Research On The Universal Approximation Of Fuzzy Systems And Its Applications

Fuzzy systems have been widely applied to complex systems modeling, prediction and control process. The theoretical basis of it lies in universal approximation property. The problem of universal approximation property of fuzzy systems has not yet been fully resolved. In this paper, universal approximation property of fuzzy systems and error estimation are discussed at first, then autonomous Lienard systems are solved by applying fuzzy systems, dual u-integrability problems on fuzzy number valued functions are investigated, and a novel family of numerical integration of closed Newton-Cotes quadrature rules is presented which uses the derivative value at the midpoint at last.The main research contents are described as follows:1. Aiming at SISO open-loop system, this paper presents the formula of a kind of fuzzy systems with CRI inference, Larsen implication operator, and gravity method defuzziflcation. These kind of fuzzy systems are capable of approximating any real continuous functions to an arbitrary degree of accuracy, and the error formula and its estimation are given. It is pointed out it can reach the high accuracy of O(Δ12). Based on the analysis of the error estimation formula, it is pointed out that the "rule number", not the "design parameters" is the decisive factor for fuzzy systems to have universal approximation property.2. By applying HX equation approximation method and marginal linearization method, simplified HX equation approximation method and simplified marginal linearization method based on autonomous Lienard systems are proposed respectively. Both proposed methods both can simplify problem solving steps and calculation. Besides, the time complexity and space complexity are reduced. The simulation results show these two methods are reliable algorithm, which can result in high precision.Next, through a series of simulation experiments and theory analysis, we found out that the reason why error becomes larger and larger with the passage of time lies in Runge-Kutta methods. It causes lower and lower accuracy of transmitted initial values. In order to improve the approximation accuracy of Runge-Kutta methods, the forecast-correction method is proposed, and the simulation results illustrate that the error has been reduced effectively. Then, in order to further improve the accuracy, parameter perturbation is introduced on the basis of extrapolation algorithm, i.e., parameter perturbation extrapolation algorithm for Runge-Kutta (PPE-R-K) methods are presented. Lastly, the numerical experiments show that PPE-R-K methods are of high precision, and it has obvious effects on inhibiting the spread of error propagation.3. On the K-quasi-additive fuzzy measure space, aiming at a kind of u-intrgrable fuzzy number valued functions, the dual K-quasi-additive fuzzy number valued integral is established by using K-quasi-additive operator and K-quasi-multiplicative operator at first, and the transformation theorem of the integral is obtained by introducing inductive operator K. Then, on the condition of quasi-minus operator K, functions’integrability are discussed, thus the condition of integrability and suffient conditions are obtained:boundary fuzzy valued function and continuous fuzzy valued function on a closed interval is integrable. At last, it illustrates the application of the integral in the field of forecasting through a specific example.4. In the process of gravity method defuzzification, Riemann sum or numerical integration formulas are often used to approximate the value of the definite integral. It is a difficult problem that how to get high precision numerical integration formula without excessive interval partition. For this purpose, a novel family of numerical integration of closed Newton-Cotes quadrature rules is presented which uses the derivative value at the midpoint in this chapter. It is proved that these kinds of quadrature rules obtain an increase of two orders of precision over the classical closed Newton-Cotes formula. And the error terms are given. The computational cost for these methods is analyzed from the numerical point of view, and it has shown that the proposed formulas are superior computationally to the same order closed Newton-Cotes formula when they reduce the error below the same level. And some numerical examples show the numerical superiority of the proposed approach with respect to closed Newton-Cotes formulas. Finally, the proposed method is applied to the problems of calculating the center of gravity and the process of gravity defuzzification method, and the numerical experiments show that it has high precision.