This paper investigates the time–domain response of nabla discrete fractional order systems by exploring several useful properties of the nabla discrete Laplace transform and the discrete Mittag–Leffler function. In particular, we find the limitation of discrete Mittag–Leffler when adopted in describing the response. Then we establish two fundamental properties of a nabla discrete fractional order system with nonzero initial instant: i) the existence and uniqueness of the system time–domain response; and ii) the dynamic behavior of the zero input response. Finally, a simulation study is provided to show the validity of the theoretical results.

中文翻译：

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nabla 离散分数阶系统时域响应的描述与分析

本文通过探索 nabla 离散拉普拉斯变换和离散 Mittag-Leffler 函数的几个有用特性，研究了 nabla 离散分数阶系统的时域响应。特别是，我们发现在描述响应时采用离散 Mittag-Leffler 的局限性。然后我们建立了具有非零初始时刻的 nabla 离散分数阶系统的两个基本性质：i) 系统时域响应的存在性和唯一性；和 ii) 零输入响应的动态行为。最后，通过仿真研究证明了理论结果的有效性。