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\1end{bmatrix},quad
C= begin{bmatrix}1 & 0end{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 $jomega$-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}