\subsection{Task 1}

\begin{figure}[H]
	\centering
	\input{sys3b_1}
	\caption{Block diagram for System 3b with values inserted}
	\label{fig:sys3bbd1}
\end{figure}

\begin{align*}
\frac{O\left(s\right)}{I\left(s\right)} & = \frac{\frac{12.5}{\left(1+s3\right)\left(1+s4\right)}}{1+\frac{0.08s\times12.5}{\left(1+s3\right)\left(1+s4\right)}}\\
	& = \frac{12.5}{\left(1+s3\right)\left(1+s4\right)+s}\\
\end{align*}

\begin{figure}[H]
	\centering
	\input{sys3b_2}
	\caption{Initial block diagram reduction}
	\label{fig:sys3bbd2}
\end{figure}

\begin{align*}
\frac{O\left(s\right)}{I\left(s\right)} & = \frac{0.5\left(\frac{1+s6}{1+s}\right)\left(\frac{12.5}{\left(1+s3\right)\left(1+s4\right)+s}\right)}{1+0.5\left(\frac{1+s6}{1+s}\right)\left(\frac{12.5}{\left(1+s3\right)\left(1+s4\right)+s}\right)}\times\frac{1+s}{1+s}\\
	& = \frac{0.5\left(1+s6\right)\left(\frac{12.5}{\left(1+s3\right)\left(1+s4\right)+s}\right)}{\left(1+s\right)+0.5\left(1+s6\right)\left(\frac{12.5}{\left(1+s3\right)\left(1+s4\right)+s}\right)}\times\frac{\left(1+s3\right)\left(1+s4\right)+s}{\left(1+s3\right)\left(1+s4\right)+s}\\
	& = \frac{6.25\left(1+s6\right)}{\left(1+s\right)\left(\left(1+s3\right)\left(1+s4\right)+s\right)+6.25\left(1+s6\right)}\\
	& = \frac{6.25\left(1+s6\right)}{\left(1+s3\right)\left(1+s4\right)\left(1+s\right)+\left(s^2+s\right)+6.25\left(1+s6\right)}\\
	\\
	\left(1+s3\right)\left(1+s4\right) & = 12s^2+7s+1\\
	\left(12s^2+7s+1\right)\left(1+s\right) & = 12s^3 + 19s^2 + 8s + 1\\
	\\
	\frac{O\left(s\right)}{I\left(s\right)} & = \frac{6.25\left(6s+1\right)}{12s^3+20s^2+9s+1+6.25\left(6s+1\right)}\times\frac{4}{4}\\
	& = \frac{25\left(6s+1\right)}{25\left(6s+1\right)+48s^3+80s^2+36s+4}\\
\end{align*}

Factorising $48s^3+80s^2+36s+4$, (Figure~\ref{fig:sys3bldiv})

\setlength{\tabcolsep}{1pt}
\begin{figure}[H]
\centering
\begin{tabular}{lllll}
& $8s^2+$ & $12s+$ & $4$\\
\cline{2-5}
$6s+1 \div$ & $48s^3+$ & $80s^2+$ & $36s+$ & $4$ \\
       & $48s^3$   & $8s^2$\\
       &           & $72s^2+$ & $36s+$ & $4$ \\
       &           & $72s^2+$ & $12s$\\
       &           &          & $24s+$ & $4$\\
       &           &          & $24s+$ & $4$\\
\end{tabular}
\caption{Long division to factorise $48s^3+80s^2+36s+4$ }
\label{fig:sys3bldiv}
\end{figure}

\begin{align*}
\frac{O\left(s\right)}{I\left(s\right)} & = \frac{25\left(6s+1\right)}{25\left(6s+1\right)+\left(6s+1\right)\left(8s^2+12s+4\right)}\\
	& = \frac{25}{8s^2+12s+29}
\end{align*}

This is the transfer function.

\subsection{Task 2}

\begin{figure}[H]
\begin{alltt}
	% System 3b: Feedback Control Systems
	S=1;
	C= 0.5 * tf( [6 1], [1 1] );
	L = tf( [0.08 0], 1 );
	P = tf( 12.5, [12 7 1] );
	M = 1;
	sys3b_tf = S * feedback ( C * feedback( P, L ), M );
	minreal( sys3b_tf )

	Transfer function:
	       3.125
	-------------------
	s^2 + 1.5 s + 3.625
\end{alltt}
\end{figure}

This is the same as the transfer function that was derived manually.

