\ A study of the performance of mass control for compartmental systems, under the presence of uncertainties, is presented. Bounds for the asymptotical mass offset are derived as a functional of the uncertainty bounds. The obtained results are illustrated by simulations for the case of neuromuscular blockade control of patients undergoing surgery

Mendonca, T

Rocha, P

Sousa, C

Open Repository of the University of Porto

Control of uncertain compartmental systems
C. Sousa∗, T. Mendonça†, P. Rocha‡
∗Escola Superior de Tecnologia e Gestão - Instituto Politécnico de Leiria/Departamento de Matemática, Leiria, Portugal
†Faculdade de Ciências - Universidade do Porto/Departamento de Matemática Aplicada, Porto, Portugal
‡Universidade de Aveiro/Departamento de Matemática, Aveiro, Portugal
Abstract— A study of the performance of mass control for
compartmental systems, under the presence of uncertainties,
is presented. Bounds for the asymptotical mass offset are
derived as a functional of the uncertainty bounds. The
obtained results are illustrated by simulations for the case
of neuromuscular blockade control of patients undergoing
surgery.
Index Terms—Compartmental systems, positive control,
uncertain systems, neuromuscular blockade control.
I. INTRODUCTION
Compartmental models have been successfully used to
model biomedical and pharmacokinetical systems, see, for
instance, [2] or [4]. Compartmental systems consist of a
finite number of subsystems, the compartments, which
exchange matter with each other and with the environ-
ment. Such systems form a subclass of positive systems
(i.e., systems for which the state and output variables
remain nonnegative whenever the input is nonnegative).
As is well-known, in this case, the design of suitable
control laws is more delicate, since one has to guarantee
the positivity of the control input. In this framework,
a nonnegative adaptive control law is proposed in [3],
in order to guarantee the partial asymptotic set-point
stability of the closed loop system. On the other hand,
a positive feedback control law is proposed in [1], in
order to stabilise the total system mass at an arbitrary
set-point. This law was used in [8] for the control of the
neuromuscular blockade (see [6], [7] and [9]) of patients
undergoing surgery, but no analysis was made of the effect
of parameter uncertainty in its performance.
In this paper, we try to fill in this gap, not only for
the neuromuscular blockade model, but for a wide class
of compartmental systems. 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 behavior of the total mass in
the controlled system. It turns out that, in this case, bounds
for the asymptotical mass offset can be easily expressed
in terms of the system uncertainties.
II. COMPARTMENTAL SYSTEMS
Compartmental systems are dynamical systems de-
scribed by a set of equations of the form
x˙i =
∑
j =i fji(x) −
∑
l =i fil(x)− fi0(x) + f0i(x)
i = 1, . . . , n
(see [2] or [10]) where x = (x1, . . . , xn)T is the state
variable and xi and fij take nonnegative values. Each
equation describes the evolution of the quantity or con-
centration of material within a subsystem, called compart-
ment. Since the compartments exchange with each other
and with the environment, in the above equation, xi is the
amount (or concentration) of material in compartment i,
fij is the flow rate from compartment j to compartment i
and the subscript 0 denotes the environment (see [2]). In
this paper, we consider the class of linear time-invariant
compartmental systems described by
x˙i =
∑
j =i
kjixj−
∑
l =i
kilxi−qixi+biu, i = 1, . . . , n, (1)
where xi and the input u take nonnegative values,
kij , qi, bi ∈ R+ and at least one bi is positive (see
Fig. 1). Note that, in this case, fji = kjixj , f0i = biu
and fi0 = qixi, and it can be easily proved that the
system is positive, this is, if we consider an input u that
remains nonnegative, then the state variable also remains
nonnegative. Moreover, (1) can also be written in matrix
form as
x˙ = Ax + bu, (2)
where A (called compartmental matrix) is so that
aii = −qi −
∑
j =i
kij and, if i = j, aij = kji,
and b = (b1, b2, . . . , bn)T .
The total mass of the system in a given state x is
defined as M(x) =
∑n
i=1 xi. For an arbitrary positive
value M∗, the set ΩM∗ =
{
x ∈ Rn+ : M(x) = M∗
}
of
all the points x in the state space with mass M∗ is called
an iso-mass.
An important issue in the context of the control of
compartmental systems is to design a control law which
yields a positive input that steers the system mass M(x)
to a desired value.
In [1], the following positive control law is proposed
to guarantee that the trajectories converge to a set ΩM∗ :
   b i   b j 
   q i    q j
to/from other 
compartments
to/from other 
compartments
k j i
k i j
i j
Fig. 1. Two compartments of a linear time-invariant compartmental
model, as described by (1).
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u(x) = max (0, u˜(x))
u˜(x) = (
∑
n
i=1 bi)
−1 (
∑
n
i=1 qixi + λ (M
∗ −M(x))) ,
(3)
where λ is an arbitrary design parameter.
In order to state the corresponding theorem, we need
to introduce the following concept of full outflow con-
nectedness. A compartmental system (1) is said to be
fully outflow connected if at every state x there is a path
i → j → k → · · · → l with positive kij ’s from every
compartment i to some compartment l such that ql > 0
(see [1]).
Theorem 1: [1] Let (2) be a fully outflow connected
compartmental system. Then, for the closed loop system
(2)-(3) with arbitrary initial conditions x(0) ∈ Rn+:
i) the iso-mass ΩM∗ is forward invariant;
ii) the state x(t) is bounded for all t > 0 and converges
to the iso-mass ΩM∗ .
The proof of this result is based on the application
of LaSalle’s invariance principle (see [5], pg.30), by
considering the Lyapunov function
V (x) =
1
2
(M∗ −M(x))2
of (2) on Rn+.
In [8], the control law (3) was applied for the control
of the neuromuscular blockade of patients undergoing
surgery, by means of the infusion of atracurium. In
fact, it is possible to model this problem as a three
compartmental model that can be described as depicted in
Fig. 2, where u is the drug infusion dose administered in
the central compartment, and k12, k21, k13, q3 are positive
micro-rate constants and q1, q2 are nonnegative micro-rate
constantes that vary from patient to patient. In this case,
the set of equations (1) becomes

