1. Jan Jantzen jj@inference.dk www.inference.dk 2013
Fuzzy PID Control
Jan Jantzen
2013

2. Summary
Reduce design choices
Tuning

3. Design Procedure Build and tune a conventional PID controller first.
Replace it with an equivalent linear fuzzy controller.
Make the fuzzy controller nonlinear.
Fine-tune the fuzzy controller.
Relevant whenever PID control is possible, or already implemented

4. Single Loop Control Load Noise
The controller should preferably be able to follow the reference r, reject load changes l and noise disturbances n, but these requirements are in conflict with each other. We would like to transfer PID tuning methods to the fuzzy controller in order to have a tuning method.

5. Rule Base With 4 Rules
1. If error is Neg and change in error is Neg then control is NB
3. If error is Neg and change in error is Pos then control is Zero
7. If error is Pos and change in error is Neg then control is Zero
9. If error is Pos and change in error is Pos then control is PB
The four rules can handle many cases, and they are sufficient for a linear controller.

6. Textbook PID Controllers
Continuous version
Discrete version
Incremental, discrete version

7. Fuzzy P controller f Rule base u GU U GE E e Gain on error
Gain on control f
Rule base u GU U GE E e
Provided that the rule base acts like the identity function
By comparison with the P controller equation

8. FP Rule Base 1. If E(n) is Pos then u(n) is 100
2. If E(n) is Neg then u(n) is -100
With a proper choice of membership functions the controller will act like a linear P controller

9. Fuzzy PD Controller e GE GCE f Rule base E CE u GU U de/dt
Provided that the rule base acts like a summation
Now we know what the gains do

10. FPD Rule Base 1. If E(n) is Neg and CE(n) is Neg then u(n) is -200
3. If E(n) is Neg and CE(n) is Pos then u(n) is 0
7. If E(n) is Pos and CE(n) is Neg then u(n) is 0
9. If E(n) is Pos and CE(n) is Pos then u(n) is 200
With a proper choice of membership functions the controller will act like a linear PD controller. Four rules are sufficient.

11. Fuzzy PD+I Controller CE e GE f PD rules GCE + GU E GIE IE u U de/dt
It is better that the integral action bypasses the rule base. It saves rules.

12. Fuzzy Incremental Controller
The output is a change to the previous state e GE
GCE f
Rule base E CE
GCU
1/s U CU cu
de/dt
This is an integrator. It could be a valve position, for instance.
The increment. It is a change to the sum of all previous signals.

13. Fuzzy - PID Gain Relation
Controller Kp
1/Ti Td FP
GE×GU
FInc
GCE×GCU
GE/GCE
FPD
GCE/GE
FPD+I
GIE/GE
It tells what each fuzzy gain does to the proportional gain, the derivative gain, and the integral gain. Conversely, given values for Kp, Ti and Td we can find one or more sets of values for the fuzzy gains. Very important table.

14. Tuning
Process gain
If we increase Kp too much, the system might oscillate or even become unstable
If we increase Kp, we suppress load changes.
If we increase Kp, the response will be more sensitive to noise.

15. Ziegler-Nichols Tuning
Increase Kp until oscillation, Kp = Ku
Read period Tu at this setting
Use Z-N table for approximate controller gains

16. Ziegler-Nichols (freq. method)
Controller Kp Ti Td P
0.5Ku PI
0.45Ku
Tu/1.2
PID
0.6Ku
Tu/2
Tu/8
Given values for Ku and Tu, the table provides the gains in the three controller cases. Easy, but often the result is a poorly damped system.

17. Z-N oscillation of 1/(1+s)3
The ultimate gain Ku = 8, and the ultimate period is Tu = 15/4 s

18. PID control of 1/(1+s)3 Response to a reference step
Response to a load step

19. Fuzzy FPD+I control of 1/(1+s)3
The response is the same as for PID control
Trajectory on the control surface, which is a plane
The membership functions are linear

20. Hand-Tuning Set Td = 1/Ti = 0
Tune Kp to satisfactory response, ignore any final value offset
Increase Kp, adjust Td to dampen overshoot
Adjust 1/Ti to remove final value offset
Repeat from step 3 until Kp large as possible

21. Quick reference to controllers
Advantage
Disadvantage FP
Simple
Maybe too simple
FPD
Less overshoot
Noise sensitive, derivative kick
FInc
Removes steady state error, smooths control signal
Slow
FPD+I
All in one
Windup, derivative kick

22. Scaling e GE GCE f Rule base E CE u GU U α 1/α de/dt
The linear controller is invariant towards scaling. In the nonlinear controller we can use it to avoid saturation in the input universes.

23. Summary Design crisp PID Replace it with linear fuzzy
Make it nonlinear
Fine-tune it

24. Advanced Section

25. Nyquist 1/(s+1)3 with PID -2 2 -1 1
Kp = 4.8, Ti = 15/8, Td = 15/32

26. Tuning Map 1/(s+1)3 -2 2 000 a) 001 b) 010 c) 011 d) 100 e) 101 f) 1102
000 a)
001 b)
010 c)
011 d)
100 e)
101 f)
110 g)
111 h)