A New Type of Fuzzy Membership Function Designed for Interval Type-2 Fuzzy Neural Network

Jiajun Wang

A New Type of Fuzzy Membership Function Designed for Interval Type-2 Fuzzy Neural Network

Jiajun Wang1

A new type of fuzzy membership function(FMF)is proposed for interval type-2 fuzzy neural network(IT2FNN)in this paper.Three types of interval type-2 FMF(IT2FMF)can be derived from the proposed type of FMF.And each type of IT2FMF has di ff erent shape of footprint of uncertainty(FOU).The derived IT2FMFs are applied to a simpli fi ed T2FNN to identify two nonlinear systems.The identi fi cation performance of the derived IT2FMFs are compared with Gaussian and ellipsoidal type of IT2FMFs through simulation.Simulation results certify that the derived IT2FMFs can achieve better identi fi cation performance than Gaussian and ellipsoidal type of IT2FMFs with elaborately tuning of the parameters for the simpli fi ed IT2FNN.

Fuzzy membership function(FMF),interval type-2 fuzzy neural network(IT2FNN),nonlinear system,system identi fi cation

DOI10.16383/j.aas.2017.e150348

1 Introduction

As the extension of the type-1 fuzzy set(T1FS)theory,the type-2 fuzzy set theory is more advanced and complex[1].The type-2 fuzzy set theory is further developed with the interval type-2 fuzzy set(IT2FS)theory[2]?[4].The IT2FS has more advanced ability to deal with the uncertainties of the system.And it is used to solve identi fi cation,control,prediction and pattern recognition problems[5],[6].Compared with T1FS,the excellent processing ability of the IT2FS originates from the interval type-2 FMF(IT2FMF).The selection of the FMF for the IT2FS has very large e ff ects on the performance of the IT2FS.The research on the FMF of the IT2FS is still an open problem.The key point of this paper is the introduction of a new type of FMF to enhance the performance of the IT2FS.

As we known,there exist six types of IT2FMFs that can be selected from the literatures at present,which are triangular,trapezoidal,sigmoidal,pi-shaped,Guassian and ellipsoidal type of FMFs[7],[8].Guassian,triangular,sigmoidal and pi-shaped type of FMFs have three parameters that need to be updated online.Trapezoidal and ellipsoidal type of FMFs have four updating parameters.At present,Guassian type of IT2FMF is widely applied in the IT2FS.And it become a standard selection in the interval type-2 fuzzy neural network(IT2FNN)[9]?[11].

The fuzzy neural network(FNN)is the hybridization of the neural network and fuzzy system,which inherits the learning ability from the neural network and the capability of fuzzy reasoning to uncertain information[12]?[14].The Takagi-Sugeno-Kang(TSK)type of fuzzy models are e ff ective in the system identi fi cation problems[15]?[18].The combination of the TSK-type with FNN can achieve superior learning accuracy than Mamdani type of FNN.The interval type-2 TSK fuzzy neural network(IT2TSKFNN)unites the IT2FS in the antecedent part and the TSK-type as the consequent parts.And it has the united advantages of the IT2FS,TSK-type fuzzy set and neural network[16],[17].In this paper,IT2TSKFNN is selected as the target FNN to test the performance of the proposed FMF.

Although the IT2FNN has superior performance inprocessing the uncertainties of the system than T1FNN,IT2FNN is computationally intensive because the typereduction procedure is very complex.And this con fi nes the application of the IT2FNN.The iterative K-M algorithm is the general method to realize the type-reduction of the IT2FNN[3].The consequent weights of almost all the IT2FNNs except TSK-type need to be rearranged in ascending order according to the iterative K-M algorithm.In this paper,we adopt the simpli fi ed IT2FNN to test the derived IT2FMFs.The simpli fi ed IT2FNN can be realized with the computation of distribution factor qrand qlwithout incurring the K-M iterative computation[19].

The main contribution of this paper can be given as following three aspects.

1)A new type of IT2FMF is proposed.Based on the proposed IT2FMF,three type of IT2FMFs can be derived.This make the selection of the IT2FMFs for the IT2FNN have larger freedom.

