Research Article Number Based Fuzzy Inference System for...

Research Article119885Number Based Fuzzy Inference System forDynamic Plant Control

Rahib H Abiyev

Applied Artificial Intelligence Research Centre Department of Computer Engineering Near East UniversityNorthern Cyprus Mersin 10 Turkey

Correspondence should be addressed to Rahib H Abiyev rahibabiyevneuedutr

Received 13 July 2016 Revised 15 September 2016 Accepted 27 September 2016

Academic Editor Ping Feng Pai

Copyright copy 2016 Rahib H Abiyev This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Frequently the reliabilities of the linguistic values of the variables in the rule base are becoming important in the modeling offuzzy systems Taking into consideration the reliability degree of the fuzzy values of variables of the rules the design of inferencemechanism acquires importance For this purpose Z number based fuzzy rules that include constraint and reliability degreesof information are constructed Fuzzy rule interpolation is presented for designing of an inference engine of fuzzy rule-basedsystem The mathematical background of the fuzzy inference system based on interpolative mechanism is developed Based oninterpolative inference process Z number based fuzzy controller for control of dynamic plant has been designed The transientresponse characteristic of designed controller is compared with the transient response characteristic of the conventional fuzzycontroller The obtained comparative results demonstrate the suitability of designed system in control of dynamic plants

1 Introduction

In industry some dynamic plants are characterized by uncer-tainty of environment and fuzziness of information Deter-ministic models used to control these dynamic plants usuallyprove to be insufficient to adequately describe the processesOne of the effective ways for modeling and controlling ofthese plants is the use of fuzzy systems based on If-ThenrulesThe rules of the fuzzy systems are basically constructedusing the knowledge of experts or experienced specialistsSometimes the people have made a rational decision usinguncertain incomplete information Fuzzy logic allows han-dling uncertain and imprecise knowledge and provides apowerful framework for reasoning The design of the rulesand also the reliabilities of the linguistic values of the variablesin these rules are an important issue in the modeling of thesystems Taking into consideration the reliability degree offuzzy values used in the fuzzy If-Then rules the design ofdecision-making module acquires importance The descrip-tion of this decision-making mechanism is very hard andits simulation is difficult Zadeh suggested Z number to deal

with uncertain information with the degree of its reliabilityZ number provides fuzzy restriction and reliability informa-tion [1]

Recently fuzzy set theory is efficiently applied for solutiondifferent real world problems These research works arebasically based on type-1 or type-2 fuzzy systems In theseresearches the reliability of information is taken into consid-eration Z number provided in [1] allows representing fuzzyconstraint and reliability of the fuzzy system Z number basedsystems have been discussed and designed in many researchworks [1ndash3] Comparing with the classical fuzzy set Z num-ber based fuzzy set more adequately describes the knowledgeof human The theory can describe both constraint andreliability of information [1] The concept of Z number isinvestigated and also applied to solve different problemsIn [3 4] Z numbers are used to solve the multicriteriadecision-making problem However the approach proposedin these researches are based on converting Z number tofuzzy numbers on the base of the approach presented in[5] Reference [6] uses Z number for multicriteria decision-making where Z number is converted to the interval-valued

Hindawi Publishing CorporationAdvances in Fuzzy SystemsVolume 2016 Article ID 8950582 7 pageshttpdxdoiorg10115520168950582

2 Advances in Fuzzy Systems

fuzzy set with a footprint of uncertainty and then by com-puting the centroid it is converted to the crisp numbers fordecision-making These researchers present the advantagesof the presented approach based on their low computationalcomplexity However the converting Z numbers in realnumbers may lead to the significant loss of information andtherefore this may affect the performance of designed fuzzysystem

The paper [7] suggests the use of Z number in theperception of the uncertainty of the information conveyed bya natural language statement and to merge human-affectiveperspectiveswithComputingwithWords (CWW) Reference[8] considers an application of Z number based approach tothe AHP The presented work is also based on the approachproposed in [5] In [9] a fair price approach for a participationin a decision-making under interval set-valued fuzzy and Znumber uncertainty is considered

As shown using Z numbers a set of research workson arithmetic operations and decision-making have beendoneThe use of Z numbers in decision-making control andmodeling needs to use efficient inference mechanism for thedesigned system Reference [2] suggests usingZ interpolationin inference process In [10]Koczy andHirota purposed inter-polative reasoning in the sparse fuzzy rule base This methodmaintains the logical interpretation of modus ponens Themethod is distance based approximate fuzzy reasoning [1011] There are many different distance definitions for fuzzysets [12] These are absolute distance Euclidian distancedisconsistency measure Hausdorff measure and Kaufmanand Gupta measure Reference [12] points that the maindeficiency of these measures is that the information of theshape of the membership function of the fuzzy sets is mostlylost Using distance measure of two fuzzy sets A and Band given fuzzy set A1015840 it is impossible to construct thesecond fuzzy set B1015840 Reference [12] mentions that Koczymeasure can be used to solve this problem Koczy distanceis based on 120572-cuts of two fuzzy sets In [10 11] using Koczymeasure a fuzzy interpolative reasoning was proposed Theinterpolative approximate reasoning is applied to the solutionof different problems [13 14] In this paper the derivationof the interpolative approximate reasoning for Z numberbased fuzzy rules is consideredThe designed method is thenapplied for solving of dynamic plants control

