Download - Target mass control for uncertain compartmental systems

This article was downloaded by: [North Dakota State University]On: 14 November 2014, At: 08:05Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

International Journal of ControlPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/tcon20

Target mass control for uncertain compartmentalsystemsClaudia Sousa a , Teresa Mendonca b & Paula Rocha ca School of Education Jean Piaget , Piaget Institute , R. Antonio Sergio, 4410-269, CanelasV.N.G., Portugalb Department of Applied Mathematics, Faculty of Sciences , University of Porto , R. doCampo Alegre, 687, 4169-007, Porto, Portugalc Department of Electrical and Computer Engineering, Faculty of Engineering , University ofPorto , R. Dr. Roberto Frias, s/n, 4200-465, Porto, PortugalPublished online: 19 May 2010.

To cite this article: Claudia Sousa , Teresa Mendonca & Paula Rocha (2010) Target mass control for uncertain compartmentalsystems, International Journal of Control, 83:7, 1387-1396, DOI: 10.1080/00207171003736311

To link to this article: http://dx.doi.org/10.1080/00207171003736311

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of theContent. Any opinions and views expressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon andshould be independently verified with primary sources of information. Taylor and Francis shall not be liable forany losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use ofthe Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

http://www.tandfonline.com/loi/tcon20

http://www.tandfonline.com/action/showCitFormats?doi=10.1080/00207171003736311

http://dx.doi.org/10.1080/00207171003736311

http://www.tandfonline.com/page/terms-and-conditions

http://www.tandfonline.com/page/terms-and-conditions

International Journal of ControlVol. 83, No. 7, July 2010, 1387–1396

Target mass control for uncertain compartmental systems

Claudia Sousaa*, Teresa Mendoncab and Paula Rochac

aSchool of Education Jean Piaget, Piaget Institute, R. Antonio Sergio, 4410-269, Canelas V.N.G., Portugal;bDepartment of Applied Mathematics, Faculty of Sciences, University of Porto, R. do Campo Alegre, 687,4169-007, Porto, Portugal; cDepartment of Electrical and Computer Engineering, Faculty of Engineering,

University of Porto, R. Dr. Roberto Frias, s/n, 4200-465, Porto, Portugal

(Received 5 October 2009; final version received 27 February 2010)

In this article we analyse the total mass target control problem for compartmental systems under the presence ofparameter uncertainties. We consider a state feedback control law with positivity constraints tuned for a nominalsystem, and prove that this law leads the value of the total mass of the real system to an interval whose boundsdepend on the parameter uncertainties and can be made arbitrarily close to the desired value of the total masswhen the uncertainties are sufficiently small. Moreover, we prove that for a class of compartmental systems in R

3

of interest, the state of the controlled system tends to an equilibrium point whose total mass lies within theaforementioned interval. Taking into account the relationship between the mass and the state components insteady state, it is possible to use the proposed mass control law to track the desired values for the steady statecomponents. This is applied to the control of the neuromuscular blockade level of patients undergoing surgery,by means of the infusion of atracurium. Our results are illustrated by several simulations and a clinical case.

Keywords: compartmental systems; positive control; uncertain systems; neuromuscular blockade control

1. Introduction

Compartmental systems form a subclass of positive

systems that consist of a finite number of subsystems,

the compartments, which exchange matter with each

other and with the environment. Such systems have

been successfully used to model biomedical and

pharmacokinetical processes; see, for instance,

Godfrey (1983) Jacquez and Simon (1993). Since one

has to guarantee the positivity of the control input, the

design of suitable control laws for such systems is more

delicate. In Haddad, Hayakawa, and Bayley (2003), for

instance, a nonnegative adaptive control law is pro-

posed in order to guarantee the partial asymptotic set-

point stability of the closed-loop system, and a positive

feedback control law is proposed in Bastin and Provost

(2002), in order to stabilise the total system mass at an

arbitrary set-point. The positive control law proposed

in Bastin and Provost (2002) was also used in

Magalhaes, Mendonca, and Rocha (2005) for the

control of the neuromuscular blockade level (Lemos,

Mendonca, and Mosca 1991; Linkens 1994; Mendonca

and Lago 1998) of patients undergoing surgery; how-

ever, no theoretical study was made of the effect of

parameter uncertainty in the controller performance.

In this article, we consider the control law referred

to in Bastin and Provost (2002) and in Magalhaes et al.

(2005), and analyse its performance for the target

control of the total mass, when the system parameters

are not exactly known. More concretely, we consider

that the control law is tuned for a nominal process

model that contains an additive uncertainty with

respect to the real model, and analyse the behaviour

of the total mass in the controlled system. It turns out

that, in this case, the asymptotical mass values lie in an

interval whose bounds can be easily expressed in terms

of the system uncertainties. Moreover, we prove that

for a class of compartmental systems in R3 of interest,

the state of the controlled system tends to an equilib-

rium point whose total mass lays within the aforemen-

tioned interval. Taking into account the relationship

between the mass and the state components in steady

state, it is possible to use the proposed mass control

law to track desired values for the steady state

components. This is applied to the control of the

neuromuscular blockade level of patients undergoing

surgery, by means of the infusion of atracurium.

