The paper presents a new definition of nabla fractional-order difference, equivalent to the Grünwald-Letnikov difference. The difference is based on the general approach of orthonormal basis functions in terms of discrete-time Takenaka-Malmquist filters. The main advantage of the proposed definition is that for finite model length, the model quickly converges to the actual difference. The paper proposes the method of selecting the poles of the Takenaka-Malmquist functions. It also proposes the implementation of the Takenaka-Malmquist-based difference in non-commensurate state-space system and fractional-order integrator. Simulation experiments show the proposed methodology’s high effectiveness in modeling fractional-order difference, integrator, and non-commensurate state-space systems.

本文提出了 nabla 分数阶差分的新定义，相当于 Grünwald-Letnikov 差分。区别在于基于离散时间 Takenaka-Malmquist 滤波器的正交基函数的一般方法。所提出定义的主要优点是，对于有限的模型长度，模型可以快速收敛到实际差异。本文提出了Takenaka-Malmquist函数的极点选择方法。它还提出了在非对称状态空间系统和分数阶积分器中实现基于 Takenaka-Malmquist 的差分。仿真实验表明，所提出的方法在对分数阶差分、积分器和非对称状态空间系统进行建模方面具有很高的有效性。