An Approach for Artificial Pancreas Control Type-I

An Approach for Artificial Pancreas to Control the Type-I DiabetesMellitus

Submitted by

Muhammad FarmanRegistration No: DMAT 0 1 143004

SESSION: 2014-2018

Supervisor

Prof. Dr. Muhammad Ozair Ahmad

Department of Mathematics and Statistics

The University of Lahore, Lahore.

Co-Supervisor

Dr. Muhammad Umer Saleem

Department of Mathematics (DS & T)

University of Education, Lahore

DEPARTMENT OF MATHEMATICS AND STATISTICSTHE T.IhIIVERSTTY OF LAHORE,

LAHORE-PAKISTAN2019

An Approach for Artificial Pancreas to Control the Type-IDiabetes Mellitus

This dissertation is submitted to the Department of Mathematics andStatistics, University of Lahore, Lahore, Pakistan, for the partial fulfillment of the

requirements for the award of the degree of

Submitted byMuhammad Farman

Registration No: DMAT 0 1 143004

APProved on )' t? ' c' ('* L-" r \

" --#Sign: . i ,', ;'" . I Sigr' \id 2l: -

Prof. Dr. Muhammad Ozair AhmadInternal Examiner (Thesis Supervisor)Deparhnent of Mathematics & Statistics

Intemal Examiner (Thesis Co-Supervisor)Department of Mathematics

University of Education, Lahore

Dr. Aziz-tr-Rehman (Extemal Examiner)Deparhnent of Mathematics

University of Engineering & Technology,

Lahore

Sign:

Prof. Dr. Muhammad Ozair AhmadHead of Departrnent of Mathematics and StatisticsThe University of Lahore, Lahore

DEPARTMENT OF MATHEMATICS AND STATISTICSTHE UNIVERSITY OF LAHORE,

LAHORE.PAKISTAl\2019

The University of Lahgre

Dr. Muhammadtlmer Saleem Dr. Anjum yrydtz (Extemal Examiner)

Acknowledgments

My highest praise is for Almighty Allah, the most Gracious and most Merciful

who guides me in difficulties all respect to Holy Prophet Muhammad (PBUH)

enlightening a conscience with essence of faith in Allah. With His grace and

guidance I have reached this stage of my academic carrier. With His blessing and

kindness, I plan, visualize and execute my dreams into reality.

I am indebted to all my teachers for the great cooperations through out the aca-

demic program. It is a matter of great pleasure to express our sincere regards to my

honorable supervisor Prof. Dr muhammad Ozair Ahmad and Dr Muhammad Umer

Saleem for affectionate supervision and masterly advice. His able guidance, valuable

suggestions and continuous encouragement helped me at every stage in completion

of my work. I am very thankful to both of you for this love, kindness and cooperation.

I wish to express my thanks to Head of Department DR. Muhammad Ozair

Ahmad, PhD coordinator Dr Maqbool Ahmad Chadury , Dr Deba Afzal and other

faculty members of department of mathematics and Statistics who were always

there to facilitate me in my research work and taught me during the course of my

study. I also very thankful to Dr Nouman Raza and Dr Muhammad Rizwan for

their piece of advice during research work. Thanks to all authors who I mentioned

in bibliography, their work enable me to step forward in the field of mathematical

biology,

I offer my gratitude to my dear parents, father in law, wife, my brothers and

sisters , whose prayers and good wishes made it possible for me to carry out this

work progressively. The immeasurable sacrifices of my parents and family led to

what I am today. Bundle of thanks to my family members specially Sara Farman

, and the little stars Huda Rameen , Noriaz Haider, Ammara, Khansa who helped

me morally and emotionally to accomplish my task. I Love you all.

I am very grateful to all my friends specially Aqeel Ahmad, Ehsan-ul Haq,

Muhammad Abdullah, Farhan and my Ph.D fellows who always encouraged and

helped me during my completion of all work.

Muhammad Farman

UOL, Pakistan, 2018

List of Publications Related Thesis

1. Saleem M.U., Muhammad Farman, Meraj M.A., Stability Analysis of

Sorensen’s Model for controllability and observability, B. Life and Environ-

mental Sciences, 54 (2): 133145, (2017) .

2. M.U Saleem, Muhammad Farman, M.O. Ahmad, Rizwan M., A Control of

Artificial Human Pancreas, Chines journal of Physics.55(6)2273-2282,(2017).

3. Muhammad Farman, M.U Saleem, M.O. Ahmad, An Approach for Artificial

Pancreas to Control the Type-1 Diabetes Mellitus, Journal of Applied Envi-

ronmental and Biological Sciences, 93-102, (2017).

4. Muhammad Farman, Muhammad Umer Saleem, M. O Ahmad, Aqeel Ahmad,

Stability Analysis and Control of Glucose Insulin Glucagon System in human,

Chines Journal of Physics, 56, 1362-1369, (2018).

5. Muhammad Farman, Muhammad Umer Saleem, Control of Augmented mini-

mal model for glucose insulin pump, Current Research in Diabetes and Obesity

Journal, DOI: 10.19080/CRDOJ.2018.08.555742 (2018).

6. M.U Saleem, Muhammad Farman, Rizwan M., M.O. Ahmad, Aqeel ahmad,

Controllability and Observability of Glucose Insulin Glucagon systems in Hu-

man, Chines journal of Physics, ,56(5), 1909-1916, (2018).

7. M.U Saleem, Muhammad Farman, Aqeel Ahmad, M. Naeem, M. O Ahmad,

Stability Analysis and Control of Fractional Order Diabetes Mellitus Model

for Artificial Pancreas, Punjab University mathematics journal, 51(4),97-113,

(2019)..

List of others Publications

1. Saleem M.U., Muhammad Farman, Meraj M.A., A linear control of Hovorka

Model, Sci. Int. (Lahore),28(1),15-18, (2016).

2. Aqeel, Muhammad Farman, M. O Ahmad, N. Raza, Abdullah, Dynamical

behavior of SIR epidemic model with non-integer time fractional derivatives: A

mathematical analysis, International journal of Advance and Applied sciences,

123-129 (2018).

3. Aqeel, Muhammad Farman, Faisal yaseen, M. O Ahmad, Dynamical trans-

mission and effect of smoking in society, International journal of Advance and

Applied sciences, 5(2),71-75, (2018).

4. Farah Ashraf, Aqeel Ahmad, Muhammad Umer Saleem, Muhammad Farman,

M.O. Ahmad, Dynamical behavior of HIV immunology model with non-integer

time fractional derivatives, International journal of Advance and Applied sci-

ences, 5(3), 39-45, (2018).

5. Muhammad Farman, Muhammad Umer Saleem, Aqeel Ahmad, M. O Ahmad,

Analysis and Numerical Solution of SEIR Epidemic Model of Measles with Non-

integer Time Fractional Derivatives by using Laplace Adomian Decomposition

Method, Ain Shams Engineering Journal, 9(4), 3391-3397 ( 2018).

6. Saleem M.U., Muhammad Farman, Aqeel Ahmad, Meraj M.A, Mathematical

model based assessment of the cancer control by Chemo-Immunotherapy, Pure

and Applied Biology vol 7, 678-683, (2018).

7. Abdullah, Aqeel, N. Raza, Muhammad Farman, M. O Ahmad, Approximate

solution and analysis of smoking epidemic model with Caputo fractional deriva-

tives, Int. J. Appl. Comput. Math 4:112 (2018).

8. Muhammad Farman, Zafer, Aqeel Ahmad, Ali Raza, Ehsan ul Haq, Numeri-

cal Simulation and Analysis of Acute and Chronic Hepatitis B, International

journal of Analysis and Application, 16(6), 842-855, (2018).

9. Aqeel Ahmad, Nouman, Muhammad Farman, M. O Ahmad, A. Hafeez, Ali,

Dynamical Behavior of Fractional Order Epidemic Model, International journal

of Analysis and Application, (accepted).

ABSTRACT

Currently diabetes mellitus is worldwide issue and challenges for diabetes com-

munity for healthy life. An effort is made to develop the idea of getting a fully

automated artificial pancreas. The artificial pancreas is a developing technology

to help patients with diabetes of type 1 diabetes mellitus (T1DM) to control au-

tomatically their blood glucose level by making available the alternative endocrine

functionality of a healthy pancreas. The concept of controllability and observability

for the linearized control system of human glucose insulin system is used so that we

can have a feedback control for artificial pancreas. For the purpose of automatic

artificial pancreas in the glucose regulatory system, we consider the Glucose Insulin

(GI) Systems and Glucose Insulin Glucagon (GIG) systems. These models includes

Augmented Meal Model (AMM), Reduced Meal Model (RMM), fractional order glu-

cose insulin system, a composite model of Glucagon-Glucose Dynamics Model and

Sorenson model being comprehensive model for Type-1 Diabetes Mellitus (T1DM).

These models can be used to simulate a glucose insulin system for the treatment of

T1DM. The Lyapunov Equation is used to check the stability analysis of the model.

A fractional-order time derivatives model is presented for comprehensive glucose

insulin regulatory model. A fractional-order state observer is designed for approx-

imating the structure of a blood glucose-insulin with glucose rate disorder to show

the complete dynamics of the glucose-insulin system with the fractional-order at

α ∈ (0 < a < 1]. The developed method provides the observer estimation algo-

rithm for a glucose-insulin system with unknown time-varying glucose rate distur-

bance. Numerical simulations are carried out to demonstrate our proposed results

and show the nonlinear fractional-order glucose-insulin systems are at least stable

as their integer-order counterpart in the existence of exogenous glucose infusion or

meal disturbance.

Controllability and observability of the linearized model are calculated under two

different cases,for case 1 insulin is taken as an input and case 2, insulin and glucagon

are taken as an input for the system. This played an important role in the devel-

opment of fully automatic artificial pancreas by stabilizing the control loop system

for the glucose-insulin glucagon pump. Proportional Integral Derivative (PID) con-

troller is designed for an artificial pancreas by using the transfer function. According

to the desire value, the algorithm of an artificial pancreas measures the glucose level

in the blood of a patient by using glucose sensor that sends a signal to an insulin

glucagon pump to adjust the basal insulin. A closed-loop system is tested in simulink

environment and simulation results show the performance of the designed controller.

We convert the Sorenson model to Sorenson model type 1 diabetes mellitus because

this is the most comprehensive model in the Glucose Insulin Glucagon dynamics for

human. This may play an important role in the development of fully automatic arti-

ficial pancreas and stabilize the control loop system for the Glucose Insulin Glucagon

pump. It would be helpful for type 1 diabetic patients to control their diseases.

The thesis is also review the state of art in hypoglycemia prevention and detection

technique in the closed-loop artificial pancreas. Hypoglycemia is the major adverse

effect of insulin therapy and therefore minimizing the risk of hypoglycemia, by apply-

ing different control and detection techniques is often considered in the development

of artificial pancreas.

Contents

Abstract 1

Table of Contents 3

List of Tables 6

List of Figures 7

1 Introduction 9

1.1 Physiology of Glucose Insulin System . . . . . . . . . . . . . . . . . . 11

1.1.1 Pancreas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.1.2 Insulin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.1.3 Glucose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.1.4 Glucagon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2 Diabetes Mellitus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2.1 Type 1 Diabetes Mellitus . . . . . . . . . . . . . . . . . . . . . 14

1.2.2 Type 2 Diabetes Mellitus . . . . . . . . . . . . . . . . . . . . . 14

1.3 Treatment of Type 1 Diabetes Mellitus . . . . . . . . . . . . . . . . . 14

1.3.1 Insulin Administration . . . . . . . . . . . . . . . . . . . . . . 15

1.3.2 Blood Glucose Measurements . . . . . . . . . . . . . . . . . . 15

1.4 Diabetes Treatment Methods . . . . . . . . . . . . . . . . . . . . . . 16

1.4.1 Standard Therapy . . . . . . . . . . . . . . . . . . . . . . . . 16

1.4.2 Basal Insulin . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.4.3 Bolus Insulin . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.4.4 Introduction to Systems and Control . . . . . . . . . . . . . . 17

1.4.5 Open-Loop Control . . . . . . . . . . . . . . . . . . . . . . . . 17

1.4.6 Closed-Loop Control . . . . . . . . . . . . . . . . . . . . . . . 18

1.5 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3

1.6 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.7 Organization of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2 Controllability and Observability 23

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.1.1 Linear Control System . . . . . . . . . . . . . . . . . . . . . . 23

2.1.2 Linear Singular Control System . . . . . . . . . . . . . . . . . 25

2.2 Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2.2 Linear, time-invariant control system . . . . . . . . . . . . . . 28

2.3 Observability (Reconstructibility) . . . . . . . . . . . . . . . . . . . . 29

2.3.1 Observability of Linear, time-invariant control system . . . . . 31

3 Glucose Insulin System 34

3.1 Augmented Minimal Model . . . . . . . . . . . . . . . . . . . . . . . 34

3.1.1 Linear Control System: . . . . . . . . . . . . . . . . . . . . . . 36

3.2 Reduced Meal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3 Linear Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.3.1 Controllability and Observability . . . . . . . . . . . . . . . . 42

3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4 Fractional Order Glucose Insulin Model 44

4.1 Fractional Order Glucose Insulin Model . . . . . . . . . . . . . . . . . 44

4.1.1 Stability Analysis and Equilibria . . . . . . . . . . . . . . . . 45

4.2 Laplace Adomian Decomposition Method . . . . . . . . . . . . . . . . 47

4.3 Case I for Normal Person . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.4 Case II for Type 1 Diabetes . . . . . . . . . . . . . . . . . . . . . . . 50

4.5 Numerical Results and Discussion . . . . . . . . . . . . . . . . . . . . 51

4.6 Input and Output Stability . . . . . . . . . . . . . . . . . . . . . . . 55


4.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5 Stability Analysis and Control of GIG System 58

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.3 Stability Theorem: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.4 Linear Control System . . . . . . . . . . . . . . . . . . . . . . . . . . 62


5.5 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6 Sorenson Model for Type 1 Diabetes Mellitus 71

6.1 Sorensen’s Model For Type 1 Diabetes . . . . . . . . . . . . . . . . . 71

6.1.1 Description of Variables . . . . . . . . . . . . . . . . . . . . . 74

6.1.2 First Subscript: Physiological Compartment . . . . . . . . . . 75

6.1.3 Second Subscript: Physiological Compartment . . . . . . . . . 75

6.1.4 Metabolic source and sink . . . . . . . . . . . . . . . . . . . . 76

6.2 Modified Form of Model in Type 1 Diabetes Mellitus . . . . . . . . . 79

6.2.1 Linearised Model . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.2.2 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 82

6.3 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.3.1 Case I: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.3.2 Case II: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

7 Control of Composite Model 86

7.1 Material and Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

7.1.1 Composite Model Glucagon-Glucose Dynamics . . . . . . . . . 87

7.2 Extended Form of Composite Model . . . . . . . . . . . . . . . . . . 89

7.2.1 Gastrointestinal Absorption Model . . . . . . . . . . . . . . . 90

7.2.2 Subcutaneous Insulin Absorption Model . . . . . . . . . . . . 90

7.2.3 Subcutaneous Glucagon Absorption Model . . . . . . . . . . . 91

7.2.4 Linearized System . . . . . . . . . . . . . . . . . . . . . . . . 94

7.3 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

7.4 Linear Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

7.4.1 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 95

7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

8 Conclusion and Future Work 99

Bibliography 102

List of Tables

3.1 Table of Parameter’s used in the Augmented Minimal Model . . . . . 35

3.2 Table of Parameters used in the Reduced Meal Model . . . . . . . . . 39

4.1 Table of Parameter’s used in Sandhya Model . . . . . . . . . . . . . . 45

5.1 Table of parameter’s value used in the model . . . . . . . . . . . . . . 60

6.1 Description of variables . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.2 Physiological Compartment . . . . . . . . . . . . . . . . . . . . . . . 75

6.3 Physiological Compartment . . . . . . . . . . . . . . . . . . . . . . . 75

6.4 Table of parameter’s and constant value of the model . . . . . . . . . 76

7.1 Table of parameters and constant value of the model . . . . . . . . . 92

6

List of Figures

1.1 Physiology of Diabetes Mellitus . . . . . . . . . . . . . . . . . . . . . 11

1.2 Illustration of open-loop control for T1DM treatment. . . . . . . . . . 18

1.3 Illustration of closed-loop control for T1DM treatment . . . . . . . . 18

2.1 Idealized diagram of a control system . . . . . . . . . . . . . . . . . . 24

2.2 control system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.1 Controllability and observability Graph . . . . . . . . . . . . . . . . . 37

3.2 Controllability and observability Graph . . . . . . . . . . . . . . . . . 38

4.1 Numerical solution of Glucose level of normal person . . . . . . . . . 52

4.2 Numerical solution of Glucose level of type 1 diabetes . . . . . . . . . 53

4.3 Behavior of insulin in normal person . . . . . . . . . . . . . . . . . . 53

4.4 Behavior of insulin in type 1 diabetes . . . . . . . . . . . . . . . . . . 54

4.5 Numerical solution of insulin concentration in plasma of normal person 54

4.6 Numerical solution of insulin concentration in plasma of type 1 diabetes 55

5.1 Controllable and observable state of model, when insulin is an input

at initial condition (80,21,900, 300,0.16) . . . . . . . . . . . . . . . . 64

5.2 Controllable and observable steady state of model, when insulin is an

input at initial condition (120,10,5, 3,0.1) . . . . . . . . . . . . . . . . 64

5.3 Controllable and observable state of model, when insulin and glucagon

are an input at initial condition (120,21,900, 300,0) . . . . . . . . . . 66

5.4 Controllable and observable state of model, when insulin and glucagon

are an input at initial condition (80,0,900, 300,0.16) . . . . . . . . . . 66

5.5 Simulink to measure Glucose, insulin and glucagon with effect of β-

cell, α-cell mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.6 Pole zero diagram of transfer function G(s) . . . . . . . . . . . . . . . 69

5.7 Frequency response transfer function G(s) . . . . . . . . . . . . . . . 69

7

6.1 Schematic representation of the Glucose Model . . . . . . . . . . . . . 72

6.2 Schematic representation of the Insulin Model . . . . . . . . . . . . . 72

6.3 Schematic representation of the Glucagon Model . . . . . . . . . . . . 73

7.1 The glucagon extended Minimal model . . . . . . . . . . . . . . . . . 87

Chapter 1

Introduction

Go down deep enough into anything and you will

find mathematics.(D Schlicter)

Mathematics is the supreme judge; from its decisions

there is no appeal.(Tobias Dantzig)

God does not care about our mathematical difficulties;

He integrates empirically.(Albert Einstein)

Mathematics has always advantages from its participation with developing sci-

ences. Each consecutive interface invigorates and improves the field. Biomedical

science is obviously the premier science of the expected future. For the health of

their subjects mathematicians have to be concerned with biology. Mathematical

biology is a fast-growing, well-known, albeit not evidently defined, subject and for

my mind, the most thrilling modern application of Mathematics. The rising use of

Mathematics in biology is predictable as biology becomes more quantitative. The

best models show how a method works and then forecast what may follow. If these

are not previously clear to the biologists and the predictions turn out to be right,

then you will have the biologist’s attention. Authentic interdisciplinary research and

the exploit of models can produce thrilling results, many of which are described [1]

The group of metabolic disorders which is caused by high blood sugar level that is

due to defects in insulin secretion, action or both. Normally, insulin controles glucose

and blood sugar level is lowered by insulin. Insulin is released from the pancreas,

when glucose level elevates from the normal level. There is no cure of diabetes but it

can be manageable. After cardiovascular and cancer, diabetes is the third fatal dis-

ease. In diabetes, insulin is not properly produced or not properly done its function.

