Full Observability Criteria for Linear Impulsive Control Systems with Application to Diabetes Type I Dynamics

The main goal of this paper is to derive a full characterization of the observability of linear time-invariant impulsive systems. Two cases are studied. The firrst case considers continuous outputs. A suitable adapted Kalman criterion is shown to characterize properly the observability. In the second case, only discrete-time measurements of the outputs are available. A new rank condition based on the structure of the impulses is shown to characterize observability. Finally, these results are tested and illustrated both on academic examples and on the dynamical model of diabetic type I patients. The latter provides a nice case study for an impulsive system with discrete time measurements as the meals may be approximated by some impulse inputs and the glycemia measurement is usually done in real life at various times through the day.