Systems Engineering and Information Technology Institute, Federal University of Itajubá (UNIFEI), CEP 37500-903 Itajubá, MG, Brazil.
IEEE Trans Cybern. 2013 Jun;43(3):858-70. doi: 10.1109/TSMCB.2012.2217323. Epub 2012 Oct 18.
An approach to obtain Takagi-Sugeno (TS) fuzzy models of nonlinear dynamic systems using the framework of orthonormal basis functions (OBFs) is presented in this paper. This approach is based on an architecture in which local linear models with ladder-structured generalized OBFs (GOBFs) constitute the fuzzy rule consequents and the outputs of the corresponding GOBF filters are input variables for the rule antecedents. The resulting GOBF-TS model is characterized by having only real-valued parameters that do not depend on any user specification about particular types of functions to be used in the orthonormal basis. The fuzzy rules of the model are initially obtained by means of a well-known technique based on fuzzy clustering and least squares. Those rules are then simplified, and the model parameters (GOBF poles, GOBF expansion coefficients, and fuzzy membership functions) are subsequently adjusted by using a nonlinear optimization algorithm. The exact gradients of an error functional with respect to the parameters to be optimized are computed analytically. Those gradients provide exact search directions for the optimization process, which relies solely on input-output data measured from the system to be modeled. An example is presented to illustrate the performance of this approach in the modeling of a complex nonlinear dynamic system.
本文提出了一种使用正交基函数 (OBF) 框架获取非线性动态系统 Takagi-Sugeno (TS) 模糊模型的方法。该方法基于一种架构，其中具有梯形结构广义 OBF (GOBF) 的局部线性模型构成模糊规则结论，并且相应 GOBF 滤波器的输出是规则前件的输入变量。所得到的 GOBF-TS 模型的特点是具有仅具有实值参数，这些参数不依赖于在正交基中使用的特定类型函数的任何用户规范。该模型的模糊规则最初是通过基于模糊聚类和最小二乘法的著名技术获得的。然后简化这些规则，并使用非线性优化算法调整模型参数（GOBF 极点、GOBF 展开系数和模糊隶属函数）。与要优化的参数相关的误差函数的精确梯度是通过分析计算的。这些梯度为优化过程提供了精确的搜索方向，该过程仅依赖于从要建模的系统测量的输入输出数据。本文提供了一个示例来说明该方法在复杂非线性动态系统建模中的性能。
