FUZZY CONTROL. Khurshid Ahmad, Professor of Computer Science, Department of Computer Science Trinity College, Dublin-2, IRELAND Nov 15-16th, 2011

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: Mamdani & akagi-sugeno Controllers Khurshid Ahmad, Professor of Comuter Science, Deartment of Comuter Science rinit College, Dublin-, IRELAND Nov 5-6th, 0 htts://www.cs.tcd.ie/khurshid.

1 : Mamdani & akagi-sugeno Controllers Khurshid Ahmad, Professor of Comuter Science, Deartment of Comuter Science rinit College, Dublin-, IRELAND Nov 5-6th, 0 htts:// Control heor? he term control is generall defined as a mechanism used to guide or regulate the oeration of a machine, aaratus or constellations of machines and aaratus. Control heor? An Inut/Outut Relationshi

Argument is that the consequent membershi function can be simlified this argument is based on a heuristic that oerators in a control environment

2 Control heor? An Inut/Outut Relationshi IDENIFIED using Fuzz Logic Control heor? An Inut/Outut Relationshi IDENIFIED using Fuzz Logic 5 Control heor? icall, rules contain membershi functions for both antecedents and consequent. Argument is that the consequent membershi function can be simlified this argument is based on a heuristic that oerators in a control environment divide the variable sace sa, error, change in error and change in control into PARIIONS; Within each artition the outut variable is a simle, often linear function of the inut variables and not membershi functions 6

3 Control heor? icall, rules contain membershi functions for both antecedents and consequent. Mamdani Controller If ek is ositivee and ek is ositive e then uk is ositive u akagi-sugeno Controllers: If ek is ositivee and ek is ositive e then uk αek+ß ek+δ; α, ß and δ are obtained from emirical observations b relating the behaviour of the errors and change in errors over a fied range of changes 7 in control LERS akagi-sugeno Controllers According to Yager and Filev, a known disadvantage of the linguistic modules is that the do not contain in an elicit form the objective knowledge about the sstem if such knowledge cannot be eressed and/or incororated into fuzz set framework' 99:9. icall, such knowledge is available often: for eamle in hsical sstems this kind of knowledge is available in the form of general conditions imosed on the sstem through conservation laws, including energ mass or momentum balance, or through limitations imosed on the values of hsical constants. 8 LERS akagi-sugeno Controllers omohiro akagi and Michio Sugeno recognised two imortant oints:. Comle technological rocesses ma be described in terms of interacting, et simler sub rocesses. his is the mathematical equivalent of fitting a iece-wise linear equation to a comle curve.. he outut variables of a comle hsical sstem, e.g. comle in the sense it can take a number of inut variables to roduce one or more outut variable, can be related to the sstem's inut variable in a linear manner rovided the outut sace can be subdivided into a number of distinct regions. akagi,., & Sugeno, M Fuzz Identification of Sstems and its Alications to Modeling and Control. IEEE ransactions on Sstems, Man and Cbernetics. Volume No. SMC-5 No

4 LERS akagi-sugeno Controllers akagi-sugeno fuzz models have been widel used to identif the structures and arameters of unknown or artiall known lants, and to control nonlinear sstems. 0 LERS akagi-sugeno Controllers Mamdani stle inference: he Good News: his method is regarded widel for caturing eert knowledge and facilitates an intuitivel-lausible descrition of knowledge; he Bad News: his method involves the comutation of a two-dimensional shae b summing, or more accuratel integrating across a continuousl varing function. he comutation can be eensive. LERS akagi-sugeno Controllers Mamdani stle inference: he Bad News: his method involves the comutation of a twodimensional shae b summing, or more accuratel integrating across a continuousl varing function. he comutation can be eensive. For ever rule we have to find the membershi functions for the linguistic variables in the antecedents and the consequents; For ever rule we have to comute, during the inference, comosition and defuzzification rocess the membershi functions for the consequents; Given the non-linear relationshi between the inuts and the outut, it is not eas to identif the membershi functions for the linguistic variables in the consequent

