Observer based nonlinear control design for glucose regulation in type 1 diabetic patients: An LMI approach

Abstract

This paper deals with the design of observer-based nonlinear control of blood glucose concentration (BGC) of Type 1 diabetes mellitus (T1DM) patients in a Linear Matrix Inequality (LMI) framework. The controller design relies on the information of the states obtained from a nonlinear observer. The control law is derived using feedback linearisation and regional pole placement technique. Further, a numerical optimisation method is proposed for the computation of the controller gains by capturing the relationship between transformed domain and original domain dynamics by iteratively tuning the circular LMI region parameters (‘q’ and ‘r’) such that the locations of closed-loop poles of the original nonlinear system are attained. The proposed controller can deliver robust closed-loop response of BGC within a specified range of parametric uncertainty and meal disturbances owing to the appropriately tuned bound of LMI region parameters. The performance of the proposed controller is tested for 100 virtual T1DM patients in the presence of parametric uncertainty and uncertain meal disturbance. Both severe hypoglycemia (<50 mg/dl) and post-prandial hyperglycemia are avoided by the proposed scheme for nominal and uncertain parameters.

Introduction

The endocrine hormone, insulin is the primary regulator of blood glucose concentration (BGC) in the human body. Type 1 diabetes mellitus (T1DM) is a chronic disease characterised by prolonged elevated BGC due to negligible insulin production by the pancreatic β-cells [1]. It is one of the leading cause of deaths around the world that accounted for about 1.5 million deaths in 2012 [2]. The regulation of BGC in the safe range (70–180 mg/dl) is a major problem in T1DM [1]. If the BGC following a meal for an individual does not return to the basal value within 180 min, then this situation is termed as post-prandial hyperglycemia [3]. Hyperglycemia (BGC > 180 mg/dl) is associated with long-term health complications that include cardiac arrests, cerebral stroke, renal failure, loss of vision, diabetes ketoacidosis, etc. Similarly, severe hypoglycemia (BGC < 50 mg/dl) leads to a serious immediate effect that may lead to diabetic coma [4]. In order to minimise the risks associated with improper dosing of external insulin, fully automated system called Artificial Pancreas system (APS) comprising of continuous glucose monitoring (CGM) devices (for glucose measurement), continuous insulin infusion pumps (for exogenous insulin infusion) and closed loop control algorithm (for computing and adjusting the insulin dosages) has evolved successfully in the last few decades [5], [6]. The physiological parameters such as insulin sensitivity, time-to-peak in BGC following a meal and the parameters related to insulin absorption vary significantly within a population of T1DM patients in a physiologically plausible range. This is termed as inter-patient variability and it is one of the major hindrances to the blood glucose regulation in T1DM patients [7].

The primary objective of the current work is to design a closed loop control algorithm that will regulate the plasma glucose concentration within the safe range (70–180 mg/dl) in the presence of uncertain meal disturbances and parametric uncertainty representing the inter-patient variability of T1DM patients.

The basis of the model-based control algorithms are the physiological models that differ in their structures and degrees of complexity in describing the glucose-insulin regulatory system of T1DM patients [6]. Physiological T1DM models can be broadly categorised into two categories: (i) IVGTT (Intravenous Glucose Tolerance Test) models where both the glucose measurements and insulin infusions are done intravenously, and (ii) subcutaneous models include subcutaneous glucose measurements and subcutaneous insulin infusions [8]. This paper deals with the design of appropriate controller based on nonlinear IVGTT Bergman's minimal model (BMM) [8]. BMM provides a very good trade-off between the model complexity and the nonlinear characteristics of the glucose-insulin regulatory system. The more complicated physiological models like the Sorensen's model [9], the Hovorka's model [10] or the Dalla Man's model [11] pose a great challenge to the control engineers in designing appropriate control laws due to the existence of a large number of state variables as well as highly complex nonlinear functions. IVGTT models find important applications for the regulation of BGC in intensive care unit (ICU) patients and for treating diabetes ketoacidosis [12].

In the last four decades, numerous control algorithms based on both linear and nonlinear models have been proposed for the blood glucose regulation of T1DM patients. Conventional PID control schemes were developed as reported in [13], while robust and adaptive control algorithms were also reported in [14]. Due to the constraint handling capability and optimal control feature, model predictive controllers are the most popular control algorithms that are incorporated in the APS [15]. Apart from the conventional control algorithms, intelligent control schemes like fuzzy logic control [16] and neural network control [17] are also designed and some of them are implemented practically. The literature reveals that a vast majority of control algorithms are either based on linearised version of the nonlinear physiological model or time-series model. The linearisation of the original nonlinear system result in significant loss of the nonlinear characteristics of the system resulting in the deterioration of the control performance at operating points other than the equilibrium point. The major difficulty with the time-series models is that it becomes difficult to express important physiological factors in terms of the model parameters directly [18]. Further, the model-based control algorithms explicitly uses the information of all the state variables (related to both blood glucose and insulin), but the measurement of all states are not practically possible in APS because of high cost and reliability issues [8], [19]. Also, the computational aspect of nonlinear control technique is another point of concern.

Thus, the main motivations behind the proposed work are to first design a nonlinear controller that utilises BGC information only. Secondly, the computational aspect of the controller gain should be simple, efficient yet robust enough.

The paper is organised into seven sections. The related works are discussed in Section 2. Section 3 clearly states the problem formulation. The mathematical model and the observer-based controller design are presented in Section 4. While Section 5 presents the simulation results, Section 6 provides the detailed analysis of the simulation experiments. Finally, the concluding remarks and the future scopes are provided in Section 7.