Download Design of H∞ Controller for Blood Glucose Regulation

DESIGN OF H ∞ CONTROLLER FOR
BLOOD GLUCOSE REGULATION
P.Satheesh kumar, T.Vinopraba, Dr.N.Sivakumaran, Dr.S.Raghavan
Dr.N.SIVAKUMARAN M.E. Ph.D.,
Assistant Professor
Modeling and Simulation Laboratory
Department of Instrumentation and Control Engineering
National Institute of Technology
Trichy-620015
[email protected]
OVERVIEW
•
•
•
•
•
•
•
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Objectives
Literature Survey
Introduction to Diabetes
Human Body Model
Identification of human body system
Robust H∞ and Predictive Controller
Conclusion
References
2
OBJECTIVES OF THE PAPER
To design a Robust H inf and predictive
controller for Diabetic model.
 To Compare the performance of the controller
for servo and regulatory problems.
3
LITERATURE SURVEY
1. Y.Ramprasad et. al.(2004), Robust PID controller was designed
using Shen tuning method, Cohen-coon tuning method and
IMC.
2. Y.Ramprasad et. al. (2006), IMC and enhanced IMC controllers
were designed to reject the meal disturbances.
3. E. Ruiz-Vellazqueza et. al. (2008), H∞ control is applied to
obtain a robust controller for the automatic insulin delivery
rate. The control action permits to prevent the
hyperglycemia levels in a type I diabetic patient.
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What is Diabetes?
Diabetes is a chronic condition that occurs when the pancreas does not produce enough
insulin or when the body cannot effectively use the insulin it produces. Hyperglycaemia and
other related disturbances in the body’s metabolism can lead to serious damage to many of
the body’s systems, especially the nerves and blood vessels.
There are two basic forms of diabetes: Type 1: people with this type of diabetes produce very
little or no insulin. Type 2: people with this type of diabetes cannot use insulin effectively.
Most people with diabetes have type 2.
A third type of diabetes, gestational diabetes mellitus (GDM), develops during some cases of
pregnancy but usually disappears after pregnancy.
People with type 1 diabetes require daily injections of insulin to survive. People with type 2
diabetes can sometimes manage their condition with lifestyle measures alone, but oral drugs
are often required, and less frequently insulin, in order to achieve good metabolic control.
Common symptoms of type 1 diabetes include: excessive thirst; constant hunger; excessive
urination; weight loss for no reason; rapid, hard breathing; vision changes; drowsiness or
exhaustion. These symptoms may occur suddenly.
People with type 2 diabetes may have similar, but less obvious, symptoms. Many have no
symptoms and are only diagnosed after many years of onset. As a consequence, almost half of
all people with type 2 diabetes are not aware that they have this life-threatening condition.
How do people get diabetes?
Type 1
• Genetic element/mutation, susceptibility to triggers:
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Viral infections
Stress
Environmental exposure - exposure to certain chemicals or drugs
White blood cells, T lymphocytes, produce immune factors called cytokines which attack and
destroy b cells of pancreas
Can take 7yrs. or longer to develop to absolute, by the time know something is wrong 80% 90% of b cells are destroyed
10% chance of inheriting if first degree relative has diabetes
Most likely to inherit from father
Increase incidences would take at least 400 years if genetic factors were the only cause
Viruses
•
•
Infection introduces a viral protein that resembles a b cell protein
T-cells and antibodies tricked by this resemblance into attacking b protein and virus
• Cases rising in certain areas of U.S. – particularly Northeastern region
• Cow’s milk – certain protein which may trigger attack on b cells
• Breast milk – hormones which protect body from attack on b cells
Type 2
• Inheritance pattern, first degree relatives with type 2 have much higher risk for developing
