Fuzzy identiication of discrete time nonlinear stochastic systems

System identification is the task of developing or improving a mathematical description of dynamic systems from experimental data (Ljung (1999); Söderström & Stoica (1989)). Depending on the level of a priori insight about the system, this task can be approached in three different ways: white box modeling, black box modeling and gray box modeling. These models can be used for simulation, prediction, fault detection, design of controllers (model based control), and so forth. Nonlinear system identification (Aguirre et al. (2005); Serra & Bottura (2005); Sjöberg et al. (1995); ?) is becomming an important tool which can be used to improve control performance and achieve robust behavior (Narendra & Parthasarathy (1990); Serra & Bottura (2006a)). Most processes in industry are characterized by nonlinear and time-varying behavior and are not amenable to conventional modeling approaches due to the lack of precise, formal knowledge about it, its strongly nonlinear behavior and high degree of uncertainty. Methods based on fuzzy models are gradually becoming established not only in academic view point but also because they have been recognized as powerful tools in industrial applications, faciliting the effective development of models by combining information from different sources, such as empirical models, heuristics and data (Hellendoorn & Driankov (1997)). In fuzzy models, the relation between variables are based on if-then rules such as IF < antecedent > THEN < consequent >, where antecedent evaluate the model inputs and consequent provide the value of the model output. Takagi and Sugeno, in 1985, developed a new approach in which the key idea was partitioning the input space into fuzzy areas and approximating each area by a linear or a nonlinear model (Takagi & Sugeno (1985)). This structure, so called Takagi-Sugeno (TS) fuzzy model, can be used to approximate a highly nonlinear function of simple structure using a small number of rules. Identification of TS fuzzy model using experimental data is divided into two steps: structure identification and parameter estimation. The former consists of antecedent structure identification and consequent structure identification. The latter consists of antecedent and consequent parameter estimation where the consequent parameters are the coefficients of the linear expressions in the consequent of a fuzzy rule. To be applicable to real world problems, the parameter estimation must be highly efficient. Input and output measurements may be contaminated by noise. For low levels of noise the least squares (LS) method, for example, may produce excellent estimates of the consequent parameters. However, with larger levels of noise, some modifications in this method are required to overcome this inconsistency. Generalized least squares (GLS) method, extended least squares (ELS) method, prediction error (PE) method, are examples of such modifications. A problem 1