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

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

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.

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

„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.

Fuzzy Sets Fuzzy set: -membership function

Examples of membership functions

Product of sets (operation „and”)

Sum of sets (operation „or”)

Illustration of the product and sum

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

Defuzzification

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.

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

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

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

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

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

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

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

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

Control rules of PD „fuzzy controller” for sltank.mdl error e derivative dh control u e=0.275 dh=0.0555 u=-0.0651 The result is obtained in the form of a nonlinear, static element described by the function:

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

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

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

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.

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

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.

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

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

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

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.

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

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

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

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.

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

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.

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.

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

„Athans Zadeh debate”