2)The derived IT2FMFs are tested with the simpli fi ed IT2FNN.The design procedure of the simpli fi ed IT2FNN is given step by step.And the parameter updating computation is demonstrated in details.

3)The derived IT2FMFs for the simpli fi ed IT2FNN can achieve better identi fi cation performance than Gaussian and ellipsoidal type of IT2FMFs in two typical nonlinear examples.

This paper is organized as follows.In Section 2,the proposed type of FMF is introduced.In Section 3,the design procedure of the simpli fi ed IT2FNN is presented.In Section 4,the parameter updating rules are derived.In Section 5,the simulation studies are given to show the e ff ectiveness of the derived IT2FMFs.In Section 6,some conclusions are given.

2 Introduction of IT2FMFs

2.1 Gaussian Type of IT2FMF

The Gaussian type of IT2FMF is given in Fig.1(a).The mathematical expression of the Gaussian type of IT2FMF can be expressed as

where m is the mean value,and σ is the standard deviation(STD),and x is the input variable.In(1),the mean valuem and the STD σ all can be seen as uncertain values.In this paper,the mean value m is selected as the uncertain value(m ∈ [m1,m2],where m1＜ m2)and the STD σ is fi xed.The footprint of uncertainty(FOU)of the Gaussian type of FMF is bounded by lower MFμand upper MF,which can be de fi ned as following equation

and

Fig.1.Gaussian and ellipsoidal type of IT2FMFs.

2.2 Ellipsoidal Type of IT2FMF

The ellipsoidal type of IT2FMF is given in Fig.1(b).The equation of the ellipsoidal type of IT2FMF can be de fi ned as following equation[8]

where m is the middle value,d is the width of the FMF,and x is the input value.The parameters a1and a2determine the area of the FOU of the ellipsoidal type of IT2FMF,which can be selected as

The boundaries of the FOU of the ellipsoidal type of IT2FMF are the lower MFμand the upper MF.The boundary FMFs are given in Table I.

From the de fi nition of the ellipsoidal type of IT2FMF,we can obtain two points.

1)There are four parameters m,σ,a1and a2that need to be updated in the identi fi cation of a system.

2)The computation of the derivation of the ellipsoidal type of IT2FMF is not an easy job.It need complex computation.

2.3 The Proposed IT2FMFs

Originating from the ellipsoidal type of IT2FMF,the proposed type of FMF can be de fi ned as following equation

where m,σ and x are the same as the de fi nition in(4).a is the parameter that can be used to adjust the shape of the FOU.According to the di ff erent value of parameter a,we can obtain three di ff erent type of IT2FMFs.

1)When a＞0 and a/=1,the obtained FMFs are called exponential type.

2)When a=1,the obtained FMF is called linear type.

According to di ff erent combination with the parameter a,we can have three types of IT2FMFs.

1)The combination of the fi rst case is that the upper MF is exponential-type FMF and the lower MF is linear-type FMF.This combination is called exponential-linear-type IT2FMF(EL-type IT2FMF).

2)The combination of the second case is that the upper MF is linear-type MF and the lower MF is exponentialtype MF.This combination is called linear-exponentialtype IT2FMF(LE-type IT2FMF).

3)The combination of third case is that the upper and lower MFs are all exponential-type MFs.This combination is called exponential-exponential-type IT2FMF(EE-type IT2FMF).

The fi gures of the derived IT2FMFs are given in Fig.2,where subgraphs(a),(b)and(c)represent EL-type,LE-type and EE-type IT2FMFs respectively.The upper and lower MFs are given in Table I respectively.Compared with the ellipsoidal type of IT2FMF,the derived IT2FMFs have the following merits.

1)The parameters that need to be updated are reduced.The ellipsoidal type of IT2FMF has four kinds of parameters that need to be updated,whereas the proposed IT2FMF has three kinds of parameters.

2)The computation complexity is simpli fi ed.Simpli fi cation of the computation is very important for the application of the IT2FNN.

3)The design freedoms of the FMFs are increased with di ff erent combination.The di ff erent combination can make them have di ff erent performance for di ff erent systems.

