$[$Ch$4$: $3]$ Consider a DC motor shown below

Using Newton's law for the rotor we obtain

$J_m{}_m^{}+b{}_m^{}=K_t{i}_a$

While using Kirchoff Law, we get

$L_a\frac{d{i}_a}{dt}+R_a{i}_a=v_a-K_e{}_m^{}$

$($a$)$

$($b$)$

If we choose the state vector as $x={\left[\begin{array}{ccc}{}_m& {}_m^{}& i_a\end{array}\right]}^T,$ develop a state space model of the DC motor system. Assume that the input is voltage, $u=v_a$ and the output is angular position, $y={}_m$