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\null\vspace{-1.75cm}

{\Large
{\bf ECE 486 \hfill Assignment \# 6}

\vspace{.15in}
Issued:  March 28 \hfill Due: April 4, 2024}
\rule{6in}{.01in}

\vspace*{.1cm}
 

\head{Reading Assignment:}   
\textbf{FPE}, Sections 6.1, 6.2, 6.4--6.6.
 
\head{Problems:} (you can use MATLAB in Problems 3 and 4, but you must explain all steps and justify all answers)

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\vspace*{.15in} 

\item For each of the transfer functions given below, draw the Bode plots (both magnitude and phase) \textit{by hand}, using the techniques discussed in class. Explain all steps in your drawing procedures. Note that the transfer functions are not given in Bode form.
\begin{align*}
	\text{(a)\,\,} KG(s) = \frac{s+8}{s(s+4)} \,\,\,  \text{(b)\,\,} KG(s) = \frac{8s}{s^2+0.2s+4} \,\,\,  \,\,\, \text{(c)\,\,} KG(s) = \frac{s^2+0.2s+1}{s(s+0.2)(s+6)}
\end{align*}
After you're done, check your results using MATLAB. (Note that the {\tt bode} command in MATLAB plots magnitude in decibels.) Turn in both the hand sketches and the MATLAB plots.

\item Show that for the transfer function $KG(s) = \dfrac{\omega^2_n}{s^2 + 2\zeta\omega_ns}$, the phase margin is independent of $\omega_n$ and is given by
\begin{align*}
	{\rm PM} = \tan^{-1}\left( \frac{2\zeta}{\sqrt{\sqrt{4\zeta^4+1}-2\zeta^2}}\right).
\end{align*}
   

\item Consider the transfer function $G(s) = \displaystyle \frac{1}{s(s^2+4s+8)}$.

\begin{romannum}
	\item Derive the value of $K$ for which the closed-loop characteristic equation $1+KG(s)$ has roots on the $j\omega$-axis.
	\item For this value of $K$, make the Bode plot of $KG(s)$ using MATLAB and explain how you can confirm the presence of $j\omega$-axis closed-loop poles using this plot.
	\item Compute the gain and phase margins for $K=12$ using the corresponding Bode plot.
	\item Determine the gain $K$ that gives the phase margin of $60^\circ$.
	\item Plot the step responses of the closed-loop systems for $K=12$ and the $K$ you found in part (iv). Which system has better damping (smaller overshoot)? Why?
\end{romannum}


\item Consider the system
\begin{align*}
	G(s) = \frac{10}{\left(\dfrac{s}{0.2}+1\right)\left(\dfrac{s}{0.5}+1\right)}.
\end{align*}
We wish to design a lead/lag controller that provides bandwidth of at least $2$, PM of at least $60^\circ$, and steady-sate tracking of constant references within $1\%$.
\begin{romannum}
	\item For the controller
	\begin{align*}
		KD(s) = 4 \frac{\dfrac{s}{0.8}+1}{\dfrac{s}{5}+1} \cdot \frac{s+0.05}{s+0.02}
	\end{align*}
	derived in class, compute the PM, bandwidth, and steady-state tracking error to verify whether the specs are met.
	\item Suppose that in addition to the above specs, the bandwidth cannot exceed $6$. Modify the design to incorporate this new spec, and verify that it indeed works.
\end{romannum}



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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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