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A Closed-Loop Artificial Pancreas Using a Proportional Integral Derivative with

Double Phase Lead Controller Based on a New Nonlinear Model of Glucose
Metabolism

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Journal of Diabetes Science and Technology ORIGINAL ARTICLE
Volume 7, Issue 3, May 2013
© Diabetes Technology Society

A Closed-Loop Artificial Pancreas Using a Proportional Integral

Derivative with Double Phase Lead Controller Based on a New
Nonlinear Model of Glucose Metabolism

Ilham Ben Abbes, M.S.1 Pierre-Yves Richard, Ph.D.,1 Marie-Anne Lefebvre, Ph.D.,1
Isabelle Guilhem, M.D.,2 and Jean-Yves Poirier, M.D.2

Abstract
Background:
Most closed-loop insulin delivery systems rely on model-based controllers to control the blood glucose (BG) level.
Simple models of glucose metabolism, which allow easy design of the control law, are limited in their parametric
identification from raw data. New control models and controllers issued from them are needed.

Methods:
A proportional integral derivative with double phase lead controller was proposed. Its design was based on
a linearization of a new nonlinear control model of the glucose–insulin system in type 1 diabetes mellitus
(T1DM) patients validated with the University of Virginia/Padova T1DM metabolic simulator. A 36 h scenario,
including six unannounced meals, was tested in nine virtual adults. A previous trial database has been used
to compare the performance of our controller with their previous results. The scenario was repeated 25 times
for each adult in order to take continuous glucose monitoring noise into account. The primary outcome was the
time BG levels were in target (70–180 mg/dl).

Results:
Blood glucose values were in the target range for 77% of the time and below 50 mg/dl and above 250 mg/dl
for 0.8% and 0.3% of the time, respectively. The low blood glucose index and high blood glucose index were
1.65 and 3.33, respectively.

Conclusion:
The linear controller presented, based on the linearization of a new easily identifiable nonlinear model, achieves
good glucose control with low exposure to hypoglycemia and hyperglycemia.

J Diabetes Sci Technol 2013;7(3):699–707

Author Affiliations: 1Hybrid System Control Team, Supelec/I.E.T.R., Cesson-Sévigné, France; and 2Department of Endocrinology, Diabetes, and
Metabolism, University Hospital of Rennes, Rennes, France

Abbreviations: (BG) blood glucose, (CHO) carbohydrate, (EMPC) extended model predictive controller, (HBGI) high blood glucose index,
(LBGI) low blood glucose index, (MPC) model predictive control, (PID) proportional integral derivative, (PIDD) proportional integral derivative
with double phase lead, (T1DM) type 1 diabetes mellitus, (UVa/Padova metabolic simulator) University of Virginia/Padova T1DM metabolic simulator

Keywords: closed-loop control, diabetes, nonlinear control model, proportional integral derivative

Corresponding Author: Ilham Ben Abbes, Supelec/I.E.T.R., Avenue de la Boulaie, CS 47601, 35576, Cesson-Sévigné Cedex, France; email address
ilham.benabbes@supelec.fr

699
A Closed-Loop Artificial Pancreas Using a Proportional Integral Derivative with
Double Phase Lead Controller Based on a New Nonlinear Model of Glucose Metabolism Ben-Abbes

Introduction

I n the literature, many closed-loop control design techniques were tested in the case of blood glucose (BG) control
[proportional integral derivative (PID), model predictive control (MPC), nonlinear MPC, H∞ control, fuzzy logic
control].1–14 Most of the controllers are model based, thus needing a control model of the BG regulation system.15–17
In this regard, simple linear models have been initially proposed,18–20 but they have shown insufficiencies to fairly
represent its behavior.20 Taking into account nonlinear features, the more refined model of Bergman and coauthors21
has evolved as a dominant model in the literature. This model can be quite appealing for control algorithms because of
its simple form. Moreover, its identification can be achieved from data stemming from the glucose monitoring system
and the insulin pump. However, some limitations in the parametric identification of this model have been underlined,
leading to difficulties in its practical use.22,23 For instance, this model needs to fix basal values of insulin and glucose
to be structurally identifiable, but these are not precisely known for type 1 diabetes patients (T1DM). Furthermore,
it has been shown that this model does not capture long-term effects of insulin delivery.24 In this article, a short
description of a new nonlinear control model easily identifiable from real patients’ data is proposed. Then a specific
PID controller from a linearized version of the new proposed model has been developed, as PID controllers have been
used to regulate BG level,13 and displayed an acceptable regulation of BG. This approach is classic in control theory,
because local properties of a nonlinear model can be deduced from the properties of the associated linearized model.25

