00740877

148 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 46, NO. 2, FEBRUARY 1999

A Model-Based Algorithm for BloodGlucose Control in Type I Diabetic Patients

Robert S. Parker, Francis J. Doyle, III,* and Nicholas A. Peppas

Abstract—A model-based predictive control algorithm is devel-oped to maintain normoglycemia in the Type I diabetic patientusing a closed-loop insulin infusion pump. Utilizing compartmen-tal modeling techniques, a fundamental model of the diabeticpatient is constructed. The resulting nineteenth-order nonlin-ear pharmacokinetic–pharmacodynamic representation is usedin controller synthesis. Linear identification of an input–outputmodel from noisy patient data is performed by filtering theimpulse-response coefficients via projection onto the Laguerrebasis. A linear model predictive controller is developed using theidentified step response model. Controller performance for un-measured disturbance rejection (50 g oral glucose tolerance test)is examined. Glucose setpoint tracking performance is improvedby designing a second controller which substitutes a more detailedinternal model including state-estimation and a Kalman filter forthe input–output representation. The state-estimating controllermaintains glucose within 15 mg/dl of the setpoint in the presenceof measurement noise. Under noise-free conditions, the model-based predictive controller using state estimation outperformsan internal model controller from literature (49.4% reduction inundershoot and 45.7% reduction in settling time). These resultsdemonstrate the potential use of predictive algorithms for bloodglucose control in an insulin infusion pump.

Index Terms—Compartmental modeling, diabetes, glucose, in-fusion pumps, insulin, Kalman filter, model identification, modelpredictive control, state estimation.

I. INTRODUCTION

DIABETES mellitus is characterized by the inability ofthe pancreas to control blood glucose concentration.

Inadequate secretion of insulin by the diabetic pancreas resultsin poor maintenance of normoglycemia (defined as bloodglucose 70–100 mg/dl) with elevated blood glucose con-centrations, sometimes upward of 300 mg/dl. It is thoughtthat most of the long-term complications associated withdiabetes, such as nephropathy and retinopathy, result fromsustained hyperglycemia (arterial blood glucose120 mg/dl).The current treatment methods for insulin dependent dia-betes, subcutaneous insulin injection or continuous infusionof insulin, can result in significant, and sometimes frequent,glucose concentration variation due to their inherently open-loop nature. Consequently, it would be beneficial to develop

Manuscript received May 21, 1997; revised June 12, 1998. This work wassupported by the Showalter Trust and by the National Science Foundation(NSF) under Grant CTS 9257059.Asterisk indicates corresponding author.

R. S. Parker is with the Department of Chemical Engineering, Universityof Delaware, Newark, DE 19716 USA.

*F. J. Doyle, III, is with the Department of Chemical Engineering, Univer-sity of Delaware, Newark, DE 19716 USA (e-mail: [email protected]).

N. A. Peppas is with the School of Chemical Engineering, Purdue Univer-sity, West Lafayette, IN 47907-1283 USA.

Publisher Item Identifier S 0018-9294(99)00823-X.

a closed-loop device capable of maintaining normoglycemiaover extended periods of time.

A device of this type would contain three major compo-nents: i) a mechanical pump; ii) anin vivo glucose sensor;and iii) a mathematical algorithm to regulate the pump givena sensor measurement. Extracorporeal and implantable insulinpumps have been in service for over 15 yr [1], [2]. Initiallythese devices had a single delivery rate, but technologicaladvances have allowed a wide variety of programmable andvariable-rate infusion pumps to be available currently [3]. Re-search shows [4], [5] continuous infusion and programmablepumps are effective for insulin therapy. By utilizing a variable-rate pump in a closed-loop framework, further improvementsin glucose control and normalization of the glucose distributionin the body are possible.

Current blood glucose monitoring is accomplished throughinvasive methods, such as a finger prick, but use of a nonin-vasive monitor would increase patient comfort and therefore,compliance to the insulin therapy. An implantable glucoseconcentration sensor would measure diabetic patient bloodglucose levels online and eliminate the patient from thefeedback loop. Significant work has been performed on thedevelopment of an implantable glucose sensor [6]–[8], andthe duration ofin vivo sensor reliability continues to increase.

A significant effort has been put forth toward the devel-opment of a closed-loop algorithm for blood glucose control[9]–[12]. These approaches have utilized almost exclusivelyfeedback control to maintain normoglycemia, even for thepurpose of disturbance rejection. This paper takes a differentapproach, specifically the use of model-based predictive con-trol (MPC). The unconstrained controller guarantees optimaldrug delivery through solution of an optimization problem ateach time step. A motivating factor for utilizing this strategyis the success of MPC when applied to other biomedicalcontrol problems, including blood pressure control [13], [14]and anesthesia delivery [15], [16]. This controller architectureis particularly well suited to the multivariable nature of thesesystems, as well as the inherent constraints involved in therespective control problems. To date, successful controller im-plementation has been bedside in nature, due to the significantcomputing power required for the calculations.

Computational power and speed aside, one benefit of usingpredictive control in place of a classical control algorithm isthe estimation of future glucose behavior based on the past in-sulin inputs, with measurement of the patient’s actual glucoselevels used as a feedback signal to correct the glucose concen-tration predictions. As a result, the MPC controller takes actionfor a predicted hypo- or hyperglycemic excursion well before

0018–9294/99$10.00 1999 IEEE

PARKER et al.: BLOOD GLUCOSE CONTROL IN TYPE I DIABETIC PATIENTS 149

it occurs, while a feedback-only controller responds after theeffect of the disturbance is manifested. Similar to the otherbiomedical applications, the human glucose-insulin controlproblem has inherent input rate and magnitude constraintsas well as an output magnitude constraint which MPC caneasily handle. The aforementioned feedback controllers requirespecial formulations to compensate for these same types ofconstraints, and performance degradation can result. Hence,the MPC controller algorithm exhibits a range of appropriatecharacteristics for the blood glucose control problem.