5 LERS akagi-sugeno Controllers Literature on conventional control sstems has suggested that a comle non-linear sstem can be described as a collection of subsstems that were combined based on a logical Boolean switching sstem function. In realistic situations such disjoint cris decomosition is imossible, due to the inherent lack of natural region boundaries in the sstem, and also due to the fragmentar nature of available knowledge about the sstem. LERS akagi-sugeno Controllers akagi and Sugeno 985 have argued that in order to develo a generic and simle mathematical tool for comuting fuzz imlications one needs to look at a fuzz artition of fuzz inut sace. In each fuzz subsace a linear inutoutut relation is formed. he outut of fuzz reasoning is given b the values inferred b some imlications that were alied to an inut. LERS akagi-sugeno Controllers akagi and Sugeno have described a fuzz imlication R as: R: if is A, k is A k then g,, k 5 5

6 A solution for the coefficients of the consequent in SK Sstems Rule : is HEN + COMPOSIION *[ 0 + ] here are two unknowns: 0 and. So we need two simultaneous equations for two values of, sa and, and two values of and ; 6 A solution for the coefficients of the consequent in SK Sstems Consider a two rule sstem: Rule : Rule : is is HEN HEN + COMPOSIION *[ + ] + *[ + *[ 0 + ] + *[ 0 + ] + *[ + ] + *[ + ] ] 0 7 A solution for the coefficients of the consequent in SK Sstems here are unknowns 0,, 0,, so we need equations. And, these can be obtained from observations comrising diffierent values of and *[ + ] + *[ + ] Four values of * * + * + * * + * 0 0 0,,, and for each, will haveavalue + * * + * * 8 + * * 6

7 7 9 A solution for the coefficients of the consequent in SK Sstems here are unknowns 0,, 0,, so we need equations. And, these can be obtained from observations comrising diffierent values of and * * * * * * * * * * * * * *,,,, ] *[ ] *[ haveavalue will each for and of Four values 0 A solution for the coefficients of the consequent in SK Sstems here are unknowns 0,, 0,, so we need equations. [ ] [ ] [ ] [ ] [ ] [ ] Y P P Y 0 0 ] [ * * * * * * * * A solution for the coefficients of the consequent in SK Sstems akagi and Sugeno have a generalised the method to an n-rule, m-arameter sstem; and b claim that this method of identification enables us to obtain just the same arameters as the original sstem, if we have a sufficient number of noiseless outut data for identification akagi and Sugeno, 985:9. [ ] [ ] [ ] [ ] [ ] [ ] Y P P Y 0 0 ] [ * * * * * * * *

8 A solution for the coefficients of the consequent in SK Sstems In order to determine the values of the arameters in the consequents, one solves the LINEAR sstem of algebraic equations and tries to minimize the difference between the ACUAL outut of the sstem Y and the simulation [] [P] : [ Y ] [ ] [ P] ε A solution for the coefficients of the consequent in SK Sstems In order to determine the values of the arameters in the consequents, one solves the LINEAR sstem of algebraic equations and tries to minimize the difference between the ACUAL outut of the sstem Y and the simulation [] [P] : [ Y ] [ ] [ P] ε LERS akagi-sugeno Controllers akagi and Sugeno have described a fuzz imlication R as: R: if is A, k is A k then g,, k, where: A zero order akagi-sugeno Model will be given as R: if is A, k is A k then k 8

9 Knowledge Reresentation & Reasoning: he Air-conditioner Eamle Consider the roblem of controlling an air-conditioner again. he rules that are used to control the airconditioner can be eressed as a cross roduct: CONROL EMP SPEED 5 Knowledge Reresentation & Reasoning: he Air-conditioner Eamle he rules can be eressed as a cross roduct of two term sets: emerature and Seed. CONROL EMP SPEED Where the set of linguistic values of the term sets is given as EMP COLD + COOL + PLEASAN + WARM + HO SPEED MINIMAL + SLOW + MEDIUM + FAS + BLAS 6 Knowledge Reresentation & Reasoning: he Air-conditioner Eamle A Mamdani Controller Recall that the rules governing the airconditioner are as follows: RULE#: IF EMP is COLD HEN SPEED is MINIMAL RULE#: IF EMP is COOL HEN SPEED is SLOW RULE#: IF EMP is PLEASEN HEN SPEED is MEDIUM RULE#: IF EMP is WARM HEN SPEED is FAS RULE#5: IF EMP is HO HEN SPEED is BLAS 7 9

