matlab - Pade Approximation of dead time

For the time delay $e^{-sT}$ I shall find the Pade Approximation for $M = 0$ and $N = 1$.

$f(s) = \sum_i^{\infty} a_is^i \approx \dfrac{\sum_{n=0}^{N} b_is^i}{\sum_{m=0}^{M} c_is^i}$

$e^{-sT} = \sum_{i=0}^{\infty}\dfrac{(-sT)^i}{i!}$

Using the Taylor approximation for $e^{-sT}$ I yield

$1 -sT \approx \frac{P_N(s)}{Q_M(s)}$

$1 -sT \approx \frac{b_0 + b_1 s}{c_0}$

with $c_0 = 1$ per definition according to my materials.

This leads me to

$1 -sT \approx b_0 + b_1 s$

$b_0 = 1$

$b_1 = -T$

If I now want to check the step response of $e^{-sT}$ and my approximation MATLAB complains about my function having more zeroes than poles, which is not surprising as I thought I would get my transfer function from $G(s) = \frac{P_N(s)}{Q_M(s)}$.

I'm sure there is a basic misunderstanding of what I am doing here on my side.

Answer

There is no misunderstanding at all. The Padé approximant you found is correct.

The "problem" is that you chose $M$ and $N$ such that the transfer function you get to approximate the delay is improper. Namely, the order of the numerator exceeds the order of the denominator.

In Control Theory, improper systems are not too useful because they cannot be realized in real life. A transfer function with more zeros than poles contains pure differentiators, so the transfer function represents a system that is non-causal (assuming stability, as Matt pointed out in the comments below), and non-causal systems cannot be physically realized.