This paper addresses the approximation properties of the smooth fuzzy models. It is widely recognized that the fuzzy models can approximate a nonlinear function to any degree of accuracy in a convex compact region. However, in many applications, it is desirable to go beyond that and acquire a model to approximate the nonlinear function on a smooth surface to gain better performance and stability properties. Especially in the region around the steady states, when both error and change in error are approaching zero, it is much desired to avoid abrupt changes and discontinuity in the approximation of the inputoutput mapping. This problem has been remedied in our approach by application of the smooth compositions in the fuzzy modeling scheme. In the fuzzy decomposition stage of fuzzy modeling, we have discretized the parameters and then calculated the result through partitioning them into a dense grid. This could enable us to present the formulations by convolution and Fourier Transformation of the parameters and then obtain the approximation properties by studying the structural properties of the Fourier Transformation and convolution of the parameters. We could show that, irrespective to the shape of the membership function, one can approximate the dynamics and derivative of the continuous systems together, using the smooth fuzzy structure. The results of the paper have been tested and evaluated on a discrete event system in the hybrid and switched systems framework.

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This paper develops a smooth model identifica-10 tion and self-learning strategy for dynamic systems taking 11 into account possible parameter variations and uncertain-12 ties. We have tried to solve the problem such that the 13 model follows the changes and variations in the system on 14 a continuous and smooth surface. Running the model to 15 adaptively gain the optimum values of the parameters on a 16 smooth surface would facilitate further improvements in 17 the application of other derivative based optimization 18 control algorithms such as MPC or robust control algo-19 rithms to achieve a combined modeling-control scheme. 20 Compared to the earlier works on the smooth fuzzy mod-21 eling structures, we could reach a desired trade-off between 22 the model optimality and the computational load. The 23 proposed method has been evaluated on a test problem as 24 well as the non-linear dynamic of a chemical process. 25 26 Keywords Fuzzy control Á Fuzzy IF-THEN systems 27 (TSK) Á Smooth compositions 28 1 Introduction 29 Soft computing methods are being used for identification of 30 non-linear and complex systems based on the input-output 31 data collected from the original system [1]. There are many

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In this article, we study the structural properties that smooth compositions bring to predictive control of TS fuzzy models and examine how they affect the uncertainties, parameter variations of the system and environmental noises to die out. We have employed the smoothness structure of compositions to convert the MPC cost function of TS fuzzy model of the nonlinear systems to an incremental iterative algorithm. Hence, the proposed algorithm does not linearize the nonlinear dynamics, neither requires solving an NP optimization problem in MPC and, therefore, is very fast and simple. The connectivist identification-MPC approach-can be employed for the systems with the long-range horizons. Therefore, the technique is beneficial to the dead-time and non-minimum phase systems. The stability analysis of the algorithm has been carried out, and the performance of the smooth TS fuzzy identification-controller scheme to the classical ones has been compared on a non-min phase test problem with different uncertainties and working environments, including (a) the normal working conditions, (b) with the additive noises, (c) with the parametric changes, (d) with the additive time-varying disturbances to demonstrate the robust behavior of the smooth compositions.

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