An augmented subcutaneous type 1 diabetic patient modelling and design of adaptive glucose control

Journal of Process Control 86 (2020) 94–105

Contents lists available at ScienceDirect

Journal of Process Control journal homepage: www.elsevier.com/locate/jprocont

An augmented subcutaneous type 1 diabetic patient modelling and design of adaptive glucose control Anirudh Nath a,∗ , Dipankar Deb b , Rajeeb Dey a a b

Department of Electrical Engineering, National Institute of Technology Silchar, Assam, India Department of Electrical Engineering, Institute of Infrastructure Technology Research and Management, Ahmedabad 380026, Gujarat, India

a r t i c l e

i n f o

Article history: Received 5 February 2019 Received in revised form 1 July 2019 Accepted 27 August 2019 Keywords: Artificial pancreas Adaptive control Type 1 diabetes Intra-patient variability Parameter estimation

a b s t r a c t In the present work, an augmented subcutaneous (SC) model of type 1 diabetic patients (T1DP) is proposed first by estimating the model parameters with the aid of nonlinear least square method using the physiological data. Next, a nonlinear adaptive controller is proposed to tackle two important issues of intra-patient variability (IPV) and uncertain meal disturbance (MD). The proposed patient model agrees quite well with the responses of one of the most popular existing nonlinear model used in the research of artificial pancreas. Further, the developed adaptive control is shown to be capable of providing desired glycemic control without feed-forward action for meal compensation or safety algorithms to avoid hypoglycemia. Due to the simple structure and capability of handling intra-patient variability of the adaptive controller, it can find immediate applicability in the development of the in-silico artificial pancreas. © 2019 Published by Elsevier Ltd.

1. Introduction In the backdrop of metabolism, the pancreas constitutes a very vital organ that is directly responsible for maintaining the blood glucose level (BGL) within the safe range (70–180 mg/dl) [1]. The failure in the normal functioning of the ˇ-cells of islets of Langerhans of the pancreas causes diabetes mellitus. According to the international diabetes atlas, the treatment of diabetes mellitus accounts for about 12% of the global health expenditure and is one of the most prevalent diseases with around 425 million people suffering from it [2]. Type 1 diabetes (T1D) is marked by negligible insulin-dependent glucose utilisation because of the antigen-antibody and leucocyte mediated destruction of the pancreatic ˇ-cells, resulting in chronic hyperglycemia (BGL > 180 mg/dl) and ultimately leading to diabetic retinopathy, neuropathy and nephropathy [1,3]. Multiple instances of prolonged hyperglycemia and severe hypoglycemia (BGL < 50 mg/dl) occur in people suffering from T1D who rely on multiple insulin dosages for blood glucose regulation [4]. Artificial pancreas (AP) provides an automated closed-loop solution to this problem by mimicking actions of the pancreas. It facilitates automated continuous insulin delivery via an insulin

∗ Corresponding author. E-mail addresses: [email protected] (A. Nath), [email protected] (D. Deb), [email protected] (R. Dey). https://doi.org/10.1016/j.jprocont.2019.08.010 0959-1524/© 2019 Published by Elsevier Ltd.

pump as suggested by the controller on the basis of current plasma glucose measurements facilitated by suitable sensor [5]. The models describing the dynamics of the glucose-insulin regulatory system (GIRS) of type 1 diabetic patients (T1DP) is crucial for the design and development of model-based control algorithms [6]. There exist two classes of mathematical models of GIRS of T1DP, namely, the empirical models and the physiological models, based on whether they have standard structures of time-series models or whether they model the various pharmacokinetic and physiological processes of GIRS, respectively [7]. Important works on the empirical models and design of controllers based on them were reported in [8]. The significance of various physiological parameters is lost in the empirical models because of their standard predefined structure and are completely data-driven [7]. On the other hand, the significance of important physiological parameters is preserved in the physiological models. But since the physiological models represent the various pharmacokinetic and physiological processes of GIRS, they are often complicated comprising of highly nonlinear differential equations and a large number of state variables [9]. It makes the process of designing nonlinear control algorithms based on such models extremely difficult. The physiological models of T1D can be further classified into (i) simple minimal models (representing the macroscopic functionality of GIRS) and (ii) the complicated maximal models (representing the minute-level functionality of GIRS) [10]. The most popular and the most extensively used minimal model (intravenous) was developed by Richard N. Bergman over 35 years ago [11]. Apart from this, important mod-

A. Nath et al. / Journal of Process Control 86 (2020) 94–105

els of the subcutaneous T1D model include the Hovorka model [12], the UVA/Padova model [13], etc. For instance, the mathematical model used in the FDA approved UVA/Padova simulator [14] has 16 states with underlying nonlinear functions and parameters. The UVA/Padova model is also implemented in GIM simulator [15]. Likewise, the Hovorka model is represented by 8 nonlinear differential equations with some of the nonlinear functions, such as renal clearance and endogenous glucose production described by discontinuous functions. This makes the design of many nonlinear controllers such as feedback linearisation, etc. impossible without additional mathematical approximations, since, for designing such nonlinear control algorithms, the nonlinear functions of the mathematical model must be differentiable [16]. Here, in the present work, an attempt is made to address the problem and find a bridging solution to this. A nonlinear subcutaneous (SC) T1DP model is proposed by augmenting the Bergman’s minimal model (BMM) with the SC glucose and insulin dynamics that is comprised of the following characteristics: (i) simple structure: it has simpler structure as compared to the UVA/Padova model [14,3] and the Hovorka’s model [12] in terms of reduced number of state variables and number as well as nature of nonlinearity, (ii) physiological significance: since the core dynamics of GIRS, i.e. the insulin action and the plasma glucose dynamics of the proposed SC T1DP model are represented by BMM, important physiological parameters like insulin sensitivity and glucose effectiveness are preserved [10], (iii) accuracy: it represents the response of the GIRS quite accurately that is statistically validated and (iv) controloriented model: the design of nonlinear controller based on this model will be much simpler as compared to its other SC counterparts because of its simple non-linearity and structural simplicity. A similarly modified version of the BMM called the identifiable virtual patient (IVP) model was proposed in [17] using the dataset of [18]. The main differences of the proposed controller with the IVP model are as follows: (i) in this work, the meal absorption dynamics adopted from [19] which is more accurate than that of the IVP model in [17] and (ii) the SC insulin absorption dynamics of the proposed model is adopted from [12] consists of two-compartments representing the SC insulin concentrations in hexameric and monomeric forms, unlike the IVP model, where a single-compartment is utilised to represent the SC insulin dynamics, thus, providing more physiological insight into the SC dynamics. Similar works on control-oriented modelling were done recently on another biological system, like microbial fuel cell [20]. In T1D patients, two important challenges exist pertaining to uncertainty and variation in the physiological parameters of GIRS. The parameter set of a physiological model of the GIRS of T1D patients varies significantly from patients to patients within a given population which is termed as inter-patient variability. Furthermore, slow time-varying nature exists within diabetic patients, which is termed as intra-patient variability [21]. The phenomena of inter-patient variability and intra-patient variability can be addressed by designing adaptive control algorithms [22]. The application of various advanced process control methods is well summarised in [23]. In the current research, the focus is on the design of an adaptive control algorithm for AP. A critical review of some of the inherent shortcomings in the existing adaptive control methods designed in the past for this problem is presented as follows: (i) the aggressive control action (insulin infusion scheme) of the minimum variance controllers may lead to hypoglycemic events [22], (ii) linearisation of nonlinear T1D models for designing controllers result in loss of nonlinear characteristics of the original system [24–30], (iii) intra-patient variability was not explicitly taken care of in the design of adaptive controllers as reported in [22,25,26,31], and moreover, these control techniques were based on time-series models that do not provide explicit information about physiologically significant parameters like insulin sensitiv-