10 Knowledge Reresentation & Reasoning: he Air-conditioner Eamle A Zero-order akagi-sugeno Controller Recall that the rules governing the airconditioner are as follows: RULE#: IF EMP is COLD HEN SPEED k RULE#: IF EMP is COOL HEN SPEED k RULE#: IF EMP is PLEASEN HEN SPEED k RULE#: IF EMP is WARM HEN SPEED k RULE#5: IF EMP is HO HEN SPEED k 5 8 Knowledge Reresentation & Reasoning: he Air-conditioner Eamle A First-order akagi-sugeno Controller Recall that the rules governing the airconditioner are as follows: RULE#: IF EMP is COLD HEN SPEED j +k * RULE#: IF EMP is COOL HEN SPEED j +k * RULE#: IF EMP is PLEASEN HEN SPEED k RULE#: IF EMP is WARM HEN SPEED j + k RULE#5: IF EMP is HO HEN SPEED k 5 9 Knowledge Reresentation & Reasoning: he Air-conditioner Eamle emerature Fuzz Sets r u th V a lu e emerature Degrees C Cold Cool Pleasent Warm Hot 0 0

11 Knowledge Reresentation & Reasoning: he Air-conditioner Eamle he analticall eressed membershi for the reference fuzz subsets for the temerature are: ' COLD ' ; COLD ' COOL ' COOL ; COOL ' PLEASEN ' PLEA ; PLEA ' WARM ' ' WARM WARM ' HO ' HO HO Knowledge Reresentation & Reasoning: he Air-conditioner Eamle Seed Fuzz Sets For FLC of Mamdani te r u t h V a l u e Seed MINIMAL SLOW MEDIUM FAS BLAS Knowledge Reresentation & Reasoning: he Air-conditioner Eamle he zero-order seed control just takes one SINGLEON value at fied values of the velocit; for all other values the membershi function is defined as zero ' MINIMAL' V V 0; ' SLOW ' MEDIUM ' ' FAS ' ' BLAS ' MINIMAL SLOW MED FAS BLAS V V V V V 0 V 50 V 70 V 00

12 Knowledge Reresentation & Reasoning: he Air-conditioner Eamle Let the temerature be 5 degrees centigrade: Fuzzification: 5 degrees means that it can be COOL and COLD; Inference: Rules and will fire: Comosition: he temerature is COLD with a truth value of COLD0.5 the SPEED will be k he temerature is COOL with a truth value of COOL 0.5 the SPEED will be k DEFUZZIFICAION : CONROL seed is COLD *k+ COOL *k/ COLD + COOL 0.5*0+0.5*0/ RPM Knowledge Reresentation & Reasoning: he Air-conditioner Eamle Zero Order akagi Sugeno Controller Membershi Function Seed MINIMAL SLOW MEDIUM FAS BLAS 5 Knowledge Reresentation & Reasoning: he Air-conditioner Eamle FUZZIFICAION: Consider that the temerature is 6 o C and we want our knowledge base to comute the seed. he fuzzification of the the cris temerature gives the following membershi for the emerature fuzz set: COLD COOL PLEASEN WAR M HO em Fire Rule # es/no # no # es # es # no #5 no 6

