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Postprandial fuzzy adaptive strategy for a hybrid proportional derivative controller for the artificial pancreas

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Abstract

This paper presents a support fuzzy adaptive system for a hybrid proportional derivative controller that will refine its parameters during postprandial periods to enhance performance. Even though glucose controllers have improved over the last decade, tuning them and keeping them tuned are still major challenges. Changes in a patient’s lifestyle, stress, exercise, or other activities may modify their blood glucose system, making it necessary to retune or change the insulin dosing algorithm. This paper presents a strategy to adjust the parameters of a proportional derivative controller using the so-called safety auxiliary feedback element loop for type 1 diabetic patients. The main parameters, such as the insulin on board limit and proportional gain are tuned using postprandial performance indexes and the information given by the controller itself. The adaptive and robust performance of the control algorithm was assessed “in silico” on a cohort of virtual patients under challenging realistic scenarios considering mixed meals, circadian variations, time-varying uncertainties, sensor errors, and other disturbances. The results showed that an adaptive strategy can significantly improve the performance of postprandial glucose control, individualizing the tuning by directly taking into account the intra-patient variability of type 1 patients.

A fuzzy inference engine was implemented within a clinically tested artificial pancreas control system. The aim of the fuzzy system was to adapt controller parameters to improve postprandial blood glucose control while ensuring safety. Results show a significant improvement over time of the postprandial glucose response due to the adaptation, thus demonstrating the usefulness of the fuzzy adaptive system.

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Funding

This work was supported by the Spanish Ministry of Science and Innovation under Grants DPI2013-46982-C2-2-R and DPI2016-78831-C2-2-R, by the Autonomous Government of Catalonia under Grant 2014 SGR 1052 and by the Formación de Professorado Universitario (FPU0244 2015) doctoral scholarship.

Author information

Authors and Affiliations

  1. Institut d’Informàtica i Aplicacions, Universitat de Girona, Campus de Montilivi, s/n, Edifici P4, 17071, Girona, Spain

    Aleix Beneyto & Josep Vehi

  2. CIBERDEM, Girona, Spain

    Josep Vehi

Authors
  1. Aleix Beneyto
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Correspondence to Josep Vehi.

Appendix

Appendix

1.1 Overview of the fuzzy system

The presented fuzzy system is a multiple-input multiple-output (MIMO) system. Particularly, the system considers five input and two output variables as shown in Fig. 14.

Fig. 14
figure 14

Overview of the fuzzy system

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The fuzzy inference engine is based on the Mamdani method. When using the Mamdani method the inference is performed in four stages: (1) fuzzification of the crisp inputs, (2) rule evaluation, (3) aggregation of the outputs, and (4) defuzzification of the output fuzzy set.

Prior to the description of the fuzzy system engine, let us make some definitions that will be used throughout the text.

Definition 1

The universe of discourse X is a collection of objects {x} that can be discrete or continuous.

Definition 2

Let X be the universe of discourse with elements {x}. Then, a fuzzy membership function is defined as

$$ {\mu}_A(x):X\to \left[0,1\right] $$
(10)

where A is a fuzzy set, and if μA(x) = 1 means that x is in A, μA(x) = 0 means x is not in A and 0 < μA(x) < 1 means that x belongs to A up to a certain degree.

Definition 3

The degree of membership is the value that takes μA(x) ∈ [0, 1] given an object {x} of the universe of discourse X.

Definition 4

The linguistic variable constitutes the range of possible values that can take the universe of discourse of that variable.

Definition 5

A fuzzy rule is a conditional statement of the form.

IF (antecedents), THEN (consequences).

where the antecedents and consequences are relations between linguistic variables and their fuzzy sets. The antecedents are referred to the input linguistic variables, and for the consequences the output linguistic variables.

1.2 Fuzzification

The fuzzification process starts by taking the input crisp linguistic variables and determining the degree of membership for their respective membership functions. Specifically, we consider five input linguistic variables: (1) meanGmin, (2) meanExcursion, (3) mean#switch04, (4) mean#switch48, and (5) meanToff. We designed three membership functions for each input variable. The membership functions are labeled by the linguistic terms Small, Normal, and High. The universes of discourse for all input variables are continuous and their elements belong to \( {\mathbb{R}}_0^{+} \). Figures 15, 16, 17, 18, and 19 show the membership functions for the five input variables.

Fig. 15
figure 15

Fuzzy sets for input linguistic variable meanGmin

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Fig. 16
figure 16

Fuzzy sets for input linguistic variable meanExcursion

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Fig. 17
figure 17

Fuzzy sets for input linguistic variable mean_#switch04

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Fig. 18
figure 18

Fuzzy sets for input linguistic variable mean#switch48

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Fig. 19
figure 19

Fuzzy sets for input linguistic variable mean_#Toff

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All of the designed membership functions are based on trapezoidal shapes with linear hedges defined as follows:

$$ {\mu}_N\left(x|a,b,c,d\right)=\left\{\begin{array}{c}0\\ {}\frac{x-a}{b-a}\\ {}1\\ {}\frac{d-x}{d-c}\\ {}0\end{array}\right.\kern0.5em {\displaystyle \begin{array}{c} for\ x\le a\\ {} for\ x\in \left[a,b\right]\\ {} for\ x\in \left[b,c\right]\\ {} for\ x\in \left[c,d\right]\\ {} for\ x\ge d\end{array}} $$
(11)

where the parameter set (a, b, c, d) define the domain of each sub-function of the piecewise trapezoidal membership function, e.g., the Normal membership function from Fig. 15 is defined by μNormal(x| 70,80,90,100).