3 The Design Procedure of IT2FNN

To test the e ff ectiveness of the derived IT2FMFs,the simpli fi ed IT2FNN is selected as the target IT2FNN[19].The structure of the simpli fi ed IT2FNN is given in Fig.3.There are six layers that need to be designed in the simplifi ed IT2FNN.

Layer 1:is the input layer.The input value xi(i=0,1,...,n,n represents the number of input)is directly transmitted to Layer 2 and Layer 4.There are no weights that need to be updated in Layer 1.

Layer 2:is the FMF layer.In this layer,the fuzzi fi cation operation is fi nished with IT2FMFs.And in Fig.3,the FMFs can be Gaussian,ellipsoidal or one of the derived IT2FMFs.After the interval type-2 fuzzi fi cation in Layer 2,the interval(i=1,...,n represents the actual input,j=1,...,m represents the fuzzy rules for each actual input)can be acquired.

Layer 3:is fi ring layer.Each node in this layer represents one fuzzy logic rule and performs a fuzzy meet operation using an algebraic product operation.The output of a rule node represents the fi ring strength Fjof the corresponding fuzzy rule Rjthat is an interval type-1 fuzzy set.The fi ring strength Fjcan be computed with the following expression

where the index m represents the fuzzy rules for each in-put variable,and the index n represents the actual input variable.

TABLE I THE MFs oF tHE ELLipsoiDAL AND DERivED IT2FMFs

Fig.2.The shape of FOU for the derived IT2FMFs.

Fig.3. The structure of the simpli fi ed IT2FNN with two input,three rules and one fi nal output.

Layer 4:is the consequent layer.The node in this layer is called TSK-type node.Each rule node in the Layer 3 has its corresponding TSK-type node in the Layer 4.The output of each node is an interval type-1 fuzzy set,denoted by[wjl,wjr],which can be called TSK-type weights.And the TSK-type weights can be computed as following expression

where cijand sijare called consequent parameter.Each TSK-type weight can be expressed as following

and

where x0≡1.

Layer 5:can be called output processing layer or type reduction layer.The distribution factor q can be designed to enable the adaptive adjustment of the upper and lower value of the output.The application of the distribution factor q can alleviate the computation in type reduction without using the K-M iterative procedure.The output of[yl,yr]can be computed as following expressions

and

where qland qrare called left and right distribution factor.

Layer 6:is the output layer.Because the output of Layer 5 is an interval set,it can not be used for the output directly.The defuzzi fi cation can be realized by computing the average of yland yr.

The simpli fi ed IT2FNN can reduce the computational complexity of the IT2FNN.The parameter updating computation is a key part to realize the simpli fi ed IT2FNN.The gradient descent method(GDM)will be used in the parameter updating computation of the simpli fi ed IT2FNN.

4 Parameter Updating Rules

In the parameter updating design of the IT2FNN,many di ff erent design method can be applied,such like GDM,extended Kalman fi lter(EKF),and particle swarm optimization(PSO)[8].In this paper,we applied the GDM in the parameter updating of single-output system identi fi cation.The cost function is de fi ned as

where yd(k)and y(k)are the desired output and the actual output of the simpli fi ed IT2FNN respectively,e(k)=y(k)?yd(k)is the identi fi cation error and k is the sample number.According to the GDM,the parameters can be updated with the following algorithm

where X(k)can represents m, σ,a,c,s or q,and η is the learning rate.

When the Gaussian type of FMF is selected in the simpli fi ed IT2FNN,there are three kinds of parameters that need to be updated,which are consequent parameters,distribution factors and antecedent parameters.

4.1 Consequent Parameter and Distribution Factor Updating Algorithm

The consequent parameters include c and s.The updating algorithm offor the consequent parameters can be given as the following expressions

and

Remark 1:In consequent parameter updating,i=0,...,n,j=1,...,m.In following antecedent parameter updating,i=1,...,n,j=1,...,m.

