Q$1$: Obtain the transfer function $E_o\frac{s}{E_i}(s)$ of the operational$-$amplifier circuit shown.

Q$2$: Determine the transfer function $x_2\frac{s}{F}(s)$ for the mechanical system shown knowing that $M=1\mathrm{.}0kg,k=10\frac{N}{m},b=1\mathrm{.}0N.\frac{s}{m},$ then use MATLAB to determine the poles of the system.

Q$3$: The equation of motion for a DC motor is given as:

${}_m^{}+0\mathrm{.}1{}_m^{}=0\mathrm{.}2v_a$

$($a$)$ Find the transfer function between the applied voltage $v_a$ and the shaft angle ${}_m.$

$($b$)$ Suppose a PD$-$controller is used $($as shown below$)$ so that the motor shaft angular displacement ${}_m$ tracks a reference command ${}_r,$ where $K_1$ and $K_2$ are the controller parameters. Determine the transfer function ${}_m\frac{s}{{}_r}(s).$

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$($c$)$ Determine the values of $K_1$ and $K_2$ such that the closed loop poles have a natural frequency equal to $10ra\frac{d}{s}$ and a damping ratio equal to $0\mathrm{.}8.$

$($d$)$ Based on the results obtained in part c $,$ determine the response for unit step input.

$($e$)$ Determine the initial value and the steady state value of the response.

Q$4$: Determine the overall transfer function for the system shown below using block diagram reduction.

Q$5$: Determine the transfer matrix for the two$-$inputs and single output system shown.