II. STATE-SPACE PATIENT MODEL

A mathematical representation of the system is necessary toimplement a model-based control scheme. Many approachesto modeling the human glucose-insulin system have beentaken [9], [17]–[19], starting from the initial glucose mod-eling work of Bolie [20]. These methods can be dividedinto semiempirical and fundamental methods. An empiricalapproach attempts to capture the behavior of a system frominput–output data alone. Usually, a model structure is chosena priori, containing a certain number of parameters to beidentified. The linearized model studied by Ackermanetal. [21] has this form. Imposing the linear structure, twoparameters for glucose effects (one for self-removal and onefor glucose effect on insulin) and two for insulin effects (onefor self-induced removal and a second for glucose removalthrough insulin induced pathways) are identified to match themodel to data. Normally, individualized empirical parametersare determined through a series of tests performed on eachpatient. This is a time and resource consuming process. Analternative model structure, which adds additional physiologicdetail, is the so called semiempirical or hybrid model. Sucha model incorporates physiologically based structure, such asthat derived in Cobelliet al. [9], where the insulin modelhas equations representing dynamic behavior and kinetics. Acompartmental approach like that of Bergmanet al. [17] canbe taken, such that a chosen number of compartments (inthis work, three) utilizing an arbitrary number of identifiableparameters (in this work, six) fully describes the diabeticpatient. The structure results from subdividing the insulincompartment into a plasma space and a compartment remoteto the plasma which affects glucose uptake. Finally, purelyfundamental models can be constructed. Nomuraet al. [18]examined the -cells of the pancreasin vitro and constructeda fundamental model for their behavior from the insulinrelease data. Fundamental models can also be generated bymathematically representing known system behavior, suchas underlying kinetics or material transport, and identifyingparameters though an extensive literature search of availabledata. This results in a model representing the average behaviorof the studied population.

In this paper a pharmacokinetic–pharmacodynamic ap-proach to fundamental modeling is taken. The model ofthe human glucose-insulin system used in this study resultsfrom initial work by Guytonet al. [22] which was updatedby Sorensen [23]. Significant modification of this model toinclude disturbances such as meals and exercise, as well as

Fig. 1. Compartmental diagram of the glucose or insulin system in a diabeticpatient.

parameters for uncertainty analysis, is ongoing work by thecurrent authors.

Utilizing compartmental modeling techniques, the diabeticpatient model is represented schematically in Fig. 1. Individualcompartment models are obtained by performing mass bal-ances around tissues important to glucose or insulin dynamics.Subcompartments, such as those in the brain and periphery,are included where significant transport resistance (e.g., timedelay) exists. In this model, the periphery represents the com-bined effects of muscle and adipose tissue while the stomachand intestine effects are lumped into the gut compartment.The controlled output for this system is the arterial glucoseconcentration, which is regulated by the manipulated variable,insulin infusion rate. A disturbance variable, glucose uptakefrom the gut compartment, is added to the model to simulatethe diabetic patient ingesting a meal. The mathematical repre-sentation of the meal submodel is described in Lehmann andDeutsch [24].

III. I NPUT–OUTPUT CONTROLLER MODEL

The linear model predictive controller utilizes an internalmodel to estimate the future output values based on a seriesof past inputs. The model form chosen for this work is thelinear step-response model. Assuming the system begins atrest, the step-response coefficients, , represent the systemresponse to a unit increase in the input variable

(1)


It is assumed that the system reaches a new steady-state aftersample times, where is known as the model memory. In

the case of the diabetic patient model, the system approachessteady-state after 180 min (1.5% of total change remaining),and the dynamic character of the response is completed.The choice of a sample time, which determines memorylength, is a combination of three factors: model dynamicsand accuracy, complexity of the identification problem, andequipment constraints. For an accurate model, the samplingrate must be rapid enough to adequately capture the fastestdynamics of the system. A heuristic bound is that the samplingrate should be no slower than 20% of the fastest time constant.The open-loop constant time of the diabetic patient model isapproximately 55 min, meaning a sample must be taken atleast once every 11 min. Fewer parameters make models easierto identify. This suggests larger sample times, yielding fewerstep response coefficients and a less complex identificationproblem. Finally, samples can be taken no faster than theequipment can determine glucose concentration; this createsa sampling rate lower bound. Researchers in the sensor fieldreport the ability to sample glucose every 4 min utilizingan electrochemical biosensor [6]. To simplify the math, anddecrease the number of parameters, the chosen sample time is5 min, and therefore the model memory,, is 180/5 36sample times.

Assuming superposition, a linear approximation of the out-put can be calculated given the past input profile

(2)

Here, the first term accounts for the response of the modelto the input change over the memory of the model, and thelatter term represents the steady-state of the process prior tothe input change. The predicted output value and input changesare given by and , respectively. Step-response coefficientsare calculated from an identified impulse-response (IR) modelof the system by

(3)

(4)

where are the identified impulse response coefficients andare the past inputs.

