Inference in Fuzzy Models of Physical Processes

Inference in Fuzzy Models of Physical ProcessesBohdan S.Butkiewicz1Warsaw University of Technology,Institute of Electronic Systems,00-665Warsaw,Poland"B.Butkiewicz"bb@.pl.pl/~bb/index.htmlAbstract.General idea of the paper is comparison of different reasoningmethods,which may be used in some types of fuzzy models.Differenttriangular norms and defuzzification methods were used.It is shownthat many reasoning methods give similar results.However,many ofthem are not very reasonable.Some simple theorems about functionsapproximated by models are presented.Special attention is applied tomodeling of physical processes.Examples of models used in reality arepresented.Some of them are build as modifications of Takagi-Sugenomodel introduced earlier by author.1IntroductionThere are many classes of fuzzy models.One of possible classification was given by Pedrycz[10].He arranged model categories in order of an increased level of structural dependencies.The least structured category appearsfirst on the list –tabular representations–fuzzy grammars–fuzzy relational equations–fuzzy neural networks–rule based models–local regression models–fuzzy regression modelsIn the paper,analyze of this categories is performed from point of view of possi-ble applications in modeling of physical processes.Fuzzy grammar[9]models are used for describing time series and signal classifiers.Fuzzy relational equation models were largely studying by Pedrycz,Nola,Hirota,and some others mainly from theoretical point of view.It seems that these models are good for system identification.Tabular,rule based,and local regression models are studying in the paper.Most popular reasoning:Mamdani[8],Larsen[7],Tsukamoto[13], and Takagi-Sugeno[12]are considered.Special attention is applied to inference methods used during approximate reasoning to obtain good results.General-ized Mamdani and generalized Larsen reasoning are used with different triangu-lar norms and some other parison of results obtained for some models of physical processes are presented.B.Reusch(Ed.):Fuzzy Days2001,LNCS2206,pp.782–790,2001.c Springer-Verlag Berlin Heidelberg2001Inference in Fuzzy Models of Physical Processes 7832Description of Models 2.1Tabular ModelsTabular model has a form of a table,where basic relations between linguistic labels of inputs and outputs are presented.Relations describing dynamics of sin-gle input single output (SISO)system of first order may be presented as a single table.Rows and colons denote linguistic values of input and derivative of input.In the table are placed linguistic values of output.The model is very popular in fuzzy modeling and control.It may presents,for example,fuzzy controller of proportional-derivative (PD)type.Tabular model may be suitable for physical processes,especially when we have no much information about process behavior.Example,steam boiler may be described by quantity of supplied water and fuel,temperature and pressure inside of boiler and quantity of outgoing ing linguistic values small,medium,large,low,medium,high steady-state behavior of boiler may be described.But good description of dynamics requires knowledge about time constant and may be other parameters,so requires analytical description of process.Tabular model is suitable also for simple discrete systems.Some theoretical and practical results obtained in fuzzy control area [2][3],and examples presented in the paper show that general Mamdani reasoning,where operations minimum and maximum were replaced by different triangular t-norms and s-norms,is good for this type of model.Moreover,for different tri-angular norms the results are similar,so it is not very important what pair of norms is used during reasoning.Also the results are identical if rules are used in aggregated form or not.Consider now a simple example of a function y =f (x )with saturation describing by seven rulesR1:if x is P L then y is P LR2:if x is P M then y is P MR3:if x is P S then y is P SR4:if x is ZE then y is ZEand symmetrically for negative values.If Mamdani,Larsen,Tsukamoto or Takagi-Sugeno model is used then rule weights equals to membership value µ{.}(x )of x for respective set {.}.Height,areas,and gravity defuzzification methods are weighted means y =i y i w i i w i(1)Also Tsukamoto model use weighted mean.Suppose that membership func-tions µ{.}(x )have symmetrical triangular shape with trapezes at the end of universum [-10,10].Thus,membership is linear function µ{.