A lack of insulin secretion or lack in its function causes impaired carbohydrate, fat

9

CHAPTER 1. INTRODUCTION 10

and glucose known as diabetes mellitus. Diabetes mellitus has two types. Insulin

dependent diabetes mellitus (IDDM) is commonly known as type 1 diabetes melli-

tus, in this deficiency of insulin secretion is present. Noninsulin dependent diabetes

mellitus (NIDDM) is commonly known as type 2 diabetes mellitus, in this insulin

is properly produced but not properly work. Type 1 diabetes is caused by impair

insulin in beta cells of pancreas [2, 3]. This insufficient level of insulin is unable

to regulate the blood glucose level to normalize the body’s hormonal system: when

someone intakes food the blood glucose level rises even more. This type of diabetes

appears with the symptoms of urination, tiredness, thirst, hunger and weight loss.

The treatment of this type is done by injecting insulin into the body, exercising and

by taking healthy food. Patients suffering from type 1 diabetes are totally dependent

on insulin injections, and, if not treated on the initial stage patients will die, because

the body cannot handle high level of glucose by itself. Type 2 diabetes is not as

common as type 1 diabetes. Patients of this type are known as insulin resistant.

Symptoms of this type appear to be same as type 1 diabetes. With the passage of

time, in patients suffering from this type of diabetes cells start to decrease and the

patients are treated like type 1 diabetes by using insulin injections. It is a condition

of diabetic patients when the blood glucose level rises above 270mg/dL and it can be

increased by eating a heavy meal or having a very low amount of insulin in the blood

to regulate the system. This serious condition is very dangerous for the patient’s

survival. A diabetic patient with a very low amount of glucose in their blood (up

to 60 mg/dL or below) causes a condition called hypoglycemia. It is caused when

the patient taking insulin treatment skips their proper meal or has laborious work

or heavy exercise and it is also very dangerous for the patients [4, 5].

In the 1970’s, the first experiment with an artificial pancreas (AP) was made with

a large device with beneficial limitations. Over the last decade, several clinical studies

were made. Insulin pumps have been used to administrate subcutaneous (SC) insulin

and subcutaneous continuous glucose monitoring (CGM) with enzymatic technology.

An artificial pancreas in glucose measurement and insulin infusion occurring in the

peritoneal cavity [6]. The artificial pancreas (AP) or automated control system has

been developed by researchers during the last decades [7]. Continuous insulin dose is

allowed by a continuous subcutaneous insulin infusion systems (CSII). The missing

feedback of glucose sensing has a fundamental drawback for closed loop control. The

idea of closed loop control is practically achieved by the development of continu-

ous glucose monitoring (CGM). Many plans were made with the available feedback

among others, proportional integral design control [8], adaptive control and fuzzy


logic control. The model predictive control is the most widely used control approach,

because of its ability to elegantly handle a broad range of scheme constraints. It is

still challenging to overcome the problems of insulin regulation in artificial pancreas

research. The main goal of an artificial pancreas system is the prevention of and

safe recovery from hypoglycemia episodes [9], Insulin-glucose dynamics of healthy

subjects as well T1DM and T2DM, with appropriate adjustment in its parameters

for glycaemia control [3, 10].

1.1 Physiology of Glucose Insulin System

This section presents the elements involved in the glucose regulation system in dia-

betes. First, the mechanism of glucose regulation in healthy person is summarized

then difference between healthy person and diabetic person explained in order to

indicate impermanence of required control scheme. Blood sugar level is balanced by

insulin and glucagon in the human body. Insulin and glucagon is known as pancre-

atic endocrine hormones because they are secreted by pancreas. The relationship of

insulin and glucagon is shown in the below figure. Diabetes, hypoglycemia and other

sugar problems are due to imbalanced pancreatic hormones.

Figure 1.1: Physiology of Diabetes Mellitus


Insulin and glucagon are hormones secreted by islet cells within the pancreas.

They are both secreted in response to blood sugar levels, but in opposite fashion.

Insulin is normally secreted by the beta cells (a type of islet cell) of the pancreas.

The stimulus for insulin secretion is a high blood glucose. Although there is always

a low level of insulin secreted by the pancreas, the amount secreted into the blood

increases as the blood glucose rises. Similarly, as blood glucose falls, the amount of

insulin secreted by the pancreatic islets goes down. As shown in figure 1.1, insulin

has an effect on a number of cells, including muscle, red blood cells, and fat cells. In

response to insulin, these cells absorb glucose out of the blood, having the net effect

of lowering the high blood glucose levels into the normal range. Glucagon is secreted

by the alpha cells of the pancreatic islets in the same manner as insulin but in the

opposite direction. If blood glucose is high, then no glucagon is secreted. When

blood glucose goes low, (such as between meals, and during exercise) more glucagon

is secreted. Like insulin, glucagon has an effect on many cells of the body, but most

notably the liver [11] .

1.1.1 Pancreas

The pancreas is present near the stomach, help in absorption of the food by making

different enzymes which work on large particles of the food and break it into smaller

particles. The main enzymes are lipase, amylase and trypsin. These enzymes are

made into small glands within pancreas. These enzymes reach into main pancreatic

tube after traveling through large tubes. Pancreatic tube connects gland and bowel.

Thus food from the stomach and enzymes from the pancreas meet with each other

through pancreatic tube. Stomach juices activate pancreatic enzymes. Gastric juice

helps lipase in fat digestion.

1.1.2 Insulin

In 1922, first time insulin separated from pancreas by Banting and Beast. Insulin

is related to blood sugar. Insulin effects carbohydrate mechanism. Beta cells of

pancreas made hormone insulin that helps glucose to enter in cells and help body to

use glucose. Insulin has two types of function. One is normal function and other is

impaired function. Normal function of insulin helps in clearing of glucose. Glycogen

is made by glucose and store in liver and muscle cells with the help of insulin. But

in impaired function blood stream is not cleared from glucose and glycogen is not

formed by glucose [2].


1.1.3 Glucose

Energy is provided by glucose (sugar) to the whole body including brain cells. Com-

mon sources of glucose is fruit, bread, pasta and cereals. Food is converted into

glucose in stomach then glucose enters into bloodstream. The value of glucose of

fasting glucose level is 100 mm per deciliter. The value of random glucose level is

140 mm per deciliter. Increase in normal level of glucose is regulated by insulin.

Decrease in normal level of glucose is regulated by glucagon [12].

1.1.4 Glucagon

Glycogen is a form of stored carbohydrate which is released from the liver by the

hormone named glucagon and so the blood glucose rises in the body. The glucagon

called the life saver as it control the blood sugar level of the blood. The function

of glycogen is to produce glucose in the blood. The main producer is comprise on

pancreas, liver and kidney. The glycogen hormone generated by pancreas secrete

glycogen that is accumulated in the form of glucose in liver. When the glucagon

receptors of liver cells binds with glucagon they change glucagon to glucose molecule

and then liver allow them to enter in the blood stream. When the concentration

of the glucose decrease into the body, glucagon encourage the liver and kidney to

synthesize more glucose in the blood to maintain the glucose level in it [12].

1.2 Diabetes Mellitus

Diabetes mellitus is a Greek word which means the excess of sugar. Diabetes mean

run through and mellitus mean honey. Mostly it can be diagnosed from the blood test

or urine test as the urine of patient become sweet, but sometime it gets difficult to

diagnose. Where diagnostic complexity exists, the accurate diagnosis can be created

with an oral glucose tolerance test (OGTT) using a 75 g anhydrous glucose load

melted in water a 2 hour value ≥ 200mg/dl instituted the analysis of diabetes. A

positive test utilizing one or other of the three method should be utilized [14].

Main diabetes is of two types

1. Type 1 Diabetes Mellitus(insulin dependent)

2. Type 2 Diabetes Mellitus(insulin independent)

Approximately 10% patient of type I. In type I hyperglycemia is at low level of

insulin and type II it is the resistance of insulin. In both cases the food stuff alters


most of the cells except brain stop using glucose and the glucose level increases in

the body and cells use fats and protein instead of glucose.

1.2.1 Type 1 Diabetes Mellitus

It is autoimmune disease in which the immune system attacks the pancreas and

destroys its cells thus pancreas stops producing insulin or its production decreases.

In T1DM, the immune system attacks the insulin-producing beta (β) cells in the

pancreas and obliterates them. The pancreas then creates little or no insulin. Ev-

erybody who is a patient of type, I need to take insulin regular basis. Scientist is

stir helpless to find the real cause of the body’s immune system to attack the beta

cells (β). They believe that this may be because of specific virus or genetic system.

The destruction of beta cells starts very earlier before the symptoms type 1 diabetes

start. This kind of diabetes mostly attack in old age. The causes of type 1 diabetes

have inherent, obesity , previous history of gestational diabetes, physical inactivity

and eating [2, 13].

1.2.2 Type 2 Diabetes Mellitus

It is common form of diabetes in which the production of insulin is normal but the role

is not effective. As body is not able to utilize the insulin due to unknown reasons.

After many years, the production of insulin in the body decreases, and person is

effected by the diabetes of T1DM and the some symptoms start appearing in the

body. It is seen that the diabetes II is caused by the genetic disorder, still the exact

reason is not known. Concordance rate in identical twins in more than non-identical

twins and it is noticed that those who have no diabetic history in their family are

much secure of this disease. The development of symptoms of T2DM are fatigue,

nausea, unusual thirst, weight loss, blurred or weak vision, frequent infection. Some

time its symptoms are not mark able even in patient [13]

1.3 Treatment of Type 1 Diabetes Mellitus

Artificial pancreas is a innovation to preserve the typical blood glucose level in dia-

betes with a substitute endocrine work to pancreas. The task is lacking to oversee

physically the blood glucose level with alone that is why the current treatment of

affront substitution (Artificial Pancreas) is appreciated for its life sparing capability.


This treatment can offer assistance in the hyperglycaemia state by catapulting more

affront by the affront pump but in case of hypoglycaemia state this treatment will

not work. Hypoglycaemia leads to neuroglycopenia and impacts can run from gentle

dysphoria to more serious issues such as seizures, obviousness, harmful for brain cells

and death.

1.3.1 Insulin Administration

Affront can be managed by several implies. In spite of the fact that syringes have

been utilized for a long time for injecting insulin boluses (single doses), they are

broadly supplanted by affront pens. For the final decades, the utilize of affront

pumps has ended up more and more widespread. These gadgets permit nearly

persistent affront implantation by giving boluses up to each miniature. Affront may

be managed (i) subcutaneously, i.e. underneath the skin, (ii) intraperitoneally,i.e.

into the membrane of the stomach depression, or (iii) intravenously, i.e directly into

the veins. The SC course is the standard for commercial affront pumps because of

the moo hazard of contaminations, but has the downside of relatively slow affront

take-up times. Since quick affront activity decreases the amplitude postprandial BG

trips (as will be shown later), speedier IP conveyance is being inquired about and

appears promising out comes, but with the complications [15]. (iv) mixture is the

speediest as it is to closest.

1.3.2 Blood Glucose Measurements

Exact blood glucose measurements are key for fitting treatment and shirking of

hypoglycemia. Two fundamental strategies are commonly being utilized: Self

Checking of Blood Glucose (SMBG) and Nonstop Glucose Observing. SMBG

comprises in measuring the glucose concentration in a little drop of blood gotten

by pricking the finger with a lancet. This strategy is by distant the most common

since of its relatively great precision at sensible fetched. The greatest disadvantage

of this method is that for each estimation, the persistent needs to extricate a blood

drop - a excruciating method. As come about most patients do not take blood

glucose estimations exceptionally habitually. Persistent Glucose Observing (CGM)

devices are an elective that gives nearly nonstop BG concentrations with less finger

pricks, at the cost of decreased precision and reliability. Also, these gadgets are


generally costly and have a time-lag that can be dangerous. These impediments

clarify its moderate movement on the advertise. This work considers both types of

measurements.

1.4 Diabetes Treatment Methods

Different approaches used for T1DM treatment, ranging from currently applied

methods to active research fields, are introduced in literature. Followings are

1.4.1 Standard Therapy

Currently, standard therapy as it will be called in this thesis is the norm when it

comes to T1DM treatment. This therapy is also referred to as basal/bolus therapy

or Multiple Daily Injections (MDI), if performed using insulin pens or syringes.

1.4.2 Basal Insulin

Basal insulin is insulin that acts relatively uniformly throughout the day and

should keep patients fasting BG concentration close to the optimum. Patients using

syringes or pens inject long-acting insulin once or twice a day while CSII-treated

patients use the insulin pump to adjust the basal rate in an ”optimal” manner. A

good overview of CSII treatment is given in [16].

1.4.3 Bolus Insulin

Bolus affront is affront that is injected in arrange to check the impact of dinners. The

carbohydrates contained in dinners are handled by the stomach related framework

and discharge glucose into the bloodstream. In arrange to dodge hyperglycemia, this

major unsettling influence needs to be checked by infusing a well-chosen amount of

fast-acting insulin using a syringe, write, or affront pump. This amount is based

on the quantity of ingested carbohydrates and the pre-meal BG concentration. To

compute the correct insulin sum, the understanding has to take an SMBG estimation

some time recently each meal [17].


1.4.4 Introduction to Systems and Control

In the setting of control, a framework, spoken to in figure 1.1, is an object of intrigued

(it can be numerous diverse things) upon which diverse actions can be taken - the

inputs u and that appears or gives diverse responses - the yields y. The inputs

are characterized by the reality that they can be manipulated from exterior the

framework, while the yields are characterized by the property that they can be

watched from exterior the framework. Additionally, disturbances may apply to the

frameworks. These are for the most part obscure, but have a quantifiable impact on

the yields. For case, in this proposal, the system is portion of the human endocrine

framework, the inputs are basically insulin injection and supper admissions, the yield

is BG concentration, and the disturbance is the estimation clamor or other obscure

excitations that have an impact on BG concentration.

Often, the behavior of a system is studied and described mathematically. This

description is called a model, and it should reproduce the outputs of a system,

based on the inputs, as accurately as possible. However, quite often, models are

not capable of capturing the whole behavior of a system, either because it is too

complex, or disturbances are too important. A system is called static if its outputs at

a given time are influenced by the inputs at that time, only. In a dynamical system,

however, the outputs are determined by current and past inputs. A controller is

used to adjust a systems inputs, in order to obtain desired outputs. This system is

called controlled system. A controller itself can be considered as a system, whose

output is the controlled systems input. If the controllers inputs depend directly on

the controlled systems outputs, then the controller is called a closed-loop controller,

otherwise it is called an open-loop controller. The output value that a closed-loop

control algorithm is intended to reach is called a set point [18].

1.4.5 Open-Loop Control

Open-loop control In control theory, an open-loop controller is a controller that com-

putes system inputs based on the current system state and a model in control theory.

In the context of T1DM treatment, open-loop control means that future insulin in-

fusions are computed using current BG measurements. Hence, standard therapy is

a good example of open-loop control applied at every SMBG measurement and us-

ing a simple static model for BG prediction. However, other implementations than

standard therapy exist for open-loop control and the use of different BG prediction

and state estimation methods may improve treatment. These improvements should


result in reduced hypo- and hyperglycemia. Open-loop control is currently not a very

active field of research, despite its potential improvements over standard therapy.

Figure 1.2: Illustration of open-loop control for T1DM treatment.

1.4.6 Closed-Loop Control

In closed-loop control, a nonstop or habitually inspected estimation is used to com-

pute the framework input ceaselessly or at the same inspecting rate, respectively.

For closed-loop T1DM treatment, persistent estimations, i.e. a CGM gadget, is re-

quired. The coming about criticism structure, appeared in figure 1.2, possibly leads

to sensational execution enhancements and better disturbance dismissal, in spite

of the fact that ensuring quiet security is still an open issue. The objective of a

closed-loop treatment is to replicate the behavior of a healthy pancreas as near as

conceivable while minimizing quiet involvement. Therefore, it is too alluded to as

the Fake Pancreas (AP) [19].

