PROGRAM LICENSE AGREEMENT LIMITED WARRANTY... 4 Trademarks Contents Preface... 11


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1 . Contents PROGRAM LICENSE AGREEMENT... LIMITED WARRANTY... 4 Trademarks Contents... 8 Preface Overview Jst one other important thing Abot the ser manal Manal System Conventions The FID Software Package Program extensions sed by this program System reqirements How to install the FID Program Description of the dialogs  Mens: The Toolbar Fzzy Resorce Stats View Fzzy Resorce Editor LingVar Editor Fzzy Set Name Editor Fzzy Set View Rle Resorce Editor Rle Definition  Karnagh Veitch Rle Definition  Expert type Defzzification Fzzy Map Dialog Define Inpts NOT sed for map Fzzy Map View PID inpt dialog Linear Plant Definition  ZeroPole representation... 4 FID  User s gide (c) FDGSystems N. Exler
2 4.6 ZeroPoleGain & Deadtime inpt Linear Plant Definition NmeratorDenominator NmeratorDenominator & Deadtime inpt Linear Plant Definition  Mltiple steps MltiStep definition Nonlinear Plant Definition Simlation Definition Connect Fzzy Inpts Connect Distrbance Simlation View Save simlation data SDF Definition Convert Fzzy to ADSPCode Abot FID FzzyLogic on an ADSP2xx Featres Fnction List Specifications Fzzification Memory Model for the fzzification interface: Inference mechanism Memory model for the inference mechanism in loopmode Memory model for the inference mechanism in seqentialmode Defzzification Memory model for the defzzification Tning Inpt variable tning Otpt variable tning Resoltions and Formats Inpt vales and scaling factors Membership fnctions for the inpt vales Membership vales (grades) Membership fnctions (singletons) for the fzzy otpt FID  User s gide (c) FDGSystems N. Exler
3 Crisp otpt vales Implementation of the generated ADSP2xx sorce code in yor specific application Examples rnning on EZLAB board Design of linear and nonlinear static fzzyelements Design of a twolevel fzzy controller Design of a simple fzzy PIcontroller compared to a classical PIcontroller Design of a simple fzzy PIDcontroller Introdction to Fzzy Logic Overview Fndamentals of Fzzy Logic Fzzy sets and membership fnctions Shape of membership fnctions Set operators Strctre of a fzzy system Fzzification Inference and Composition Defzzification Fzzy Control Control strctre Design steps of a fzzy controller Bibliography Index FID  User s gide (c) FDGSystems N. Exler
4 Preface Fzzy logic was first introdced in 965 by Lofti A. Zadeh bt it was ridicled and not taken seriosly b y many of Zadeh's contemporaries. The following years, therefore, saw limited work on fzzy logic and only a small nmber of people concentrated on developing the first applications of it. Inspite of this thogh, the firs t fzzy controller was realized by Mamdani and hi s colleages in the 97's and was sccessflly sed fo r the controller of a rotary cement kiln in Denmark. At the end of the 98's and beginning of the 9's Japanes e engineers showed the world that they had taken fzz y logic very seriosly indeed and were able to presen t nmeros applications of fzzy logic, many of whic h were to be fond in hosehold appliances and consmer articles. The world woke p and sddenly there was a worldwide fzzy boom! Many hman decision processes can be reprodce d relatively accrately with fzzy logic. This means it i s possible to reprodce those processes carried ot b y hmans for which there is no control model. Thi s reprodction is generally not very difficlt. Mor e problematic, however, is the implementation of fzz y logic onto hardware, especially for cheap and fast single processor soltions. There are some special fzzy processors bt these ar e mostly employed as jst coprocessors and are therefor e not likely to become standard. Reglar microprocessors mostly lack the necessary speed for fzzy logi c FID  User s gide (c) FDGSystems N. Exler
5 applications. As fzzy algorithms are principly complex and systematic signal processing problems, the signa l processor is the best sited hardware for dealing wit h them. The Analog Devices ADSP 2xx series has lon g since not jst processed signals bt also has a timer, serial ports, host interface ports, ADC/DAC along wit h mch more on chip. ADSP's, therefore cover both fzzy logic as well as reglar signal processing tasks sch a s FFT, digitial filters, PIDcontrollers and so on. Thi s means the combination of the processor family ADSP  2xx, with one development tool and varied softwar e packages along with their libaries can be sed in a ver y wide range of development areas. The FID Application Software is one of the 3rd part y software packages for the ADSP2xx family tha t spports Fzzy development on DSP's. The FI D Application Software is an easy to se Microsof t Windows based application software for developing, writing and testing fzzy programs based on the ADSP 2xx system. This makes yo independent of specia l fzzy controllers. Vienna, Agst 994 FID  User s gide (c) FDGSystems N. Exler