95

ity, glucose effectiveness, directly, unlike the physiologically based models and (iv) presence of integral action in the model predictive control (MPC) [32] may lead to hypoglycemia during fasting due to glycemic variability as mentioned in [33]. In the present work, a nonlinear adaptive controller designed for the BMM [34] is further extended for the SC T1DP model. The reason behind this choice of the control algorithm is that it can handle uncertainty due to inter-patient as well as intra-patient variability without any sort of linear approximations, thus retaining the nonlinear characteristics of the nonlinear model. The main highlights of the proposed adaptive control technique for the proposed SC T1DP model are as follows: (i) The time-varying adaptive control law along with the parametric updating laws facilitate an online adaptation to the parametric variations due to intra-patient variability and uncertainty in the external disturbance (meal intake). (ii) The output of the uncertain T1DP model, i.e. BGL follows the output of the nonlinear reference system (representing the desired dynamics) despite parametric uncertainty. Unlike conventional model-based adaptive controllers [26,32,33] the reference system here consists of estimated parameters (in place of known/fixed parameters), since there exist no nominal/universal sets of parameters for a specific T1DP model that can represent the ideal characteristics of GIRS of diabetic subjects. (iii) The earlier method is extended for an uncertain system with exogenous disturbance unlike the original work as reported in [34]. The current method does not require explicit information about the meal disturbance model in contrast to the controller proposed in [34]. (iv) Unlike, the control techniques in [35,36], here the proposed feedback control law do not require prior meal announcement or meal estimation and safety algorithms like insulin-on-board (IOB) to avoid post-prandial hyperglycemia [35] and severe hypoglycemia [36], respectively. The adaptive feedback control law achieves the desired performance without any of these additional schemes. (v) The proposed control law do not require additional feedforward disturbance compensation [37] or IOB safety scheme to avoid severe hypoglycemia when the patient parameters vary (i.e. intra-patient variability). (vi) Two sets of controller parameters, ci and  i facilitate a convenient way to maintain the closed-loop stability (expressed in terms of ci ) and improve the transient response, respectively, by tuning their values separately. 2. Methods A novel philosophy of closed-loop regulation of the BGL in T1DPs has been proposed in this research work as discussed below. The philosophy is a two-step strategy comprising of (i) representation of the nonlinear and complex SC dynamics of T1D/ complex physiological models of T1D (with large number of states and complex nonlinear functions) by a simple nonlinear T1D model (with less number of states and minimum non-linearity) using the input–output data-set (data of the T1D physiological model/clinical) and (ii) designing a nonlinear adaptive control algorithm for the AP for maintaining BGL in the safe range in the presence of inter-patient and intra-patient variability. The mathematical model of SC dynamics of T1D should provide a good trade-off between the model complexity and response accuracy. The controller should provide an effective blood glucose regulation without any hypoglycemic episodes where the intra-patient variability exists.

96

A. Nath et al. / Journal of Process Control 86 (2020) 94–105

insulin infusion at the SC level. It has the control input (exogenous insulin infusion rate) that is modelled via 3 ODEs as I˙ sc1 (t) =

1 Isc1 (t) + u(t) Ti

(2a)

I˙ sc2 (t) =

1 (Isc1 (t) − Isc2 (t)) Ti

(2b)

I˙ p (t) = −ke Ip (t) +

Isc2 (t) , Ti Vi

(2c)

where Isc1 (t), Isc2 (t) are the insulin masses in the first and second SC compartments, respectively and Ip (t) is the plasma insulin concentration. Model parameters tmax , ke and Vi represent parameters of the SC insulin absorption model. The parameter, Ti denotes the time-constant for the absorption dynamics of subcutaneous insulin. 2.1.3. Subcutaneous glucose-absorption dynamics An inherent physiological lag exists in the SC measurement of the glucose concentration that can be modelled by a first-order ODE [12]. The dynamics of appearance of glucose in the SC compartment from where the CGM devices take the glucose readings, is described by a first order differential equation Fig. 1. Block diagram of the proposed model comprising of the sub-systems: (i) plasma glucose dynamics, (ii) delayed insulin action dynamics (remote insulin), (iii) subcutaneous insulin absorption dynamics and (iv) meal dynamics.

1 G˙ sc (t) = (Gp (t) − Gsc (t)), Tsc

2.1. T1D patient model

where Gsc (t), Gp (t) are the states for glucose concentrations in the interstitial fluid (ISF) and blood plasma, respectively, and Tsc is the time constant for the lag between the appearance of glucose from plasma to the interstitial fluid [12].

In this present study, the proposed SC mathematical model of T1DP captures the macroscopic behaviour of the GIRS dynamics of T1DPs while having a simplistic model structure. Here, the BMM [11] has been augmented with the SC insulin absorption dynamics, the meal absorption dynamics and the SC glucose dynamics (Fig. 1). As stated earlier in Section 1, the main motivation behind modelling is to develop an appropriate nonlinear control-oriented model with the help of simple ordinary differential equations (ODEs) and nonlinear functions without actually losing the physiological significance in terms of the structural organisation of the model ([16]). The various subsystems of the proposed SC T1DP model are stated below: 2.1.1. Plasma glucose dynamics This is the core-dynamics of the glucose-insulin interaction representing the utilisation, appearance and production of the plasma glucose in the blood. These dynamics are adopted from the BMM [11] with an introduction of a weighting factor, k useful for parameter estimation. The BGL dynamics and the delayed (remote) insulin action of insulin are G˙ p (t) = −p1 Gp (t) − RI(t)Gp (t) + p1 Gb + kD(t)

(1a)

˙ RI(t) = −p2 RI(t) + p3 Ip (t),

(1b)

where Gp (t), RI(t) are the BGL (mg/dL) and active insulin (min−1 ) p in the remote compartment, respectively, p3 denotes insulin sensi2 tivity. Gb represents the basal value (steady-state value) Gp (t) and Ip (t) denotes the plasma insulin concentration (mU/l) in the blood. Remark 1. The purpose of introducing the gain factor k in (1a) is to scale the rate of glucose appearance in the plasma glucose compartment in such a way that, the increase in the plasma glucose concentration (following a meal disturbance) is reflected in the model response. 2.1.2. Insulin-absorption dynamics This sub-system is typical of the SC dynamics of a standard GIRS. This sub-system is adopted from the Hovorka model [12] that tell us how the insulin is being absorbed into the blood after an exogenous

(3)

2.1.4. Meal-absorption dynamics The model that describes the meal absorption dynamics is a three compartmental model having three states representing the mass of glucose in the stomach, Q1 (t), Q2 (t) in solid as well as liquid phases, respectively and Qgut (t) as the glucose mass in the intestine. Ra (t) represents the rate of glucose appearance in the blood. k21 , kempt , and kabs are the rate constants for elimination of glucose in the three compartments [19]. The coefficient D denotes the amount of ingested glucose (mg) and ı(t) is an unit impulse input to the meal disturbance model indicating the time of meal intake. Q˙ 1 (t) = −k21 Q1 (t) + Dı(t)

(4a)

Q˙ 2 (t) = −kempt Q2 (t) + k21 Q1 (t)

(4b)

Q˙ g (t) = −kabs Qg (t) + kemp Q2 (t)

(4c)

Ra (t) =

kabs f Qg (t). Vi

2.1.5. State space representation of the proposed T1D model The proposed T1DP model with the underlying dynamics (subsystems) as presented above can be expressed into standard state space formulation as follows: x˙1 (t) = −ˆp1 (t)(x1 (t) − Gb ) − x1 (t)x2 (t) + d(t)

