I am trying to get my head around the following problem: I have a unit-gain stable LTI system (we can assume it is a discrete-time one) of which I need to calculate the delay in tracking a ramp signal (we can assume the ramp signal $u[k] = kT$). This can easily be done applying the final value theorem. I end up with the same result by calculating the group delay of the system at frequency (or pulsation) zero.
Is there a general theorem stating this equivalence or a general argument to conclude that they must be the same? Or is this just a coincidence?
As an example let's consider $$y[k]=a y[k-1] + (1-a) u[k-1]$$ so we can write $Y(z) = G(z)U(z)$ where $G(z) = (1-a)/(z-a)$ with a sampling period $T$.
Now the group delay $$\tau_g(\omega) = - \frac{d}{d\omega} \phi(\omega)$$ where $\phi(\omega)$ is the phase of $G(e^{j\omega T})$.
When calculating the group delay at $\omega = 0$ one gets $\tau_g(0) = T/(1-a)$. On the other hand one can calculate the error $\Delta(z)$ between the input ramp $U(z) = T z/(z-1)^2$ and the output $Y(z)$ and take the limit for $z \rightarrow 1$ of $(z-1) \Delta(z)$ which will end up being equal to $T/(1-a)$.
This equivalence (not the value $T/(1-a)$ itself) is true also for other first order stable systems with unity gain like $G(z) = \frac{(1-a)z}{z-a}$ or $G(z) = \frac{1-a}{2} \frac{z+1}{z-a}$. I also checked with a second order system $G(z) = \frac{1-2a + a^2+b^2}{z^2 -2 a z + a^2+b^2}$ that has a pair of complex conjugate poles $a \pm j b$. So I wonder if this is valid in general for unity gain stable causal systems (both continuous time and discrete time). Can anybody help?