Figure 1.3: Illustration of closed-loop control for T1DM treatment

1.5 Motivation

Diabetes is a disease with an enormous human and economic impact, but its current

treatment is suboptimal, as it does not fully embrace the possibilities offered by

insulin pumps and CGM devices. Thus, research to improve the treatment has a lot of


potential to positively affect patient’s life while reducing the health care burden. For

these reasons, this thesis aims at making the treatment of patients with T1DM more

successful. An adequate control calculation must be competent of dealing with these

physiological and specialized challenges while still giving satisfactory performance.

Currently, the fundamental challenges is that the advance of manufactured pancreas

faces are the advancement of a solid closed-loop control calculation and availability

of a strong and exact glucose sensor [20].

The issue is to discover a criticism control for the framework so that it can work

automatically. The address is that can we control states of the framework so that

we can impact the framework by choosing a criticism control. For the purpose we

require to check the property, controllability, of the mathematical models for the

GIG frameworks. At that point the next address is to know that what kind of data

we can have from the yield of the framework. The property to answer this address

is the discernible of the framework. With respect to a control system there are two

questions one ordinarily came over. To begin with one is that in what way can we

impact the framework by choosing a fitting control? Second is what data almost the

framework can we get from the yield of the framework? The concept of controllability

answer the first question where as the observability concept answer the second one.

Controllability and observability are main issues in the analysis of a system before

deciding the best control strategy to be applied, or whether it is even possible to

control or stabilize the system. Controllability is related to the possibility of forcing

the system into a particular state by using an appropriate control signal. If a state is

not controllable, then no signal will ever be able to control the state. Observability

instead is related to the possibility of observing, through output measurements, the

state of a system. If a state is not observable, the controller will never be able to

determine the behaviour of an unobservable state and hence cannot use it to stabilize

the system [21, 22, 23, 13].

Refinements in calculations (relative indispensably subordinate, show predictive

control, fluffy rationale), stages (manufactured pancreas framework, Diabetes Assis-

tant), nonstop glucose screens (CGMs) and affront pumps, along with continuing

miniaturization and movability of these gadgets, have enhanced the common sense

and convenience of modern close-loop control frameworks by adults and teenagers

with sort 1 diabetes mellitus [24]. Furthermore, recent studies [25] incorporating

both insulin and glucagon infusions have extended the entire concept from an arti-

ficial β-cell closer to an artificial endocrine pancreas system [26].


1.6 Objectives

The goal of the present work is to get the answer of the question that why we still

can not have a fully automated artificial pancreas? In this work we tried to find the

reason of the problem of having no fully automatic artificial pancreas. The first step

in this direction is to find that do we have a controllable and observable system for

GIG. For the purpose some of the well known GIG systems are treated. If a linear

system is controllable then the nonlinear system is locally controllable but if the

linear system is not controllable then in general no conclusion can be drawn for the

nonlinear system. The most of the systems for GIG are nonlinear due to the action

of glucose, insulin and glucagon thus the systems treated here are linearised about

the equilibrium point. If system is controllable and observable then we can design a

feedback control. For an uncontrollable systems an attempt is made to answer the

reason on the basis of our results.

The aims of this research project are to implement and assess several differential

methods, with emphasis on procedures that do not require gradient information.

These techniques are referred to as derivative the mathematical solution of glucose

insulin-glucagon system and fully automatic artificial pancreas for diabetic patients.

Because other way of treatment like transparent of pancreas or liver is the chance

of infection, costly and very difficult process. The existing artificial pancreas is very

expensive and not reliable for every patient because it is not fully controllable. Imple-

ment the model of differential equation and Model of non-liner ordinary differential

equations for possible solutions.

1.7 Organization of Thesis

The thesis is organized as follows

Chapter 1: This chapter is an introductory chapter. The physiological background

is provided to familiarize the reader with subject. The general background of the

work, the research motivation, thesis objective and outline are given in this chapter.

Chapter 2: This chapter contains the basics about the linear control systems for

linear, continuous time control systems and linear time invariant control systems.

Chapter 3: This chapter contains the glucose insulin models. Controllability and

observability are discussed the Augmented minimal model and Reduced meal model

for type 1 diabetes mellitus for an artificial pancreas. These models can be used to

simulate a glucose insulin system for the treatment of type 1 diabetes and stabilizes


the control loop system for the glucose insulin pump.

Muhammad Farman, M.U Saleem, Control of Augmented minimal model

for glucose insulin pump, Current Research in Diabetes and Obesity

Journal, DOI: 10.19080/CRDOJ.2018.08.555742 (2018).

Chapter 4: In this chapter, we proposed the fractional order glucose insulin model

for healthy and type 1 diabetes and numerical simulations are carried out to support

the analytical results. Also discussed the controllability and observability of the

linearized system to design the close loop for automatic artificial pancreas.

M.U Saleem, M. Farman, Aqeel Ahmad, M.A Meraj Stability Analysis

and Control of Fractional Order Diabetes Mellitus Model for Artificial

Pancreas, Punjab University mathematics journal, 51(4),97-113, (2019)..

Chapter 5: Stability analysis of the model of glucose-insulin and glucagon system

in humans is made which is one of the important factors of study for healthy life.

We show the numerical simulation of the model for type 1 diabetes mellitus for

the controllability and observability matrix according to different initials conditions

on state vector. Model is used for this purpose and consists of glucose, insulin

and glucagon function in human body. Equilibrium points for different levels of

concentration of glucose are calculated. Proportional-Integral-Derivative (PID)

controller designed for an artificial pancreas. This paper is published

Muhammad Farman, Muhammad Umer Saleem, M. O Ahmad, Aqeel

Ahmad, Stability Analysis and Control of Glucose Insulin Glucagon

System in human, Chines Journal of Physics, 56 (4), 1362-1369, (2018).

Chapter 6: chapter 6 includes the Glucose insulin glucagon model like Sorensons

model for type 1 diabetes mellitus. Models are used for this purpose and consists

of glucose, insulin and glucagon function in human body. Equilibrium points for

different case of concentration of glucose are calculated by using Mathematica

software for stability of the system. Results are refined by using Jacobean linearized

method to check the stability of the model to design feedback control for artificial

pancreas. Publication include in this chapter are

Saleem M.U., Muhammad Farman, Meraj M.A., Stability Analysis

of Sorensen’s Model for controllability and observability, B. Life and

Environmental Sciences, 54 (2): 133145, (2017).

Muhammad Farman, M.U Saleem, M.O. Ahmad, An Approach for


Artificial Pancreas to Control the Type-1 Diabetes Mellitus, Journal of

Applied Environmental and Biological Sciences, 93-102, (2017).

Chapter 7: In this chapter composite model and its extension is treated for

type 1 diabetes. Equilibrium points for different case of concentration of glucose

are calculated and check the stability analysis of the system by using Lyapunov

function. Also checked the controllability and observability of the linearized system

to design the close loop for automatic artificial pancreas.

M.U Saleem, Muhammad Farman, Rizwan M., M.O. Ahmad, Aqeel

ahmad, Controllability and Observability of Glucose Insulin Glucagon

systems in Human , Chines journal of Physics, ,56(5), 1909-1916, (2018).

Chapter 8: Finally, a conclusion is drawn in chapter 7 and an outlook on possible

future work is given.

Chapter 2

Controllability and Observability

2.1 Introduction

A control system design deals with the problem of making a concrete physical system

behave according to certain desired specifications. The ultimate product of a control

system design problem is a physical device that, if connected to the to be controlled

physical system, makes it behave according to the specifications. This device is

called a controller. To get from a concrete to be controlled physical system to a

concrete physical device to control the system, the following intermediate steps are

often taken. First, a mathematical model of the physical system is made. Such a

mathematical model can take many forms. For example, the model could be in the

form of a system of ordinary and/or partial differential equations, together with a

number of algebraic equations, relating the relevant variables of the system. The

model could also involve difference equations, some of the variables could be related

by transfer functions, etc. The usual way to get a model of an actual system is

to apply the basic laws that the system satisfies. Often, this method is called first

principles modeling [27].

2.1.1 Linear Control System

Mathematically a linear control system is written in the form of following two equa-

tions,

x(t) = D(t)x(t) + E(t)u(t), t ∈ I (2.1)

y(t) = F (t)x(t), t ∈ I (2.2)

23

CHAPTER 2. CONTROLLABILITY AND OBSERVABILITY 24

Where x(t) ∈ Rn, u(t) ∈ Rp and y(t) ∈ Rk for t ∈ I. The matrices D(t) ,

E(t) and F (t) are defined on I and have correct dimensions (i.e., D(t)) is n × n, E(t) is n × p and F (t) is k × nmatrix) . I is closed interval, I = [to, te],

t0 < te < ∞ , respectively. I = [t0,∞). We suppose the elements of the matrices

D(.), E(.) and F (.) are in L2(I;R).

The function u(.) is also suppose to be in L2(I;Rp) is called the input respectively

the control of the system. For given initial value x0 ∈ Rn and input u(.) ∈ L2(I;Rp)

of equation x(t) = D(t)x(t)+E(t)u(t), t ∈ I also called the state equation of system.

This system has a unique solution of x(.),in the sense of Caratheodory, i.e., x(.) is

absolutely continuous on I with x(t0) = x0 and the derivative x(.) is exist almost

every where on I. Furthermore equation (2.1) is satisfied on I , x(t) , t ∈ I is called

state of system at time t. If we have the solution of x(.) of equation (2.1) with initial

value xo and equation (2.2) establish y(.) ∈ L2(I;Rk) is called output of the system.

Figure 2.1: Idealized diagram of a control system

Figure 2.2: control system

In general we have not access the state x(.) itself but only some function of

the state. We can think the coordinate y(.) as those quantities of the system we

can measure. The diagram given in Figure (2.1)and (2.2) represents an idealized


situation. In reality we have distinguish between the system (or plant) and model

for the system, which is given by equation (2.1) and (2.2) for instance. The real

situation is representation in Figure 3.2. Usually one expects that the model for the

system describe the dynamics of the system sufficiently well, so that input-output

behavior of the model is sufficiently close to that of the system.

The output-equation (2.2) can also have the more general form

y(t) = F (t)x(t) +G(t)u(t), t ∈ I,

Which reflects a situation where the input influences the output also directly.

2.1.2 Linear Singular Control System

The equation

H(t)x(t) = D(t)x(t) + E(t)u(t), t ∈ I,

Where H(t) is an n × n matrix which is not invertible everywhere on I. such

system are called linear descriptor or linear singular control system.

In the system equation (2.1), (2.2) is called linear continues time system or (con-

trol system). The time set for such a system is an bounded or unbounded close

interval in R. A class of system are discrete time systems, where time set is finite

and infinite sequence t0 < t1 ... < te respectively t0 < t1 ... < te , and instead of

equation (2.1) and (2.2) we have

x(tk+1) = D(tk)x(tk) + E(tk)u(tk), k = 1, 2, 3, ...,

y(tk) = F (tk)x(tk), k = 0, 1, 2, 3, ...,

The main part of this work will be concerned with time-invariant , continuous-

time systems.This class of system is the special case of class characterized by

equation (2.1) and (2.2) where the matrices are constant in time:

x(t) = Dx(t) + Eu(t), t ∈ R (2.3)

y(t) = Fx(t), t ∈ R (2.4)

Lemma 1. a) Let X be a subspace of Rn. Then we have

Rn = X ⊕ X⊥.


b) For a subspace X1, X2 of Rn the following is true:

(X1 ∩X2)⊥ = X⊥1 +X⊥2 .

Let S be a real m× n matrix. Then we have

ker S = (range S>)⊥

(and consequently Rn = rangeS> ⊕ kerS).

2.2 Controllability

We denote x(.; t0, x0, u(.)) is the solution of equation (2.1) with initial data (t0, x0)

and input function u(.). Let φ(t, s) t, s ∈ I, denote the matrix solution of the

homogenous equation

x = D(t)x(t)

Then we have

x(t; t0, x0, u(.)) = φ(t, t0) +∫ t

t0φ(t, τ)dτ t ∈ I (2.5)

2.2.1 Definition

Assume that the control system is given by equations (2.1) and (2.2)

a) Let x0, x1 ∈ Rn and t0 ∈ I be given. The state x0 is at time t0 controllable

to x1, if and only if there exit t1 > t0 and a u ∈ L2([t0, t1];Rp) such that

x(t; t0, x0, u(.)) = x1

b) System (2.1) and (2.2) is completely controllable at t0, iff every state x0 ∈ Rn is

at time t0 ∈ I controllable to any state x1.

A state x1 is at t1 ∈ I reachable from x0 if and only if t0 < t1 ∈ I and a

u ∈ L2([t0, t1];Rp) such that x(t; t0, x0, u(.)) = x1. The System (2.1) and (2.2) is

completely reachable at time t1 ∈ I if and only if there exist a t0 < t1, t0 ∈ I, such

that every x1 ∈ Rn ia at time t1 reachable from every other state x0 ∈ Rn.

Using equation (2.3), x0 is controllable to x1 at time t0 iff t1 > t0 and a

u ∈ L2([t0, t1];Rp) such that

φ(t1, t0)x0 +∫ t

t0φ(t1, τ)E(τ)u(τ)dτ = x1


or equivalently

−x0 + φ(t0, t1)x1 =∫ t

t0φ(t0, τ)E(τ)u(τ)dτ.

Theorem 1. a) A State x0 is at time t0 controllable to the state x1 iff there exists

a t1 > t0x0 − φ(t0, t1)x1 ∈ range W (t0, t1),

where

W (t0, t1) =∫ t1

t0φ(t0, τ)E(τ)E(τ)>φ(t0, τ)>dτ

b) System (2.1) and (2.2) is completely controllable at some time t0 iff there exists

t1 > t0 such that

rangeW (t0, t1) = n (orequivalently range(t0, t1) = Rn).

Proof: The proof for a) follows directly from the previous lemma and the con-

sideration stated above that lemma. For the proof of b) let us first assume that

rankW (t0, t1) = n i.e., rangeW (t0, t1) = Rn. Then it follows immediately from a)

that any state x0 is at time t0 controllable to any x1 and in particular to 0. This

proves that the system is completely controllable at t0. We should remark that any

x0 can be controlled to any x1 on the fixed time interval [t0, t1].

If any x0 is at time t0 controllable to 0, then this is true for the elements b1, ..., bnof a basis for Rn, i.e., there exist ti > t0 and u ∈ L2([t0, ti];R

p) such that

bi = rangeW (t0, ti), i = 1, 2, ..., n,

If we can prove that rangeW (t0, s) ⊂ rangeW (t0, t) for t > s > t0, then

bi ∈ rangeW (t0, t1), i = 1, 2, ..., n. This implies rankW (t0, t1) = n. Now let

x ∈ rangeW (t0, s) and t > s. According to previous lemma this means that there

exists a u ∈ L2(t0, s;Rp) such that

x =∫ s

t0φ(t0, τ)E(τ)u(τ)d(τ) =

∫ t1

t0φ(t0, τ)E(τ)u(τ)d(τ)

Here u(τ) = u(τ) for t0 ≤ τ ≤ s and u(τ) = for s < τ ≤ t. Since u ∈ L2(t0, t;Rp)

we see, again using previous lemma, that x ∈ rangeW (t0, t) [30, 22, 13].


2.2.2 Linear, time-invariant control system

We consider controllability for a system given by the equation (2.3) and (2.4). The

fundamental matrix solution of the homogenous equation x(t) = Dx(t) + Eu(t), is

given by

φ(t, s) = eD(t−s) =∞∑k=0

1

k!Dk(t− s)k

If we take t0 = 0, Then controllability Grammian is given by

W (0, t) =∫ t

t0e−DτEE>e−D

>τdτ (2.6)

In view of Theorem 1 we are interested in range W (0.t)) for t > 0. Accord-

ing to (Lemma 1, c), we have range W (0.t) = (kerW (0, t))⊥. We first inves-

tigate kerW (0, t). Since W (0, t) is symmetric,x ∈ kerW (0, t) is equivalent to

x>W (0.t)x = 0. Using this and (2.2) we see that x ∈ kerW (0, t) if and only if

x>W (0, t)x =∫ t

t0x>e−DτEE>e−D

>τdτ =∫ t

t0‖E>e−D>τx‖2

2dτ

i.e.if and only if E>e−D>τx ≡ 0. by analyticity of τ → e−D

>τ this in turn is equivalent

to

E>(D>)kx = 0, k = 0, 1, ...

By Cayley-Hamilton theorem we can restrict k to k = 0, 1, 2, ..., n− 1 and get

kerW (0, t) =n−1⋂k=0

kerE>(D>)k for all t ∈ R.

Using Lemma 1 this implies

rangW (0, t) =n−1∑k=0

(kerE>(D>)k)⊥ =n−1∑k=0

rangeDkE for all t ∈ R.

From Theorem 1 we get

Theorem 2: Let control system be given by equation (2.3) and (2.4)

a) A State x0 is at time 0 (equivalently at any time t0 ∈ R)controllable to zero

if and only if

x0 ∈ range(E,DE, ..., Dn−1E)


b) System (2.3) and (2.4) is completely reconstructible at some time 0 (equivalently

at any time t0 ∈ R) if and only if

rank(E,DE, ..., Dn−1E) = n

Proof: We only have to verify that range

(E,DE, ..., Dn−1E) =n−1∑k=0

rangeDkE.

But this is clear, because range AkB is generated by the columns of AkB and if

a1, ..., ar, b1, ..., br generate X + Y .

If the rank condition in statement b) of the theorem is satisfied for the matrices

D and E, then we say that the pair (D;E) is controllable. Since the condition given

in statement b) of Theorem 2 is independent of t0 and t1, completely controllability

for a time-invariant linear system is equivalent to the following property: For any

x0, x1 ∈ R and any times t0 < t1 there exists a control u(.) such that the solution

of (2.3) with initial values (t0, x0) and control function u(.) reaches x1 at time t1.

