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\begin{document}

\null\vspace{-1.75cm}

{\Large
{\bf ECE 486 \hfill Assignment \#  3}

\vspace{.15in}
Issued:  February 1 \hfill Due: February 8, 2024}
\rule{6in}{.01in}

\vspace*{.1cm}
 

\head{Reading Assignment:} 
\textbf{FPE},  Sections 3.3-3.6.

\head{Problems:} (unless otherwise noted, you can use a calculator/computer to arrive at numerical answers)

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

\item Consider the system given by the block diagram below:
\Ebox{.95}{problem2.pdf}
Compute the transfer function from the input $U$ to the output $Y$.

\smallskip 

(Your answer should be an expression involving the transfer functions $K_1,K_2,P_1,P_2,G$.)

\item Consider two systems in a negative feedback configuration shown below:
\Ebox{.6}{problem1.pdf}
All state variables, all inputs, and all outputs are scalars. 
%\\[5pt]
\begin{romannum}
\item Find the transfer function from the input $R$ to the output $Y$.

\smallskip

(Your answer should be a ratio of two polynomials in $s$ with coefficients expressed in terms of the system parameters $a,b,c,k,\ell,m$.)
\item Write down the conditions that must be satisfied by the system parameters $a,b,c,k,\ell,m$ for this transfer function to be stable (i.e., for all poles to  have negative real parts).
\end{romannum}

\newpage

\item Consider the mass-spring system shown in the diagram below:
\Ebox{0.4}{mass_spring_system}
As shown in class, this system has the following state-space model with $x_1 = x$ and $x_2 = \dot{x}$:
\begin{align*}
\left(\begin{matrix}
	\dot{x_1} \\ \dot{x_2}
\end{matrix}\right) &= \left(
\begin{matrix}
	0 & 1 \\
	-\frac{k}{m} & -\frac{\rho}{m} 
\end{matrix}\right)
\left(\begin{matrix} x_1 \\ x_2\end{matrix}\right) 
+ \left(\begin{matrix} 0 \\ \frac{1}{m} \end{matrix}\right)u, \qquad y = \left( \begin{matrix} 1 & 0 \end{matrix}\right) \left( \begin{matrix} x_1 \\ x_2 \end{matrix}\right),
\end{align*}
where $k$ is the spring constant and $\rho$ is the friction coefficient.

\begin{romannum}
	\item Determine the transfer function of this system.
	\item Now suppose that the output equation is replaced by
	$$
	y = \left( \begin{matrix} c_1 & c_2 \end{matrix}\right) \left( \begin{matrix} x_1 \\ x_2 \end{matrix}\right),
	$$
	where $c_1$ and $c_2$ are arbitrary constants. Find the values of $c_1$ and $c_2$ to guarantee that the transfer function of the resulting system has the form of the prototype 2nd-order response discussed in class. Write down the expressions for the parameters $\omega_n$ and $\zeta$ in terms of $k$, $\rho$, and $m$.
	\item Determine the steady-state response of the system from part (ii) to the sinusoidal external force $u(t) = \cos (\omega t)$.
	\item Sketch the plots of the magnitude and the phase shift of the steady-state response from part (iii) as functions of the input frequency $\omega$. What happens when the input frequency $\omega$ matches the system's natural frequency $\omega_n$? This is the phenomenon known as \textit{resonance}.
\end{romannum}

\item Consider the transfer function
\begin{align*}
	H(s) = \frac{16}{s^2 + 4s + 16}
\end{align*}
\\[-30pt]
\begin{romannum}
	\item Suppose that you are given the following time-domain specs: $t_r \le 0.6, t_s \le 1.6$ (where $t_s = t^{5\%}_s$). Plot the admissible pole locations in the $s$-plane corresponding to these two specs. Does the given system satisfy these specs?
	\item Suppose that in addition to the specs from (i), we have the following spec on the overshoot: $M_p \le 1/e^2$. Plot the admissible pole locations in the $s$-plane corresponding to all three specs. Does the given system satisfy the new spec?
	\item Now suppose that you are given the two specs from (i) {\em plus} the following spec on the peak time: $t_p \le 1$ ({\em instead of} the overshoot spec). Plot the admissible pole locations in the $s$-plane corresponding to these three specs. (Pole locations for peak time were not discussed in class, so you need to derive this.) Does the given system satisfy the new spec?
\end{romannum}




\end{arabnum}



\end{document}