13 Knowledge Reresentation & Reasoning: he Air-conditioner Eamle INFERENCE: Consider that the temerature is 6 o C and we want our knowledge base to comute the seed. Rule # & are firing and are essentiall the fuzz atches made out of the cross roducts of COOL SLOW PLEASAN MEDIUM 7 Knowledge Reresentation & Reasoning: he Air-conditioner Eamle COMPOSIION: he COOL and PLEASAN sets have an outut of 0. and 0. resectivel. he singleton values for SLOW and MEDIUM have to be given an alha-level cut for these outut values resectivel: Zero-order akagi-sugeno Model 0.5 Membershi Function alha-cut Seed MINIMAL SLOW MEDIUM FAS BLAS 8 Knowledge Reresentation & Reasoning: he Air-conditioner Eamle DEFUZZIFICAION: he roblem of finding a single, cris value is no longer a roblem for a akagi-sugeno controller. All we need is the weighted average of the singleton values of SLOW & MEDIUM. Recall the Centre of Area comutation for the Mamdani controller η Σ outut Σ.... n outut.... n 9

14 Knowledge Reresentation & Reasoning: he Air-conditioner Eamle DEFUZZIFICAION: For akagi-sugeno, the comutation for η is restricted to the singleton values of the SPEED linguistic variable we do not need to sum over all values of the variable : η Σ Σ Σ SLOW *.... n outut.... n outut SLOW _ ' _ ' SLOW. + MEDIUM *. + MEDIUM _ ''. MEDIUM _ ''. 0 Knowledge Reresentation & Reasoning: he Air-conditioner Eamle DEFUZZIFICAION: For akagi-sugeno, the comutation for η is restricted to the singleton values of the SPEED linguistic variable we do not need to sum over all values of the variable : SLOW Σ SLOW * _. + MEDIUM * ' η SLOW MEDIUM _. + _. ' '' 0.* * MEDIUM _ ''. Knowledge Reresentation & Reasoning: he Air-conditioner Eamle DEFUZZIFICAION: Recall the case of the Mamdani equivalent of the fuzz airconditioner where we had fuzz sets for the linguistic variables SLOW and MEDIUM: he Centre of Area COA comutations involved a weighted sum over all values of seed between.5 and 57.5 RPM: in the akagi-sugeno case we onl had to consider values for seeds 0RPM and 50 RPM.

15 Knowledge Reresentation & Reasoning: he Air-conditioner Eamle DEFUZZIFICAION: Recall the case of the Mamdani equivalent of the fuzz air-conditioner where we had fuzz sets for the linguistic variables SLOW and MEDIUM: he Centre of Area COA comutations involved a weighted sum over all values of seed between.5 and 57.5 RPM: in the akagi-sugeno case we onl had to consider values for seeds 0RPM and 50 RPM. SPEED SLOW MEDIUM OUPU OF RULES WEIGHED SPEED he seed is 6.9 RPM SUM Knowledge Reresentation & Reasoning: he Air-conditioner Eamle DEFUZZIFICAION: For Mean of Maima for the Mamdani controller, we had to have an alha-level cut of 0., and the summation ran between RPM, leading to a seed of 50 RPM. We get the same result for akagi-sugeno controllers: η 0.*50/0.50 RPM Knowledge Reresentation & Reasoning: he Air-conditioner Eamle DEFUZZIFICAION: Comaring the results of two model identification eercises Mamdani and akagi-sugeno- we get the following results: Controller akagi- Sugeno RPM Mamdani RPM Centre of Area. 6.9 Mean of Maima

16 Knowledge Reresentation & Reasoning: he Air-conditioner Eamle DEFUZZIFICAION: Comaring the results of two model identification eercises Mamdani and akagi-sugeno- we get the following results: Mamdani Controller and the use of COA is the best result Error Controller akagi-sugeno Mamdani RPM RPM Centre of Area % 0% Mean of Maima 5% 5% 6 LERS akagi-sugeno Controllers A formal derivation Consider a domain where all fuzz sets are associated with linear membershi functions. Let us denote the membershi function of a fuzz set A as A,. All the fuzz sets are associated with linear membershi functions. hus, a membershi function is characterised b two arameters giving the greatest grade and the least grade 0. he truth value of a roosition is m A and is m B is eressed as is A and is B A Λ B 7 LERS- An eamle A worked eamle Consider an FLC of Mamdani te: which eresses rules like: ek N Z P ek N N N Z Z N Z P P Z P P Rule : If ek is negative AND ek is negative then uk is negative ALSO Rule 9: If ek is ositive AND ek is ositive then uk is ositive 8 6

