Electromechanical System Modeling

We can model an armature-controlled direct current (DC) motor with a load.

• Input: Armature voltage $v_a$
• Output: Angular position of the motor shaft $\theta (t)$ for position control, or angular velocity of the motor shaft $\omega (t)$ for speed control.
• Intermediate variables: Armature current $i_a(t)$, angular position $\theta (t)$ or angular velocity $\omega (t)$

KVL applied to the armature circuit gives:

$$R_a{i}_a(t)+L_a\frac{d{i}_a}{dt}+K\frac{d\theta (t)}{dt}=v_a(t)$$

Rotational version of Newton’s second law applied to motor shaft:

$$J\frac{d^2\theta (t)}{d{t}^2}=K{i}_a(t)-K_f\frac{d\theta (t)}{dt}$$

Position Control Case

We choose the state variables, input and output variables

$$\begin{array}{rl}{x}_1(t)& =i_a(t)\\ x_2(t)& =\theta (t)\\ x_3(t)& =\omega (t)\\ u(t)& =v_a(t)\\ y(t)& =\theta (t)\end{array}$$

State-space model:

$$\left[\begin{array}{c}{\dot{x}}_1\\ {\dot{x}}_2\\ {\dot{x}}_3\end{array}\right]=\left[\begin{array}{ccc}-\frac{R_a}{L_a}& 0& -\frac{K}{L_a}\\ 0& 0& 1\\ \frac{K}{J}& 0& -\frac{K_f}{J}\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]+\left[\begin{array}{c}\frac{1}{L_a}\\ 0\\ 0\end{array}\right]u(t)$$

and

$$y=\left[\begin{array}{ccc}0& 1& 0\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]$$

Speed Control Case

If we replace $\frac{d\theta (t)}{dt}$ with $\omega (t)$, we get the differential equation of a DC motor system in the speed control case:

$$\begin{array}{r}{R}_a{i}_a(t)+L_a\frac{d{i}_a(t)}{dt}+K\omega (t)=v_a(t)\\ J\frac{d\omega }{dt}=K{i}_a(t)-K_f\omega (t)\end{array}$$

State-space model:

$$\left[\begin{array}{c}{\dot{x}}_1\\ {\dot{x}}_2\end{array}\right]=\left[\begin{array}{cc}-\frac{R_a}{L_a}& -\frac{K}{L_a}\\ \frac{K}{J}& -\frac{K_f}{J}\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+\left[\begin{array}{c}\frac{1}{L_a}\\ 0\end{array}\right]u(t)$$

and

$$y(t)=\left[\begin{array}{cc}0& 1\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]$$