target mass control for uncertain compartmental systems
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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
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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: cnsousa@gaia.ipiaget.org
ISSN 0020–7179 print/ISSN 1366–5820 online
� 2010 Taylor & Francis
DOI: 10.1080/00207171003736311
http://www.informaworld.com
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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).
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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 ¼ max 0, ~uðxÞð Þ
~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
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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)
System massDesired massBounds of I(M*)
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.
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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)
System massDesired massBounds of I(M*)
(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)
System massDesired massBounds of I(M*)
(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)
System massDesired massBounds of I(M*)
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].
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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,
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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 ¼ max 0, ~uðxÞð Þ
~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 ¼ max 0, ~uðxÞð Þ
~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,
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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)
System massDesired massInitial asymptotical mass
(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)
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)
System massDesired massInitial asymptotical mass
(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.
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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).
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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.
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