1. Whether the Spreaded Good Opinion About Fuzzy Controllers is Justified
Ryszard Gessing
Silesian Technical University
Gliwice, Poland

2. Outline of Presentation
Preamble
„Fuzzy Controllers”-an introduction
„Fuzzy” or conventional PID?
„Controllers” or nonlinearities?
Simpler implementation of nonlinearities
The linear fuzzy block may be replaced by usual summing junction
Conclusion

3. Why a such a paper about „fuzzy controllers”?
Because I do not share the opinion about
superiority of „fuzzy controllers”, which
to frequently appears in literature;
To give automatic control educators and
indirectly - to students the arguments
supporting different point of view.

4. Outline of Presentation
Preamble
„Fuzzy Controllers”-an introduction
„Fuzzy” or conventional PID?
„Controllers” or nonlinearities?
Simpler implementation of nonlinearities
The linear fuzzy block may be replaced by usual summing junction
Conclusion

5. „Fuzzy Controllers”-an introduction
They base on „fuzzy logic” making it possble to account nonunique verbal formulations;
The primary goal was to make it possible to utilize the experts knowledge expressed in linguistic form;
They base on such notions as: fuzzy sets, membership functions, control rules, fuzzyfication, activation, aggregation, accumulation, defuzzyfication.

6. Fuzzy Sets
Fuzzy set:
-membership function

7. Examples of membership functions

8. Product of sets (operation „and”)

9. Sum of sets (operation „or”)

10. Illustration of the product and sum

11. Control rules – „fuzzy controller”
error e
control u
If error neg. then control neg.
If error zero then control zero
If error pos. then control pos.
The result: „fuzzy controller” P
described by nonlinearity:
fuzzyfication activation accumulation
defuzzyfication

12. Defuzzification

13. Remarks
In the expert control rules, without membership functions, it is contained little information;
The experts usually are not able to specify neither the membership functions nor the operations „and”, „or”;
In connection with this, the membership functions, as well as the operations „and”, „or” are chosen for control rules intuitively.

14. Outline of Presentation
Preamble
„Fuzzy Controllers”-an introduction
„Fuzzy” or conventional PID?
„Controllers” or nonlinearities?
Simpler implementation of nonlinearities
The linear fuzzy block may be replaced by usual summing junction
Conclusion

15. Example sltank.mdl from MATLAB demo: non-linear plant with actuator containing integrator
sltank.mdl, sltankcorr.mdl

16. The membership functions for sltank.mdl
derivative dh
error e
valve control u

17. Conventional controller PD
„Fuzzy controller” PD
Conventional controller PD
Control constraints:

18. Conventional controller PD
„Fuzzy controller” PD
Conventional controller PD
Control constraints:

19. The membership functions for sltank2.mdl
error e
derivative dh
valve control u
sltank2.mdl sltank2corr.mdl

20. Conclusions MATLAB demos: sltank.mdl sltankrule.mdl sltank2.mdl
speak for PID controller.

21. Outline of Presentation
Preamble
„Fuzzy Controllers”-an introduction
„Fuzzy” or conventional PID?
„Controllers” or nonlinearities?
Simpler implementation of nonlinearities
The linear fuzzy block may be replaced by usual summing junction
Conclusion

22. Control rules of PD „fuzzy controller” for sltank.mdl
error e
derivative dh
control u
e=0.275
dh=0.0555 u=
The result is obtained in the form of a nonlinear, static element
described by the function:

23. Example of the surface described by the function u=f(e,dh) for sltank
Example of the surface described by the function u=f(e,dh) for sltank.mdl u
u=f(e,0) u e u e dh
u=f(0,dh) dh

24. u=f(e,0)
u=f(e,0)
u=f(0,dh)
u=f(0,dh)

26. Example of the surface described by the function u=f(e,dh) for sltank2
Example of the surface described by the function u=f(e,dh) for sltank2.mdl u u e u e dh dh

27. Conclusions As the result we obtain not a controller but
a nonlinearity described by the function u=f(e,de) – or static nonlinear block, with difficult for shaping nonlinear characteristic;
The parts P and D of the „fuzzy controller” are implemented outside this block;
The static fuzzy block may be interpreted as nonlinear summing junction.

28. Fuzzy and conventional controller
Nonlinear summing
junction
„Fuzzy controller” PD
Conventional controller PD

29. Additional remark
A serious drawback is nonanalitical description of the function u=f(e,de) obtained using fuzzy approach, which creates additional difficulties for stability and quality analysis; only the methods based on simulations may be used.

30. Outline of Presentation
Preamble
„Fuzzy Controllers”-an introduction
„Fuzzy” or conventional PID?
„Controllers” or nonlinearities?
Simpler implementation of nonlinearities
The linear fuzzy block may be replaced by usual summing junction
Conclusion

31. Table of Control Rules – Compact notation of the Control Rules
Notation used for control u in the table
NB - negative big
NM - negative medium
PM - positive big
PB - positive medium

32. Table of the controls (look-up table):
determines the function u=f(e,de) in discrete points

33. Conclusions
Look up tables with an appropriate interpolation are significantly better and simpler method of implementation of the function u=f(e,de);
They make it possible to shape nonlinearity locally and suite it to nonlinearity of the plant;
Implementation of an appropriate nonlinearity by means of fuzzy method is an ineffective way with drudgery.

34. Outline of Presentation
Preamble
„Fuzzy Controllers”-an introduction
„Fuzzy” or conventional PID?
„Controllers” or nonlinearities?
Simpler implementation of nonlinearities
The linear fuzzy block may be replaced by usual summing junction
Conclusion

35. For given below membership functions and appropriate operations used for design of the fuzzy block we obtain the linear function u=f(e,de)

36. Really, we obtain the plain described by the function: u=f(e,de)=e+de
and the fuzzy block may be replaced by the usual
summing junction.
startPID.m

37. Other systems with „fuzzy controllers” which have been researched by our students:
Inverted pendulum with swing-up action;
Regulation of a position of a ball in a pipe (a laboratory system);
Industrial controller FM355 (Siemens) with original auto-tuning for fuzzy and conventional PID controllers.
For these systems the „fuzzy controllers”
became also worse than conventional ones.

38. Outline of Presentation
Preamble
„Fuzzy Controllers”-an introduction
„Fuzzy” or conventional PID?
„Controllers” or nonlinearities?
Simpler implementation of nonlinearities
The linear fuzzy block may be replaced by usual summing junction
Conclusion

39. The arguments presented say that this opinion is not justified at all.
Whether the Spreaded Good Opinion About Fuzzy Controllers is Justified?
The arguments presented say that this opinion is not justified
at all.

40. Theoretically, the fuzzy block described by u=f(e,de) gives some limited possibility of improving controller, but:
The same possibility gives nonlinear static element
described by the function u=f(e,de), which may be
easier realized using other mentioned method;
Local shaping of the demanded nonlinearity using fuzzy approach is very difficult and ineffective;
The problem of needed nonlinearity u=f(e,de) is weakly recognized in nonlinear control theory;
Great disadvantage is nonanalitical description of the nonlinearity u=f(e,de) obtained by using fuzzy approach.

42. M. Athans http://fuzzy.iau.dtu.dk/debate.nsf

48. „Athans Zadeh debate”