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 LTVLO proposed 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 selection of the asymptotic eigenvalues where the observability matrix is assured. The estimation of the unmeasurable variables is done using Ackermann's methodology. Additionally, it is shown the Lyapunov approach to prove the stability of the time-varying proposal. In order to evaluate the proposed methodology, we designed three experiments: A) VP obtained with the Bergman minimal model; B) VP obtained with the compartmental model presented by Hovorka in 2004; and C) real patients data set. For experiments A) and B), it is applied a meal plan to the VP, where the dynamic response of each state model is compared to the response of each variable of the time-varying observer. Once the observer is evaluated in experiment B), the proposal is applied to experiment C) 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 instant of time to estimate the states under a known structure. The results are obtained using simulation with MatlabTM and SimulinkTM. 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 fit parameter.