The organization of this article is as follows. We begin by a brief description of the nonlinear model and the synthesis
of the controller. We then detail the experiment and the results of simulations from the publicly available version of the
University of Virginia/Padova T1DM metabolic simulator using a scenario proposed by Cameron and coauthors.26
Finally, the performances of our controller are compared with the controllers tested by Cameron and coauthors.26

Methods
Simulation Model of Type 1 Diabetes Mellitus Patients
The UVa/Padova T1DM metabolic simulator (UVa/Padova metabolic simulator)27,28 is approved by the Food and Drug
Administration as an in silico model of diabetes29 for closed-loop algorithm preclinical tests. Yet only 10 adult subjects
are included in the publicly available version of this software. They were the virtual subjects used to generate our
experimental data for the identification process and to validate our controller.

Presentation of the New Nonlinear Control Model of the Glucose Metabolism

The minimal model of Bergman and coauthors21 imposes the knowledge of two parameters Gb (glucose basal value)
and Ib (insulin basal value), corresponding to a particular steady state, to be structurally identifiable.30 For a nondiabetic
subject, Ib represents the value of the insulin produced by the pancreas in steady state, and Gb is the measured glucose
value associated. Therefore, they can be measured. In case of T1DM patients, they are unknown. Thus, in the design
of this new model, we did not consider a model in variation around basal values, but we set instead a generic condition
about equilibrium states.

This condition was built by studying a mathematical relation at equilibrium between the injected insulin and the
measured glucose. The data necessary to determine this mathematical relationship were obtained from the 10 virtual
adults of the UVa/Padova metabolic simulator. The form of this equilibrium curve is supposed to be continuous and
decreasing from a glycemia maximum at 0 U insulin to a glycemia that converges to zero as the associated insulin
value is getting higher. Equation (1) details the mathematical equation at equilibrium validated on 10 virtual adults:

kgo
Geq = (1)
P1exp(SIUeq)

This leads to the nonlinear form of our model.

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A Closed-Loop Artificial Pancreas Using a Proportional Integral Derivative with
Double Phase Lead Controller Based on a New Nonlinear Model of Glucose Metabolism Ben-Abbes

The model itself consists of three differential equations. The input is the insulin delivery rate and the output is the
measured glycemia. The glucose coming from meals is considered as a disturbance. The form of Equation (2.1) is
deduced from Equation (1) and is nonlinear. It represents the glucose compartment. Equations (2.2) and (2.3) model
the diffusion of insulin by two first-order equations. More details about the synthesis of the model can be found
elsewhere.31,32 The equations of the proposed new nonlinear model of the glucose metabolism are

Ġ(t) = –P1 exp(SIX2(t))G(t) + D(t) + kg0 (2.1)

Ẋ2(t) = –wi (X2(t) – X1(t)) (2.2)

Ẋ1(t) = –wi (X1(t) – Ui(t)) (2.3)

where Ui (pmol/min) is the injected insulin and D (mg/min) is the glucose issued from meals and considered as an
unknown disturbance in this work. The variable G (mg/dl) denotes the glycemia. X2 and X1 (pmol/min) represent
the insulin in distant compartments. The parameter wi (min-1) represents the reverse time constant associated with the
diffusion of insulin in the organism. The parameter P1 (min-1) represents a gain on the joint action of the pair insulin–
glucose on glucose. The parameter SI (pmol/min) represents a gain on the action of insulin on glucose. The parameter
kg0 (mg/(dl/min)) corresponds to the endogenous glucose production. The physiological meaning of these parameters
implies that they are strictly positive. The only measured output of the system is G which corresponds to the
BG values.