Our results are illustrated by several simulations and

a clinical case.

*Corresponding author. Email: [email protected]

ISSN 0020–7179 print/ISSN 1366–5820 online

� 2010 Taylor & Francis

DOI: 10.1080/00207171003736311

http://www.informaworld.com

Dow

nloa

ded

by [

Nor

th D

akot

a St

ate

Uni

vers

ity]

at 0

8:05

14

Nov

embe

r 20

14

2. Compartmental systems

Compartmental systems are dynamical systemsdescribed by a set of equations of the form

_xi ¼Xj6¼i

fjiðxÞ �Xl6¼i

filðxÞ � fi0ðxÞ þ f0iðxÞ

i ¼ 1, . . . , n

(Sandberg 1978; Godfrey 1983) where x¼ (x1, . . . , xn)T

is the state variable and xi and fi j take nonnegativevalues. Each equation describes the evolution of thequantity or concentration of material within a sub-system, called compartment. Since the compartmentsexchange matter with each other and with the envi-ronment, in the above equation, xi is the amount(or concentration) of material in compartment i, fi j isthe flow rate from compartment i to compartment jand the subscript 0 denotes the environment (Godfrey1983). In this article, we consider the class of lineartime-invariant compartmental systems described by

_xi ¼Xj6¼i

kjixj �Xl6¼i

kilxi � qixi þ biu, i ¼ 1, . . . , n,

ð1Þ

where xi and the input u take nonnegative values, therate constants, kij, as well as qi, bi are nonnegative andat least one bi is positive (Figure 1).

Note that, in this case, fji¼ kjixj, f0i¼ biu andfi0¼ qixi, and it can be easily proved that the systemverifies the following properties:

. it is positive, that is, if we consider an input uthat remains nonnegative, then the state var-iable also remains nonnegative;

. (1) can be written in matrix form as

_x ¼ Axþ bu, ð2Þ

where A (called compartmental matrix) is suchthat

aii ¼ �qi �Xj 6¼i

kij and, if i 6¼ j, aij ¼ kji,

and b¼ [b1 b2 . . . bn]T;

. if u� 0 and if the system is fully outflowconnected, (i.e. for every compartment i with

qi¼ 0 there is a chain i! j! k!� � �! l withpositive rate constants at each step andwith ql40), then x¼ 0 is a globally asymptot-ically stable equilibrium point of the system(Bastin and Provost 2002).

In this article, we shall only consider fully outflowconnected systems.

2.1 Mass control – the exact case

The total mass of a compartmental system in a givenstate x is defined as MðxÞ ¼

Pni¼1 xi. For an arbitrary

positive value M �, the set �M � ¼ fx 2 Rnþ :

MðxÞ ¼M �g of all the points x in the state spacewith mass M� is called an iso-mass.

If u� 0, the third property presented in the previ-ous section guarantees that the mass does not accu-mulate inside the system. However, this may nothappen if u is not always zero. This undesired situationis avoided if one is able to stabilise the total mass ofthe system in a given positive target value M �, orequivalently, lead the state trajectories to the iso-mass�M �.

This leads us to an important issue in the contextof the control of compartmental systems: to design acontrol law which yields a positive input that steers thesystem mass M(x) to a desired target value M �.

In Bastin and Provost (2002), the positivecontrol law:

uðxÞ ¼ max 0, ~uðxÞð Þ

~uðxÞ ¼Xni¼1

bi

!�1 Xni¼1

qixi þ � M � �MðxÞð Þ

!,ð3Þ

where � is an arbitrary design parameter, is proposedfor this purpose and the desired convergence propertiesare obtained through the following result.

Theorem 2.1 (Bastin and Provost 2002): Let (2) be afully outflow connected compartmental system. Then, forthe closed-loop system (2)–(3) with arbitrary initialconditions xð0Þ 2 R

nþ:

(i) the iso-mass �M � is forward invariant;(ii) the state x(t) is bounded for all t40 and

converges to the iso-mass �M �.

The proof of this theorem is based on the applica-tion of LaSalle’s invariance principle (LaSalle 1976,p. 30), by considering the Lyapunov function

VðxÞ ¼1

2M � �MðxÞð Þ

2

of (2) on Rnþ. We shall make use of an adapted version

of this function in our convergence analysis.

bi bj

qi qj

To/from other compartments

To/from other compartmentskji

kij i j

Figure 1. Two compartments of a linear time-invariantcompartmental model, as described by (1).

1388 C. Sousa et al.

Dow

nloa

ded

by [

Nor

th D

akot

a St

ate

Uni

vers

ity]

at 0

8:05

14

Nov

embe

r 20

14

In Magalhaes et al. (2005), the control law (3) was

applied for the target control of the neuromuscular

blockade level of patients undergoing surgery, by

means of the infusion of atracurium. However, even

after a satisfactory identification of the patient’s

characteristics, it was necessary to consider an addi-

tional integrator, in order to achieve good results. This

might be explained by the fact that (contrary to what

happens, for instance, with state feedback stabilisers,

which are not uniquely defined from the system

matrices) the control law (3) strongly depends on the

system parameters. Since parameter uncertainty is

present not only in this case, but in most of the

applications, it is relevant to analyse the robustness of

that control law.

