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Complexity

Volume 2018, Article ID 4670159, 11 pages

https://doi.org/10.1155/2018/4670159

## Receding Horizon Control of Type 1 Diabetes Mellitus by Using Nonlinear Programming

^{1}Doctoral School of Applied Informatics and Applied Mathematics, Óbuda University, Bécsi Street 96/B, Budapest 1034, Hungary^{2}Mathematical Sciences Research Center, Karachi, Pakistan^{3}Antal Bejczy Center for Intelligent Robotics (ABC iRob), Óbuda University, Bécsi Street 96/B, Budapest 1034, Hungary^{4}Physiological Controls Research Center, Óbuda University, Bécsi Street 96/B, Budapest 1034, Hungary

Correspondence should be addressed to György Eigner; uh.adubo-inu.kin@ygroyg.rengie

Received 23 November 2017; Accepted 26 February 2018; Published 23 April 2018

Academic Editor: Thierry Floquet

Copyright © 2018 Hamza Khan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Receding Horizon Controllers are one of the mostly used advanced control solutions in the industry. By utilizing their possibilities we are able to predict the possible future behavior of our system; moreover, we are able to intervene in its operation as well. In this paper we have investigated the possibilities of the design of a Receding Horizon Controller by using Nonlinear Programming. We have applied the developed solution in order to control Type 1 Diabetes Mellitus. The nonlinear optimization task was solved by the Generalized Reduced Gradient method. In order to investigate the performance of our solution two scenarios were examined. In the first scenario, we applied “soft” disturbance—namely, smaller amount of external carbohydrate—in order to be sure that the proposed method operates well and the solution that appeared through optimization is acceptable. In the second scenario, we have used “unfavorable” disturbance signal—a highly oscillating external excitation with cyclic peaks. We have found that the performance of the realized controller was satisfactory and it was able to keep the blood glucose level in the desired healthy range—by considering the restrictions for the usable control action.

#### 1. Introduction

The advanced control solutions have inevitable role in today’s medical practice regarding the control of physiological processes [1]. Many control solutions are under development that can be used for various kinds of control problems. Advanced control methods have been successfully applied for physiological regulation problems, for example, control of anesthesia [2, 3], angiogenic inhibition of cancer [4, 5], immune response in presence of human immunodeficiency virus [6], and regulation of blood glucose (BG) level [7–10] as well.

Diabetes Mellitus (DM) is the collective name of several chronic diseases connected to the metabolic system of the human body. In most of the cases, the DM condition appears due to the issues related to the insulin hormone [11]. The insulin is the key hormone which makes the glucose molecules possible to enter from the blood into the glucose consuming cells through the insulin-dependent gates on the cell-wall [12].

There are many types of DM. The most dangerous is the Type 1 DM (T1DM) where the metabolic system is not able to function normally due to the lack of insulin. Type 2 DM (T2DM) is the most widespread kind of DM and it occurs mostly because of the lifestyle. In this case usually the blood glucose and insulin levels continuously increase over a long period of time. Due to the extreme glucose and insulin load the cells become resistant to the insulin over time. In order to compensate this condition the body produces more and more insulin that leads to the “burnout” of the pancreatic -cells that produce the hormone. At this point the T2DM turns into T1DM. Other frequently occurring type is the Gestational DM (GDM) from which women may suffer during pregnancy. Usually, this condition is temporary; however, sometimes it turns into T2DM and becomes permanent [13–15].

In case of DM the application of these kinds of advanced control techniques has high importance. Due to the nature of the phenomenon to be controlled the researchers on the field have to face many challenges such as high nonlinearities, model and parameter uncertainties, and even time-delay effects, as well. However, regardless the type of DM a few common control goals can be defined: keeping the glycemia (the BG level) in a the healthy range; totally avoiding the hypoglycemic periods; and avoiding the high BG variability as much as possible [16–18].

In this research we have investigated the T1DM. As we mentioned, T1DM is the most dangerous condition because the patients need external insulin intake in order to keep their metabolic status on appropriate level. In this case the patient’s pancreatic -cells are terminated by the immune system of the patient during an autoimmune reaction. As a consequence, these patients are not able to produce insulin—which leads to short term starving, coma, or even death [11]. Furthermore, the appropriate treatment—how the insulin is administered—is also important to avoid long term side effects, for example, the chronic failure of peripheric vasculature [13].

By using advanced control techniques not just acceptable control action but higher treatment quality can be obtained. The selected control methods have to handle the already mentioned unfavorable effects—such as the nonlinearities and so on—as well. In case of T1DM many solutions are available; however, all of them have their own limitations, simplifications, and restrictions—thus, none of them are general [10]. In these days from control point of view the most beneficial approach is the Artificial Pancreas (AP) concept. This idea aims to imitate the regular operation of the pancreas from the insulin production point of view, namely, administering insulin demands on the needs determined by the BG level [19]. Thus, we have to face contradictory requirements: the generalization and personalization as well.