6 7. Introdction to Fzzy Logic 7. Overview Fzzy logic is a generalization of binary logic (Boolean logic), and therefore, a fzzy logic based system can completely represent a crisp logic system. The converse is not tre. Fzzy logic is based on fzzy set theory and provides a rigoros framework for representing noncrisp sitations. Similarly, the fzzy set theory is a generalization of classical set theory. In 965, Lofti A. Zadeh, a professor of electrical engineering at the University of California, Berkeley expanded the concept of a classically defined set to that of a fzzy set. The basic fzzy rlebased strctre can be sed in many different types of applications, inclding control, process control, decision making, schedling, prediction, and estimation. By allowing high flexibility in the definition of fzzy logic operations, and especially in how the cobination of the firing strength of all rles and the defzzification is performed, the area of applications is even frther increased. 7.2 Fndamentals of Fzzy Logic Unlike binary logic and binary algebra sed in all standard compters, fzzy logic is mltivaled. Instead of an element being a member of a set or not, fzzy logic allows degrees of membership so that something FID  User s gide (c) FDGSystems N. Exler Page: 3
7 can be partially tre and partially false at the same time or in other words, something can be a member of two or more fzzy sets. Fzzy sets which are therefore an essential part of the fzzy set theory, can be defined as ncertain qantities of objects or vales. First of all the meaning of ncertain qantities (fzzy sets) will be explained, starting with crisp qantities Fzzy sets and membership fnctions The separation of people into grops (sets) according to their height can give an idea of crisp and ncertain (fzzy) sets. If, for example, three crisp sets { small, medim, tall} are defined and the appropriate people are allocated to them, classical set theory (crisp sets) demands the definition of two thresholds for three sets. These sets are assigned by the system designer, and are given labels sch as small, medim, tall. e.g.: threshold :,7 m; threshold 2:,8 m The three sets have then the following ranges: small <.7m.7m medim <.8m.8m tall This is shown graphically in figre 36. The circles arond the ranges along the height axis are jst to visalize these ranges as in the classical set theory. Six people with different heights are marked along the height axis (person A:.65m; person B:.85m, person C:.73m, person D:.78m, person E:.7m, person F: FID  User s gide (c) FDGSystems N. Exler Page: 4
8 .8m). A E C D F B height [m] small medim tall crisp sets Fig. 36: Crisp sets for three grops of people. The degree (TRUE or FALSE) of membership µ to a crisp or fzzy set {small, medim, tall} can be described by sing membership f nctions (see figre 37). For crisp sets it isn't really necessary to define membershi p fnctions, bt it makes it easier to nderstand th e difference between membership fnctions and crisp o r fzzy sets later on. µ... degree of membership membership fnctions A E C D F B height [m] small medim tall crisp sets Fig. 37: Membership fnctions describing the degree of membership µ of crisp sets. The great disadvantage of this division (see figre 3 6 and 37) is that a person with a height of,79 m belongs FID  User s gide (c) FDGSystems N. Exler Page: 5
9 to the grop of medim people, bt a person who is jst centimeter taller than the person before belongs to the grop of tall people. Normally we wold say that th e second person is jst a little taller than the first person. One soltion cold be t he definition of more sets, bt the discrimination between the sets remains still very crisp. The division of hman beings and also of natre i n general is mch smoother than in the Boolean logic and therefore mltilevels of separation are sed (e.g. in th e brain for different signal levels are known). The only soltion to this problem is ncertain qantities called fzzy sets in the fzzy set theory. The fzzy se t theory can be seen as a generalization of the ordinary set theory. To extend the crisp sets to fzzy sets it i s necessary to overlap the crisp sets (see figre 38). Th e overlapping region is the fzzy part of the overlappin g sets. A E C D F B height [m] small medim tall fzzy sets Fig. 38: Fzzy sets for three grops of people. This overlapping region of the fzzy sets can b e described by sing membership fnctions (MFs) wit h different shaped edges. e.g.: In figre 39 trapezi m shaped MFs (straight lines with a finit slope) are sed to FID  User s gide (c) FDGSystems N. Exler Page: 6