3. Mass control in uncertain compartmental systems

3.1 Robustness

We now prove that the control law presented in the

previous section leads the total mass of an uncertain

system to an interval whose bounds depend on the

parameter uncertainties.Since the control law (3) does not depend on the

interactions between compartments (i.e. it does not

depend on the kij’s) and assuming that it is possible to

precisely measure what is injected from the outside

into the system (i.e. the parameters bi are not subject

to uncertainties), we shall consider the case where the

only uncertain parameters are q1, . . . , qn. Therefore, we

shall assume that a control law (3) is designed for a

nominal system

_x ¼ Aþ DAð Þxþ bu, ð4Þ

while the real system is given by

_x ¼ Axþ bu, ð5Þ

and the matrix DA of parameter uncertainties is

diagonal. Moreover, we assume that the relative error

in the parameters is smaller than 100%, i.e.

Dqi¼�DAii is such that jDqij � qi, i¼ 1, . . . , n. Tuning

the control law (3) for this nominal system yields


~uðxÞ ¼Xni¼1

bi

!�1 Xni¼1

qi þ Dqið Þxi þ � M � �MðxÞð Þ

!:

ð6Þ

It is proved in Sousa, Mendonca, and Rocha (2007)

that, for suitable values of the design parameter �,when the control law (6) is applied to (5), the

asymptotical values of the system mass lie in an

interval which is related to M � as stated in the nexttheorem.

Theorem 3.1: Let (5) be a fully outflow connectedcompartmental system, Dq¼max{jDqij} and take thedesign parameter � in (6) larger than Dq. Then, the statetrajectories x(t) of the closed loop system (5)–(6), witharbitrary initial conditions xð0Þ 2 R

nþ, converge to the

forward invariant set

� ¼ x 2 Rnþ : MðxÞ 2 IðM �Þ

� �,

with IðM �Þ ¼ ��þDqM

�, ��DqM

�h i

.

Remark 1:

(i) The set I(M �) is a neighbourhood of M �. Thisbounds the absolute mass offset by

max M� ��

�þDqM�,

�

��DqM� �M�

� �¼

Dq��Dq

M�,

leading to the bound Dq��Dq for the relative mass

offset. Clearly, this bound tends to zero whenDq goes to zero; this means that the controllaw is robust with respect to parameteruncertainty. Moreover, increasing the param-eter � contributes to increasing the robustnessof the control law.

(ii) When Dqi¼ 0, i¼ 1, . . . , n, we recover theresult stated in Theorem 2.1.

(iii) Using the same kind of techniques as inTheorem 3.1, it is possible to show that,under the same conditions, the state trajecto-ries x(t) of the closed-loop system (5)–(6), witharbitrary initial conditions xð0Þ 2 R

nþ, con-

verge to the forward invariant set

~� ¼ x 2 Rnþ : MðxÞ 2 ~IðM �Þ

� �,

where ~IðM �Þ ¼ ½ ��Dqmin

M �, ��Dqmax

M �� and

Dqmin¼min{Dqi}, Dqmax¼max{Dqi}, if thedesign parameter � is larger than Dqmax.This allows us to conclude that, in particular,if Dq1¼Dq2¼ � � � ¼Dqn¼D, since Dqmin¼

Dqmax¼D, the state trajectory x(t) convergesto the iso-mass �M, with M ¼ �

��DM�.

(iv) Other bounds for the relative mass offset canbe derived from set ~IðM �Þ), namely

maxDqmin

�� Dqmin

,Dqmax

�� Dqmax

� �:

3.2 Mass control in neuromuscular relaxantadministration

In this subsection, some simulation examples arepresented for the control of the administration of the

International Journal of Control 1389

Dow

nloa

ded

by [

Nor

th D

akot

a St

ate

Uni

vers

ity]

at 0

8:05

14

Nov

embe

r 20

14

neuromuscular relaxant drug atracurium to patients

undergoing surgery. It is possible to model thisproblem as a three-compartmental model that can be

described as depicted in Figure 2, where u is the druginfusion dose administered in the central compartment,

and k12, k21, k13, q3 are positive micro-rate constantsand q1, q2 are nonnegative micro-rate constants that

vary from patient to patient. In this case, the set ofequations (1) becomes

_x1 ¼ �ðk12 þ k13 þ q1Þx1 þ k21x2 þ u

_x2 ¼ k12x1 � ðk21 þ q2Þx2

_x3 ¼ k13x1 � q3x3

,

8><>: ð7Þ

where x1, x2 and x3 are the drug amounts in the central,peripheral and effect compartments, respectively. The

drug effect is given by ye¼ x3.We consider that the patient’s real model is

given by (7), with the following values for the

parameters (units¼min�1): k12¼ 4.3157, k13¼ 0.0017,k21¼ 15.1814, q1¼ 0.1047, q2¼ 0.1, q3¼ 0.0836.