}(x )=a i x +b i where a i ,b i are constant in some regions.If height defuzzification method is used then values y i are independent on weight w i ,except trapezes.So,y = y i (a i x +b i )/ (a i x +b i ).Similar situation is observed for Larsen model.The results are presented in the Fig.1.Generally,a theorem may be easy proofed.Theorem 1Suppose that membership functions are symmetric and described in different784 B.S.Butkiewiczregions by polynomials of n-th order.If Mamdani or Larsen models are used with height defuzzification method then models are described piecewise by ratio-nal functions y =a n x n +a n −1x n −1+...+a 0b n x n +b n −1x n −1+...+b 0(2)Now,consider Tsukamoto method.Let similarly µ{.}(x )=a i x +b i .FunctionsFig.1.Mamdani model with height defuzzification,logic (left),Yager (right)opera-tionsµ{.}(y )=c i |y |and values y i =µ{.}−1(w i )=w i /c i .So,y =[ (a i x +b i )2/c i ]/ (a i x +b i ).Generally,one obtains theorem.Fig.2.Mamdani (left)and Larsen (right)models with area defuzzification and logic operationsTheorem 2If membership functions are symmetric and described in different regions by poly-nomials of n-th order and Tsukamoto model is used then model is describedInference in Fuzzy Models of Physical Processes 785piecewise by rational functionsy =a n x 2n +a 2n −1x n −1+...+a 0b n x n +b n −1x n −1+...+b 0(3)Finally,consider Mamdani and Larsen models with areas and gravity methods.For symmetrical triangular shapes values y i are constant but areas S i depend on the weights w i in square S i =A (1−w i /2)w i .Thus,for areas method y = S i w i / S i = Ay i (a i x +b i )[1−(a i x +b i )2/2]/ A (a i x +bi )[1−(a i x +bi )2/2].General dependence is more complicated.An example is presented in the Fig.2.Two other examples of Mamdani and Tsukamoto models are shown in the parison of models with Mamdani,Larsen and TsukamotoFig.3.Mamdani model with gravity defuzzification (left)and Tsukamoto model (right)both with algebraic operationsreasoning with different triangular norms and defuzzification methods shows that choice of triangular norm has no big influence on the result,curves obtained are very similar.Grater influence has defuzzification method,especially when height method is compared with areas and gravity.Mamdani and Larsen models give similar results.Tsukamoto model is different.Some interesting results concerning equivalence of approximated reasoning using different interpretation of fuzzy if-then rules and aggregation problem are presented in [6].Sometimes one uses Wang model.It was proofed,(Wang theorem [14])that any continuous function may be exactly approximated by fuzzy tabular model with gaussian membership functions for input and consequences.Gaussian func-tions are not very convenient to use and may have nothing common with physical process behavior.If we have some knowledge about functional relations between process vari-ables it is better to use Takagi Sugeno model.786 B.S.Butkiewicz2.2Rule-based ModelsThis type of models is the most popular.This approach to modeling seems more general.The rules may contain heterogeneous form,and different statements Rules can be graduated by word quantifiplex analyze is impossible,because of possible system and rules diversity.It seems that rule based models are good for situations where non numerical values are expected as decision or model output,example possibility that output take some linguistic value.It may describe some sociological,medical and other decision problems.An example of model supporting human decision for personnel selection in tourist agency is presented below.Suppose that chief of agency looks for a can-didate,which can work as a guide.The candidate ought to:–known at least two languages among English,French,German,Spain in very high level–known history and geography of a region in high level–know-how to use telefax,xserox,computer–have pleasant sight at least in satisfactory level–be responsible and patient in high level–be able to resolve unexpected problems at least in medium level–................Let the candidate fulfill each feature with some level.For example he has a note from an exam or something like this.There are several candidates.Who is the best?Any numerical value of the candidate feature can be treated as fuzzy number with membership µC (x ).Any requirement can be described by fuzzy set with membership µR (x ).Overlapping membership functions it is possible to find level l =max {min [µC (x ),µR (x )]}of feature satisfaction.Finding weighted sum w i l i ,where w i are weight of i −th feature choused before,it is possible to find the best candidate.Other good solution is put l =max [µC (x )µR (x )].2.3Local Regression ModelsThis models are the best if we have some,may be not exact,mathematical de-scription for physical process.However,the