The distribution factor include the left factor qland right factor qr.The updating algorithms offor the distribution factor can be computed with the following expressions

and

4.2 Antecedent Parameter Updating Algorithm

The antecedent parameters include m,σ and a.The common updating algorithm offor the antecedent parameters can be given as following expression

where X can be m,σ or a.For the simpli fi ed IT2FNN,the partial derivativecan be computed with the following expressions

The computation of the partial derivativesin(21)is relation to the selection of the IT2FMFs.Their computation with di ff erent type of IT2FMFs are given as following parts.

4.2.1 When the Gaussian Type of IT2FMF is Selected

There are three kinds of antecedent parameters that need to be updated in Gaussian type of IT2FMF,which are m1,m2and σ.The computation of the partial derivativesandcan be given as following expressions

Equations(24)?(29)give the computation of the partial derivatives of the Gaussian type of IT2FMF with respect of the parameters m1,m2and σ.Whenare acquired,then the partial derivativecan be obtained with(22)and(23).

4.2.2 When the Ellipsoidal Type of IT2FMF Is Selected

When the ellipsoidal type of IT2FMF is selected,there are four kinds of antecedent parameters that need to be updated,which are m,σ,a1and a2.The computation of the partial derivativesfor the IT2FMF can be given in(30)?(35)at the bottom of this page.and=0.

4.2.3 When the Derived IT2FMFs Are Selected

When the derived IT2FMFs are selected,there are three kinds of antecedent parameters that need to be updated,which are m,σ and a.The computation of the partial derivativesfor the derived IT2FMFs are given with the following three cases.

1)EL-type IT2FMF

In the EL-type IT2FMF,the parameter a＞1.The computation of the partial derivativeof the EL-type IT2FMF can be given in Table II.

TABLE IIoF EL-typE IT2FMF

TABLE IIoF EL-typE IT2FMF

?X mij?σij＜xi≤mij mij＜xi≤mij+σij?μ?mij ?aij?μij σij )(aij?1)?μσij(mij?xi σij )(aij?1) aij σij(xi?mij?mij ?1 ij 1 σij?σij σij )aij?μμij aij σij(mij?xi σij )aij aij?σij σij(xi?mij ij mij?xi xi?mij?σij σ2 i j σ2ij?μij σij )?μ?aij ?(mij?xi σij )aijln(mij?xi σij ) ?(xi?mij σij )aijln(xi?mij?aij 0 0 ij

2)LE-type IT2FMF

In the LE-type IT2FMF,the parameter 0＜a＜1.The computation of the partial derivativeof the LE-type IT2FMF are given in Table III.

TABLE IIIoF LE-typE IT2FMF

TABLE IIIoF LE-typE IT2FMF

?X mij?σij＜xi≤mij mij＜xi≤mij+σij?μ?mij ?1?μij 1 σij?mij ?aij σij?μσij )(aij?1)?ij σij(mij?xi σij )(aij?1) aij σij(xi?mijμij mij?xi xi?mij?σij σ2 i j σ2ij?μσij )aij?ij aij?σij σij(mij?xi σij )aij aij σij(xi?mij?aij 0 0?μμij?aij ?(mij?xi ij σij )aijln(mij?xi σij ) ?(xi?mij σij )aijln(xi?mij σij )

3)EE-type IT2FMF

In the computation of the partial derivativeandfor the LE-type IT2FMF,there are two case need to be considered.One is when 0＜a＜1,and the other is when a＞1.The computation ofandfor two cases is given in Table IV and Table V.

TABLE IVoF EE-typE IT2FMF WHEN 0＜a＜1

TABLE IVoF EE-typE IT2FMF WHEN 0＜a＜1

?X mij?σij＜xi≤mij mij＜xi≤mij+σij?μ(1?aij)(1?aij)?mij ? 1?μij aijσij(mij?xi σij )aij 1 aijσij(xi?mij σij )aij?mij ?aij?μij σij(mij?xi σij )(aij?1) aij σij(xi?mij σij )(aij?1)aijσij(mij?xi?μij 1 σij )1 aijσij(xi?mij aij ? 1 σij )?σij aij 1?μij σij(mij?xi aij σij )aij ?aij σij(xi?mij?σij σij )aij?μij (mij?xi aijln(mij?xi 1 aijln(xi?mij 1 1?aij a2ij σij )σij ) ? 1 a2ij(xi?mij σij )σij )?μ?aij ?(mij?xi ij σij )aijln(mij?xi σij ) (xi?mij σij )aijln(xi?mij σij )