(5)

x˙2 (t) = −ˆp2 (t)x2 (t) + pˆ 3 (t)x3 (t) x˙3 (t) = −ˆp4 (t)x3 (t) + pˆ 5 (t)x4 (t) x˙4 (t) = −ˆp6 (t)x4 (t) + pˆ 6 (t)x5 (t) x˙5 (t) = −ˆp6 (t)x5 (t) + u(t) x˙6 (t) = −ˆp7 (t)x6 (t) + pˆ 7 (t)x1 (t) where xi (t), i = 1, . . ., 6 represents Gp (t), RI(t), Ip (t), Isc2 (t), Isc1 (t) and Gsc (t), respectively. The term d(t) represents the rate of appearance of glucose (Ra (t)) into the plasma glucose compartment. The

A. Nath et al. / Journal of Process Control 86 (2020) 94–105

97

model parameters are given as pˆ 1 (t), pˆ 2 (t), pˆ 3 (t), pˆ 4 (t) = ke (t), V (t) pˆ 5 (t) = T i (t) , pˆ 6 (t) = T 1(t) and pˆ 7 (t) = T 1(t) . Note that the parame-

For nonlinear systems, the parameters  i are estimated via iterative approximation given by

ters, pˆ i , i = 1, . . ., 7 of the uncertain T1D model in (5) are uncertain and time-varying in nature that need to be estimated. The above mathematical model of T1D will be termed as augmented subcutaneous minimal model (ASMM) from now. The output of the proposed SC T1DP model in (5) is the glucose concentration in the ISF which is given by

i  ik+1 = ik + i ,

i

i

sc

(10)

where k and  are iteration number and change vector, respectively. By applying Taylor series the approximation is made Gm (ti , )  Gm (ti ,  k ) +

 ∂ Gm (t ,  k ) i j

y(t) = x6 (t).

(6)

2.1.6. Scope of inter-patient and intra-patient variability The inter-patient variability is introduced in the current work by parameter estimation of different T1D subjects provided in the simulators like GIM and UVA/Padova T1D Simulator. Apart from the inter-patient variability, the parameters of the T1D subjects also vary within a day. As mentioned in [38], the within-day variability in the model parameters is introduced by superimposition of sinusoidal oscillations on the nominal parameter value of varying amplitude and three and nineteen hours time period which accounts for the variability in the insulin sensitivity and insulin absorption dynamics.

 Gm (ti ,  k ) +



∂ j

(j − jk )

(11)

Jij jk ,

j

where J denotes Jacobian matrix. The Taylor series expansion is used to express the original nonlinear equations in an approximate, linear-in-parameter form in order to obtain the residual errors between input–output data set and the nonlinear dynamical equations of the system. The error can be represented as eri = yi −

p 

Jjs s ,

(12)

s=1

where yi = Gi − Gm (ti ,  k ). Substituting in (9), we obtain 2.1.7. Open-loop parameter estimation The objective is to estimate the parameters of the ASMM whose output is the plasma glucose concentration (Gp (t)) for the inputs: (i) exogenous meal intake and (ii) subcutaneous insulin infusions using the data of the T1D simulator, GIM [15]. Least squares are largely used in dynamical model parametric estimation [39,40]. Here, a nonlinear least squares (NLS) method is employed for estimating the parameters of the ASMM in (5). It is essentially an iterative method for the approximation of the minimum of a curved hyper-plane. The principal reason behind the choice of NLS as the parameter estimation algorithm is that the ASMM is essentially a nonlinear model. Parameter estimation methods like maximum likelihood estimator is not used in this work because one needs to specify the form of the distribution of the random error for the calculation of the likelihood function which is avoided in the current study. However, comparative studies of parameter estimation via both NLS and maximum likelihood algorithms may be investigated further. Other reasons behind this are stated as follows: (i) the BMM can accurately represent the intravenous data for T1DPs [11]. Hence, it can be augmented with linear dynamics of SC insulin absorption to represent the dynamics of SC glucose due to SC exogenous insulin infusion [17]. (ii) NLS algorithm is simple and effective when the one know that a particular structure of the model is capable of representing the data-set. The error is defined as the difference between the measured blood glucose concentration (BGC) from GIM and the predicted BGC (output of the proposed model) at time ti is denoted by eri = Gi − Gmi (ti , p )

(7)

where eri is the residual error, Gi is the measured BGC at time ti and Gmi (ti , p ) is the predicted BGL at ti utilising the parameters  p = [p1 p2 p3 Ti Vi ke k], p = 7. The error function, SOS er , is given by SOS er =

N 

er2i .

(8)

i=1

The minima of SOS er is reached once the gradient, zero.

 ∂er ∂ SOS er i =2 eri . ∂ i ∂i i

∂ SOS er becomes ∂ i

(9)

N 

−2

Jij (yi −

m 

(13)

Jis s ).

s=1

i=1

In the current work, NLS regression discussed above is solved using the MATLAB toolbox, “SIMULINK DESIGN OPTIMIZATION” [41] along with the ‘trust-region-reflective’ algorithm [42]. For the performance assessment of the proposed model, the coefficient of determination (R2 ) statistic is chosen which will indicate the proportion of the variance in the plasma glucose concentration (dependent variable) that is predictable from the given inputs as the meal and external insulin infusions (independent variables)

no

2

R =1−

2

j=1

no

j=1

(yoj − yˆ oj ) (no − 1)

(yoj − y¯ o )2 (no − mc )

,

(14)

where no and mc denote the number of observations and fitted coefficients, respectively, yoj are the observed values, yˆ oj are the model output values and y¯ o is the mean of the observations. 2.2. Adaptive controller design In this section, an adaptive controller (for exogenous insulin infusion) is proposed for regulating the BGL in the safe range (70–180 mg/dl) based on the new SC T1DP model (ASMM) based on the adaptive parametric compensation (APC) technique as introduced in [34]. The primary objective of the closed loop adaptive system is to regulate the BGL within 70–180 mg/dl without any severe hypoglycemic (BGL < 50 mg/dl) episodes. Also, the controller should bring the BGL under 180 mg/dl within 120 min to prevent postprandial hyperglycemia. All these control specifications should be achieved when both inter-patient, as well as intra-patient variability, exist and exogenous meal disturbance affects the GIRS of the T1D subjects (Fig. 2). Using the APC technique presented in [43–45], the regulation of BGL is performed by ensuring that the output (BGL) of the uncertain system (T1DP model with parametric uncertainty) follow the output of a reference system (desired behaviour) by choosing a suitable adaptive control law along with some adaptive update laws. Unlike conventional MRAC [30], the reference system is considered to be nonlinear in this case. This choice of the reference system to be a representative of desired system behaviour is very important as it determines the actual behaviour of the uncertain T1DP

98

A. Nath et al. / Journal of Process Control 86 (2020) 94–105

∼

−x1 (t)e2 (t) + d(t) e˙ 2 (t)

=

e˙ 3 (t) = e˙ 4 (t) = e˙ 5 (t)

=

e˙ 6 (t) = Fig. 2. Scheme of the proposed adaptive control system comprising of the adaptive control law, u(t); the reference system, xˆ (t), the adaptive laws, pˆ i , the reference signal, r(t) and the uncertain T1DP model, x(t).

model under inter-patient and intra-patient variability. The control technique in [34] is extended for the uncertain nonlinear system with exogenous disturbance input, with the assumption that the disturbance model (meal model) [19] is completely known.