Note that the the difference t1 − t0 can be arbitrary small. Of course, if t1 − t0is small, then the values of u(t) and also x(t) on the interval (t0, t1) can become large.

Since the conditions in Theorem 2 are independent of t0 and t1, it can be expected

that in this theorem we can replace ’controllable’ respectively ’completely control-

lable’ by ’reachable’ respectively ’completely reachable’. It is indeed an easy exercise

to prove the analogous results for ’reachable at time t1 from x0’ and ’completely

reachable’. The subspace

(E,DE, ..., Dn−1E) =n−1∑j=0

rangeDjE.

is called the controllable subspace for system (2.3), (2.4) [30, 22, 23].

2.3 Observability (Reconstructibility)

Let the control system is given by equations (2.1) and (2.2)


a) Let x0 ∈ Rn non-observable and t0 ε I if and only if there exit t1 < t0 ∈ I,

such that

F (t)x(t; t0, x0, u(.)) = F (t)x(t; t1, x0, u(.)) (2.7)

for all t ∈ [t1, t0 and u ∈ L2([t1, t0];Rp).

b) System (2.1) and (2.2) is completely observable at time t0 ∈ I, if and only if

0 is the only state which at time t0 non-observable.

A state x0 is non-observable at t0 ∈ I iff t1 > t0 , t1 ∈ I such that

F (t)x(t; t0, x0, u(.)) = F (t)x(t; t1, x0, u(.)) for all t ε [t0, t1] and a u ∈ L2([t0, t1];Rp).

The System (2.1) and (2.2) is completely observable at time t0 εI if and only if 0 is

the only state which at time t0 is non-observable.

Using the representation (2.5) then equation (2.6) equivalent to

F (t)φ(t; t0)x0 = 0, t ∈ [t1, t0] (2.8)

For t0, t1 ∈ I t1 < t0, we define the mapping Ft0,t1 : Rn → L2([t1, t0];Rk) by

Ft0,t1x = F (.)φ(., t0)x, x ∈ Rn (2.9)

Then condition (2.5) is equivalent to

x0 ∈ KerFt0,t1 (2.10)

If we choose complementary subspace X of KerFt0,t1 , i.e.,

Rn = X ⊕KerFt0,t1

, then any state x0 ∈ Rn has a unique representation as

x0 = x01 + x02, x01 ∈ X, x02KerFt0,t1

The component x01 is reconstructible at t0 (equivalently any time) in the following

sense: If, for some t∗ < t0, we have yt = Fx(t; t0, x01, 0) = 0 on [t∗, t0], then x01 =

0. Note, that by linearity of the system this is equivalent to Fx(t; t0, x0, u(.) =

Fx(t; t0, 0, u(.)), t ∈ [t∗, t0] for all u ∈ L2([t∗, t0];Rp). By linearity of the system

we could phrase this also as follow: Let x0, x1 be two states of the system. If,

for some t∗ < t0, we have Fx(t; t0, x0, u(.) = Fx(t; t0, 0, u(.)), t ∈ [t∗, t0] for all

u ∈ L2([t∗, t0];Rp), then x1 − x0 ∈ kerFt0,t1 . A special complementary subspace

forkerFt0,t1 is (kerFt0,t1)⊥. The consideration from above motivate the following


definition: we call a state x0 ∈ Rn observable at time t0 ∈ I if and only if there

exit a t1 < t0,t0 ∈ I, such that

x0 ∈ (KerFt0,t1)⊥

. Observe that ’reconstructible’ is not the negation of ’non-reconstructible’

Using lemma , equation (2.7) is equivalent to

x0 ∈∫ t0

t1φ(τ, t0)>F (τ)>F (τ)φ(τ, t0)dτ

With

N(t0, t1) =∫ t0

t1φ(τ, t0)>F (τ)>F (τ)φ(τ, t0)dτ, t1 < t0, (2.11)

Theorem 3. Let control system be given by equation (2.3) and (2.4)

a) A State x0 ∈ Rn is at t0 ∈ I non-reconstructible iff there exits such that

t1 < t0, t1 ∈ I, such that

x ∈ ker N(t0, t1).

b) The linear system (2.1) and (2.2) is completely reconstructible at t1 ∈ I iff there

exists a t1 < t0, t1 ∈ I, such that

rank N(t0, t1) = n

Proof:

We only have to prove statement b). Using a) and the definition of complete

reconstructibility. We immediately get that for any x ∈ Rn there exists a t1 < t0such that x ∈ kerN(t0, t1). Let e1, ..., en be a basis for Rn. Then there exist si < t0such that ei ∈ kerN(t0; si), i = 1, ..., n. Since, for s < t0, x ∈ kerN(t0; s) is

equivalent to x ∈ kerFt0;s, i.e., to F (τ)Φ(τ, t0)x = 0 i.e. on [s, t0], we easily see

that kerN(t0, s) ⊂ kerN(t0; t) for s < t. Therefore we get ei ∈ kerN(t0, t1), i =

1, ..., n, with t1 = min(s1, ..., sn) < t0. The matrix N(t0, t1) is usually called the

reconstructibility Grammian of system (2.1), (2.2). The same remarks apply to this

matrix as the ones given for the controllability Grammian [31, 21, 23].

2.3.1 Observability of Linear, time-invariant control system

In this subsection we consider observability for linear , time-invariant control sys-

tem given by equation (2.3) and (2.4). Since most of the consideration concerning

observability for linear , time-invariant control system.


Equation (2.7) for a system (2.5) and (2.6) which is equivalent to

FeDτx0 ≡ 0. (2.12)

By analyticity of τ → eDτ , equation (3.3.6) is equivalent to

FDjx0 = 0, j = 0, ..., n− 1

respectively to

x0 ∈ kerF, (2.13)

where F =

F

FD

.

.

.

FDn−1

which in view lemma 1 is equivalent to

x0 ∈ (range(F>, D>F>, ..., (D>)n−1F>)⊥

The condition rankN(t0, t1) = n is equivalent to kerN(t0, t1) = 0 which in turn for

arbitrary t0, t1 is equivalent to rankF = n

Thus we have established the following results:

Theorem 4. Let control system be given by equation (2.3) and (2.4)

a) A State x0 ∈ Rn is non-reconstructible at time t0 (or, equivalently , at any

time) if and only if

x0 ∈ ker F, (2.14)

b) System (2.3) and (2.4) is completely reconstructible at some time t0 (or, equiva-

lently , at any time) if and only if

rank F = n, (2.15)

Proof: It is easily seen that for linear, time-invariant systems the notions ’non-

reconstructible’ respectively ’completely reconstructible’ are equivalent to ’observ-

able’ respectively ’completely observable’. For linear, time-invariant systems one

usually uses the latter terms. We call the pair (F,A) observable if and only if

condition rankF = n is satisfied. The subspace kerF is called the unobservable

subspace for system (3.1.3), (3.1.4). If we compare conditions rankF = n and


rank(E,DE, ..., Dn−1E) = n we see that complete reconstructibility of the system

(2.3), (2.4) is equivalent to complete controllability of the system

˙x(t) = −DTx(t)− F Tv(t)

z(t) = ETx(t), t ∈ R,

and that complete observability of the above system is equivalent to the complete

observability of the system (2.5), (2.6). This follows immediately from rankC =

rank(F T , DTF T , ..., (DT )n−1F T )

rank(E,DE, ..., Dn−1E) = rank

ET

ETDT

.

.

.

ET (DT )n−1

.

This indicates that there exists a relation between control systems such that a

system property like reconstructibility is equivalent to another system property, as

for instance controllability, for the related systems [22, 23, 13, 21].

Chapter 3

Glucose Insulin System

In this chapter, Augmented Minimal model and Reduced meal model for type 1

diabetes mellitus are discussed. Controllability and observability of these system are

treated for control purpose for close loop feedback design.

3.1 Augmented Minimal Model

First time, augmented minimal model glucose kinetics was described for type 1 di-

abetes mellitus. In human homeostasis augmented minimal model is most accurate

and most useful model. Transfer of insulin from the injected site to the central

circulation and glucose from the stomach to the central circulation is described in

augmented minimal model [28].

G = −Sg(G−Gb)−XG+Ra(t)

Vg(3.1)

Gi = −1

τ(Gi −G) (3.2)

X = −p2X + p3(Ip − IPbasal) (3.3)

I = −nI +ka1ISQ1 + ka2ISQ2

Vi(3.4)

ISQ1 = −(ka1 + kd)ISQ1 + J(t) (3.5)

ISQ2 = −ka2ISQ2 + kdISQ2 (3.6)

Ra = − 1

τmeal(Ra−D(t)) (3.7)

34

CHAPTER 3. GLUCOSE INSULIN SYSTEM 35

In these equations, t is the independent model variable time, G(t) is the plasma

glucose concentration [mg/dL], I(t) is the plasma insulin level [U/mL] and X(t) is

the interstitial insulin activity. Gb is the basal plasma glucose concentration [mg/dL]

and Ipbasal is the basal plasma insulin concentration [µU/mL]. Basal plasma con-

centrations of glucose and insulin are typically measured before administration of

glucose (or sometimes 180 minutes after).ISQ1 and ISQ2 are the subcutaneous insulin

transport after injection (J), Ra is the glucose rate appearance from a meal D. These

are unknown parameters in this model: Sg, Vg, p2, p3, ka1, ka2, kd and n [28].

Parameter value Parameters values

n 0.2 Vi 0.125

Sg 0.0094 J(t) 0.690656

Vg 96000000 D(t) 247.76

p2 0.0265 ka1 0.002

p3 0.00005 ka2 0.0211

Ip 15 kd 0.0166

IPbasal 2.7648 τmeal 0.055

τ 0.2 Gb 142

Table 3.1: Table of Parameter’s used in the Augmented Minimal Model

Model after substitution parameter value becomes

G = −0.0094(G− 142)−XG+Ra(t)

2.5(3.8)

Gi = − 1

0.2(Gi −G) (3.9)

X = −0.0265X + 0.00005(15− 2.7648) (3.10)

I = −0.2I +0.002ISQ1 + 0.02ISQ2

0.125(3.11)

ISQ1 = −(0.002 + 0.0166)ISQ1 + 0.690656 (3.12)


ISQ2 = −0.0211ISQ2 + 0.0166ISQ2 (3.13)

Ra = − 1

0.055(Ra− 247.76) (3.14)

Put left hand side of the system (3.8− 3.14) equal to zero and get the Equilibrium

point of the model is

(G,Gi, X, I, ISQ1, ISQ2, Ra) = (3091.85, 3091.85, 0.023085, 2.97056, 37.132, 0, 247.76)

Hence, the linearized model according to the equilibrium point is

G = −0.032465G− 3091.81X + 0.4Ra (3.15)

Gi = 5G− 5Gi (3.16)

X = −0.0265X (3.17)

I = −0.2I − 0.016ISQ1 + 0.1688ISQ2 (3.18)

ISQ1 = −0.0186ISQ1 (3.19)

ISQ2 = −0.0045ISQ2 (3.20)

Ra = −18.18Ra (3.21)

3.1.1 Linear Control System:

Consider the linear system

x(t) = Ax(t) +Bu(t) (3.22)

y(t) = Cx(t) (3.23)

In this case we have

x = [G Gi X I ISQ1 ISQ2 Ra]T

Where

A =

−0.032485 0 −3091.85 0 0 0 0.4

5 −5 0 0 0 0 0

0 0 −0.0265 0 0 0 0

0 0 0 −0.2 −0.016 0.1688 0

0 0 0 0 −0.0186 0 0

0 0 0 0 0 −0.0045 0

0 0 0 0 0 −18.18


If we consider the glucose concentration in plasma is the only measured output and

the insulin concentration in plasma is only input then B = [0 0 1 0 0 0]T and

C = [1 0 0 0 0 0 0 ]. The controllability matrix is R = [B AB A2B ... A6B] and

its rank is 1.So the system is not controllable. The observability matrix is O =

[C;CA;CA2;CA3; ...;CA6]T and its rank is 3. So the system is not observable. At

equilibrium point (3091.85; 3091.85; 0.023085; 2.97056; 37.132; 0; 247.76) and at dif-

ferent initial conditions like (100; 0; 11.01; 0; 0; 0; 0), (100; 0; 11.01; 11.01; 11.01; 0; 0),

(110; 0; 15.01; 0; 0; 0; 0), (130; 0; 20.01; 0; 0; 0; 0) the controllability and observability

of the graph is approaches to zero, so system not work like close loop for feedback

design are shown in figures 3.1 to 3.3.

Figure 3.1: Controllability and observability Graph


Figure 3.2: Controllability and observability Graph

3.2 Reduced Meal Model

This model is designed to tell about control glucose on insulin and insulin on glucose

and also about their subsystems. Glucose and insulin subsystems are the part of

this model. Insulin dependent utilization and insulin independent utilization in the

glucose kinetics model is the subsystem of glucose. Plasma and fat equilibrating tis-

sue is represented by insulin independent. Peripheral tissue is represented by insulin

dependent partition. Plasma and liver is described in inulin subsystem [29]. This

departs from the initial model, which included a strongly nonlinear modeling of the

rate of appearance of glucose. The model is consists of 11 nonlinear differential equa-

tions, different parameters and constants their values given in [29, 28]. Followings

are the equation of the model


Gp = −(k2 + kp2)Gp + k1Gt − Uii − kp3Id +Ra(t) + kp1 (3.24)

Gt = −k1Gt + k2Gp −Vm0 + VmX

X

Kmo +Gt

Gt (3.25)

Gi = − 1

τIG(Gi −

Gp

Vg) (3.26)

Id = −ki(Id − I1) (3.27)

I1 = −ki(I1 −IpVi

) (3.28)

Ip = −(m2 +m4)Ip +m1Il + ka1ISQ1 + ka2ISQ2 (3.29)

Il = −(m1 +m3)Ip +m2Ip (3.30)

X = −p2u(X −IpVi

+ Ib) (3.31)

ISQ1 = −(ka1 + kd)ISQ1 + J(t) (3.32)

ISQ2 = −ka2ISQ2 + kdISQ2 (3.33)

Ra = − 1

τmeal(Ra −D(t)) (3.34)

Following is the table of parameter’s value

Parameter value Parameter value Parameter value

k1 0.0702 k2 0.1151 Vg 1.834

kp1 5.1207 kp2 0.0061 kp3 0.0087

Ib 104 Uii 1 τIG 0.2

J(t) 0.690656 D(t) 247.76 Vi 0.0503

ka1 0.002 ka2 0.0211 kd 0.0166

Vmo 5.3263 VmX0.0417 τmeal 0.055

Km0 234.0043 m1 0.0312 m2 0.3616

m4 0.1446 ki 0.0075 m3 0.306

p2u 0.0276

Table 3.2: Table of Parameters used in the Reduced Meal Model


After Substitution parameter values, we get

Gp = −0.1212Gp + 0.0702Gt − 0.0087Id +Ra(t) + 4.1207 (3.35)

Gt = −0.0702Gt + 0.1151Gp −5.3263 + 0.0417X

234.0043 +Gt

Gt (3.36)

Gi = −5Gi + 2.7262Gp (3.37)

Id = −0.0075Id + 0.0075I1 (3.38)

I1 = −0.0075Id + 0.1491I1 (3.39)

Ip = −0.5062Ip + 0.0312Il + 0.002ISQ1 + 0.0211ISQ2 (3.40)

Il = 0.0244Ip (3.41)

X = −0.0276X + 0.5487Ip − 2.8704 (3.42)

ISQ1 = −0.0638ISQ1 + 0.690656 (3.43)

ISQ2 = −0.00449ISQ2 (3.44)

Ra = −18.1818Ra + 4504.7272 (3.45)

Substitute left hand side of the system equal to zero, we find equilibrium points of

(Gp, Gt, Gi, Id, I1, Ip, Il, X, ISQ1 , ISQ2 , Ra) are

(41197.8, 67539.4, 22462.7, 0, 0, 0,−0.69391,−104, 10.8253, 0, 247.787) and

(1947.56,−225.976.04, 1061.89, 0, 0, 0,−0.69391,−104, 10.8253, 0, 247.787)

In this model both the equilibrium point includes two negative values one for

Il and other for X. Il is the insulin kinetics of glucose production. X insulin

concentration in plasma are taken in units so it cannot have a negative value. This

is the most comprehensive model in the glucose insulin system dynamics for human

but the result show that this model has some deficiency in it since the equilibrium

point is not in feasible region.

Gp = −0.1212Gp + 0.0702Gt − 0.0087Id +Ra(t) (3.46)

Gt = −0.7145Gt + 0.1151Gp − 6.15× 10−7X (3.47)

Gi = −5Gi + 2.7262Gp (3.48)

Id = −0.0075Id + 0.0075I1 (3.49)

I1 = −0.0075Id + 0.1491I1 (3.50)

Ip = −0.5062Ip + 0.0312Il + 0.002ISQ1 + 0.0211ISQ2 (3.51)


Il = 0.0244Ip (3.52)

X = −0.0276X + 0.5487Ip (3.53)

˙ISQ1 = −0.0638ISQ1 (3.54)

˙ISQ2 = −0.00449ISQ2 (3.55)

Ra = −18.1818Ra (3.56)

Theorem 3.1: The linear x(t) = Ax(t), where A continuous and bounded for t ≥ t0, is uniformly asymptotically stable if and only if given a positive definite real matrix

A, there exists a symmetric positive definite real matrix P , which satisfies

P (t) + AT (t)P (t) + P (t)A(t) = −Q(t), t ≥ t0

The linear time invariant system x(t) = Ax(t)the corresponding equation to be used

as ATP + PA+Q = 0 this is called Lyapunov equation [32, 33, 34].

