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\begin{document}

\null\vspace{-1.75cm}

{\Large
{\bf ECE 486 \hfill Assignment \#  9 (Extra Credit)}

\vspace{.15in}
Issued: April 25 \hfill Due:  May 2, 2024}
\rule{6in}{.01in}


  

\head{Reading Assignment:}   
\textbf{FPE}, Sections 7.1--7.5, 7.7--7.8.
 
\head{Problems: \ \rm (all computations using Matlab)}

 \textit{Modeling \&\ full-state feedback:}
\begin{arabnum}
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\item 
Consider the SISO model,
\[
Y(s) = \frac{s+1}{s^2+2s +2} U(s)
\]
Obtain a second-order state-space model (without Matlab). 
\\
Obtain a state-feedback compensator $u=-Kx+r$, placing the closed loop poles at $-4$ and $-25$.
 
\item 
Consider the satellite position model with delay,  
\[
G_p(s) = \frac{1-  s/2 }{1+ s/2 } \frac{1}{s^2}.
\]
Obtain a third-order state-space model (without Matlab). 
\\
Obtain a state-feedback compensator $u=-Kx+r$, placing the closed loop poles at $-4$, $-13$ and $-25$.

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\textit{Observers \&\ sensitivity}
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\item 
Return to the feedback system considered in Problem~1:
\begin{alphanum}
\item
Construct a stable observer to estimate $x$  based on measurements of $(u,y)$.   
\item
Obtain a state-feedback compensator $u=-K\hat{x}+K_r r$,  where $K$ was obtained in your prior work, and $K_r$ is chosen
so that the DC gain $Y/R$ is equal to unity.
\item
Obtain a step response using Matlab. 
\end{alphanum}
\item 
Return to the feedback system considered in Problem~2: 
\begin{alphanum}

\item
Construct a stable observer, and using this obtain a compensator of
the form $U=-G_cY + G_rR$.
\item
Obtain a Nyquist plot for $G_cG_p$ using Matlab, and estimate the gain and phase margins.  
 \end{alphanum}
 

 


 
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%\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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