Comparing Closed-loop Control of Drug Infusion using MPC and PID

Tiro Setati

, Willem Jacobus Perold

, Pieter Rousseau Fourie

and Daniel Withey

Council for Scientiﬁc and Industrial Research (CSIR), Pretoria, South Africa

Department of Electrical and Electronic Engineering, University of Stellenbosch, Stellenbosch, South Africa

innovation4life Intellectual Property (Pty) Ltd., Stellenbosch, South Africa

Council for Scientiﬁc and Industrial Research (CSIR) Pretoria, South Africa

Keywords:

Control Systems, Model-Predictive Controller (MPC), Proportional-Integral-Derivative (PID) Controller,

Anesthesia, Intravenous Infusion.

Abstract:

Continuous intravenous drug delivery can require careful dosage and the maintenance of consistent blood

plasma concentration levels. Here, the closed-loop control of drug concentration in simulated patients’ blood-

streams was investigated. During the investigation, the closed-loop controller performance of a Proportional-

Integral-Derivative (PID) and a Model Predictive Controller (MPC) were compared. From the simulation

results, the MPC has shown to be a better controller because of its shorter settling time and smaller step-

response overshoot, which are desirable in a clinical setting. However, the MPC has shown to be more

resource-intensive on the controller system. Through the simulations conducted, it can therefore be con-

cluded that a closed-loop controller containing an MPC control block adequately controls the drug infusion to

a simulated patient.

1 INTRODUCTION

The function of human blood is to transport oxygen,

nutrients, sugars and hormones to parts of the hu-

man body that need them the most, and to carry away

waste materials (Felman, 2020). Blood is also used to

carry drugs for medical purposes, some of which re-

quire continuous drug supply and careful monitoring

of blood concentration levels. Methods for analyzing

blood using electrochemical sensors, and also by non-

invasive sensing, exist and are the subject of ongoing

research (Shokrekhodaei and Quinones, 2020).

This paper will be focused on the design of the

closed-loop control system delivering a drug into a

patient intravenously based on a set-point of drug

plasma concentration. Experiments will involve sim-

ulating a closed-loop control system which will be

able to administer the drug to maintain drug plasma

concentrations to the simulated patient. This paper

will also be comparing the use of a PID controller and

an MPC controller to ﬁnd which controller is more

suited to deliver a drug to a patient autonomously.

For similar drug infusion systems, like insulin in-

fusion systems, it was proven that MPC closed-loop

control systems are more efﬁcient than the standard

open-loop infusion of insulin, and that patients’ in-

sulin levels remained within the set-point range more

often than in the open-loop case (Clarke et al., 2009)

(Bruttomesso et al., 2009). The artiﬁcial pancreas

has also been designed and compared with the MPC

and the PID controllers, and it has been shown that

the MPC outperforms the PID in this application, al-

though both controllers provided adequate glucose

control (Pinsker et al., 2016).

Unlike (Clarke et al., 2009) and (Bruttomesso

et al., 2009), which both make use of glucose-insulin

dynamics models, this paper uses the paediatric Paed-

fusor patient model used in anesthesia. Additionally,

the design of the models used in (Pinsker et al., 2016)

were personalized models for the MPC, but for this

paper, a general patient model was designed and ap-

plied for the entire patient group to study the efﬁcacy

of the system controlling the drug infusion based off

of a general model.

There have been a number of studies conducted

making use of closed-loop control strategies in anes-

thesia (Nas¸cu et al., 2014) (De Keyser et al., 2015)

(Padula et al., 2017) (Ntouskas and Sarimveis, 2021).

To detect the level of hypnosis experienced by the pa-

tient, these papers make use of the bispectral (BIS)

index commonly used in practice (Rampil, 1998), al-

though other measures, such as the mid-latency audi-

126

Setati, T., Perold, W., Fourie, P. and Withey, D.

Comparing Closed-loop Control of Drug Infusion using MPC and PID.

DOI: 10.5220/0010783700003123

In Proceedings of the 15th International Joint Conference on Biomedical Engineering Systems and Technologies (BIOSTEC 2022) - Volume 1: BIODEVICES, pages 126-133

ISBN: 978-989-758-552-4; ISSN: 2184-4305

tory evoked potentials (MLAEP) could also be used

to form an indication of the depth of anesthesia ex-

perienced by a patient (Mantzaridis and Kenny, 1997)

(Kuhnle et al., 2013).