The structure of the impulse-response model in (3) issimilar to that of the first-order Wiener functional, . Toidentify Wiener functionals, Gaussian white noise (GWN)input sequences are typically used. However, identification ofa physical system model using GWN is unrealistic due to thetremendous strain placed on the input regulator (e.g., pump orvalve). Lee and Schetzen [25] derive a cross-correlation for-mula for computing Wiener kernels in the case of a Gaussianwhite-noise input, which can be modified to account for non-GWN inputs [26]. This method is mean squared error (MSE)optimal, meaning that it minimizes the MSE between the actualoutput and the predicted output. The non-GWN sequence used

for diabetic patient model identification is given by (5) andcan be treated as a special constant-switching-pace symmetricrandom signal (CSRS)

with probability

with probability

with probability (5)

If the input is in deviation form, is the maximum symmetricinput deviation from its steady-state value, and is thenumber of data points in the record. In the diabetic patientmodel, the nominal insulin delivery rate is 22.33 mU/minand the minimum delivery of insulin is 0 mU/min whichyields a value of 22.33 mU/min for . Accurate parameteridentification requires the pulses to be separated by at leastpoints, and that the second pulse is at leastpoints from theend of the data record. Based on the earlier choice of ,the minimum number of data points required is . Thecalculation using the modified Wiener functional method isoutlined next.

Using the special CSRS, the th-order discrete modifiedfunctional form is given by

(6)

Here, is the modified Wiener functional and is theorder of the functionals used to estimate. Truncating after thelinear terms , can be described by the followingequations:

(7)

(8)

Therefore, in trying to identify the functionals from in-put–output data, the following structure is utilized:

(9)

which can be restructured to yield

(10)

Equation (10) results from the knowledge that the zeroth-ordermodified functional, , is the mean of .Using this description of , the output data in deviation formis given by . Note that (10) has a formidentical to that of (3). The impulse-response coefficients,represented by the functional , are identified using theaforementioned cross-correlation method, accounting for thestatistical properties of the input signal through division bythe second-order moment, [equivalent to the variance for


the mean-zero process ]. Therefore, the impulse-responsecoefficients are given by

(11)

Using this cross-correlation method, the developed coefficientsminimize the prediction error variance, ,between the predicted and the measured output value.

Impulse response coefficients developed by minimizingMSE are accurate if the output signal is noise-free. If theoutput contains noise, as a signal from any biological systemwould, performance of the identification algorithm degrades,and the coefficients show significant noise effects. Filteringprovides a smoother set of coefficients which should im-prove the quality of any prediction by reducing noise-inducedvariations. The impulse-response coefficients are filtered byprojection onto the Laguerre basis, utilizing smooth Laguerrefunctions to approximate the noisy coefficients. Expansionof the Laguerre functions returns smoothed impulse-responsecoefficients, which are an optimal estimate of the noise-corrupted coefficients in the MSE sense [27].

In discrete time, theth-order Laguerre function is [28]

(12)

such that is the unit step function, for. The Laguerre pole, denoted, ,

determines the rate of exponential asymptotic decay of theLaguerre functions [29]. By comparing the actual modelimpulse response to that generated by the Laguerre functions,

is chosen to minimize the residual error, and is set to 0.84.The set is orthonormal in the interval , and iscomplete in . Hence, the least-squares generated impulse-response coefficients , satisfying

and can be represented in termsof Laguerre functions as [27]

(13)

where is the number of Laguerre functions chosen todescribe the least-squares impulse-response coefficients, andthe Laguerre parameters,, are unknown. These Laguerrecoefficients are the least-squares solution to

(14)

Here, is the impulse-response co-efficient vector, and is the Laguerrefunction matrix. The smoothed impulse-response coefficientsare generated by substituting the calculated’s into (13).Fig. 2 shows the impulse-response coefficients identified usingleast-squares and Laguerre projection. Although the Laguerregenerated coefficients have a decreased gain, they are notcorrupted by the noise present on the measurement signal

Fig. 2. Impulse response models relating glucose concentration (output) toinsulin delivery rate (input); identification by linear least-squares (solid) andLaguerre projection (dashed) compared with the noise-free impulse responsecoefficients of the nonlinear model (dash-dot).

(variance 1.45 mg/dl) which affects the least-squares derivedcoefficients.

A random binary sequence (RBS) is driven through theidentified model and the full nonlinear model for validationpurposes, and the results are shown in Fig. 3. Clearly, theidentified model does not include all of the gain information ofthe diabetic patient model, but it does succeed in capturing thedynamic behavior. This second component is more important,as the controller will be updated at 5 min intervals withnew information, making the dynamic component significantlymore important in the control computation than the steady-state behavior [30]. This same RBS produces much pooreroutput when run using the least-squares identified parameters(output not shown). By comparison, the sum of residuals fromthe Laguerre coefficient validation is 12.1 mg/dl, while thatfor the least-squares validation is 35.4 mg/dl.

IV. L INEAR MODEL PREDICTIVE CONTROL

A linear MPC algorithm is now constructed to control bloodglucose concentration based on arterial glucose sampling andintravenous insulin delivery. The optimization problem solvedby MPC is given by

(15)The goal is to minimize the error in setpoint tracking and themanipulated input movement, respectively, over the sequenceof future input moves, . Here, is thevector of future reference values while is thevector of predicted future glucose concentrations. Vectors usethe standard statistical notation of predicting a value at time

given information up to time . Weighting matricesfor the setpoint tracking penalty and insulin move penaltyare given by and , respectively. These two matricescan be used as tuning parameters for the controller, tradingoff output performance, and manipulated variable movement.


Fig. 3. Validation of the input–output model using a random binary se-quence. Top: comparison of the actual nonlinear diabetic patient model (�)with the predicted output (+). Middle: residuals between the Volterra–Laguerremodel and the actual nonlinear model for the forcing sequence. Bottom: ran-dom binary sequence used to validate the Laguerre derived impulse-responsemodel.

Additionally, the input move penalty serves to regulate themagnitude of noise-induced manipulated variable movement.When there is no noise in the diabetic patient simulationis set equal to zero, while is used when band-limitedwhite noise (variance 1.45 mg/dl) corrupts the output signal.

