Feedback systems are usually described in the frequency domain since it is simpler to articulate. Typically we are given a \emph{plant} and a \emph{controller} in a feedback system.

% \begin{center}
% \includegraphics{images/feedback/feedback.ps}
% \end{center}

\begin{center}
\begin{pspicture}(0,-3)(8,3)

\rput(0,0){$x$}
\rput(7.2,0){$y$}
\rput(3.75,0){$F$}
\rput(3.75,-2){$G$}
\rput(1.5,-2.5){$H$}

\psframe(1,-3)(6,1)
\psframe(3,-0.5)(4.5,0.5)
\psframe(3,-2.5)(4.5,-1.5)

% plus or minus for adder
\rput(2.1,-0.5){$-$} % input from loop
\rput(1.45,0.35){$+$} % input signal

\psline[linewidth=1.25 pt, arrowscale=1.5]{->}(0.25,0)(1.5,0)
\psline[linewidth=1.25 pt, arrowscale=1.5]{->}(1.75,-2)(1.75,-0.25)
\psline[linewidth=1.25 pt, arrowscale=1.5]{->}(4.5,0)(7,0)
\psline[linewidth=1.25 pt, arrowscale=1.5]{->}(5.5,-2)(4.5,-2)
\psline[linewidth=1.25 pt](3,-2)(1.75,-2)
\psline[linewidth=1.25 pt, arrowscale=1.5]{->}(2,0)(3,0)
\psline[linewidth=1.25 pt](5.5,0)(5.5,-2)

\pscircle(1.75,0){0.25}
\psline(1.5,0)(2,0)
\psline(1.75,0.25)(1.75,-0.25)

\end{pspicture}
\end{center}

Lets trace $x$ through the loop where $F$ is the plant, and $G$ is the controller, and the signal is negated after $G$ before added to the input signal for the loop. Let $x(t) = e^{i\omega t}$, then $y(t) = H(\omega)e^{i\omega t}$. Lets call $q(t)$ the signal that we trace through. This signal goes through the system, and $y(t)$ is fed back into $G$ and becomes $q(t)$:


\begin{align*}
q(t) &= G(\omega)H(\omega)e^{i\omega t} \\
q(t) &= e^{i\omega t} - G(\omega)H(\omega)e^{i\omega t} \quad \mbox{picks up a new input and negation and addition occurs} \\
q(t) &=  e^{i\omega t} \lr{1-G(\omega)H(\omega)} \\
q(t) &=  F(\omega)\lr{1-G(\omega)H(\omega)} e^{i\omega t} \quad \mbox{ goes through $F$}\\
q(t) &= y(t) = H(\omega)e^{i\omega t} =  F(\omega)\lr{1-G(\omega)H(\omega)} e^{i\omega t} \\
\end{align*}

Now after the loop has finished, we can solve for $H(\omega)$:

\begin{align*}
H(\omega)e^{i\omega t} &=  F(\omega)\lr{1-G(\omega)H(\omega)} e^{i\omega t} \\
H(\omega) &=  F(\omega)-G(\omega)F(\omega)H(\omega) \\
H(\omega)+G(\omega)F(\omega)H(\omega) &=  F(\omega) \\
H(\omega)\lr{1+G(\omega)F(\omega)} &=  F(\omega) \\
H(\omega) &=  \frac{F(\omega)}{1+G(\omega)F(\omega)} \\
\end{align*}

This is called \emph{Black's equation}. If we had two positive signs for the adder, the equation would only change slightly:
$$ H(\omega) =  \frac{F(\omega)}{1-G(\omega)F(\omega)} $$ 
To generalize this, we can state that the frequency response to the system is $\frac{\mbox{Forward Gain}}{1 - \mbox{Loop Gain}}$. Note that when in a loop, $F(\omega)$ and $G(\omega)$ are in cascade (a series), therefore the frequency response during the loop (the Loop Gain) is $F(\omega)G(\omega)$. 