(1) The control processes are too complex to analyze by conventional quantitative techniques.

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1 Chapter 0 Fuzzy Control and Fuzzy Expert Systems The fuzzy logc controller (FLC) s ntroduced n ths chapter. After ntroducng the archtecture of the FLC, we study ts components step by step and suggest a desgn procedure of the FLC. An example of the desgn procedure s also gven. The structure and functon of the fuzzy expert systems are smlar to those of the FLC, and thus the desgn procedure of FLC can be used n the fuzzy expert systems. 0. Fuzzy logc controller 0.. Advantage of fuzzy logc controller Fuzzy logc s much closer n sprt to human thnkng and natural language than the tradtonal (classcal) logcal systems. Bascally, t provdes an effectve means of capturng the approxmate, nexact nature of the real world. Therefore, the essental part of the fuzzy logc controller (FLC) s a set of lngustc control strategy based on expert knowledge nto an automatc control strategy. The FLC s consdered as a good methodology because t yelds results superor to those obtaned by conventonal control algorthms. In partcular the FLC s useful n two cases. () The control processes are too complex to analyze by conventonal quanttatve technques. (2) The avalable sources of nformaton are nterpreted qualtatvely, nexactly, or uncertanly. Indeed, the advantage of FLC can be summarzed as follows. () Parallel or dstrbuted control: n the conventonal control system, a control acton s determned by sngle control strategy lke µ = f(x, x 2,, x n ). But n FLC, the control strategy s represented by multple fuzzy rules, and thus t s easy to represent complex systems and nonlnear systems. (2) Lngustc control: the control strategy s modeled by lngustc terms and thus t s easy to represent the human knowledge. (3) Robust control: there are more than one control rule and thus, n general, one error of a rule s not fatal for the whole system Confguraton of fuzzy logc controller There s no systematc procedure for the desgn of an FLC. However we can present here a basc confguraton of FLC as shown n Fg 0.. The confguraton conssts of four man components: fuzzfcaton nterface, knowledge base, decson-makng logc, and defuzzfcaton nterface. Knowledge base Input Fuzzfcaton nterface Defuzzfcaton nterface Output Inference State varable Control output Controlled system (process) Control varable Control nput Fg 0. Confguraton of FLC

2 () The fuzzfcaton nterface transforms nput crsp values nto fuzzy values and t nvolves the followng functons. - receves the nput values - transforms the range of values of nput varable nto correspondng unverse of dscourse - converts nput data nto sutable lngustc values (fuzzy sets). Ths component s necessary when nput data are fuzzy sets n the fuzzy nference. (2) The knowledge base contans a knowledge of the applcaton doman and the control goals. It conssts of a data base and a lngustc rule base. - The data base contans necessary defntons whch are used n control rules and data manpulaton. - The lngustc rule base defnes the control strategy and goals by means of lngustc control rules. (3) The decson-makng logc performs the followng functons - smulates the human decson-makng procedure based on fuzzy concepts - nfers fuzzy control actons employng fuzzy mplcaton and lngustc rules. (4) The defuzzfcaton nterface the functons - a scale mappng whch converts the range of output values nto correspondng unverse of dscourse - defuzzfcaton whch yelds a nonfuzzy control acton from an nferred fuzzy control acton. In the followng sectons, the control components wll be developed n detal Choce of state varables and control varables Before startng the detaled procedure of the FLC desgn, we have to choose the varables. A fuzzy control system s desgned to control a process, and thus t s needed to determne state varables and control varables of the process. The state varables become nput varables of the fuzzy control system, and the control varables become output varables. Selecton of the varables depends on expert knowledge on the process. In partcular, varables such as state, state error, state error devaton, and state error ntegral are often used. 0.2 Fuzzfcaton nterface component In the fuzzfcaton component, there are three man ssues to be consdered: scale mappng of nput data, strategy for nose and selecton of fuzzfcaton functons. () Scale mappng of nput data: We have to decde a strategy to convert the range of values of nput varables nto correspondng unverse of dscourse. When an nput value s come through a measurng system, the values must be located n the range of nput varables. For example, f the range of nput varables was normalzed between and +, a procedure s needed whch maps the observed nput value nto the normalzed range. (2) Strategy for nose: When observed data are measured, we may often thnk that the data were dsturbed by random nose. In ths case, a fuzzfcaton operator should convert the probablstc data nto fuzzy numbers. In ths way, computatonal effcency s enhanced snce fuzzy numbers are much easer to manpulate than random varables. Otherwse, we assume that the observed data do not contan vagueness, and then we consder the observed data as a fuzzy sngleton. A fuzzy sngleton s a precse value and hence no fuzzness s ntroduced by 2

