Nonlinear Function Approximation Using Fuzzy Functional SIRMs Inference Model

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1 Nonlinear Function Approximation Using Fuzz Functional SIRMs Inference Model HIROSATO SEKI Osaka Institute of Technolog Department of Technolog Management 5-6-, Omia, Asahi-ku, Osaka JAPAN h seki@ieee.org Abstract: Since the single input rule modules connected fuzz inference model (SIRMs model) is proposed b Yubazaki, Yi et al., man researches on the extension of the SIRMs model have been reported. Moreover, the fuzz functional SIRMs inference model, in which the consequent parts of the functional-tpe SIRMs model are generalized to fuzz function, has proposed as one of various extension SIRMs models. In this paper, we appl the fuzz functional SIRMs inference model to identification of nonrinear funcions, and show the applicabilit of the model. Ke Words: Soft Computing, Approximate Reasoning, Fuzz Inference, SIRMs Connected Fuzz Inference Model, Fuzz Functions, Nonlinear Modeling. Introduction Since Mamdani [] applied the concept of fuzz inference to steam engine experimental device, relevant research and applications have been executed in various fields. Especiall, researches on the T S inference model [2], which is widel used as fuzz control method and so on, are reported in man papers. Fuzz inference plas a significant role in fuzz applications. However, as for the fuzz rules in the traditional fuzz inference models, all the input items of the sstem are set to the antecedent part, and all output items are set to the consequent part. Therefore, the problem is that the number of fuzz rules becomes ver huge; hence, the setup and adjustment of fuzz rules become difficult. On the other hand, the single input rule modules connected tpe fuzz inference model (SIRMs model) [3 8] that unifies the inference outputs from fuzz rule modules of one input tpe IF-THEN form can reduce the number of fuzz rules drasticall. The model has been applied to nonlinear function identification, control of a first order lag sstem with dead time, orbital pursuit control of a nonrestrained object, stabilization control of a handstand sstem [4], etc., and good results are obtained. However, since the number of rules of the SIRMs model is limited as compared to the traditional inference models, inference results gained b the SIRMs model are simple in general. Therefore, we have proposed the functional-tpe SIRMs model [9 ] in which the consequent parts are generalized to functions from real numbers, and have shown the equivalence conditions [2] between the functional-tpe SIRMs model and T S inference model. The functional-tpe SIRMs model has obtained good results regarding identification of nonlinear functions and a medical diagnosis []. However, although the antecedent parts of the above SIRMs models are fuzz sets, the consequent parts are function. Thus, this structure is hard to understand from the difference of antecedent parts and consequent parts. Rather, it ma be completel natural that the consequent parts are also fuzz sets [3]. Therefore, Seki has proposed a fuzz functional SIRMs model [4, 5] in which the consequent parts of the functional-tpe SIRMs model are generalized to fuzz function, and shown its theoretical properties. However, the applicabilit of this model has remained to be full elucidated et. In this paper, we appl the fuzz functional SIRMs model to approximation of nonlinear functions, and show the applicabilit of this model b comparing it with the conventional SIRMs model and simplified fuzz inference model. 2 Fuzz Inference Models In this section we review the min max gravit model, product sum gravit model and fuzz functional SIRMs inference model. ISBN:

2 2. Min Max Gravit Model We firstl explain the min max gravit model as Mamdani s fuzz inference model [] for the fuzz inference form (see Fig. ). The rules of the min max gravit model are given as follows: { x = A Rule R i = i,x 2 = i,...,x n = A n i = B i () where x,x 2,...,x n are variables of the antecedent part, i,a2 i,...,an i fuzz sets, B i fuzz sets of the consequent part, i =, 2,...,M and M is the total number of rules. Each inference result B i which is infered from the fact x 0,x0 2,...,x0 n and the fuzz rule i,a2 i,...,an i B i is given in the following. The degree of fitness, h i, of the fact x 0,x0 2,...,x0 n to the antecedent parts i,a2 i,...,an i is given as h i =min{ i (x 0 ), i (x 0 2),...,A n i (x 0 n)} (2) Thus, the inference result B i is given as B i() = min{h i,b i ()} (3) The final consequence B of () is aggregated from B,B 2,..., B M b using the max. Namel, B () =max{b (),B 2(),...,B M()} (4) The representative point 0 for the resulting fuzz set B is obtained as the center of gravit of B : B ()d 0 = (5) B ()d 2.2 Product Sum Gravit Model We secondl explain a fuzz inference model called product sum gravit model [6] for the fuzz inference form (see Fig. 2). The rules of the product sum gravit model are also given as (). Each inference result B i which is infered from the fact x 0,x0 2,...,x0 n and the fuzz rule i,a2 i,...,an i B i is given in the following. The degree of fitness, h i, of the fact x 0,x0 2,...,x0 n to the antecedent parts i,a2 i,...,an i is given as h i = i (x 0 ) i (x 0 2) A n i (x 0 n) (6) where stands for the algebraic product. Thus, the inference result B i is given as B i() = i (x 0 ) A n i (x 0 n) B i () = h i B i () (7) The final consequence B of () is aggregated from B, B 2,..., B M b using the algebraic sum (+). Namel, B () =B ()+B 2()+ + B M() (8) The representative point 0 for the resulting fuzz set B is obtained as the center of gravit of B : B ()d 0 = (9) B ()d 3 Fuzz Functional SIRMs Inference Model Seki [9 ] has proposed the functional-tpe SIRMs model in which the consequent part of the SIRMs B' a 2 b B min{a,b} (= h ) 2 a b B B' ab (= h ) x x 2 2 B 2 B'2 x x 2 2 B 2 B'2 x 0 x x 0 2 x 2 x 0 x x 0 2 x 2 B' = B' B' U 2 B' = B' + B' Figure : Min max gravit model. Figure 2: Product sum gravit model. ISBN:

3 model is a function, where the sstem has n inputs and output, and each rule module corresponds to one of the n input items and has onl the input item in its antecedent. However, although the antecedent parts of the functional-tpe SIRMs model are fuzz sets, the consequent parts are function. Thus, this structure is hard to understand from the difference of antecedent parts and consequent parts. Rather, it ma be completel natural that the consequent parts are also fuzz sets [3]. Thus, Seki [4, 5] has proposed a fuzz functional SIRMs model in which the consequent parts of the functional-tpe SIRMs model are generalized to fuzz function. This model can make it easier to understand the structure of rules compared with the ordinar fuzz inference models. The rules of the fuzz functional SIRMs model are given as follows. Rules- :{x = j = F j (x )} m j=. Rules-i : {x i = A i j i = F i j (x i)} m i j=. Rules-n : {x n = A n j n = F n j (x n)} mn j= (0) where the Rules-i stands for the ith single input rule module, x i corresponding to the ith input item is the sole variable of the antecedent part of the Rules-i, and i is the variable of its consequent part. A i j and F j i(x i) are, respectivel, fuzz set and fuzz function of the jth rule of the Rules-i, where i =, 2,...,n; j =, 2,...,m i, and m i stands for the number of rules in the Rules-i. An example as one-dimensional case of fuzz function is shown in Fig. 3 [3]. Given an input x 0 i to the Rules-i, the degree of the antecedent part in the jth rule in the Rules-i is given F (x) 2 F (x ) 0 b (). h i j = A i j(x 0 i ) () The consequent parts Fj i(x i) constitutes fuzz sets as well as the fuzz functional inference model [3]. Therefore, the inference results i 0 from rule modules of the fuzz functional SIRMs model should be also obtained from the product sum gravit model or min-max gravit model. Namel, the inference result i 0 from rule modules Rules-i is given as F i 0 j i (x) ()d = (2) Fj i (x) ()d Final inference result 0 of the fuzz functional SIRMs model is given b n 0 = w i i 0 (3) i= where w i stands for the importance degree for each input item x i (i =, 2,...,n). 