10 describe fzzy sets. The main other shapes sed ar e trianglar, Gaß and cosshaped MFs. With this concept a person with a height of,79 m belongs with a degree of.75 to the fzzy set medim and with a degree of.25 to the fzzy set tall. In a lingistic interpretation this means, this person is a little bit more medim height than tall. It is now clear that the fzzy concept describes the decision making process o f hmans mch better than the classical mathematica l concepts. µ (S ) i i membership fnctions A E C D F B height [m] small medim tall fzzy sets Fig. 39: Membership fnctions for fzzy sets. Now, it is necessary to give some definitions befor e going frther on. The fzzy sets are described b y fnctions called membership fnctions as alread y defined in figre 37 and 39 which describe the degree of membership of every inpt vale to a fzzy set between the vales and (or and %). In this case th e fzzy sets are normalized to the range of to. Mathematically the fzzy sets can be described i n different kinds of definitions. The most commo n FID  User s gide (c) FDGSystems N. Exler Page: 7
11 definitions are: A fzzy set A in X is a set of ordered pairs: A = { (x,µ A(x)) x X } where µ A(x) is the membership fnction of x in A, which maps the collection of objects in the range of definitio n X (niverse of discorse) to the membership space M. That means in discrete form for the fzzy set medi m before: medim = { (.68,.), (.7,.5), (.72,.), (.74,.), (.76,.), (.78,.), (.8,.5), (.82,.) height ( m) } and in continos form:. height.68 m 25*height m < height <.72 m medim =..72 m height.78 m 25*height m < height <.82 m. height.82 m Beside the first definition the following definition i s qite common: A = µ A (x ) / x + µ A (x 2 ) / x = i µ A(x i) / xi i =.. n or A = x µ (x) / x A That means in discrete form for the fzzy set medim: A=./ /.7 +./ Frther definitions and examples abot fzzy sets can be FID  User s gide (c) FDGSystems N. Exler Page: 8
12 fond in nmeros books and papers describing fzz y logic (see chapter 8) Shape of membership fnctions The shape of membership fnctions has an inflence on the "weighting" of the inpt vales in the range o f definition. e.g. A trianglar shaped MF has only on e inpt vale where the de gree of membership (DOM) µ is. If it is necessary to have a range of inpt vales where the DOM vales shold be it is necessary to s e trapezim shaped MFs. Gaß shaped MFs are often sed for statistically clstered inpt vales. For contro l prposes jst shaped MFs are sed to map the inp t vales to the fzzy sets. In the example of chapter 7.2. jst rectanglar an d trapezim shaped membership fnctions (MFs) hav e been taken into consideration. These shapes of MFs are called shaped. Special cases of shaped MFs are shaped (or trianglar MFs), Zshaped and Sshape d MFs (figre 4). Other shapes for membership fnctions which are often sed are Gaß and cosshaped MF s (figre 4 and 42). All these MFs can be represente d only by for corner points p, p, p, p. and the shape of MF. The vales betwee n these points can be interpolated very easily. For fzzy otpts a more simpler representation of MFs are often sed. The sed "Mfs" are singletons of rea l membership fnctions (figre 43). which can b e described by one or two vales. The first vale is always a crisp vale in the otpt definition range (niverse o f FID  User s gide (c) FDGSystems N. Exler Page: 9
13 discorse) and is called c (cen ter) and the second vale is a weighting on that singleton (a...area). The weightin g vale a (can be nderstood as the area nder the MF ) describes the inflence of a singleton on the crisp otpt vale compared to the other singletons. µ() µ() Zshape /\shape µ() µ() Sshape Fig. 4: Different pishape shaped membership fnctions. µ() µ() Fig. 4: Gassshaped membership fnctions. µ() µ() Fig. 42: Cosshaped membership fnctions. FID  User s gide (c) FDGSystems N. Exler Page: 2