17 LERS- An eamle A worked eamle Consider an FLC of Mamdani te: ek N Z P ek N N N Z Z N Z P P Z P P he nine rules eress the deendence of change in the value of control outut on the error the difference between eected and outut values and the change in error. his deendence will cature some ver comle non-linear, and linear relationshis between e and e and u. 9 LERS- An eamle A zero-order akagi-sugeno Controller ek N Z P ek N α α α Z α α 5 α 6 P α 7 α 8 α 9 which eresses rules like: Rule :If ek is negative AND ek is negative then uk α Rule 9:If ek is ositive AND ek is ositive then uk α9 50 LERS- An eamle A first-order akagi-sugeno Controller ek N Z P ek N α e+ß e +δ α e+ß e+ δ α e+ß e+ δ Z α e+ß e +δ α 5 e+ß 5 e+ δ 5 α 6 e+ß 6 e+ δ 6 P α 7 e+ß 7 e+ δ 7 α 8 e+ß 8 e+ δ 8 α 9 e+ß 9 e+ δ 9 which eresses rules like: Rule :If ek is negative AND ek is negative then uk α e+ß e+δ Rule 9:If ek is ositive AND ek is ositive then uk α 9 e+ß 9 e+δ 9 5 7

18 LERS akagi-sugeno Controllers akagi and Sugeno have described a fuzz imlication R is of the format: R: if is A, k is A k then g,, k, where:,, k m A,.. m A f g Variable of the consequence whose value is inferred Variables of the remise that aear also in the art of the consequence Fuzz sets with linear membershi functions reresenting a fuzz subsace in which the imlication R can be alied for reasoning. Logical function connects the roositions in the remise. Function that imlies the value of when,. 5 k satisfies the remise. LERS akagi-sugeno Controllers In the remise if A i is equal to i for some i where i is the universe of discourse of i, this term is omitted; i is unconditioned; otherwise i is regarded as conditioned. 5 LERS akagi-sugeno Controllers In the remise if A i is equal to i for some i where i is the universe of discourse of i, this term is omitted; i is unconditioned. he following eamle will hel in clarifing the argumentation related to 'conditioned' and 'unconditioned' terms in a given imlication: R: if is small and is big then + +. he above imlication comrises two conditioned remises, and, and one unconditioned remise,. he imlication suggests that if is small and is big, then the value of would deend uon and be equal to the sum of,, and., where is unconditioned in the remise. 5 8

19 LERS akagi-sugeno Controllers icall, for a akagi-sugeno controller, an imlication is written as: R: if is and and k is k then k k. he assumtion here is that onl and connectives are used in the antecedants or remises of the rules. And, that the relationshi between the outut and inuts is strictl a LINEAR weighted average relationshi. he weights here are 0,.. k. 55 LERS akagi-sugeno Controllers Reasoning Algorithm Recall the arguments related to the membershi functions of the union and intersection of fuzz sets. he intersection of sets A and B is given as: A B min A, B We started this discussion b noting that we will elore the roblems of multivariable control MultileInutSingleOutut. Usuall, the rule base in a fuzz control sstem comrises a number of rules; in the case of multivariable control the relevant rules have to be tested for what the iml. 56 LERS akagi-sugeno Controllers Consider a sstem with n imlications rules; the variable of consequence,, will have to be notated for each of these imlications, leading to i variables of consequence. here are three stages of comutations in akagi-sugeno controllers: FUZZIFICAION: Fuzzif the inut. For all inut variables comute the imlication for each of the rules; INFERENCE or CONSEQUENCES: For each imlication comute the consequence for a rule which fires. Comute the outut for the rule b using the linear relationshi between the inuts and the outut k k.. AGGREGAE & DEFUZZIFICAION: he final outut is inferred from n-imlications and given as an average of all individual imlications i with weights i : Σ i * i / Σ i where i stands for the truth value of a given roosition. 57 9

