The Application Of B-spline Functions In Fuzzy Systems

The problem to design a fuzzy system can be considered as a function approximation prob-lem. So it is reasonable to design fuzzy system by the numerical approximation method. In this paper, the B-spline function method is introduced to design fuzzy system, and two classes of B-spline fuzzy systems (B-FSs) are proposed, both of which can approximate functions and their derivatives simultaneously. In fact, we can regard the B-FSs as the curves (surfaces) in computational geometry. When the data to construct fuzzy system is inexact, we generalize the energy method of computational geometry. And then we design the faired B-FSs by the general-ized energy method and the wavelet fairing method. The simulation results show that the faired B-FSs can improve the B-FSs, especially when the data for fuzzy systems is inexact. Finally, the B-FSs and faired B-FSs are used in the variable universe adaptive fuzzy controllers of3D crane, so the performance of the above fuzzy systems is verified further. The details are as follows.1. Two classes of SISO B-FSs are constructed. Firstly, the first and second class of B-FSs (1-B-FSs and2-B-FSs) are designed by the extrapolated data and the original one, respectively. The two classes of SISO B-FSs are proved to approximate functions and their derivatives simul-taneously. At last, we use them in fuzzy system modelling and variable universe adaptive fuzzy controllers. The simulation results show that the two classes of SISO B-FSs are feasible.2. Two classes of MISO B-FSs are constructed. For the MISO case, if we want to obtain a B-FS like the SISO1-B-FS, the data for fuzzy system must be well-posed. So, we preprocess the original data by a linearly extrapolated method. Since this method ensures that the extrapolated data is a properly posed set of nodes, we can design the MISO1-B-FS by the extrapolated data and the MISO1-B-FS is an interpolation system like the SISO one. Likewise, the MISO2-B-FS is available by the original data. By the blend function techniques and Taylor formula, we prove that the two classes of MISO B-FSs can approximate functions and their derivatives simultaneously. At last, the sinc-FSs and the two classes of MISO B-FSs are used in fuzzy system modelling and variable universe adaptive fuzzy controllers. It is shown that the1-B-FS is better than other fuzzy systems in most cases.3. Two classes of faired MISO B-FSs are designed using the energy method in computa-tional geometry for reducing adverse effects of the inexact data. Towards this goal, we generalize the energy method to high-dimension cases so that the energy method which is only suitable for SISO and DISO B-FSs is extended to fair the MISO ones. Then the problem to construct a faired MISO B-FS is transformed into solving an optimization problem with a strictly convex quadratic objective function. And a faired MISO B-FS is obtained by solving the optimization problem. Furthermore, the faired B-FSs are used in variable universe adaptive fuzzy controllers of the double inverted pendulum. The simulation results show that the controllers by faired B-FSs perform better than those by B-FSs, especially when the data for fuzzy systems are inexact.4. Two classes of faired MISO B-FSs are designed using a quasi-uniform B-spline wavelet decomposition method in computational geometry. We note that, the energy faired B-FSs have the same number of rules before and after fairing, and the processing time will increase rapid-ly with the increasing of rules and input variables. In fact, among a lot of of fairing methods in computational geometry, the wavelet method can fair the curves (surfaces) and reduce the control points at the same time and it is not sensitive to the number of control points. So we design the wavelet faired B-FSs. First, the multi-resolution of B-FSs is transformed into the multi-resolution of quasi-uniform B-splines. Then the corresponding quasi-uniform B-splines are decomposed by the quasi-uniform B-spline wavelet method, and a series of fuzzy systems with gradually increasing fairness and gradually reducing rules are available. Those fuzzy sys-tems are all called faired B-FSs by wavelet method. At last, the simulation results show that the faired B-FSs by wavelet method can improve the original B-FSs and considerably reduce their running time simultaneously.5. The simulation and physical experiments of3D crane are used to verify the performance of B-FSs and faired B-FSs further. And the simulation and physical experiments results show that the faired B-FSs by wavelet method are more favorable for physical realization.