Proving a transfer function is proper or not

Question

I'm trying to prove (not just show) whether the following transfer function is proper or not $$ G(s) = \frac{1}{1+se^{-s}} $$

A transfer function $G(s)$ is said to be proper if there is $\alpha \geq 0$ such that $$\sup_{s\in\mathbb{C}_{\alpha}}\left\Vert G(s)\right\Vert<\infty$$ where $\mathbb{C}_{\alpha}:=\left\{ \text{Re}\left(s\right)>\alpha\right\} $

Thanks

Edit:

The $\left\Vert .\right\Vert$ operator is used (rather than $\left| .\right|$) because this definition holds for a MIMO transfer function as well.

So far I've arrived at $$\sup_{s\in\mathbb{C}_{\alpha}}\left\Vert \frac{1}{1+se^{-s}}\right\Vert =\inf_{s\in\mathbb{C}_{\alpha}}\left|1+se^{-s}\right|\underset{s=\sigma+j\omega}{=}\inf_{s\in\mathbb{C}_{\alpha}}\left|1+\left(\sigma+j\omega\right)e^{-\sigma-j\omega}\right|$$

if I use the triangle inequality I have $$\inf_{s\in\mathbb{C}_{\alpha}}\left|1+\left(\sigma+j\omega\right)e^{-\sigma-j\omega}\right|\leq\inf_{s\in\mathbb{C}_{\alpha}}1+\left|\left(\sigma+j\omega\right)e^{-\sigma-j\omega}\right|=\inf_{s\in\mathbb{C}_{\alpha}}1+\left|\sigma+j\omega\right|\left|e^{-\sigma}\right|{\left|e^{-j\omega}\right|}=\left(\inf_{s\in\mathbb{C}_{\alpha}}1+\left|\sigma+j\omega\right|\left|e^{-\sigma}\right|\right)_{\omega\to\infty}=\infty$$

which is not very helpful

Answer 1

I think it should be strictly proper because the delay part has no excess of poles or zeros, so the s multiplying it must make the complete transfer function tend to zero at infinity. But I'll leave it for you try other forms of verifying the limits formally.

Answer 2

Applying your definition of a proper transfer function to rational transfer functions (i.e. transfer functions with only polynomial expressions in $s$ for the numerator and denominator) is equivalent to that the order of the numerator polynomial should be equal or less than the order of the denominator polynomial and that the denominator only has roots (i.e. poles) with negative real part. Usually, I encounter the definition of proper without the constraints on the poles and thus only with the constraint on the order of the numerator.

The constraint on the order of the numerator would ensure that $\|G(s)\|<\infty$ when $|s|\to\infty$. This can be shown to be the case for your transfer function. Now it only has to be checked if all poles of your transfer function are stable, for which we need to solve

$$ 1+s\,e^{-s}=0. \tag{1} $$

It can be shown that the solutions to $(1)$ are $s=-W_n(1)$, with $W_n(x)$ the n$^\text{th}$ Lambert W-function. However, $Re(-W_1(1))=1.5339>0$ which would imply at least one unstable pole with positive real part and thus not proper.

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It would also be possible to derive that your transfer function is unstable (at least one pole with positive real part) by considering that your transfer can also be obtained by applying unit negative feedback to $s\,e^{-s}$ (your transfer function would be the associated closed loop sensitivity transfer function). Namely, from the Nyquist plot of $s\,e^{-s}$, using the Nyquist stability criterion, it would follow that the closed loop is unstable.