**Taylor and Bi-local Piecewise Approximations with Neuro-Fuzzy Systems**

**Horia-Nicolai TEODORESCU ^{1,2}**

^{1 }Institute of Computer Science of the Romanian Academy, Iaşi Branch,

11, Carol I Blvd., 700505, Iaşi, Romania

^{2 }Gheorghe Asachi Technical University of Iasi,

67, Dimitrie Mangeron Blvd., 700050, Iaşi, Romania,

hteodor@etti.tuiasi.ro

**Abstract**: A fuzzy neuron (linear combiner of fuzzy systems) with piecewise polynomial characteristic function is defined and analyzed. The linear fuzzy neuron uses a pre-specified type of fuzzy logic systems with complementary pairs of input membership functions. Taylor local approximations are built using the described fuzzy neuron. Moreover, local Taylor approximations are obtained using single fuzzy systems. Hence, the linear combiner fuzzy neurons are universal local approximators implementing truncated Taylor series. Moreover, they represent continuous piecewise approximators. Bi-local approximation with fuzzy logic systems are also introduced and demonstrated.

**Keywords**: Local approximation, TS system, fuzzy neuron, universal local approximator, approximation algorithm.

**>>Full text**

**CITE THIS PAPER AS**:

H.-N. TEODORESCU, **Taylor and Bi-local Piecewise Approximations with Neuro-Fuzzy Systems**, *Studies in Informatics and Control*, ISSN 1220-1766, vol. 21 (4), pp. 367-376, 2012. https://doi.org/10.24846/v21i4y201202

**1. Introduction**

Methods of local approximations based on truncated Taylor series are well understood; several methods to compute bounds for the approximation errors are available (Christensen and Christensen, 2006), (Powell, 1981), (Shahriari, 2006). The use of Taylor approximations is convenient, among others, because methods to determine the approximation accuracy are well established. Taylor approximations are widely used in solving differential equations (Kloeden, Platen, 1995), (Tachev, 2009), control applications (Hedjar et al., 2005), circuit modeling (Jridi and Alfalou, 2009), data processing, image generation, signal prediction (Hedjar et al., 2005), and economic modeling (Judd, 1998).

Another class of widely used and well understood approximation methods is based on piecewise polynomials, see for example (Powel, Chapter 18, p. 212). The use of piecewise polynomials approximations is convenient, because methods to determine the approximation accuracy are well established. Proving that some kind of fuzzy systemsare identical with the class of polynomials on a specified interval is important because polynomials constitute a core family of functions that stand at the basis of numerous mathematical constructs and applications.

In addition, because Taylor approximations and piecewise polynomials approximations are so widely used, showing that some kind of fuzzy system may implement such approximating functions is both of intellectual and practical interest.

It is well known that fuzzy logic systems (FLS) endowed with defuzzification perform a mapping from the input **R ^{n}** spaceto the output space, the latter being typically the real line

**R**. When the fuzzy system has a single input taking values in

**R**, the mapping performedis

**R**

**R**. The input-output functionsare also named characteristic functions.

Recall that a Sugeno fuzzy system, also named TSK (standing for Takagi-Sugeno-Kang) or T-S fuzzy system, of order zero, is defined by the input-output function

where S_{i} Rare called singletons and : R [0, 1] are the corresponding membership functions. In case of TSK systems with several inputs, each fuzzified independently, with max-type inference and weighted sum defuzzification, the output is

while for product-type inference the characteristic function is (Tanaka & Wang, 2001), (Yu and Li, 2004),

with the membership functions related to the *i*^{th} rule denoted by , i = 1..n, the index *h* denoting the *h*^{th} input variable. TSK fuzzy systems of higher orders are defined in a similar way, but instead of conclusions represented by constants, they have polynomial conclusions. The rules describing a TSK system of order *n* have the form *If input x is A, then output is P _{n}(x)*, where

*P*is a polynomial of order

_{n}(x)*n*.

