1. (thanks to Gary Fedder)
PID Controls of Motors
Howie Choset
(thanks to Gary Fedder)

2. Controls Review of Motor Model Open Loop (controller-free) Response
Proportional Control
Stable
Faster Response = Bigger Overshoot
Steady State Error
PI Control
Maintain Stability
Decrease Steady State Error = Bigger Overshoot
PID Control
Derivative term reduces overshoot, settling time
Feed Forward
Overcome damping

3. Mass-Spring-Damper Model (Analogy)
Model of mass spring damper system
z(t) position, z(t) velocity
t0 initial time, z(t0), z(t0) initial position & velocity

4. Review of Motor Model
moment of inertia of the rotor (J) [kg.m^2/s^2] * damping ratio of the mechanical system (b) [Nms] * electromotive force constant
Ke is volt (electromotive force) per radians per second (V/ rad/sec)
torque constance
Kt is torque amp (Nm/Amp)
* electric resistance (R) = [ohm]
electric inductance (L) = [H]
(VL = L di/dt) * input (V): Source Voltage * output (theta): position of shaft * The rotor and shaft are assumed to be rigid

5. Review of Motor Model Mechanical Equation of Motion
Torque is proportional to current (Lenz’s Law)
Back emf is proportional to motor speed (Faraday’s Law)
Mechanical Equation of Motion
Electrical Equation of Motion
Assume (K=Ke=Kt)
Solve for sq/V
Speed, theta dot

6. Transfer Function of Motor (with Approximations). =
Open Loop Transfer Function . =
Can rewrite function in terms of an
electrical and mechanical behavior
Electrical time constant on motor is much smaller
example motor with
equivalent time constants . =
For small motors, the mechanical
behavior dominates (electrical transients die faster).

7. Open Loop Response (to a Step)
Apply constant voltage
Slow response time (lag)
Weird Apples-to-Orange relationship between input and output
If you want to set speed, what voltage do you input?
Weird type of steady state error
No reaction to perturbations
Plant
Input
Voltage
Output
Speed

8. Closed Loop Controller
Controller Evaluation
Steady State Error
Rise Time (to get to ~90%)
Overshoot
Settling Time (Ring) (time to steady state)
Stability
Give it a velocity command
and get a velocity output
Plant
Controller - +
Ref
error
voltage

9. Close the loop analogy

10. Asymptotic Stability:

11. Closed Loop Response (Proportional Feedback)
Proportional Control
Easy to implement
Input/Output units agree
Improved rise time
Steady State Error (true)
P: Rise Time vs.  Overshoot
P: Rise Time vs.  Settling time R +
error
voltage
Controller
Plant -
Voltage = Kp error

12. Closed Loop Response (PI Feedback)
Proportional/Integral Control
No Steady State Error
Bigger Overshoot and Settling
Saturate counters/op-amps
P: Rise Time vs.  Overshoot
P: Rise Time vs.  Settling time
I: Steady State Error vs. Overshoot
Ref +
error
voltage
Plant -
Voltage = (Kp+1/s Ki) error

13. Closed Loop Response (PID Feedback)
Proportional/Integral/Differential
Quick response
Reduced Overshoot
Sensitive to high frequency noise
Hard to tune
P: Rise Time vs.  Overshoot
P: Rise Time vs.  Settling time
I: Steady State Error vs. Overshoot
D: Overshoot vs. Steady State Error R +
error
voltage
Plant -
Voltage = (Kp+1/s Ki + sKd) error

14. Feed Forward Volt Decouples Damping from PID To compute
Try different open loop inputs and measure output velocities
For each trial i,
Tweak from there. . +
error R +
volt
Plant
Controller + -

15. Follow a straight line with differential drive
Error can be difference in wheel velocities or accrued distances
Make both wheels spin the same speed
asynchronous – false start
wheels can have slight differences (radius, etc)
Make sure both wheels spin the same amount and speed
false start
More complicated control laws – track orientation
m1vref = vref + K1 * thetaerror + K2 * offset error
modeling kinematics of robot
dead-reckoning

16. Encoders

17. Encoders – Incremental
Photodetector
Encoder disk
LED Photoemitter

18. Encoders - Incremental

19. Encoders - Incremental
Quadrature (resolution enhancing)

20. Where are we?
If we know our encoder values after the motion, do we know where we are?

21. Where are we?
If we know our encoder values after the motion, do we know where we are?
What about error?

22. Encoders - Absolute More expensive Resolution = 360° / 2N
where N is number of tracks
4 Bit Example