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\begin{document}

\null\vspace{-1.75cm}

{\Large
{\bf ECE 486 \hfill Assignment \# 7}

\vspace{.15in}
Issued:  March 31 \hfill Due: April 7, 2022}
\rule{6in}{.01in}

\vspace*{.1cm}
 

\head{Reading Assignment:}   
\textbf{FPE}, Sections 6.3, 6.7.1--6.7.6.
 
\head{Problems:} 

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\vspace*{.15in} 

\item Consider the transfer function
$$
H(s) = \frac{1}{s+a},
$$
where $a > 0$. Prove that the Nyquist plot of $H$ is a circle of radius $\dfrac{1}{2a}$ centered at the point $\left(\dfrac{1}{2a},0\right)$.

\item  For the two plant transfer functions given below, use the Nyquist stability criterion to determine all values of the feedback gain $K$ that stabilize the closed-loop system.
\begin{align*}
	\text{(a)\,\,} G(s) &= \frac{1}{(s+2)(s+5)} \qquad \text{(b)\,\,} G(s) = \frac{1}{(s+2)(s^2 + 2s+ 5)}
	\end{align*}
\textit{Instructions}: To draw the Nyquist plot, use the Bode plots of $G(s)$. Explain all steps in arriving at the Nyquist plot. Start with hand-sketched Bode plots. You can then generate more accurate Bode plots with MATLAB to get exact numerical values if necessary. \textit{The use of MATLAB for drawing the Nyquist plot is not allowed} (except to check your work at the end). It is also recommended that you check your results with Routh-Hurwitz stability criterion.

\item For the two transfer functions and gain values given below, use the Nyquist plot to find the gain and the phase margins:
\begin{align*}
		 \text{(a)} \,\, G(s) &= \frac{1}{(s-1)(s+2)(s+4)},\,\, K = 10; \\
		\text{(b)} \,\, G(s) &= \frac{1}{(s+1)^3},\,\, K = 3.
\end{align*}
\textit{Instructions}: Follow the same instructions as in the previous problem. \textit{You may (and should) use  MATLAB for drawing a more accurate Nyquist plot to find the GM and the PM, but you must explain all your work on the hand-sketched Nyquist plot.}




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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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