PID control of speed and torque

A PID controller produces a control signal from the error $e(t)$ between reference and measured value using proportional, integral and derivative actions:

$$u(t) = K_pe(t) + K_i\int_0^te(\tau)\,d\tau + K_d\frac{de(t)}{dt}$$ $$G_c(s) = \frac{U(s)}{E(s)} = K_p + \frac{K_i}{s} + K_ds$$

PID speed and torque control of a DC motor (cascade control)

Motor equations (separately excited DC):

$$V_a = R_aI_a + L_a\frac{dI_a}{dt} + K\omega, \qquad T = KI_a = J\frac{d\omega}{dt} + B\omega + T_L$$

Torque is proportional to armature current, so torque control = current control.

 w*->(+)->[Speed PID]-Ia*->(+)->[Current PI]
      ^-   (limit)          ^-       |
      |                     |   firing/duty
      |                     |        v
      |                     |   [Converter]
      |                     |        | Va
      |                     |   [DC motor]--> w
      |                     |     |  |
      |  current sensor <---+-----+  |
      +---- tacho/encoder <----------+

1. Outer speed loop: speed error $\omega^* - \omega$ goes to the speed PID (usually PI). Its output is the current (torque) reference $I_a^*$, limited to the maximum allowed current, so the motor accelerates at maximum safe torque.
2. Inner current (torque) loop: current error $I_a^* - I_a$ goes to the current PI controller, which sets the converter firing angle (rectifier) or duty cycle (chopper). This loop is fast; it quickly corrects supply-voltage changes and limits current during starting, overload and stall.
3. Converter supplies $V_a$ to the armature.
4. Feedback: tachogenerator/encoder for speed, CT/Hall sensor/shunt for current.

For an induction motor the same structure is used in vector (field-oriented) control: an outer speed PI produces a torque-current command $i_{qs}^*$, and inner current PI loops control the inverter.

Behaviour

• On a sudden load increase, speed drops; the speed PID raises $I_a^*$, torque increases and speed returns to the reference with zero steady-state error (integral action).
• During starting, the speed error is large, $I_a^*$ saturates at the limit, so the motor accelerates at constant maximum torque.
• Derivative action is used sparingly (filtered) because of noise in speed measurement.

Advantages: accurate speed, fast torque response, inherent current protection, simple tuning of each loop separately.