Relating transfer functions with step responses

Question

Relate the transfer function to its' corresponding step response.

First, I tried setting up the poles and zeros of the transfer functions. This helped a bit since I know that $G_A (s)$, $G_B(s)$ and $G_C (s)$ should all have oscillatory step responses (because of conjugate poles). The problem arose when trying to determine the differences between $G_A (s)$ and $G_B (s)$, and we thought that there has to be a better way of understanding this. Working with poles and zeros, we didn't really understand what we were doing.

So, I'm now thinking that I need to use the fact that the inverse laplace of the transfer function equals the impulse response, and the integral of the impulse response equals the step response. This is how we relate the step response with the transfer function, but we don't know how to work with it.

Edit: For example, when I take the inverse laplace of $G_D (s)$, I get $\frac{1}{5}$, and the integral of this is $t/5$, which can be translated to the first curve above.

But when I take the inverse laplace of $G_F (s)$, I get $9te^{-3t}$, and the integral of this is $-9e^{-3t}(3t+1)$, but this is too much of a hassle, how can I relate this to one of the step responses in an easy way?

Maybe I should go back to using the poles and zeros, any tips would be greatly appreciated.

Answer 1

In the given example you have 3 types of systems:

1. the ideal integrator (D) with a step response that grows infinitely (fig. 1); that's the obvious one, as you've found out by yourself.
2. underdamped systems with complex conjugate poles (A, B, and C): their step response oscillates
3. critically damped systems with a double real pole (E and F): no oscillation in the step response

So systems A, B, and C must correspond to figures 2, 4, and 5, whereas systems E and F must correspond to figures 3 and 6. You had already arrived at that conclusion.

But we have more cues. Clearly, system C has less damping than systems A and B, which have the same damping. Smaller damping corresponds to more oscillation (i.e. higher overshoot and undershoot). So we can conclude that system C corresponds to figure 4. The difference between systems A and B can be seen by using the Initial Value Theorem:

$$f(0^+)=\lim_{s\rightarrow \infty}sF(s)\tag{1}$$

For system A we get

$$g_A(0^+)=1$$

whereas for system B we have

$$g_B(0^+)=-1$$

These are the values of the impulse responses, i.e., the derivatives of the step response. We see that the step response in figure 5 has a negative derivative at $t=0^+$, from which we can conclude that it corresponds to system B. Then we're left with figure 2 for system A.

The difference between systems E and F is the distance of the double pole from the imaginary axis. The further away from the imaginary axis, the faster will the impulse response decay, and consequently, the faster will the step response rise. So system E must correspond to figure 6, and system F corresponds to figure 3.