Question $1$:

Based on the following equations, linearize the system around the given trimming points! $(3$p$)$

$\{$:$[x_1^{}(t),=x_1^2(t)+\sqrt[\phantom{2}]{2}sin(u(t))],[x_2^{}(t),=x_1(t)*x_2(t)-4x_2^3(t)+cos(2u(t))],[y(t),=x_2^2(t)*u(t)]\},{\stackrel{}{x}}_1=1$;$,{\stackrel{}{x}}_2=\frac{1}{2}\frac{\text{;}}{b}ar(u)=-\frac{}{4}$

Question $2$:

a$)$ Using Newton's $2$nd Law give the following mechanical systems equation! $(2$p$)$

b$)$ Calculate the natural frequency, damping ratio, and damped natural frequency! What can you say about the system's stability, give reasoning! $(1$p$)$

c$)$ Reformulate the system as a state space model with the position as the output! $(1$p$)$

$m=2[kg],k=6[\frac{N}{m}],c=3[\frac{Ns}{m}],=1[\frac{Ns}{m}],g=9\mathrm{.}81[\frac{(m)}{s^2}]$

Input: $F(t)$

Output: $x(t)$

Friction formula: $F_{\text{f}\text{r}\text{i}\text{c}}=x^{}$

Question $3$:

a$)$ Given the following system, design a P controller to reach a time constant of $0\mathrm{.}05[s]$ what can you say about the stability? Give reasoning! $(1$p$)$

$x^{}(t)=-12x(t)+4u(t)$

b$)$ What will be the steady$-$state value and the $5\%$ settling time for the unit step response? $(1$p$)$

c$)$ Is this controller enough to eliminate steady$-$state error? If not, what would you suggest? $(1$p$)$