3 fuzzfcaton n ths case. In control applcatons, the observed data are usually crsp and used as fuzzy sngleton nputs n the fuzzy reasonng. (3) Selecton of fuzzfcaton functon: A fuzzfcaton operator has the effect of transformng crsp data nto fuzzy sets. x = fuzzfer (x 0 ) Where x 0 s a observed crsp value and x s a fuzzy set, and fuzzfer represents a fuzzfcaton operator. Fg 0. shows a fuzzfcaton functon whch transforms crsp data nto a fuzzy sngleton value. µ F (x) x 0 x Fg 0. Fuzzfcaton functon for fuzzy sngleton Fg 0.2 shows a fuzzfcaton functon transformng a crsp value nto a trangular fuzzy number. The peak pont of ths trangle corresponds to the mean value of a data set, whle the base s twce the standard devaton of the data set. µ F (x) x 0 x base Fg 0.2 Fuzzfcaton functon for fuzzy trangular number 3

4 0.3 Knowledge base component 0.3. Data base The knowledge base of an FLC s comprsed of two parts: a data base and a fuzzy control rule base. We wll dscuss some ssues relatng to the data base n ths secton and the rule base n the next secton. In the data base part, there are four prncpal desgn parameters for an FLC: dscretzaton and normalzaton of unverse of dscourse, fuzzy partton of nput and output spaces, and membershp functon of prmary fuzzy set. () Dscretzaton and normalzaton of unverse of dscourse The modelng of uncertan nformaton wth fuzzy sets rases the problem of quantfyng such nformaton for dgtal computers. A unverse of dscourse n an FLC s ether dscrete or contnuous. If the unverse s contnuous, a dscrete unverse may be formed by a dscretzaton procedure. A data set may be also normalzed nto a certan range of data. ) Dscretzaton of a unverse of dscourse: It s often referred to as quantzaton. The quantzaton dscretzes a unverse nto a certan number of segments. Each segment s labeled as a generc element and forms a dscrete unverse. A fuzzy set s then defned on the dscrete unverse of dscourse. The number of quantzaton levels affects an mportant nfluence on the control performance, and thus t should be large enough to gve adequate approxmaton. That number should be determned n consderng both the control qualty and the memory storage n computer. For the dscretzaton, we need a scale mappng, whch serves to transform measured varables nto values n the dscretzed unverse. The mappng can be unform (lnear), nonunform (nonlnear), or both. Table 0. shows an example of dscretzaton, where a unverse of dscourse s dscretzed nto 3 levels (-6, -5, -4,, 0,,, 5, 6). Table 0. An example of dscretzaton Range x 2.4 Level No < x < x < x < x < x < x < x < x < x < x

5 +. < x < x 6 2) Normalzaton of a unverse of dscourse: It s a dscretzaton nto a normalzed unverse. The normalzed unverse conssts of fnte number of segments. The scale mappng can be unform, nonunform, or both. Table 0.2 shows an example, where the unverse of dscourse [-6.9, +4.5] s transformed nto the normalzed closed nterval [-, ]. Table 0.2 An example of normalzaton Range Normalzed segments Normalzed unverse [ 6.9, 4.] [ 4., 2.2] [.0, 0.5] [ 0.5, 0.3] [ 2.2, 0.0] [ 0.3, 0.0] [.0, +.0] [ 0.0, +.0] [0.0, +0.2] [+.0, +2.5] [+0.2, +0.6] [+2.5, +4.5] [+0.6, +.0] (2) Fuzzy partton of nput and output spaces A lngustc varable n the antecedent of a rule forms a fuzzy nput space, whle that n the consequent of the rule forms a fuzzy output space. In general, a lngustc varable s assocated wth a term set. A fuzzy partton of the space determnes how many terms should exst n a term set. Ths s the same problem to fnd the number of prmary fuzzy sets (lngustc terms). There are seven lngustc terms often used n the fuzzy nference: NB: negatve bg NM: negatve medum NS: negatve small ZE: zero PS: postve small PM: postve medum PB: postve bg A typcal example s gven n Fg 0.3 representng two fuzzy partton n the same normalzed unverse [, +]. Membershp functons wth trangle and trapezod shapes are used here. 5