Design of the Proportional Integral Derivative with Double Phase Lead Controller
The objective of the controller is to maintain glycemia in the range 70–180 mg/dl to avoid both low (<50 mg/dl) and
high (>250 mg/dl) levels, despite meal disturbances and measurement noises. A PID-type controller is considered to
be robust to disturbances and is easy to implement from a linear or linearized model. The design of such a controller
requires a particular set point, and an efficient regulation will imply to respect the defined control range. The control
variable is the injected insulin. The controller will produce an output that varies proportionally to the error between
the measured glycemia and its set point desired value (proportional action). It will react to the rate of change of this error
occurring during a meal (derivative action). Eventually, a cancellation of the static error is expected so that glycemia
stays in the normal range (integral action). The linearization of the new model around a set point and the design of
the controller are described here.

Model Linearization
In the present approach, meals are considered as disturbances and then the input D is neglected in the control design
process. The equilibrium state (G 0, X20, X10) for a constant insulin input Ui0 is given by

X20 = X10 = Ui0 (3.1)

kg0
G0 = (3.2)
P1exp(SIUi0)
Let g = G – G0, x2 = X2 – Ui0, x1 = X1 – Ui0, and u = Ui – Ui0, then the linearized model around this equilibrium point
(G 0, X20, X10, Ui0) is given by
kgo
⎛ġ⎞ – ⎛
–SIkg0 0 ⎛g⎞ ⎛0⎞ ⎞
⎜ ẋ2 ⎟ = G 0 ⎜ ⎟
⎜ x2 ⎟ + ⎜ 0 ⎟ u (4)
⎝ ẋ1 ⎠ 0 –wi wi ⎝ x ⎠ ⎝ ⎠
0 ⎝ 0 –wi
1
⎠w 1

Since the state matrix is upper triangular, its eigenvalues are displayed on its diagonal. They are strictly negative,
because the parameters of the model are strictly positive. The equilibrium state is thus exponentially stable. In the
frequency domain, the following transfer function between u and g is obtained:

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A Closed-Loop Artificial Pancreas Using a Proportional Integral Derivative with
Double Phase Lead Controller Based on a New Nonlinear Model of Glucose Metabolism Ben-Abbes

Km
H(p) = (5)
(1 + T1p)(1 + T2p)2

where p denotes the symbolic Laplace variable.

The gain is defined by Km, and its value is given by (–SIG0). The time constants are defined by T1 and T2. T2 is equal to
(1/wi), and T1 is equal to (G0/kg0). The values of T1 and Km depend on the linearization point.

Choice of the Type and Form of the Proportional Integral Derivative Controller
The system presents an instability risk in closed loop, because it is a third-order system. Thus we may expect from
a proposed controller to increase the bandwidth while keeping the stability of the system. Several solutions can
be proposed to answer this point. From an analysis of the form of transfer function of the system [Equation (5)],
the simplest solution to decrease the instability risk in closed loop is to compensate the double-order pole linked to
the time constant T2. As a consequence, a proportional integral derivative with double phase lead (PIDD) controller
is proposed to ensure the feasibility of the controller. The transfer function of this controller, corresponding to an
implementation in a series form, is given by

1 ⎞ ⎛ 1 + Tdp ⎞
2
C(p) = KC⎛ 1 + with a < 1 (6)
⎝ Tip ⎠ ⎝ 1 + aTdp⎠

Its factors correspond respectively to the proportional, integral, and derivative actions, the latter being filtered to reduce
the noise sensitivity and to be physically implementable. The value of a was chosen equal to 0.1, which corresponds
to a classical tuning value of this parameter.

Its structure, as detailed earlier, allows some poles of the system to be cancelled in the calculation of the controlled
system open-loop transfer function by setting the time constants Ti = T1 and Td = T2. The resulting transfer function in
open loop, denoted OL(p), is then given by

KmKc
OL(p) = H(p)C(p) = (7)
T1p(1 + aT2p)2

Tuning of the Controller

Tuning of the proposed controller is constrained by intrinsic characteristics of the system and desired specifications.
Indeed, the physical control variable has to be positive because it is the delivery rate of insulin. Such a constraint cannot
be taken into account in the computation of the control, as it would be done in an optimization process since a PID
provides an explicit expression of its output. As a result, a smooth control is required in order to maintain the system
in the linearity zone. This can be ensured by specifying a sufficiently large phase margin for the controlled system.
Let Dj be the desired phase margin, and the corresponding cutoff frequency should be

tan⎛ 90º – Dj
⎞
⎝ 2 ⎠
wc = (8)
aT2

By setting ⎮OL(wc)⎮ = 1, this provides the value of the controller gain:

T1wc
Kc = (9)
Km

Another constraint is that the measured glycemia is noisy. In the case of the UVa/Padova metabolic simulator, this
noise signal is a low-frequency signal with a bandwidth equivalent to that of the noise-free measured glucose. To filter
the noise, a very high value of the phase margin is chosen (Dj is 85° or 87.5°, depending on the insulin sensitivity of
the patient).

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A Closed-Loop Artificial Pancreas Using a Proportional Integral Derivative with
Double Phase Lead Controller Based on a New Nonlinear Model of Glucose Metabolism Ben-Abbes

Implementation of the Controller

The controller has been designed for a model in variation around the equilibrium point. So the actual insulin delivery
corresponds to the sum of the controller output and the equilibrium control value (Ui0). In the implementation,
the integral action was initialized at the insulin value corresponding to the linearization set point so that the output of
the implemented controller is the total insulin delivery.

Tuning of the controller has been performed to minimize the risk of computing negative values of insulin delivery.
Yet a saturation of this variable is mandatory to ensure its positivity in any case. Furthermore, a specific implementation
of the integral action, as displayed in Figure 1, is performed to avoid the phenomenon of integral windup.
The implementation of the controller is represented in Figure 1.

Figure 1. Implementation of the controller in the simulator.

Results
Identification of the Parameters of the Model
The input was the injected insulin and the output was the noise-free BG values; no meal disturbance was considered
in the scenario. The identification protocol used to obtain data was as follows:

• An open-loop scenario was applied, consisting of a basal insulin step (increase of its basal value), with a patient-
specific magnitude depending on the insulin sensitivity.

• The step magnitude was chosen by using available information from each subject (the maximum drop in mg/dl/U
insulin and the basal insulin value) so that the BG values of the patient stay in their normal range (BG > 70 mg/dl).

• Twelve hours of data were considered, as the duration of the insulin action is lower than 6 h.

• The sampling time was that of the simulator (i.e., 1 min).

At the beginning of the experiment, the previous basal insulin values were supposed to be constant so that the model
initial states could be supposed at steady states. The states of X1 and X2 were therefore considered equal to initial
basal value (Ub), and the initial glucose value was fixed to the glucose value measured (Gb). From the data provided by
this protocol, the identification toolbox of Matlab33 was used to obtain the four parameters of the new nonlinear model.
The optimization algorithm “nonlinear least squares” was selected. An indicator of the goodness of the identified
model is given by the fit function:

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A Closed-Loop Artificial Pancreas Using a Proportional Integral Derivative with
Double Phase Lead Controller Based on a New Nonlinear Model of Glucose Metabolism Ben-Abbes

⎮⎮ y – ŷ⎮⎮ ⎞
Fit = ⎛ 1 – – ⎠
× 100%,
⎝ ⎮⎮ y – y⎮
⎮
where y is the vector of measured glucose values, ŷ is
the vector of glucose values estimated by the model, and
y– is the mean of the measured glucose values. The closer
its value to 100%, the better the estimated model.

The mean fit obtained by our model on the 10 virtual

patients is 99.5%, with a standard deviation of 0.5%.
A cross validation using another noise-free scenario with-
out meal disturbance was performed. The new scenario
consists of variations of basal values during one day.
The mean fit obtained on the 10 virtual patients is 71.8%.
Figure 2 represents a cross validation result for the
virtual adult 3. This indicates that the new control model Figure 2. Cross-validation results of the identified model of a virtual
patient on another data set.
can provide a good approximation of the simulator
model when considering BG values.