Bikdash (Bikdash, 1999), (Bikdash et al., 2001) have introduced and applied a modified Sugeno (TSK) systems, named Interpretable Sugeno Approximators (ISA), which use outputs in the form u^{k} = b_{0}^{k} + b_{1}^{k}(x – r_{1}^{k}) + b_{2}^{k}(x – r_{2}^{k}) + … + b_{n}^{k}(x- r_{n}^{k}), “where r^{k} is the rule center, i.e., centers of the membershipfunctions of all inputs tested by the *k*^{th} rule, […] and the coefficients *b* are interpreted as Taylor series coefficients.”

One of the first papers to use neuro-fuzzy systems in relation to Taylor approximation was (Yu and Li, 2004). However, that approach endeavored only to approximate a Taylor approximating function, not to implement the Taylor approximant. Herera et al. (Herrera et al., 2004)(Herrera et al., 2005a,b) analyzed TSK systems with second order polynomial consequent and connected them to Taylor approximations. (Wang, Li, Niemann, & Tanaka, 2000) provided a detailed analysis of the TSK systems with linear consequent and their approximation power, for the general case. (Sonbol and Fadali 2002) and (Sonbol and Fadali, 2006) also provided an interesting approach for Taylor-like approximation with TSK systems, which contains some of the ideas presented in this paper, while (Fadali, 2002) furthers the analysis for general polynomial approximations.

We show that a constructive approach is possible that allowsbuilding a direct, true Taylor approximation using either a simpler neuro-fuzzy system or a modified TSK system. Also, bi-local approximations are introduced, as well as neuro-fuzzy systems for implementing them. A specific choice of the input membership functions allows us the direct implementation of the polynomial approximations, with zero-order TS systems, thus simplifying computations in the implementation of the approximators.

For brevity, we discuss only the case of single input single output (SISO) FLS. Also, to keep the paper focused, we deal here only with Taylor mono- or bi-local approximations, but the methods presented herein are readily extensible to Tchebychev and to other polynomial approximations. These will be dealt with elsewhere.

The flow of the paper is as follows. In the second section we introduce the linear combiner fuzzy neuron (LCFN) and the piecewise polynomial approximations using LCFNs. The piecewise Taylor approximant is derived as a particular case of the piecewise polynomial approximants. The notions of bi-local approximation and piecewise approximant are defined in Section 3 in relation to representations of the approximant by fuzzy logic systems. The fourth section is devoted to a specific class of fuzzy systems able to implement Taylor approximants and bi-local approximants. The general outline of an algorithm for building Taylor-like approximations with fuzzy systems is presented in the fifth section. The final section is devoted to a general discussion and to conclusions.

A preliminary, partial version of this paper was presented in (Teodorescu, 2010).