6 N Z P 0 + (a) N: negatve, Z: zero, P: postve NB NM NS ZE PS PM PB 0 + (b) NB, NM, NS, ZE, PS, PM, PB Fg 0.3 Example of fuzzy partton wth lngustc terms The number of fuzzy terms n a nput space determnes the maxmum number of fuzzy control rules. Suppose a fuzzy control system wth two nput and one output varables. If the nput varables have 5 and 4 terms, the maxmum number of control rules that we can construct s 20 (5 4) as shown n Fg 0.4. Fg 0.5 shows an example of system havng 3 fuzzy rules. x 2 PS ZO NS NB NB NS ZO PS PB x Fg 0.4 A fuzzy partton n 2-dmenson nput space 6

7 x 2 bg small small bg x Fg 0.5 A fuzzy partton havng three rules A fuzzy control system could always nfer a proper control acton for every state of process. Ths property s concerned wth the supports on whch prmary fuzzy sets are defned. The unon of these supports should cover the related unverse of dscourse n relaton to some level. For example, n Fg 0.3, any nput value s ncluded to at least one lngustc term wth membershp value greater than 0.5. (3) Membershp functon of prmary fuzzy set There are varous types of membershp functons such as trangular, trapezod, and bell shapes. Table 0.3 shows an example defnng trangular membershp functons on the dscretzed unverse of dscourse n Table 0.. For example, term NM s defned such as: µ NM ( 6) = 0.3 µ NM ( 5) = 0.7 µ NM ( 4) =.0 µ NM ( 3) = 0.7 µ NM ( 2) = 0.3 Table 0.3 Defnton of trangular membershp functon Level No. NB NM NS ZE PS PM PB

8 An example of bell shaped membershp functon s gven n Table 0.4 and Fg 0.6, where fuzzy sets are defned on the normalzed unverse of dscourse [, +] gven n Table 0.2. They have the shapes of functon of parameter mean m f and standard devaton σ f. µ ( x m x) = exp{ 2σ f f ( 2 f ) 2 } Table 0.4 Defnton of bell-shaped membershp functon Normalzed unverse Normalzed segments m f σ f Fuzzy sets [.0, 0.5] NB [ 0.5, 0.3] NM [ 0.3, 0.0] NM [.0, +.0] [ 0.0, +0.2] ZE [+0.2, +0.6] PS [+0.6, +0.8] PM [+0.8, +.0] PB NB NM NS ZE PS PM PB 0 + Fg 0.6 Example of bell-shaped membershp functon 8

