Linking stability of nonlinear sampled-data systems and their continuous-time limits

Abstract

Discrete-time (DT) models of sampled-data systems can be regarded as predictions of the future state value given current values for the state, input and sampling period. A DT model is exact if its prediction coincides with the true value and is otherwise approximate. Conditions under which Semiglobal Exponential Stability under nonuniform sampling (SES-VSR) is preserved among different (exact or approximate) DT models when fed back with the same sampling-period dependent control law exist. The current paper proves that the exact DT model is SES-VSR if and only if its corresponding continuous-time (CT) limit (for infinitesimally small sampling period) is Globally Exponentially Stable (GES), which is restrictive. The contribution of this note consists in extending and relaxing the assumptions of previous results, as follows: (i) only mild conditions are given in order to link stability properties between DT models fed back with different control laws, and (ii) the DT model properties making the CT limit Globally Asymptotically and Locally Exponentially Stable (GALES), instead of GES, are provided.