Problem #$2$: $(17$pts$)$

A mathematical model of a second order system is given by following linear time invariant ordinary differential equation along with its initial states and forcing function.

$\frac{d^{2}y(t)}{d{t}^2}+10\frac{dy(t)}{dt}+34y(t)=3\frac{dr(t)}{dt}+2r(t)$;$,r(t)=3e^{-2t}$;

$y(0^{-})=2,y^{\text{'}}(0^{-})=0,r(0^{-})=1$

a$)$ Find out the transfer function and zero$-$state response.

b$)$ Find out the zero$-$input response and complete complex response.

c$)$ Find out the complete time response and its steady state value $($if exists$).$

d$)$ Find out the natural response and forced response.

e$)$ Draw the pole$-$zero plot of complete response of the system in $s-$plane.

f$)$ Plot the representative time response and comment about the stability according to the polezero plot.

g$)$ Specify type of the system relative to damping and plot the representative time response of part c$).$

h$)$ Find out rise time, peak time, percent overshoot, settling time and number of extrema before response settles down.

i$)$ Find out the relative stability, error transmittance and system type number.

j$)$ Find out transient response decay time.

k$)$ Find out the steady$-$state errors $($if exist$)$ for unit impulse, unit step, unit ramp and unit parabolic inputs.