This paper compares an MPC controller and a

PID controller in controlling the delivery of anesthe-

sia. This is done by comparing the step responses of

both types of control strategies, along with analysing

how both operate in the presence of sensor noise and

comparing the computation time taken by both con-

trollers. This would provide some useful information

in considering the practicality of each controller in a

surgical context.

This paper contains a total of ﬁve sections. Sec-

tion 1 is the introduction of the paper, Section 2 con-

tains the background material related to the paper,

Section 3 expands on the methods used to run the ex-

periments, Section 4 gives the experiment results and

discusses the results, and Section 5 is the conclusion.

2 BACKGROUND

2.1 Control Systems in Anesthesia

For the successful implementation of a control sys-

tem in anesthesia, there are a few key components that

must be included. There must be a measurable con-

trol variable used to determine the effect of the drugs

administered, an established set-point for the control

variable, a controller and a control actuator to admin-

ister the drug, and the system must ﬁt the appropriate

pharmacokinetics and pharmacodynamics for the ap-

plication (Struys et al., 2006).

The use of control systems for drug delivery in

medical contexts dates as far back as the 1980s. They

were initially implemented in insulin infusion sys-

tems for people with diabetes (Deckert et al., 1980)

(Doyle et al., 2014) and the applications have ex-

panded to other uses, including the control of neuro-

muscular systems (Solomonow, 1984) and anesthesia

(Westenskow, 1987) (Dumont, 2012).

For medical control systems, especially drug ad-

ministration systems, PID controllers pose stability

risks in the administration. PID controllers need to be

meticulously tuned for speciﬁc scenarios and without

proper tuning, they run the risk of oscillating before

reaching, or during, steady state (Struys et al., 2006).

This is because the design of PID controllers does

not incorporate information about drug metabolism,

which could potentially allow the administration of

dangerously high drug concentrations.

Figure 1: The three-compartment mamillary PK model.

2.2 Pharmacokinetics and

Pharmacodynamics

Pharmacokinetics (PK) deﬁne the movement of a

drug within a patient. This involves the absorption,

distribution, metabolism and elimination of the drug

and this movement is highly dependent on patient-

speciﬁc parameters, such as age, weight, height, and

sex. Pharmacodynamics (PD) deﬁne the inﬂuence of

the drug on the patient. This involves receptor bind-

ing, post-receptor effects and chemical interactions

(Bibian, 2006).

For intravenous drug infusion, the pharmacoki-

netics of the patient are best described by the three-

compartment mamillary PK model, depicted in Fig-

ure 1. This model consists of three compartments in

which the drug can be distributed. The ﬁrst compart-

ment (V1) is the blood, where the drug is directly in-

jected. The second compartment (V2) is the fast com-

partment, which is the muscle. Lastly, the third com-

partment (V3) is the slow compartment, which repre-

sents the fat (Hull, 1979).

To simulate the pharmacokinetics of a patient, a

state-space model is commonly used. The state-space

model is deﬁned as:





˙q

(t)

˙q

(t)

˙q

(t)









−(k

+ k

) k

−k

0 −k









(t)













u (1)



1 0 0







(t)





+ 0 · u (2)

with q

as the mass of the drug in the i−th com-

partment,, and the k

i j

parameters represent the trans-

fer rate constants from compartment i to compartment

j, u is the infusion rate and C

is the drug plasma con-

centration in the patient (Hull, 1979).

Comparing Closed-loop Control of Drug Infusion using MPC and PID

127

2.3 Control Strategies

The vast number of controllers all have unique ap-

proaches to the control problem and have varied lev-

els of complexities, but the end goal of each type of

control system is to drive the behaviour of the sys-

tem output. The two controllers used in this study are

listed below.

2.3.1 PID

The Proportional-Integral-Derivative (PID) controller

is one of the most popular control algorithms used

(Kumar et al., 2011). The PID controller is made of

three separate controllers and each controller serves a

speciﬁc purpose in the control trajectory of the con-

troller and each of these controllers is dependent on

the system’s error signal e(t), which is the difference

between the measured output and the system’s refer-

ence signal.