x˙1 = −(k12 + k13 + q1)x1 + k21x2 + u
x˙2 = k12x1 − (k21 + q2)x2
x˙3 = k13x1 − q3x3
(4)
where x1, x2 and x3 are the drug amounts in the central,
peripheral and effect compartments, respectively. How-
ever, even after a satisfactory identification of the patients
characteristics, it was necessary to consider an additional
integrator, in order to achieve good results. This might be
explained by the fact that (contrary to what happens, for
   k 
1 3
   q 
1
   q 
2
k 
2 1
k 
1 2
Central Compartment 
(C1)
Peripheral Compartment 
(C2)
b 
1
=1
Effect Compartment 
(C3)
   q 
3
Fig. 2. Compartmental model for the neuromuscular blockade effect
of the drug atracurium.
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.
III. UNCERTAIN COMPARTMENTAL SYSTEMS
In this section, we analyse the performance of the
control law (3), proposed in [1], in the presence of
parameter uncertainties.
If we consider that we can measure precisely what is
injected from the outside into the system, the parameters
bi are not subject to uncertainties. On the other hand, since
the control law does not depend on the kij’s, we may
assume that the only uncertain parameters are q1, . . . , qn.
Therefore, we shall assume that a control law (3) is
designed for a nominal system
x˙ = (A + ∆A)x + bu, (5)
while the real system is given by
x˙ = Ax + bu, (6)
being
∆A = diag(−∆q1,−∆q2, . . . ,−∆qn)
the matrix of parameter uncertainties. Note that, in this
case, the control law (3) becomes
u(x) = max (0, u˜(x))
u˜(x) = (
∑
n
i=1 bi)
−1 (
∑
n
i=1 (qi + ∆qi)xi+
+λ (M∗ −M(x))) .
(7)
It turns out that, for suitable values of the design
parameter λ, when the control law (7) is applied to (6), the
asymptotical values of the system mass lay in an interval
which is related to M∗ as stated in the next theorem.
Theorem 2: Let (6) be a fully outflow connected com-
partmental system, ∆qmin = min {∆qi}, ∆qmax =
max {∆qi} and take the design parameter λ in (7) larger
than ∆qmax. Then, the state trajectories x(t) of the
closed loop system (6)-(7), with arbitrary initial condi-
tions x(0) ∈ Rn+, converge to the forward invariant set
Ω =
{
x ∈ Rn+ : M(x) ∈ I(M∗)
}
,
with I(M∗) =
[
λ
λ−∆qminM
∗, λλ−∆qmaxM
∗
]
.
Proof: Let Mmin = λλ−∆qminM∗ and Mmax =
λ
λ−∆qmaxM
∗. Consider the function V : Rn → R defined
by
V (x) =


