# Discrete Blood Glucose Control in Diabetic Göttingen Minipigs

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## Abstract

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## 1. Introduction

- The controller design procedure is based on the Göttingen Minipig model and is introduced in Section 3.
- Section 4 describes the animal experimental study and the results of the model identification procedure that is the basis for the controller validation.
- Discussion of the results and a summary are given in Section 6.

## 2. Göttingen Minipig Model

#### 2.1. Nonlinear State Space Model

#### 2.2. Model Structure

## 3. Controller Design

#### 3.1. Model Linearisation

#### 3.2. Discrete Controller Design Prerequisites

#### 3.3. Disturbance Rejection Design

#### 3.3.1. Initial Controller Gain for Disturbance Rejection

#### 3.3.2. Integral Action

#### 3.3.3. Lead–Lag Compensator

#### 3.4. Robust Loop-Shaping ${\mathcal{H}}_{\infty}$-Controller

## 4. Animal Experimental Study and Model Identification

#### 4.1. Experimental Animal Study

#### 4.2. Model Identification

## 5. In Silico Feedback Control Study

#### 5.1. Controller Implementation

#### 5.2. Disturbance Rejection Performance

## 6. Conclusions and Discussion

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Overview of the artificial pancreas components with reference $r\left(t\right)$ and disturbance $d\left(t\right)$.

**Figure 4.**Block diagram of the discrete disturbance rejection controller $C\left(z\right)$ with hold transfer function $H\left(s\right)$ and δ-sampler in the feedback branch.

**Figure 5.**Frequency response of the plant and the disturbance dynamics, comparing full order models $\overline{G}\left(z\right)$ and ${\overline{G}}_{d}\left(z\right)$ and reduced order models ${\widehat{G}}^{r}(z$ and ${\widehat{G}}_{d}^{r}\left(z\right)$.

**Figure 6.**Frequency response of the compensated open loop (OL) $L\left(z\right)$ with classical controller and compensated loop ${L}_{\infty}\left(z\right)$ with the robust controller.

**Figure 7.**Comparison of minipig #4 in vivo measurements to in silico results of the identification procedure with individualised Göttingen Minipig data set. The orange triangle indicates a carbohydrate (CHO) uptake of 16 g a the beginning of the experiment. Indicated in the figure is a ±10% confidence interval by the red area.

**Figure 8.**Block diagram of the discrete disturbance rejection controller $C(z,\mathit{\theta})$ implementation with gain scheduling depending on estimated process parameters insulin sensitivity ${\widehat{k}}_{IS}\left(k{T}_{s}\right)$ and blood glucose concentration ${\widehat{G}}_{p}\left(k{T}_{s}\right)$.

**Figure 9.**Controller disturbance rejection test with nonlinear minipig model #2. 50 g CHO are given to the minipig at 15 h, indicated by an orange triangle.

**Figure 10.**Comparison of controller disturbance rejection test with nonlinear minipig model #4. In this figure, 50 mg CHO are given to the minipig at 15 h, indicated by an orange triangle.

**Figure 11.**Comparison of controller disturbance rejection test with nonlinear minipig model #5 and time-varying insulin sensitivity ${k}_{IS}\left(t\right)$. 50 mg carbohydrates (CHO) are given to the minipig at 15 h, indicated by an orange triangle.

**Figure 12.**Disturbance rejection statistics with single robust gain-scheduled controller that stabilises the family of Göttingen Minipig model. In this figure, 50 mg CHO are given to the minipig at a time $t=900$ min, indicated by an orange triangle (μ—average mean, σ—standard deviation).

**Table 1.**Overview of achieved results for 36 h in silico trial with the robust controller (with percentage of time of the blood glucose concentation above 120 mg/dL or below 70 mg/dL).

Göttingen Minipig | Perc. Time in Hyperglycaemia | Perc. Time in Hypoglycaemia |
---|---|---|

#03 | 5.5% | 0% |

#04 | 4.8% | 0% |

#05 | 4.1% | 0% |

#07 | 6.1% | 0% |

#08 | 6.0% | 1.3% |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Misgeld, B.J.E.; Tenbrock, P.G.; Lunze, K.; Leonhardt, S.
Discrete Blood Glucose Control in Diabetic Göttingen Minipigs. *Processes* **2016**, *4*, 22.
https://doi.org/10.3390/pr4030022

**AMA Style**

Misgeld BJE, Tenbrock PG, Lunze K, Leonhardt S.
Discrete Blood Glucose Control in Diabetic Göttingen Minipigs. *Processes*. 2016; 4(3):22.
https://doi.org/10.3390/pr4030022

**Chicago/Turabian Style**

Misgeld, Berno J.E., Philipp G. Tenbrock, Katrin Lunze, and Steffen Leonhardt.
2016. "Discrete Blood Glucose Control in Diabetic Göttingen Minipigs" *Processes* 4, no. 3: 22.
https://doi.org/10.3390/pr4030022