# Commuted PD Controller for Nonlinear Systems: Glucose-Insulin Regulatory Case
###### Abstract  
As an option to deal with insulin-dependent disease, a recently commuted PD control strategy is designed and carefully analyzed for different clinic diabetic patients. This controller approach is mainly conceived to stabilize the glucose blood concentration in a diabetic patient around its basal value; hence, avoiding extreme situations such as hypoglycemia and hyperglycemia. This control strategy receives two inputs carefully tuned to actuate when the measured variable is out of a prescribed healthy zone. Therefore, one of these variables is invoked to decrease the glucose concentration to insulin injection, and the other is employed to increase the glucose absorption, both by using a proper PD controller. According to our numerical experiments, our controller approach performs well, even when there is an external disturbance in the controlled system. glucose-insulin system; commuted proportional derivative controller; nonlinear system; exogenous perturbation +
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In literature, several approaches have been used to design artificial pancreas. For instance, a robust controller using a higher-order sliding mode control was studied in [6]. Using optimal H\({}_{\infty}\) control theory, an insulin injection control was analyzed in [5]. And employing a parametric programming approach for the control design was considered in [4]. On the other hand, in [10], control algorithms using standard linear control techniques, like the proportional-derivative controller, were studied, as in [11; 12; 13]. In this paper, the feedback measurement signal was assumed to be available at discrete-time moments which resulted in an interesting and useful technological fact for an artificial pancreas design. In contrast, we claim that no significant benefit is obtained from using a nonlinear model-based control design strategy. For instance, in [14], an impulsive model predictive control is presented, but the mathematical model has to be linear, and in [15], miss undertaken internal dynamics. On the other hand, PID controllers in the artificial pancreas have been studied in [16], but only for hypergly comditions, and in [17] a PID robust control is presented for the hypoglycemic situation, but both conditions are not studied simultaneously, as in the present work. However, the nonlinear techniques can offer new ways of control implementation that can face some nonlinearities ignored in by the linear controller tools. Some of these nonlinear control design strategies involve sliding mode control [6; 18], delay control [19], optimal control [20], switched LPV control [21], sub-optimal control [22], model predictive control [23], fuzzy control [24], reinforcement learning [25], etc. Therefore, the use of these control design tools allows us to innovate new developments for artificial pancreas approaches. Our proposal aims to act upon the control part of an insulin-dependent diabetic system to develop an effective and simple solution to avoid nondesired clinical problems. The main objective of the present work is to design a robust control for an artificial pancreas to minimize the effect of extreme situations such as hyperglycemia or hypoglycemia. Additionally, we evaluate our artificial pancreas performance under different scenarios. Our control strategy uses a novel switched strategy: when a peak value on glucose blood concentration is detected, the controller tries to minimize its level by injecting insulin into the system, and when the glucose blood concentration is lower than a healthy level, glucose ingestion is administrated by the controller. The objective is to maintain a healthy level of glucose. The control strategy is based on a proportional-derivative (PD) theory, where the input signal is the detected peak value of the blood glucose concentration. The PD controller format is considered due to its simple and easy realization [11], as pointed out by [12; 20]. We avoid using the PID controller because its integral action may be useless [13].