The proportional feedback controller, or P-

controller, linearly adjusts the controller’s output to

the error signal. The integral feedback controller,

or I-controller, linearly adjusts the controller propor-

tional to the integral of the error signal. This is

used to decrease the steady-state error signal of the

system between the output and the reference signal.

The derivative feedback controller, or D-controller,

decreases the overshoot of the system step response

and improves the system’s overall stability. The

D-controller’s output is proportional to the rate of

change of the system error. (Gene F. Franklin, 2015).

Therefore, the controller output can be written as

PID

= k

e(t) + k

e(τ)dτ + k

˙e(t) (3)

with k

as the proportional gain, k

as the integral

gain, and k

as the derivative gain.

2.3.2 MPC

Model Predictive Control (MPC) is an advanced con-

trol strategy that solves an open-loop optimal control

problem with deﬁned constraints and system-state dy-

namics on-line. A unique feature of the MPC is that

it incorporates a model of the system within the con-

trol algorithm to optimize the control actions that are

determined .

The MPC uses the system model to make predic-

tions of the future behaviour of the system and its dy-

namics over a deﬁned period, called the “prediction

horizon.” From these predictions, the MPC will im-

plement the ﬁrst few predictions made in the deﬁned,

shorter period called the “control horizon.” Once the

Figure 2: The closed-loop control system.

control manoeuvre is implemented, the MPC will pre-

dict the possible control actions for the next prediction

horizon (Findeisen and Allg

ower, 2002) (Rakovi

c and

Levine, 2018).

3 METHODS

The designed closed-loop control system will be sim-

ulated as depicted in Figure 2. The closed-loop con-

trol system is made up of the following elements.

3.1 The Control Loop

The system’s reference deﬁnes the value the system

aims to achieve. The reference signal will be the

plasma concentration within the patient. In the simu-

lation, this will be speciﬁed by a selected concentra-

tion.

The plant to be used for the system is the patient.

To understand the inﬂuence of the drug on the patient,

a thorough understanding of pharmacokinetics (PK)

is needed. For the closed-loop control system, the pa-

tient will be simulated as a three-compartmental state-

space Paedfusor PK model of the patient (Absalom

et al., 2003). The same model used for the plant will

be used as the internal model of the MPC. The mea-

sured output from the simulated patient model will be

the drug concentration in the patient’s plasma. This

output will be fed back into the controller to close the

loop.

The controllers used, the MPC and PID con-

trollers, will be implemented separately to compare

the performance given by each controller.

3.1.1 MPC

To determine a step response with no overshoot and

with the shortest settling time possible, the MPC con-

troller was designed with the parameters listed in Ta-

ble 1.

The sampling time was selected assuming that the

plasma concentration measurements will be taken ev-

ery 10 s, and then the drug will be delivered. The pre-

diction horizon sample number was chosen as a value

that would allow for an adequate amount of future

BIODEVICES 2022 - 15th International Conference on Biomedical Electronics and Devices

128

predictions, without calculating too many future pre-

dictions, saving computational resources. The con-

trol horizon is chosen as 10% of the prediction hori-

zon. The weights were determined by designing the

MPC controller with the MATLAB MPC Designer by

adjusting the robustness of the controller to have no

overshoot and a short settling time (The MathWorks,

a).

Table 1: MPC controller parameters.

Parameter Value

Sample time (T

) 10 s

Prediction horizon 20

Control horizon 2

Weights

MV 0

MV Rate 0.146

OV 0.412

MV: Manipulated Variable

OV: Output Variable

3.1.2 PID

The parameters of the PID controller were determined

by using MATLAB’s PID Tuner to automatically de-

termine the parameters based on the desired response

(Xue et al., 2007). The parameters are listed in Ta-

ble 2. Unfortunately, a system response with no over-

shoot was unattainable, therefore a system response

with as small an overshoot as possible was consid-

ered.

Table 2: PID controller parameters.

PID parameter Value

P (proportional term) 2.789

I (integral term) 0.06746

D (derivative term) -5.0887

3.2 Added Measurement Noise

To analyse how the systems will respond in the pres-

ence of noise, measurement noise was added accord-

ing to Table 3. The signal-to-noise ratio (SNR) is the

ratio of the power of a desired signal relative to the

power of the noise within the signal (Kieser et al.,

2005), in decibels (dB). The SNR is deﬁned as:

SNR

= 10log

Signal

Noise

dB (4)

with P

Signal

as the power of the desired signal, re-

spectively.

