Linear Time-Varying Luenberger Observer Applied to Diabetes

We present a linear time-varying Luenberger observer (LTVLO) using compartmental models to estimate the unmeasurable states in patients with type 1 diabetes. The proposed LTVLO is based on the linearization in an operation point of the virtual patient (VP), where a linear time-varying system is obtained. LTVLO gains are obtained by the selection of the asymptotic eigenvalues where the observability matrix is assured. The estimation of the unmeasurable variables is done using Ackermann's methodology. The Lyapunov approach is used to prove the stability of the time-varying proposal. In order to evaluate the proposed methodology, we designed three experiments: 1) VP obtained with Bergman's minimal model; 2) VP obtained with Hovorka's model; and 3) real patient data set. For both experiments 1) and 2), it is applied a meal plan to the VP, where the dynamic response of each state model is compared with the response of each variable of the time-varying observer. Once the observer is obtained in experiment 2), the proposal is applied to experiment 3) with data extracted from real patients, and the unmeasurable state space variables are obtained with the LTVLO. LTVLO methodology has the feature of being updated each time instant to estimate the states under a known structure. The results are obtained using simulation with MATLAB and Simulink. The LTVLO estimates the unmeasurable states from in silico patients with high accuracy by means of the update of Luenberger gains at each iteration. The accuracy of the estimated state space variables is validated through thefit parameter.