Fixed Point Iteration-based Adaptive Control for a Delayed Differential Equation Model of Diabetes Mellitus

Abstract

In natural sciences, especially in life sciences, controller designers frequently meet the problem that though the controlled system is modeled by a set of nonlinearly coupled Ordinary Differential Equations (ODE) containing various independent variables, only a single control input is available by the use of which the propagation of only one variable has to be controlled. Normally the controlled state variable can be observed by direct measurements, while no sensors are available for obtaining information on the propagation of the other ones. Though in the possession of a reliable system model one has good odds to develop state observers, in the practice just the reliable model used to be missing. While the traditional control design methodologies normally need some state estimation, the Fixed Point Iteration-based (FPI) Adaptive Controller was developed to evade this difficulty. In this design instead modeling the effects of the propagation of the various state variables, these effects are directly observed and compensated on this basis. This approach can be used without structural modification if certain effects appear through some time-delay (Delayed Differential Equations-DDE). In many cases simple and effective models can be developed that contain only pure delay effects. In this paper it is shown that a recent DDE model of Diabetes Mellitus can be used in the FPI-based adaptive blood glucose concentration level control even if the available model is very imprecise. This statement is substantiated by numerical simulations.