Digital velocity control of DC motors

Digital control

The controller models described previously are continuous. The actual controllers are discrete approximations of these continuous models. For accuracy of the approximation, we need all of the following to be the case:

• The sampling frequency is much larger than the natural frequency of the system.
• Time delays caused by computation are insignificant.
• The control value does not saturate.
• The mechanical damping \(B\) is small in comparison with the effects of the proportional term \(K_P\) in the controller.

Applying Tustin’s method to the continuous PI controller transfer function \(K_P+\frac{K_I}{s}\), the corresponding discrete transfer function is \[ \begin{aligned} \frac{U_a(z)}{E(z)}=\frac{b_{0}+b_{1}z^{-1}}{a_{0}+a_{1}z^{-1}}, \end{aligned} \qquad{(1)}\] where \[ \begin{aligned} a_{0} = 1,\quad a_{1} = -1, \quad b_{0}=K_P+\frac{1}{2}K_I T,\text{ and}\quad b_{1}=-K_P+\frac{1}{2}K_I T, \end{aligned} \] where \(T\) is the sample time. The measured error is \[ E(n)=\Omega_R(n)-\Omega_J(n). \]

For more on Tustin’s method, see subsection 6.4.2. In lab 7, we will implement the corresponding difference equation using the general-purpose algorithm developed in lab 6.