It is straightforward to show that an analytical solution to theunconstrainedproblem can be constructed [31]. One merelycalculates , the difference between the vector ofpredicted future glucose concentrations from past inputs andthe vector of future reference values. The analytical solutioninvolves a multiplication of by , whereis given by

(16)Here, the leading vector results from implementing only thefirst calculated move for . This structure demonstratesthe ease of implementing linear MPC. Since the gain matrixcan be precalculated, the on-line computation reduces to asimple multiplication. Therefore, the calculation could beperformed without the need for a high-powered computer, andinstead on a single digital chip, allowing for straightforwardimplementation in a microprocessor-based pump.

However, the analytical solution can not be implementedwithout accounting for the constraints present in the system.An input rate constraint, , guarantees the pump doesnot undergo changes in insulin delivery rate that are greaterthan the mechanism can handle. As such, a rate constraint thatis conservative with respect to pump dynamics shown in theliterature [3], and can span the full range of insulin deliveryrates in a maximum of four moves is chosen. The maximumchange in insulin delivery rate is given by

mU/min per sample time (17)

Physiological conditions imply a magnitude constraint whichmust be applied to the insulin delivery rate. The plasma insulinconcentration in the healthy patient is rarely above 100 mU/l.Since the goal is to return the diabetic patient to as normal astate as possible, the maximum insulin delivery rate from thepump should not result in a plasma insulin concentration inexcess of 100 mU/l. Additionally, it is impossible to removeinsulin once it has been delivered to the patient. Therefore, theinput magnitude is constrained as follows:

mU/min mU/min (18)

This constraint, as well as the input rate constraint in (17),are implemented by clipping the rate and magnitude of thecalculated pump action.

Low blood glucose concentrations in the diabetic patientare dangerous and starve the cells of fuel. Therefore, anappropriate output constraint is given by

mg/dl (19)

However, the inclusion of hard output constraints in theproblem statement can lead to infeasible programming prob-lems, and will require more computational power than canbe delivered on a digital chip, given current technology. Analternative is to include the output constraint in a “soft”form, by adding another term to the objective function [32].This would modify the analytic solution to the unconstrainedproblem, and solution of a nonlinear programming problemmay be required depending on the formulation of the addedterm. To avoid the potential problems with including anoutput constraint, it is treated through careful selection ofthe controller tuning weights to yield the soft constraintformulation

subject to: mU/min mU/min

mU/min per sample time

mg/dl checked (20)

Tuning this controller is anad hocprocedure, using the twoavailable parameters: , the move horizon, and, the predic-tion horizon. These parameters are determined by performinga two-dimensional search over and , while subjecting thediabetic patient to an unmeasured meal disturbance with nomeasurement noise. The criterion used to evaluate performanceis sum of squared error (SSE) over the time-course of thesimulation, providing the output tracks the reference. Thecontroller settings minimizing SSE over the simulation lengthwhile eliminating output oscillations are , ,

, and .Using the SSE optimal tuning parameters as a starting

point, the controller is detuned to accommodate a measurementsignal with noise of variance 1.45 mg/dl. This detuning isnecessary due to violation of the glucose concentration lowerbound in response to a 50 g OGTT. Increasing the predictionhorizon to ten successfully detunes the controller yieldingsatisfactory performance. In order to reduce the chatter in themanipulated input signal, the weighting matrices,and ,


are adjusted until the constraints are satisfied and chatter inthe manipulated input is reduced.

V. MPC WITH STATE ESTIMATION

Model predictive control with state estimation, MPC/SE, hasseveral advantages over standard MPC. The increased amountof information provided to the controller yields tighter control,and additional tuning parameters (a Kalman filter and thereference filter) are included to adjust closed-loop performance[33].

The internal model structure in MPC/SE is changed fromthe input–output form of (2) to the linear state-space form

(21)

(22)

This model is constructed from the continuous nonlineardiabetic patient model discussed in Section II in two steps,the latter using commands from the MPC toolbox. First, thenonlinear model is linearized analytically to yield a linearcontinuous-time model of the diabetic patient. Then, the con-tinuous linear model is converted to a discrete-time minimum-phase representation for use in simulation, yielding, , and

.Using the internal model of (21) and (22), the controller

can estimate the state of the plant and the output using thefollowing equations:

(23)

(24)

The Kalman filter, , has several practical applications inthe MPC/SE algorithm developed in this paper. By updat-ing the internal controller model with current measurementinformation using the Kalman filter, mismatch between theactual patient and the internal controller model is significantlyreduced. Therefore, the predictions using the updated modelare more accurate than those of the static input–output model.The Kalman filter can also be tuned to infer an unmeasureddisturbance value so that the controller takes appropriateaction to counteract the detected disturbance. Noise filteringis achieved by adjusting the Kalman filter gain based on thereliability of the measurement signal, which reduces noiseinduced manipulated input movement. The tradeoff betweendisturbance inference and noise filtering will be addressedbelow.

Kalman filter design is accomplished using the discretelinear system model and the known noise characteristics ofdisturbances [33]. This formulation of the MPC/SE algorithmutilizes the steady-state Kalman filter, , which is calculatediteratively off-line, to minimize controller algorithm computa-tion requirements as

(25)

(26)

Here, and are from (21) and (22), above. Tuning theKalman filter is accomplished using the matrices, ,and [33]. The initial state covariance matrix,

, is the expected value of thesquare of the initial deviation between the actual state and thebest linear estimate of that state. The matrix is cho-sen to represent an initial uncertainty of 1 mg/dl in the glucoseconcentrations or 1 mU/l in the insulin concentrations, andsatisfies the requirement . The measurement noisecovariance matrix, , is dependent on the statistics of themeasurement noise. When incorporated, band-limited whitenoise with power 0.1, gain 0.85 mg/dl, and sample timeequal to 0.05 min, corrupts the glucose measurement signal.These noise characteristics define a nearly white sequence withmean 0 mg/dl, and maximum deviation of 5 mg/dl yielding

. The third parameter of the steady-state Kalmanfilter is , the process noise covariance matrix.