The paper is organized as follows Section 2 presentsthe interpolative approximate reasoning method Section 3considers the use of the interpolative approximate reasoningmethod for Z numbers Section 4 presents the applicationof the designed interpolative approximate reasoning for thedevelopment of control system of the dynamic plant

2 Fuzzy Rule Interpolation

The inference technique of fuzzy logic resembles humanreasoning capabilities In the literature various fuzzy rea-soning methods are purposed to process uncertain infor-mation and increase the performance of the designed sys-tems These fuzzy reasoning methods are mainly based onthe compositional inference rule analogy and similarity

120583(x)

120572

A

xinf(A120572) sup(A120572)

Figure 1Membership function infimum and supremum for 120572-cut

interpolation and the concept of distance The speed pro-cessing capabilities and complexity of these reasoning meth-ods are important issues

In the paper inference based on fuzzy interpolation ofrules is used Fuzzy rule interpolation is proposed by Koczyand Hirota [10 11] and is designated for use for sparserule base The inference engine requires the satisfactionof the following conditions the used fuzzy sets should becontinuous convex and normal with bounded support

The proposed inference method is based on distancemeasure Here the distance measure of two fuzzy sets isdescribed using a fuzzy set defined in the interval [0 1] Theused distancemeasure is based on120572-cut Let us consider someconcepts used for two fuzzy sets 1198601 and 1198602 120572-cut of fuzzysets 1198601 and 1198602 will be denoted as 1198601120572 and 1198602120572 We say thatfuzzy set 1198601 is less than 1198602 that is 1198601 lt 1198602 iff

inf 1198601120572 lt inf 1198602120572 sup 1198601120572 lt sup 1198602120572 (1)

where inf1198601120572 and inf1198602120572 are infimum of 1198601 and 1198602sup1198601120572 and sup1198602120572 are supremum of 1198601 and 1198602 (Fig-ure 1)

Let us consider the following case Assume that wehave fuzzy rule base for the controller and in the result ofobservation we detect that input variable X is 119860lowast Let usdetermine the output119884 of the rule-based systemAssume that119860lowast is located between fuzzy sets1198601 and1198602 Let us determinethe output of fuzzy system using the rules which includes 1198601and 1198602 fuzzy sets

RulesGiven that 119883 is 119860lowast

If 119883 is 1198601 Then 119884 is 1198611If 119883 is 1198602 Then 119884 is 1198612Find Conclusion 119884 = 119861lowast

(2)

If 1198601 lt 119860lowast lt 1198602 and 1198611 lt 1198612 then using linearinterpolation [10] shows that

Advances in Fuzzy Systems 3

119889 (119860lowast 1198601)119889 (119860lowast 1198602) = 119889 (119861lowast 1198611)119889 (119861lowast 1198612) (3)

Here 119889(lowast) is the distance between two fuzzy sets Variousformulas are applied to calculate 119889(lowast) distance between twosets These are the following Hamming distance Euclidiandistance Hausdorff distance and Kaufman-Gupta distance

119889 (119883 119884) = |119883 minus 119884| 119889 (119883 119884) = radic(119883 minus 119884)2119889 (119883 119884) = maxsup

119909isin119883

inf119910isin119884

119889 (119909 119910) sup119910isin119884

inf119909isin119883

119889 (119909 119910) 119889 (119883 119884) = (10038161003816100381610038161199091 minus 11991011003816100381610038161003816 + 10038161003816100381610038161199092 minus 11991021003816100381610038161003816)2 (1205732 minus 1205731)

(4)

Here [1198861 1198862] and [1198871 1198872] are supports of A and B respec-tively [1205731 1205732] is the support of both A and B Reference [12]notices that these distance measures do not give informationabout the shape of themembership functions of the fuzzy setsKoczy andHirota [10 11] introduced distance based on120572-cutsof the two fuzzy sets Using this distance the final fuzzy set can

easily be constructed Koczy and Hirota [10 11] based on 120572-cut calculated lower 119889119871 and upper 119889119880 distances between 1198601120572and 1198602120572

