n}} \sqrt{1-\zeta^{2}}t)$

Note that the time constant depends on the product of the damping and the frequency!

The denominator is called chacateristic equation:

$\hspace{2.5em}$ $s^{2}+\zeta{\omega_{n}} s+{{\omega_{n}}}^{2}$

$\hspace{2.5em}$ $r_{1,2} = \frac{-\zeta{\omega_{n}}\pm \omega_{n}\sqrt{1-\zeta^{2}}}{2}$

We have three forms for the solution:

enter image description here

Overdamped: $r_{1} \neq r_{2}$ $\in$ $\Re$

Critically damped: $r_{1} = r_{2}$ $\in$ $\Re$

Underdamped: $ r_{1} = {r_{2}}^{*}$. Where ${r_{2}}^{*}$ is the complex conjugate of $r_{1}$

In the s plane, should look like this:

enter image description here

The figure above show us a complex response (the conjugate is implicated). Note that $-\zeta\omega_{n}$ its in the real part of the solution! So, is responsible for the time response and, the imaginary part, $\omega_{n}\sqrt{1-\zeta^{2}}$ is responsible for the oscilation.

Note that the real part is the exponential term in the solution!

  • $\begingroup$ I'm not really familiar with that notation, in process control we use G(s) = \frac{K}{\tau^2 +2\zeta \tau + 1} and zeta is my damping factor $\endgroup$ Jan 17 '16 at 23:12
  • $\begingroup$ For your G(s) , the omega^{2} is 1, and the 's' is tau . I forgot to multiply by the K in my H(s). $\endgroup$ Jan 17 '16 at 23:25
  • $\begingroup$ Ah, gotcha, isn't that solution invalid if zeta is > 1 since we'll end up with a root of a negative number? $\endgroup$ Jan 17 '16 at 23:31
  • 2
    $\begingroup$ For a second order system, you have 3 types of solutions. When the root is a negative number, you will have a complex answer. This implies in oscillation. $\endgroup$ Jan 17 '16 at 23:37
  • $\begingroup$ More about second order solutions here $\endgroup$ Jan 17 '16 at 23:43

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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