Equivalence between uniform- and varying-sampling stability properties for discrete-time exact and approximate models

Digital control strategies for continuous-time (CT) nonlinear systems often face the problem of determining an appropriate discrete-time (DT) model. The exact DT model for the nonlinear system is unavailable in general, and thus approximate models must be employed for control design. Conditions that ensure practical stability of the exact DT model for all sufficiently small sampling periods based on the stability of the approximate model exist in the literature. Most of these existing results assume that the sampling period is kept constant during operation, and show that stability holds if the sampling period is small enough. Since such results may be not valid in a varying sampling rate (VSR) setting, previous work focused on a suitable extension, specifically aimed at controller-driven sampling (CDS). In this paper, we show that a specific stability property named (β, R
ⁿ
)-stability, and its CDS extension, named (β, R
ⁿ
)-stability under VSR, are equivalent. The equivalence of these two properties is not trivial, in the sense that it cannot be deduced directly from their definitions. We also provide an ϵ-δ characterization of (β, R
ⁿ
)-stability under VSR which may be useful in establishing the equivalence between other stability properties and their corresponding CDS extensions.