1
2
(
M(x)−Mmin
)2 if M(x) < Mmin
1
2
(
M(x)−Mmax
)2 if M(x) > Mmax
0 otherwise.
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Note that V is a Lyapunov function of the system on Rn+
because it is continuous and V˙ (x) ≤ 0, ∀x ∈ Rn+ (see
[5], pg.30). In fact, if x ∈ Rn+,
V˙ (x) =


(
M(x)−Mmin
) dM(x)
dt if M(x) < Mmin(
M(x)−Mmax
) dM(x)
dt if M(x) > Mmax
0 otherwise
is nonpositive, as we next show.
• Suppose that M(x) < Mmin. Since∑n
i=1 (qi + ∆qi)xi + λ (M
∗ −M(x)) ≥
≥∑n
i=1 qixi + ∆qminM(x) + λ (M
∗ −M(x))
=
∑
n
i=1 qixi + (λ−∆qmin)
(
Mmin −M(x)
)
> 0,
it follows that u˜(x) > 0. Thus,
u(x) = u˜(x)
= (
∑
n
i=1 bi)
−1 (
∑
n
i=1 (qi + ∆qi) xi+
+λ (M∗ −M(x))) .
In this case,
dM(x)
dt =
∑n
i=1 ∆qixi + λ (M
∗ −M(x))
≥ ∆qminM(x) + λ (M∗ −M(x))
= (λ−∆qmin)
(
Mmin −M(x)
)
> 0
and
V˙ (x) =
(
M(x)−Mmin
) dM(x)
dt
< 0.
• Suppose that M(x) > Mmax. If u˜(x) < 0, u(x) = 0
and
dM(x)
dt = −
∑n
i=1 qixi ≤ 0.
Thus, in this case,
V˙ (x) =
(
M(x)−Mmax
) dM(x)
dt
≤ 0.
If u˜(x) ≥ 0, since u(x) = u˜(x), it follows that
dM(x)
dt =
∑n
i=1 ∆qixi + λ (M
∗ −M(x)) .
≤ ∆qmaxM(x) + λ (M∗ −M(x))
= (λ−∆qmax)
(
Mmax −M(x)
)
< 0
and
V˙ (x) =
(
M(x)−Mmax
) dM(x)
dt
< 0.
Applying LaSalle’s invariance principle (see [5], pg.30),
it turns out that x(t) converges to the largest invariant set
contained in{
x ∈ Rn+ : V˙ (x) = 0
}
= I1 ∪ I2,
where
I1 =
{
x ∈ Rn+ : M(x) ∈ I(M∗)
}
and
I2 =
{
x ∈ Rn+ : u(x) = 0,
∑n
i=1 qixi = 0 and
M(x) > Mmax
}
.
It can be shown that the set I2 has no invariant subset.
Indeed, if this would be the case, u would be zero and
the assumption of full outflow connectedness would imply
that M(x) → 0, meaning that x would leave the subset
(see [1]). On the other hand, I1 is an invariant set, which
concludes the proof of the theorem.
Remark 3:
i) If ∆q1 = ∆q2 = . . . = ∆qn = ∆q, since
∆qmin = ∆qmax = ∆q, the state trajectory
x(t) converges to the iso-mass ΩM , with M =
λ
λ−∆q M
∗
.
ii) When ∆qi = 0, i = 1, . . . , n, we recover the
result from Theorem 1.
Using the same kind of techniques as in Theorem 2, it
is possible to show that:
Proposition 4: Let (6) be a fully outflow connected
compartmental system. Then, the state trajectories x(t)
of the closed loop system (6)-(7), with arbitrary initial
conditions x(0) ∈ Rn+, converge to the forward invariant
set
Ω˜ =
{
x ∈ Rn+ : M(x) ∈ I˜(M∗)
}
,
where I˜(M∗) =
[
λ
λ+∆qM
∗, λλ−∆qM
∗
]
and ∆q =
max {|∆qi|}, if the design parameter λ is taken to be
larger than ∆q.
Note that, contrary to what happens with the set I(M∗)
in Theorem 2 (which may even not contain the value M∗),
the set I˜(M∗) in Proposition 4 is a neighborhood of M∗.
This allows to bound the absolute mass offset by
max
{
M∗ − λλ+∆qM∗, λλ−∆qM∗ −M∗
}
=
= ∆qλ−∆qM
∗,
leading the bound ∆qλ−∆q for the relative mass offset.