\subsection{Task 3}

To get the function $o(t)$, output in the time domain.

\begin{align*}
\frac{O\left(s\right)}{I\left(s\right)}\times\frac{1}{s} &  = \frac{1}{s}\times\frac{25}{8s^2+12s+29}\\
	& = \frac{25}{s\left(8s^2+12s+29\right)}\\
	& = \frac{A}{s} + \frac{Bs+C}{8s^2+12s+29}\\
     25 & = A\left(8s^2+12s+29\right) + Bs^2 + C\\
\end{align*}
Let $s = 0$:
\begin{align*}
25 & = 29A\\
A & = \frac{25}{29}
\end{align*}
Let $s = 1$:
\begin{align*}
25 & = \frac{1225}{29} + B + C\\
\end{align*}
Let $s = 2$:
\begin{align*}
25 & = \frac{2124}{29} + 4B + 2C\\
\frac{-500}{29} & = B + C\\
\frac{-1400}{29} & = 4B + 2C\\
 B & = \frac{-200}{29}\\
 C & = \frac{-300}{29}\\
\end{align*}

These values can then be put back into the origin output function.
\begin{figure}[H]
	\begin{align*}
	 O\left(s\right) & = \frac{25}{29s} + \frac{\frac{-200}{29}s + \frac{-300}{29}}{8s^2+12s+29}\\
	  & = \frac{\frac{-200}{29}s + \frac{-300}{29}}{8s^2+12s+29}\times\frac{\frac{1}{8}}{\frac{1}{8}}\\
	  & = \frac{\frac{-25}{29}s + \frac{-37.5}{29}}{s^2+\frac{3}{2}s+\frac{29}{8}}\\
	  & = \frac{-\frac{25}{29}\left(s+\frac{3}{2}\right)}{\left(s+\frac{3}{4}\right)^2+\frac{49}{16}}\\
	  & = \frac{-\frac{25}{29}\left(s+\frac{3}{4}\right)}{\left(s+\frac{3}{4}\right)^2+\frac{49}{16}}+\frac{-\frac{25}{29}\times\frac{3}{4}}{\left(s+\frac{3}{4}\right)^2+\frac{49}{16}}\\
	  \frac{-\frac{25}{29}\times\frac{3}{4}}{\left(s+\frac{3}{4}\right)^2+\frac{49}{16}} & = \frac{-\frac{75}{203}\times\frac{7}{4}}{\left(s+\frac{3}{4}\right)^2+\frac{49}{16}}\\
	  & = \frac{\frac{7}{4}\left(\frac{3}{7}\times-\frac{25}{29}\right)}{\left(s+\frac{3}{4}\right)^2+\left(\frac{7}{4}\right)^2}\\
	 \\
	O\left(s\right) & = \frac{25}{29s} + \frac{-\frac{25}{29}\left(s+\frac{3}{4}\right)}{\left(s+\frac{3}{4}\right)^2 + \left(\frac{7}{4}\right)^2} + \frac{\frac{7}{4}\left(\frac{3}{7}\times-\frac{25}{29}\right)}{\left(s+\frac{3}{4}\right)^2 + \left(\frac{7}{4}\right)^2}\\
	\\
	o\left(t\right) & = \frac{25}{29} - \frac{25}{29}e^{-\frac{3}{4}t}\cos{\frac{7}{4}t} - \frac{3}{7}\frac{25}{29}e^{-\frac{3}{4}t}\sin{\frac{7}{4}t}\\
	  & = \frac{25}{29}\left(1-e^{-\frac{3}{4}t}\cos{\frac{7}{4}t}-\frac{3}{7}e^{-\frac{3}{4}t}\sin{\frac{7}{4}t}\right)\\
	  \\
	  & = \frac{25}{29}\left(1-e^{-\frac{3}{4}t}\left(\cos{\frac{7}{4}t}+\frac{3}{7}\sin{\frac{7}{4}t}\right)\right)\\
	\end{align*}
	\caption{Obtaining the output function in the time domain for System 3b}
	\label{fig:sys4outfunc}
\end{figure}
\subsection{Task 4}

\begin{figure}[H]
	\centering
	\input{sys3b_mat}
	\caption{MATLAB code for System 3b}
	\label{fig:sys3mlab}
\end{figure}

\begin{figure}[H]
	\centering
	\includegraphics[height=100mm]{plots_sys3b.eps}
	\caption{Graph and Bode Plot for System 3b}
	\label{fig:sys3bplots}
\end{figure}