**REFERENCES**:

- BIKDASH, M.,
**Modeling and Control of a Bergey-Type furling Wind Turbine**. http://wind.nrel.gov/furling/bikdash.pdf - BIKDASH, M., D. CHENG, M. HARB,
**A Hybrid Model of a Small Autofurling Wind Turbine**. Journal of Vibration and Control, Vol. 7, No. 1, 2001, pp. 127-148. - CHRISTENSEN, O., K. L CHRISTENSEN,
**Approximation Theory: From Taylor Polynomials to Wavelets**. 3rd Edition. Birkhäuser, 2006, Boston. ISBN 0817636005, 162 pages. - FADALI, M. S.,
**Stability Testing for Systems with Polynomial Uncertainty**, Proceedings of the 2002 American Control Conference, Anchorage, AK, May, 2002. - HEDJAR, R., R. TOUMI, P. BOUCHER, D. DUMUR,
**Finite Horizon Nonlinear Predictive Control by the Taylor Approximation: Application to Robot Tracking Trajectory**. International Journal of Applied Mathematics and Computer Science, Vol. 15, No. 4, 2005, pp. 527-540. - HERRERA, L. J., H. POMARES, J. GONZÁLEZ, O. VALENZUELA,
**Function Approximation through Fuzzy Systems Using Taylor Series Expansion-Based Rules: Interpretability and Parameter Tuning**. In MICAI 2004: Advances in Artificial Intelligence, Lecture Notes in Computer Science, Vol. 2972/2004, Springer Berlin / Heidelberg, 2004, pp. 508-516, ISBN978-3-540-21459-5 - HERRERA, L. J., H. POMARES, I. ROJAS, O. VALENZUELA, A. PRIETO,
**TaSe, a Taylor Series-based Fuzzy System Model that Combines Interpretability and Accuracy**. Fuzzy Sets and Systems, vol. 153, 2005, pp. 403-427. - HERRERA, L. J., H. POMARES, I. R. ALBERTO,
**Analysis of the TaSe-II TSK-Type Fuzzy System for Function Approximation**. L. Godo (Ed.): Symbolic and Quantitative Approaches to Reasoning with Uncertainty – ECSQARU 2005, Lecture Notes in Computer Science. Vol. 3571/2005 (LNAI 3571) Springer-Verlag Berlin Heidelberg, 2005, pp. 613-624. - JRIDI, M., A. ALFALOU,
**Direct Digital Frequency Synthesizer with CORDIC Algorithm and Taylor Series Approximation for Digital Receivers**. European Journal of Scientific Research, Vol. 30 No. 4, 2009, pp. 542-553, http://www.eurojournals.com/ejsr_30_4_03.pdf. - JUDD, K. L.,
**Numerical Methods in Economics**, MIT Press, Boston, MA, 1998, ISBN 9780262100717, 633 pages. - POWELL, M. J. D.,
**Approximation Theory and Methods**, Cambridge University Press, Cambridge UK, 1981, 339 pages. - SHAHRIARI, S.,
**Approximately Calculus**. AMS Bookstore, 2006, Providence, Rhode Island. ISBN 0821837508, 292 pages. - SONBOL, A., FADALI, M. S.,
**A New Approach for Designing TSK Fuzzy Systems from Input-Output Data**. Proceedings of the 2002 American Control Conference, Anchorage, AK, May, 2002. http://www.unr.nevada.edu/~fadali/Papers/FuzzyACC02.pdf - SONBOL, A., M. S. FADALI,
**TSK Fuzzy Systems Types II and III Stability Analysis: Continuous Case**, IEEE Transactions on Systems, Man & Cybernetics, Section B, Vol. 36, No. 1, January 2006, pp. 2-12. - TACHEV, G.,
**Approximation, Numerical Differentiation and Integration Based on Taylor Polynomial**. Journal of Inequalities in Pure and Applied Mathematics, (Victoria University) vol. 10, issue 1, article 18, 2009, http://www.emis.de/journals/JIPAM/images/310_08_JIPAM/310_08_www.pdf - TANAKA, K., H. O. WANG,
**Fuzzy Control Systems Design and Analysis: ALinear Matrix Inequality Approach**. Edition: 2. John Wiley and Sons, Hoboken, New Jersey, 2001, ISBN 0471323241, 305 pages - TEODORESCU, H. N.,
**Neuro-fuzzy Models for bi-Taylor Approximation and Applications**. 6^{th}European Conference on Intelligent Systems and Technologies, Iasi, October 7-9, 2010 (oral communication). - YU, W., X. LI,
**Fuzzy Identification using Fuzzy Neural Networks with Stable Learning Algorithms**. IEEE Transactions on Fuzzy Systems, Vol. 12, No. 3, June 2004, pp. 411-420. - WANG, H. O., J. LI, D. NIEMANN, K. TANAKA,
**T-S Fuzzy Model with Linear Rule Consequence and PDC Controller: A Universal Framework for Nonlinear Control Systems**. Proceedings 9th IEEE International Conference on Fuzzy Systems, 7-10 May 2000, San Antonio, Texas http://people.ee.duke.edu/~jingli/research/fuzz224.pdf.