From the mathematically rigorous derivation of the discretetime Kalman filter, the matrix is constructed using theavailable knowledge of the unmeasured disturbances. Thesedisturbances are assumed to be random, with zero mean andknown covariance, and quasi-stationary. However, inaccuracyintroduced by the model (through plant-model mismatch)and unknown disturbance characteristics force a change informulating . It is utilized as a tunable parameter, whichis adjusted until output performance is satisfactory. The in-dividual matrix entries are varied on an element by elementbasis, under the assumption that the process noise is diagonal,as off-diagonal components lack a physical basis. Excellentperformance in the noise-free case results from combining twoeffects. First, the elements of are weighted according torelative importance in detecting a glucose meal disturbance.The glucose states (1–8) will vary more significantly thanthe insulin, glucagon, or auxiliary equation states, hencelarger weights are used for those elements. Knowledge of themismatch between the linear internal model and the actualnonlinear model of the diabetic patient is also accountedfor. The linear controller estimates of the first 17 states arevery close to the actual output of the nonlinear model, butthe glucagon states showed significant deviation between theestimates and the actual values. Therefore, the weighting inthe matrix is increased for these two elements, resulting in

diag (27)

where is an -length vector of ones.The approach used here for controller development is sim-

ilar to that of Ricker [34] for MPC with state estimation.Constraints identical to those of the linear MPC algorithm,(17)–(19), are enforced by clipping the manipulated inputand checking the output constrainta posteriori. The objectivefunction is modified due to the existence of reference modeldynamics, which adjust the reference value based on thedesired closed-loop performance for the system. The math-ematical derivation can be found in [34]. It is worth notingthat the unconstrained version of the state-estimating MPCalgorithm also admits a convenient analytical solution whichcan be easily implemented on a digital chip.


The problem now reduces to tuning the controller. A three-dimensional (3-D) search is performed over, , and thereference filter , such that sum-squared error is minimized,and the constraints are satisfied. By utilizing the Kalman filterin the controller, a more aggressive formulation withand is possible, taking . To account formeasurement noise, the controller is detuned by increasing theprediction horizon to as well as retuning the matrixsuch that all weights are less than 1.0, yielding

diag (28)

VI. NONLINEAR QUADRATIC DYNAMIC

MATRIX CONTROL WITH STATE ESTIMATION

An alternate control methodology which takes greater ad-vantage of the nonlinear model of the diabetic patient isnonlinear quadratic dynamic matrix control with state esti-mation (NLQDMC/SE). This is a logical extension of linearMPC with state estimation, with compensation for the knownnonlinearity of the controlled process, as explained in Gattuand Zafiriou [35]. These changes include the use of thenonlinear model by the controller and updating of the linearmodel based on the current operating point.

When calculating the effects of the past inputs on the outputprediction in NLQDMC/SE, the nonlinear model is used inplace of the linear discrete model. This yields a more accurateprediction of future behavior since the nonlinearity is included.In solving the optimization problem, the effect of future insulindelivery rates on future glucose behavior is determined usingthe linear model, as in linear MPC/SE, thereby requiring thesolution of only one quadratic programming problem online[36]. The sum of the nonlinear and linear effects is then usedto calculate the vector of future input moves necessary to drivethe system to track the reference value.

NLQDMC/SE takes advantage of the nonlinear model oncemore during the controller computation sequence. The estima-tion of the future plant states given the current information,including the newly calculated insulin delivery rate, is accom-plished through integration of the nonlinear model updated bythe correction term, .

A 3-D search over the move horizon , the predictionhorizon , and the reference filter is performed to tunethe controller. The criterion is to minimize the sum-squarederror over the time course of the simulation. Under noise-freeconditions, the resulting parameters are , , and

. Similar to MPC/SE, the weighting matrix valuesare and . Simulation results are compared tothose of the other controllers in Section VII.

VII. RESULTS AND DISCUSSION

A blood glucose controller will be expected to operate inthe presence of measurement noise, and the ability of theaforementioned linear algorithms to reject a simulated mealdisturbance is shown in Fig. 4. The minimum arterial glucoseconcentration during postprandial hypoglycemia under linearMPC is 65.6 mg/dl, which is only slightly above the outputconstraint of 60 mg/dl. The maximum glucose concentration

Fig. 4. Response of the initially controlled diabetic patient to a 50 g OGTTadministered at time= 50 min, with measurement noise of variance= 1.45.Solid: linear MPC controller with parametersm = 2, p = 10, �y = 3, and�u = 1. Dashed: linear MPC/SE controller with parametersm = 2, p = 8,�y = 1, and�u = 1.

resulting from the OGTT is 95.5 mg/dl which lies wellwithin the normoglycemic range. Superior performance can beachieved using MPC/SE, where the postprandial hypoglycemiareaches 68.5 mg/dl. This represents a 19% decrease in under-shoot from the 81-mg/dl reference when compared to linearMPC.

Simulation study results are promising for both the lin-ear MPC and MPC/SE algorithms. To provide a basis forcomparison, the internal model controller (IMC) controlleremployed by Sorensen [23] is considered. The nonlinearmodel in Section II is approximated by a first-order plustime-delay (FOTD) transfer function, and the IMC controlleris designed from the linear approximation. For simulationusing the MATLAB/SIMULINK environment, this controlleris implemented in a discrete framework for comparison againstthe inherently discrete MPC formulations. Fig. 5 shows thedisturbance rejection simulation results.