∼

∼

∼

∼

p2 (t)x2 (t) − p2 e2 (t) − p3 (t)x3 (t) + p3 e3 (t) p4 (t)x3 (t) − p4 e3 (t) − p5 (t)x4 (t) + p5 e4 (t)

∼

p6 (t)(x4 (t) − x5 (t)) − p6 (e4 (t) − e5 (t))

∼

p6 (t)x5 (t) − p6 e5 (t) − r(x6 (t)) + u(t)

∼

∼

p7 (t)x6 (t) − p7 e6 (t) − p7 (t)x1 (t) + p7 e1 (t)

where ei (t) = xi (t) − xˆ i (t), i = 1, . . ., 6 are the error signals (difference between actual states and the states of the reference system). The parameters of closed loop error dynamics are denoted by ˜ p˜ i (t) = pi − pˆ i (t), i = 1, . . ., 7. The term d(t) represents the differˆ ence between d(t) and d(t) (estimate of d(t)) as ˆ ˜ = d(t) − d(t) d(t)

(19)

2.2.1. Uncertain system The ASMM is considered to be the uncertain system representing the dynamics of T1D patient as provided in (5).

Remark 2. It is to be noted that the proposed APC control algorithm does not need exact information about the meal disturbance model in (4). Only the boundedness of the exogenous disturbance signal, d(t) is required which is quite practical since the meals (amount of carbohydrate) taken by the T1DPs are always finite in quantity.

2.2.2. Reference system The structure of the reference system with known parameters is chosen similar to the original nonlinear and is system described by the following ODEs:

2.2.4. Adaptive control law The adaptive controller structure is chosen as

ˆ xˆ˙ 1 (t) = −p1 (ˆx1 (t) − Gb ) − xˆ 1 (t)ˆx2 (t) + d(t)

u(t) = r (x6 (t)) −

(15)

xˆ˙ 2 (t) = −p2 xˆ 2 (t) + p3 xˆ 3 (t) xˆ˙ 3 (t) = −p4 xˆ 3 (t) + p5 xˆ 4 (t) xˆ˙ 4 (t) = −p6 xˆ 4 (t) + p6 xˆ 5 (t) xˆ˙ 5 (t) = −p6 xˆ 5 (t) + r(x6 (t)) xˆ˙ 6 (t) = −p7 xˆ 6 (t) + p7 xˆ 6 (t), where xˆ i (t), i = 1, . . ., 6 represents the states of the reference system in (15) representing the desired dynamics. pi , i = 1, . . ., 7 represent the known nominal parameters. The output of the desired dynamics represented by the reference system is yˆ (t) = xˆ 6 (t).

(16)

It is desired that the BGL, i.e. output (6) of the uncertain system in (5) should follow the desired output (16) of the reference system in (15). The input to the reference system in (15) is r (x6 (t)) whose role is to produce desired response of the reference system (15). The choice of r(x6 (t)) may be any standard continuous insulin infusion profile (as suggested by the diabetes experts). In this case, we have chosen r(x6 (t)) to be continuous function of the SC glucose level given below: r(x6 (t)) = 0.01 x6 (t) + 0.9.

(17)

2.2.3. Error dynamics The error dynamics should be stabilised by the control law. It is obtained by differentiating the difference between (5) and (15) with respect to time, as provided below e˙1 (t) = p˜ 1 (t)(x1 (t) − Gb ) − p1 e1 (t) − xˆ 2 (t)e1 (t)

(18)

e5 (t)ˆp6 (t) . 2

(20)

Remark 3. It is important to note that the adaptive control law in (20) is a function of the reference signal in (17), the estimated parameter pˆ 6 and error e5 that have direct influence on the control law appearing in the fifth state of the system (5). In the case of perfect tracking of the reference states of the reference system (15) by the states of the uncertain system (5), the error term e5 becomes zero and the control law exactly follows the reference signal resulting in desired system behaviour. Any deviations from the nominal characteristics of the system (5) due to parametric uncertainty is compensated by the change in the adaptive control law from the reference signal, due to the presence of the estimated parameter pˆ 6 and e5 . 2.2.5. Parameter update laws The parametric update laws (derived from the Lyapunov stability analysis) which provide the estimated parameters of the reference system are provided below: pˆ˙ 1 (t) = 1−1 c1 e1 (t)(x1 (t) − Gb )

(21)

pˆ˙ 2 (t) = 2−1 c2 e2 (t)x2 (t) pˆ˙ 3 (t) = −3−1 c2 e2 (t)x3 (t) pˆ˙ 4 (t) = 4−1 c3 e3 (t)x3 (t) pˆ˙ 5 (t) = −5−1 c3 e3 (t)x4 (t)





pˆ˙ 6 (t) = 6−1 c4 e4 (t)(x4 (t) − x5 (t)) + c5 xˆ 5 (t)e5 (t) pˆ˙ 7 (t) = 7−1 c6 e6 (t)(x6 (t) − x1 (t)) ˆ˙ = d−1 c1 e1 (t) d(t)

A. Nath et al. / Journal of Process Control 86 (2020) 94–105

where  i , i = 1, . . ., 7, d and ci , i = 1, . . ., 6 represents two sets of constant parameters that need to be tuned to maintain the closed loop stability and shape of the transient response, respectively. Remark 4. It is important to note that the update laws in (21) and the feedback control law in (20) are derived from the closed-loop Lyapunov stability analysis. Remark 5. From here, the argument ‘t’ of xi (t)s, xˆ i (t)s, ei (t)s, pˆ j (t)s and p˜ j (t)s will be dropped for easy representation, unless stated otherwise they are functions of time in this paper. Theorem 1. If the adaptive feedback control law u(t) in (20) in conjunction with the adaptive laws in (21) is applied to the nonlinear error dynamics in (18), then it can be guaranteed that all the closed-loop signals within the controlled system are bounded and the error signals ei (t), i = 1, . . ., 6 asymptotically approach zero i.e. lim ei (t) = 0, i = 1, . . ., 6.

The expression for V˙ in (26) can be written as V˙ = V˙ 1 + V˙ 2 + V˙ 3 + V˙ 4 + V˙ 5 + c5 e5 (r(x6 ) − u(t)) −

V˙ 1

6

7

i=1

(22)

j=1

where ci ,  j > 0, i = 1, . . . 6, j = 1, . . ., 7. Differentiating (22) and then substituting error dynamics (18) and update parameter laws (21), we get







1 c4 p6 e4 = − c5 pˆ6 e5 − 2 pˆ 6 c5

+c4 e4 (˜p6 (x4 − x5 ) − p6 (e4 − e5 )) − 4 p˜ 4 pˆ˙ 4

Similarly, V˙ 2

+c5 e5 (˜p6 x5 − p6 e5 − r(t) + u(t)) − 5 p˜ 5 pˆ˙ 5



1 e3 c3 p5 = − c4 p6 e4 − 2 c4 p6

Similarly, V˙ 3 =

−p1 c1 e12 − c1 xˆ 2 e22 − c1 x1 e1 e2 − p2 c2 e22 +p3 c2 e2 e3 − p4 c3 e32 + p5 c3 e3 e4 − p6 c4 e42





(24)



+p6 c4 e4 e5 + p7 c6 e1 e6 + c5 e5 u(t) − c5 e5 r(x6 ). Note: Let us consider the under-braced term of (24) which is modified as follows: −p6 c5 e52 + p˜ 6 c5 e5 x5 − 6

1 p˜ 6 c5 e5 xˆ 5 6

2

+

e22 p23 c22 2c3 p4

−

1 c3 e32 p4 2 (30)

(32)

1 c2 e22 p2 2

= −ˆp6 c5 e52 . Eq. (24) is modified with the replacement of the under-braced term by (25) as provided below −p1 c1 e12 − c1 xˆ 2 e22 − c1 x1 e1 e2 − p2 c2 e22 + p3 c2 e2 e3 (26)

e2 x1 p1

2

+

1 c2 e22 p2 2 e22 x1 2 c1 p1

−

1 c2 e22 p2 . 2

(34)

Remark 6. It is important to note that the model parameters, pj and the estimated parameters, pˆ j , j = 1, . . ., 7 are all positive quantities that represent time-constants of various states of the system. The first state variable, x1 represents the plasma glucose concentration which is always a positive nonzero quantity and varies in the range 30–400 mg/dl. = −c1 e12 xˆ 2 + c6 e5 e1 p7 − c6 e62 p7 = −c6 p7 e6 − c1 p7 > , c6 4ˆx2

−p7 c6 e62 + p7 c6 e1 e6 + c5 e5 u(t) − c5 e5 r(x6 ).