119889119871 (1198601120572 1198602120572) = 119889 (inf 1198601120572 inf 1198602120572) 119889119880 (1198601120572 1198602120572) = 119889 (sup 1198601120572 sup 1198602120572) (5)

Here

119889119871 (119860lowast120572 1198601120572) = 119889 (inf 119860lowast120572 inf 1198601120572)= inf 119860lowast120572 minus inf 1198601120572

119889119880 (119860lowast120572 1198602120572) = 119889 (sup 119860lowast120572 sup 1198602120572)= sup 119860lowast120572 minus sup 1198602120572



(6)

It is needed to mention that Hamming or Euclidianformula can be used in (5) to measure the distances Taking(5) into (3) we can obtain the following

inf 119861lowast120572 = (1119889120572119871 (1198601120572 119860lowast120572)) inf 1198611120572 + (1119889120572119871 (1198602120572 119860lowast120572)) inf 11986121205721119889120572119871 (1198601120572 119860lowast120572) + 1119889120572119871 (1198602120572 119860lowast120572)sup 119861lowast120572 = (1119889120572119880 (1198601120572 119860lowast120572)) sup 1198611120572 + (1119889120572119880 (1198602120572 119860lowast120572)) sup 11986121205721119889120572119880 (1198601120572 119860lowast120572) + 1119889120572119880 (1198602120572 119860lowast120572)

(7)

Then the output 120572-level set in the consequent part for every120572 will be

119861lowast120572 = [inf 119861lowast120572 sup 119861lowast120572] (8)

Reference [10] extended the formula for 2k rules

inf 119861lowast120572 = sum2119896119894=1 (1119889120572119871 (119860 119894120572 119860lowast120572)) inf 119861119894120572sum2119896119894=1 (1119889120572119871 (119860 119894120572 119860lowast120572)) sup 119861lowast120572 = sum2119896119894=1 (1119889120572119880 (119860 119894120572 119860lowast120572)) sup 119861119894120572sum2119896119894=1 (1119889120572119880 (119860 119894120572 119860lowast120572))

(9)

In the next section we are applying the above formulas toobtain output of fuzzy system based on Z numbers

3 Interpolative Reasoning Using 119885 NumberBased Fuzzy Rules

Let us consider Z If-Then rules and its inference mechanismAssume that multiinput-single output fuzzy Z rules are givenas follows

Z Number Based If-Then Rule Base is as follows

If 1199091 is (11986011 11987711) and and 119909119898 is (1198601119898 1198771119898)Then 119910 is (1198611 1198771)



(10)


Let us consider the application of interpolation approachgiven in Section 2 to the inference process of the fuzzysystem based on Z numbers At first iteration using 120572-cut thedifferences between coming input signals and fuzzy values ofvariables in antecedent parts are calculated For this purpose120572-cuts can be used with Hamming Euclidian or one ofthe distances given in (4) The distances will be calculatedfor constraint and reliability variables in antecedent partseparately

Here formulas (5) and (6) are used to find lower andupper distances of membership function After finding thedistances using formula (9) in consequent part the lower andupper parts of the fuzzy output signal are calculated Theseoperations can be expressed using the following formulasalsoWe can obtain lower and upper parts through120572-cuts Forthis purpose at first iteration the distances are calculated as

119889119871 (119860120572119883119894119895 119883120572119895 ) = 100381610038161003816100381610038161003816119860120572119883119894119895 minus 119883120572119895 100381610038161003816100381610038161003816 119889120572119894 = 119898sum

119895

119889119871 (119860120572119883119894119895 119883120572119895 ) (11)

where 119895 = 1 119898 and 119898 is a number of input signals 119894 =1 119899 and 119899 is a number of rules For special case 120572 = 0 1119889119871(119860120572119883119894 119883120572119894 ) is a distance between two fuzzy sets Formula (11)is used to find distance 119889119888120572119894 for constraint parameter A andalso the same formula can be adapted to find distance 119889119903120572119894for reliability parameterThe total distance will be sum of twodistances computed for constraint and reliability

119889120572119894 = 119889119888120572119894 + 119889119903120572119894 (12)

Here 119889119903120572119894 is the distance computed for the reliabilityparameterThe fuzzy set in the output of the rule is calculatedby using the following equation

(119884120572 119877120572119884) = sum119899119894=1 (1119889prop119894 ) (119861119884119894 119877119884119894)prop1sum119899119894=1 (1119889prop119894 ) (13)

Formula (13) is applied for finding inf sup and centervalues in output fuzzy sets If we combine (11) and (13) thenwe can get the following

(119884120572 119877120572119884) = sum119899119894=1 (1sum119898119895 119889119871 (119860120572119883119894119895 119883120572119895 )) (119861119884119894 119877119884119894)prop1sum119899119894=1 (1sum119898119895 119889119871 (119860120572119883119894119895 119883120572119895 )) (14)

