Positively invariant sets of a T1DM model: Hypoglycemia prediction and avoidance

In this work, maintaining glycemia of a Type-1 Diabetes Mellitus (T1DM) system model within or above the normal level is studied. The largest Positively Invariant Set (PIS) such that glycemia error is nonnegative is found under (i) basal insulin delivery during fasting phase (open loop) and under (ii) positive state feedback control. The main results are: (i) the largest Positively Invariant Set (PIS) during fasting phase under basal injection is determined such that the state variables remain non negative for all future times, (ii) hypoglycemia is predicted whenever the set of initial conditions is outside the open-loop PIS, (iii) these results can be used in both open and closed-loop control to avoid hypoglycemia. (iv) A positive state feedback control is designed for the first time to regulate glycemia ensuring that glycemia remains invariant above or within the desired set point. (v) The largest PIS is computed which ensures the stability of the system and the positivity of the state and the control. The significance of the results are related to (i) hypoglycemia-free blood glucose regulation for any trajectory initiated in the PIS, (ii) outside the PIS future hypoglycemic events are predicted in advance, (iii) a general hypoglycemia prediction algorithm is implemented taking benefit from the largest open-loop and closed-loop PIS, and (iv) in closed-loop, the state feedback control is positive everywhere in the PIS. The positive invariant sets to regulate glycemia have not been taken into account in most of the research in the area of glycemia regulation. The result is a control strategy which ensures a positive control action and which keeps glycemia above a specified level and thus prevents hypoglycemia. Hypoglycemia, the major challenge and open problem is either prevented in closed-loop or can be avoided using the prediction algorithm.