Proof: By using Lyapunov equation ATP + PA+Q = 0 , then we have A =

−0.12 0.07 0 −0.0087 0 0 0 0 0 0 0

0.11 −0.71 0 0 0 0 0 −6.15× 10−7 0 0 0

2.72 0 −5 0 0 0 0 0 0 0 0

0 0 0 −0.0075 0.0075 0 0 0 0 0 0

0 0 0 −0.0075 0.1491 0 0 0 0 0 0

0 0 0 0 0 −0.50 0.03 0 0.002 0.02 0

0 0 0 0 0 0.02 0 0 0 0 0

0 0 0 0 0 0.54 0 −0.0276 0 0 0

0 0 0 0 0 0 0 0 −0.063 0 0

0 0 0 0 0 0 0 0 0 −0.004 0

0 0 0 0 0 0 0 0 0 0 −18.18

and substitute Q = I11×11 in Lyapunov equation ATP + PA + Q = 0, and by

using matlab we find matrix P but P is not symmetric positive-definite solution P so


system is not asymptotically and uniformly stable. In other technique eigne values of A are

(−5.0000,−0.7228,−0.1072,−0.0071, 0.1487,−0.5012, 0.0012,−0.0276,−0.0630,−0.0040,−18.1800),

so the system is not again asymptotically and uniformly stable.

3.3 Linear Control

Consider the linear system

x(t) = Ax+Bu (3.57)

y(t) = Cx (3.58)

In this case we have

x = [Gp, Gt, Gi, Id, I1, Ip, Il, X, ISQ1 , ISQ2 , Ra]T

andA =

−0.12 0.07 0 −0.008 0 0 0 0 0 0 0

0.11 −0.71 0 0 0 0 0 −6.15× 10−7 0 0 0

2.72 0 −5 0 0 0 0 0 0 0 0

0 0 0 −0.007 0.0075 0 0 0 0 0 0

0 0 0 −0.007 0.1491 0 0 0 0 0 0

0 0 0 0 0 −0.50 0.03 0 0.002 0.02 0

0 0 0 0 0 0.02 0 0 0 0 0

0 0 0 0 0 0.54 0 −0.0276 0 0 0

0 0 0 0 0 0 0 0 −0.063 0 0

0 0 0 0 0 0 0 0 0 −0.004 0

0 0 0 0 0 0 0 0 0 0 −18.18

3.3.1 Controllability and Observability

Here we take the only measured output of glucose and the only input is insulin then

B = [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0]T and C = [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

R = [B,AB,A2B,A3B ... A10B]


The system is not controllable because its rank (i.e.rank(R) 6= 11). The observability

matrix is given by

O = [C,CA,CA2, CA3 ... CA10]T

The system is not observable because its rank (i.e.rank(R) 6= 11). By using the Matlab

function found that system is not controllable nor observable. At equilibrium point the

controllability and observability of the graph is approaches to zero, so system not work

like close loop for feedback design.

3.4 Conclusion

Consider Augmented Minimal model of Glucose Kinetics and reduced meal model which

had a simple but comprehensive models. The controllability and observability for linearized

system are treated. In first model,we consider that only measured output is concentration

of glucose in plasma that we can measure easily. The system is not controllable neither

observable. The controllability and observability of the graph is approaches to zero, so

system not work like close loop for feedback design. Reduced meal model which had

a simple but comprehensive models are also treated for glucose insulin pump. For this

purpose, In the model the equilibrium point includes two negative values one for Il and

other for X. Il is the insulin kinetics of glucose production. X insulin concentration in

plasma are taken in units so it cannot have a negative value. The system is not stable at

these points results are also verified with Lyapunov function for stability analysis. Consider

that only measured output is concentration of glucose in plasma that we can measured

easily. The system is not controllable neither observable.

Chapter 4

Fractional Order Glucose InsulinModel

4.1 Fractional Order Glucose Insulin Model

We determine a model for all plasma glucose concentration, generalized insulin and plasma

insulin. Diabetes dynamics is a mathematical model. There are two other models of

glucose/insulin used to explain interaction. These are valid to predict blood glucose because

these are inherent requirement of frequently updated information. In this model we take

glucose level G, glucose uptake X, insulin level I. This model also include the basal values

Gband Ib [4]. The model is given in the followings equations

G(t) = −m1G+m2I +m1Gb (4.1)

X(t) = −m2X +m3I −m3Ib +m6Ib (4.2)

I(t) = −m3I +m4G+m4m5 −m6I +m6Ib (4.3)

with initial conditions

G(0) = p1 = 100, X(0) = p2 = 0, I(t) = p3 = Ib (4.4)

Where G(t) is plasma glucose concentration, X(t) is plasma insulin variable for remote

compartment, I(t) is plasma insulin concentration, Gb is the basal preinjection value of

44

CHAPTER 4. FRACTIONAL ORDER GLUCOSE INSULIN MODEL 45

plasma glucose, Ib is basal preinjection value of plasma insulin, m1 is the insulin indepen-

dent rate uptake in liver, muscle and adipose tissue, m2 is the rate of decrease in tissue

glucose uptake ability, m3 is the insulin independent increase in glucose uptake ability in

Ib, m4 is the rate of pancreatic cells which are released after the glucose injection and

glucose concentration above system, m5 is the threshold value of glucose, m6 is the decay

rate for insulin in plasma [4]. The values of parameter are given as follows.

Parameter Normal person Type 1 Diabetes Units

m1 0.0317 0 min−1

m2 0.0123 0.017 min−1

m3 4.92× 10−6 5.3× 10−6 min−2(µU/ml)

m4 0.0039 0.0042 (µU/ml)min−2(mg/dl)−2

m5 79.053 80.25 mg/dl

m6 0.2659 0.264 min−1

Gb 80 80 mg/dl

Ib 7 15 µU/ml

Table 4.1: Table of Parameter’s used in Sandhya Model

4.1.1 Stability Analysis and Equilibria

Model after substituting parameter values given in table 4.1, we get

G(t) = −0.0317000G+ 0.0123I + 2.536 (4.5)

X(t) = −0.0123X + 0.00000492I − 0.00000492 + 1.8613 (4.6)

I(t) = −0.26590492I + 0.0039G+ 2.1695 (4.7)


Substituting the left hand side of the system equal to zero and we get the values of G, X

and I . Hence the equilibrium points are

(83.6417, 151.3288, 9.3817)

Theorem 4.1: The linear x(t) = Ax, where A continuous and bounded for t ≥ t0 , is

uniformly asymptotically stable if and only if given a positive definite real matrix A, there

exists a symmetric positive definite real matrix P , which satisfies

P (t) +AT (t)P (t) + P (t)A(t) = −Q(t), t ≥ t0

The linear time invariant system x(t) = Ax the corresponding equation to be used as

ATP + PA+Q = 0 this is called Lyapunov equation [32, 33].

Proof:

Here

A =

−0.0317 0.0123 0

0 −0.0123 1.1876

0.0039 −0.2659 0

and substitute Q = I3×3 in ATP + PA+Q = 0, get P

P =

16.0010 0.5880 −2.2467

0.5880 219.0273 1.8475

−2.2467 1.8475 49.0567

The Lyapunov equation has symmetric positive-definite solution P , then the eigen

values of A are (−0.0315,−0.0062 + 0.5619i,−0.0062 − 0.5619i) has negative real parts,

so the system is asymptotically and uniformly stable. Now, we give some fundamental

results and definitions from fractional calculus. For detailed over view of the topic readers

are referred to [35, 36, 37, 38].


Definition 4.1 The definitions of Laplace transform of Caputo’s derivative and Mittag-

Leffler function in two arguments is written as

L {Dαf(t), s} = sαF (s)− Σn−1i=0 s

α−i−1f (i)(0), (n− 1 < α ≤ n); n ∈ N.

The fractional order extension of this model have been first studied in [40, 39] and show

the realistic biphasic decline behavior of infection of diseases but at a slower rate. The new

diabetes mellitus model described in the form of fractional differential equations (FDEs)

given as

Dα1G(t) = −m1G+m2I +m1Gb (4.8)

Dα2G(t) = −m2X +m3I −m3Ib +m6Ib (4.9)

Dα3I(t) = −m3I +m4G+m4m5 −m6I +m6Ib (4.10)

with initial conditions G(0) = p1 = 100, p2 = X(0) = 0, I(0) = p3 = Ib.

4.2 Laplace Adomian Decomposition Method

Consider the fractional-order epidemic model (4.8 − 4.10) subject to the initial condition

(4.4). For α1 = α2 = α3 = 1 the fractional order model converts to the classical diabase

model. Applying the Laplace transform on equation (4.8− 4.10), we get

L {Dα1t G(t)} = −m1L {G(t)}+m2L {I(t)}+ L {m1Gb} (4.11)

L {Dα2t X(t)} = −m2L {X(t)}+m3L {I(t)} −L {(m3Ib −m6Ib)} (4.12)

L {Dα3t I(t)} = −(m3 +m6)L {I(t)}+m4L {G(t)}+ L {(m6Ib +m4m5)} (4.13)

By applying the rule of Laplace transform, we get

Sα1L {G} − Sα1−1G(0) = −m1L {G(t)}+m2L {I(t)}+ L {m1Gb} (4.14)


Sα2L {X} − Sα1−1X(0) = −m2L {X(t)}+m3L {I(t)} −L {m3Ib −m6Ib} (4.15)

Sα3L {I} − Sα1−1I(0) = −(m3 +m6)L {I(t)}+m4L {G(t)}+ L {m6Ib +m4m5}(4.16)

Sα1L {G} = Sα1−1G(0)−m1L {G(t)}+m2L {I(t)}+ L {m1Gb} (4.17)

Sα2L {X} = Sα1−1X(0)−m2L {X(t)}+m3L {I(t)} −L {m3Ib −m6Ib} (4.18)

Sα3L {I} = Sα1−1I(0)− (m3 +m6)L {I(t)}+m4L {G(t)}+ L {m6Ib +m4m5} (4.19)

by using the initial conditions (4.4), we get

L {G} =p1

S+m1GbSα1+1

+m2

Sα1L {I(t)} − m1

Sα1L {G(t)} (4.20)

L {X} =p2

S− m3Ib −m6Ib

Sα2+1− m2

Sα2L {X(t)}+

m3

Sα2L {I(t)} (4.21)

L {I} =p3

S+m6Ib +m4m5

Sα3+1− m3 +m6

Sα3L {I(t)}+

m4

Sα3L {G(t)} (4.22)

4.3 Case I for Normal Person

First of all we study the glucose, plasma concentration and insulin for non-diabetic person

for the period of 10 hours. The model show that when we give glucose to normal man then

the level of glucose concentration is very high but after time passing it become stable [4].

The model after substituting the parameters values for case I.

L {G} =p1

S+

2.536

Sα1+1+

0.0123

Sα1L {I(t)} − 0.0317

Sα1L {G(t)} (4.23)

L {X} =p2

S+

1.861

Sα2+1− 0.0123

Sα2L {X(t)}+

4.92× 10−6

Sα2L {I(t)} (4.24)


L {I} =p3

S+

2.1696

Sα3+1− 0.2659

Sα3L {I(t)}+

0.0039

Sα3L {G(t)} (4.25)

with initial conditions G(0) = p1 = 100, p2 = X(0) = 0, I(0) = p3 = 7.

It should be assumed that method gives the solution as an infinite series

G =∞∑k=0

Gk, X =∞∑k=0

Xk, I =∞∑k=0

Ik (4.26)

Substitute equations (4.26) in (4.23− 4.25), we have the followings results

L {G0} =p1

S+

2.536

Sα1+1, L {X0} =

p2

S+

1.861

Sα2+1, L {I0} =

p3

S+

2.1696

Sα3+1(4.27)

Similarly we have

L {G1} =0.0123

Sα1L {I0} −

0.0317

Sα1L {G0}, ..., L {Gk+1} =

0.0123

Sα1L {Ik}−

0.0317

Sα1L {Gk} (4.28)

L {X1} =−0.0123

Sα2L {X0}+

0.00000492

Sα2L {I0}, ...,

L {Xk+1} =−0.0123

Sα2L {Xk}+

0.00000492

Sα2L {Ik} (4.29)

L {I1} =−0.2659

Sα3L {I0}+

0.0039

Sα3L {G0}, ...,L {Ik+1} =

−0.2659

Sα3L {Ik}+

0.0039

Sα3L {Gk} (4.30)

The purpose of the work is to analysis the mathematical behaviour of the solution

G(t), X(t), I(t) for the different values of α. By applying the inverse laplace transform

to both sides of the equation (4.27), we get the values of G0, X0, I0 and used for further

process. Putting the values of G0, X0, I0 into the equations (4.28 − 4.30) and get the


values of G1, X1, I1. Similarly we find the remaining term G2, G3, G4, ...., X2, X3, X4, ....

and I2, I3, I4, .... in the same manners. Solution can be written as

G(t) = G0 +G1 +G2 +G3 +G4, ... (4.31)

X(t) = X0 +X1 +X2 +X3 +X4, ... (4.32)

I(t) = I0 + I1 + I2 + I3 + I4, ... (4.33)

by substituting the values of G0, X0, I0, G1, X1, I1 and G2, G3, G4, .... we get

G(t) = 100− 0.547tα1

α1!+ 0.0267

tα3

α3!+ 0.0174

t2α1

2α1!− 0.0190

tα1+α3

(α1 + α3)!− 0.0071

tα1+2α3

(α1 + 2α3)!

+ 0.00012t2α1+α3

(2α1 + α3)!+ 0.0025

t3α1

3α1!(4.34)

X(t) = 1.86103tα2

α2!− 1.92× 10−9 t

2α2

2α2!+ 3.5× 10−6 tα2+α3

(α2 + α3)!+ 0.00028

t3α2

3α2!+

1.3× 10−7 t2α2+α3

(2α2 + α3)!− 2.8× 10−6 tα2+2α3

(α2 + 2α3)!+ 4.87× 10−8 tα1+α2+α3

(α1 + α2 + α3)!(4.35)

I(t) = 7 + 0.6983tα3

α3!− 0.1856

t2α3

2α3!− 0.0021

tα1+α3

(α1 + α3)!− 0.1539

t3α3

3α3!−

0.0026tα1+2α3

(α1 + 2α3)!− 0.00031

t2α1+α3

(2α1 + α3)!(4.36)

4.4 Case II for Type 1 Diabetes

Now we study the model for diabetic patient for the period of 10 hours. The model show

that at start time the level of glucose is very high but when we give glucose then his level

of glucose does not fall. After time passing from 250mg/dl it fall only about 275mg/dl [4].

The fractional model after substituting the parameters values for case II.

L {G} =p1

S+

0.017

Sα1L {I(t)} (4.37)


L {X} =p2

S+

3.9599

Sα2+1− 0.017

Sα2L {X(t)}+

5.3× 10−6

Sα2L {I(t)} (4.38)

L {I} =p3

S+

4.297

Sα3+1− 0.264

Sα3L {I(t)}+

0.0042

Sα3L {G(t)} (4.39)

with initial conditions G(0) = p1 = 240, p2 = X(0) = 0, I(0) = p3 = 15. We computed

first three terms by using the L-ADM for the equations (4.37− 4.39) . We have followings

series solution

G(t) = 240 + 0.255tα1α1!

+ 0.0228tα1+α3

(α1 + α3)!− 0.0193

tα1+2α3

(α1 + 2α3)!, (4.40)

X(t) = 3.9606tα2

α2!− 0.067314

t2α2

(2α2)!+ 6.8× 10−5 tα2+α3

(α2 + α3)!+ 0.0011

t3α2

3α2!

−0.0000039t2α2+α3

(2α2 + α3)!− 0.00006

tα2+2α3

(α2 + 2α3)!, (4.41)

I(t) = 15 + 1.645tα3

α3!− 0.3551

t2α3

2α3!+ 0.2994

t3α3

3α3!+ 0.0011

tα1+α3

(α1 + α3)!+

0.00031tα1+2α3

(α1 + 2α3)!(4.42)

4.5 Numerical Results and Discussion

The analytical solution of fractional order model consist of nonlinear system of fractional

differential equation has been presented by using Laplace Adomian decomposition method.

To observe the effects of the parameter on the dynamics of the fractional-order model

for Case I and Case II, we conclude several numerical simulations varying the value of

parameter given in table 4.1 with time 20 to 40 minutes. These simulations reveals that a

change of the value affects the dynamics of the model . In figures 4.1 and 4.2 clearly shows

the bounded solution according to normal values of glucose level for normal person and


type 1 diabetes. In figures 4.3 and 4.4 basal values of insulin properly rise and bounded

according to inial conditions and approach to zero when no insulin injected nor produced

in human body. Figure 4.5 represent the insulin concentration in plasma with effect of

glucose level. Figure 4.6 represents no insulin produce in human body caused by type 1

diabetes with passage of time. The system gives the solution at fractional derivative on non

integer values which are more appropriate and accurate values in given domain. Glucose

level increase in figure 4.4 due to cause of type 1 diabetes which control with suitable input

values of insulin to normalize the glucose level.

