Characterization of semiglobal stability properties for discrete-time models of non-uniformly sampled nonlinear systems

Abstract

Discrete-time models of non-uniformly sampled nonlinear systems under zero-order hold relate the next state sample to the current state sample, (constant) input value, and sampling interval. The exact discrete-time model, that is, the discrete-time model whose state matches that of the continuous-time nonlinear system at the sampling instants may be difficult or even impossible to obtain. In this context, one approach to the analysis of stability is based on the use of an approximate discrete-time model and a bound on the mismatch between the exact and approximate models. This approach requires three conceptually different tasks: i) ensure the stability of the (approximate) discrete-time model, ii) ensure that the stability of the approximate model carries over to the exact model, iii) if necessary, bound intersample behaviour. Existing conditions for ensuring the stability of a discrete-time model as per task i) have some or all of the following drawbacks: are only sufficient but not necessary; do not allow for varying sampling rate; cannot be applied in the presence of state-measurement or actuation errors. In this paper, we overcome these drawbacks by providing characterizations of, i.e. necessary and sufficient conditions for, two stability properties: semiglobal asymptotic stability, robustly with respect to bounded disturbances, and semiglobal input-to-state stability, where the (disturbance) input may successfully represent state-measurement or actuation errors. Our results can be applied when sampling is not necessarily uniform.