Universal Approximation Of Type-2 Fuzzy Systems And Its Applications

The related theory of type-2 fuzzy sets, universal approximation of type-2 fuzzy systems and its applications are studied in this paper. Some properties of type-2 fuzzy implications are obtained and the construction of type-2 fuzzy reasoning relations is simplified by the related methods about solving fuzzy relation equations. And then, the functional expressions of interval and non-interval type-2 fuzzy systems are given respectively and both their interpolation proper-ty and universal approximation are proved. Moreover, a variable universe adaptive type-2 fuzzy controller is designed by the interval and non-interval type-2 fuzzy systems we construct for a class of uncertain nonlinear systems. In order to verify the performance of type-2 fuzzy sys-tems we construct, all of them are applied in approximating a dynamic system and controlling a chaotic system. The details are as follows:1. Some properties of extended fuzzy implication based on arbitrary t-norm are obtained. It is pointed out that if a fuzzy implication is continuous, then its extended fuzzy implication is a type-2 fuzzy implication. Moreover, the extended fuzzy implication could keep the normality and fuzzy convexity of fuzzy truth values and the monotonicity of the original fuzzy implica-tions.2. The construction of type-2 fuzzy reasoning relations is simplified by the related methods about solving fuzzy relation equations. Through the analysis of the expression of type-2 fuzzy reasoning relations, the prototype of fuzzy relation equation are found. By using the properties of type-2 meet and join operations, type-2 fuzzy reasoning relations are divided into three parts. Then the calculation of each part is simplified respectively by using the related methods about solving fuzzy relation equations. At last, the possibility of applying the proposed method in the construction of type-2 fuzzy reasoning relations is illustrated on several examples.3. The function expressions of interval and non-interval type-2 fuzzy systems are given respectively and both their interpolation property and universal approximation are proved. T-wo kinds of construction methods for the antecedents and consequents of reasoning rules of interval type-2 fuzzy systems and four kinds of construction methods for the antecedents and consequents of reasoning rules of non-interval type-2 fuzzy systems are designed. Based on KM algorithm and the method about solving the centroid of general type-2 fuzzy sets, the in-terpolation formulas of interval and non-interval type-2 fuzzy systems and the proofs of their universal approximation are obtained respectively under some conditions. It can be seen that the output of a non-interval type-2 fuzzy system is equivalent to a weighted summation of those of a class of interval type-2 fuzzy systems. Finally, the interval and non-interval type-2 fuzzy systems we construct are applied in approximating a dynamic system. Quantum-behaved parti-cle swarm optimization algorithm is used to optimize the system parameters in order to improve the approximation performance of the type-2 fuzzy systems. The simulation results show that both the interval and non-interval type-2 fuzzy systems we construct outperform type-1 fuzzy system.4. A kind of variable universe stable adaptive type-2 fuzzy controllers are designed by using the interval and non-interval type-2 fuzzy systems we construct. Both the formulas of interval and non-interval type-2 fuzzy systems are turned into a unified form with an inner product of two vectors. For a class of uncertain nonlinear systems, on the basis of the theory of variable universe adaptive type-2 fuzzy control and universal approximation of type-2 fuzzy systems, a kind of variable universe adaptive type-2 fuzzy controller is designed with Lyapunov synthetic analysis method. What's more, the controllers we design are applied in controlling a chaotic system and quantum-behaved particle swarm optimization algorithm is used to optimize the parameters of type-2 fuzzy systems, which can improve the control performance of the type-2 fuzzy controllers. The simulation results show that the controllers by the interval and non-interval type-2 fuzzy systems we construct outperform that by type-1 fuzzy system.