Using (14) we can derive the output Z fuzzy signal of thesystem It is needed to note that for finding output signalwe use 119861119884119894 for finding the reliability we use 119877119884119894 variables inright side of (13) (or (14)) After getting output signals theconverting of Z number to the crisp number is performedFormula119884 = ((119884119897 +4lowast119884119898 +119884119903)6)lowast ((119877119884119897 +4lowast119877119884119898 +119877119884119903)6)given in [4 5] is used to drive the crisp value of output signalHere Y is fuzzy output value and 119877119884 is the reliability It isneeded to mention that in the antecedent and consequentparts of the rule base we are using triangular type fuzzy setsfor the input and output parameters If we are considering120572 = 0 and 120572 = 1 levels then we can get left 119884119897 middle 119884119898

e(k)

PlantD

g(k)

y(k)Fuzzy

system based Zon

number

e998400(k)

sumsum

e(k)

minus+

Figure 2 Structure of type-2 FNS based control system

and right 119884119903 values of the output signal Left (119884119897 119877119884119897) andright (119884119903 119877119884119903) values are corresponding to 120572 = 0 level middle(119884119898 119877119884119898) value is corresponding to 120572 = 1 level4 Simulation Studies Control of

Dynamic Plant

In industry many dynamic plants are operating in an un-certain environment and characterized by the fuzziness ofinformation These dynamic plants are characterized withuncertainties related to the structure and the parametersThe deterministic models cannot adequately describe thesedynamic plants and using these models it was difficult toobtain the required control performance The use of fuzzyset theory for constructing control systems can be a valuablealternative to solve the problem [15ndash19] In this paper thefuzzy Z number based system described in the above sectionis used for the control of dynamic plants

The proposed fuzzy Z number based control system isused for control of dynamic plants The structure of thecontrol system is given in Figure 2 In this structure thedifference 119890(119896) between the plantrsquos output signal 119910(119896) and theset-point signal 119892(119896) is determined Using 119890(119896) the change oferror 1198901015840(119896) and the sum of error sum119890(119896) are determined Inthe figure 119863 indicates differentiation operation and sum indi-cates integration operation Using these input signals thefuzzy Z number based system is used for closed-loop controlsystem

The kernel of fuzzy controller is the knowledge base thathas If-Then form The accuracy of a fuzzy control systemdepends on input variables as well as the expert rules andthe membership functions therefore it is important forthem to be chosen carefully The antecedent part of If-Thenrules include input and the consequent part of the rules isthe control signal to be applied to the plant The analyseshave shown that the control of dynamic plants is basicallyimplemented using input variables error and change-in-errorAt first iteration using these parameters the knowledge basethat has If-Then form is developed In knowledge base the If-Then rules demonstrate the association between input param-eters and output control signal On the base analysis of thedynamic plants theKBdescribing input-output association isdesigned Using error and change of error the correspondingvalue of control signal is determined

During designing KB the input variables are describedby linguistic values The forms of these linguistic valuesare taken in triangle form Each of them is representedby triangle membership functions During designing these


01 x09070503

NL NS Z PS PL 120583(x)

Figure 3 Membership functions of input variables

Table 1 Knowledge of the expert

Control u Change-in-error e1015840

NL 119880 NS 119880 Z 119880 PS 119880 PL 119880Error e

NL 119880 PL 119880 PL 119880 PL 119880 PS 119880 Z 119880NS 119880 PL 119880 PL 119880 PS 119880 Z 119880 NS 119880Z 119880 PL 119880 PS 119880 Z 119880 NS 119880 NL 119880PS 119880 PS 119880 Z 119880 NS 119880 NL 119880 NL 119880PL 119880 Z 119880 NS 119880 NL 119880 NL 119880 NL 119880

triangle membership functions for the linguistic values itis necessary to determine the universe of discourse andnumber of linguistic values for each input parameter In thepaper the number of membership functions is taken to beequal to 5 The linguistic values for the input and outputparameters are denoted as NL (negative large) NS (negativesmall) Z (zero) PS (positive small) and PL (positive large)For simplicity we scale the range of variables between 0and 1 The membership functions used to describe inputand output variables are shown in Figure 3 After definingmembership functions for each parameter the constructionof fuzzy If-Then rules is performed Each rule has two-inputand one-output variables Table 1 describes the relationshipsbetween input parameters error and change-in-error andoutput parameter control signal of the controller Using thetable fuzzy production rules can be obtained In (15) usingTable 1 the fragment of rule base is given

Fragment of Fuzzy Rule Base is as follows

IF error is (NL 119880) and change-in-error is (NL 119880)Then Control signal is (PL 119880)

IF error is (Z 119880) and change-in-error is (PS 119880)Then Control signal is (NS 119880)