model can be used also without this knowledge.Takagi-Sugeno model may be considered under some conditions [15]as universal approximator.Very popular is Takagi-Sugeno-Kang model (TSK)[11]use linear functions as local approximations.Author experience shown that using other than linear functions,example polynomials of second order,with-out real knowledge of local system behavior,not gives better approximation.Contrary,the results can be considered as bad,i.e.not justified by any reason.Modification of TSK modelThe TSK model may describe sufficiently well any continuouse function y =f (x 1,x 2,...,x n ).However,the model has one very important inconvenience.Sup-pose that model describe a function y =f (x )and is composed with two rules if x is Small then y =a 1x +b 1Inference in Fuzzy Models of Physical Processes787 if x is Large then y=a2x+b2If conventional triangular or trapezoidal functions are use for membership func-tions of fuzzy sets of x,here for Large and Small,see Fig.4(left),then non expected effect arise,see Fig.4(right)and5,and[1].In intermediate region parabolic distortion appears.Thus,the model is worth than conventional crisp model with two linear functions and two separated regions.Of course,it is pos-Fig.4.Membership functions for x(left),TSK model and chord(right)sible use conventional spline convolution model for better approximation,but it is complicated.Author proposed in[4]some simple modification of TSK model which can avoid inconvenience of TSK model.Suppose that we have some exper-imental data forming two straight lines with different slopes and an intermediate region.TSK model gives function g(x).Two lines in separate regions may be joined directly in intermediate region by chord line c(x),but this solution is not very good.However,is is possiblefind better solution.In intermediate region the weighted meanu(x)=g(x)+λc(x)1+λ(4)can be taken as model value,whereλis a constant choosed experimentally for good approximation of the data.The result is shown in the Fig.5.The data represent strange effect of optical property relaxation observed in chalkogenide viteouse semiconductor glasses after gamma irradiation[5].Very interesting property of Takagi-Sugeno model is possibility of knowledge discovering.If mathematical description of some phenomena is known,building this model for unknown process we may verify what phenomena are observed in this process.Presented example shown that two effects are discovered using model built for relaxation process of optical properties in chalcogenides.After gamma irradiation along the time T elapse transparency of semiconductor glass changes in accordance with two different lows.Membership functions of the model give appropriate regions for the lows.2.4Fuzzy Regression ModelsSometimes it is not possible to introduce in the model all variables,which have influence on physical process.Simply,these variables are not measured or are not788 B.S.ButkiewiczFig.5.TSK model without modification(left)and after modification(right) possible to measure.However,the model must take often in consideration influ-ence of these variables.An example of such situation may be sintering process.It depends on actual total mass of details in the furnace,ambient temperature not possible to preview many days before etc.Reasonable solution is to build fuzzy regression model where conclusion is a fuzzy number or fuzzy function or/and to build fuzzy-probability model where conclusion is random variable or function. In this way additional uncertainty may be introduced.Numerical considerations concerning inference method are limited to an ex-ample of model of sintering process.Different components,as Cu,MnS,C (graphite),StZn are added to iron powder.Exact description of physical and chemical changes during sintering is unknown,because of their complexity.Thus, mathematical model can not be built.Main task of model was preview geomet-rical changes of detail dimensions after sintering process,taking in consideration proportion of powder components,temperature in sintering zone of the furnace, velocity of tape transporting details in the furnace,and initial density of pressed powder.It was impossible to gathered data in special way.Production process can not be interrupted.However,some specimens with different components were prepared and put in the furnace together with produced details.Thus,some data for the model were very difficult to compare,and results were sometimes dis-crepant.First,rule-based model was built.Mamdani and Larsen methods are compared.After,modified Takagi Sugeno model is proposed and accepted.The rules have formif Cu is S and C is S