0 5 10 15 20 2580

85

90

95

100

105

110

115

120Glucose Level measured for normal person

Time (min)

G(t)

Gluc

oaw

Leve

l

Glucose level at α

1= 0.9,α

3= 0.8

Glucose level at α1=α

3=0.75

Glucose level at α1= α

3=0.5

Figure 4.1: Numerical solution of Glucose level of normal person


0 5 10 15 20 25 30 35 400

50

100

150

200

250

300

350Gluocse Level measured for Type 1 Diabetes person

Time (min)

G(t)

Gluc

ose

Leve

l

Glucose level at α

1= α

3=0.7


3=0.8


3=0.5

Figure 4.2: Numerical solution of Glucose level of type 1 diabetes

0 5 10 15 20 250

2

4

6

8

10

12

14

16

18

20 Monitoring of insulin basal values for normal person

Time (min)

I(t)I

nsuli

n Le

vel

Insulin level at α

1=α

3=0.15

Insulin level at α1=α

3=0.09


3=0.05

Figure 4.3: Behavior of insulin in normal person


0 5 10 150

5

10

15

20

25

30Monitoring of insulin basal values for Type 1

Time (min)

I(t)

Insu

lin L

evel

Insulin level at α

1=α

3=0.3


3=0.6


3=0.2

Figure 4.4: Behavior of insulin in type 1 diabetes

0 5 10 15 20 250

5

10

15

20

25

30

35

40Monitoring of Insulin concentration in plasma for normal person

Time (min)

X(t)

Insu

lin co

ncen

tratio

n in

plasm

a

Insulin level at α

1=0.6,α

2=0.4,α

3=0.5


2=α

3=0.6

Insuline level at α1=α

2=α

3=0.5

Figure 4.5: Numerical solution of insulin concentration in plasma of normal person


0 2 4 6 8 100

1

2

3

4

5

6

7

8

9

10Monitoring of Insulin concentration in plasma for Type 1

Time (min)

X(t)

Insu

lin C

once

ntra

tion

in pla

sma

Insulin level at α

2= 0.02α

3=0.1


3=0.006


3=0.01

Figure 4.6: Numerical solution of insulin concentration in plasma of type 1 diabetes

4.6 Input and Output Stability

Stability is a main anxiety in feedback control design in engineering for automatic control

systems because a feedback control law can stabilize a and also destabilize a system. We use

Lyapunov’s indirect method [33] to examine the system stability (4.1)-(4.3). The system

equilibrium points depends on the steady state of glucose and insulin concentration in

plasma. The blood glucose level has the steady state 100(mg/dl) and the steady state

of insulin in the system (4.1)-(4.3) with the feedback infusion rates and the values of

parameters are given in Tables 4.1 . We find the following equilibrium points by using

Matlab as (83.6417, 151.3288, 9.3857). The linear control system is

x(t) = Ax+B, y(t) = Cx (4.43)

where x = [G, I,X]T B = [0, 0, 1]T and C = [1, 0, 0]. A is the Jacobian matrix at the

equilibrium. We find the following eigenvalues of A by using the MATLAB are as follows


(−0.0315,−0.0062 + 0.5619i,−0.0062 − 0.5619i), since the eigen values of the system are

negative real parts, it satisfied the inputoutput stability theorem.


The dynamical system has physical properties Controllability and observability and rep-

resents the effect of regulatory system of blood glucose in human. The system (4.43) is

controllable if for any initial state x0 and any desired state xf , there exists a control

of insulin such that x(T ) = xf for some T > 0. The system (4.43) is observable if any

initial state can be uniquely determined by the output Glucose (blood glucose) over (0, T )

for some T > 0. To check the controllability of (4.43), it suffices to examine the rank

of the Kalman controllability matrix [41, 42]. C = [B|AB|A2B]. Here we take the only

measured output of glucose and the only input is insulin. we compute the determinant of

the matrix det(C) = 0.0245. Therefore C has the full rank of 3 and then the system 4.43

is controllable. In the same way, O = [CT |ATCT |(AT )2C] and det(O) = −2.3574e005. We

can have the full rank of 3 and then the system (4.43) is observable. Hence the system is

controllable and observable. For case II type 1 diabetes mellitus, the system is controllable

but not observable.

4.7 Conclusions

In this chapter a theoretical and numerical investigation of the bio-medical glucose insulin

model is presented. It shows the controllability and applicability of the model for the

control of the blood glucose concentration in normal person and T1DM. For the purpose

of automatic artificial pancreas in the glucose regulatory system, we discuss fractional

order glucose insulin model. The model is stable by using the lyapunov equation and

input/output stability is also satisfied for automatic control. Hence the model is stable


in each case to design the close loop for artificial pancreas. System is controllable and

observable for case 1. For case II, the system is controllable but not observable according

to the given parameters values.

It is important to note that Laplace adomian decomposition method for mathemat-

ical models based on system of fractional order differential equations are more powerful

approach to compute the convergent solutions. The convergence analysis is provided to

demonstrate the efficiency of the method. The model provide the continues glucose mea-

suring in limited time and solutions are bounded in normal values for healthy person and

type 1 diabetes. This is perhaps due to development in the biological approach for the new

model; e.g the hypothesis associated to internal insulin creation through a time dependent

model. Our results show that the fractional-order models can give enhanced turns to the

data than integer-order models in some cases, it is clear that for the satisfactory turns, the

models need additional improvement and insertion of these changes should greatly improve

future models [58].

Chapter 5

Stability Analysis and Control ofGIG System

5.1 Introduction

The goal for treatment of type-1 diabetes mellitus is the development of an artificial pan-

creas to regulate the blood glucose level in the closed loop design. A system of nonlinear

differential equations having five variables (glucose, insulin, β-cell mass, α-cell mass, and

glucagon) with seventeen parameters is considered. The Lyapunov function is used to

check the stability analysis of the model. Controllability and observability of the linearized

model are discussed under two different cases, in case 1 insulin is taken as an input and in

case 2, insulin and glucagon are taken as an input for the system. This played an impor-

tant role in the development of fully automatic artificial pancreas by stabilizing the control

loop system for the glucose-insulin glucagon pump. Proportional Integral Derivative (PID)

controller is designed for an artificial pancreas by using the transfer function. According

to the desire value, the algorithm of an artificial pancreas measures the glucose level in the

blood of a patient by using glucose sensor that sends a signal to an insulin glucagon pump

to adjust the basal insulin. A closed-loop system is tested in Simulink environment and

simulation results show the performance of the designed controller.

58

CHAPTER 5. STABILITY ANALYSIS AND CONTROL OF GIG SYSTEM 59

5.2 Mathematical Model

The level of blood glucose is mainly controlled by two hormones having opposite effects.

While insulin clears out blood glucose by stimulating its uptake by muscles and adipose

tissues and storing it as glycogen in the liver, glucagon supplies bloodstream by glucose

produced through liver gluconeogensis and glycogenolysis. Consequently, any dysfunction

in the secretion of insulin or glucagon will lead to problems in the control of glycaemia.

Despite the important of glucagon in the control of glycaemia,the literature on glucagon and

α-cells is less important than the available information on insulin and β-cells [43]. Based

on two existing mathematical models by Boutayeb et al. [44] described the dynamics

of glucose, insulin and β-cell mass in presence and absence of genetic predisposition to

diabetes and Ruby Celecte et al. who studied the role of glucagon in the regulation of

blood glucose [45]. A mathematical model considering the dynamics of glucose, insulin ,

β-cells mass, α-cells mass and glucagon. We assumed that the glucagon is released by the

α-cells at low glucose concentrations (Gl = 80mg/dl) in order to stimulate hepatic glucose

production that raises the concentration of glucose in the blood. Where as the insulin is

secreted by the α-cells to reduce the elevation of glucose levels in the blood [46].

dG

dt= a− bG− cIG

αG+ 1+ cjJ (5.1)

dI

dt=

dβG2

e+G2− fI (5.2)

dβ

dt= rβ(1− β

K) (5.3)

dα

dt= rjα(1− α

Kj) (5.4)

dJ

dt= −djα(G−Gt)− fjJ (5.5)

Following is table of parameter’s value used in model


Parameter value Units Description

a 864 mg/(dId) glucose production rate by liver when G=0

b 1.44 d−1 glucose clearance rate independent of insulin

c 0.85 mI/(mUd) insulin induced glucose production rate

cj 1350 d−1 glucagon induced glucose production rate

d 43.2 mU/(mIdmg) β − cell maximum insulin secretory rate

dj 0.05 1/(mgd) α− cell maximum glucagon secretory rate

e 20000 mg2/dI2 gives inflection point of sigmoid function

f 432 d−1 whole body insulin clearance rate

fj 0.3 d−1 whole body glucagon clearance rate

K 900 mg environmental capacity of β − cell mass

Kj 300 mg inverse of half saturation constant

Gt 80 mg/dI minimum level of glycaemia

r 0.01 mg growth rate of β − cell

rj 0.001 mg growth rate of α− cell

Table 5.1: Table of parameter’s value used in the model

The mathematical model given in [46], after substituting the values of parameter, we get

the following system of equations

dG

dt= 854− 1.44G− 0.85IG

αG+ 1+ 1350J (5.6)

dI

dt=

43.2βG2

20000 +G2− 432I (5.7)

dβ

dt= 0.01β(1− β

900) (5.8)


dα

dt= 0.001α(1− α

300) (5.9)

dJ

dt= −0.05α(G− 80)− 0.3J (5.10)

For equilibrium point, set left hand side of the system (5.6−5.10) equal to zero; we get

values of the variables (G, I, β, α, J) are P1(600, 0, 0, 0, 0), P2(80, 21, 900, 300, 0.16). The

first point is an unstable point describing a severe hyperglycaemia with zero levels of β-cell

mass and α-cell mass leading to zero levels of insulin and glucagon and hence raising the

level of glucose. The second point is a stable point corresponding to a normal state with

normal values of glucose, insulin, β-cell mass, α-cell mass and glucagon. By using the

Jacobian technique, the linearized model about the stable equilibrium point is

dG

dt= −1.44G− 0.002833I + 0.000198α+ 1350J (5.11)

dI

dt= 178.512G− 432I + 10.4727β (5.12)

dβ

dt= −0.01β (5.13)

dα

dt= −0.001α (5.14)

dJ

dt= −15G− 0.3J (5.15)

5.3 Stability Theorem:

The linear x(t) = Ax(t), where A(t) continuous and bounded for t ≥ t0 , is uniformly

asymptotically stable if and only if given a positive definite real matrix A, there exists a

symmetric positive definite real matrix P , which satisfies


The linear time invariant system x(t) = A(t)x(t)the corresponding equation to be used as



Here A =

−1.44 −0.002833 0 0.0001983 1350

178.512 −432 10.4727 0 0

0 0 −0.01 0 0

0 0 0 −0.001 0

−15 0 0 0 −0.3

and Q = I5×5 By using this Lyapunov equation ATP + PA+Q = 0 on matlab, we get

P =

26.1330 9.7451 −5.25664× 10−7 1.4737536× 10−5 0.0275

9.7451 4.0574 1.2121 6.089863× 10−6 −0.3268

−5.25664× 10−7 1.2121 50.0000 0 2.5435368× 10−5

1.4737536× 10−5 6.089863× 10−6 0 500.0000 −0.0001

0.0275 −0.3268 2.5435368× 10−4 −0.0001 0.2904

The eigen values of P are (0.0012, 0.6342, 29.8105, 50.0350, 500.0000) and which shows

that P is symmetric positive definite real matrix P. Hence prove that the system is

uniformly asymptotically stable. Similarly at point (600,0,0,0,0), eigne values of A, is

(−1.44,−4320.0100,−0.300, 0.001) so, system is not stable at that point.

5.4 Linear Control System

A linear control system is

x(t) = Ax(t) +Bu(t), y(t) = Cx(t) (5.16)


We have two cases, for case I: Insulin as the input only and glucose as an output and in

case II: insulin and glucagon as an input and glucose as output only.


Case1: Here we take the only measured output of glucose and the only input is insulin

then

B = [0, 1, 0, 0, 0]T C = [1, 0, 0, 0, 0]

where

A =

−1.44 −0.002833 0 0.0001983 1350

178.512 −432 10.4727 0 0

0 0 −0.01 0 0

0 0 0 −0.001 0

−15 0 0 0 −0.3

The controllability matrix is

R = [B; AB; A2B; A3B; A4B]

rank(R) = rank([B; AB; A2B; A3B; A4B]) = 2

The observability matrix is

O = [C; CA; CA2; CA3; CA4]

rank(O) = rank([C; CA; CA2; CA3; CA4]) = 5

Hence the system is observable but not controllable. Figures 5.1 and 5.2 represent the

controllable and observable state at the equilibrium point and for different initial conditions

on the state variable values, when insulin is an input and the glucose level for that state

system is observable but not controllable. Note that our given requirements are not met,

specifically, the steady-state error is much too large.


Figure 5.1: Controllable and observable state of model, when insulin is an input at

initial condition (80,21,900, 300,0.16)

Figure 5.2: Controllable and observable steady state of model, when insulin is an

input at initial condition (120,10,5, 3,0.1)


For Case 2:Here we take the only measured output of glucose and the only input is

insulin and α− cellthen

B = [0, 1, 0, 1, 0]T C = [1, 0, 0, 0, 0]

where

A =

−1.44 −510 0 0 1350

0 −432 40.926 0 0

0 0 0.01 0 0

0 0 0 0.001 0

0 0 0 4 −0.3

The controllability matrix is

R = [B; AB; A2B; A3B; A4B]

rank(R) = rank([B; AB; A2B; A3B; A4B]) = 4

The observability matrix is

O = [C; CA; CA2; CA3; CA4]

rank(O) = rank([C; CA; CA2; CA3; CA4]) = 5

The rank of controllability matrix and observability matrix is 4 and 5 respectively. Hence

the system is observable but not controllable. Figures 5.3 and 5.4 represents the controllable

and observable state with different initial conditions of state variables, when glucagon is

an input with insulin its form close loop to stabilize the glucose level.


Figure 5.3: Controllable and observable state of model, when insulin and glucagon

are an input at initial condition (120,21,900, 300,0)

Figure 5.4: Controllable and observable state of model, when insulin and glucagon

are an input at initial condition (80,0,900, 300,0.16)


5.5 Controller Design

Tracking step reference signal with zero steady state error is possible if and only if loop

transfer function contain one transfer function contains at least one integral. Figure 5.5

represents the simulink constructed on linear model (5.11− 5.15), which shows the contin-

uous measuring of glucose, insulin and glucagon with effect of β-cell, α-cell mass. Glucose,

insulin and glucagon show the positive behaviour in measurement on its approaching val-

ues.

Figure 5.5: Simulink to measure Glucose, insulin and glucagon with effect of β-cell,

α-cell mass

The transfer function can be determined by the system identification in the MATLAB


after importing these signals into identifying the Graphical User Interface (GUI) in the

MATLAB workspace. Also, the design control for this transfer function will be easy and

accurate. S−function in this paper shows artificial pancreas [47, 48]. The transfer function

of this linear system thus will be a rational function

G(s) =c(s)

d(s)

where

c(s) = c0sn + c1s

n−1 + ...+ cn−1s+ cn

and

d(s) = sn + d1sn−1 + ...+ dn−1s+ dn

are polynomials function. In facts we obtain results for control system

x(s) = (sI −A)−1(B(u(s) + x0))

y(s) = C(sI −A)−1Bu(s) + C(sI −A)−1x0) +Du(s)

where matrix

G(s) = C(sI −A)−1B +D

is the system of transfer matrix. We have used MATLAB symbolic math toolbox, following

transfer functions are obtained, when insulin is input only We have

G(s) =−1.705e−13s2 − 0.04045s− 0.01213

s3 + 433.6s2 − 1.954e4s− 8.748e6

continuous-time transfer function. When insulin and glucagon are input

G1(s) =1.705e−13s2 + 1350s+ 5.832e5

s3 + 433.6s2 − 1.954e4s− 8.748e6

continuous-time transfer function. Pole-zero diagram of these transfer functions and fre-

quency response are shown in Figures 5.6 and 5.7 for open loop.


Figure 5.6: Pole zero diagram of transfer function G(s)

Figure 5.7: Frequency response transfer function G(s)


5.6 Conclusions

The obtained results will have some impact on attempts to construct an artificial pancreas.

Until now primarily one tries to implement a feedback control by taking glucose as the only

input providing information on the system state and by using administration of insulin as

the only control input. The use of insulin as the only control input may be insufficient

in case of hypoglycemia because a low glucose level cannot be raised by insulin action.

It seems to be that in order to restore the desired glucose level that one has to include

administration of glucagon as another control input. For instance, in this model, we can

see that the system improves its controllability when we consider glucagon as second input

along with insulin. In our opinion, one has to develop models for the glucose-insulin-

glucagon system which pay special attention to the control mechanisms of the systems

first for the health system in order to understand the working of the control loops in the

health system which should be restored by the artificial pancreas. Our results indicate

that such a feedback control will have to use administration of insulin and glucagon by

a combined insulin-glucagon pump. An algorithm for an artificial pancreas in simulink

and used the transfer function of the model for Proportional-Integral-Derivative (PID)

controller is designed. Controllability and observability of linear system are discussed for

this purpose. The system is not controllable but observable for case 1 and 2. When we

considered a glucagon as input with insulin rank of controllability matrix is improved and

its rank is 4. Only one variable is not controllable. To overcome that problem we discussed

the state of the model where the model is completely controllable. A control system can

only be used in the form of closed-loop control to stabilize the system [49].

Chapter 6

Sorenson Model for Type 1Diabetes Mellitus

In this chapter, convert the Sorenson’s model for T1DM because this is the most compre-

hensive model in the Glucose Insulin Glucagon dynamics for human but the result show

that this model has some deficiency in original model since the equilibrium point is not in

feasible region. We treated Sorensen’s Model for T1DM to check the linear controllability

and observability.

6.1 Sorensen’s Model For Type 1 Diabetes

The physiological compartments of the human body are classified in six category: brain,

heart, periphery, gut, liver and kidney. Arrows joining the physiological compartments

represent the direction of blood flow. The heart and lungs compartment serves to close

the circulatory loop, representing simply the blood volume of the cardiopulmonary system

and the major arteries. Figure 6.1, which represents the mass balance of 8 ODE’s in

each compartment results with linear and nonlinear terms that are related to each specific

metabolic rate. In Insulin model, the mass balance in each compartment results in 7 ODE’s

with linear and nonlinear terms represents in figure 6.2, which are related to each specific

71

CHAPTER 6. SORENSON MODEL FOR TYPE 1 DIABETES MELLITUS 72

metabolic rate and glucagon mode is 1 ODE with linear and nonlinear terms, which are

related to each specific metabolic rate represents in figure 6.3.