Our aim is to stabilise the system mass on the value

M �¼ 72.0513 (which, in an exact modelling situation,

can be shown to correspond to the typical 10% level

of neuromuscular blockade, cf. Section 3.5), using thecontrol law (6). We start by taking the design param-

eter �¼ 0.2.In the first simulation, depicted in Figure 3, it is

assumed that the nominal patient model coincides withthe real one, i.e. Dqi¼ 0, i¼ 1, 2, 3. As expected, the

system mass converges to M �.Figure 4 shows the result of a simulation scenario,

where the Dqi’s are taken to be all equal, namely

Dq1¼Dq2¼Dq3¼ 0.03. In this case, illustratingRemark 1 (iii), the system mass reaches the set-point

M ¼ ��DM

� ¼ 84:7662.The simulations in Figure 5 correspond to the case

where the Dqi’s are different.Finally, Figure 6 illustrates the behaviour of the

mass of the controlled system for different values of the

parameter �, under fixed uncertainties for the systemparameters. In accordance with our theoretical results,one observes that the increasing of � corresponds tothe decrease of the final mass offset.

These simulations suggest that the asymptoticalvalues of the drug mass reach a constant value withinthe range which is expected according to Theorem 3.1.This issue will be investigated in the next subsection.

3.3 Convergence analysis

We shall now give a theoretical explanation for theresults obtained in the simulations of the

0 20 40 60 80 100 1200

10

20

30

40

50

60

70

80

90

100

110

Time (min)

Mas

s (µ

gkg

–1)

System massDesired massBounds of I(M*)

Figure 3. Simulation for the neuromuscular blockade con-trol, considering Dqi¼ 0, i¼ 1, 2, 3.

0 20 40 60 80 100 1200

10

20

30

40

50

60

70

80

90

100

110

Time (min)

Mas

s (µ

gkg

–1)


Figure 4. Simulation for the neuromuscular blockade con-trol, considering Dqi¼ 0.03, i¼ 1, 2, 3.

Figure 2. Compartmental model for the effect of drugadministration.


Dow

nloa

ded

by [

Nor

th D

akot

a St

ate

Uni

vers

ity]

at 0

8:05

14

Nov

embe

r 20

14

previous subsection. It is stated in Magalhaes et al.(2005) that, in the absence of uncertainties, the appli-cation of the control law (6) to a system of the form (7)not only leads the mass to a certain value M �, but alsoleads the whole system state to an equilibriumpoint xM

�

. Here, we shall prove that a similar resultstill holds under the presence of uncertainties. Moreconcretely, under certain conditions, the nominal con-trol law under consideration leads the state of the realsystem to an equilibrium point �x and consequently itstotal mass to a constant valueMwhich are, respectively,related to the equilibrium state xM

�

and the desiredmassvalue M �. As we shall later see, the convergence to aspecific state equilibrium point is an important propertyin the context of the control of the neuromuscularblockade level.

Although we focus on the administration of amuscular relaxant, our results are also valid for othercompartmental systems with the same structure.The next proposition is useful in the proof of ourmain result.

Proposition 3.2: Let Dq be defined as in Theorem 3.1,

Dqmin and Dqmax be defined as in Remark 1 and

define qmin¼min{qi}, qmax¼max{qi}. Assume that

qminþDqmin4Dq and take the design parameter �in (6) larger than qmaxþDqmax. Then, when the control

law (6) is applied to (7), there exists an instant t140

such that, for t� t1,

uðxðtÞÞ ¼euðxðtÞÞ � 0:

Proof: According to Theorem 3.1, when the control

law (6) is applied to (7), the asymptotical values of the

system mass lay in the interval IðM �Þ ¼

½ ��þDqM

�, ��DqM

��, provided that the design parameter

� in (6) is larger than Dq. This implies that, for every

"40, there exists an instant t"40 such that

MðxðtÞÞ 2�

�þ DqM � � ",

�

�� DqM � þ "

� �,

for t� t".

0 20 40 60 80 100 1200

10

20

30

40

50

60

70

80

90

100

110

Time (min)

Mas

s (µ

gkg

–1)

system massdesired massbounds of I(M*)

0 20 40 60 80 100 1200

10

20

30

40

50

60

70

80

90

100

110(b)(a)

Time (min)

Mas

s (µ

gkg

–1)

system massdesired massbounds of I(M*)

Figure 6. Simulations for the neuromuscular blockade control. These simulations where obtained considering Dq1¼�0.02,Dq2¼ 0 and Dq3¼�0.01, and different values of �. (a) Simulation for �¼ 0.2. (b) Simulation for �¼ 2.