4 Learning Algorithms for Fuzz Functional SIRMs model Generall speaking, the setup of the membership functions and fuzz rules in the fuzz inference sstems is difficult. Hence, we expect to automaticall optimize the membership functions and fuzz rules based on input-output data for the sstems. From the reason, several learning algorithms for membership function and fuzz rules are proposed [7 22]. Therefore, we review the learning algorithm [5] b the steepest descent method in the case of triangulartpe fuzz sets. When the training input output data (x, x 2,..., x n ; T ) are given for a fuzz sstem model, it is usual to use the following objective function E for evaluating an error between T and 0, which can be regarded as an optimum problem: 0 F (x) x 0 F (x ) 2 0 h A h 2 2 x E = 2 (T 0 ) 2 (4) where T is the desired output value, and 0 the corresponding fuzz inference result. The triangular-tpe fuzz sets are used, where let center of gravit f i j (x i) of fuzz set in the consequent part of fuzz functional SIRMs model be a linear expression as follows. Figure 3: Fuzz function. f i j(x i )=c i j + d i jx i (5) ISBN:

4 The parameters of center a i j and width bi j of the fuzz sets, the bottom lj i and the center of gravit f j i(x i) of consequent part, and importance degree w i are obtained b the steepest descent method as follows. We consider the following triangular-tpe fuzz set A i j (x i). A i xi a j(x i )={ i j /bi j ; ai j bi j x i a i j + bi j 0; otherwise (6) where a i j and bi j (i =, 2,...,n; j =, 2,...,m i) stand for the center and width, respectivel. From (6), the learning algorithms at t +step of each parameter are obtained as follows. a i j(t +) = a i j(t) α E a i (7) j b i j(t +) = b i j(t) β E b i j c i j(t +) = c i j(t) γ E c i j d i j(t +) = d i j(t) δ E d i j l i j(t +) = l i j(t) ɛ E l i j w i (t +) = w i j(t) ζ E w i j (8) (9) (20) (2) (22) where α, β, γ, δ, ɛ and ζ are the learning rates in the learning process, t means the learning iteration number. 5 Approximation of nonlinear functions b fuzz functional SIRMs model In this section, we appl the fuzz functional SIRMs model and the above learning algorithms to the following two nonlinear functions with two input variables and one output variable in order to compare them with the conventional SIRMs model and neurofuzz method using simplified fuzz inference model [7] for identifing and evaluating the problems, and show the effectiveness of the fuzz functional SIRMs model. The following two nonlinear functions are used: where x, x 2 [, ] are input variavles, and [0, ] is a normalized output variable. In identifing nonlinear functions, there are five membership functions for the inputs x and x 2, where the centers of the membership functions A i,ai 2,...,Ai 5 for i =, 2 are, 0.5, 0, 0.5,, and each width of membership functions is 0.5. Moreover, all of consequent parts and importance degrees for input item are set to be 0 and 0.5, respectivel. Here, we obtain the error of evaluation regarding desired model and inference model where the error of evaluation is mean square error for checking data. In our case, 260 checking data (x,x 2 ) are emploed from (, ) to(, ), and 49 training data are used from 260 checking data in a random order. The learning rates are α =0., β =0., γ = 0., δ=0.0 and ɛ =0.0. In the following, we identif Functions and 2 b using the fuzz functional SIRMs model (FF-SIRMs, for short in the tables). Moreover, this modelis also compared with conventional SIRMs model and neurofuzz method (NF, for short in the tables). For nonlinear functions and 2, learninig iterations are executed 000 times, and 0 simulations are run. Tables and 2 show the error evaluation of average of 0 simulations using the checking data for identifing Functions and 2, respectivel. The fuzz functional and conventional SIRMs models do not necessaril obtain good results comapred with the neuro-fuzz method based on the simplified fuzz inference model for the Function as a multiplicative function, as shown in Table. On the other hand, all models give good results regarding Function 2 as an additive function, as shown in Table 2. From these results, we have clarified that the fuzz functional and conventional SIRMs models can obtain good results for additive functions even if a few rules are used. Moreover, since the fuzz functional SIRMs model is extended to fuzz functions from real numbers for the consequent parts of the conventional SIRMs model, it can obtain ingenious results compared with the conventional SIRMs model. Moreover, the fuzz functional SIMRs model and Table : Error of evaluation for Function of (23). FF-SIRMs SIRMs [] NF [] Average Function : = (2x +4x ) Function2 : = (2 sin(πx )+cos(πx 2 )+3) 6 (23) (24) Table 2: Error of evaluation for Function 2 of (24). FF-SIRMs SIRMs [] NF [] Average ISBN:

