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):

and this system with damping factor 1.19 and it's response:

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?

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:

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:

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!

• 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 – MathsIsHard Jan 17 '16 at 23:12
• 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). – leCrazyEngineer Jan 17 '16 at 23:25
• Ah, gotcha, isn't that solution invalid if zeta is > 1 since we'll end up with a root of a negative number? – MathsIsHard Jan 17 '16 at 23:31
• 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. – leCrazyEngineer Jan 17 '16 at 23:37
• More about second order solutions here – leCrazyEngineer Jan 17 '16 at 23:43