then∆h=f1(Cu,C,σ1)if Cu is S and C is M then∆h=f2(Cu,C,σ2) ...................................if StZn is S then∆h=f10(StZn,σ10) ...................................if T emp is L and T ime is L then∆h=f17(T emp,T ime,σ17)where∆h describes changes of any parameter,example the height of detail,vari-ables Cu,C,StZn,...contents of the powder components,and variables T emp, T ime the temperature and time in sintering zone of the furnace.VariablesσiInference in Fuzzy Models of Physical Processes789 have a special task.Functions f i are fuzzy functions.Its values are fuzzy num-bers with trapezoidal shapes of membership functions.Valueσi desribe width of trapezes,so it describe uncertainty.It is approximate model of the process. Any rule may be considered as multidimensional cloud with different density. Example of this fuzzy surface is presented in the Fig.6.Fig.6.Example of fuzzy surface for rule1.Stars represent center value of fuzzy num-bers,points minimal and maximal values3ConclusionPhysical processes may be modeled in two ways depending on knowledge of mathematical relations between process variables.Without any knowledge,rule-based fuzzy Mamdani and Larsen models or Takagi-Sugeno-Kang fuzzy regres-sion model with linear functions can be build.In many types of models Mamdani reasoning with logic operation(minimum and maximum)is used.Author expe-rience shows that also other triangular norms,ex.algebraic operations,can be used with success.However,many operations give similar results.In the paper only a few examples are presented,but author have tried many other triangular rsen reasoning is underestimated.It gives also good results and often is easier in numerical applications then Mamdani.If we have some knowledge in the form of functional dependencies for relations between variables,we may use these functions as conclusions in some rules in Takagi-Sugeno fuzzy regres-sion model.Theoretically it is the best solution in this case,but practically it is difficult tofind the best shape of membership functions,so estimation of model parameters is complicated.If standard triangular or trapezoidal shape for mem-bership functions is used then strange effect arises in intermediate regions,shown as example presented in the paper.Therefore,author proposed new solution for Takagi-Sugeno model.In intermediate region weighted mean of standard model and chord joining ends of intermediate region is introduced.790 B.S.ButkiewiczReferences1.Babuska R.:Fuzzy Modeling for Control,Kluwer Academic Publisher(1998).2.Butkiewicz B.S.:Steady-State Error of a System with Fuzzy Controller,IEEETransactions on System,Man,and Cybernetics,Part B:Cybernetics,Vol.28,No.6, (1998)855–860.3.Butkiewicz B.S.:About Robustness of Fuzzy Logic PD and PID Controller un-der Changes of Reasoning Methods,European Symp.on Intelligent Techniques, Aachen,Germany(2000)350–356.4.Butkiewicz B.S.:Fuzzy Reasoning Methods,its Properties and Applications(inpolish),accepted for Prace Naukowe Politechniki Warszawskiej,Elektronika.5.Butkiewicz B.S.,Golovchak R.,Kovalskiy A.,Shpotyuk O.,and Vakiv M.:Onthe Problem of Relaxation for Radiation-Induced Optical Effects in Some Ternary Chalcogenide Glasses,Radiation Effects and Deffects in Solids,(2000).6.Czogala E.,Leski J.,On Equivalence of Approximate Reasoning Results UsingDifferent Interpretations of Fuzzy if-then Rules,Fuzzy Sets and Systems,Vol.117, (2001)279–296.rsen P.M.:Industrial application of fuzzy logic control,Int.J.Man MachineStudies,Vol.12,No.1(1980)3–10.8.Mamdani E.H.:Application of fuzzy algorithm for control of simple dynamic plant,Proc.IEE,Vol.121,No.12(1974)158—1588.9.Mizumoto M.,Toyoda J.,Tanaka K.:General formulation of formal grammars,Information Science,Vol.4(1972)87–100.10.Pedrycz W.:Fuzzy Models:Methodology,Design,Applications and Challenges,inPerdycz W.(ed.)Fuzzy Modelling Paradigms end Practice,Kluwer Acad.Publ.pp3-22,1996.11.Sugeno M.,Kang G.T.:Structure Identification of Fuzzy Model,Fuzzy sets andSystems,Vol.28(1988)15–33.12.Takagi T.,Sugeno M.:Fuzzy Identification of Systems and its Application to Mod-eling and Control,IEEE Trans.on Systems,Man,and Cybernetics,Vol.15(1985) 116-132.13.Tsukamoto Y.:An approach to fuzzy reasoning method,in Fuzzy Set Theory andApplications,Gupta M.M.,Ragade R.K.,Yager R.R.,(eds.),Amsterdam,North-Holland(1979).14.Wang X.L.:Fuzzy Systems are Universal Approximators,Proc.IEEE Int.Conf.on Fuzzy Systems,San Diego,CA(1992)1163–1169.15.Ying H.,Ding Y.,Li S.,Shao S.:Comparison of Necessary Conditions for TypicalTakagi-Sugeno and Mamdani Fuzzy System as Universal Approximators,IEEE Trans.on Systems,Man,and Cybernetics-Part A,Vol.29,No.5(1999)508–514.。