The IMC algorithm shows superior performance to thelinear MPC algorithm for undershoot (11% decrease) but is in-ferior in terms of settling time (9% increase). These controllersare comparable for the noise-free case, but are both inferiorto the more advanced algorithms. The increased informationavailable to the MPC/SE algorithm, in combination with theKalman filter, yields greater than 40% performance improve-ment in both undershoot and settling time. In a comparisonof the state-estimating algorithms, the nonlinear controller(NLQDMC/SE) offers 18% less undershoot with a nearlynegligible (6%) increase in settling time when compared tothe linear state-estimating controller. Tabulated results can befound in Table I. Clearly, the utilization of state-estimationcombined with a more detailed controller model providesignificant performance improvement.

There are important aspects of the problem other thancontroller algorithm selection that need to be addressed. One istime delay in the sensor measurement. The assumption to this


Fig. 5. Disturbance rejection comparison of four controllers with no mea-surement noise for a 50 g OGTT: 1) linear MPC (solid) (m = 2, p = 8,�y = 1, and�u = 0), 2) MPC/SE (dashed) (m = 2, p = 7, �r = 0:65,�y = 1, and�u = 0), 3) NLQDMC/SE (dotted) (m = 2, p = 5,�r = 0:57,�y = 1, and�u = 0), and 4) Discrete IMC (dash-dot).

TABLE IDISTURBANCE REJECTION RESULTS

Controller Undershoot(mg/dl)

Settling Time(min)

IMC w/filter 8.7 376linear MPC 9.7 343

MPC/SE 4.4 204NLQDMC/SE 3.6 216

point has been that an accurate measurement of the arterial glu-cose concentration is instantaneously available at each sampletime. Existence of delay in a system has a destabilizing effect,and large enough delays result in unstable closed-loop systems.In addition, measurement delay causes a loss in performance.To determine the degree of performance loss, the linear MPCand MPC/SE controllers are subjected to measurement delaysof 5, 10, and 15 min. Two cases are analyzed: i) unknownmeasurement delay, where the delay is not accounted for inthe internal model and ii) known measurement delay. For thelatter case, a new internal model is identified for each increasein time delay. A typical result for unknown delay is shownin Fig. 6. The MPC controller without state estimation utilizesa model identified from input–output data, while the state-estimating controller is adjusted by placing rows of zeros inthe step response matrix, such that a row of zeros is added foreach sample time of delay. For either controller and knownmeasurement delay, the prediction horizonis increased bythe number of sample times of delay in the measurement (e.g.,time delay sample timerounded to the next positive integer).

All results are tabulated in Table II.Clearly, small dead times (5 min) do not significantly

affect performance, as only slight degradation occurs (a max-imum of 21% increase in undershoot). Both controllers main-tain glucose concentration well above the constraint of 60mg/dl in this case. As described earlier, the large dead time

Fig. 6. Disturbance rejection using the linear MPC controller (m = 2,p = 10, �y = 3, and �u = 1). Comparison of unknown measurementdead times. No dead time (solid, for reference), 5 min dead time (dashed),10 min dead time (dash-dot), and 15 min dead time (dotted). Feedback signalcontains noise with variance= 1.45. Top: Noise-free glucose concentrationdata. Bottom: Insulin delivery rate.

TABLE IITIME-DELAY DISTURBANCE REJECTION RESULTS. AN M A FTER THE NUMBER

DENOTES THEINTERNAL MODEL INCLUDES DELAY (KNOWN DELAY). VARIANCE

= 1.45 mg/dl MEASUREMENT NOISE PRESENT ON OUTPUT SIGNAL

TimeDelay(min)

Undershoot(ml/dl)

SettlingTime(min)

ConstraintEquations(17)-(19)

0 15.4 315 Reference5 17.7 316 Satisfied5 M 18.8 358 Satisfied

10 21.1 327 Satisfied10 M 21.9 378 Violated15 39.0 551+ Violated

MPC

15 M 25.1 364 Violated

0 12.5 262 Reference5 15.3 494 Satisfied5 M 15.7 551+ Satisfied

10 24.8 551+ Violated10 M 17.6 551+ Satisfied15 29.7 551+ Violated

MPC/SE

15 M 25.6 536 Violated

of 15 min has a destabilizing effect, inducing sustained os-cillation in most controllers (except for a known delay in thelinear MPC controller, plot not shown). In addition, each ofthe simulations with 15-min measurement delay resulted inviolation of the glucose lower bound. Hence, it is necessaryto keep the measurement delay less than or equal to 10 min(as described in Table II).

VIII. C ONCLUSIONS

Model-based predictive control of insulin infusion pumpsrequires development of an accurate model of the humanglucose-insulin system and design of a constrained controller.


A nonlinear model of the diabetic patient is developed us-ing compartmental modeling theory and literature data. Anovel identification algorithm is applied to develop a linearinput–output representation of the nonlinear system in thepresence of measurement noise. Model-based predictive al-gorithms for insulin infusion pump control are developed. Theconstraint handling and prediction capabilities of MPC providean excellent framework for the glucose control problem. LinearMPC is sufficient for controlling blood glucose, but results inglucose concentrations near the output lower bound. LinearMPC with state estimation, utilizing a Kalman filter and amore accurate internal model, yields improved control whencompared to the linear MPC scheme and a discretized IMCcontroller from literature. Additionally, the digital nature ofthese control algorithms allows potential implementation ontochip technology when sensors can guarantee longin vivolifetimes.

