Show me the steps to solve A simplified schematic diagram of a quarter$-$car system is shown in Figure $1,$ where $x_1(t)$ is the

road profile $($input to the system$),$ and $x_2(t)$ and $x_3(t)$ are the vertical positions of the unsprung

mass $m_u$ and sprung mass $m_s,$ respectively. Let $v_2(t)$:$=x_2^{}(t)$ and $v_3(t)$:$=x_3^{}(t)$ denote the

vertical velocities of the unsprung mass and sprung mass, respectively. Let $F_{k_s},F_{k_t}$ and $F_{b_s}$ be

the forces in the spring $k_s,$ spring $k_t$ and damper $b_s,$ respectively. The equations of motion are

given by:

$F_{k_s}-k_s(x_3-x_2)=0$

$F_{k_t}-k_t(x_2-x_1)=0$

$F_{b_s}-b_s(x_3^{}-x_2^{})=0$

$m_s{x}_3^{}+F_{b_s}+F_{k_s}=0$

$m_u{x}_2^{}-F_{b_s}-F_{k_s}+F_{k_t}=0$

Find a state space representation with the following sets of state variables:

$($a$)q_1(t)=x_2(t),q_2(t)=x_2^{}(t),q_3(t)=x_3(t),q_4(t)=x_3^{}(t),u(t)=x_1(t),$ and $y(t)=x_3(t).$

$($b$)q_1(t)=F_{k_s}(t),q_2(t)=v_3(t),q_3(t)=F_{k_t}(t),q_4(t)=v_2(t),u(t)=x_1(t),$ and $y(t)=x_3(t).$