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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.

Description and analysis of the time–domain response of nabla discrete fractional order systems