Figure 6.1: Schematic representation of the Glucose Model

Figure 6.2: Schematic representation of the Insulin Model


Figure 6.3: Schematic representation of the Glucagon Model

Followings are the equations of the model

V GBV GBV = QGB(GH −GBV )− VBI

TB(GBV −GBI) (6.1)

VBIGBI =VBITB

(GBV −GBI)− rBGU (6.2)

V HG GH = QGBGBV +QGLGL +QGKGK +QGPGPV +QGHGH − rRBGU (6.3)

V GG GG = QGG(GH −GG)− rGGU (6.4)

V GL GL = QGAGH +QGGGG −QGLGL + rHGP − rHGU (6.5)

V GK GK = QGK(GH −GK)− rKGE (6.6)

V GPV GPV = QGP (GH −GPV )− VPI

TGP(GPV −GPI) (6.7)

VPIGPI =VPITGP

(GPV −GPI)− rPGU (6.8)


V IB IB = QIB(IH − IB) (6.9)

V IH IH = QIBIB +QILIL +QIKIK +QIP IPV +QIHIH (6.10)

V IGIG = QIG(IH − IG) (6.11)

V IL IL = QIAIH +QIGIG −QILIL − rPIR − rLIC (6.12)

V IK IK = QIK(IH − IK)− rKIC (6.13)

V IPV IPV = QIP (IH − IPV )− VPI

T IP(IPV − IPI) (6.14)

VPI IPI =VPIT IP

(IPV − IPI)− rPIC (6.15)

V ΓΓ = rPΓR − rPΓC (6.16)

6.1.1 Description of Variables

G Glucose concentration (mg/dl) T Diffusion rate (min)

Q Vascular Plasma flow rate (dl/min) V Volume (dl)

r Metabolic source and sink rate (mg/min) M Multiplier of basal MR

τ Time constant (min) I Insulin concentration (mg/dl)

F Fractional clearance (dimensionless) t Time constant (min)

Γ Glucagon concentration (pg/ml) B Brain

Table 6.1: Description of variables


6.1.2 First Subscript: Physiological Compartment

G Gut H Heart and Lung

L Liver K Kidney

P Periphery A Hepatic artery

G Glucose I Insulin

Γ Glucagon B Basal value

I Interstitial fluid space V Vascular plasma space

N Normalized value B Brain

Table 6.2: Physiological Compartment

6.1.3 Second Subscript: Physiological Compartment

BGU Brain glucose uptake GGU Gut glucose utilization

HGP Hepatic glucose production HGU Hepatic glucose uptake

KGE Kidney glucose excretion PGU Peripheral glucose uptake

RBCU Red blood cell glucose uptake KIC Kidney Insulin clearance

LIC Liver insulin clearance PIR Peripheral insulin release

PIC Peripheral insulin clearance PΓC Plasma glucagon clearance

MΓC Metabolic glucagon clearance PΓR Pancreatic glucagon release

Table 6.3: Physiological Compartment

Following is the table of parameter’s and constant values given in [50, 13] used in model


Table 6.4: Table of parameter’s and constant value of the model

Parameter Value Parameter Value

QGR 5.9dlmin−1 QG

H 43.7dlmin−1

QGA 2.5dlmin−1 QG

L 12.6dlmin−1

QGG 10.1dlmin−1 QG

K 10.1dlmin−1

QGP 15.1dlmin−1 QG

BV 3.5dl

V GH 13.8dl V G

L 25.1dl

V GG 11.2dl V G

K 6.6dl

V GPV 10.4dl VPI 67.7dl

VBI 4.5dl TGP 5.0min

TR 2.1min V Γ 11310ml

V IB 0.26l QI

A 0.18l/min

T IP 20min V IH 0.99l

QIR 0.45l/min V I

G 0.94l

QIH 3.12l/min V I

L 1.14l

QII 0.90l/min V I

K 0.51l

QIK 0.72l/min V I

PV 0.74l/min

V IPI 6.74l V I

G 0.72min−1

QIP 1.05l/min α 0.00482min−1

β 0.931lmin−1 K 0.00794min−1

M1 0.00747l/min M2 0.0958U

γ 0.575l/min Q0 6.33dlmin−1

6.1.4 Metabolic source and sink

rBGU = 70mg/min(Constant), rRBCU = 10mg/min(Constant),

rGGU = 20mg/min(Constant), rPGU = M IPGUM

GPGUτ

BPGU ,


τBPGU = 35mg/min, τBHGP = 155mg/min, τΓ = 65min, FLIC = 0.40, FKIC = 0.30,

FPIC = 0.415, rBPIR = 4mU/min, rMΓC = 9.10,MGPGU = GNPI

M IPGU = 7.03 + 6.52tanh[0.338(INPI − 5.82)]

rHGP = M IHGPM

ΓHGPM

GHGP τ

BHGP

M IHGP =

1

τI(M I∞

HGP −MIHGP ) (6.17)

M I∞HGP = 1.21− 1.14tanh[0.62(GLN − 0.89)]

MΓHGP = MΓ0

HGP − f2

MΓ0HGP = 2.7tanh[0.39ΓN ]

f2 =1

τΓ[(MΓ0HGP − 1

2)− f2] (6.18)

MGHGP = 1.42− 1.41tanh[1.66(INL − 0.497)]

rHGU = M IHGUM

GHGUτ

BHGU

τBHGU = 20mg/min

M IHGU =

1

τI[M I∞

GHU −MIGHU ] (6.19)

M I∞HGU = 2.0tanh[0.55INL ]


MGHGU = 5.66 + 5.66tanh[2.44(GNL )− 1.48]

rKGE = {71 + 71tanh[0.11(GK − 460)], 0 < GK < 460mg/min

−330 + 0.83GK , GK > 460mg/min

rLIC = FLIC [QIAIH +QIGIG + rPIR]

rKIC = FKIC [QIKIK ]

rPIC =IIP

[(1−FPICFPIC

)( 1QI

P

− T IP

VPI)]

rPIR =S(GH)

S(GBH)rBPIR

P = α[P∞ − P ] (6.20)

I = β[X − I] (6.21)

Q = K[Q−Q0] + [γP − S] (6.22)

S = [M1Y +M2(X − I)0+]Q

S =(GH)3.27

(132)3.27 + 5.93(GH)3.02

P∞ = Y = (X)1.11

rPΓC = rMΓCΓ

rPΓR = MGPΓRM

IPΓRτ

BPΓR

τBPΓR = rMTCΓB

MGPΓR = 2.93− 2.10tanh[4.18(GNH − 0.61)]

M IPΓR = 1.31− 0.61tanh[1.06(INH − 0.47)][6]


6.2 Modified Form of Model in Type 1 Diabetes

Mellitus

In this section model (6.1− 6.22), convert Sorenson’s model into Type 1 Diabetes Mellitus

[51]. The nomenclature, basal values, parameters values and metabolic rates are same like

non diabetic model. After eliminating pancreatic insulin released model and rPIR due to

type 1 diabetes mellitus and substitution of parameters and basal values given in table 6.1,

the model ends up 19 ordinary differential equations and takes the form

GBV = 1.685GH − 2.297GBV + 0.612GBI (6.23)

GBI = 0.476GBV − 0.476GBI − 15.555 (6.24)

GH = 0.427GBV + 0.913GL + 0.731GK + 1.094GPV − 3.166GH − 0.724 (6.25)

GBI = 0.901(GH −GG)− 1.785 (6.26)

GL = 0.099GH + 0.402GG − 0.501GL + 6.175M IHGP (2.7tanh(0.389Γ)− f2) (6.27)

(1.42− 141tanh((0.006GL − 0.31))− 4.5M IHGU (1 + tanh(0.024GL − 3.61)))

GK = 1.53GH − 1.53GK − 10.721− 10.721(0.11GK − 50.6) (6.28)

GPV = 1.451GH − 2.748GPV + 1.296GPI (6.29)

GPI = 0.2GPV − 0.2GPI − 0.005GPI(7.03 + 6.52tanh(0.015IPI − 1.967)) (6.30)

M IHGP = −0.04M I

HGP + 0.048− 0.045tanh(0.077IL − 1.477) (6.31)


M IHGU = −0.04M I

HGU + 0.08tanh(0.025IL) (6.32)

f2 = −0.015f2 − 0.007 + 0.02tanh(0.389Γ) (6.33)

IB = 1.73IH − 1.73IB (6.34)

IB = 0.454IB + 0.909IL + 0.727IK + 1.06IPV − 3.151IH (6.35)

IG = 0.765IH − 0.765IG (6.36)

IL = 0.094 + 0.378IG − 0.789IL (6.37)

IK = 1.411IH − 1.835IK (6.38)

IPV = 1.418IH − 1.874IPV + 0.455IPI (6.39)

IPI = 0.05IPV − 0.111IPI (6.40)

Γ = −0.08Γ + 0.08(2.93− 2.10tanh(0.041GH − 2.55))

(1.31− 0.61tanh(1.31− 0.61tanh(0.049IH − 0.429) (6.41)

For equilibrium the left hand side of the equations (6.23− 6.41) are substituted zero. By

algebraic manipulations we can represent all the equations as a function of either GK or IB

separately. The model represent a type 1 diabetes mellitus subject so it is not surprising to

take insulin concentration in all compartments zero. The equation for kidney compartment

provides GK = 197.10mg/d1. The uniqueness of values provide in [52]. Hence we get a

unique point of equilibrium for the Sorenson Model in type 1 diabetes mellitus.

(185.2, 152.5, 197.1, 195.1, 207.7, 197.1, 193.6, 189.9, 2.33, 0, 0.1, 0, 0, 0, 0, 0, 0, 0, 1.3).


6.2.1 Linearised Model

Linearized model is given as

GBV = 1.685GH − 2.297GBV + 0.612GBI

GBI = 0.476GBV − 0.476GBI

GH = 0.427GBV + 0.913GL + 0.731GK + 1.094GPV − 3.166GH

GBI = 0.901(GH −GG)

GL = 0.099GH + 0.402GG − 0.563GL + 2.755M IHGP − 8.467M I

HGU − 5.299f2 + 4.354Γ

GK = 1.53GH − 1.53GK

GPV = 1.451GH − 2.748GPV + 1.296GPI

GPI = 0.2GPV − 0.204GPI − 0.007IPI

M IHGP = −0.04M I

HGP + 0.007IL

M IHGU = −0.04M I

HGU + 0.002IL

f2 = −0.015f2 − 0.006Γ

IB = 1.73IH − 1.73IB

IB = 0.454IB + 0.909IL + 0.727IK + 1.06IPV − 3.151IH


IG = 0.765IH − 0.765IG

IL = 0.094IH + 0.378IG − 0.789IL

IK = 1.411IH − 1.835IK

IPV = 1.411IH − 1.874IPV + 0.455IPI

IPI = 0.05IPV − 0.111IPI

Γ = −0.08Γ + 0.0016IH − 0.00000069GH

6.2.2 Stability Analysis

The linear x(t) = A(t)x(t), where A(t) continuous and bounded for t ≥ t0, is uniformly

asymptotically stable if and only if given a positive definite real matrix A(t), there exists a

symmetric positive definite real matrix P (t), which satisfies P (t)+Aτ (t)P (t)+P (t)A(t) =

−Q(t),t ≥ t0. The linear time invariant system x(t) = A(t)x(t) the corresponding equation

to be used as AτP + PA+Q = 0. This is called Lyapunov equation [32, 33].

Here A is matrix of coefficients of above linearized model and Q = I, where I is an

identity matrix with the same order of A. By using the equation AτP + PA + Q = 0 on

Matlab. We find matrix P and its det(P ) = 4.8878e11 which shows that P is symmetric

positive definite real matrix P . The eigen values of A are

(-4.6799,-2.5946,-2.0129,-1.0195 + 0.3193i,-1.0195 - 0.3193i,-0.3518,-0.1949, -0.0118,-

0.0152,-0.0800,-0.0400,-4.4302,-1.7618,-1.8580,-0.9559 + 0.3134,-0.9559 - 0.3134i,-0.2154,-

0.0779,-0.0400) negative real roots. Hence prove that the system is uniformly asymptoti-

cally stable.


6.3 Results and Discussions

Achieving and maintaining normal blood glucose concentrations are critical issues for suc-

cessful long term care of patients with diabetes mellitus. Serious attention is needed to

maintain blood glucose level as close to the nondiabetic range as possible in individuals

with T1DM to reduce the development and progression of micro-vascular and cardiovascu-

lar complications. Till now we don’t have a fully automated artificial pancreas. An effort

is made to answer the hurdles of having a fully automated artificial pancreas, since we need

to have a feedback control for the system. First step in this direction is to check if we can

stabilize the system by choosing an appropriate feedback control.

A mathematically linear control system is given by the following two equations

x = Ax+Bu,

y = Cx,

Here x = [GBV GBI GH GG GL GK GPV GPI MIHGP M

IHGU F2 IB IB IH IL IK IPV IPI Γ]τ

and B = [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0] and C = [0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0].

The n × np controllability matrix is given by R = [B,AB,A2B,A3B, ..., An−1B]. The

rank(i.e. rank(R)=n), so the system is said to be controllable. The nk×np the observability

matrix is given by O = [C,ACA;CA2;CAn − 1]τ . The rank (i.e. rank(R)=n) , so the

system is said to be observable [13, 21, 53].

6.3.1 Case I:

If we consider the insulin concentration in periphery vascular blood space as in-

puts and glucose concentration in extracellular fluid as the only output than B =

[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0] and C = [0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0]. R is

the controllability matrix, where R = [B,AB,A2B,A3B, ..., A18B] and its rank is 12. The


observability matrix is O = [C,CA;CA2;CA3;CA4, ..., CA18] and its rank is 11. Hence

the linear system is neither controllable nor observable.

6.3.2 Case II:

If we consider the insulin concentration in periphery vascular blood space and glucagon

concentration as inputs and glucagon concentration in extracellular fluid as the only output

than B = [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0] and C = [0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0].

In this case rank of controllability and observability matrix is 13 and 11 respectively.

Stability analysis of A is checked by using Lyapunov equation and found that the system

is stable because P is definite positive matrix and eigen values of A are negative real roots.

So the system is uniformly asymptotically stable. Two cases are discussed according to

input value, for case 1, insulin is an input only and glucose is output. For case 2, insulin

and glucagon are input and glucose as an output only.

6.4 Conclusion

The results obtained in this paper will have some impact on attempts to construct the

artificial pancreas. Until now primarily one tries to implement a feedback control by

taking glucose as the only input providing information on the system state and by using

administration of insulin as the only control input. The use of insulin as the only control

input may be insufficient in case of hypoglycemia, because a low glucose level cannot be

raised by insulin action. In our opinion one has to develop models for the glucose-insulin-


first for the healthy system in order to understand the working of the control loops in the

healthy system which should be restored by the artificial pancreas. Our results indicate

that such a feedback control will have to use administration of insulin and glucagon by a


combined insulin-glucagon pump.

The system is uniformly asymptotically stable by using the Lyapunov equation because

eigen values of A are all negative real roots. We treat the linear system because if a

linear system is controllable and observable, then a nonlinear system may or may not be

controllable and observable. If a nonlinear system is neither controllable nor observable

then linear may or may not be. This is the most comprehensive model in the history of

Glucose Insulin Glucagon systems. Results show that the deficiency of the model can be

improved if glucagon is used as input along with insulin. For this purpose, we check the

controllability and observability of the system which further can be treated to design the

feedback control for fully automatic artificial pancreas. It was concluded that controllability

and observability is improved by considering glucagon as another input since glucagon plays

an important role in glucose regulatory systems. The system is neither controllable nor

observable in both cases. The situation is improved when we consider insulin and glucagon

as inputs than only insulin but still system is not controllable and observable thus we

cannot design the feedback control for fully automatic artificial pancreas.

The discussion in the conclusion show that the following tasks should be considered for

future research. Development and validation of a comprehensive model for the system

including the important control mechanisms involving insulin and glucagon for the healthy

system [54].

Chapter 7

Control of Composite Model

Man may be the captain of his fate, but he is also

the victim of his blood sugar.(Wilfrid Oakley)

Artificial pancreas is one of the solutions to control the type1 diabetes mellitus now a day.

Effort is being made to develop the desired artificial pancreas for this purpose. Article

presents the controllability and observability of the glucose insulin glucagon dynamical

model which is the modified form of composite model of glucose insulin glucagon dynam-

ics for type 1 diabetes mellitus. The concept of controllability and observability for the

linearized control system of human glucose insulin systems is used so that we can have

a feedback control. This model completely describes the glucagon effect in the safety of

artificial pancreas to overcome the risk of hyperglycemia. Composite model of glucagon-

glucose dynamics and its extension is treated for type 1 diabetes mellitus to check the

linear controllability and observability. Two cases discussed in system according to the

input. For case I, insulin as the input only and glucose as an output and in case II insulin

and glucagon as an input and glucose as output only.

86

CHAPTER 7. CONTROL OF COMPOSITE MODEL 87

7.1 Material and Method

7.1.1 Composite Model Glucagon-Glucose Dynamics

Consider the composite model having combined form of minimal model, Hovorka model

and glucagon sub model. Figure 7.1 graphically shows the structure of proposed glucagon-

extended minimal model in composite model [55].

