Closed-Loop Global Fuzzy TSK Modeling: A Case Study H. Azimian * , A. Fatehi * , B. N. Araabi ** * KN Toosi University of Technology/Department of Electrical Engineering, Tehran, Iran ** University of Tehran/ School of Electrical and Computer Engineering, Tehran, Iran Abstract— a comprehensive step-by-step approach to the construction of a global fuzzy TSK model for a real SISO nonlinear system in closed loop is proposed. First the overall existing nonlinear distortions are evaluated, and then sub optimal experiments are designed so that the impacts of distortion are minimized. Such experiments can result in models which are consistent local estimations of the true underlying linear system at different operating points. Finally by suitable fuzzy combination of extracted open loop local models, a global fuzzy simulation model is constructed. Employing a quasi tailor-made parameterization, a model refinement can be carried out by trimming the membership functions through any optimization algorithms. In opposite of the conventional tailor-made parameterization algorithm, the stability of this modified algorithm is guaranteed. Keyword: Closed-Loop System Identification, Fuzzy Modeling, Tailor-Made Parameterization I. INTRODUCTION Simplicity and versatility of linear system theory tools for analysis and synthesis, have invoked a great enthusiasm for translation of difficult and probably real world’s intrinsically nonlinear problems into a set of tractable linear counterparts, among the system researchers. However, it is not always easy to present a precise classical framework for stating a nonlinear system in terms of a set of meaningful linear systems. In recent decades few fundamental concepts have been employed to facilitate such a fantasy. Piecewise linear (affine) system theory [11], multiple model theory [13], Linear Parameter Varying (LPV) system theory and Fuzzy modeling [18] are among the most common theories in related area. [18] offered the construction of global nonlinear models by fuzzy combination of linear models. The approach presented in this work conveyed a novel insight about the modeling of the system but it was still naive in analyzing the interpretation of the local models. In [9], [10] and [13] the interpretation of local models is studied more extensively. In [10], a global fuzzy TSK model is constructed by concurrent identification of local models out of the collected data. In this approach regularization is employed so that the fuzzy consequents are more interpretable locally. In [3] another data-driven global fuzzy modeling approach based on fuzzy clustering is proposed by Babuska. One of the most common fuzzy modeling structures called LOLIMOT, proposed by Nelles in [14], [15], also is based on simultaneous identification of both local linear models and basis functions by neuro-fuzzy optimization methods. Indeed LOLIMOT is like a Radial Basis Function whose nodes are local linear models. No precise a priori knowledge about the premise space partitions is required. Indeed the partitioning of the premise space evolutes over several iterations and from this point of view, LOLIMOT is similar to the fuzzy modeling based on clustering, proposed by Babuska. Although, all these approaches have proved vigorous properties in providing nice global fuzzy models for the nonlinear systems, some drawbacks at specific situations can be contributed to these methods. For instance most of the judicious applications of these methods are reported for constructing prediction models rather than simulation models. Strictly speaking, constructing a precise nonlinear simulation model is a difficult task and in some practical cases may be impossible specifically when all parameters of a global fuzzy model including the membership functions and model parameters are going to be identified simultaneously. Fortunately, prediction models are more applicable and easy-to-implement as well. Another bottleneck in construction of a global nonlinear simulation model is the collected data itself. Persistent excitation of signal for linear system identification stresses on the frequency contents of the excitation, whereas for nonlinear system identification not only the frequency contents but also amplitude of the excitation should be considered. In reality, in many situations, it is impossible to excite the nonlinear system ideally. The other situation in which a full automated global modeling algorithm may not work properly is nonlinear identification of unstable systems in loop. No general comment can be given about the stability of the algorithm even when the system is at the stability boundaries. In general form of Tailor-made parameterization, presented for closed loop identification of nonlinear plants through nonlinear optimization algorithms, the connectedness of the parameter space is necessary. With lack of connectedness the optimum parameters set may not be reached if the initial parameters set resides in a region other than that of the optimum parameter set [20]. In this paper we aim at construction of a fuzzy TSK model for elevation channel of Humusoft’s CE150 twin- rotor helicopter as an unstable nonlinear SISO system. In section II, understudy system is explained briefly. In section III some common closed loop system Rtqeggfkpiu qh vjg 37vj Ogfkvgttcpgcp Eqphgtgpeg qp Eqpvtqn ( Cwvqocvkqp. Lwn{ 49 / 4;. 4229. Cvjgpu / Itggeg V34/227