IF error is (PS 119880) and change-in-error is (NL 119880)

Then Control signal is (PS 119880)

(15)

Example 1 The presented fuzzy system based on Z numbersusing rule base is applied for control of dynamic plants Thedesigned system is validated using plant models in order tocheck if the system performance is acceptable In the firstexample the designed fuzzy controller based on Z number

10 20 30 40 50 60 70 80 90 1000Time

0

2

4

6

8

10

12

14

Y

Figure 4 Transient response characteristic of Z number based(solid curve) and conventional fuzzy (dashed line) controller

is applied for the control of dynamic plant given by thefollowing equation

119910 (119896) = 119910 (119896 minus 1) 119910 (119896 minus 2) (119910 (119896 minus 1) + 25)(1 + 119910 (119896 minus 1)2 + 119910 (119896 minus 2)2) + 119906 (119896) (16)

where 119910(119896minus1) 119910(119896minus2) are one- and two-step delayed outputof the dynamic plant and 119906(119896) is input control signal

The control of the plant is performed using differentvalues of the reference signal The period of the referencesignal is 200 samples and the mathematical expression forthe reference signal is given as follows

Reference (119896) =

10 0 le 119896 lt 5015 50 le 119896 lt 10010 100 le 119896 lt 15015 150 le 119896 lt 200

(17)

In the paper the root-mean-square error (RMSE) is used as aperformance criterion

RMSE = radicsum119870119894=1 (119910119889119894 minus 119910119894)2119870 (18)

where119870 is the total number of the samplesThe simulation of the fuzzy Z number based control sys-

tem has been performed for the plant (16) The performanceof Z number based fuzzy control system is compared with theperformance of conventional fuzzy control system using thesame initial conditions Figure 4 depicts the time responsecharacteristics of the conventional fuzzy and Z number basedcontrol system In the figure the solid line is the responsecharacteristic of the control system with Z number basedfuzzy controller dashed line is the response characteristicof the conventional fuzzy control system As shown fromthe figure the values of static errors for both controllers arezero transient overshoots are the same (near 19) Sums of


20 40 60 80 100 120 140 160 180 2000Time

0

2

4

6

8

10

12

14

16

18

Y

Figure 5 Time response characteristic of control systems withZ number based controller for different values of set-point inputsignals Dashed line is set-point signal solid line is plant output

Table 2 RMSE values

ControllerRMSE values

100 iterations(Figure 4)


Fuzzy 702697 2626501988Z number based controller 6449811 2614209756

RMSE values of control systems for 100 iterations are given inTable 2 RMSE value of Z number based control system is lessthan conventional fuzzy control system

In next simulation the performance fuzzy Z numberbased control system has been tested using different set-point signals given by formula (14) Figure 5 depicts the timeresponse characteristics of Z number based control systemusing different values of set-point signals The RMSE valuesfor both simulations are given in Table 2 As can be seen theRMSE value for Z number based fuzzy control system is lessthan that of the conventional fuzzy controller

Example 2 In the second experiment the simulation of thecontrol system of the temperature of rectifier column 119870-2 inoil refinery plant is performed The process is described bythe following differential equations

1198860119910 (2) (119905) + 1198861119910 (1) (119905) + 1198862119910 (119905) = 1198870119906 (119905) (19)

where 1198860 = 0072min2 1198861 = 0056min and 1198862 = 1 1198870 =60∘C(kgfcm2) here 119910(119905) is regulation parameter of object119906(119905) is output of fuzzy controller

The rule base given in Table 2 is used for the design ofZ number based controller for the plant (18) Analogouslyto Example 1 the designed control system is tested usinginput signals given in Example 1 for 100 and 200 iterationsSimulation results of designed fuzzy control system basedon Z number are compared with the simulation results ofcontrol systems based on the conventional fuzzy controller InTable 3 the results of comparative estimation of RMSE valuesof the response characteristic of control systems are given

Table 3 RMSE values





The simulation results demonstrate the efficiency of using Znumber based controller in control of dynamic plants

5 Conclusions

The paper presents the design of Z number based fuzzyinference system The interpolative inference mechanism ofthe fuzzy system has been presented and the same inferencemechanism is developed for Z number based fuzzy system Zfuzzy rule base is designed and the reliability degrees of thefuzzy values of the variables are estimated Interpolative fuzzyinference engine process using Z number based system hasbeen designed The designed algorithm is used for control ofdynamic plants The If-Then rule base using the knowledgeof experts is designed The reliability degrees of the fuzzyvalues of the variables are estimated Using rule base andfuzzy interpolative reasoning the control of dynamic plants isperformed The obtained results demonstrate the suitabilityof designed system in control of dynamic plants