Clearly, increasing the parameter λ contributes to increase
the robustness of the control law.
Remark 5: Note that other bounds for the relative
mass offset can be derived from set I(M∗), namely
max
{ |∆qmin|
λ−∆qmin ,
|∆qmax|
λ−∆qmax
}
.
IV. SIMULATIONS
In this section, some simulation examples are presented
for the neuromuscular blockade control. We consider that
the patient’s real model is given by (4), with the following
values for the parameters (units = min−1): k12 =
0.1928, k13 = 0.0017, k21 = 0.1556, q1 = 0.1047, q2 =
0, q3 = 0.0836.
Our aim is to stabilise the system mass on the value
M∗ = 1.1169 (corresponding to a 10% level of neuro-
muscular blockade), using the control law (7). We start
by taking the design parameter λ = 0.2.
In the first simulation, depicted in Fig. 3, it is assumed
that the nominal patient model coincides with the real
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one, i.e., ∆qi = 0, i = 1, 2, 3. As expected, the system
mass converges to M∗.
Fig. 4 shows the result of a simulation scenario, where
the ∆qi’s are taken to be all equal, namely ∆q1 = ∆q2 =
∆q3 = 0.03. In this case, illustrating Remark 3i), the
system mass reaches the set-point M = λλ−∆q M
∗ =
1.3141.
The simulations in Fig. 5 correspond to the case where
the ∆qi’s are different. The three illustrated cases corre-
spond to the following situations: M∗ lies outside I(M∗),
M∗ lies in the interval I(M∗) and M∗ coincides with one
of the bounds of this interval.
Fig. 6 illustrates the case where not only the qi’s are
subject to uncertainties, but also the other parameters,
showing that the controller performance only depends on
the ∆qi’s.
Finally, Fig. 7 illustrates the behavior of the mass of
the controlled system for different values of the parameter
λ, under a fixed uncertainty for the system parameters.
According to the definition of I(M∗), one observes that
the increasing of λ corresponds to the decrease of the
final mass offset.
0 20 40 60 80 100 120
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Time (min)
M
as
s (
µg
/k
g)
 
 
system mass
desired mass
bounds of I(M*)
Fig. 3. Simulation for the neuromuscular blockade control, considering
∆qi = 0, i = 1, 2, 3.
0 20 40 60 80 100 120
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Time (min)
M
as
s (
µg
/k
g)
 
 
system mass
desired mass
bounds of I(M*)
Fig. 4. Simulation for the neuromuscular blockade control, considering
∆qi = 0.03, i = 1, 2, 3.
0 20 40 60 80 100 120
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Time (min)
M
as
s (
µg
/k
g)
 
 
system mass
desired mass
bounds of I(M*)
(a)
0 20 40 60 80 100 120
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Time (min)
M
as
s (
µg
/k
g)
 
