Applied Soft Computing 2 (2002) 89–103 Design of adaptive Takagi–Sugeno–Kang fuzzy models Dragan Kukolj ∗ Faculty of Engineering, University of Novi Sad, Trg D. Obradovica 6, 21000 Novi Sad, Yugoslavia Received 13 December 2001; received in revised form 18 September 2002; accepted 20 September 2002 Abstract The paper describes a method of fuzzy model generation using numerical data as a starting point. The algorithm generates a Takagi–Sugeno–Kang fuzzy model, characterised with transparency, high accuracy and small number of rules. The training algorithm consists of three steps: partitioning of the input–output space using a fuzzy clustering method; determination of parameters of the consequent part of a rule from over-determined batch least-squares (LS) formulation of the problem, using singular value decomposition algorithm; and adaptation of these parameters using recursive least-squares method. Three illustrative well-known benchmark modelling problems serve the purpose of demonstrating the performance of the generated models. The achievable performance is compared with similar existing models, available in literature. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Fuzzy model; On-line learning; Benchmark modelling problems; Prediction 1. Introduction Fuzzy models that one designs are expected to be characterised with: (1) an accurate and fast procedure for calculation of the desired quantity; (2) a general and flexible procedure, applicable to a wide class of very diverse problems. Fuzzy model generation based on learning by examples approach is achieved during the structure identification and model parameter iden- tification step [1,2]. In cases when the input quantities are known, structure identification step reduces to determination of the required number of rules and determination of the conditional part of the rule. The required number of rules can be determined using cluster validity measures [3,4], or cluster removal and/or merging techniques [5,6]. Determination of the conditional part of the rule necessitates defining of the fuzzy sets of input variables. This can be done ∗ Tel.: +381-21-350-701; fax: +381-21-350-610. E-mail address: dragan.kukolj@micronasnit.com (D. Kukolj). using grid-type space partitioning [7] and fuzzy clus- tering methods [8,9]. During the model parameter identification step, various learning methods are ap- plied. Gradient descent methods (GD) represent an efficient method for simpler problems [10–12]. How- ever, they can easily end up in local minima and are sensitive with respect to the form of the selected ob- jective function, fuzzy set shapes and the type of the fuzzy operator. GD methods are frequently applied in combination with the least-squares (LS) method [7,13]. Recursive or batch variants of LS algorithms are characterised with faster convergence [14,15]. Genetic algorithms (GA) search a wide space of pos- sible solutions, so that there is a high probability that the found optimum is global or near-global [16–20]. The most successful realisations of fuzzy models are adaptive fuzzy models, frequently obtained using clustering methods for partitioning the input–output space, combined with GA [20], GD [8], LS [6,14] or GD–LS [7,13] optimisation methods for model parameter adaptation. Structural identification is a 1568-4946/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII:S1568-4946(02)00032-7