• Perhaps inheriting a tendency towards obesity since 85% obese
Gestational
• Genetically predisposed, have greater chance for developing type 2 later in life
Being in Control
Non-diabetic
Generally between 80mg/dL-120mg/dL
•
Fasting glucose level: <110mg/dL
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2 hours after a 75g carb meal: <140mg/dL
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110mg/dL-125mg/dL: impaired fasting glucose
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By definition 2 fasting glucose above 126 mg/dL – positive for diabetes
Diabetic Goals
•
90mg/dL-130mg/dL before meals
•
110mg/dL-150mg/dL bedtime
HbA1c (glycosylated hemoglobin) – measures the level of glucose irreversibly bound to hemoglobin, 90 day
measure of average blood sugar – can be misleading
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<6.0% for non diabetics = 114mg/dL
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<7.0% for diabetics = 147mg/dL
•
•
•
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Control best obtained with pre-meal testing,
2 hour post meal testing, and bed time = 7x per day
Lows more frequent in controlled diabetics,
can’t feel them as well
Long term diabetics, may not feel lows as well
Lows can occur more in less educated diabetics
Exercise – increases insulin sensitivity
Feelings of High
Blood Sugars
Feelings of Low
Blood Sugars
Frequent Urination
Shakes
Increased Thirst
Dizzy
Lethargy
Feeling of confusion,
disorientation
Irritability
Sweaty
Anxiety
Headache
3D Structure of Insulin
Insulin Secretion
• Glucose transported into b cell by a glucose
transporter
• Results in membrane depolarization and an influx of
extracellular calcium
• Fusion of insulin storage vesicle in plasma occurs
• Hexamer released from cell as crystal and dissolves
to monomer
Reasons for monomer transformation:
– Change in pH
– Loss of ligands due to dilution, dissociation of allosteric ligands
– Endogenous chelator removes the His B10 Zn2+ ions
The Good News…
• By managing the ABCs of diabetes, people
with diabetes can reduce their risk for
heart disease and stroke.
A stands for A1C
B stands for Blood pressure
C stands for Cholesterol
Ask About Your A1C
• A1C measures average blood glucose
over the last three months.
• Get your A1C checked at least twice a
year.
A1C Goal = less than 7%
Key Steps for Lowering A1C
• Eat the right foods.
• Get daily physical activity.
• Test blood glucose regularly.
• Take medications as prescribed.
Need for Blood Glucose (BG) regulation
• A high glucose concentration exerts an osmotic pressure in the
extracellular fluid, and causes cellular dehydration. This excessive BG level
causes loss of glucose through urination (glycosuria), leading to osmotic
diuresis that depletes the body further of fluids and electrolytes.
• Too low a BG level carries the risk of hypoglycaemic coma. The BG level
should not drop below a certain level because glucose is the only nutrient
that can be used for energy by the brain, retina, and germinal epithelium
of the gonads.
• Too high a glucose concentration (>11.1 mmol/l) can affect wound healing
and interfere with human neutrophil function.
• Therapy that maintains BG level at below 11.9 mmol/l improves the
longterm outcome in diabetic patients with acute myocardial infarction.
Block diagram of feedback control system
Desired glucose
concentration
81.1mg/dL
controller
+
Insulin infusion
pump
-
Glucose
sensor
Glucose concentration
of the patient
patient
MATHEMATICAL MODEL OF HUMAN BODY
• Parker Model.
• Bergman Model.
• Sorenson model.
• Puckett model.
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SCHEMATIC REPRESENTATIONS OF COMPARTMENTS
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GLUCOSE MODEL
BRAIN:
q
vT
G BC  (GHC  GBC ) CB  (GBC  GBT ) B C
vB
TB v B
1 BU
G BT  (GBC  GBT )
 T
TB
vB
(1)
(2)
HEART AND LUNGS:
GUT:
1
 C  (G C q  G C q  G C q  G C q  G C q  
G
H
B B
L L
K K
P P
H H
RBCU ) C
vH
q