Table 3: Magnitude of Measurement Noise Added for each

Simulation.

Simulation SNR [dB]

1 40

2 30

3 20

4 15

5 10

6 5

7 0

8 -3

3.3 Patient Group

To analyse the inﬂuence that inter-patient variability

would have on the system, the designed system was

tested on a simulated patient group. The 14 simu-

lated patients used in this paper was randomly gener-

ated using the patient characteristic data from (Kuhnle

et al., 2013), which is also listed in Table 4.

Table 4: Simulated Patient Group (Kuhnle et al., 2013).

Patient characteristic data Mean (SD) (range)

Number of patients (n) 14

Age (yr) 8.6 (4.3) (4.0 - 16.5)

Weight (kg) 29.2 (14.6) (15.0 - 60.0)

Height (cm) 125.9 (26.2) (67.0 - 160.0)

3.4 Measuring Plasma Concentration

To measure the anesthetic plasma concentration in

a patient directly is undesirable because it would

be an invasive procedure, where the patient’s blood

would need to be drawn and the anesthetic would

need to be measured in real time. Therefore, mea-

sures of hypnosis have been found that can correlate

the level of anesthesia a patient experiences to the

amount of plasma in a patient’s plasma. These mea-

sures include electroencephalogram (EEG) modelling

(Rampil, 1998), power spectrum analysis (Pichlmayr

et al., 2012), bispectral (BIS) index (Ontario et al.,

2004) and evoked potentials (Walsh et al., 2005).

The work conducted in (Kuhnle et al., 2013) found

a strong correlation between plasma concentration

and mid-latency auditory evoked potentials (MLAEP)

(Plourde, 2006) which suggests that it may be pos-

sible to estimate blood plasma concentration from

MLAEP, in the case of the anesthesia application.

Comparing Closed-loop Control of Drug Infusion using MPC and PID

129

Figure 3: Comparison of the plasma concentrations from

the MPC and PID control loops (in µg.ml

−1

4 SIMULATION RESULTS

The following section expands on the results achieved

from the simulations. The simulations were run using

MATLAB and Simulink.

4.1 Step Response

The objective of the control system is to reach the

reference plasma concentration as quickly as possible

without overshooting the reference value because in

a practical system, overshooting the reference value

could lead to complications as a result of drug over-

dose. This will simulate a drug being delivered intra-

venously to a patient to reach a desired plasma con-

centration. The reference plasma concentration is set

to 3 µg.ml

−1

To compare the performance of the MPC con-

troller to the PID controller, the step responses of both

controllers are plotted together. The step response

of the plasma concentration from the patient group’s

average patient is depicted in Figure 3. The step re-

sponse of the MPC is better than that of the PID. The

2% settling time of the MPC controller is at 47.9 s and

that of the PID is at 146.2 s. Additionally, the step re-

sponse of the MPC has no overshoot, while that of

the PID has a slight overshoot, reaching a maximum

of 3.04 µg.ml

−1

4.2 Inter-patient Variability

To determine how the systems would operate with a

wide variety of patients, it was tested on the patient

group described in Table 4. The results labelled “Cp:

Design standard” are the results of the average patient,

who was used to design the system and whose results

are depicted in Figure 3.

Figure 4: Step responses of the patient groups’ plasma con-

centration with the MPC controller (in µg.ml

−1

4.2.1 MPC

The results of the step response of the MPC closed-

loop controller for the patient group is shown in Fig-

ure 4. The results vary according to the age of the

patient. Patients that are younger than the average pa-

tient tend to have shorter settling times and overshoot

the reference plasma concentration. Patients that are

older than the average patient do not overshoot the

reference value, but their 2% settling time is longer

than that of the average patient.

The reason for younger patients overshooting the

reference plasma concentration is likely because they

have more rigorous pharmacokinetics because of their

smaller compartmental models and the same rate

transfer constants as those of older patients.

To analyse the inﬂuence of the internal patient

model on the control system, an MPC controller was

implemented with the average patient as the internal

patient model, with an age of 8.6 years, weight of 29.2

kg and height of 125.9 cm. This MPC controller de-

sign was used to control the anesthetic infusion for the

same patient population, and the performance of this

control system is depicted in Figure 5.