 
system mass
desired mass
bounds of I(M*)
(b)
0 20 40 60 80 100 120
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Time (min)
M
as
s (
µg
/k
g)
 
 
system mass
desired mass
bounds of I(M*)
(c)
Fig. 5. Simulations for the neuromuscular blockade control. (a) Simula-
tion with ∆q1 = 0.08,∆q2 = 0.07 and ∆q3 = 0.05; the system mass
lays asymptotically in the interval I(M∗) = [1.4893, 1.8616]. (b) Sim-
ulation with ∆q1 = −0.09, ∆q2 = 0.02 and ∆q3 = 0.01; the system
mass lays asymptotically in the interval I(M∗) = [0.7703, 1.2411] .
(c) Simulation with ∆q1 = −0.08,∆q2 = 0 and ∆q3 = −0.01;
the system mass lays asymptotically in the interval I(M∗) =
[0.7978, 1.1169] .
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0 20 40 60 80 100 120
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Time (min)
M
as
s (
µg
/k
g)
 
 
system mass
desired mass
bounds of I(M*)
(a)
0 20 40 60 80 100 120
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Time (min)
M
as
s (
µg
/k
g)
 
 
system mass
desired mass
bounds of I(M*)
(b)
0 20 40 60 80 100 120
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Time (min)
M
as
s (
µg
/k
g)
 
 
system mass
desired mass
bounds of I(M*)
(c)
Fig. 6. Simulations for the neuromuscular blockade control when all
the parameters are subject to uncertainties. These simulations where
obtained considering the same ∆qi’s as in the simulations in Fig. 5
and other parameters uncertainties, namely ∆k13 = 0.02, ∆k12 =
0.04 and ∆k21 = −0.03. Notice that the simulation results are exactly
the same as the corresponding ones in Fig. 5.
0 20 40 60 80 100 120
0
0.5
1
1.5
2
2.5
Time (min)
M
as
s (
µg
/k
g)
 
 
system mass
desired mass
bounds of I(M*)
(a)
0 20 40 60 80 100 120
0
0.5
1
1.5
2
2.5
Time (min)
M
as
s (
µg
/k
g)
 
 
system mass
desired mass
bounds of I(M*)
(b)
0 20 40 60 80 100 120
0
0.5
1
1.5
2
2.5
Time (min)
M
as
s (
µg
/k
g)
 
 
system mass
desired mass
bounds of I(M*)
(c)
Fig. 7. Simulations for the neuromuscular blockade control. These
simulations where obtained considering ∆q1 = −0.02, ∆q2 =
0.01 and ∆q3 = −0.03, and different values of λ. (a) Simulation for
λ = 0.02. (b) Simulation for λ = 0.2. (c) Simulation for λ = 2.
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CONCLUSION
This paper presents a study of the performance of
the control law (3) proposed in [1] when applied to the
mass control of compartmental systems with parameters
uncertainties.
It turns out that the asymptotical mass values lay in
an interval whose limits can be expressed in terms of the
uncertainty bounds. This allows to derive bounds for the
asymptotical mass offset.
In order to illustrate the obtained results, simulations
for the case of neuromuscular blockade automatic control
by atracurium infusion were carried out. These simula-
tions suggest that, in this case, the asymptotical values
of the drug mass reach a constant value in the afore-
mentioned interval. This issue is currently under further
investigation.
ACKNOWLEDGMENT
This work was partially supported by FCT through
the Unidade de Investigação Matemática e Aplicações
(UIMA), Universidade de Aveiro, Portugal.
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English