2c4 p6

To render V˙ 4 < 0 in (34), the following stability condition need to be satisfied c1 p2 p1 < (35) , x1 > 0. c2 x1 2

(25)

= −p6 c5 e52 + p˜ 6 c5 e52

 

−

= −c1 e12 p1 − c1 e1 e2 x1 −



−p4 c3 e32 + p5 c3 e3 e4 − p6 c4 e42 + p6 c4 e4 e5 −ˆp6 c5 e52

e32 p5 2 c32

1 1 c2 e22 p4 + c2 e2 e3 p3 − c3 e32 p2 2 2

= −c1 p1 e1 +

Similarly, V˙ 5

= −p6 c5 e52 + p˜ 6 c5 e5 (x5 − xˆ 5 )

V˙ =

−



1 ∼ p6 c5 e5 xˆ 5 −p7 c6 e62 6

+

To render V˙ 3 < 0 in (32), the following stability condition need to be satisfied c2 p4 p2 < . (33) c3 p2 Similarly, V˙ 4

Substituting the adaptation laws in (21) in the above expression of V˙ in (23), one can obtain

∼

2

To render V˙ 2 < 0 in (30), the following stability condition need to be satisfied c3 p6 p4 < (31) . c4 p3 2

˙ −1 p˜ 1 pˆ˙ 1 − d d˜ dˆ − 7 p˜ 7 pˆ˙ 7 .

−p6 c5 e52 + p6 c5 e5 x5 − 6

2c5 pˆ 6

3

+c6 e6 (˜p7 x6 − p7 e6 − p˜ 7 x1 + p7 e1 ) − 6 p˜ 6 pˆ˙ 6

V˙ =

+

(28) 1 − c4 e42 p6 . 2

1 1 = − c3 e32 p4 + c3 e3 e4 p5 − c4 e42 p6 2 2



(23)

2

c42 pˆ6 e42

(29)

1 c2 e2 p3 = − c3 p4 e3 − 2 c3 p4

+c3 e3 (˜p4 x3 − p4 e3 − p˜ 5 x4 + p5 e4 ) − 3 p˜ 3 pˆ˙ 3

2

c4 pˆ 6 < . c5 p6

˜ V˙ = c1 e1 p˜ 1 (x1 − Gb ) − p1 e1 − xˆ 2 e1 − x1 e2 + d(t) +c2 e2 (˜p2 x2 − p2 e2 − p˜ 3 x3 + p3 e3 ) − 2 p˜ 2 pˆ˙ 2

(27)

1 1 = − c4 e42 p6 + c4 e4 e5 p6 − c5 e52 pˆ6 2 2

Proof. Let us consider a positive definite, radially unbounded and decrescent function given by 1  2 1  2 1 ˜2 ci ei + j p˜ j + d d 2 2 2

1 pˆ 6 c5 e52 , 2

where V˙ i , i = 1, . . ., 5 are defined as follows:

t→∞

V=

99

xˆ 2 > 0.

e1 2

2

+

c6 p7 e12 4

(36) − c1 e12 xˆ 2 (37)

Remark 7. The second state variable of the system (5) and its second state of the reference system (15) xˆ 2 , respectively, represents the delayed effect of the insulin action and is always a positive quantity. And since the T1DP are always under insulin therapy, the initial conditions of xi , i = 1, . . ., 6 are always non-zero and are also assumed in the present work. Hence, all the stability conditions (29)–(37) will be always satisfied for certain tuned values of controller parameters, ci , i = 1, . . ., 6.

100

A. Nath et al. / Journal of Process Control 86 (2020) 94–105

If the above stability conditions in (29)–(37) are satisfied, then V˙ 1 , V˙ 2 , V˙ 3 , V˙ 4 and V˙ 5 are guaranteed to be negative definite. From the left over terms in (27), one can chose the adaptive control law as provided in (20) to ensure the negative semi-definiteness of the V˙ (t) in (27) follows V˙ ≤ 0 ⇒ c5 e5 (u(t) − r(x6 )) − ⇒ u(t) = r(x6 ) −

1 c5 e52 pˆ 6 ≤ 0 2

1 e5 pˆ 6 . 2

(38)

Remark 8. It is important to note that for highly uncertain artificial pancreas system, it is extremely difficult to model the dynamics with a unique set of parameters since the parameters of individual patients keep changing over time. Parameter convergence for such system consisting of complicated uncertain T1D dynamics is not justified since it is not possible to derive a reference system that will represent ideal behaviour. In addition to this, there is restriction in the nature of the reference signal or the exogenous insulin infusion, which needs to be extremely smooth and should be approved by the physicians, since artificial pancreas deals directly with human life. So any arbitrary signal (sufficiently rich with distinct different frequencies) [46] cannot be utilised for this problem. For this reason, no claim is made on parameter convergence that demands that the adaptive system should be persistently exciting [47,48], in the present work.

Fig. 3. Profile of (a) exogenous insulin injection, uOL (t) and (b) rate of glucose in plasma, Ra (t) in virtual T1D subject [15] for parameter estimation.

3. Results The parameter estimation problem of the proposed ASMM in (5) (open-loop) and performance evaluation of the proposed adaptive controller in (20) (closed loop) are presented below. 3.1. Scenario 1: Open-loop parameter estimation of the proposed T1D model Parameter estimation: The main objective of this scenario is to estimate the model parameters of the proposed ASMM in (5) utilising the open-loop input–output data-set of the virtual T1D subject of the GIM simulator [15]. The mathematical model of the virtual T1DP of the GIM simulator is the Dalla Man model [13] which are structurally more complicated and possess more states and model parameters. A virtual simulation scenario is created to capture the effect of discrete exogenous SC insulin injections, uOL (t) along with a single meal disturbance input of 300 g of carbohydrate at 08:00 (equivalent to 480 min). It is assumed that the virtual T1D subject is in fasting state initially. An external insulin injection, uOL (t) of 3 U is administered subcutaneously at time t = 480 min (08:00 h) from the time of the start of the simulation (00:00 h). The maximum number of iterations, parameter tolerance and function tolerance of the NLS parameter estimation algorithm are 100, 0.001 and 0.001, respectively. The input data represents the time-profile of the external insulin dosage, uOL (t) as well as the rate of appearance of glucose in the bloodstream, Ra (t). The output data represents the plasma glucose concentration profile as generated by GIM for the considered inputs. The profile of the two inputs, i.e. uOL (t) and Ra (t) are depicted in Fig. 3. The result of this scenario is presented in Fig. 4. Model validation: Estimated parameters of the ASMM in scenario 1 are utilised for the validation of the profile of BGL of the virtual T1DP under multiple meal disturbances. The input data-set consists of uOL (t) and Ra (t) and are depicted in Fig. 5. The details of the experimental protocol consisting of the timing and dosages of exogenous SC insulin injections and the exogenous carbohydrate intakes are given in Table 1. Simulation is carried out for a time period of 24 h

Fig. 4. 24 h plasma glucose concentration profile for GIM and the simulated response of the model for single meal and single insulin injection.

Fig. 5. Profile of (a) exogenous insulin injection, uOL (t) and (b) rate of appearance of glucose in the plasma, Ra (t) in the virtual T1D subject of the GIM simulator [15] for the protocol in Table 1 for model validation.