Competing Interests

The author has no competing interests

References

[1] L A Zadeh ldquoA note on Z-numbersrdquo Information Sciences vol181 no 14 pp 2923ndash2932 2011

[2] R A Aliev O H Huseynov R R Aliyev and A A AlizadehThe arithmetic of Z-numbers World Scientific Publishing 2015

[3] R A Aliev and L M Zeinalova ldquoDecision making under Z-informationrdquo in Human-Centric Decision-Making Models forSocial Sciences (Studies in Computational Intelligence) P Guoand W Pedrycz Eds pp 233ndash252 Springer Berlin Germany2014

[4] B Kang D Wei Y Li and Y Deng ldquoDecision making usingZ-numbers under uncertain environmentrdquo Journal of Compu-tational Information Systems vol 8 no 7 pp 2807ndash2814 2012

[5] B Kang D Wei Y Li and Y Deng ldquoA method of convertingz-number to classical fuzzy numberrdquo Journal of ComputationalInformation System vol 9 no 3 pp 703ndash709 2012

[6] Z-QXiao ldquoApplication of Z-numbers inmulti-criteria decisionmakingrdquo in Proceedings of the International Conference onInformative and Cybernetics for Computational Social Systems(ICCSS rsquo14) pp 91ndash95 IEEE Qingdao China October 2014

[7] S K Pal R Banerjee S Dutta and S S Sarma ldquoAn insight intothe Z-number approach to CWWrdquo Fundamenta Informaticaevol 124 no 1-2 pp 197ndash229 2013

[8] A Azadeh M Saberi N Z Atashbar E Chang and P Pazho-heshfar ldquoZ-AHP a Z-number extension of fuzzy analytical hier-archy processrdquo in Proceedings of the International Conference


on Digital Ecosystems and Technologies (DEST rsquo13) pp 141ndash147Menlo Park Calif USA July 2013

[9] J Lorkowski V Kreinovich and R A Aliev ldquoTowards deci-sion making under interval set-valued fuzzy and Z-numberuncertainty a fair price approachrdquo in Proceedings of the IEEEInternational Conference on Fuzzy Systems (FUZZ-IEEE rsquo14) pp2244ndash2253 Beijing China July 2014

[10] L T Koczy and K Hirota ldquoApproximate reasoning by linearrule interpolation and general approximationrdquo InternationalJournal of Approximate Reasoning vol 9 no 3 pp 197ndash2251993

[11] L T Koczy and K Hirota ldquoInterpolative reasoning with insuffi-cient evidence in sparse fuzzy rule basesrdquo Information Sciencesvol 71 no 1-2 pp 169ndash201 1993

[12] Z C Johanyak and S Kovacs ldquoDistance based similaritymeasures of fuzzy setsrdquo in Proceedings of the 3rd Slovakian-Hungarian Joint Symposium onAppliedMachine Intelligence pp265ndash276 Herlrsquoany Slovakia January 2005

[13] S Kovacs and L T Koczy ldquoApplication of interpolation-basedfuzzy logic reasoning in behaviour-based control structuresrdquoin Proceedings of the IEEE International Conference on FuzzySystems UZZIEEE vol 6 pp 25ndash29 Budapest Hungary July2004

[14] Z Huang and Q Shen ldquoA new fuzzy interpolative reasoningmethod based on center of gravityrdquo in Proceedings of the 12thIEEE International Conference on Fuzzy Systems (FUZZ rsquo03) vol1 pp 25ndash30 IEEE May 2003

[15] R H Abiyev and O Kaynak ldquoType 2 fuzzy neural structure foridentification and control of time-varying plantsrdquo IEEE Trans-actions on Industrial Electronics vol 57 no 12 pp 4147ndash41592010

[16] RHAbiyev andOKaynak ldquoFuzzywavelet neural networks foridentification and control of dynamic plantsmdasha novel structureand a comparative studyrdquo IEEE Transactions on IndustrialElectronics vol 55 no 8 pp 3133ndash3140 2008

[17] R Abiyev ldquoFuzzy inference system based on neural networkfor technological process controlrdquo Mathematical and Compu-tational Applications vol 8 no 2 pp 245ndash252 2003

[18] R H Abiyev and O Kaynak ldquoIdentification and control ofdynamic plants using fuzzy wavelet neural networksrdquo in Pro-ceedings of the IEEE International Symposium on IntelligentControl (ISIC rsquo08) pp 1295ndash1301 IEEE San Antonio Tex USASeptember 2008

[19] R Abiyev ldquoNeuro-fuzzy system for technological processescontrolrdquo in Proceedings of the 6th World Multi-Conference onSystemics Cybernetics and Informatics8th International Confer-ence on Information Systems Analysis and Synthesis (SCI-ISASrsquo02) vol 12 pp 249ndash253 Orlando Fla USA July 2002

