# Commuted PD Controller for Nonlinear Systems: Glucose-Insulin Regulatory Case
## 3 Results
### Decaying Exponential Disturbance Simulations  
As mentioned previously, the term \(D(t)\) (4) represents the rate at which glucose enters the blood from intestinal absorption following a meal. In oral glucose tests with normal subjects, the aim is for the model to produce the desired effect of the plasma glucose level rising quite rapidly (from the rest level) to a maximum in less than 30 min and then falling to the base level after 2-3 h [20]. There is some evidence to suggest that the exact form of \(D(t)\) for nondiabetics is not important provided the previously stated aim is met. The exponential function adequately models this situation.  
Figure 2a pictures the open-loop system response (1)-(4) with \(u(t)=0\). Figure 2b presents the signal output of the closed-loop system (2)-(5) with the commuted PD controls (9) and (10) showing the effectiveness of the proposed commuted PD controller. Also, a comparison is presented in Figure 3, where the behavior of classic PD control [11] does not perform better than the proposed control strategy. Furthermore, the commuted PD controller stabilizes the blood glucose concentration at the basal value \(G_{b}\) in the three cases. As Figure 3 shows, no hypoglycemia is attained when commuted PD control (9) and (10) is chosen. That is not the case when \(u_{g}(t)\) (10) is not considered, as \(u_{c}(t)\) behavior shows. In Figure 3c, considering commuted controller, in just a few minutes, hypoglycemia occurs. Then, the blood glucose concentration is normalized, but the basal glucose concentration is not reached.  
Analysing Figure 4, we see how commuted PD controller works. We comment on the Patient 1 case. During the first minutes after meal ingestion, when \(D(t)\) is biggest, the system needs insulin injection to reduce the glucose concentration (graphic of \(u(t)\) in Figure 4a). To normalize its level, glucose ingestion is needed an hour after (graphic of \(u_{g}(t)\) in Figure 4a). This is a common situation difficult to deal with. This might indicate that the insulin administration could be too strong for a pancreatic system. To mitigate this effect, we suggest ingesting some controlled quantity of glucose. This is done by \(u_{g}(t)\) (10) action. The same can be seen for Patients 2 and 3 given in Figure 4b and Figure 4c, respectively.  
Figure 4: Control input \(u(t)\) (9) and \(u_{g}(t)\) (10), with \(D(t)\) defined in (4), for each patient under study. To prevent hypoglycemia, after a few minutes of the end of its injection, glucose ingestion is needed. (**a**) Patient 1. (**b**) Patient 2. (**c**) Patient 3.  
Figure 3: Closed-loop simulations. Classic model (16)–(18) with \(u_{c}(t)\) (19) from [11], is compared to proposed system (2)–(5) with commuted PD controller (9) and (10). It shows how under the commuted PD controller (9) and (10) the glucose blood concentration is stabilized at \(G_{b}\). (**a**) Patient 1. (**b**) Patient 2. (**c**) Patient 3.  
Figure 2: From (**a**), we notice that Patient 2 presents strong hypoglycemia (reaching 30 mg/dL), and Patient 3 hyperglycemia (reaching 120 mg/dL). From the closed-loop system (**b**), only Patient 3 presents hypoglycemia for one hour approximately, reaching the lowest glucose level 52 mg/dL (considered healthy in [4]). (**a**) Open-loop system (1)–(4). (**b**) Closed-loop system (2)–(5), with control (9) and (10).