TABLE VoF EE-typE IT2FMF WHEN a＞ 1

TABLE VoF EE-typE IT2FMF WHEN a＞ 1

?X mij?σij＜ xi≤ mij mij＜ xi≤ mij+σij?μ(1?aij)?μij 1 aijσij(xi?mij aijσij(mij?xi(1?aij)aij ? 1?mij σij )σij )aij?μij aij?mij σij(mij?xi σij )(aij?1) ?aij σij(xi?mij σij )(aij?1)?μij aijσij(mij?xi aij?σij ? 1 1 aijσij(xi?mij 1 σij )aij 1 σij )?μ?σij ?aij ij σij(mij?xi σij )aij aij σij(xi?mij σij )aij?μij (mij?xi aijln(mij?xi 1?aij ? 1(xi?mij σij )1 a2ij σij ) 1 a2ij σij )aijln(xi?mij σij )?μ?aij (mij?xi ij σij )aijln(mij?xi σij ) ?(xi?mij σij )aijln(xi?mij σij )

Remark 2:From the above computation,we can obtain the following conclusions.

1)The di ff erence realization between the FMFs mainly focuses on the computation of.

2)Consequent parameter and distribution factor updating algorithm are all the same for Gaussian,ellipsoidal,and the derived IT2FMFs in the simpli fi ed IT2FNN.

3)The computation of the ellipsoidal type of IT2FMF is more complex than Gaussian and the derived IT2FMFs.And the computation of EL-type and LE-type IT2FMFs are more easy to be realized than the EE-type IT2FMF.

5 Simulation Results and Analysis

To test the e ff ectiveness of the derived IT2FMFs,the IT2FMFs are applied in the simpli fi ed IT2FNN to identify two typical nonlinear time-varying systems[5],[19],[20].The structure of the system identi fi cation constructed with MATLAB/Simulink is given in Fig.4.

Fig.4. The structure of the system identi fi cation with simplified IT2FNN.

To compare the performance of the derived IT2FMFs with the selected IT2FMFs,the rules of each node in the second layer of the simpli fi ed IT2FNN is set to be m=3,and the input of the simpli fi ed IT2FNN is set to be n=2.In the simulation of the identi fi cation with simpli fi ed IT2FNN,the common initialization data are given as the following data

where i=0,1,2 and j=1,2,3.

The antecedent parameters of di ff erent type of IT2FMFs in this paper are given in Table VI,where i=1,2 and j=1,2,3.

TABLE VI THE INitiAL ANtECEDENt PARAMEtERs FoR DiFFERENt TypE oF IT2FMFs

The integral of the absolute value of the error(IAE)is selected as the performance criterion,and which is given as following expression

where Tsis the sample time,and e(k)=y(k)?yd(k)is the identi fi cation error.In the simulation,the sample time Tsis set to be 0.001s.

5.1 Example 1

The fi rst nonlinear system to be identi fi ed is given as the following expression where k is the sample number.The input variables of the simpli fi ed IT2FNN is u(k)and yd(k).The input signal is generated with u(k)=sin(2πk/1000).

The identi fi cation results of the system in(38)are given in Fig.5(a)with di ff erent type of IT2FMFs.The identi fication errors and IAEs are given in Fig.5(b)and Fig.5(c).In Fig.5,the subscript 1,2,3,4 and 5 present Gaussian,ellipsoidal,EL-type,LE-type and EE-type IT2FMFs,respectively.When the output of the system contains the uniform random noise(between[?0.1,0.1]),the simulation results are given in Fig.6(a).And the comparison of identi fi cation errors and IAEs with disturbance are given in Fig.6(b)and Fig.6(c).The comparison data of the IAEs for Example 1 without and with disturbance are given in Table VII at 2 second.

Fig.5. Identi fi cation of Example 1 with di ff erent FMFs.

Fig.6. Identi fi cation of Example 1 with disturbance.