0 20 40 60 80 100 1200

10

20

30

40

50

60

70

80

90

100

110

Time (min)

Mas

s (µ

gkg

–1)


(a)

0 20 40 60 80 100 1200

10

20

30

40

50

60

70

80

90

100

110

Time (min)

Mas

s (µ

gkg

–1)


(b) (c)

0 20 40 60 80 100 1200

10

20

30

40

50

60

70

80

90

100

110

Time (min)

Mas

s (µ

gkg

–1)


Figure 5. Simulations for the neuromuscular blockade control. (a) Simulation with Dq1¼ 0.02, Dq2¼ 0 and Dq3¼ 0.05; thesystem mass lays asymptotically in the interval I(M �)¼ [57.6410, 96.0684]. (b) Simulation with Dq1¼�0.01, Dq2¼�0.04 andDq3¼�0.02; the system mass lays asymptotically in the interval I(M �)¼ [60.0427, 90.0641]. (c) Simulation with Dq1¼ 0.06,Dq2¼ 0.01 and Dq3¼�0.01; the system mass lays asymptotically in the interval I(M �)¼ [55.4241, 102.9304].


Dow

nloa

ded

by [

Nor

th D

akot

a St

ate

Uni

vers

ity]

at 0

8:05

14

Nov

embe

r 20

14

Note that

~uðxÞ � 0,X3i¼1

qi þ Dqið Þxi þ � M � �MðxÞð Þ � 0

,X3i¼1

�� qi � Dqið Þxi � �M�:

Since �4qmaxþDqmax, it follows that �� qmin�

Dqmin40. Thus, for t� t",X3i¼1

�� qi � Dqið ÞxiðtÞ

� �� qmin � Dqmin

�

�� DqM � þ "

� �,

and

�� qmin � Dqmin

�

�� DqM � þ "

� �� M �

, " ��

�� qmin � DqminM � �

�

�� DqM �:

Since qminþDqmin4Dq and �4qmaxþDqmax, it is easy

to show that ��qmin�Dqmin

M � � ��DqM

�4 0. Thus, if

we take " ¼ ��qmin�Dqmin

M � � ��DqM

� and t1¼ t", then

~u(x(t))� 0, for t� t1. œ

Let b in Equation (4) be given by b¼ [1 0 0]T and

consider the following point:

�x ¼ �1 �x3 �2 �x3 �x3½ �T,

where

�1 ¼q3k13

�2 ¼q3k12

q2 þ k21ð Þk13

�x3 ¼�M �

�� Dq1ð Þ�1 þ �� Dq2ð Þ�2 þ �� Dq3ð Þ:

Proposition 3.3: Let Dq, qmin and qmax be defined as

in Proposition 3.2. If qminþDqmin4Dq and the design

parameter � in (6) is larger than qmaxþDqmax, then �x

is the only equilibrium point of the closed-loop

system (6)–(7) that belongs to the set

fx 2 R3þ : dMðxÞ

dt ¼ 0g:

Proof: According to Proposition 3.2, there exists an

instant t140 such that, for t� t1, u(x(t))¼ ~u(x(t))� 0.

Thus, for t� t1,

dMðxðtÞÞ

dt¼ 0,

X3i¼1

DqixiðtÞ þ � M � �MðxðtÞÞð Þ ¼ 0

, �� Dq1ð Þx1ðtÞ þ �� Dq2ð Þx2ðtÞ

þ �� Dq3ð Þx3ðtÞ ¼ �M�

and the equilibrium points of the controlled

closed-loop system (6)–(7) that belong to the set fx 2

R3þ : dMðxÞ

dt ¼ 0g are the solutions of the system

Axþ bu ¼ 0dMðxÞ

dt¼ 0

8<: : ð8Þ

But, in this case,

uðxÞ ¼ ~uðxÞ

¼X3i¼1

bi

!�1 X3i¼1

qiþDqið Þxiþ� M� �MðxÞð Þ

!

¼X3i¼1

qixi

¼ q1q2 q3½ �x,

thus, (8) becomes

Aþb q1 q2q3½ �ð Þx¼ 0

��Dq1ð Þx1ðtÞþ ��Dq2ð Þx2ðtÞþ ��Dq3ð Þx3ðtÞ ¼ �M�

�:

and it is easy to verify that x is the only solution of this

system. œ

Theorem 3.4: Let Dq, qmin and qmax be defined as in

Proposition 3.2. Assume that qminþDqmin4Dq and take

the design parameter � in (6) larger than qmaxþDqmax.

Then, the state trajectories x(t) of the closed-loop

system (6)–(7), with arbitrary initial conditions

xð0Þ 2 R3þ, converge to the equilibrium point x.

Corollary 3.5: Under the same conditions of the

previous result, the system mass converges to the total

mass M of the system in �x, that is, to:

M ¼ �1þ�2þ1ð Þ�M �

��Dq1ð Þ�1þ ��Dq2ð Þ�2þ ��Dq3ð Þ:

In the following, we prove Theorem 3.4.