Control of uncertain compartmental systems

C. Sousa

T. Mendonca

P. Rocha

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Control of uncertain compartmental systemsC. Sousa∗, T. Mendonça†, P. Rocha‡∗Escola Superior de Tecnologia e Gestão - Instituto Politécnico de Leiria/Departamento de Matemática, Leiria, Portugal†Faculdade de Ciências - Universidade do Porto/Departamento de Matemática Aplicada, Porto, Portugal‡Universidade de Aveiro/Departamento de Matemática, Aveiro, PortugalAbstract— A study of the performance of mass control forcompartmental systems, under the presence of uncertainties,is presented. Bounds for the asymptotical mass offset arederived as a functional of the uncertainty bounds. Theobtained results are illustrated by simulations for the caseof neuromuscular blockade control of patients undergoingsurgery.Index Terms—Compartmental systems, positive control,uncertain systems, neuromuscular blockade control.I. INTRODUCTIONCompartmental models have been successfully used tomodel biomedical and pharmacokinetical systems, see, forinstance, [2] or [4]. Compartmental systems consist of afinite number of subsystems, the compartments, whichexchange matter with each other and with the environ-ment. Such systems form a subclass of positive systems(i.e., systems for which the state and output variablesremain nonnegative whenever the input is nonnegative).As is well-known, in this case, the design of suitablecontrol laws is more delicate, since one has to guaranteethe positivity of the control input. In this framework,a nonnegative adaptive control law is proposed in [3],in order to guarantee the partial asymptotic set-pointstability of the closed loop system. On the other hand,a positive feedback control law is proposed in [1], inorder to stabilise the total system mass at an arbitraryset-point. This law was used in [8] for the control of theneuromuscular blockade (see [6], [7] and [9]) of patientsundergoing surgery, but no analysis was made of the effectof parameter uncertainty in its performance.In this paper, we try to fill in this gap, not only forthe neuromuscular blockade model, but for a wide classof compartmental systems. More concretely, we considerthat the control law is tuned for a nominal process modelthat contains an additive uncertainty with respect to thereal model, and analyse the behavior of the total mass inthe controlled system. It turns out that, in this case, boundsfor the asymptotical mass offset can be easily expressedin terms of the system uncertainties.II. COMPARTMENTAL SYSTEMSCompartmental systems are dynamical systems de-scribed by a set of equations of the formx˙i =∑j =i fji(x) −∑l =i fil(x)− fi0(x) + f0i(x)i = 1, . . . , n(see [2] or [10]) where x = (x1, . . . , xn)T is the statevariable and xi and fij take nonnegative values. Eachequation describes the evolution of the quantity or con-centration of material within a subsystem, called compart-ment. Since the compartments exchange with each otherand with the environment, in the above equation, xi is theamount (or concentration) of material in compartment i,fij is the flow rate from compartment j to compartment iand the subscript 0 denotes the environment (see [2]). Inthis paper, we consider the class of linear time-invariantcompartmental systems described byx˙i =∑j =ikjixj−∑l =ikilxi−qixi+biu, i = 1, . . . , n, (1)where xi and the input u take nonnegative values,kij , qi, bi ∈ R+ and at least one bi is positive (seeFig. 1). Note that, in this case, fji = kjixj , f0i = biuand fi0 = qixi, and it can be easily proved that thesystem is positive, this is, if we consider an input u thatremains nonnegative, then the state variable also remainsnonnegative. Moreover, (1) can also be written in matrixform asx˙ = Ax + bu, (2)where A (called compartmental matrix) is so thataii = −qi −∑j =ikij and, if i = j, aij = kji,and b = (b1, b2, . . . , bn)T .The total mass of the system in a given state x isdefined as M(x) =∑ni=1 xi. For an arbitrary positivevalue M∗, the set ΩM∗ ={x ∈ Rn+ : M(x) = M∗}ofall the points x in the state space with mass M∗ is calledan iso-mass.An important issue in the context of the control ofcompartmental systems is to design a control law whichyields a positive input that steers the system mass M(x)to a