From Figure 5, the plasma concentration of the pa-

tients that are younger than the internal patient model

oscillate before reaching steady state. This would be

undesirable in practice as it would result in irregular,

and potentially dangerous, infusion to the patient. It

is therefore better to incorporate an internal patient

model which corresponds to the patient undergoing

anethesia to allow for safer sedation for the patient

population.

4.2.2 PID

The step responses of the patient group simulated

with the PID closed-loop control system are depicted

in Figure 6.

From the results, it is clear to see that there is

also inter-patient variability at play with the PID con-

troller. Patients that are younger than the average pa-

tient have quicker settling times when compared to

BIODEVICES 2022 - 15th International Conference on Biomedical Electronics and Devices

130

Figure 5: Step responses of the patient groups’ plasma con-

centration for the MPC controller with constant internal

model (in µg.ml

−1

Figure 6: Step responses of the patient groups’ plasma con-

centration with the PID controller (in µg.ml

−1

the average patient. This could be for the same reason

that younger patients overshoot in the MPC simula-

tions.

4.3 Execution Time

To compare how resource intensive each system is,

the two controller’s execution times were measured

for the simulation of the patient group over a 2000 s

window. Simulink is a model-based design software

package and it is not optimized for real time opera-

tion. However, the simulation times are still useful as

a basis for comparison (The MathWorks, b).

The execution times for each patient and the av-

erage execution time are shown in Table 5. For the

MPC control loop, the average execution time was

6.777 s and the PID control system executed for an

average time of 5.691 s. The execution times also in-

dicate that the PID controller remains more consistent

in terms of computation time when controlling infu-

sion for the different patients with varying PK mod-

els, indicated by a smaller standard deviation. This

indicates that the PID system would be less resource

intensive in a practical setting as it does not need as

much processing power to execute the control loop as

the MPC controller would need.

Table 5: Execution Times for all 14 Patients According to

the MPC or PID Controller used.

MPC [s] PID [s]

Average 6.78 5.69

Standard deviaton 2.25 0.73

Figure 7: Simulated patient’s plasma concentrations with

measurement noise in the MPC control loop (Upper plot:

SNR 40 dB to 15 dB. Lower plot: SNR 10 dB to -3 dB.)

4.4 Measurement Noise

For the measurement noise simulations, the MPC

control system’s performance is depicted in Figure 7

and the PID controller’s performance is depicted in

Figure 8 and the root mean squared error (RMSE) of

the various noise levels for both controllers are listed

in Table 6.

According to Table 6, the MPC controller has bet-

ter noise suppression than the PID controller. In a

real-system, noise is an inevitable factor to consider,

so this justiﬁes that the MPC controller is a better con-

troller to use.

Table 6: RMSE of the MPC and PID controllers at different

SNR.

SNR MPC PID

40 0.711 0.721

30 0.587 0.697

20 0.594 0.580

15 0.555 0.579

10 0.574 0.547

5 0.575 0.582

0 0.585 0.568

-3 0.775 0.588

Comparing Closed-loop Control of Drug Infusion using MPC and PID

131

Figure 8: Simulated patient’s plasma concentrations with

measurement noise in the PID control loop (Upper plot:

SNR 40 dB to 15 dB. Lower plot: SNR 10 dB to -3 dB.)

Figure 9: Changing the reference plasma concentrations in

the MPC and the PID controller.

4.5 Changing Reference

To fully demonstrate the set-point following of

closed-loop control, the system is tested under the

condition of changing the reference concentration

during the simulation.

The results of the changing reference are depicted

in Figure 9. The results obtained from these simu-

lations are similar to those in the step response sim-

ulations, with the PID controller’s step response de-

picting longer settling time and slight overshoots. For

larger reference changes, the settling time of the PID

controller is longer than that of the MPC controller

and the overshoot is greater than that of the MPC con-

troller. For smaller reference changes, the MPC and

PID step responses are similar.

5 CONCLUSIONS

From the data gathered from both the MPC and PID

controllers, it can be concluded that the MPC con-

troller is the more suited controller to use in the con-

trol of drug plasma concentration. Even though the

MPC requires more processing power to operate than

the PID, it is more appropriate for this setting because

of its faster settling time, no overshoot, and shows bet-

ter operation in noise. Additionally, based on the re-

sults from the changing reference tests, the MPC con-

troller’s response follows the set point changes more

consistently than that of the PID controller.