Proof: Let �x be the equilibrium point mentioned in

Proposition 3.3 and

M ¼ �1þ�2þ1ð Þ�M �

��Dq1ð Þ�1þ ��Dq2ð Þ�2þ ��Dq3ð Þ

be the total mass of the system in �x.Take t140 such that, for t� t1, u(x(t))¼ ~u(x(t))� 0

(the existence of such an instant is guaranteed by

Proposition 3.2). Taking into account that

M � ¼�� Dq1ð Þ�1 þ �� Dq2ð Þ�2 þ �� Dq3ð Þ

�1 þ �2 þ 1ð Þ�M

¼M�Dq1�1 þ Dq2�2 þ Dq3

�1 þ �2 þ 1ð Þ�M,


Dow

nloa

ded

by [

Nor

th D

akot

a St

ate

Uni

vers

ity]

at 0

8:05

14

Nov

embe

r 20

14

for t� t1 we have

uðxðtÞÞ ¼ ~uðxðtÞÞ¼X3i¼1

qiþDqið ÞxiðtÞþ� M� �MðxðtÞÞð Þ

¼X3i¼1

qixiðtÞþX3i¼1

Dqi xiðtÞ� �xið Þþ� M�MðxðtÞÞ

¼ q1q2 q3½ �xðtÞþ Dq1Dq2Dq3½ � xðtÞ� �xð Þ

þ� 111½ � �x�xðtÞð Þ

and

xðtÞ� �xð Þzfflfflfflfflffl}|fflfflfflfflffl{:

¼AxðtÞþbu� Aþb q1q2 q3½ �ð Þ �x

¼AxðtÞþb q1 q2 q3½ �xðtÞ½

þ Dq1Dq2Dq3½ � xðtÞ� �xð Þþ� 111½ � �x�xðtÞð Þ�

� Aþb q1 q2 q3½ �ð Þ �x

¼ Aþb q1 q2 q3½ �þb Dq1Dq2Dq3½ �ð

�b� 111½ �Þ xðtÞ� �xð Þ

¼ �A x� �xð ÞðtÞ:

Since it can be easily seen (using the Routh–Hurwitzstability criterion) that, for �4qmaxþDqmax, all theeigenvalues of �A lie in C

-, it turns out that �A isasymptotically stable and hence x� �xð ÞðtÞ ! 0 or,equivalently, xðtÞ ! �x. œ

3.4 Convergence improvement

Assuming that the conditions of Theorem 3.4 aresatisfied, the application of the nominal control law tothe system (7) makes the system mass converge to aconstant positive value

M ¼�1 þ �2 þ 1ð Þ�M �

�� Dq1ð Þ�1 þ �� Dq2ð Þ�2 þ �� Dq3ð Þð9Þ

that may be different from M �.In case the system mass reaches the value (9) in a

finite time t�, this value will be known from that timeinstant on. This fact can be used in order to redesignthe controller and achieve convergence of the mass tothe value M �. Indeed, if for t� t�, the new control law


~uðxÞ ¼X3i¼1

qi þ Dqið Þxi þ � K�MðxÞð Þ,ð10Þ

with K ¼ M �ð Þ2

M, is applied, the system mass converges

to M �, because, in this case, (9) becomes

�1 þ �2 þ 1ð Þ�K

�� Dq1ð Þ�1 þ �� Dq2ð Þ�2 þ �� Dq3ð Þ

¼�1 þ �2 þ 1ð Þ� M �ð Þ

2

M

�� Dq1ð Þ�1 þ �� Dq2ð Þ�2 þ �� Dq3ð Þ¼M �:

If the system mass does not reach the value (9) in afinite time, it will not be possible to follow the previousprocedure. Nevertheless, it is still possible to adapt ourcontrol law, in order to guarantee that the asympto-tical mass will be closer to the desired value, byreplacing M by ~M ¼MðtÞ, with t large enough toensure that ~M is close to M. Thus, if the control law


~uðxÞ ¼X3i¼1

qi þ Dqið Þxi þ � L�MðxÞð Þ,

with L ¼ M �ð Þ2

~M, is applied, Corollary 3.5 guarantees

that, in this case, the total system mass converges to

�1 þ �2 þ 1ð Þ� M �ð Þ2

~M

�� Dq1ð Þ�1 þ �� Dq2ð Þ�2 þ �� Dq3ð Þ

and, since ~M M, we have

�1 þ �2 þ 1ð Þ� M �ð Þ2

~M

�� Dq1ð Þ�1 þ �� Dq2ð Þ�2 þ �� Dq3ð ÞM �:

In the sequel, some simulation examples arepresented for the control of the administration of theneuromuscular relaxant drug atracurium to patientsundergoing surgery. The values of the parameters areexactly the same as in Section 3.2.

Our aim is to stabilise the system mass on the valueM �¼ 72.0513 and we start by taking the design

parameter �¼ 0.2. Figure 7 illustrates the behaviourof the mass of the controlled system for this value of �and different values for Dqi. Figure 8 illustrates thebehaviour of the mass of the controlled system fordifferent values of the parameter �, under fixeduncertainties for the system parameters.

According to the previous study, when we changeour control law the system mass is brought near to thedesired value M �.