14 µ() µ() c Fig. 43: Singletons for membership fnctions c a In practice, the system designer choose the membership fnctions and the fzzy sets are a reslt of this choice. The same labels which are sed for the membershi p fnctions are sed for the fzzy sets and vice versa. Each of the labels represent a fzzy set positioned in th e operational domain of possible crisp vales. The representation in form of nmbers can be therefor e simplified qite a lot sing shaped membershi p fnctions with only 4 corner points, and a linea r interpolation between these points. Another possibilit y for the representation of the membership fnctions with arbitrary shapes cold be the representation by arrays of nmbers, bt that consmes a lot of memory spac e Therefore, PIshaped MFs are well sited for practica l implementations on microprocessors especially DSP's Set operators Up to now, the differences between fzzy sets an d membership fnctions are described. Therefore th e membership fnction is the mo st important part for fzzy set operations. Operations with fzzy sets (ncertai n sets) are defined via their membership fnctions. The basic operators for the Boolean logic are AND, OR FID  User s gide (c) FDGSystems N. Exler Page: 2
15 and NOT and these operations can also be sed for se t operations. Bt these basic operators are jst sed fo r fzzy set operations in order to describe the problems in a lingistical form. For the actal calclation of fzz y vales the basic operators AND and OR are sbstitte d by different set operators. The most sed set operators for fzzy contro l applications are the minimm an d the maximm operator and sometimes the complement operator (negation). These operators were origionally sed by Zadeh an d Mamdani. Intersection of 2 fzzy sets: C = A B can be described by the minimm operator: µ (x) = min{µ (x), µ (x)} x X C A B Union of 2 fzzy sets: C = A B can be described by the maximm operator: µ (x) = max{µ (x), µ (x)} x X C A B Complement of a fzzy set: A' A can be described by the complement operator: µ (x) =  µ (x) x X A' A e.g.: µ A (x) =.5; µ B(x) =.8 A B min{.5;.8} =.5 A B max{.5;.8} =.8 A' µ A' (x) =  µ A(x) =.5 The binary ANDoperator and the minoperator are very closely related to each other. Similarly, the binary O R operator and the ma xoperator are very closely related to each other. The reason for the first relation is that th e reslt of an ANDoperation is only if both or all inpt FID  User s gide (c) FDGSystems N. Exler Page: 22
16 vales are. For the set operator minimm the sam e conclsion can be sed. The same relation is valid fo r the OR and the maximm operator. The three basic fzzy operations are graphically show n in figre 46. µ() µ () µ () 2 µ() µ() µ () µ () 2 Fig. 46: Minimm, maximm and complement operation µ() µ() µ () µ() The following mathematical characteristics are valid for the min and maxoperators: commtative: op(µ A, µ B ) = op(µ B, µ A) associative: op(µ, µ, µ ) = op(op(µ, µ ), µ ) A B C A B C This two characteristics are mainly important if mor e than two inpt vales are sed in a lingistical rle. If the minoperator is compared to hman decisio n making then it can be interpreted as a pessimisti c operator which always takes the smallest vale. Therefore, this operator has no compensation featre. Sometimes, this featre i s desirable, bt on the other side it is often an ndesi rable featre. e.g. People compensate some weaknesses in certain areas by their strengths i n other areas. The maxoperator is very optimistic an d therefore it is an extremely compensative operator. Besides the min and maxoperator a lot of othe r operators can be sed for the AND and OR operation. (e.g.: algebraic prodct, algebraic sm, drastic prodct, drastic sm,..) FID  User s gide (c) FDGSystems N. Exler Page: 23
17 FID  User s gide (c) FDGSystems N. Exler Page: 24
18 7.3 Strctre of a fzzy system All fzzy logic systems se a rlebase (knowledge base) as their central strctre. Rles, typically cast in an IF... THEN... syntax, represent system operation an d mapping inpts to otpts. Measred and calclate d crisp inpt vales are fzzified, sing membershi p fnctions, into fzzy trth vales (or degrees o f membership). These are then applied as conditions to the rles contained in the rlebase, with triggered rle s specifying necessary act ions, again as fzzy trth vales. These actions are combined and defzzified into crisp, exectable system o tpts. Where inpts and otpts are continos (as in control a pplications), this fzzificationinferencedefzzification process is performed on a n ongoing basis, at reglar sampling intervals. Conceptally this process is similar to the se of a Fas t Forier Transform (FFT), to transform time domai n signals into the freqency domain, to process th e reslting freqencies, and then to transform the reslt s back into time domain. The added expense o f transforming from the time to the freqency domain i s jstified becase the system model is easier t o nderstand and to maniplate in terms of freqencies. Similarly, a fzzy system "transforms" signals from th e "crisp domain" to the "fzzy domain", makes decision s based on these fzzy vales and a knowledge of th e desired system operation cast in fzzy terms (rles), and then transforms the resl ts back into the crisp domain for exection (figre 47). The jstification is, as wit h FID  User s gide (c) FDGSystems N. Exler Page: 25