For a less costly implementation, the PID control

system could be investigated further as the maximum

overshoot it produces may be within an acceptable

range and it is a less resource intensive control sys-

tem, as shown by of the more consistent computa-

tion time achieved when running the simulations for a

number of patients. However, PID controllers need to

be meticulously tuned for applications such as this be-

cause they run the risk of oscillating before or during

steady state if they are not properly tuned.

As mentioned in (Ntouskas and Sarimveis, 2021)

and (Gonzalez-Cava et al., 2021), for optimal con-

trol, irrespective of the control strategy used, the inter-

patient variability must be considered further as the

patients that lie at the edges of the PK models do not

follow the references given as precisely as the average

patients. This could be done by adapting the patient

models to make more considerations for patients that

lie at the edges of these PK models. Although the

plasma concentrations of these patient models do not

deviate too wildly from the reference points provided,

it would be desirable to have them follow the refer-

ence point as closely as possible for safer and more

predictable plasma concentration levels.

REFERENCES

Absalom, A., Amutike, D., Lal, A., White, M., and Kenny,

G. (2003). Accuracy of the ’paedfusor’in children un-

dergoing cardiac surgery or catheterization. British

Journal of Anaesthesia, 91(4):507–513.

Bibian, S. (2006). Automation in clinical anesthesia. PhD

thesis, University of British Columbia.

Bruttomesso, D., Farret, A., Costa, S., Marescotti, M. C.,

Vettore, M., Avogaro, A., Tiengo, A., Dalla Man, C.,

Place, J., Facchinetti, A., et al. (2009). Closed-loop

artiﬁcial pancreas using subcutaneous glucose sensing

and insulin delivery and a model predictive control al-

gorithm: preliminary studies in padova and montpel-

lier.

Clarke, W. L., Anderson, S., Breton, M., Patek, S., Kash-

mer, L., and Kovatchev, B. (2009). Closed-loop artiﬁ-

BIODEVICES 2022 - 15th International Conference on Biomedical Electronics and Devices

132

cial pancreas using subcutaneous glucose sensing and

insulin delivery and a model predictive control algo-

rithm: the virginia experience.

De Keyser, R., Copot, D., and Ionescu, C. (2015). Estima-

tion of patient sensitivity to drug effect during propo-

fol hypnosis. In 2015 IEEE International Conference

on Systems, Man, and Cybernetics, pages 2487–2491.

IEEE.

Deckert, T., Bojsen, J., Christiansen, J. S., Kølendorf, K.,

Svendsen, P. A., and Andersen, A. R. (1980). 24-hour

blood glucose proﬁles in insulin-dependent diabetics

treated with intravenous insulin infusion systems: A

comparison between closed-and open-loop systems.

Acta Medica Scandinavica, 208(1-6):451–458.

Doyle, F. J., Huyett, L. M., Lee, J. B., Zisser, H. C., and

Dassau, E. (2014). Closed-loop artiﬁcial pancreas

systems: engineering the algorithms. Diabetes care,

37(5):1191–1197.

Dumont, G. A. (2012). Closed-loop control of anesthesia-a

review. IFAC Proceedings Volumes, 45(18):373–378.

Felman, A. (2020). Blood: Components, functions, groups,

and disorders. https://www.medicalnewstoday.com/

articles/196001.

Findeisen, R. and Allg

ower, F. (2002). An introduction to

nonlinear model predictive control. In 21st Benelux

meeting on systems and control, volume 11, pages

119–141. Technische Universiteit Eindhoven Veld-

hoven Eindhoven, The Netherlands.

Gene F. Franklin, J. David Powell, A. E.-N. (2015). Feed-

back Control of Dynamic Systems, chapter 4, pages

216–237. Pearson Education Limited, 7 edition.

Gonzalez-Cava, J. M., Carlson, F. B., Troeng, O., Cervin,

A., van Heusden, K., Dumont, G. A., and Soltesz,

K. (2021). Robust pid control of propofol anaesthe-

sia: uncertainty limits performance, not pid structure.

Computer Methods and Programs in Biomedicine,

198:105783.

Hull, C. (1979). Pharmacokinetics and pharmacodynamics.