REFERENCES

[1] J.-L. Selam and M. A. Charles, “Devices for insulin administration,”Diabetes Care,vol. 13, pp. 955–979, 1990.

[2] F. P. Kennedy, “Recent developments in insulin delivery techniques,current status, and future potential,”Drugs,vol. 42, pp. 213–227, 1991.

[3] A. Cohen, “New disposable electronic micropump for parenteral drugdelivery,” in Pulsatile Drug Delivery: Current Applications and FutureTrends,R. Gurny, H. E. Junginger, and N. A. Peppas, Eds. Stuttgart,Germany: Wissenschaftliche, 1993, pp. 151–161.

[4] J.-L. Selam, D. Raccah, N. Jean-Didier, J. L. Lozano, K. Waxman,and M. A. Charles, “Randomized comparison of metabolic controlachieved by intraperitoneal insulin infusion with implantable pumpsversus intensive subcutaneous insulin therapy in type I diabetic patients,”Diabetes Care,vol. 15, pp. 53–58, 1992.

[5] H. Buchwald, T. D. Rohde, and K. Kernstine, “Insulin delivery byimplanted pump, a chronic treatment for diabetes,”ASAIO Trans.,vol.35, pp. 5–7, 1989.

[6] W. Preidel, S. Saeger, I. Von Lucadou, and W. Lager, “An elec-trocatalytic glucose sensor for in-vivo application,”Biomed. Instrum.,Technol.,vol. 25, pp. 215–219, 1991.

[7] K. Nishida, M. Sakakida, K. Ichinose, T. Uemura, M. Uehara, K.Kajiwara, T. Miyata, M. Shichiri, K. Ishihara, and N. Nakabayashi,“Development of a ferrocene-mediated needle-type glucose sen-sor covered with newly designed biocompatible membrane, 2-methacryloyloxyethylphosphorylcholine-co-n-butyl methacrylate,”Med.Prog. through Technol.,vol. 21, pp. 91–103, 1995.

[8] C. Meyerhoff, F. Bischof, F. Sternberg, H. Zier, and E. F. Pfeiffer, “Online continuous monitoring of subcutaneous tissue glucose in men bycombining portable glucosensor with microdialysis,”Diabetologia,vol.35, pp. 1087–1092, 1992.

[9] C. Cobelli and A. Mari, “Validation of mathematical models of complexendocrine-metabolic systems. A case study on a model of glucoseregulation,”Med. Biol. Eng. Comput.,vol. 21, pp. 390–399, 1983.

[10] M. E. Fisher, “A semiclosed-loop algorithm for the control of bloodglucose levels in diabetics,”IEEE Trans. Biomed. Eng.,vol. 38, pp.57–61, 1991.

[11] E. Salzsieder, G. Albrecht, U. Fischer, and E.-J. Freyse, “Kineticmodeling of the glucoregulatory system to improve insulin therapy,”IEEE Trans. Biomed. Eng.,vol. BME-32, pp. 846–855, 1985.

[12] H. M. Broekhuyse, J. D. Nelson, B. Zinman, and A. M. Albisser,“Comparison of algorithms for the closed-loop control of blood glucoseusing the artificial beta cell,”IEEE Trans. Biomed. Eng.,vol. BME-28,pp. 678–687, 1981.

[13] K. Y. Kwok, S. L. Shah, A. S. Clanachan, and B. A. Finegan, “Eval-uation of a long-range adaptive predictive controller for computerizeddrug delivery systems,” inProc. IFAC Adaptive Signals in Control andSignal Processing Conference,1992, pp. 317–322.

[14] R. Gopinath, B. W. Bequette, R. J. Roy, H. Kaufman, and C. Yu, “Issuesin the design of a multirate model-based controller for a nonlinear druginfusion system,”Biotechnol. Prog.,vol. 11, pp. 318–332, 1995.

[15] D. A. Linkens and M. Mahfouf, “Generalized predictive control (GPC)in clinical anaesthesia,” inAdvances in Model-Based Predictive Control,

D. Clarke, Ed. Oxford, U.K.: Oxford Univ. Press, 1994, ch. 5, pp.429–445.

[16] D. R. Wada and D. S. Ward, “Open loop control of multiple drug effectsin anesthesia,”IEEE Trans. Biomed. Eng.,vol. 42, pp. 666–677, 1995.

[17] R. N. Bergman, L. S. Phillips, and C. Cobelli, “Physiologic evaluationof factors controlling glucose tolerance in man,”J. Clin. Invest.,vol.68, pp. 1456–1467, 1981.

[18] M. Nomura, M. Shichiri, R. Kawamori, Y. Yamasaki, N. Iwama, andH. Abe, “A mathematical insulin-secretion model and its validation inisolated rat pancreatic islets perfusion,”Comput., Biomed. Res.,vol. 17,pp. 570–579, 1984.

[19] W. R. Puckett, “Dynamic modeling of diabetes mellitus,” Ph.D. thesis,Dept. Chem. Eng., Univ. of Wisconsin-Madison, Madison, 1992.

[20] V. W. Bolie, “Coefficients of normal blood glucose regulation,”J. Appl.Physiol., vol. 16, pp. 783–788, 1961.

[21] E. Ackerman, L. C. Gatewood, J. W. Rosevear, and G. D. Molnar,“Model studies of blood-glucose regulation,”Bull. Math. Biophys.,vol.27, Suppl., pp. 21–37, 1965.

[22] J. R. Guyton, R. O. Foster, J. S. Soeldner, M. H. Tan, C. B. Kahn, L.Koncz, and R. E. Gleason, “A model of glucose-insulin homeostasis inman that incorporates the heterogeneous fast pool theory of pancreaticinsulin release,”Diabetes,vol. 27, pp. 1027–1042, 1978.