9 0.3.2 Rule base A fuzzy system s characterzed by a set of lngustc statements usually represented by n the form of f-then rules. In ths secton, we examne several topcs related to fuzzy control rules. ) Source of fuzzy control rules There are two prncpal approaches to the dervaton of fuzzy control rules. The frst s a heurstc method n whch rules are formed by analyzng the behavor of a controlled process. The dervaton reles on the qualtatve knowledge of process behavor. The second approach s bascally a determnstc method whch can systematcally determne the lngustc structure of rules. We can use four modes of dervaton of fuzzy control rules. These four modes are not mutually exclusve, and t s necessary to combne them to obtan an effectve system. - Expert experence and control engneerng knowledge: operatng manual and questonnare. - Based on operators control actons: observaton of human controller s actons n terms of nput-output operatng data. - Based on the fuzzy model of a process: lngustc descrpton of the dynamc characterstcs of a process. - Based on learnng: ablty to modfy control rules such as self-organzng controller. 2) Types of fuzzy control rules There are two types of control rules: state evaluaton control rules and object evaluaton fuzzy control rules. () State evaluaton fuzzy control rules: State varables are n the antecedent part of rules and control varables are n the consequent part. In the case of MISO (multple nput sngle output), they are characterzed as a collecton of rules of the form. R : f x s A, and y s B then z s C R 2 : f x s A 2, and y s B 2 then z s C 2 R n : f x s A n, and y s B n then z s C n where x, y and z are lngustc varables representng the process state varable and the control varable. A, B and C are lngustc values of the varables x, y and z n the unverse of dscourse U, V and W, respectvely =, 2,, n. That s, x U, A U y V, B V z W, C W In a more general verson, the consequent part s represented as a functon of the state varable x, y. R : f x s A, and y s B then z = f (x, y) The state evaluaton rules evaluate the process state (e.g. state, state error, change of error) at 9

10 tme t and compute a fuzzy control acton at tme t. In the prevous secton concerned wth the fuzzy partton of nput space, we sad that the maxmum number of control rules s defned by the partton. In the nput varable space, the combnaton of nput lngustc term may gve a fuzzy rule. When there s a set of fuzzy rules as follows R : f x s A, and y s B then z s C =, 2,, n The rules can be represented as the form of table n Fg 0.7. y B n C 6 C n C 5 C 5 B 2 C 3 C 4 C 4 B C C 2 C 7 A A 2 A n x Fg 0.7 Fuzzy rules represented by a rule table (2) Object evaluaton fuzzy control rules: It s also called predctve fuzzy control. They predct present and future control actons, and evaluate control objectves. A typcal rule s descrbed as R : f (z s C (x s A and y s B )) then z s C. R 2 : f (z s C 2 (x s A 2 and y s B 2 )) then z s C 2. R n : f (z s C n (x s A n and y s B n )) then z s C n. A control acton s determned by an objectve evaluaton that satsfes the desred states and objectves. x and y are performance ndces for the evaluaton and z s control command. A and B are fuzzy values such as NM and PS. The most lkely control rule s selected through predctng the results (x, y) correspondng to every control command C, =, 2,, n. In lngustc terms, the rule s nterpreted as: f the performance ndex x s A and ndex y s B when a control command z s C, then ths rule s selected, and the control command C s taken to be the output of the controller. 0.4 Inference (Decson makng logc) In general, n decson makng logc part, we use four nference methods descrbed n the prevous 0

11 chapter: Mandan method, Larsen method, Tsukamoto method, and TSK method Mandan method Ths method uses mnmum operator as a fuzzy mplcaton operator, and max-mn operator for the composton as shown n secton 9.4. Suppose fuzzy rules are gven n the followng form. R : f x s A and y s B then z s C, x U, A U y V, B V z W, C W =, 2,, n ) When nput data are sngleton such as x = x 0 and y = y 0 (n ths case, the nput data are not fuzzfed.), the matchng degrees (frng strength) of A and B are µ A ( x 0 ) and µ B ( y 0 ), respectvely. Therefore the matchng degree of rule R s α = µ A µ B ( x0) ( y0) Then µ C ( z) ( z) = α µ C where derved from ndvdual control rules s defned as follows: µ ( z) = V[ α µ C n = n = C = U C C ( z)] 2) When nput data are fuzzy sets, A and B =, 2,, n ( z) C s the result of rule R. The aggregated result C α = mn[max( µ ( x) µ ( x)), max( µ ( y) µ ( y))] µ ( z) = α µ C C x A The aggregate result C s defned by µ ( z) = V[ α µ C n = n = C = U C C ( z)] A Larsen method Ths method uses the product operator ( ) for the fuzzy mplcaton, and the max-product operator for the composton. Suppose fuzzy rules are gven n the followng form. R : f x s A and y s B then z s C, =, 2,, n ) When nput data are sngleton, x = x 0 and y = y 0. The matchng degrees s v B B