The simulations presented in this section as well asin Section 3.2 were also performed under the presenceof noise, with a log-normal distribution with zero meanand variance equal to 0.1, leading similar results, thatis, a good tracking of the total mass reference.

3.5 Control of the neuromuscular blockade level

In Magalhaes et al. (2005), the control law (3) wasapplied for the control of the neuromuscular blockadelevel of patients undergoing surgery, by means of theinfusion of atracurium. In the absence of uncertainties(i.e. if Dqi¼ 0, i¼ 1, 2, 3), it follows from Theorem 3.4that, considering

M � ¼ �1 þ �2 þ 1ð Þ yeref,


Dow

nloa

ded

by [

Nor

th D

akot

a St

ate

Uni

vers

ity]

at 0

8:05

14

Nov

embe

r 20

14

the application of the control law (3)–(5) leads the

system state to an equilibrium point

xM�

¼ xM�

1 xM�

2 xM�

3

� �T,

where the third component, that corresponds to the

neuromuscular blockade effect, is given by xM�

3 ¼ yeref.

Furthermore, notice that the relation between the

effect concentration ye and the neuromuscular block-

ade level r(t) is given by the Hill equation (Magalhaes

et al. 2005)

rðtÞ ¼100C �

50

yeðtÞð Þ�þC �

50

,

where C50 and � are patient dependent parameters.

In order to obtain a set-point r(t)¼ ref, the corre-

sponding effect concentration yeref is given by

yeref ¼ C50100

ref� 1

� �1=�

:

Thus, it is possible to control the neuromuscular

blockade level via total mass control.

In the presence of uncertainties, a control law (6)

for the nominal system is designed considering

M � ¼ �1nominal þ �2 nominal þ 1ð Þ yeref,

where �inominal refers to the value of �i for the nominal

system. Clearly, if this control law was applied to the

nominal system, we would be able to guarantee that

the state trajectories would converge to an equilibrium

point whose third component would be equal to the

desired reference. However, according to Theorem 3.4,

the application of this control law to the real system

will make the state trajectories converge to the equi-

librium point

�x ¼ �1 �x3 �2 �x3 �x3½ �T,

with

�1 ¼q3k13

�2 ¼q3k12

q2k13 þ k21k13

�x3 ¼�M �

�� Dq1ð Þ�1 þ �� Dq2ð Þ�2 þ �� Dq3ð Þ:

0 100 200 300 400 500 600 700 800 90010000

10

20

30

40

50

60

70

80

90

100

Time (min)

Mas

s (µ

gkg

–1)

System massDesired massInitial asymptotical mass

(a)

0 100 200 300 400 500 600 700 800 90010000

10

20

30

40

50

60

70

80

90

100

Time (min)

Mas

s (µ

gkg

–1)


(b)

Figure 8. Simulations for the neuromuscular blockade control. These simulations where obtained considering Dq1¼�0.02,Dq2¼ 0 and Dq3¼�0.01, and different values of �. (a) Simulation for �¼ 0.2. (b) Simulation for �¼ 2.

0 100 200 300 400 500 600 700 800 90010000

10

20

30

40

50

60

70

80

90

100

Time (min)

Mas

s (µ

gkg

–1)


(a)

0 100 200 300 400 500 600 700 800 90010000

10

20

30

40

50

60

70

80

90

100

Time (min)

Mas

s (µ

gkg

–1)


(b)

Figure 7. Simulations for the neuromuscular blockade control. (a) Simulation with Dqi¼ 0.03, i¼ 1, 2, 3. (b) Simulation withDq1¼ 0.02, Dq2¼ 0 and Dq3¼ 0.05.


Dow

nloa

ded

by [

Nor

th D

akot

a St

ate

Uni

vers

ity]

at 0

8:05

14

Nov

embe

r 20

14

Since M � ¼ �1 nominal þ �2 nominal þ 1ð Þ yeref, it fol-lows that

�x3 � yeref�� ¼

� �1 nominal þ �2 nominal þ 1ð Þ yeref�� Dq1ð Þ�1 þ �� Dq2ð Þ�2 þ �� Dq3ð Þ

� yeref

�� ¼ yeref

� �1 nominal þ �2 nominal � �1 � �2ð Þ

þ Dq1�1 þ Dq2�2 þ Dq3

� �� Dq1ð Þ�1 þ �� Dq2ð Þ�2 þ �� Dq3ð Þ

��

��and clearly �x3! yeref when the parameter uncertaintiesgo to zero; moreover, increasing the parameter �contributes to increase the robustness of thecontrol law.

3.6 Clinical case

A study was approved by the Ethics Committee ofHospital Geral de Santo Antonio (Porto, Portugal) inorder to evaluate and compare the performance ofdifferent control strategies. At this moment, 60 patients(with health features levels I to IV according to theAmerican Society of Anesthesiology (ASA)) haveundergone elective surgery with automatic control ofneuromuscular blockade.

