Engineering Question

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}

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