Overview
Problem 1
Consider the system described by the following transfer function:
\begin{equation*}
\frac{Y(s)}{R(s)} = \frac{6}{(s+1)^2(s+4)}.
\end{equation*}
\begin{enumerate}[label=\alph*.]
\item \textbf{Find the differential equation of the system.}
\item \textbf{Derive a state-space representation of the system.}
\end{enumerate}
Problem 2
For the system represented by:
\[
A = \begin{bmatrix} -3 & 1 \\ -1 & 2 \end{bmatrix},\quad
B= \begin{bmatrix}1\\1\end{bmatrix},\quad
C= \begin{bmatrix}1 & 0\end{bmatrix},\quad D=[0]
\]
Find the transfer function of the system.
Problem 3
Consider the following block diagram representation of a feedback control system as shown in Figure~\ref{fig:prob3}:
\begin{enumerate}[label=\alph*.]
\item \textbf{Determine the transfer function of the system.}
\item \textbf{Derive the state-space representation, $K=5$, $\alpha=0.5$.}
\item \textbf{Confirm the transfer function using the state-space approach.}
\end{enumerate}
Problem 4
Consider the system described by:
\begin{equation*}
\ddot{y} + \dot{y} + \tfrac{1}{2}y = \tfrac{1}{2}u
\end{equation*}
\begin{enumerate}[label=\alph*.]
\item \textbf{Find the state transition matrix $\Phi(t)$.}
\item \textbf{Find the homogeneous state vector.}
\item \textbf{Find the homogeneous output response.}
\item \textbf{Find the transfer function of the system.}
\item \textbf{Find the forced state vector.}
\item \textbf{Find the forced output response.}
\item \textbf{Find the complete output response of the system.}
\end{enumerate}
Problem 5
Consider the following electrical circuit shown in Figure~2:
\begin{enumerate}[label=\alph*.]
\item \textbf{Obtain the transfer function $E_o(s)/E_i(s)$.}
\item \textbf{Find the state-space representation of the system (i.e., the matrices A, B, C, and D).}
\end{enumerate}
Problem 6
Consider the following state-space system:
\[
\dot{x}(t) =
\begin{bmatrix}
-1 & 0 & 1 \\
1 & -2 & 0 \\
0 & 0 & -3
\end{bmatrix}
x(t)
+
\begin{bmatrix}
0 \\
0 \\
1
\end{bmatrix}
u(t),
\quad
y(t) =
\begin{bmatrix}
1 & 1 & 0
\end{bmatrix}
x(t).
\]
\subsection*{(a) State transition matrix $\Phi(t)$}
\subsection*{(b) Homogeneous output response with $x(0) = [1 \;\; 0 \;\; 1]^T$}
\subsection*{(c) Transfer function}
Problem 7
Verify if the following matrices can be state transition matrices $\Phi(t)$.
\subsection*{(a)}
\[
\Phi(t) =
\begin{bmatrix}
-e^{-t} & 0 \\
0 & 1-e^{-t}
\end{bmatrix}.
\]
\subsection*{(b)}
\[
\Phi(t) =
\begin{bmatrix}
1-e^{-t} & 0 \\
1 & e^{-t}
\end{bmatrix}.
\]
\subsection*{(c)}
\[
\Phi(t) =
\begin{bmatrix}
1 & 0 \\
1-e^{-t} & e^{-t}
\end{bmatrix}.
\]
\subsection*{(d)}
\[
\Phi(t) =
\begin{bmatrix}
e^{-2t} & te^{-2t} & \tfrac{t^2}{2}e^{-2t} \\
0 & e^{-2t} & te^{-2t} \\
0 & 0 & e^{-2t}
\end{bmatrix}.
\]
Problem 8
Consider the system with the following closed-loop transfer function:
\[
\frac{Y(s)}{R(s)} = \frac{K}{0.5s^5+7s^4+3s^3+42s^2 + 4s +56}.
\]
\begin{enumerate}[label=\alph*.]
\item \textbf{Apply the Routh-Hurwitz criterion to study the stability of the system.}
\item \textbf{Is there any pole on the $j\omega$-axis? If yes, find them.}
\end{enumerate}
Problem 9
Determine whether the system is stable or not, and if not, determine the number of unstable poles for the following characteristic equations:
\begin{enumerate}[label=\alph*.]
\item $C.E_1 = s^5+2s^4+2s^3+2s^2+3s+4$
\item $C.E_2 = s^6+s^5+3s^4+4s^3+s^2+s+1$
\end{enumerate}