Figure 7.1: The glucagon extended Minimal model

Pharmacodynamics on the net endogenous glucose production, insulin absorption

model to add the effects of subcutaneous glucagon, insulin delivery on plasma insulin

and glucagon concentration is described in glucagon glucose composite model. The model

describe in following equations

G(t) = −[SG +X(t)− Y (t)]G(t) + SGGb +RaV

(7.1)


Y (t) = −p3Y (t) + p3SN [N(t)−Nb] (7.2)

X(t) = −p2X(t) + p2S1[I(t)− Ib] (7.3)

F (t) =1

tmaxG(−F (t) +AGDG) (7.4)

Ra(t) =1

tmaxG(−Ra(t) + F (t)) (7.5)

I(t) = −keI(t) +S2(t)

V1tmax1(7.6)

S1(t) = u(t)− S1(t)

tmax1(7.7)

S2(t) =S1(t)− S2(t)

tmax1(7.8)

N(t) = −kNN(t) +Z2(t)

VN tmaxN(7.9)

Z1(t) = w(t)− Z2(t)

tmaxN(7.10)

Z2(t) =Z1(t)− Z2(t)

tmaxN(7.11)

Where G, I,N,X and Y represents the plasma glucose concentration, plasma insulin

and plasma glucagon concentration, insulin and glucagon action on glucose production

respectively with time t. Where P3 = k8 , SN = k7/k8k9 are constant parameters

in the model. Where Ra and F are the plasma appearance of glucose and glucose

appearance in the first compartment respectively. Where V is measure as the glucose

distribution volume (dI/kg). ke, u1(t), VI and tmaxI represent the insulin clearance in


the plasma, subcutaneous insulin infusion rate, the distribution volume of plasma insulin

and absorption time constant of insulin. S1 and S2 represents the two compartments

subcutaneously administrated insulin. Where KN is the first order decay for glucagon

in plasma, w is glucagon infusion rate, VN is the distribution volume of plasma, Z1 and

Z2 are two-compartment resented absorption of subcutaneously administered glucagon [55].

The data is classified into three subjects 117-1, 126-1, and 128-1 according to different

parameters values in each case. In each subject we have two cases, for case I: Insulin as the

input only and glucose as an output and in case II: insulin and glucagon as an input and

glucose as output only. The only measured output is concentration of glucose in plasma

that we can easily measure. The system is neither controllable nor observable in both

cases for every subject. But the rank of the matrix is improved in each subject when we

consider insulin and glucagon as inputs than only insulin but still system is not controllable

and observable thus we cannot design the feedback control for fully automatic artificial

pancreas. Now considering the modified form of that composite model for controllability

and observability of the model for artificial pancreas.

7.2 Extended Form of Composite Model

The glucose kinetics is described by a system of differential equation in the form

G(t) = −[SG +X(t)− Y (t)]G(t) + SGGb +D2(t)

tmaxGV(7.12)

X(t) = −p2X(t) + p2SI [I(t)− Ib] (7.13)

Y (t) = −p3Y (t) + p3SN [N(t)−Nb] (7.14)


Where G, I,N,X and Y represent the plasma glucose concentration, plasma insulin

and plasma glucagon concentration, insulin and glucagon action on glucose production

respectively with time t. SG represents the fractional glucose effectiveness, SI and SN are

insulin and glucagon sensitivities. p2, p3 are the constant parameters describing the insulin

and glucagon action. Ib and Nb are basal values of insulin and glucagon. V is the volume

of glucose distribution and tmaxG rate appearance in glucose plasma [56].

7.2.1 Gastrointestinal Absorption Model

The glucose rate of appearance into systematic circulation after meal is described by gas-

trointestinal absorption model as well as glucagon extended minimal model [57]. The

equation representing such a model is

D1(t) =1

tmaxG(−D1(t) +AGDG) (7.15)

D2(t) =1

tmaxG(−D2(t) +D1(t)) (7.16)

Where D1(t) and D2(t) represent the first and second compartment of glucose. AG

and DG show the carbohydrate bioavailability and intake of carbohydrates per kg of body

weight respectively.

7.2.2 Subcutaneous Insulin Absorption Model

The equations of such a model are

I(t) = −keI(t) +S2(t)

V1tmaxI(7.17)

S1(t) = u1(t)− S1(t)

VItmaxI(7.18)


S2(t) =S1(t)− S2(t)

VItmaxI(7.19)

Where ke represent the insulin clearance in the plasma, u1(t) is subcutaneous insulin

infusion rate, VI describes the distribution volume of plasma insulin and tmaxI absorption

time constant of insulin. S1 and S2 represents the two compartments subcutaneously

administrated insulin [56].

7.2.3 Subcutaneous Glucagon Absorption Model

Pharmacokinetics of subcutaneous glucagon absorption has same structure as Hovarka and

coauthors proposed in the subcutaneous insulin absorption. Plasma glucagon concentration

N was estimated as

N(t) = −kNN(t) +Z2(t)

VN tmaxN(7.20)

Z1(t) = u2(t)− Z1(t)

tmaxN(7.21)

Z2(t) =Z1(t)− Z2(t)

tmaxN(7.22)

Where KN is the first order decay for glucagon in plasma, u2 is glucagon infusion

rate, VN is the distribution volume of plasma, Z1 and Z2 are two-compartment resented

absorption of subcutaneously administered glucagon [55]. Hence the model takes the form

in the following equations

G(t) = −[SG +X(t)− Y (t)]G(t) + SGGb +D2(t)

tmaxGV(7.23)

X(t) = −p2X(t) + p2SI [I(t)− Ib] (7.24)

Y (t) = −p3Y (t) + p3SN [N(t)−Nb] (7.25)


D1(t) =1

tmaxG(−D1(t) +AGDG) (7.26)

D2(t) =1

tmaxG(−D2(t) +D1(t)) (7.27)

I(t) = −keI(t) +S2(t)

V1tmaxI(7.28)

S1(t) = u1(t)− S1(t)

VItmaxI(7.29)

S2(t) =S1(t)− S2(t)

VItmaxI(7.30)

N(t) = −kNN(t) +Z2(t)

VN tmaxN(7.31)

Z1(t) = u2(t)− Z1(t)

tmaxN(7.32)

Z2(t) =Z1(t)− Z2(t)

tmaxN(7.33)

This is the table of parameters use in each subject data for close loop artificial pancreas

are given in [56, 55].

Table 7.1: Table of parameters and constant value of the model

Parameter Value Parameter Value

Ke 0.1381/min VI 0.12ml/kg

p2 0.0283441/min SN 0.02179min−1/pg/ml

tmaxI 55Min p3 0.000051/min

Ib 11.01U/ml tmaxG 69.6Min

KN 0.621/min u2(t) 70ng/kg

VN 16.06ml/kg u1(t) 4U/kg

tmaxN 32.46Min DG 20Mg

Nb 46.30pg/ml AG 0.8unitless

SI 6.40e−4min− 1/U/ml V 1.7dl/kg

Gb 100mg/dl SG 0.0141/min


After Substitution Parameter value, we have

G(t) = −0.0896G− 6.4XG+ 6.4Y G+D2 + 8.96 (7.34)

X(t) = −0.028344X + 0.003324I − 0.036589 (7.35)

Y (t) = −0.00005Y + 1.0895× 10−7N − 5.044385× 10−5 (7.36)

D1(t) =1

69.6(−D1 + 16) (7.37)

D2(t) =1

69.6(−D2(t) +D1) (7.38)

I(t) = −0.138I +S2

6.6(7.39)

S1(t) = 4− S1

55(7.40)

S2(t) =S1 − S2

55(7.41)

N(t) = −0.62N +Z2

520.0092(7.42)

Z1(t) = 70− Z1

32.46(7.43)

Z2(t) =Z1 − Z2

32.46(7.44)

For equilibrium points are in the above system (7.34− 7.44) put L.H.S. equal to zero

G(t) = X(t) = Y (t) = D1(t) = D2(t) = I(t) = S1(t) = S2(t) = N(t) = Z1(t) = Z2(t) = 0

By using the Mathematica software tools find the equilibrium points. Hence the equilibrium

point is

(G,X, Y,D1, D2, I, S1, S2, Z1, Z2, N) = (0.1398, 27.0273,−0.855691, 16, 16,

241.546, 220, 220, 2272.2, 2272.24, 7.0301)

In equilibrium point glucose, insulin and glucagon are concentrations in plasma are positive,

so the system is stable and feasible for controllability and observability.


7.2.4 Linearized System

The linearized system about equilibrium point is

G(t) = −178.5407G− 0.89472X + 0.8947Y +D2

X(t) = −0.028344X + 0.003324I

Y (t) = −0.00005Y + 1.0895× 10−7N−

D1(t) =D1

69.6

D2(t) =1

69.6(−D2(t) +D1)

I(t) = −0.138I +S2

6.6

S1(t) = −S1

55

S2(t) =S1 − S2

55

N(t) = −0.62N +Z2

520.0092

Z1(t) = − Z1

32.46

Z2(t) =Z1 − Z2

32.46

7.3 Stability Analysis

Theorem 7.1: The linear x(t) = Ax, where A continuous and bounded for t ≥ t0 , is

uniformly asymptotically stable if and only if given a positive definite real matrix A, there

exists a symmetric positive definite real matrix P (t), which satisfies


The linear time invariant system x(t) = Ax the corresponding equation to be used as



Proof:

Here A =

178.58 −0.8947 0.8972 0 1 0 0 0 0 0 0

0 −0.028 0 0 0 0.0033 0 0 0 0 0

0 0 .00005 0 0 0 0 0 1.089× 10−7 0 0

0 0 0 −0.014 0 0 0 0 0 0 0

0 0 0 0.014 −0.0144 0 0 0 0 0 0

0 0 0 0 0 −0.138 0 0.1515 0 0 0

0 0 0 0 0 0 −0.0181 0 0 0 0

0 0 0 0 0 0 0.0181 −0.0181 0 0 0

0 0 0 0 0 0 0 0 −0.62 0 0.00192

0 0 0 0 0 0 0 0 0 −0.0308 0

0 0 0 0 0 0 0 0 0 0.0308 −0.0308

coefficients of above linearized model and Q = I, where I is an identity ma-

trix with the same order of A. By using the equation ATP + PA + Q = 0

on Matlab, compute matrix P and det(P ) = 1.4221e10 which shows that

P is symmetric positive definite real matrix. The eigen values of A are

(−178.5400,−0.0283,−0.0001,−0.1430,−0.1430,−0.1380,−0.0181,−0.0181,−0.6200,−0.0308,−0.0308)

negative real roots. Hence prove that the system is uniformly asymptotically stable.

7.4 Linear Control

7.4.1 Results and Discussion

Achieving and maintaining normal blood glucose concentrations are critical issues for

successful long term care of patients with diabetes mellitus. Serious attention is needed to

maintain blood glucose level as close to the non-diabetic range as possible in individuals

with type 1 diabetes mellitus to reduce the development and progression of micro-vascular

and cardiovascular complications. Till now we don’t have a fully automated artificial


pancreas. An effort is made to answer the hurdles of having a fully automated artificial

pancreas, since we need to have a feedback control for the system. First step in this

direction is to check if we can stabilize the system by choosing an appropriate feedback

control. A mathematically linear control system is given by the following two equations

x = Ax+Bu

y = Cx

Here x = [G,X, Y,D1, D2, I, S1, S2, Z1, Z2, N ]T and B = [00000100000]T

and C = [100000000000]. The n × np controllability matrix is given by

R = [BABA2BA3B...An−1B]. The rank (i.e. rank(R) = n), so the system

is said to be controllable. The nk × n is the observability matrix is given by

O = [C;CA;CA2;CA3;CA(n− 1)]T . The rank (i.e. rank(R) = n) , so the system is said

to be observable [21, 22, 53].

Theorem 7.2: let ρ(B) = p, The pair (A,B) are said to be controllable if

Gcn−p+1 = [BABA2BA3B...An−pB] has full row rank. This is equivalent to

Gcn−p+1.G(n−p+1)c being nonsingular and G(n−p+1)c.G(n−p+1)c > 0 (positive definite).

Theorem 7.3: let ρ(C) = p, The pair (A,C) are said to be observable if

Gcn−p+1 = [CCACA2CA3BCA(n − p)C]T has full row rank. This is equivalent to

Gcn−p+1.Gcn−p+1 being nonsingular and Gcn−p+1.G(n−p+ 1)c > 0 (positive definite)[21, 53]

We have two cases in the system, for case I: Insulin as the input only and glucose as

an output and in case II: insulin and glucagon as an input and glucose as output only.

For Case I: If we consider the insulin concentration in periphery vascular


blood space as inputs and glucose concentration in extracellular fluid as the

only output than B = [00100000000]T and C = [100000000000] The control-

lability matrix is R = [BABA2BA3BA4B...A10B] and the observability matrix

is O = [C;CA;CA2;CA3;CA4; ..., CA10]. By using the property of controlla-

bility Det(R × R) = 0, so the controllability matrix is not controllable and

det(O × O) = 1.76614e−019, matrix is nonsingular so the observability matrix is

observable. Hence the system is observable but not controllable.

For case II: If we consider the insulin concentration in periphery vascular blood space

and glucagon concentration as inputs and glucagon concentration in extracellular fluid as

the only output than B = [00100000100]T and C = [100000000000]. By using theorem

7.3 of observability Det(O × O) = 0, so the controllability matrix is not controllable and

det(O×O) = 1.76614e−019, matrix is nonsingular so the observability matrix is observable.

Hence the system is observable but not controllable.

7.5 Conclusion

For instance, in case of extended form of composite model for type 1diabetes mellitus, we

can see that the system improves its controllability when we consider glucagon as second

input along with insulin. In our opinion one has to develop models for the glucose-insulin-


first for the healthy system in order to understand the working of the control loops in the

healthy system which should be restored by the artificial pancreas. The system is uniformly

asymptotically stable by using the lyapunov equation because eigen values of matrix A are

all negative real roots. In each subject we have two cases, for case I: Insulin as the input

only and glucose as an output and in case II: insulin and glucagon as an input and glucose


as output only. First model is neither controllable nor observable in both cases. But the

second modified from of the composite model is observable but not controllable. The only

measured output is concentration of glucose in plasma that we can easily measure. But the

rank of the matrix is improved in model when we consider insulin and glucagon as inputs.

It means glucagon play an important role in the development of an artificial pancreas for

peoples types 1 diabetes. Our results indicate that such a feedback control will have to use

administration of insulin and glucagon by a combined insulin-glucagon pump.

Chapter 8

Conclusion and Future Work

Long term treatment plan of diabetic patient for regulation of blood glucose level is

critical. To overcome the complications of T1DM, blood sugar level must be maintained.

Unfortunately, therapy to overcome the complications is difficult to handle by patients

because it includes self-monitoring of BGL, multiple daily injections of insulin and use

of insulin pump. Hyperglycemia and hypoglycemia is most common complications in

type 1 diabetes patients even after insulin bolus therapy or continuous delivery of insulin

throughout the day. Hypoglycemia is dangerous condition. Artificial pancreas is used to

regulate BGL and have no negative consequences in T1DM patients.

The results of our calculations strongly indicate that it will not be possible to design a

stabilization feedback control if insulin is the only one input variable and glucose the only

output variable. Since the controllability and observability are sufficient conditions for the

existence of a stabilizing feedback control (obtained as a solution linear-quadratic regulator

problem), further investigations should be also the lowest detectability and stabilizability

properties for the models, which provide a set of necessary and sufficient conditions exis-

tence of a stabilizing control.

The results obtained in this thesis will have some impact on attempts to construct

99

CHAPTER 8. CONCLUSION AND FUTURE WORK 100

the artificial pancreas. Until now primarily one tries to implement a feedback control by

taking glucose as the only output providing information on the system state and by using

administration of insulin as the only control input. Glucose concentration is measured

subcutaneously by an implanted sensor which measures glucose concentration in the inter-

stitial fluid and not directly in blood. Therefore one will need a submodel describing the

relation between glucose concentrations in blood and in interstitial fluid. The use of in-

sulin as the only control input may be insufficient in case of hypoglycaemia, because a low

glucose level cannot be raised by insulin action. It seems to be that in order to restore the

desired glucose level that one has to include administration of glucagon as another control

input. For instance, in case of Sorensen’s model, GIG model for T1DM we can see that the

system improves its controllability when we consider glucagon as second input along with

insulin. In our opinion one has to develop models for the glucose-insulin-glucagon system

which pay special attention to the control mechanisms of the systems first for the healthy

system in order to understand the working of the control loops in the healthy system which

should be restored by the artificial pancreas. This means that in a second step one has

to reduce such a model to the case of T1DM in order to construct a stabilizing feedback

control for the model which presumably also works for the patient. Our results indicate

that such a feedback control will have to use administration of insulin and glucagon by

a combined insulin-glucagon pump. Despite the increased complexity of such a system

we see that this approach is already pursued by some groups as for instance at Boston

University and Massachusetts General Hospital and at Pancreum, The Wearable Artificial

Pancreas Company.

Framework provided by biologist in the field of mathematics for understand complex

phenomena. While the inclusion of delays but one approach among many, the theory

behind it should be continue to developed, with a particular eye towards practical results

and the ability to draw applicable conclusions.

CHAPTER 8. CONCLUSION AND FUTURE WORK 101

The discussion in the previous section shows that the following tasks should be con-

sidered for future research: (i) Development and validation of a comprehensive model for

the system including the important control mechanisms involving insulin and glucagon for

the healthy system. (ii) Adapting the model developed according to (i) to the case of

T1DM and investigation of the T1DM-model with respect to its control properties. Of

particular interest are the properties of the model obtained by linearisation around the

equilibrium state which should be preserved. The linearised model with glucose as the ob-

servable output and both, insulin and glucagon, as the control input should be controllable

and observable. (iii) Proportional integral controller design can be developed for Sorenson

Model and Composite model for type 1 diabetes mellitus to monitoring the glucose insulin

and glucagon measurement because these are the comprehensive model for physiology of

diabetes mellitus.

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