SU
G SC  (G HC  G SC ) CS  meal

vS
v SC
v SC
(3)
(4)
LIVER:
HGU
1 
G LC  (G HC q A  G SC q S  G LC q L ) C  HGP

vL
v LC
v LC
(5)
KIDNEY:
q

G KC  (G HC  G KC ) CK  KE
vK
v KC
(6)
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PERIPHERY:
q
vT
C
G PC  (G H
 G PC ) CP  (G PT  G PC ) GP C
vP
TP v P
(7)
 T  (G C  G T ) 1  PGU
G
P
P
P
TPG
vT
P
(8)



I LC
1 
A



1
.
2088

1
.
138
tanh
1
.
669

0
.
8885

A


IHGP

 HGP 
25 
21
.
43



A NHGP
A IHGU
1  2.7 tanh(0.388 N )  1

 
 ANHGP 
65 
2



I LC 
1 

 2 tanh 0.549
 AIHGU 

25 
21 .43 


(9)
(10)
(11)
INSULIN MODEL:
BRAIN:


C
C
C QB
I B
 IH
 IB
V BC
(12)
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HEART AND LUNGS:

IHC  I BC QB  I LC QL  I KC QK  I PC QP  I HC QH  IVI
GUT:

ISC  I HC  I SC
LIVER:
V
C
S

KIDNEY:

PERIPHERY:

IPT  I PC  I PT
V
QP
C
P
 T1
I
P

 I PC  I PT

(15)
(16)

Q
IKC  I HC  I KC KC  KC
VK
V KC

(13)
(14)

IPC  I HC  I PC
C
H
QS
1
 
ILC  I HC QA  I SC QS  I LC QL C  PIR C LC
VL
VL

V1
T
(17)
V PT
I C
P VP
(18)
SIA  PC
V PT
GLUCAGON MODEL:
(19)
F
N  PNR  N  PNC
VN
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Open Loop Response
130
120
50%
22.5%
5%
glucose concentration(mg/dL)
110
100
90
80
70
60
50
40
0
200
400
600
time(min)
800
1000
1200
Transient response of a perturbed patient model with step change in
insulin from its nominal value of 22.3 mU/min.
Stabilizing set of Controller parameters
For
K P  1.3916
KI  0
K D  141 .5126 K I  13.096
K D  19.3878 K I  19.8796
Stabilizing region of (KI,KD)
Steady state I/O Plot for the system
Non-Linearity I/O Checking
110
Blood Glucose
Level(mg/dL)
100
90
80
70
60
50
14
16
18
20
22
24
26
Insulin Infusion rate(mU/min)
28
30
32
22
IDENTIFICATION OF HUMAN BODY SYSTEM
Using the ident box in MATLAB a linear ARX model was
identified and the transfer function is
2.077 s 2  0.5253s  90
s 3  0.07842s 2  70s  1.032
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MODEL VALIDATION
Step responses of actual and predicted model
170
predicted
actual
160
150
Blood Glucose
Level(mg/dl)
140
130
120
110
100
90
80
0
500
1000
1500
Time(min)
2000
2500
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ROBUST H∞ CONTROLLER
• A Controller is said to be robustly stable if it controls the
process at all uncertainties.
• H∞ methods are used in control theory to synthesize
controllers achieving robust performance or stabilization.
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STEPS TO DESIGN A ROBUST H∞ CONTROLLER
1. The system along with uncertainties is modeled.
2. Designing of weighting functions is most important in
Robust H∞ controller.
3. Open loop system is designed so that we can get TF of
uncertainties to disturbance.
4. Sub-optimal controller is designed in MATLAB.
5. Controller is tested for both nominal and worst case
uncertainties.
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Uncertainties in three parameters are considered
.
• Effect of Glucose on Hepatic Glucose Uptake (40%)
• Effect of Glucose on Hepatic Insulin (40%)
• Fraction of Hepatic Insulin clearance (20%)
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For closed loop stability it is necessary to satisfy the
below condition
 W p I  GK 1 