19 freqency domain processing, that the system model i s easier to nderstand and to maniplate in the fzz y domain than in the crisp domain. Inpts Otpts Crisp to Fzzy Transform Rlebase Fzzy to Crisp Transform "Fzzification" "Inference and Composition" "Defzzification" Fig. 47: FzzificationInferenceDefzzification process This basic fzzy rlebased strctre can be sed i n many different types of applications, inclding control, process control, decisio n making, schedling, prediction, and estimation. By allowing high flexibility in th e definition of fzzy logic operations, and especially i n how the combination of the firing strength of all rle s and the defzzification is performed, the area o f applications is even frther increased Fzzification Consider a fzzy temperatre controller with two inpts. The two inpts are the temperatre T and the change i n temperatre T. The actal measred temperatre T and the calclated change in temperatre T = T old  T are crisp vales. FID  User s gide (c) FDGSystems N. Exler Page: 26
20 For a classical temperatre controller (e.g. PID) only the error in temperatre is sed as an inpt. The otpt o f this PIDcontroller is the actal control action for a control nit (e.g. heating system). If a fzzy temperatre controller, represented by fzz y terms, is sed, then it is necessary to convert (o r transform) the crrent i npts into fzzy inpts by finding the degrees of membership (fzzy trth vales) for al l inpt membership fnctions. This conversion from th e crisp inpt space into the fzzy inpt space is called fzzification. The two inp t variables T and T are called lingistic variables, an d each of these variables consist of a few membershi p fnctions (fzzy terms). The membership fnction s (MFs) are assigned by the system designer (figre 48), and are given labels sch as cold, warm for T and negative, zero, positive for T. Each of these label s represent a fzzy set positioned in the operationa l domain (niverse of discorse) of possible crisp vales. In figre 48 the fzzification process for inpt vales of T = C and T = 2 C/time nit (shown on the x  axis). The degree of membersh ip is the grade vale at the intersection the system inpt vale makes with a membership fnction. FID  User s gide (c) FDGSystems N. Exler Page: 27
21 µ cold warm µ negative zero positive T= 4  ) T= 2 T / [ C] ) T / [ C/time] Fig. 48: Fzzification for two inpt variables. In figre 48 this yields a fzzy inpt vale for T of.75 for MF 'cold', and of.25 for MF 'warm'. This process is repeated for the change of temperatre inpt yielding.2 for MF 'negative',.8 for 'zero', and zero for M F 'positive' Inference and Composition The inference processing (rle evalation) is the central part of the knowledge based decision making, and i s expressed by lingistical rles. Rles are statement s expressing a dependancy relation among system inpt s and system otpts. Individal rles represent parts o f the soltion to a problem. All rles considered togethe r determine the final soltion. Rle evalation takes the fzzy inpts (degrees o f membership) from the fzzification step and the rle s from the knowledge base and calclates fzzy otpts. A typical rle base for a temperatre controller belo w shows in principle the rle evalation. The membership fnctions for the otpt heating are shown in figre 47. FID  User s gide (c) FDGSystems N. Exler Page: 28
22 RULE STRENGTH IF T is cold (.75) AND T is zero (.8) THEN heating is high (.75) IF T is cold (.75) AND T is negative (.2) THEN heating is high (.2) IF T is cold (.75) AND T is positive (.) THEN heating is medim (.) IF T is warm (.25) AND T is negative (.2) THEN heating is medim (.2) IF T is warm (.25) AND T is zero (.8) THEN heating is off (.25) IF T is warm (.25) AND T is positive (.) THEN heating is off (.) Next to each of the membership labels in the IFpart s (antecedants) the corresponding fzzy inpts obtained in the fzzification step are shown in brackets. To the right of each rle the reslting firing strength of the rles ar e shown. These vales can be calclated by sing a se t operator like the minoperator for the AND operation. If more than one r le fires at the same fzzy otpt, then the rle that is most tre will dominate. This can b e implemented by taking the maximm of these rl e strengths and assigning the vale to the correspondin g fzzy otpt