1. PID Control Systems (Proportional, Integral, Derivative)
A presentation to the Robotics Society of Southern California
May 11, 2013
By Alex Brown

2. Introduction PID is a method of controlling a dynamic system.
It consists of 3 simple calculations which are added together to
generate a control signal to make your system track to a reference
value. A tuned system gives the fastest response to an error.
It doesn’t take a lot of math. Just testing to determine gains.
It really only works well for simple linear systems; but its concepts
can be applied to more complex systems.

3. Closed Loop Control System
Signal
Controller
Process
Reference
E.g. motor,
arm, temp, etc.
E.g. PID
Feedback

4. Control System for Robot Drive Motors
PWM
MOTOR
Reference
Forward Speed
PID
Feedback
Tachometer or position encoder

5. Motor characteristics
No Load
“Normal” load
Speed
Heavy load
Stall
PWM
100%
PWM
No load
with load
Motor
Speed
Friction
Inertia

6. P term (Proportional) PWM Reference MOTOR Forward Speed+ Kp
MOTOR _
Feedback (Actual Speed)
Bump 8 8
Speed Ref
Speed Ref
Step change 4 4 2
PWM
PWM 1 8
Actual
Speed
Actual
Speed 4 2 4 1
Hits bump

7. I term (Integral) Reference MOTOR Forward Speed
Error
PWM
Reference
Forward Speed + + KP
MOTOR _ + KI
Feedback (Actual Speed) 8
Turn integrator on here 8
Speed Ref
Speed Ref 4 4
Error 2
Error 2
Integrator
Integrator
PWM
PWM
Actual
Speed
Actual
Speed 4 2 4 2
Hits bump

8. Integrator Windup Speed Ref Integ PWM Actual Speed Kp 4 300+ 50 100 50Kp
Integ
300+ 50
PWM
100 50
Actual
Speed 4 2

9. Integrator Windup (cont.)
Rate KD _
PWM
Reference
Forward Speed + KP +
MOTOR _ + KI
Stop
LOGIC
Example logic:
stop if total PWM is >= 100 and error is polarity to increase integrator.
Stop if actual speed is converging with reference value
stop if it is known that integrator will cause subsequent error.

10. D term (Derivative) Reference MOTOR Forward Speed
Rate KD _
PWM
Reference
Forward Speed + KP +
MOTOR _ + KI 8
Speed Ref 4
Feedback (Actual Speed)
Error 2 Kp
Rate
PWM
Actual
Speed 4 2

11. PID PROs CONs Kp KI KD PID Removes short term errors quickly
Doesn’t remove long term errors Kp
Has a fixed time constant
Removes long term errors
Removes errors very slowly.
May cause integrator windup. KI KD
Improves short term stability.
Sometimes is NECESSARY
depending on what is being
controlled.
PID
Acts as above providing
optimally fast response to
errors
Don’t have to use all the terms.
We don’t always want
optimally fast responses.

12. Is that all there is?
In theory, a PID controller can succeed for any system that is linear.
However, it may not perform just the way we would like.
A pure PID system works best on a linear system with a slow moving
reference or small reference changes.
That’s no fun! We want our robots to respond quickly but smoothly
to large changes of command.
And our robot systems usually have many non-linearities.
I worked on autopilot design for 30 years and never heard the word
“PID” until I retired and started on robots. Our autopilots used one or
more of the three PID terms at appropriate times usually with logic
and/or other devices to achieve good performance in the real world.

13. Slow response to speed reference change
As indicated earlier, a PI drive motor system will not quickly achieve the
new reference speed until the integrator runs long enough to command the
necessary additional PWM.
We know that under normal conditions, there is a proportion between
speed and PWM.
Hence another term may be added (a “FeedForward” term) that approximates
the PWM required at any commanded speed. This will usually provide much
faster approach to the new reference and leave the integrator to only pick up
any residual error .
KFF
Rate KP _ +
PWM
Reference
Forward Speed + KP +
MOTOR _ + KI

14. KP + KI + KFF KP Only KP + KI KFF 8 Speed 4 Reference error 100
error
100
Integrator 40
100
KFF 40
100
PWM

15. Loops within Loops Outer loop Middle loop Servo loop or inner loop.
PID
PID
PID
Motor
Motor feedback
feedback
E.g. Navigation
This may be
the slowest
response loop.
Maneuvers vehicle
smoothly. E.g
speed and steering.
Can be slower than
servo loop.
Fastest loop. Moves
motor/actuator to
desired speed/position
quickly.

