Determining final value of the output of a discrete system

I'm going through an exam question where I've been told that the samples f(kT) of the following function \begin{equation}{\text{F}\left(z\right)=\frac{1}{1-0.819z^{-1}}} \end{equation} are applied to a discrete system with the pulse transfer function

\begin{equation}{\text{G}\left(z\right)=\frac{C(z)}{F(z)}=\frac{T}{2}{(\frac{1+z^{-1}}{1-z^{-1}})}} \end{equation}

where T = 0.1s. I'm then asked to determine the final value of the output of the system. I know that the final value theorem states

\begin{equation}{\lim_{k\to\infty}\text{f}\left(kt\right)=\lim_{\text{z}\to1}(1-z^{-1})\cdot\text{F}\left(\text{z}\right)} \end{equation}

but how do I obtain a value of F(z)? I'm not sure how the two equations above are manipulated to obtain the F(z) I should be using the final value theorem with.

• You will have a better answer on the signal processing forum – M.Ferru May 16 '17 at 10:38
• The steady state gain ('DC gain') is found by letting z=1 in the z-TF. Equivalent to s=0 in Laplace. – Chu May 16 '17 at 15:27