One of the mostly used algorithms is the modified proportional-integral-derivative (PID) solutions due to their simplicity and flexibility. Moreover, several clinical trials have been done by using this methodology and investigate its effectivity [20–22]. Linear Parameter Varying (LPV) model-based solutions have high importance, since the uncertainties can be handled with high efficiency by them [7, 9, 23]. The Tensor Product (TP) model-based techniques also represent interesting directions, since they can be combined by Linear Matrix Inequality (LMI) based control and LPV methodology as well [24, 25].

The most frequently used method is the Model Predictive Control (MPC) regarding the control of DM [19, 26–28]. MPC is a widely used approach in various research fields as well [29–32]. In general the goals of the MPC applications are tracking and stabilization [33]. In case of an MPC the actual value of the control signal is obtained by solving an open-loop control problem over a finite horizon. Of course, some feedback is present because the starting point of the next horizon is the last realized point of the previous one. The type of the optimal control problem depends on the type of the control task. The classical MPC is realized in a framework of cost-function-based optimal control where the dynamics of the system to be controlled can be considered as a set of constraints. Furthermore, the cost function regularly contains terms which depend on the tracking error and the control signal itself. Optionally, a cost contribution that depends on the tracking error at the end of the horizon can be added as well.

In case of the Receding Horizon Control (RHC) via Nonlinear Programming (NP) based approximation the state variables and the control inputs of the system are considered over a discrete time grid. At each point of the grid the Lagrangian multipliers determine the reduced gradient which is driven into zero numerically to find the optimal solution. This solution consists of the estimated values of the state variables and control signals over the finite horizon. In that case if the dynamic model of the system is not precise, the optimal design can be used only for consecutive finite horizons since the actual state of the controlled system propagates according to its exact dynamics. In order to minimize the effects coming from the inaccuracies the actual measured state variable from the end of the previous horizon is applied as starting point for the next one in the next horizon-length design [34]. The RHC framework can be hardly combined with the Lyapunov function based control. However, certain approaches can be found in the literature where the Lyapunov stability [35] and RHC were successfully combined for specific cases [36–39].

Alternative solutions also exist which can be used instead of Lyapunov’s stability theorem. The Robust Fixed Point Transformation (RFPT) based control [40, 41] uses Banach’s fixed point theorem [42] to transform the control problem into a fixed point problem which can be solved iteratively. This method allows designing a robust iterative adaptive controller which can avoid the main limitations of RHC if these are combined.

In this study we present the first part of our research, namely, the design of an appropriate RHC controller on NP basis which can be completed by RFPT in our further work.

The paper is structured as follows. First, the applied diabetes model is introduced. After that, the RHC design based on the NP approach is presented. Then, the results are introduced with their discussion. Finally, we conclude our work and provide an outline regarding our further research.

#### 2. Type 1 Diabetes Mellitus Model

During our research we have applied a modified Minimal Model [43] which originates from the model of Bergman [46]. This model has several beneficial properties, such as simplicity, good transformability, and flexibility and it is based on simpler biological considerations. The main goal of the model is to describe the glucose-insulin dynamics, namely, to define the connection between the blood glucose and insulin levels. However, in order to characterize the daily life of a T1DM patient this model has to be extended with additional submodels. These submodels are the absorption of the external glucose and insulin intakes. During the daily routine these substances are not directly injected to the blood stream—however, this can occur in case of persistent hospitalization. Instead, the carbohydrate is consumed via food intake and the insulin is entered through the extracellular tissue matrix under the skin [13]. Thus, their appearance does not contain sharp peaks: it happens through longer dynamics.

The glucose and insulin absorption are described by (1)–(4), respectively. These submodels originate from the Cambridge model [44], but we applied them in appropriate dimensions to insert them into the core model. The core model is described by (5)–(7).

The state variables in (1)–(7) have the meaning and purpose as follows. mg/dL and mg/dL are the primary and secondary compartments belonging to glucose, where the time constant determines how long it takes for the meal to be absorbed after consumption in time. mU/L and mU/L are the primary and secondary compartments belonging to insulin, where the time constant determines how long it takes the insulin to be absorbed after injection (to the extracellular space) in time. Variable mg/dL is the blood glucose (BG) concentration—the so-called glycemia—and mU/L is the blood insulin concentration and 1/min is the insulin-excitable tissue glucose uptake activity—which describes the connection between the blood’s glucose and insulin levels, respectively.

From system engineering point of view the external glucose, namely, the food intake, can be handled as disturbance. In this case g/min is the disturbance input. It can be inserted to via complex which describes the bioavailability of the glucose from complex carbohydrates. The control signal mU/min—the injected insulin—is directly connected to . More detailed description of the used model parameters can be found in Table 1 and in [43–45].