1 
Wu K I  GK  
1

where
Wp, Wu are the weighting functions.
K is the controller.
G is the process along the uncertainties
G = FU(Gmds,Δ)
W p s  
1
10 s  1
Wu s  
1
100
28
Controller designed is
5s 4  9876s 3  9414s 2  2492s  6839
Gc ( s)  5
s  4350s 4  5499s 3  3696s 2  6388s  1518
The controller is tested for full order non-linear
model for both nominal and worst case models.
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Sensitivity and inverse weighting functions
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Model Predictive Control
 Modified form of classical optimal control problem
 Can systematically and optimally handle
Multivariable interactions
Operating input and output constraints
Process nonlinearities
Basic Idea
Given a model for plant dynamics, possible consequences of the
current input moves on the future plant behavior (such as possible
constraint violations in future etc.) can be forecasted on-line and used
while deciding the input moves.
•
•
•
Explicit use of a model to predict the process output at future instants.
Constraints on input and outputs( Physical constraints and Safety constraints)
can be integrated in the calculation of control signal.
Calculation of a control sequence by minimizing an objective function
FUTURE
PAST
SET POINT
PREDICTED PLANT
OUTPUT
PLANT
OUTPUT
CONTROL LEVEL
T
T+1
T+2
CONTROL
HORIZON
T+C
T+
P
PREDICTION
HORIZON
MPC Formulation
-Utilize a model to predict the output in future and minimize the difference between the predicted output and the
desired one by computing appropriate control actions.
Camacho and Bordons,1999
Model Predictive Control…..
The optimization cost function is given by:
N
N
J    xi (ri  xi )   ui u i
2
i 1
2
i 1
without violating constraints (low/high limits).
where
xi :
ith control variable (e.g. measured temperature)
ri :
ith reference variable (e.g. required temperature)
ui :
ith manipulated variable (e.g. control valve)
:
weighting coefficient reflecting the relative importance of xi
:
weighting coefficient penalizing relative big changes in ui etc.
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Parameters for the MPC
Parameters
Variation
xi=8,ui=2
xi=6,ui=1
Settling Time
(min)
9.8
13.8
Peak Overshoot
(%)
4
2.3
From the table it is quite clear that top parameters will
give good response for the system.
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36
For the nominal case non-linear model
100
H-inf
MPC
Blood Glucose
Level(mg/dL)
95
90
85
80
75
20
30
40
50
60
70
80
90
100
Time(min)
37
For the worst case non-linear model
120
MPC
H-inf
Blood Glucose
Level(mg/dL)
110
100
90
80
70
20
40
60
80
100
120
Time(min)
140
160
180
200
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CONCLUSION
•
Thus, the human body model is constructed in MATLAB
software using 19 differential equations.
• The MPC controller eliminates the undershoots and Robust
optimal H∞ controller settles faster.
• When uncertainties are introduced into the system, the
performance of MPC are not satisfactory.
• As the nominal parameters vary from patient to patient,
Robust H∞ controller is best suitable.
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REFERENCES
1. Y.Ramprasad, G.P.Rangaiah, S.Lakshminarayanan, “Robust PID
Controller for Blood Glucose Regulation in Type I Diabetics”,
Industrial Engineering & Chemical Research, vol.43, pp.82578268, 2004.
2. R.S.Parker, F.J.Doyle, J.H.Ward, N.A.Peppas, “Robust H∞
Glucose Control in Diabetes using a Physiological Model” ,
AIChE J., vol.46, pp.2537-2549, 2000.
3. C.Fredrick, F.Tyrone, Closed-Loop Control of Blood Glucose,
Springer, 2007.
4. Da-Wei Gu, Petko Hristov Petkov ,Mihail Mihaylov
Konstantinov,Robust
control
Design
in
MATLAB,
Springer,2005.
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5. T.Sorensen, “A Physiologic Model of Glucose Metabolism in
Man and its use to Design and Assess improved Insulin
Therapies for Diabetes”, Ph.D thesis, Department of Chemical
Engineering, Massachusetts Institute of Technology,
Cambridge, 1985.
6. Parker, R. S.; Doyle, F. J., III; Ward, J. H.; Peppas, N. A. “Robust
H∞ Glucose Control in Diabetes Using a Physiological Model”,
AIChE J. 2000, 46, 2537-2549.
7. T.Vinopraba, N. Sivakumaran, T.K.Radhakrishnan, S.Raghavan,
Optimal Control of Blood Glucose Regulation for Type-I
Diabetics, Proc. International Conference on TIMA, MIT, Anna
University, 2009.
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THANK YOU
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