A. Nath et al. / Journal of Process Control 86 (2020) 94–105

101

Table 1 Parameter estimation protocol for the nonlinear least squares problem. Insulin dosage 1 (time)

Insulin dosage 2 (time)

Insulin dosage 3 (time)

3 U (480 min)

4.5 U (720 min)

4.5 U (1200 min)

Breakfast (time)

Lunch (time)

Dinner (time)

45 g (480 min)

70 g (720 min)

70 g (1200 min)

Fig. 7. Profiles of the output of the reference system, xˆ 6 (blue solid line) and the actual blood glucose level (BGL), x6 (red solid line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 6. 24 h plasma glucose concentration profile for GIM simulator (blue solid line) and the response of the proposed T1D subcutaneous model (red solid line) for 3 exogenous meal inputs and 3 insulin injections. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Table 2 Tuned values of controller parameters. Parameters c1 c3 c5 1 3 5 7

Values −3

1 × 10 1.094 × 101 1 × 101 1 × 1010 1 × 107 1 × 107 1 × 104

Fig. 8. Blood glucose trajectories for virtual T1D patients.

Parameters

Values

c2 c4 c6 2 4 6 d

1.5099 × 10 7.6 × 10−3 4 × 10−5 1 × 104 1 × 104 1 × 106 1 × 10−3

6

and the result is presented in Fig. 6 displaying the predicted blood glucose level for multiple meals multiple injection scenarios. 3.2. Scenario 2: Results of APC controller for the ASMM Simulation scenario for the T1D subject: A 24 h virtual simulation scenario of the daily variation of the glucose profile is considered here for the assessment of the controller’s performance. The model parameters of the system (15) are randomly chosen from Table 3 during each simulation, to represent parametric uncertainty. Three meals representing breakfast (50–75 g carbohydrate at 90–150 min), lunch (40–60 g carbohydrate at 330–510 min) and dinner (40–60 g carbohydrate at 960–990 min) are provided during the 24 h period where both the carbohydrate content of meals and meal timings are uncertain and vary randomly in the specified ranges. The initial BGL of the T1DPs are assumed to be at the hyperglycemic state of 180 mg/dl before the start of the numerical simulations. Results for the APC controller for ASMM: The controller gains ci in (21) are tuned to satisfy the stability constraints in (29)–(37), and then the other set of controller gains,  j are tuned to improve transient performance, and are provided in Table 2. The numerical simulations are designed for the uncertain system in (5), the reference system in (15), parameter update laws in (21) and adaptive feedback control law in (20). Fig. 8 illustrates the trajectories of BGL of 100 virtual T1DPs whose parameters are randomly varied in a specified range. The corresponding exogenous insulin infusion rate as prescribed by the APC state-feedback control law are shown in Fig. 9. The stability conditions in (35), (37) and (29) that depend on time varying parameters are shown in Fig. 10. The trajectories of the closed-loop error signals in (18) and the estimated parameters in (21) are depicted in Fig. 12 and 15, respectively.

Fig. 9. Subcutaneous insulin infusion rate for virtual T1D patients.

Results for CVGA: Control variability grid analysis (CVGA) provide the efficacy about the computation of the proposed control technique. The CVGA of the virtual simulation results is depicted in Fig. 11 where it is observed that the occurrence of the severe hypoglycemic event (BGL mg/dl) is completely eliminated during the random simulations. 3.3. Scenario 3: Random simulation with different initial conditions of insulin Simulation scenario for the T1D subject: A 24 h simulation scenario is considered for the assessment of the closed loop performance for different initial conditions of insulin. The model parameters, the administration of meal disturbances of varying carbohydrate content and timing are same as in Scenario 3.2. The only difference in the Scenario 3.3 from Scenario 3.2 lies in the initial conditions of glucose and insulin. It is assumed that the initial glucose level of the virtual T1DPs is x1 = 90 mg/dl, x6 = 90 mg/dl and the subjects have received prior insulin dosages which is reflected in high initial conditions of insulin, x2 ∈ [0.001, 0.01] (min−1 ) and x3 ∈ [10, 100] (mU/l). It is assumed that the virtual T1DPs have not received any subcutaneous insulin injections prior to the start of the simulation (x4 = 0 mU and x5 = 0 mU). A total of 100 numerical simulations considering random parameters, uncertain meal disturbance are carried out for the closed loop validation. Results for the APC controller for ASMM: The closed loop blood glucose trajectories of the virtual T1DPs in the above mentioned

102

A. Nath et al. / Journal of Process Control 86 (2020) 94–105 Table 3 Estimated parameters, p1 , p2 , p3 , p4 , p5 , p6 and k along with the range of parametric variability for the proposed ASMM in (5).

Fig. 10. Stability conditions in (a) (37), (b) (35) and (c) (29).

Fig. 11. CVGA for parametric variability of ±30%.

simulation scenario setup is illustrated in Fig. 13. The corresponding CVGA plot is shown in Fig. 14. 4. Discussion As depicted in Fig. 4, the predicted (simulated) plasma glucose concentration provided by the proposed ASMM in (5) fits the data set of GIM for a period of 24 h (1440 min) for a single meal and single subcutaneous insulin injection. This result indicates the predictive ability of the proposed model to represent the nature of the response when a single meal disturbance is provided to the T1D patient in a fasting condition. The results of the parameter estimation for this scenario is provided in the first row of Table 3. It can be inferred from Fig. 6 that the ASMM can reproduce the response of the T1D model of GIM simulator efficiently. The esti-

Parameters

Estimated values

Variability range

p1 p2 p3 p4 p5 p6 k

4.9 × 10−3 2.13 × 10−2 8.8033 × 10−5 5.027 × 10−2 4.519 × 10−1 1.912 × 10−2 22.752 × 10−2

[3.43 × 10−3 , 6.37 × 10−3 ] [1.491 × 10−2 , 2.769 × 10−2 ] [6.162 × 10−2 , 11.44429 × 10−5 ] [3.519 × 10−2 , 6.535 × 10−2 ] [3.163 × 10−1 , 5.874 × 10−1 ] [1.338 × 10−2 , 2.486 × 10−2 ] [15.926 × 10−2 , 29.577 × 10−2 ]

mated parameters of the proposed model for Scenario 1 is used to investigate the capability of the proposed T1D model to reproduce the response of the T1D patient model of the GIM simulator to multiple exogenous meal disturbances and insulin injections. The coefficient of determination (R2 ) is found to be 0.9772 indicating a good fit of the data set with the predicted output of the model. The results of the closed-loop simulations illustrated in Fig. 8, indicate that the BGL of the T1DP is regulated successfully in the safe range for cases with no severe hypoglycemia (BGL < 50 mg/dl). Despite glucose excursions above the hyperglycemic level of 180 mg/dl due to the effect of uncertain high meal disturbances, the BGL trajectories are brought below this level safely by the APC feedback controller and thus minimising the time spent by the virtual T1DPs in the hyperglycemic state. The glucose concentration in the ISF, x6 (t) successfully tracks the output of the reference system, xˆ 6 (t) despite the presence of uncertainty in model parameters and exogenous meal disturbances. This confirms the achievement of the control objective of making the output of the uncertain system to follow the reference system output. The results confirm the efficiency of the proposed APC controller for the ASMM in dealing with the parametric uncertainties. The control signal representing the insulin infusion rates for the random numerical simulations are illustrated in Fig. 9. The exogenous insulin infusion ensures tight post-prandial glycemic control in the presence of bounded uncertain parameters as well as uncertain meal disturbances during different times of the 24 h period. It can be observed that the control signal (rate of insulin infusion) is very high when there is a high glucose appearance in the blood following an exogenous meal disturbance thus compensating the effect of disturbance in the system. The control signal gradually falls when the BGL starts falling after its peak. The nature of the control signal is smooth and is devoid of any aggressive insulin infusion thus making the control action safe, thereby avoiding hypoglycemia. One can conclude from the discussion on the characteristics of the control signal above that, that the feedback control action has the potential to minimise the risks of postprandial glycemic control without any feed-forward action or meal estimation algorithm. The estimated parameters in (21) are illustrated in Fig. 15 which are responsible for the adaptive nature of the control system. As depicted in Fig. 15 the trajectories of the estimated parameters converge to stable values in finite time despite the presence of multiple meal disturbances in the system. The error signals in (18) remain bounded during the simulation and their corresponding time profile are illustrated in Fig. 12 which shows that these errors converge to zero despite the presence of parametric uncertainties and exogenous disturbances affecting the closed-loop dynamics. The choice of the controller parameters, ci s in (21) are tuned such that the stability conditions in (29)–(37) that are provided in Table 2 are satisfied during the whole simulation period. The stability conditions are satisfied during the whole simulation period as illustrated in Fig. 10. As illustrated in Fig. 11, the CVGA analysis reveal that only 3% of the closed-loop simulations are located in Grid Lower D and the

A. Nath et al. / Journal of Process Control 86 (2020) 94–105

103

Fig. 12. Closed-loop error signals in (18).