Testing the Proportional Integral Derivative with Double Phase Lead Controller
To compare the performances of the new PIDD controller with existing controllers, the Cameron and coauthors26
scenario was used for the closed loop. According to this scenario, the 10 adult patients of the UVa/Padova metabolic
simulator were simulated during 36 h. Six unannounced meals were planned, lasting 15 min each and respectively
measuring 50 g carbohydrate (CHO) at 9:00 am, 70 g at 1:00 pm, 90 g at 5:30 pm, 25 g at 8:00 pm, 50 g at 9:00 am, and
70 g at 1:00 pm. The glycemia was measured through the continuous glucose monitoring system. The controller used
a sample time of 5 min. The glycemia set point value was 140 mg/dl. To take into account the effect of noise, the
scenario was repeated 25 times for each patient. Just as Cameron and coauthors,26 we have excluded adult 9 from
the average results. Indeed, they studied this adult and showed that the suppression of the endogenous glucose
production of this virtual patient following a meal was still active after 6 h, which led to hypoglycemia after meals.
They then concluded that it was not representative of
a normal diabetes patient and should be considered as Table 1.
an outlier. Furthermore, even with an optimal bolus Main Objectives Performance Measuresa
correction associated with meals, this virtual patient is Mean BG %BG %BG %BG %BG
ID
subject to hypoglycemia while other adults are not. (± SEM) 70–180 < 50 (n)b > 250 50–70
Adult 1 131 (±6) 72% 0.8% (4) 0% 11.8%
Table 1 provides the performance measures for the 10 adults Adult 2 131 (±4) 89% 0% (1) 0% 3.5%
and the average values for the 9 valid adults. The mean
Adult 3 148 (±4) 70% 0.3% (1) 0% 3.1%
time within the target range (70–180 mg/dl) was 77%.
Percentages of time in hypoglycemia (BG < 50 mg/dl) and Adult 4 133 (±5) 78% 0.3% (1) 2.4% 3%
hyperglycemia (BG > 250 mg/dl) were 0.8% and 0.3%, Adult 5 125 (±5) 77% 1.8% (5) 0% 10.7%
respectively. There was nearly no reading above 250 mg/dl Adult 6 135 (±6) 73% 0.8% (2) 0.1% 6.5%
(adult 4 and adult 6). As a result, both the means of Adult 7 141 (±4) 82% 0% (0) 0% 0.6%
low blood glucose index (LBGI) and high blood glucose
Adult 8 120 (±6) 95% 1.3% (3) 0% 2.7%
index (HBGI)34 were low (Table 2), indicating that the
controller minimized the risk of low and high glucose Adult 9 128 (±16) 55% 21.3% (25) 0% 3%
excursion. Adult 10 147 (±4) 57% 0.6% (3) 0% 12.2%
Average
135 (±5) 77% 0.8% 0.3% 6%
Figure 3 shows the average values and the 95% and 70% values
confidence intervals of the BG values for the nine virtual a
BG in mg/dl. SEM, standard error of the mean.
patients. The analysis of this figure shows that the meal
b
n refers to the number of the tests (among the 25 tests) with
BG < 50 mg/dl.
disturbances are rejected. We recall that the proposed

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A Closed-Loop Artificial Pancreas Using a Proportional Integral Derivative with
Double Phase Lead Controller Based on a New Nonlinear Model of Glucose Metabolism Ben-Abbes

controller receives no information concerning the time Table 2.

and quantity of CHO in meals. Yet some patients suffer Performance Measures (Quality Indicators)
from hypoglycemia events during nighttime. It mostly
Interquartile Premeal Postmeal
concerns the patients for whom the duration of the meal ID range LBGI HBGI BG BG
action is higher than 4 h, which is the time between two (mg/dl) (mg/dl) (mg/dl)
meals. Furthermore, no upper limit on the maximum Adult 1 65 2.64 3.09 138 172
quantity of injected insulin was considered in this case. Adult 2 67 1.28 2.50 139 165
Thus adding a safety system limiting the daily insulin dose
Adult 3 58 1.08 4.68 151 188
to the controller would improve the presented results.
Adult 4 65 1.25 3.52 128 200

