Ashley Brandon, Liu Weijiu
Department of Mathematics, University of Central Arkansas, 201 Donaghey Avenue, Conway, AR 72035, USA.
Department of Mathematics, University of Central Arkansas, 201 Donaghey Avenue, Conway, AR 72035, USA.
Math Biosci. 2017 Jul;289:78-88. doi: 10.1016/j.mbs.2017.05.001. Epub 2017 May 8.
Type 1 diabetes patients need external insulin to maintain blood glucose within a narrow range from 65 to 108 mg/dl (3.6 to 6.0 mmol/l). A mathematical model for the blood glucose regulation is required for integrating a glucose monitoring system into insulin pump technology to form a closed-loop insulin delivery system on the feedback of the blood glucose, the so-called "artificial pancreas". The objective of this paper is to treat the exogenous glucose from food as a glucose disturbance and then develop a closed-loop feedback and feedforward control system for the blood glucose regulation system subject to the exogenous glucose disturbance. For this, a mathematical model for the glucose disturbance is proposed on the basis of experimental data, and then incorporated into an existing blood glucose regulation model. Because all the eigenvalues of the disturbance model have zero real parts, the center manifold theory is used to establish blood glucose regulator equations. We then use their solutions to synthesize a required feedback and feedforward controller to reject the disturbance and asymptotically track a constant glucose reference of 90 mg/dl. Since the regulator equations are nonlinear partial differential equations and usually impossible to solve analytically, a linear approximation solution is obtained. Our numerical simulations show that, under the linear approximate feedback and feedforward controller, the blood glucose asymptotically tracks its desired level of 90 mg/dl approximately.
1型糖尿病患者需要外部胰岛素将血糖维持在65至108毫克/分升（3.6至6.0毫摩尔/升）的狭窄范围内。为了将葡萄糖监测系统集成到胰岛素泵技术中，形成基于血糖反馈的闭环胰岛素输送系统，即所谓的“人工胰腺”，需要一个血糖调节的数学模型。本文的目的是将食物中的外源性葡萄糖视为葡萄糖干扰，然后为受外源性葡萄糖干扰的血糖调节系统开发一种闭环反馈和前馈控制系统。为此，在实验数据的基础上提出了一个葡萄糖干扰的数学模型，然后将其纳入现有的血糖调节模型。由于干扰模型的所有特征值的实部均为零，因此使用中心流形理论建立血糖调节方程。然后，我们利用这些方程的解来合成所需的反馈和前馈控制器，以抑制干扰并渐近跟踪90毫克/分升的恒定血糖参考值。由于调节方程是非线性偏微分方程，通常无法解析求解，因此获得了一个线性近似解。我们的数值模拟表明，在线性近似反馈和前馈控制器下，血糖渐近地近似跟踪其期望水平90毫克/分升。
