Finding the maximum output with step function based on a transfer function

Question

I have an open and closed loop transfer functions as below:

$$ \begin{align*} L(s) & = \dfrac{k}{s^2 + as + b} \\ G(s) & = \dfrac{k}{s^2 + as + (b + k)} \end{align*} $$

Now, I want to find the maximum output with a unit step as input. It might look like below:

$$ y_{max} = \mathscr{L^{-1}}[G(s) * U(s)] = \mathscr{L^{-1}}[\dfrac{k}{s^2 + as + (b+k)} * \dfrac{1}{s}] $$

or

$$ y_{max} = \lim_{s\to0}sG(s)*U(s) $$

In most cases, we may be curious about its initial and final value to check a system. Typically, in a second-order lag system, the process before the steady-state might have oscillation based on the transfer function, and the maximum can also be higher or lower than the steady-state value.

How can I find the maximum value of the transfer function in this case? It seems not going to work with the initial and final theorem. Besides, if $a = 4, b = 2, k = 6$, what result could it be?

Any help is appreciated! Thank you for your time and advice.

Note: I am not sure why people devote this, if there is anything not mentioned, please comment below and let me know. I mean, you can comment and devote...

Answer 1

There are three types of damping ratios in a second-order system:

1. $0 < ζ < 1$(under-damping)
2. $ζ = 1$(critical-damping)
3. $ζ > 1$(over-damping)

Each case can derive different forms of output with step function as input. Here, I will skip the whole procedure to derive three situations.

Now, with $a = 4, b = 2, k = 6$, we check the damping ratio from the general formula $ζ = \dfrac{1}{\sqrt 2}$ which implies that the system oscillates and thus overshoot will happen before the steady-state.