Abstract
The article proposes a nonlinear optimal control method for the administered infusion of insulin, aiming at regulation of patients’ blood glucose levels. The dynamic model of blood glucose concentration which receives as control input the insulin’s infusion rate, undergoes approximate linearization through Taylor series expansion and through the computation of the associated Jacobian matrices. The linearization takes place round a temporary equilibrium which is updated at each iteration of the control method and which is defined by the last value of the system’s state vector and the last value of the control inputs vector exerted on it. For the linearized model, a robust H-infinity feedback controller is designed. To find the feedback control gain of the controller, an algebraic Riccati equation is solved at each iteration of the control algorithm. Through Lyapunov stability analysis it is first proven that the control loop satisfies the H-infinity tracking performance criterion, which signifies elevated robustness to model uncertainty and external perturbations. Moreover, the global asymptotic stability of the control loop is proven. The proposed control method can contribute to the treatment of diabetes patients and to blood glucose levels regulation in patients of intensive care units.
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Appendix: Differential Flatness of the Blood Glucose–Insulin Infusion Model
Appendix: Differential Flatness of the Blood Glucose–Insulin Infusion Model
By exploiting the differential flatness properties of the blood glucose–insulin infusion model the problem of setpoints seletion in the system’s nonlinear optimal control can be easily solved. Actually, one can define reference setpoint only for the flat output of the system (blood glucose concentration) and can find the reference values for the rest of the state variables of the models through the differential relations connecting them to the flat output. It holds that:
The flat output of the system is \(y=x_1\) that is the blood glucose concentration. Next from the first row of the state-space model of the system, given in Eq. (44)
one solves with respect to \(x_2\). This gives
From the second row of the state-space model of the system, given in Eq. (44), one solves with respect to \(x_3\). This gives
Finally, from the third row of the state-space model of the system, given in Eq. (44), one solves with respect to the control input u. This gives
Since all state variables and the control inputs of the blood glucose–insulin infusion model can be written as differential functions of its flat output (y concentration of blood glucose) it can be confirmed that the model is a differentially flat one.
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Rigatos, G., Siano, P. & Melkikh, A. A Nonlinear Optimal Control Approach of Insulin Infusion for Blood Glucose Levels Regulation. Intell Ind Syst 3, 91–102 (2017). https://doi.org/10.1007/s40903-016-0063-8
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DOI: https://doi.org/10.1007/s40903-016-0063-8