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Q. 7.EX.5

Modeling a DC Motor in State-Variable Form
Find the state-space equations for the DC motor with the equivalent electric circuit shown in Fig. 2.32(a).

Verified Solution

Recall the equations of motion [Eqs. (2.52) and (2.53)] from Chapter 2:

$J_m \ddot{θ}_m + b\dot{θ}_m = K_ti_a, \\ L_a \frac{di_a}{dt} + R_ai_a = v_a − K_e \dot{θ}_m.$

$J_m \ddot{θ}_m + b \dot{θ}_m = K_t i_a .$         (2.52)

$L_a \frac{di_a}{dt} + R_a i_a = v_a – K_e \dot{θ}_m .$         (2.53)

A state vector for this third-order system is $\pmb x \triangleq \left[\begin{matrix} θ_m & \dot{θ}_m & i_a \end{matrix} \right]^T,$ which leads to the standard matrices

$\pmb F = \left[\begin{matrix}0 & 1 & 0 \\ 0 & -\frac{b}{J_m} & \frac{K_t}{J_m} \\ 0 & -\frac{K_e}{L_a} & -\frac{R_a}{L_a} \end{matrix} \right] , \ \ \ \pmb G = \left[\begin{matrix} 0 \\ 0 \\ \frac{1}{L_a} \end{matrix} \right] , \ \ \ \pmb H = \left[\begin{matrix} 1 & 0 & 0 \end{matrix} \right] , \ \ \ J =0 .$

where the input $u \triangleq v_a.$