Impulsive MPC schemes for biomedical processes. Application to type 1 diabetes

During the last decades, dynamic systems considering short-duration inputs have been extensively studied, mainly in the field of biomedical research (being the problem of administration of pills or injections in chronic disease treatments a typical example). Mathematically, inputs are modeled as a sequence of impulses, whose amplitude and frequency must be selected by appropriated control laws. In this chapter, advanced estimation/control schemes are developed for such kind of hybrid systems. In a first stage, the equilibrium of impulsive systems is studied, showing that the formal concept of equilibrium (of continuous or discrete-time systems) is no longer valid and, so, a more general definition (accounting for orbits instead of points) is needed. Then, in a second stage, and based on the aforementioned characterization, two underlying discrete-time systems are proposed to describe the state evolution just before and after the impulsive times. Finally, in a third stage, an impulsive zone model predictive control is formulated by means of the use of artificial/intermediary variables that ensures feasibility for any change of the target zone and provides an enlarged domain of attraction. To complement the model predictive control formulation, an enhanced Kalman observer is designed, able to adequately estimate the state even in cases when some of the system inputs are unknown. To assess the proposed controller, the problem of the glucose regulation in type 1 diabetes patients under challenging conditions is tackled. The example includes parameters variation, non-Gaussian sensor noise, unknown disturbances (i.e., meal intakes), and input constraints. The simulations are performed in a virtual type 1 diabetes mellitus patient extracted from the metabolic UVA/Padova metabolic simulator, and the obtained results show to be promising form both theoretical and practical points of view.