approximate floor function with a linear one

Question

How to linearize floor function in form of transfer function?

suppose you have input and out put to the floor function as r and y.

y = floor(r),

I want to replace floor or approximate it with a linear transfer function, so I can have it as:

G = y/r;

where G has the same functionality as floor.

G can be a linear Laplace transformation of floor function, as you know a nonlinear Laplace transformation of floor function would be,

e^(-s)/s(1-e^(-s))

what about linear transformation?

Any ideas would be appreciated.

Answer 1

The Laplace transform of it has been derived from other question.

LaplaceTransform[Floor[t], t, s]

Plot

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To get better asymptotic approximation, we're going to use Padé approximant. For better intuitive explanation, see the question titled: The unreasonable effectiveness of Padé approximation.

I'll compute it with WolframAlpha.

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Its first-order approximation will be:

PadeApproximant[LaplaceTransform[Floor[t], t, s], {s, 1, 1}]

Plot

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Its second-order approximation will be:

PadeApproximant[LaplaceTransform[Floor[t], t, s], {s, 2, 2}]

Plot

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Play around with any order you need. To me, 2/2 is good enough.