The control system is easy to set up in a clinicalenvironment and consists of a sensor (Datex AS/5NMT– NeuroMuscular Transmission monitor to measurethe neuromuscular blockade level), a delivery device(perfusion compact B – Braun) and a PC compatiblecomputer with two serial ports for connecting the NMTsensor and the perfusion pump. After calibration of theNMT module, a 500 mg kg�1 bolus of atracurium wasadministered. The atracurium infusion rate was updatedevery 20 s based on the r(t) input signal (via RS232Cserial port). The control actions begins a few minutesafter the bolus administration. In order to ensure

patient safety, the whole procedure has been followedby an anaesthesiologist who also guided the induction(first minutes under open-loop control) and recoveryphases using standard practices.

Figure 9 illustrates the performance of the pro-posed control scheme in a clinical situation. Theclinical performance was considered very good, sincethere is a very good tracking of the reference.

4. Conclusion

This article presents an analysis of the performance ofthe control law (3) proposed in Bastin and Provost(2002) when applied to the control of compartmentalsystems with parameter uncertainties.

This study was motivated by the need to apply thislaw to the control of physiological variables, namelythe control of neuromuscular blockade in patientsunder general anaesthesia. This is a situation wererobustness is a relevant issue, since patient models arehighly subject to uncertainties.

As major results, we showed that the asymptoticalmass values converge to an interval whose bounds canbe expressed in terms of the parameter uncertaintiesand of the desired mass, which allowed to concludethat the considered control law is robust underparameter uncertainty. Although mass convergencedoes not necessarily imply the convergence of the stateof the system, we were able to prove that for a class ofcompartmental systems in R

3 of interest, the state ofthe controlled system tends to an equilibrium pointwhose total mass lies within the aforementionedinterval. Taking into account the relationship betweenthe mass and the state components in steady state, weused the proposed mass control law to track desiredvalues for the steady state components. In this way weobtained a novel method for steady-state control viamass control. This work contributes to setting a

Figure 9. Control of the administration of the neuromuscular relaxant drug atracurium to patients undergoing surgery(clinical situation).


Dow

nloa

ded

by [

Nor

th D

akot

a St

ate

Uni

vers

ity]

at 0

8:05

14

Nov

embe

r 20

14

theoretical foundation for the practical implementa-tion of control procedure. Indeed, this was applied tothe control of the neuromuscular blockade level ofpatients undergoing surgery, by means of the infusionof atracurium. Our results are illustrated by severalsimulations and a clinical case.

Since the considered control law relies on the statecomponents, in order to fully empower our controlmethod, it is necessary to consider the situation wherealso the state needs to be estimated. This is a subjectunder current investigation.

Acknowledgements

This work was partially supported by FCT through theUnidade de Investigacao Matematica e Aplicacoes (UIMA),Universidade de Aveiro, Portugal.

References

Bastin, G., and Provost, A. (2002), ‘Feedback Stabilisationwith Positive Control of Dissipative Compartmental

Systems’, in Proceedings of the 15th InternationalSymposium on Mathematical Theory of Networks and

Systems, Notre Dame, Indiana, USA, Paper 14900-3 inSession MA5 .

Godfrey, K. (1983), Compartmental Models and Their

Application, London: Academic Press.

Haddad, W., Hayakawa, T., and Bayley, J. (2003), ‘AdaptiveControl for Non-negative and Compartmental Dynamical

Systems with Applications to General Anesthesia’,International Journal of Adaptive Control and SignalProcessing, 17, 209–235.

Jacquez, J., and Simon, C. (1993), ‘Qualitative Theory of

Compartmental Systems’, SIAM Review, 35, 43–79.LaSalle, J.P. (1976), The Stability of Dynamical Systems,Bristol, England: SIAM.

Lemos, J., Mendonca, T., and Mosca, E. (1991), ‘Long-rangeAdaptive Control with Input Constraints’,International Journal of Control, 54, 289–306.

Linkens, D. (1994), Intelligent Control in Biomedicine,London: Taylor and Francis.

Magalhaes, H., Mendonca, T., and Rocha, P. (2005),‘Identification and Control of Positive and

Compartmental Systems Applied to NeuromuscularBlockade’, in Preprints of the 16th IFAC World Congress,Prague, Czech Republic, pp. 1–6.

Mendonca, T., and Lago, P. (1998), ‘PID Control Strategiesfor the Automatic Control of Neuromuscular Blockade’,Control Engineering Practice, 6, 1225–1231.

Sandberg, W. (1978), ‘On the Mathematical Foundationsof Compartmental Analysis in Biology, Medicine andEcology’, IEEE Transactions on Circuits and Systems, 25,

273–279.Sousa, C., Mendonca, T., and Rocha, P. (2007), ‘Control ofUncertain Compartmental Systems’, in Proceedings of the15th Mediterranean Conference on Control and Automation,

Atheens, Greece, pp. 1–6.


Dow

nloa

ded

by [

Nor

th D

akot

a St

ate

Uni

vers

ity]

at 0

8:05

14

Nov

embe

r 20

14

Download - Target mass control for uncertain compartmental systems

Top Related