12 α = µ A µ B ( x0) ( y0) The resultc of rule R s defned by µ ( z) = α ( z) C µ C The aggregated result C s µ C ( z) = V[ α µ C or n = n = C = U C ( z)] 2) When nput data are gven as the form of fuzzy sets, A and B, we have matchng degrees as α = mn[ max( µ A ( x) µ A( x)), max( µ B ( y) µ B( y))] The resultc of rule R s defned by µ ( z) = α ( z) C µ C The aggregate result C s µ C ( z) = V[ α µ or n C = U C = n = x C ( z)] Tsukamoto method Ths method s used when the consequent part of each rule s represented by fuzzy set wth a monotonc membershp functon. The nferred output of each rule s defned as a crsp value nduced by the rule s matchng degree (frng strength). We suppose fuzzy rules are gven n the followng form and the set C has a monotonc membershp functon µ (z) C R : f x s A and y s B then z s C, =, 2,, n The matchng degree α of each rule s defned lke n the prevous methods n the cases of both sngleton nput and fuzzy set nput. The result of z rule R s obtaned by (Fg 0.8) z = µ C ( α ) v 2

13 The aggregated result z s taken as the weghted average of each rule s output αz + α 2z z = α + α 2 2 Ths method gves a crsp value as an aggregated result and thus there s no need to defuzzfy t. If x x 2 then y µ = 2 z 2 R : z µ = 2 (z ) 2 R 2 : z R 3 : µ = z z R 4 : µ = z z Fg 0.8 Example of Tsukamoto control rules If x, x 2 then y R : y =.0x + 0.5x R 2 : y 2 = 0.x + 4.0x R 3 : y 3 = 0.9x + 0.7x R 4 : y 4 = 0.2x + 0.x Fg 0.9 Example of TSK fuzzy control rules 3

14 0.4.4 TSK method Ths method s used when the consequent part s gven as a functon of nput varables. R : f x s A and y s B then z s f (x, y) Where z = f(x, y) s a crsp functon of nput varables x and y. Usually f(x, y) has a polynomal form (Fg 0.9). Suppose nput data are sngleton x 0 and y 0, then the nferred result of rule R s f (x 0, y 0 ). The matchng degree α of R s same wth the prevous one. Therefore the fnal aggregated result z s the weghted average usng the matchng degree α α f z = x y + f α + α ( 0, 0) α2 2( 0, 0) 2 The fnal result s a crsp value and thus there s no need to defuzzfy t. x y 0.5 Defuzzfcaton In many practcal applcatons, a control command s gven as a crsp value. Therefore t s needed to defuzzfy the result of the fuzzy nference. A defuzzfcaton s a process to get a non-fuzzy control acton that best represents the possblty dstrbuton of an nferred fuzzy control acton. Unfortunately, we have no systematc procedure for choosng a good defuzzfcaton strategy, and thus we have to select one n consderng the propertes of applcaton case. The three commonly used strateges are descrbed n ths secton Mean of maxmum method (MOM) The MOM strategy generates a control acton whch represents the mean value of all control actons, whose membershp functons reach the maxmum (Fg 0.0). In the case of a dscrete unverse, the control acton may be expressed as z 0 = k j = z j k z j : control acton whose membershp functons reach the maxmum. k: number of such control actons. µ(z) z 0 z Fg 0.0 Mean of maxmum (MOM) Center of area method (COA) The wdely used COA strategy generates the center of gravty of the possblty dstrbuton of a fuzzy set C (Fg 0.). In the case of a dscrete unverse, thus method gves 4

15 n C j = 0 = n z µ ( z ) z j = µ ( z ) C j j j Where n s the number of quantzaton levels of the output, C s a fuzzy set defned on the output dmenson (z). µ(z) z 0 z Fg 0. Center of area (COA) Bsector of area (BOA) The BOA generates the acton (z 0 ) whch parttons the area nto two regons wth the same area (Fg 0.2). z0 µ ( ) = β C z dz α z0 µ ( z) dz where α = mn{z z W} C β = max{z z W} Fg 0.2 Bsector of area (BOA) 5

