Electronic – Instantaneous DC output using Laplace transform

I believe settling time is not really easy to define. Mathematically speaking, the settling time is infinite, because the filter tends asymptotically to its steady state value. In terms of engineering, it's usually the time needed for the filter to be in a certain margin from the steady state. Commonly 2% to 5% according to a source in Wiki, but it's very arbitrary.

A common rule-of-thumb to estimate it is to take 4 to 8 times of the dominant time constant of the system as the settling time, being the time constant defined exactly on the time taken to reach steady-state value (1 time constant is about 63%). But this is less intuitive with higher-than-2nd-order systems.

An even more common rule-of-work is to look at the response and just decide for yourself (as arbitrarily as you like as the definition is loose) when you're there.

To answer your main question, I simulated that transfer function using MATLAB and found your answer is correct.

To answer your question about definition, the settling time is absolutely not 'the time when the output of the filter will become equal to the step input magnitude'. In neither of the two possible ways to read it:

1. It is not the time when they become equal, as that would mean that an overshoot, or even an unstable behaviour, is mistaken by an exact settling time
2. It is not the time the output becomes equal to the step input magnitude, but rather the time it becomes almost equal to its steady state value. Unless you are treating a closed-loop system's transfer function it will be coincidential to have your system match the input's step magnitude.

If my way of doing is makes sense, to obtain the Laplace transforms of V(in) and V(out) which interval should I use? As you see the plots start from 0sec to 1sec. And the step ipout starts at 0.1sec. Can I just take the Laplace transform's of Vin and Vout for the entire 1 seconds and do the math? Is there any restriction here?

A better way to do this in lt spice is to do a .AC analysis, and then plot V(Vout)/V(Vin) (or whatever net names you have). AC analysis runs a sine sweep through the circuit, choose the source and the source's amplitude (the source below is set to 1V and then you can monitor any output. The green line in the below plot is \$ H(s)\$

That being said, if you really want to export time series data use the tfest command in matlab OR if you want to do system identification manually (which is not trivial and is too broad to explain all of it in an answer here, since it is a science unto itself) you could probably do this by taking the FFT of both signals to get the data into the frequency domain and then dividing the magnitude of the output by the input, you should get something that looks like a bode plot. Then use polynomial fitting to fit the data to a two pole transfer function model. (I've only done that once, usually its easier to estimate the poles by 'eyeballing' it, and much easier to use LT spice for all the computation so I only use that)

Here is the general idea (there is some translation you'll have to do from sampled to continuous and freq to laplace):

$$ \frac{|FFT(Vout)|}{|FFT(Vin)|} = |H(s)| = \frac{K}{(\tau_1 s+1)(\tau_2 s+1)}$$

$$ \frac{\angle FFT(Vout)}{\angle FFT(Vin)} = \angle H(s) = \exp^{-as}$$