ON-LINE EVOLUTION OF TAKAGI-SUGENO FUZZY MODELS Plamen Angelov 1 , José Victor 2,3 , António Dourado 3 and Dimitar Filev 4 1 Department of Communications Systems Lancaster Universiy, UK p.angelov@lancaster.ac.uk 2 School of Technology and Management Polytechnic Institute of Leiria, Portugal zevictor@estg.ipleiria.pt 3 Adaptive Computation Group - CISUC University of Coimbra, Portugal {zevictor, dourado}@dei.uc.pt 4 Ford Motor Co.,USA Dfilev@Ford.com Abstract: Evolving Takagi-Sugeno (eTS) fuzzy models and the method for their on-line identification has been recently introduced for both MISO and MIMO case. In this paper, the mechanism for rule-base evolution, one of the central points of the algorithm together with the recursive clustering and modified recursive least squares (RLS) estimation, is studied in detail. Different scenarios are considered for the rule base upgrade and modification. The radius of influence of each fuzzy rule is considered to be a vector instead of a scalar as in the original eTS approach, allowing different areas of the data space to be covered by each input variable. Simulation results using a well-known benchmark (Mackey-Glass chaotic time-series prediction) are presented. Copyright © 2004 IFAC Keywords: evolving Takagi-Sugeno fuzzy models, rule-base evolution, recursive clustering, RLS algorithm. 1. INTRODUCTION For several centuries the so-called first principles models have dominated the natural sciences. However, for a number of practical engineering problems they are difficult or even impossible to build (Angelov, 2002; Yager and Filev, 1994). Another alternative is to use so-called "black-box" models (polynomial, regression models, neural networks). They can fit the data with arbitrary precision, but they are not transparent enough: their coefficients and structure is not directly related to the system being modelled (Yager and Filev, 1994). Fuzzy rule-based models and especially Takagi-Sugeno (TS) fuzzy models have gained significant impetus due to their flexibility and computational efficiency (Takagi and Sugeno, 1985; Yager and Filev, 1994). They have a quasi-linear nature and use the idea of approximation of a nonlinear system by a collection of fuzzily mixed local linear models. The TS fuzzy model is attractive because of its ability to approximate nonlinear dynamics, multiple operating modes and significant parameter and structure variations (Takagi and Sugeno, 1985). On-line learning of TS fuzzy models involves recursive, non-iterative clustering responsible for model structure (rule base) learning and recursive consequent parameter estimation (Angelov, 2002; Angelov and Filev, 2004). eTS is based on the assumption that the model structure evolves gradually instead of being known a priori (Angelov and Filev, 2004). It is important to note that this evolution is much slower than the evolution of the model parameters. For the eTS the notion of informative potential of the new data sample (accumulated spatial proximity measure) is very important. It has been first introduced in the mountain clustering approach (Yager and Filev, 1993) and then refined in the subtractive clustering approach (Chiu, 1994). It is used as a trigger to update the rule-base (Angelov, 2002; Angelov and Filev, 2004). It is a great advantage of this approach that the learning can start without a priori information and only a single data sample. This interesting feature makes the approach potentially very useful in autonomous, robotic, and smart adaptive systems (Angelov, 2002).