16 0.5.4 Lookup table Even wth the many advantages, t s ponted out that the FLC has the problem of tme complexty. It takes much tme to compute the fuzzy nference and defuzzfcaton. Therefore a lookup table s often used whch smply shows relatonshps between nput varables and control output actons. But the lookup table can be constructed after makng the FLC and dentfyng the relatonshps between the nput and output varables. In general, t s extremely dffcult to get an acceptable lookup table of a nonlnear control system wthout constructng a correspondng FLC.E Example 0. Table 0.5 shows an example of lookup table for the two nput varables error (e) and change of error (ce), and control varable (v). The varables are all dscretzed and normalzed n the range [, +]. For example, when e =.0 and ce = 0.5, we can obtan v = 0.5 by usng the lookup table nstead of by executng the full fuzzy controller. Table 0.5 Example of lookup table e ce Desgn procedure of fuzzy logc controller When we decded to desgn a fuzzy logc controller, we can follow the followng desgn procedure ) Determnaton of state varables and control varables In general, the control varable s determned dependng on the property of process to be controlled. But we have to select the state varables. In general, state, state error and error dfference are often used. The state varables are nput varables, and the control varables are output of our controller to be developed. 2) Determnaton of nference method We select one method among four nference methods descrbed n the prevous secton. The decson s dependent upon the propertes of process to be studed. 3) Determnaton of fuzzfcaton method It s necessary to study the property of measured data of state varables. If there s uncertanty n the data, the fuzzfcaton s necessary, and we have to select a fuzzfcaton method and membershp functons of fuzzy sets. If there s no uncertanty, we can use sngleton state varables. 4) Dscretzaton and normalzaton of state varable space In general, t s useful to use dscretzed and normalzed unverse of dscourse. We have to decde whether t s necessary and how we can do. 5) Partton of varable space. 6

17 The state varables are nput varables of our controller and thus the partton s mportant for the structure of fuzzy rules. At ths step, partton of control space (output space of the controller) s also necessary. 6) Determnaton of the shapes of fuzzy sets It s necessary to determne the shapes of fuzzy sets and ther membershp functons for the parttoned nput spaces and output spaces. 7) Constructon of fuzzy rule base Now, we can buld control rules. We determned the varables and correspondng lngustc terms n antecedent part and consequent part of each rule. The archtecture of rules s dependent upon the nference method determned n step 2). 8) Determnaton of defuzzfcaton strategy In general, we use sngleton control values and thus we have to determne the method. 9) Test and tunng It s almost mpossble to obtan a satsfactory fuzzy controller wthout tunng. In general t s necessary to verfy the controller and tune t untl when we get satsfactory results. 0) Constructon of lookup table If the controller shows satsfactory performance, we have to decde whether we use a lookup table nstead of usng the nference system. The lookup table s often used to save computng the tme of the nference and defuzzfcaton. The lookup table shows the relatonshps between a combnaton of nput varables and control actons. 0.7 Applcaton example of FLC desgn Servomotors are used n many automatc system ncludng drvers for prnters, floppy dsks, tape recorders, and robot manpulatons. The control of such servomotors s an mportant ssue. The servomotor process shows nonlnear propertes, and thus we apply the fuzzy logc control to the motor control. The task of the control s to rotate the shaft of the motor to a set pont wthout overshout. The set pont and process output n measured n degree. ) Determnaton of state varables and control varable () State varables (nput varable of controller): - Error equals the set pont mnus the process output (e). - Change of error (ce) equals the error from the process output mnus the error from the last process output. (2) Control varable (output varable of the controller): - Control nput (v) equals the voltage appled to the process. 2) Determnaton of nference method The Mandan nference method s selected because t s smple to explan. 3) Determnaton of fuzzfcaton method We can measure the state varables wthout uncertanty and thus we use the measured sngleton for the fuzzy nference 7