16. Trapezoidal Speed Profile
PID tracking
Speed
Reference
Stopped
Constant
acceleration
Constant
speed
Constant
deceleration
Stopped
You can eliminate much of the PID tracking lag by adding an acceleration feed-forward term.
Acceleration using this method works well. Deceleration is challenging due to the difficulty
of determining when to begin slowing at a constant acceleration to stop at a target distance.
My method is to continuously calculate the accel required to stop by the target distance. When
that accel is >= desired deceleration, switch to controlling to the calculated accel and vary plus
or minus to stop at target distance.

17. Application to real robots.
Differential:
MOTOR
Reference
Forward Speed
PWM
PID
MOTOR
Actual Speed
Average
PWM
PID
MOTOR
Reference
Forward Speed
PWM
PID
MOTOR
Both will work to hold forward speed, but both will have poor directional control
with nothing to keep them in sync.

18. Or, a downstream “equalization” system as below
Need an upstream steering system like the below
Reference
Forward Speeds
PWM
PID
MOTOR
Reference
Fwd Speed +
Steering
Left wheel
PWM
PID
MOTOR
Right wheel
Or, a downstream “equalization” system as below
MOTOR _
Reference
Forward Speed
PWM +
PID _ +
MOTOR
Actual Speed
Average
But, it is still going to require steering someday.

19. Ackerman Steering forward speed control PID
Reference
Forward Speed
PWM
PID
MOTOR
Actual Speed
Forward speed control is independent of steering

20. Or, as I prefer, an upstream steering system AND modifying the PID loop to
control position (or distance) rather than speed.
Reference
Forward Distance
PWM
Common
Reference
Fwd Dist +
Steering
PID
MOTOR
Left wheel
PWM
PID
MOTOR
Right wheel
Advantages:
Simple PID loop. Only requires Kp.
Very accurate dead reckoning of distance and steering.
No integrator windup concerns. Minor distance offset will
supply any PWM needed up to 100%
Disadvantages:
Requires feedback to Distance loop to ensure distance commands
do not exceed motors ability to follow.

21. Generate Trapezoidal speed profile
Reference
Forward Distance
Speed and Accel
Left wheel
AccelRef
KFFacc _ +
Common
Reference
Fwd Dist +
Steering
DistRef +
PWM Kp +
MOTOR +
SpeedRef
KFFspd
Right wheel
PWM
Same as above
MOTOR
Maximum
accel
Reference
accel
Reference
speed
Reference
distance
Speed
error
Target
fwd speed
Accel
limiter
Generate Trapezoidal speed profile
Calculate accel
to stop at
target distance
Logic to
begin decel

22. Steering Control We have two type of steering systems commonly used.
Differential Drive Ackerman Steering

23. Steering Task Wall Following Target distance Target distance Measured
Error
Error

24. +/- commands to L & R motor control loops Lateral KP Error KD Lateral
Steering
displacement
+/- commands to L & R
motor control loops
Lateral
Error KP +
Rate KD
Steering
displacement _
Lateral
Error KP +
Angle command to
steering servo

25. Steering
displacement
Lateral
Error
Limiter
45 deg + KP
Error

26. Rate KD ? _
Lateral
Error KP +
Error

27. Conclusion
Pure PID is best for inner loop where you want fast response. Even there
just use the terms necessary to get the performance you want.
Use the P, I & D terms freely in designing outer loop navigation when
you want your control to converge on a steady reference exponentially.
For more info on pure pid, google pid tutorials
For more of my slant on it, try abrobotics.tripod.com. It includes a
simulation program written by (ex) clubmember Lior Elazary.