MFs. This inference method is called MINMAX inference. i.e. Take the minimm of each condition (antecedant ) and then combine all otpt vales which are assigned to the same otpt MF by sing the maxoperator. Normally, these reslting va les are applied to the otpt MFs by sing Mamdani's minimm operation rle. Another method oft en sed is Larsen's prodct operation rle. The reslting fzzy otpt MF is a reslt o f sperimposing all fired fzzy otpt MFs. This proces s is called composition of otpt MFs to one otpt MF. The rlebase above can also be represented in a KarnaghVeitch (KV) type diagram. The followin g diagram shows all membership levels for the inpts and FID  User s gide (c) FDGSystems N. Exler Page: 29
23 the otpts. Each table entry represents a fzzy rle. T heating cold warm negative high medim T zero high off positive medim off In figre 49 the who le process of fzzification, inference processing, composition and defzzification i s graphically shown for two different rles (one rle i s intentionally different to the rle base above). R l e IF ( T = warm ) AND (dt = zero ) THEN ( H = off ) cold T warm negativezero positive dt MIN off med. high H Inference T dt Fzzification off Composition med. high Defzzification H H R l e cold T warm negativezero positive dt MAX off med. high Inference H 2 IF ( T = cold ) OR (dt = positive ) THEN ( H = medim ) Fig. 49: A fzzy controller with 2 inpts, otpt and 2 rles. FID  User s gide (c) FDGSystems N. Exler Page: 3
24 7.3.3 Defzzification Finally, the defzzification process converts the fzz y otpts from the rle evalation step into crisp syste m otpts. There are several possible methods o f performing the defzzification. A commo n defzzification method, especially for control prposes, is the center of gravity (COG) or centroid defzzification method.there are also a lot of other popla r defzzification methods like the mean of maxim m (MOM), maximm left (ML) or maximm right (MR ) defzzification. The fzzy otpts obtained in the inference an d composition step are sed to trncate the corresponding otpt membership fnction by the appropiate trt h vales as shown in figre 5. Then, the center of gravity of the reslting fzzy set is fond by finding the balance point of the reslting membership fnction and only the vale along the heating axis, which is a reslt of th e projection of the center of gravity to the heating axis, is sed as the crisp otpt (shown in figre 5 with a n arrow). off med. high % % COG heating Fig. 5: Center of gravity defzzification FID  User s gide (c) FDGSystems N. Exler Page: 3
25 The reslt of the defzzification sing other methods than COG is shown in figre 5. off med. high % heating % ML MR MOM Fig. 5: MOM, ML and MR defzzification. Instead of sing discrete membership fnctions for the otpt fzzy sets, singletons are often sed for embedded control applications. Then, the following simple comptation formla can be sed: Centroid = all x µ(x)*x / all x µ(x) where µ(x) is the fzzy otpt vale and x are the centroids in the niverse of discorse of X. For the example mentioned in chapter 2.2, the otpt becomes: heating = (.25 * % +.25 * 5% +.75 * %) / ( ) = 7 % This crisp otpt vale can be sed directly to perform a control action or some other data post processing can be performed. FID  User s gide (c) FDGSystems N. Exler Page: 32
26 7.4 Fzzy Control 7.4. Control strctre The standard control loop strctre is shown in figr e 52. The ADSP performs all the data pre and post  processing and the fzzy c ontrol on a single processor. If it is necessary the fzzy controller can be combined with classical controllers like PID's to realize mixed contro l soltions with a very high performance at low cost ADSP S ca l i n g F z if y Fzzy  Controller S M B S NB NS PS M NS NS PS B Rlebase PS PS PB D e f zz i f y S ca l i n g Data reference vales Data preprocessing (e.g. FFT, filtering,..) postprocessing D A control action Plant otpt vales measred vales D A Fig. 52: Fzzy controller with 2 inpts, otpt and 9 rles Design steps of a fzzy controller Start with a definition of the problem. Normally, this is not a mathematical model, jst how the system to be controlled shold work. Definition of all inpt (which can be measred or calclated) and otpt variables (lingistic variables) and their inpt and otpt ranges. Definition of membership fnctions for each inpt and otpt variable. FID  User s gide (c) FDGSystems N. Exler Page: 33
27 stats 3 type 3 Simlation controller 39 definition 46, 47 Distrbance 44, 5 examples 46, 48, 49 fzzy 48 fzzy closedloop 49 Fzzy PID 5 PID 2 plant 39 save 52 SDF definition 52, 53 view 5 FID  User s gide (c) FDGSystems N. Exler