TABLE VII THE CoMpARisoN oF tHE IAEs FoR ExAMpLE 1

5.2 Example 2

The second nonlinear system to be identi fi ed is given as the following expression

where the parameters f a,b and c are time-varying parameters,and which are given as following expressions

where T is the samples per period.To test the identi fication performance,the input signal is given as following expression

where

The identi fi cation results of the system in(39)are given in Fig.7(a)with di ff erent FMFs.The identi fi cation errors and IAEs are given in Fig.7(b)and Fig.7(c).And the identi fi cation results with uniform noise between[?0.1,0.1]are given in Fig.8(a).The comparison of identi fi cation errors and IAEs with disturbance are given in Fig.8(b)and Fig.8(c).The comparison data of the IAEs for Example 2 without and with disturbance are given in Table VIII at 2 second.

5.3 Analysis and Discussion

From the simulation results and comparison,we can acquire the following fi ve conclusions.

1)The proposed type(including EL-type,LE-type and EE-type)of IT2FMFs are e ff ective and can be applied in the system identi fi cation with simpli fi ed IT2FNN.

2)The derived IT2FMFs can achieve better performance than Gaussian and ellipsoidal type of IT2FMFs with elaborately tuning of the parameters of the simpli fi ed IT2FNN.

TABLE VIII THE CoMpARisoN oF tHE IAEs FoR ExAMpLE 2

Fig.7. Identi fi cation of Example 2 with di ff erent IT2FMFs.

Fig.8. Identi fi cation of Example 2 with disturbance.

3)Ellipsoidal type of IT2FMF can be used in the static parameter system.And it is more robustness than Gaussian type of IT2FMF under disturbance environment.While when it is used in the time-varying parameter system,the identi fi cation error is larger than Gaussian andthe derived IT2FMFs.

4)In static system identi fi cation,the EL-type IT2FMF has better identi fi cation accuracy than LE-type IT2FMF with disturbance.While in time-varying system identi fi cation,the LE-type IT2FMF has better identi fi cation accuracy than EL-type IT2FMF with disturbance.

5)Among the derived FMFs,the EE-type IT2FMF has stronger identi fi cation ability than EL-type and LE-type IT2FMFs,with or without considering the time-varying or disturbance characteristics of the actual system.

Remark 3:Although the derived IT2FMFs can achieve better identi fi cation performance than Gaussian and ellipsoidal type of IT2FMFs in the above two examples,we can not say that the derived IT2FMFs can guarantee better performance in all kinds of environment.Because uncertainty can appear di ff erent for di ff erent system,one type of IT2FMF can not fi t all the condition.This paper gives more freedom in the selection of the IT2FMFs that can be use in the IT2FNN design.

6 Conclusions

In this paper,a new type of FMF is proposed for the IT2FNN.And three type of IT2FMFs can be derived with the proposed type of FMF.The whole paper can be summarized with the following three conclusions.

1)The derived three types of IT2FMFs are simpler than ellipsoidal type of IT2FMF and have better identi fi cation ability in system identi fi cation.

2)The derived IT2FMFs and the adoption of the distribution factor q can simplify the computation of the type reduction problem of the IT2FNN.And this combination can make the realization of the IT2FNN an easy job.

3)The proposed IT2FMFs can give the selection of the IT2FMFs more freedom in IT2FS.This is very meaningful for the research of the IT2FNN.

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Jiajun Wang graduated from Shandong Institute of Light Industry(Qilu University of Technology),China,in 1997.He received the M.Sc.degree and the Ph.D.degree from Tianjin University,China,in 2000 and 2003.He is currently a Professor at the School of Automation,Hangzhou Dianzi University,Hangzhou,China.His research interests include backstepping control,sliding mode control,neural networks and their applications in motion control system.E-mail:wangjiajun@hdu.edu.cn

Jiajun Wang.A new type of fuzzy membership function designed for interval type-2 fuzzy neural network.Acta Automatica Sinica,2017,43(8):1425?1433

November 18,2015;accepted April 1,2016.This work was supported by the National Natural Science Foundation of China(61273086).

Recommended by Associate Editor Huaguang Zhang.

1.School of Automation,Hangzhou Dianzi University,Hangzhou 310018,China