18 4) Dscretzaton and normalzaton The shaft encoder of the motor has a resoluton of 000. The unverses of dscourse are as follows: 000 e ce 00 The servo amplfer has an output range of 30 V and thus the control varables (v) are n the range 30 v 30 We dscretze and normalze the nput varables n the range [, +] as shown n Table 0.6. The control varable v s normalzed n the range [, +] wth the equaton. v = 30 v Table 0.6 Dscretzaton and normalzaton error (e) error change (ce) quantzed level 000 e ce < e < ce < e < ce <e < ce < e < ce < e 00 0 < ce < e < ce < e < ce < e < ce < e < ce < e < ce ) Partton of nput space and output space We partton space of each nput and output varable nto seven regons, and each regon s assocated wth lngustc term as shown n Fg 0.3. Now we know the maxmum number of possble fuzzy rules s 49. NB NM NS ZE PS PM PB 0 + e ce v 8

19 Fg 0.3 Partton of space 6) Determnaton of the shapes of fuzzy sets We normalzed the nput and output varables on the same nterval [, +] and parttoned the regon nto seven subregons, and thus we defne the prmary trangular fuzzy sets for the all varables as shown n Table 0.7 and Fg 0.4. Table 0.7 Defnton of prmary fuzzy sets Level NB NM NS ZE PS PM PB NB NM NS ZE PS PM PB Fg 0.4 Graphcal representaton of prmary fuzzy sets 9

20 7) Constructon of fuzzy rules We ntervewed wth an expert of the servomotor control, and we collect knowledge such as: If the error s zero and the error change s postve small, then the control nput s negatve small. Ths type of rules are rewrtten n the followng form () If e s PB and ce s any, then v s PB. (2) If e s PM and ce s NB, NM, or NS, then v s PS. (3) If e s ZE and ce s ZE, PS, or PM, then v s ZE. (4) If e s PS and ce s NS, ZE, or PS, then v s ZE. (5) If e s NS and ce s NS, ZE, PS, or PM, then v s NS (6) If e s NS or ZE and ce s PB, then v s PS. The full set of fuzzy rules s summarzed n the rule table n Fg 0.5 e ce NB NM NS ZE PS PM PB NB NB NM NM NS NS PS ZE NS ZE PS PM PS PM PB PB Fg 0.5 Fuzzy rule table 8) Determnaton of defuzzfcaton strategy We take the COA (center of area) method because t s most commonly used. 9) Test and tunng We checked the performance of the developed controller and refned some fuzzy rules. 0) Constructon of lookup table 20

21 After verfyng the controller showng good performance, we decded to use a lookup table. We extended the nference for every combnaton of dscretzed nput varables c and ce. For example, for c = 0.2 and ce = 0, v s 0.4 for c = 0.4 and ce = 0.4, v s 0.2 The correspondng lookup table s gven n Table 0.8. Now we can use ths lookup table n order to save the nference tme and defuzzfcaton tme. Table 0.8 Lookup table e ce Fuzzy expert systems An expert system s a program whch contans human expert s knowledge and gves answers to the user s query by usng an nference method. The knowledge s often stored n the form of rule base, and the most popular form s that of f-then. A fuzzy expert system s an expert system whch can deal uncertan and fuzzy nformaton. In our real world, a human expert has hs knowledge n the form of lngustc terms. Therefore t s natural to represent the knowledge by fuzzy rules and thus to use fuzzy nference methods. The structure of a fuzzy expert system s smlar to that of the fuzzy logc controller. It s confguraton s shown n Fg 0.6. As n the fuzzy logc controller, there can be fuzzfcaton nterface, knowledge base, and nference engne (decson makng logc). Instead of the defuzzfcaton module, there s the lngustc approxmaton module fuzzfcaton nterface Ths module deals user s request, and thus we have to determne the fuzzfcaton strategy. If we want to make the fuzzy expert system receve lngustc terms, ths module has to have an ablty to 2