[23] J. T. Sorensen, “A physiologic model of glucose metabolism in man andits use to design and assess improved insulin therapies for diabetes,”Ph.D. thesis, Dept. Chem. Eng., Massachusetts Inst. Technol. (MIT),Cambridge, 1985.

[24] E. D. Lehmann and T. Deutsch, “A physiological model of glucose-insulin interaction in type I diabetes mellitus,”J. Biomed. Eng.,vol. 14,pp. 235–242, 1992.

[25] Y. W. Lee and M. Schetzen, “Measurement of the Wiener kernelsof a nonlinear system by cross-correlation,”Int. J. Contr., vol. 2, pp.237–254, 1965.

[26] P. Z. Marmarelis and V. Z. Marmarelis,Analysis of PhysiologicalSystems: The White-Noise Approach.New York: Plenum, 1978.

[27] M. Schetzen,The Volterra and Wiener Theories of Nonlinear Systems.New York, NY: Wiley, 1980.

[28] Y. Fu and G. A. Dumont, “An optimum time scale for discrete Laguerrenetwork,” IEEE Trans. Automat. Contr.,vol. 38, pp. 934–938, 1993.

[29] V. Z. Marmarelis, “Identification of nonlinear biological systems usingLaguerre expansions of kernels,”Ann. Biomed. Eng.,vol. 21, pp.573–589, 1993.

[30] I.-L. Chien and B. A. Ogunnaike, “Modeling and control of high-puritydistillation columns,” presented atAIChE Annu. Meet.,1993.

[31] M. Morari and N. L. Ricker,Model Predictive Control Toolbox. Natick,MA: The MathWorks, Inc., 1994.

[32] E. Zafiriou and A. L. Marchal, “Stability of SISO quadratic dynamicmatrix control with hard output constraints,”AIChE J., vol. 37, pp.1550–1560, 1991.

[33] K. J. Astrom and B. Wittenmark,Computer-Controlled Systems Theoryand Design,2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1990.

[34] N. L. Ricker, “Model predictive control with state estimation,”Ind. Eng.Chem. Res.,vol. 29, no. 3, pp. 374–382, 1990.

[35] G. Gattu and E. Zafiriou, “Nonlinear quadratic dynamic matrix controlwith state estimation,”Ind. Eng. Chem. Res.,vol. 31, pp. 1096–1104,1992.

[36] C. E. Garc´ıa, “Quadratic dynamic matrix control of nonlinear processes.An application to a batch reaction process,” presented atAIChE Annu.Meet., 1984.

[37] R. S. Parker, F. J. Doyle III, J. E. Harting, and N. A. Peppas, “Modelpredictive control for infusion pump insulin delivery,” inProc. 18thAnnu. Conf. IEEE—Engineering in Medicine and Biology Society,1996,no. 265.

Robert S. Parker was born in Rochester, NY, in 1972. He received the B.S.degree in chemical engineering from the University of Rochester, Rochester,NY, in 1994, and began Ph.D. research in chemical engineering at PurdueUniversity, West Lafayette, IN, in the fall of that year under Prof. F. J. Doyle,III, and Prof. N. A. Peppas. In September 1997, he moved with Prof. Doyleto the University of Delaware to complete the doctoral degree.

His research focuses on model-based control of biomedical and biologicalsystems.


Francis J. Doyle, III , was born in Philadelphia, PA, in 1963. He received theB.S.E. degree from Princeton University, Princeton, NJ, in 1985, the C.P.G.S.degree from Cambridge University, Cambridge, U.K., in 1986, and the Ph.D.degree from the California Institute of Technology, Pasadena, in 1991, all inchemical engineering.

From 1991–1992, he was a Visiting Scientist in the strategic processtechnology group at the DuPont Company, Wilmington, DE. From 1992–1997,he was an Assistant and Associate Professor at the School of ChemicalEngineering at Purdue University, West Lafayette, IN. Since 1997, he hasbeen an Associate Professor at the Department of Chemical Engineering atthe University of Delaware, Newark. His research interests include nonlineardynamics and control with applications in process and biosystems control,nonlinear model reduction, and the reverse engineering of biological controlsystems.

Dr. Doyle received the National Young Investigator Award from theNational Science Foundation in 1992, an Office of Naval Research YoungInvestigator Award in 1996, an ASEE Section Outstanding Teacher Award in1996, and a Tau Beta Pi Section Teaching Award in 1996.

Nicholas A. Peppasreceived the Dipl.Eng. degree in chemical engineeringat the National Technical University of Athens, Greece, in 1971 and theSc.D., degree at the Massachusetts Institute of Technology in 1973. He didpostdoctoral research at the Arteriosclerosis Center of MIT.

He is the Showalter Distinguished Professor of Chemical and BiomedicalEngineering in the School of Chemical Engineering of Purdue University,West Lafayette, IN. He joined Purdue as an Assistant Professor in 1976 andwas promoted to Associate Professor in 1978 and Professor in 1982. Hisresearch contributions have been in several areas of polymers and biomedicalengineering, especially in controlled delivery of drugs, peptides and proteins,development of novel biomaterials, biomedical transport phenomena, andbiointerfacial problems.

Dr. Peppas has been recognized by various awards including the 1995 APV-International Pharmaceutical Technology Medal, the 1994 Pharmaceutical andBioengineering Award of AIChE, and the 1988 Curtis McGraw Award ofASEE for best engineering research under the age of 40. He is a foundingFellow of the American Institute of Medical and Biological Engineering, aFellow of the American Physical Society, a Fellow of the Society for Bioma-terials, a fellow of the American Association of Pharmaceutical Scientists, andan honorary member of the Italian Society of Medicine and Natural Sciences.

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