Description of the Positively Invariant Sets of a Type 1 Diabetic Patient Model
Résumé
Positive invariant sets of a biological mathematical model of type-1 diabetes are characterized in this paper. Glycemia is physically a non-negative variable but the presented mathematical model does not guarantee positive trajectories of this state variable for t > 0.
Therefore, to give a better mathematical representation of the blood glucose-insulin interaction, invariance conditions of the positivity of the open-loop free-system response are firstly found.
Secondly, closed-loop positive invariance is investigated for the ultimate goal of regulating glycemia within normal levels using automated exogenous insulin infusions. The conditions for a standard state feedback control that guarantees a positiveness and stability of the input/state solutions are presented. The resulting non-negative control is virtually tested on diabetic patients to illustrate the theoretical results.