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\begin{document}

\null\vspace{-1.75cm}

{\Large
{\bf ECE 486 \hfill Assignment \#  5}

\vspace{.15in}
Issued:  February 23 \hfill Due:  March 1, 2024}
\rule{6in}{.01in}

\vspace*{.1cm}
 

\head{Reading Assignment:}   
\textbf{FPE},     Sections 5.1 - 5.3
\\ \\
Note:    \textit{It is in your interest to consult MATLAB only after
completing a hand sketch}.
\\ \\
You are encouraged to verify your results using MATLAB, and
you will need help from MATLAB to complete Problem 2: 
{ \tt rlocus} and  
{ \tt rltool} are useful and fun!
 
\head{Problems:}

\begin{arabnum}
\setcounter{arabnum}{0}  

\vspace*{.15in} 
   


\item
Sketch the root loci below by hand, applying rules A---F.
\\
$KL(s) = $
\[
\hbox{(a)}\ \
K \frac{1}{s^2 + 2s + 10}
\qquad
\hbox{(b)}\ \ 
K \frac{s-2}{s^2+2s+10}
\qquad
\hbox{(c)}\ \ 
K \frac{(s+1)(s+2)}{s(s^2+4)(s^2+5)}
\qquad
\hbox{(d)}\ \
K\frac{s+2}{s^5+1}
\]
%\hbox{(d)}\ \
%K\frac{(s+2)^4}{s^5+1}
%\]
%\begin{eqnarray*}
%KG(s) &=& \frac{K(s+2)}{s^5+1}
%\\
%KG(s) &=& \frac{K(s^2+2s+12)}{s^2+2s+10}
%\\
%KG(s) &=& \frac{K(s-0.2)}{s(s+3)(s^2+5)}
%\end{eqnarray*} 



\item
Consider the feedback system shown below.

\centerline{\includegraphics[width=0.4\textwidth]{hw5_unity_feedback}}

Determine the transfer function $Y(s)/R(s)$, and from this the characteristic equation.   Sketch the root locus for $c>0$. 

\item
Consider the Evans-style feedback configuration

\centerline{\includegraphics[width=0.45\textwidth]{Evans_canonical_form}}

Suppose that $L(s)$ has
\begin{romannum}
	\item $3$ LHP poles and $1$ LHP zero;
	\item $1$ RHP pole, $2$ LHP poles, and $3$ LHP zeros;
	\item $5$ LHP poles, $4$ LHP zeros, $1$ RHP zero.
\end{romannum}
In each of these three cases, use your knowledge of the effect of open-loop poles and zeros on the root locus to explain whether or not the corresponding closed-loop system can always be stabilized by choosing large enough gain $K$. Be sure to give detailed justification for your answers.



\end{arabnum}






%\bigskip
%
%\textit{Just for fun}:  Recall during lecture on September 21 we pondered on the possibility of $j\omega$-axis crossings for a PID compensator applied to the DC motor.   Experiment with zero selection in the PID compensator using {\large\tt rltool}.   
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