Is a filter/control transfer function with positive phase "causal"?
In general, from the positiveness of the phase nothing can be concluded about the causality of the corresponding system. Note that the phase of the given system is

$$\phi(\omega)=\arctan(\alpha\omega\tau)-\arctan(\omega\tau)\tag{1}$$

which is positive for all $\omega&gt;0$ if $\alpha&gt;1$. Yet the system is causal, which can be easily seen by calculating its impulse response:

$$\begin{align}G(s)&amp;=\frac{\alpha s+\frac{1}{\tau}}{s+\frac{1}{\tau}}=\alpha\frac{s+\frac{1}{\tau}-\frac{1}{\tau}}{s+\frac{1}{\tau}}+\frac{1}{\tau}\frac{1}{s+\frac{1}{\tau}}=\alpha+\frac{1-\alpha}{\tau}\frac{1}{s+\frac{1}{\tau}}\\\Longleftrightarrow g(t)&amp;=\alpha\delta(t)+\frac{1-\alpha}{\tau}e^{-t/\tau}u(t)\tag{2}\end{align}$$

where $u(t)$ is the unit step function. Obviously, we have $g(t)=0$ for $t&lt;0$, i.e., the system is causal.

Note that the system's phase is not only positive, but it also has a positive slope for $\omega&lt;1/(\tau\sqrt{\alpha})$. This means that its group delay (the negative derivative of the phase) is negative in that frequency range, as shown in the figure ($\alpha=2$, $\tau=1$):



So even a negative group delay in some frequency range says nothing about the causality of a system.

A rational transfer function with all its poles in the left half-plane (as is the case in your example) and a region of convergence $\text{Re}\{s\}&gt;0$ always corresponds to a causal and stable system, even if its phase is positive or its group delay is negative in a certain frequency range.