Fig. 13. Blood glucose trajectories for virtual T1D patients under varying insulin initial conditions.

rest 97% are confined to Grid B and Grid Upper B indicating the attainment of a significant closed-loop performance. This results validate the efficiency of the proposed APC controller in achieving tight glycemic control under parametric uncertainty. The percentage of time spent by the blood glucose of T1DP in physiologically significant zones are reported in Table 4. Comparing the results of the CVGA analysis of the proposed adaptive controller, as illustrated in Fig. 11, with the Figure 10(a) of [33], one can find that lesser percentage of dots are present in Grid Lower C. This indicates that the risks associated with hypoglycemia is less likely to occur in the proposed methodology. It is important to note that, in both the results in the present work and the [33], there is a variation of ±30% variation in insulin sensitivity. Results for Scenario 3: The closed loop blood glucose regulation of the virtual T1DPs under random initial conditions of insulin, as depicted in Fig. 13, shows that severe hypoglycemia does not occur during the random numerical simulations under

Fig. 14. CVGA for parametric variability of ±30% under varying insulin initial conditions. Table 4 Performance metrics. Time

Mean

Standard deviation

% in range (70–180 mg/dl) % in hyperglycemia (>180 mg/dl) % in severe hypoglycemia (<50 mg/dl)

87.58 12.41 0

4.08 4.087 0

intra-patient variability. The time spent in hyperglycemia is also significantly reduced compared to Scenario 3.2 where initial conditions of insulin is considered to be almost negligible. This fact is validated by performing CVGA analysis as illustrated in Fig. 14. Now, one can easily observe that the dots are distributed in Grid B, Grid Lower B and Grid C as well, which is not the case in Scenario 3.2. All the above-mentioned closed-loop simulation results establish the fact that the proposed APC controller is very effective

104

A. Nath et al. / Journal of Process Control 86 (2020) 94–105

Fig. 15. The estimated parameters in (21).

in achieving a tight post-prandial glycemic control despite intrapatient variability and uncertain meal disturbances. Theorem 1 guarantees the convergence of the closed-loop error signals to zero that ensure accurate tracking of the desired dynamics (reference system) by the uncertain T1DP model. The estimated parameters have the capability of self-adjustment with the changing conditions due to uncertainty in parameters as well as disturbances that is crucial for the achievement of efficient blood glucose regulation under intra-patient variability. 5. Conclusion This work discusses the modelling and adaptive controller design of T1D patients. The proposed subcutaneous patient model is simple and can capture the glucose-insulin dynamics agreeable with the existing model. The model is validated against the input–output data of the UVa Padova T1D model using GIM simulator. Further, the proposed control strategy for blood glucose regulation is achieved for the proposed augmented subcutaneous minimal model in the presence of inter-patient and intra-patient parametric variations as well as exogenous meal disturbance. The closed-loop statistical analysis via CVGA confirmed the efficacy of the adaptive control algorithm under ±30% parametric variability. No severe hypoglycemia and hyperglycemia after meal are recorded during the random numerical simulations. The practical validation of the proposed control scheme can be a future direction of this research. Conflicts of interest The authors declare no conflicts of interest.

References [1] A. Cinar, Artificial pancreas systems: an introduction to the special issue, IEEE Control Syst. 38 (1) (2018) 26–29. [2] International diabetes federation, IDF Diabetes Atlas, 8th ed., International Diabetes Federation, Brussels, Belgium, 2017. [3] P. Herrero, J. Bondia, N. Oliver, P. Georgiou, A coordinated control strategy for insulin and glucagon delivery in type 1 diabetes, Comput. Methods Biomech. Biomed. Eng. 20 (13) (2017) 1474–1482. [4] B.W. Bequette, F. Cameron, B.A. Buckingham, D.M. Maahs, J. Lum, Overnight hypoglycemia and hyperglycemia mitigation for individuals with type 1 diabetes: how risks can be reduced, IEEE Control Syst. 38 (1) (2018) 125–134. [5] A. El Fathi, M.R. Smaoui, V. Gingras, B. Boulet, A. Haidar, The artificial pancreas and meal control: an overview of postprandial glucose regulation in type 1 diabetes, IEEE Control Syst. 38 (1) (2018) 67–85. [6] B.W. Bequette, Challenges and recent progress in the development of a closed-loop artificial pancreas, Ann. Rev. Control 36 (2) (2012) 255–266. [7] A. Nath, S. Biradar, A. Balan, R. Dey, R. Padhi, Physiological models and control for type 1 diabetes mellitus: a brief review, IFAC-PapersOnLine 51 (1) (2018) 289–294. [8] N. Jayanthi, B.V. Babu, N.S. Rao, Survey on clinical prediction models for diabetes prediction, J. Big Data 4 (1) (2017) 26. [9] M. Messori, G.P. Incremona, C. Cobelli, L. Magni, Individualized model predictive control for the artificial pancreas: in silico evaluation of closed-loop glucose control, IEEE Control Syst. 38 (1) (2018) 86–104. [10] C. Cobelli, C. Dalla Man, G. Sparacino, L. Magni, G. De Nicolao, B.P. Kovatchev, Diabetes: models, signals, and control, IEEE Rev. Biomed. Eng. 2 (2009) 54. [11] R.N. Bergman, Minimal model: perspective from 2005, Horm. Res. Paediatr. 64 (Suppl. 3) (2005) 8–15. [12] R. Hovorka, V. Canonico, L.J. Chassin, U. Haueter, M. Massi-Benedetti, M.O. Federici, T.R. Pieber, H.C. Schaller, L. Schaupp, T. Vering, et al., Nonlinear model predictive control of glucose concentration in subjects with type 1 diabetes, Physiol. Meas. 25 (4) (2004) 905. [13] C. Dalla Man, R.A. Rizza, C. Cobelli, Meal simulation model of the glucose-insulin system, IEEE Trans. Biomed. Eng. 54 (10) (2007) 1740–1749. [14] C.D. Man, F. Micheletto, D. Lv, M. Breton, B. Kovatchev, C. Cobelli, The uva/padova type 1 diabetes simulator: new features, J. Diabetes Sci. Technol. 8 (1) (2014) 26–34. [15] C. Dalla Man, D.M. Raimondo, R.A. Rizza, C. Cobelli, Gim, Simulation Software of Meal Glucose-Insulin Model, 2007.