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fuzzy set with a footprint of uncertainty and then by com-puting the centroid it is converted to the crisp numbers fordecision-making These researchers present the advantagesof the presented approach based on their low computationalcomplexity However the converting Z numbers in realnumbers may lead to the significant loss of information andtherefore this may affect the performance of designed fuzzysystem

The paper [7] suggests the use of Z number in theperception of the uncertainty of the information conveyed bya natural language statement and to merge human-affectiveperspectiveswithComputingwithWords (CWW) Reference[8] considers an application of Z number based approach tothe AHP The presented work is also based on the approachproposed in [5] In [9] a fair price approach for a participationin a decision-making under interval set-valued fuzzy and Znumber uncertainty is considered

As shown using Z numbers a set of research workson arithmetic operations and decision-making have beendoneThe use of Z numbers in decision-making control andmodeling needs to use efficient inference mechanism for thedesigned system Reference [2] suggests usingZ interpolationin inference process In [10]Koczy andHirota purposed inter-polative reasoning in the sparse fuzzy rule base This methodmaintains the logical interpretation of modus ponens Themethod is distance based approximate fuzzy reasoning [1011] There are many different distance definitions for fuzzysets [12] These are absolute distance Euclidian distancedisconsistency measure Hausdorff measure and Kaufmanand Gupta measure Reference [12] points that the maindeficiency of these measures is that the information of theshape of the membership function of the fuzzy sets is mostlylost Using distance measure of two fuzzy sets A and Band given fuzzy set A1015840 it is impossible to construct thesecond fuzzy set B1015840 Reference [12] mentions that Koczymeasure can be used to solve this problem Koczy distanceis based on 120572-cuts of two fuzzy sets In [10 11] using Koczymeasure a fuzzy interpolative reasoning was proposed Theinterpolative approximate reasoning is applied to the solutionof different problems [13 14] In this paper the derivationof the interpolative approximate reasoning for Z numberbased fuzzy rules is consideredThe designed method is thenapplied for solving of dynamic plants control

The paper is organized as follows Section 2 presentsthe interpolative approximate reasoning method Section 3considers the use of the interpolative approximate reasoningmethod for Z numbers Section 4 presents the applicationof the designed interpolative approximate reasoning for thedevelopment of control system of the dynamic plant

2 Fuzzy Rule Interpolation

The inference technique of fuzzy logic resembles humanreasoning capabilities In the literature various fuzzy rea-soning methods are purposed to process uncertain infor-mation and increase the performance of the designed sys-tems These fuzzy reasoning methods are mainly based onthe compositional inference rule analogy and similarity

120583(x)

120572

A

xinf(A120572) sup(A120572)

Figure 1Membership function infimum and supremum for 120572-cut

interpolation and the concept of distance The speed pro-cessing capabilities and complexity of these reasoning meth-ods are important issues

In the paper inference based on fuzzy interpolation ofrules is used Fuzzy rule interpolation is proposed by Koczyand Hirota [10 11] and is designated for use for sparserule base The inference engine requires the satisfactionof the following conditions the used fuzzy sets should becontinuous convex and normal with bounded support

The proposed inference method is based on distancemeasure Here the distance measure of two fuzzy sets isdescribed using a fuzzy set defined in the interval [0 1] Theused distancemeasure is based on120572-cut Let us consider someconcepts used for two fuzzy sets 1198601 and 1198602 120572-cut of fuzzysets 1198601 and 1198602 will be denoted as 1198601120572 and 1198602120572 We say thatfuzzy set 1198601 is less than 1198602 that is 1198601 lt 1198602 iff

inf 1198601120572 lt inf 1198602120572 sup 1198601120572 lt sup 1198602120572 (1)

where inf1198601120572 and inf1198602120572 are infimum of 1198601 and 1198602sup1198601120572 and sup1198602120572 are supremum of 1198601 and 1198602 (Fig-ure 1)

Let us consider the following case Assume that wehave fuzzy rule base for the controller and in the result ofobservation we detect that input variable X is 119860lowast Let usdetermine the output119884 of the rule-based systemAssume that119860lowast is located between fuzzy sets1198601 and1198602 Let us determinethe output of fuzzy system using the rules which includes 1198601and 1198602 fuzzy sets

RulesGiven that 119883 is 119860lowast

If 119883 is 1198601 Then 119884 is 1198611If 119883 is 1198602 Then 119884 is 1198612Find Conclusion 119884 = 119861lowast

(2)

If 1198601 lt 119860lowast lt 1198602 and 1198611 lt 1198612 then using linearinterpolation [10] shows that





119909isin119883

inf119910isin119884

119889 (119909 119910) sup119910isin119884

inf119909isin119883


(4)