Discussion Adult 5 63 2.71 2.35 130 172

Adult 6 81 1.80 3.67 142 185
The PIDD controller, based on a linearization of our
Adult 7 60 0.44 3.47 145 183
new nonlinear model, achieves satisfactory glycemic
Adult 8 46 1.45 1.21 122 148
regulation in a group of virtual adult T1DM patients
from the UVa/Padova metabolic simulator. To compare Adult 10 92 2.17 5.48 154 202
the performances with those of other controllers, we Average
66 1.65 3.33 139 179
chose to use the same scenario and experimental values
conditions as Cameron and coauthors.26 They developed
a novel “extended model predictive controller” (EMPC)
with a modified cost function to take into account the
uncertainty of the prediction in the future BG values
and to minimize the combined risk of hypoglycemia
and hyperglycemia. They demonstrated the improved
performance of EMPC against a PID controller and a
basic MPC controller. The PID controller was actually
a proportional derivative controller. The weights for
proportional and derivative terms were optimized to
minimize the average BG risk index (LBGI + HBGI).
The MPC controller used a prediction horizon of 300 min
and a specific prediction algorithm to detect and to
estimate the meals. The performance comparisons (adult
9 excluded) of the controllers are summarized Table 3.
Figure 3. Average control performance and confidence intervals for
The time spent with glucose levels in target range
the nine virtual patients. The first confidence interval is at ±σ (70%)
(70–180 mg/dl) and the hyperglycemic range are not and the second one at ±1.96σ (95%). The lower graph indicates the
different between MPC (79.6% and 19.9%, respectively) time of the meals and quantities of CHO ingested each minute (the
duration of the meals is 15 min).
and PIDD (77% and 16%, respectively). Severe hypo-
glycemic events are similar, whereas minor hypo-
glycemic readings are more frequent with the PIDD controller (6% versus 0.2%). In our simulation, the sensor noise was
taken into account by repeating the scenario 25 times. Such a process allows the robustness to noise to be evaluated.

Table 3.
Performance Indicators of the Different Controllers (Blood Glucose in mg/dl)
Interquartile %BG %BG %BG
Algorithm Mean BG %BG < 50 %BG > 250
range 70–180 50–70 180–250
135 77% 0.8% 6% 0.3% 16%
PIDD 66
(standard deviation ± 10) (±13%) (±3%) (±5.8%) (±0.9%) (±9%)
PID 156 54.9 72.6% 0% 0% 0.6% 26.8%
MPC 151 54.8 79.6% 0% 0.2% 0.3% 19.9%
EMPC 147 45.0 84.3% 0% 0.7% 0% 15%

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A Closed-Loop Artificial Pancreas Using a Proportional Integral Derivative with
Double Phase Lead Controller Based on a New Nonlinear Model of Glucose Metabolism Ben-Abbes

In a noise-free case, there are no events of hypoglycemia, as the minimum glycemia measured is 69.55 mg/dl. Furthermore,
in the simulator, the implemented noise has a positive mean, which explains the incidence of hypoglycemia.

The results of the controllers from Cameron and coauthors26 are better than ours in terms of hypoglycemic events.
However, they ran their scenario only one time, which questions the use of noisy sensor measurements. In Table 1,
we indicate the number of tests (among the 25 scenarios) with severe hypoglycemia events. The maximum number
of this instance is five for adult 5. Furthermore, in the implementation of our controller, no upper limit on the
injected insulin was considered, but Cameron and coauthors26 had used a fixed limit. Thus, our presented results are
more robust.

Finally, as compared with MPC, our PIDD controller is more easily implementable. The design of the controller is
straightforward once the parameters of the model are identified. Furthermore, such controllers are particularly robust
without meal announcement.

Conclusion
In this article, a PIDD controller based on a linearization of a new nonlinear control model was proposed. In comparison
to previously published results of PID and MPC controllers,21 our PIDD controller achieves good regulation
performance. Its performance indicators are quite similar to those of the MPC controller, with the benefit of a simpler
implementation. The results obtained are better than previously proposed PID controllers, even though we considered
a less constrained case (noise effect and no upper limit on injected insulin). Then, even if the controller exhibits some
severe hypoglycemia events, considering a safety limit on the total injected insulin dose would improve the results as
well as working on a better adjustment of the controller gain. The identification process is subject to further research
to consider data obtained from a continuous glucose monitoring sensor. The range of data used in the identification
process should be larger to take into account the noise and should include meal data. Consequently, we are currently
working to add a model of diffusion of the ingested glucose in the new nonlinear model to address this issue.

Funding:
This work was supported by Supelec, Cesson-Sévigné, France, and University of Rennes 1, France.

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