A. Nath et al. / Journal of Process Control 86 (2020) 94–105 [16] J. Bondia, S. Romero-Vivo, B. Ricarte, J.L. Diez, Insulin estimation and prediction: a review of the estimation and prediction of subcutaneous insulin pharmacokinetics in closed-loop glucose control, IEEE Control Syst. 38 (1) (2018) 47–66. [17] S.S. Kanderian, S.A. Weinzimer, G.M. Steil, The identifiable virtual patient model: comparison of simulation and clinical closed-loop study results, J. Diabetes Sci. Technol. 6 (2) (2012) 371–379. [18] S.S. Kanderian, S. Weinzimer, G. Voskanyan, G.M. Steil, Identification of intraday metabolic profiles during closed-loop glucose control in individuals with type 1 diabetes, J. Diabetes Sci. Technol. 3 (5) (2009) 1047–1057. [19] C. Dalla Man, M. Camilleri, C. Cobelli, A system model of oral glucose absorption: validation on gold standard data, IEEE Trans. Biomed. Eng. 53 (12) (2006) 2472–2478. [20] R. Patel, D. Deb, Parametrized control-oriented mathematical model and adaptive backstepping control of a single chamber single population microbial fuel cell, J. Power Sources 396 (2018) 599–605. [21] A. Haidar, The artificial pancreas: how closed-loop control is revolutionizing diabetes, IEEE Control Syst. 36 (5) (2016) 28–47. [22] U. Fischer, W. Schenk, E. Salzsieder, G. Albrecht, P. Abel, E.-J. Freyse, Does physiological blood glucose control require an adaptive control strategy? IEEE Trans. Biomed. Eng. (8) (1987) 575–582. [23] F. Doyle, L. Jovanovic, D. Seborg, I. Glucose control strategies for treating type 1 diabetes mellitus, J. Process Control 17 (7) (2007) 572–576, http://dx.doi. org/10.1016/j.jprocont.2007.01.013. [24] A.K. Patra, P.K. Rout, Adaptive sliding mode gaussian controller for artificial pancreas in tidm patient, J. Process Control 59 (2017) 13–27, http://dx.doi. org/10.1016/j.jprocont.2017.09.005. [25] M. Eren-Oruklu, A. Cinar, C. Colmekci, M.C. Camurdan, Self-tuning controller for regulation of glucose levels in patients with type 1 diabetes, in: American Control Conference, 2008, IEEE, 2008, pp. 819–824. [26] K. Turksoy, L. Quinn, E. Littlejohn, A. Cinar, Multivariable adaptive identification and control for artificial pancreas systems, IEEE Trans. Biomed. Eng. 61 (3) (2014) 883–891. [27] M. Eren-Oruklu, A. Cinar, L. Quinn, D. Smith, Adaptive control strategy for regulation of blood glucose levels in patients with type 1 diabetes, J. Process Control 19 (8) (2009) 1333–1346. [28] S. Coman, C. Boldisor, Simulation of an adaptive closed loop system for blood glucose concentration control, Bull. Transilv. Univ. Bras. Eng. Sci. Ser. I 8 (2) (2015) 107. [29] Z. Tashakorizade, N. Naghavi, S.H. Sani, Glucose regulation in type 1 diabetes mellitus with model reference adaptive control and modified smith predictor, Iran. J. Biomed. Eng. 8 (2014) 159–171. [30] M. Tárník, E. Mikloviˇcová, J. Murgaˇs, I. Ottinger, T. Ludwig, Model reference adaptive control of glucose in type 1 diabetics: a simulation study, IFAC Proc. Vol. 47 (3) (2014) 5055–5060. [31] B. Pagurek, J. Riordon, S. Mahmoud, Adaptive control of the human glucose-regulatory system, Med. Biol. Eng. 10 (6) (1972) 752–761. [32] G.P. Incremona, M. Messori, C. Toffanin, C. Cobelli, L. Magni, Model predictive control with integral action for artificial pancreas, Control Eng. Pract. 77 (2018) 86–94, http://dx.doi.org/10.1016/j.conengprac.2018.05.006.

105

[33] D. Boiroux, A.K. Duun-Henriksen, S. Schmidt, K. Nørgaard, N.K. Poulsen, H. Madsen, J.B. Jørgensen, Adaptive control in an artificial pancreas for people with type 1 diabetes, Control Eng. Pract. 58 (2017) 332–342, http://dx.doi.org/ 10.1016/j.conengprac.2016.01.003. [34] A. Nath, Blood glucose regulation in type 1 diabetic patients: an adaptive parametric compensation control-based approach, IET Syst. Biol. 12 (2018), 219–225(6). [35] K. Turksoy, I. Hajizadeh, S. Samadi, J. Feng, M. Sevil, M. Park, L. Quinn, E. Littlejohn, A. Cinar, Real-time insulin bolusing for unannounced meals with artificial pancreas, Control Eng. Pract. 59 (2017) 159–164, http://dx.doi.org/ 10.1016/j.conengprac.2016.08.001. ˜ Automatic regulatory [36] P. Colmegna, F. Garelli, H.D. Battista, R. Sánchez-Pena, control in type 1 diabetes without carbohydrate counting, Control Eng. Pract. 74 (2018) 22–32, http://dx.doi.org/10.1016/j.conengprac.2018.02.003. [37] G. Marchetti, M. Barolo, L. Jovanovic, H. Zisser, D.E. Seborg, A feedforward-feedback glucose control strategy for type 1 diabetes mellitus, J. Process Control 18 (2) (2008) 149–162, http://dx.doi.org/10.1016/j.jprocont. 2007.07.008. ˜ Intra-patient dynamic [38] M. Moscoso-Vásquez, P. Colmegna, R.S. Sánchez-Pena, variations in type 1 diabetes: a review, in: 2016 IEEE Conference on Control Applications (CCA), IEEE, 2016, pp. 416–421. [39] Z. Cao, Y. Yang, J. Lu, F. Gao, Constrained two dimensional recursive least squares model identification for batch processes, J. Process Control 24 (6) (2014) 871–879, http://dx.doi.org/10.1016/j.jprocont.2014.04.002. [40] D. Robertson, J. Lee, A least squares formulation for state estimation, J. Process Control 5 (4) (1995) 291–299, http://dx.doi.org/10.1016/09591524(95)00021-H, iFAC Symposium: Advanced Control of Chemical Processes. [41] T. Coleman, M.A. Branch, A. Grace, Optimization Toolbox User’s Guide, Matlab The Mathworks Inc., 2012. [42] T.F. Coleman, Y. Li, An interior trust region approach for nonlinear minimization subject to bounds, SIAM J. Optim. 6 (2) (1996) 418–445. [43] D. Deb, G. Tao, J.O. Burkholder, An adaptive inverse compensation scheme for signal-dependent actuator nonlinearities, in: 2007 46th IEEE Conference on Decision and Control, IEEE, 2007, http://dx.doi.org/10.1109/cdc.2007. 4434743. [44] D. Deb, G. Tao, J.O. Burkholder, D.R. Smith, Adaptive synthetic jet actuator compensation for a nonlinear aircraft model at low angles of attack, IEEE Trans. Control Syst. Technol. 16 (5) (2008) 983–995. [45] D. Kapoor, D. Deb, A. Sahai, H. Bangar, Adaptive failure compensation for coaxial rotor helicopter under propeller failure, in: American Control Conference (ACC), 2012, IEEE, 2012, pp. 2539–2544. [46] N. Kumpati, S.A.M. Annaswamy, Persistent excitation in adaptive systems, Int. J. Control 45 (1) (1987) 127–160. [47] V. Adetola, M. Guay, D. Lehrer, Adaptive estimation for a class of nonlinearly parameterized dynamical systems, IEEE Trans. Autom. Control 59 (10) (2014) 2818–2824, http://dx.doi.org/10.1109/TAC.2014.2318080. [48] V. Adetola, M. Guay, Excitation signal design for parameter convergence in adaptive control of linearizable systems, Proceedings of the 45th IEEE Conference on Decision and Control (2006) 447–452.