British Journal of Anaesthesia, 51(7):579–594.

Kieser, R., Reynisson, P., and Mulligan, T. J. (2005). Def-

inition of signal-to-noise ratio and its critical role in

split-beam measurements. ICES Journal of Marine

Science, 62(1):123–130.

Kuhnle, G., Hornuss, C., Lenk, M., Salam, A., Wiepcke, D.,

Edelmann-Gahr, V., Flake, G., Daunderer, M., Ober-

hauser, M., M

uller, H.-H., et al. (2013). Impact of

propofol on mid-latency auditory-evoked potentials in

children. British journal of anaesthesia, 110(6):1001–

1009.

Kumar, V., Nakra, B., and Mittal, A. (2011). A review on

classical and fuzzy pid controllers. International Jour-

nal of Intelligent Control and Systems, 16(3):170–181.

Mantzaridis, H. and Kenny, G. (1997). Auditory evoked

potential index: a quantitative measure of changes in

auditory evoked potentials during general anaesthesia.

Anaesthesia, 52(11):1030–1036.

Nas¸cu, I., Krieger, A., Ionescu, C. M., and Pistikopoulos,

E. N. (2014). Advanced model-based control stud-

ies for the induction and maintenance of intravenous

anaesthesia. IEEE Transactions on biomedical engi-

neering, 62(3):832–841.

Ntouskas, S. and Sarimveis, H. (2021). A robust model pre-

dictive control framework for the regulation of anes-

thesia process with propofol. Optimal Control Appli-

cations and Methods.

Ontario, H. Q. et al. (2004). Bispectral index monitor: an

evidence-based analysis. Ont Health Technol Assess

Ser, 4(9):1–70.

Padula, F., Ionescu, C., Latronico, N., Paltenghi, M., Vi-

sioli, A., and Vivacqua, G. (2017). Optimized pid

control of depth of hypnosis in anesthesia. Computer

methods and programs in biomedicine, 144:21–35.

Pichlmayr, I., Lips, U., and K

unkel, H. (2012). The elec-

troencephalogram in anesthesia: fundamentals, prac-

tical applications, examples. Springer Science &

Business Media.

Pinsker, J. E., Lee, J. B., Dassau, E., Seborg, D. E., Bradley,

P. K., Gondhalekar, R., Bevier, W. C., Huyett, L.,

Zisser, H. C., and Doyle, F. J. (2016). Randomized

crossover comparison of personalized mpc and pid

control algorithms for the artiﬁcial pancreas. Diabetes

Care, 39(7):1135–1142.

Plourde, G. (2006). Auditory evoked potentials. Best Prac-

tice & Research Clinical Anaesthesiology, 20(1):129–

139.

Rakovi

c, S. V. and Levine, W. S. (2018). Handbook of

model predictive control. Springer.

Rampil, I. J. (1998). A primer for eeg signal processing in

anesthesia. Anesthesiology: The Journal of the Amer-

ican Society of Anesthesiologists, 89(4):980–1002.

Shokrekhodaei, M. and Quinones, S. (2020). Review

of non-invasive glucose sensing techniques: Optical,

electrical and breath acetone. Sensors, 20(5):1251.

Solomonow, M. (1984). External control of the neuromus-

cular system. IEEE transactions on Biomedical Engi-

neering, (12):752–763.

Struys, M. M., Mortier, E. P., and De Smet, T. (2006).

Closed loops in anaesthesia. Best Practice & Research

Clinical Anaesthesiology, 20(1):211–220.

The MathWorks, I. Matlab. https://www.mathworks.com/

products/matlab.html.

The MathWorks, I. Optimize performance.

https://www.mathworks.com/help/simulink/

performance performance.html.

Walsh, P., Kane, N., and Butler, S. (2005). The clinical

role of evoked potentials. Journal of neurology, neu-

rosurgery & psychiatry, 76(suppl 2):ii16–ii22.

Westenskow, D. R. (1987). Closed-loop control of blood

pressure, ventilation, and anesthesia delivery. Interna-

tional journal of clinical monitoring and computing,

4(2):69–74.

Xue, D., Chen, Y., and Atherton, D. P. (2007). Linear

feedback control: analysis and design with MATLAB.

SIAM.

Comparing Closed-loop Control of Drug Infusion using MPC and PID

133