B样条函数在模糊系统中的应用
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摘要
模糊系统的设计可以看成是一类函数逼近问题,从而可以利用数值逼近方法来设计模糊系统.本文将把B样条函数引入到模糊系统设计中,给出两类B样条模糊系统,并证明它们均能逼近函数及其导函数.将上述两类B样条模糊系统看成是计算几何中的曲线曲面,当构造模糊系统的数据不准确时,我们推广了计算几何中的能量法,并利用推广的能量法和小波方法设计了光顺的B样条模糊系统.仿真结果表明,光顺的B样条模糊系统能够改善B样条模糊系统的性能,尤其是构造模糊系统的数据不准确的时候.最后,我们将B样条模糊系统和光顺的B样条模糊系统应用到三维吊车的变论域自适应模糊控制器中,通过实物实验进一步检验了它们的性能.具体工作及结果总结如下：
1.构造了两类单输入单输出的B样条模糊系统.首先利用外推的数据构造了单输入单输出的第一类B样条模糊系统(1-B-FS),接着直接利用原始数据构造了第二类B样条模糊系统(2-B-FS).我们证明了这两类B样条模糊系统均能逼近函数及其导函数.最后,将它们分别应用到模糊推理建模和变论域自适应模糊控制器中,仿真实验表明这两类B样条模糊系统在实际应用中的可行性.
2.构造了两类多输入单输出的B样条模糊系统.对于多输入单输出的情形,为了得到类似单输入单输出情形的第一类B样条模糊系统,必须保证构造模糊系统的数据集是插值适定结点组.因此,我们首先给出一种线性外推的方式对数据进行了预处理.由于利用线性外推方法得到的数据集是插值适定结点组,进而我们得到了多输入单输出的第一类B样条模糊系统,它与单输入单输出的情形一样,依然具有插值性质.同样,利用原来的数据构造了多输入单输出的第二类B样条模糊系统.接着分别利用混合函数技巧(the blend function techniques)和Taylor公式,我们证明了这两类B样条模糊系统均能逼近函数及其导函数.最后,将sine模糊系统及两类B样条模糊系统应用到模糊推理建模和变论域自适应模糊控制器中,仿真结果表明,大多数情形下,第一类B样条模糊系统的性能优于其他模糊系统.
3.设计了能量法光顺的B样条模糊系统.为了减少不准确数据对模糊系统的影响,我们将计算几何中的能量法引入到模糊系统的设计中,构造了光顺的多输入单输出B样条模糊系统.首先,我们将仅适用于单输入单输出和双输入单输出B样条模糊系统的能量法推广到高维情形.由推广的能量法,我们将构造光顺B样条模糊系统的问题转化为求解一个严格凸二次函数的优化问题.将优化问题的最优解作为B样条模糊系统的线性组合系数就得到了光顺的B样条模糊系统.最后,将光顺的B样条模糊系统应用到二级倒立摆的变论域自适应模糊控制器中,仿真结果表明,利用光顺B样条模糊系统的控制器的控制效果优于利用B样条模糊系统的控制器,尤其在数据不准确的情形.
4.设计了小波方法光顺的B样条模糊系统.由于利用能量法光顺的B样条模糊系统,光顺前后模糊系统的规则个数相同,且随着规则和输入变量个数的增加,处理的时间将迅速增加.而在计算几何的众多光顺方法中,小波方法在光顺曲线(面)的同时具有减少控制顶点的作用,且其运行时间对控制顶点个数不敏感.由此我们设计了小波方法光顺的B样条模糊系统.首先将B样条模糊系统的多分辨率表示转化为准均匀B样条函数的多分辨率表示,接着利用准均匀B样条小波分解方法对相应的准均匀B样条函数进行分解就得到了一系列光顺性逐渐增强、规则个数逐渐减少的模糊系统,即基于小波方法光顺的B样条模糊系统.最后,仿真结果表明,小波方法光顺的B样条模糊系统在改善原来B样条模糊系统性能的同时,大大提高了运行效率.
5.利用三维吊车的仿真和实物实验进一步验证了B样条模糊系统和光顺B样条模糊系统的性能并指出：小波方法光顺的B样条模糊系统更有利于实物实现.
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.
1.构造了两类单输入单输出的B样条模糊系统.首先利用外推的数据构造了单输入单输出的第一类B样条模糊系统(1-B-FS),接着直接利用原始数据构造了第二类B样条模糊系统(2-B-FS).我们证明了这两类B样条模糊系统均能逼近函数及其导函数.最后,将它们分别应用到模糊推理建模和变论域自适应模糊控制器中,仿真实验表明这两类B样条模糊系统在实际应用中的可行性.
2.构造了两类多输入单输出的B样条模糊系统.对于多输入单输出的情形,为了得到类似单输入单输出情形的第一类B样条模糊系统,必须保证构造模糊系统的数据集是插值适定结点组.因此,我们首先给出一种线性外推的方式对数据进行了预处理.由于利用线性外推方法得到的数据集是插值适定结点组,进而我们得到了多输入单输出的第一类B样条模糊系统,它与单输入单输出的情形一样,依然具有插值性质.同样,利用原来的数据构造了多输入单输出的第二类B样条模糊系统.接着分别利用混合函数技巧(the blend function techniques)和Taylor公式,我们证明了这两类B样条模糊系统均能逼近函数及其导函数.最后,将sine模糊系统及两类B样条模糊系统应用到模糊推理建模和变论域自适应模糊控制器中,仿真结果表明,大多数情形下,第一类B样条模糊系统的性能优于其他模糊系统.
3.设计了能量法光顺的B样条模糊系统.为了减少不准确数据对模糊系统的影响,我们将计算几何中的能量法引入到模糊系统的设计中,构造了光顺的多输入单输出B样条模糊系统.首先,我们将仅适用于单输入单输出和双输入单输出B样条模糊系统的能量法推广到高维情形.由推广的能量法,我们将构造光顺B样条模糊系统的问题转化为求解一个严格凸二次函数的优化问题.将优化问题的最优解作为B样条模糊系统的线性组合系数就得到了光顺的B样条模糊系统.最后,将光顺的B样条模糊系统应用到二级倒立摆的变论域自适应模糊控制器中,仿真结果表明,利用光顺B样条模糊系统的控制器的控制效果优于利用B样条模糊系统的控制器,尤其在数据不准确的情形.
4.设计了小波方法光顺的B样条模糊系统.由于利用能量法光顺的B样条模糊系统,光顺前后模糊系统的规则个数相同,且随着规则和输入变量个数的增加,处理的时间将迅速增加.而在计算几何的众多光顺方法中,小波方法在光顺曲线(面)的同时具有减少控制顶点的作用,且其运行时间对控制顶点个数不敏感.由此我们设计了小波方法光顺的B样条模糊系统.首先将B样条模糊系统的多分辨率表示转化为准均匀B样条函数的多分辨率表示,接着利用准均匀B样条小波分解方法对相应的准均匀B样条函数进行分解就得到了一系列光顺性逐渐增强、规则个数逐渐减少的模糊系统,即基于小波方法光顺的B样条模糊系统.最后,仿真结果表明,小波方法光顺的B样条模糊系统在改善原来B样条模糊系统性能的同时,大大提高了运行效率.
5.利用三维吊车的仿真和实物实验进一步验证了B样条模糊系统和光顺B样条模糊系统的性能并指出：小波方法光顺的B样条模糊系统更有利于实物实现.
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.
引文
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