5 conventional SIRMs model use 0 (= 5 2) rules though the conventioanl fuzz inference model uses 25 (= 5 2 ) rules. Yet, the fuzz functional SIRMs model obtains good results compared with the traditional fuzz inference models. Therefore, we have shown the applicabilit of the fuzz functional SIRMs model from the nonlinear functions. 6 Conclusion This paper has shown the applicabilit of the fuzz functional SIRMs inference model b appling to the nonlinear modeling. The fuzz functional SIRMs model has few rules, and can obtain good results. Therefore, the fuzz functional SIRMs model will be compact and useful. References: [] E. H. Mamdani, Application of Fuzz Algorithms for Control of Simple Dnamic Plant, Proc. IEE 2, 974, pp [2] T. Takagi and M. Sugeno, Fuzz Identification of Sstems and Its Applications to Modeling and Control, IEEE Trans. Sst., Man, and Cbern. SMC-5, 985, pp [3] N. Yubazaki, J. Yi and K. Hirota, SIRMs (Single Input Rule Modules) Connected Fuzz Inference Model, Journal of Advanced Computational Intelligence and Intelligent Informatics, 997, pp [4] J. Yi, N. Yubazaki and K. Hirota, Stabilization Control of Seriestpe Double Inverted Pendulum Sstems Using the SIRMs Dnamicall Connected Fuzz Inference Model, Artificial Intelligence in Engineering 5, 200, pp [5] J. Yi, N. Yubazaki and K. Hirota, Upswing and Stabilization Control of Inverted Pendulum Sstem Based on the SIRMs Dnamicall Connected Fuzz Inference Model, Fuzz Sets and Sstems 22, 200, pp [6] J. Yi, N. Yubazaki and K. Hirota, A new fuzz controller for stabilization of parallel-tpe double inverted pendulum sstem, Fuzz Sets and Sstems 26, 2002, pp [7] J. Yi, N. Yubazaki and K. Hirota, A Proposal of SIRMs Dnamicall Connected Fuzz Inference Model for Plural Input Fuzz Control, Fuzz Sets and Sstems 25, 2002, pp [8] J. Yi, N. Yubazaki and K. Hirota, Anti-Swing and Positioning Control of Overhead Traveling Crane, Inf. Sci. 55, 2002, pp [9] H. Seki, H. Ishii and M. Mizumoto, On the Generalization of Single Input Rule Modules Connected Tpe Fuzz Reasoning Method, IEEE Trans. Fuzz Sst. 6, 2008, pp [0] H. Seki, H. Ishii and M. Mizumoto, On the Monotonicit of Fuzz-Inference Methods Related to T S Inference Method, IEEE Trans. Fuzz Sst. 8, 200, pp [] H. Seki, H. Ishii and M. Mizumoto, Nonlinear Identification and Medical Diagnosis Sstem Using Functional-Tpe SIRMs Connected Fuzz Inference Method, International Journal of Innovative Computing, Information and Control 6, 200, pp [2] H. Seki and M. Mizumoto, On the Equivalence Conditions of Fuzz Inference Methods Part : Basic Concept and Definition, IEEE Trans. Fuzz Sst. 9, 20, pp [3] H. Seki and M. Mizumoto, Fuzz Functional Inference Method, Proc. 200 IEEE World Congress on Computational Intelligence, FUZZ- IEEE 200, Barcelona, Spain, 200, pp [4] H. Seki, Fuzz Functional SIRMs Inference Model, Proc. Joint 5th International Conference on Soft Computing and Intelligent Sstems and th International Smposium on Advanced Intelligent Sstems (SCIS&ISIS200), Okaama, Japan, 200, pp [5] H. Seki, An Expert Sstem for Medical Diagnosis Based on Fuzz Functional SIRMs Inference Model, Proc. Joint 5th International Conference on Soft Computing and Intelligent Sstems and th International Smposium on Advanced Intelligent Sstems (SCIS&ISIS200), Okaama, Japan, 200, pp [6] M. Mizumoto, Fuzz Controls under Various Fuzz Reasoning Methods, Inf. Sci. 45, 988, pp [7] J. R. Jang, Self-Learning Fuzz Controllers Based on Temporal Back Propagation, IEEE Trans. Neural Networks 3, 992, pp [8] H. Ichihashi, Iterative Fuzz Modeling and A Hierarchical Network, Proc. 4th IFSA World Congress of Engineering, Brussels, Belgium, 99, pp [9] L. X. Wang and J. M. Mendel, Back-Propagation Fuzz Sstem as Nonlinear Dnamic Sstem Identifiers, Proc. 992 IEEE Int. Conf. Fuzz Sst., San Diego, USA, 992, pp [20] X. Cui and K. G. Shin, Direct Control and Coordination Using Neural Networks, IEEE Trans. Sst. Man Cbern. 23, 993, pp ISBN:

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