基于非对称Lanczos 算法的线性分数阶系统模型降阶方法

采用非对称Lanczos 算法研究线性分数阶系统的模型降阶问题, 提出一种保持系统传递函数一定数量的分数阶矩的模型降阶方法. 根据Caputo 导数的运算法则给出线性分数阶系统的分数阶矩的计算方法; 利用非对称Lanczos 算法构造对应的非对称三对角矩阵; 根据非对称三对角矩阵的性质证明降阶系统与原系统具有相同的一定数量的分数阶矩; 给出降阶系统与原系统传递函数的误差估计, 为合理选择降阶系统的阶次提供理论依据. 数值实例的计算结果验证了所提出方法的有效性.

Reduced order modeling method for linear fractional-order systems based on unsymmetric Lanczos algorithm

The problem of reduced order modeling for linear fractional-order systems is investigated based on the unsymmetric Lanczos algorithm, and a reduced order modeling method is proposed to retain a certain number of fractional- order moments of transfer functions. According to the operation principle of Caputo derivative, the computing method of fractional-order moment of the linear fractional-order system is given. By using the unsymmetric Lanczos algorithm, the corresponding unsymmetric tridiagonal matrix is constructed. Based on the properties of the unsymmetric tridiagonal matrix, it is proved that the reduced order systems and the original systems have a certain number of fractional-order moments. Besides, the error estimation with respect to the the transfer functions between the reduced order system and the original system is given, providing the theoretical basis for choosing the order of the reduced order system. Finally, a numerical example is given to illustrate the effectiveness of the proposed method.