22 handle such fuzzy nformaton. The fuzzfcaton strategy, f necessary, s smlar to that of the fuzzy logc controller. Contrary to the fuzzy logc controller, t s not needed to consder the dscretzaton or normalzaton. But the fuzzy partton and assgnng fuzzy lngustc terms to each subregon are necessary. The expert s knowledge may be represented n the form of f-then by usng fuzzy lngustc terms. Each rule can have ts certanty factor whch represents the certanty level of the rule. Ths certanty factor s used n the aggregaton of the results from each rule Inference engne (Decson makng logc) The fuzzy expert systems can use the nference methods of the fuzzy logc controller. The system does not deal wth a machne or process, and thus t s dffcult to have a fuzzy set wth monotonc membershp functon n the consequent part of a rule. Therefore especally, Mamdan method and Larsen method are often used Lngustc approxmaton As we stated before, a fuzzy expert system does not control a machne nor a process, and thus, n general, the defuzzfcaton s not necessary. Instead of the defuzzfcaton module, sometmes we need a lngustc approxmaton module. Ths module fnds a lngustc term whch s closest to the obtaned fuzzy set. To do t, we may use a measurng technque of dstance between fuzzy sets Scheduler Ths module controls all the processes n the fuzzy expert system. It determnes the rules to be executed and sequence of ther executons. It may also provde an explanaton functon for the result. For example, t can show the reason how the result was obtaned. Knowledge base User nput Fuzzfcaton nterface Scheduler Lngustc approxmaton Output Inference engne Fg 0.6 Confguraton of fuzzy expert system 22

23 [Summary] The fuzzy logc controller (FLC) s good when - the control process s too complex - the nformaton s qualtatve Advantage of the fuzzy logc controller - Parallel and dstrbuted control - Lngustc control - Robust control Components of the FLC - Fuzzfcaton nterface - Knowledge base - Decson-makng logc - Defuzzfcaton nterface Fuzzfcaton nterface components - Scale mappng of nput data - Strategy of nose - Selecton of fuzzfcaton functon Data base components - Dscretzaton of unverse of dscourse - Normalzaton of unverse of dscourse - Fuzzy partton of nput and output spaces - Membershp functon of prmary fuzzy set Rule base - Choce of state varables and control varables - Source of fuzzy control rules - Type of fuzzy control rules: state evaluaton rules and object evaluaton rules. Decson makng logc - Mamdan nference method - Larsen nference method - Tsukamoto nference method - TSK nference method 23

24 Defuzzfcaton methods - Mean of maxmum method (MOM) - Center of area method (COA) - Bsector of area (BOA) Lookup table - Dsadvantage of the FLC s the hgh tme complexty. - Savng of computaton tme - Drect relatonshp between nput varable and control output actons Desgn procedure of the FLC - Determnaton of varables - Determnaton of nference method - Determnaton of fuzzfcaton method - Dscretzaton and normalzaton of varables - Partton of space - Determnaton of fuzzy sets - Constructon of fuzzy rule base - Determnaton of defuzzfcaton strategy - Test and tunng - Lookup table 24

25 [Exercses] 0. Descrbe advantage of the fuzzy logc controller (FLC). 0.2 In whch case the FLC s superor to the conventonal control algorthm? 0.3 Explan the followngs: - state varable - control output - control nput - control varable 0.4 Explan the followng components n the FLC. - fuzzfcaton nterface - knowledge base - data base - decson-makng logc - defuzzfcaton nterface 0.5 Explan the three man ssues n the fuzzfcaton nterface component 0.6 Explan the three man ssues n the data base. 0.7 What s the dfference between the dscretzaton and normalzaton of unverse of dscourse. 0.8 Explan the relatonshp between the fuzzy partton of nput varable and the number of fuzzy rules. 0.9 Why and how membershp functons can be defned n a table when the unverse of dscourse s dscretzed? 0.0 What s the objectve of the normalzaton of unverse of dscourse? 25

26 0. What are the man crtera to determne the state varables and control varables? 0.2 Explan the two-types of control rules: - State evaluaton fuzzy control rules R : f x s A, and y s B then z s C - Object evaluaton fuzzy control rules R : f (z s C (x s A and y s B )) then z s C 0.3 Explan the four nference methods: - Mamdan method - Larsen method - Tsukamoto method - TSK method 0.4 What s the property of the membershp functon n consequent part n Tsukamoto method? 0.5 Explan the three defuzzfcaton methods: - Mean of maxmum method - Center of area method - Bsector of area 0.6 What s the lookup table? Why s t often used? 0.7 Show the desgn procedure of the FLC. 26