119889119871 (1198601120572 1198602120572) = 119889 (inf 1198601120572 inf 1198602120572) 119889119880 (1198601120572 1198602120572) = 119889 (sup 1198601120572 sup 1198602120572) (5)

Here





(6)



(7)





(9)








(10)




119889119871 (119860120572119883119894119895 119883120572119895 ) = 100381610038161003816100381610038161003816119860120572119883119894119895 minus 119883120572119895 100381610038161003816100381610038161003816 119889120572119894 = 119898sum

119895

119889119871 (119860120572119883119894119895 119883120572119895 ) (11)


119889120572119894 = 119889119888120572119894 + 119889119903120572119894 (12)






e(k)

PlantD

g(k)

y(k)Fuzzy

system based Zon

number

e998400(k)

sumsum

e(k)

minus+



Dynamic Plant






01 x09070503













(15)


10 20 30 40 50 60 70 80 90 1000Time

0

2

4

6

8

10

12

14

Y








(17)






20 40 60 80 100 120 140 160 180 2000Time

0

2

4

6

8

10

12

14

16

18

Y


Table 2 RMSE values








1198860119910 (2) (119905) + 1198861119910 (1) (119905) + 1198862119910 (119905) = 1198870119906 (119905) (19)



Table 3 RMSE values






5 Conclusions


Competing Interests


References





























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in







119909isin119883

inf119910isin119884

119889 (119909 119910) sup119910isin119884

inf119909isin119883


(4)



119889119871 (1198601120572 1198602120572) = 119889 (inf 1198601120572 inf 1198602120572) 119889119880 (1198601120572 1198602120572) = 119889 (sup 1198601120572 sup 1198602120572) (5)

Here





(6)



(7)





(9)








(10)




119889119871 (119860120572119883119894119895 119883120572119895 ) = 100381610038161003816100381610038161003816119860120572119883119894119895 minus 119883120572119895 100381610038161003816100381610038161003816 119889120572119894 = 119898sum

119895

119889119871 (119860120572119883119894119895 119883120572119895 ) (11)


119889120572119894 = 119889119888120572119894 + 119889119903120572119894 (12)






e(k)

PlantD

g(k)

y(k)Fuzzy

system based Zon

number

e998400(k)

sumsum

e(k)

minus+



Dynamic Plant






01 x09070503













(15)


10 20 30 40 50 60 70 80 90 1000Time

0

2

4

6

8

10

12

14

Y








(17)






20 40 60 80 100 120 140 160 180 2000Time

0

2

4

6

8

10

12

14

16

18

Y


Table 2 RMSE values








1198860119910 (2) (119905) + 1198861119910 (1) (119905) + 1198862119910 (119905) = 1198870119906 (119905) (19)



Table 3 RMSE values






5 Conclusions


Competing Interests


References





























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






119889119871 (119860120572119883119894119895 119883120572119895 ) = 100381610038161003816100381610038161003816119860120572119883119894119895 minus 119883120572119895 100381610038161003816100381610038161003816 119889120572119894 = 119898sum

119895

119889119871 (119860120572119883119894119895 119883120572119895 ) (11)


119889120572119894 = 119889119888120572119894 + 119889119903120572119894 (12)






e(k)

PlantD

g(k)

y(k)Fuzzy

system based Zon

number

e998400(k)

sumsum

e(k)

minus+



Dynamic Plant






01 x09070503













(15)


10 20 30 40 50 60 70 80 90 1000Time

0

2

4

6

8

10

12

14

Y








(17)






20 40 60 80 100 120 140 160 180 2000Time

0

2

4

6

8

10

12

14

16

18

Y


Table 2 RMSE values








1198860119910 (2) (119905) + 1198861119910 (1) (119905) + 1198862119910 (119905) = 1198870119906 (119905) (19)



Table 3 RMSE values






5 Conclusions


Competing Interests


References





























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




01 x09070503













(15)


10 20 30 40 50 60 70 80 90 1000Time

0

2

4

6

8

10

12

14

Y








(17)






20 40 60 80 100 120 140 160 180 2000Time

0

2

4

6

8

10

12

14

16

18

Y


Table 2 RMSE values








1198860119910 (2) (119905) + 1198861119910 (1) (119905) + 1198862119910 (119905) = 1198870119906 (119905) (19)



Table 3 RMSE values






5 Conclusions


Competing Interests


References





























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




20 40 60 80 100 120 140 160 180 2000Time

0

2

4

6

8

10

12

14

16

18

Y


Table 2 RMSE values








1198860119910 (2) (119905) + 1198861119910 (1) (119905) + 1198862119910 (119905) = 1198870119906 (119905) (19)



Table 3 RMSE values






5 Conclusions


Competing Interests


References





























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



Research Article Number Based Fuzzy Inference System for...

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