We used trapezoidal shapes to lower the computational burden of more complex shapes, and also because they describe well enough the knowledge of the experts who participated in the system design.

The fuzzification process is performed by taking a specific crisp value of the input variables and evaluating the degree of membership. For example, for an input mean_Excursion = 70 mg/dl, the membership functions tell us that the particular excursion belongs 50% to Small, 50% to Normal and 0% to High. This process is performed for each input over all the membership functions. The output of the fuzzification process is the degree of membership of all the input variables to each of their fuzzy membership functions.

1.3 Rule evaluation

In this step, the rules stored at the rule database are used to apply the fuzzified inputs to the rule antecedents. The rule database of the system is composed by the following rules

  1. 1.

    IF meanGmin is Small, THEN, IOBmax is Decrease.

  2. 2.

    IF meanGmin is High AND meanExcursion is High, THEN, IOBmax is IncreaseLittle.

  3. 3.

    IF meanGmin is High, THEN, kp is IncreaseLittle.

  4. 4.

    IF meanTOFF is Small AND meanGmin is Small, THEN, IOBmax is DecreaseLittle.

  5. 5.

    IF meanTOFF is High AND meanGmin is High, THEN, IOBmax is IncreaseLittle.

  6. 6.

    IF meanSwitch04 is Small, THEN, kp is IncreaseLittle.

  7. 7.

    IF meanSwitch04 is High, THEN, kp is DecreaseLittle.

  8. 8.

    IF meanSwitch48 is High, THEN, kp is Decrease.

  9. 9.

    IF meanSwitch48 is Small OR meanSwitch48 is Normal, THEN, kp is NoChange.

  10. 10.

    IF meanGmin is Normal AND meanExcursion is Normal, THEN IOBmax is NoChange AND kp is NoChange.

  11. 11.

    IF meanTOFF is High, THEN, IOBmax is Increase.

  12. 12.

    IF meanTOFF is High AND meanGmin is Normal, THEN, IOBmax is IncreaseLittle and kp is NoChange.

  13. 13.

    IF meanTOFF is Small AND mean#switch04 is High, THEN IOBmax is Increase.

  14. 14.

    IF meanTOFF is High AND mean#switch04 is Small, THEN IOBmax is Decrease.

  15. 15.

    IF meanExcursion is Normal OR meanExcursion is High, THEN, IOBmax is IncreaseLittle.

Both the antecedents and consequences can have multiple parts. These parts can use the fuzzy union (OR) and intersection (AND) operators. The union and intersection of two fuzzy sets A and B are defined as follows:

$$ {\mu}_{A\bigcup B}(x)=\max \left[{\mu}_A(x),{\mu}_B(x)\right] $$
(12)
$$ {\mu}_{A\cap B}(x)=\min \left[{\mu}_A(x),{\mu}_B(x)\right] $$
(13)

Using these operations, the system evaluates the antecedents of the rule to obtain the truth number. Then, this number is used to obtain the degree of membership of the output variables by applying the rule consequences.

For the output variables kp and IOBmax, we designed identical membership functions as shown in Figs. 20 and 21.

Fig. 20
figure 20

Fuzzy sets for output linguistic variable kp

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Fig. 21
figure 21

Fuzzy sets for output linguistic variable IOBmax

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By maintaining a sufficient overlap between adjacent membership functions, we tried to ensure a smooth response of the fuzzy system. Here, we clipped the consequent membership functions based on the degree of membership of the antecedent.

1.4 Aggregation of the outputs

Once all the rules have been evaluated, the clipped consequent sets are all combined into a single fuzzy set. The union operator is used to that aim. This generates an overall fuzzy output set.

1.5 Defuzzification

The defuzzification is a process that allows us to obtain a single value out of the overall fuzzy output set. In this application, the single value is contained in the universe of discourse of the output linguistic variables, i.e., [− 0.15, 0.15]. The system uses the mCoA defined as follows:

$$ mCoA=\frac{\int f(x)\cdot x\; dx}{\int f(x) dx} $$
(14)

where f(x) represents the overall fuzzy output set and x belongs to the output domain. Therefore, the system takes the centroid of gravity of the overall fuzzy output. The x that corresponds to the centroid of gravity is taken as the output of the fuzzy inference engine.

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Beneyto, A., Vehi, J. Postprandial fuzzy adaptive strategy for a hybrid proportional derivative controller for the artificial pancreas. Med Biol Eng Comput 56, 1973–1986 (2018). https://doi.org/10.1007/s11517-018-1832-1

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