20 LERS akagi-sugeno Controllers Consider the following fuzz imlications or rules R,R, R used in the design of a akagi-sugeno controller: R If is small & is small then + R If is big then R If is big then where i refers to the consequent variable for each rule labelled R i and and refer to the inut variables that aear in remise of the rules. 58 LERS akagi-sugeno Controllers he membershi function for small, small, big and big are given as follows Small Small Big Big LERS akagi-sugeno Controllers he membershi function for small, small, big and big are given as follows akagi-sugeno Eamle 7 Membershi Function Inut Small Small Big Big 60 0

21 LERS akagi-sugeno Controllers An eamle Let us comute the FINAL OUPU for the following values: & 5 using akagi and Sugeno s formula: Σ i * i / Σ i where i stands for the truth value of a given roosition. 6 LERS akagi-sugeno Controllers An eamle FUZZUFICAION: We have the following values of the membershi functions for the two values & 5: Small Small Big Big LERS akagi-sugeno Controllers An eamle INFERENCE & CONSEQUENCE: & 5 Rule Premise Premise Consequenc e R Small 0.5 Small ruth Value Min Premise & Premise Min0.5, R Big 0. R Big

22 LERS akagi-sugeno Controllers An eamle AGGREGAION &DEFUZZIFICAION: & 5 Σ i * i / Σ i Using a Centre of Area comutation for we get: i, i, i * i i 0.5 * * * LERS Ke difference between a Mamdani-te fuzz sstem and the akagi-sugeno-kang Sstem? A zero-order Sugeno fuzz model can be viewed as a secial case of the Mamdani fuzz inference sstem in which each rule is secified b fuzz singleton or a re-defuzzified consequent. In Sugeno s model, each rule has a cris outut, the overall inut is obtained b a weighted average this avoids the timeconsuming rocess of defuzzification required in a Mandani model. he weighted average oerator is relaced b a weighted sum to reduce comutation further. Jang, Sun, Mizutani 997:8. 65 LERS akagi-sugeno Controllers akagi,., & Sugeno, M Fuzz Identification of Sstems and its Alications to Modeling and Control. IEEE ransactions on Sstems, Man and Cbernetics. Volume No. SMC-5 No

23 LERS akagi-sugeno Controllers akagi and Sugeno 985 have argued that in order to develo a generic and simle mathematical tool for comuting fuzz imlications one needs to look at a fuzz artition of fuzz inut sace. In each fuzz subsace a linear inut-outut relation is formed. he outut of fuzz reasoning is given b the values inferred b some imlications that were alied to an inut. 67 Knowledge Reresentation & Reasoning: he Air-conditioner Eamle akagi-sugeno model is an aroimation of a Mamdani controller in that the akagi- Sugeno model ignores the fuzziness of linguistic variables in the consequent, but accounts for the fuzziness of variables in the antecedants; whereas Mamdani controller takes into the fuzziness of variables aearing both in the antecdents and the consequent. 68 Knowledge Reresentation & Reasoning: he Air-conditioner Eamle Model Identification: Given a choice between two models, sa Mamdani and akagi- Sugeno, we have to first identif wh to chose a fuzz logic sstem will a cris descrition not work as it is simler to comute and second whether to use an elaborate model sa Mamdani rather than an aroimation to the model sa, akagi-sugeno. he choice can be based on the relative erformance of the two models 69

Financial Informatics XI: Fuzzy Rule-based Systems

Financial Informatics XI: Fuzzy Rule-based Systems Khurshid Ahmad, Professor of Computer Science, Department of Computer Science Trinity College, Dublin-2, IRELAND November 19 th, 28. https://www.cs.tcd.ie/khurshid.ahmad/teaching.html