desired value.In [1], the following positive control law is proposedto guarantee that the trajectories converge to a set ΩM∗ :   b i   b j    q i    q jto/from other compartmentsto/from other compartmentsk j ik i ji jFig. 1. Two compartments of a linear time-invariant compartmentalmodel, as described by (1).???????????????????????? ??????????????????????????????????????????????????????????????????????????????????????????u(x) = max (0, u˜(x))u˜(x) = (∑ni=1 bi)−1 (∑ni=1 qixi + λ (M∗ −M(x))) ,(3)where λ is an arbitrary design parameter.In order to state the corresponding theorem, we needto introduce the following concept of full outflow con-nectedness. A compartmental system (1) is said to befully outflow connected if at every state x there is a pathi → j → k → · · · → l with positive kij ’s from everycompartment i to some compartment l such that ql > 0(see [1]).Theorem 1: [1] Let (2) be a fully outflow connectedcompartmental system. Then, for the closed loop system(2)-(3) with arbitrary initial conditions x(0) ∈ Rn+:i) the iso-mass ΩM∗ is forward invariant;ii) the state x(t) is bounded for all t > 0 and convergesto the iso-mass ΩM∗ .The proof of this result is based on the applicationof LaSalle’s invariance principle (see [5], pg.30), byconsidering the Lyapunov functionV (x) =12(M∗ −M(x))2of (2) on Rn+.In [8], the control law (3) was applied for the controlof the neuromuscular blockade of patients undergoingsurgery, by means of the infusion of atracurium. Infact, it is possible to model this problem as a threecompartmental model that can be described as depicted inFig. 2, where u is the drug infusion dose administered inthe central compartment, and k12, k21, k13, q3 are positivemicro-rate constants and q1, q2 are nonnegative micro-rateconstantes that vary from patient to patient. In this case,the set of equations (1) becomesx˙1 = −(k12 + k13 + q1)x1 + k21x2 + ux˙2 = k12x1 − (k21 + q2)x2x˙3 = k13x1 − q3x3(4)where x1, x2 and x3 are the drug amounts in the central,peripheral and effect compartments, respectively. How-ever, even after a satisfactory identification of the patientscharacteristics, it was necessary to consider an additionalintegrator, in order to achieve good results. This might beexplained by the fact that (contrary to what happens, for   k 1 3   q 1   q 2k 2 1k 1 2Central Compartment (C1)Peripheral Compartment (C2)b 1=1Effect Compartment (C3)   q 3Fig. 2. Compartmental model for the neuromuscular blockade effectof the drug atracurium.instance, with state feedback stabilisers, which are notuniquely defined from the system matrices) the controllaw (3) strongly depends on the system parameters. Sinceparameter uncertainty is present not only in this case, butin most of the applications, it is relevant to analyse therobustness of that control law.III. UNCERTAIN COMPARTMENTAL SYSTEMSIn this section, we analyse the performance of thecontrol law (3), proposed in [1], in the presence ofparameter uncertainties.If we consider that we can measure precisely what isinjected from the outside into the system, the parametersbi are not subject to uncertainties. On the other hand, sincethe control law does not depend on the kij’s, we mayassume that the only uncertain parameters are q1, . . . , qn.Therefore, we shall assume that a control law (3) isdesigned for a nominal systemx˙ = (A + ∆A)x + bu, (5)while the real system is given byx˙ = Ax + bu, (6)being∆A = diag(−∆q1,−∆q2, . . . ,−∆qn)the matrix of parameter uncertainties. Note that, in thiscase, the control law (3) becomesu(x) = max (0, u˜(x))u˜(x) = (∑ni=1 bi)−1 (∑ni=1 (qi + ∆qi)xi++λ (M∗ −M(x))) .(7)It turns out that, for suitable values of the designparameter λ, when the control law (7) is applied to (6), theasymptotical values of the system mass lay in an intervalwhich is related to M∗ as stated in the next theorem.Theorem 2: Let (6) be a fully outflow connected com-partmental system, ∆qmin = min {∆qi}, ∆qmax =max {∆qi} and take the design parameter λ in (7) largerthan ∆qmax. Then, the state trajectories x(t) of theclosed loop system (6)-(7), with arbitrary initial condi-tions x(0) ∈ Rn+, converge to the forward invariant setΩ ={x ∈ Rn+ : M(x) ∈ I(M∗)},with I(M∗) =[λλ−∆qminM∗, λλ−∆qmaxM∗].Proof: Let Mmin = λλ−∆qminM∗ and Mmax =λλ−∆qmaxM∗. Consider the function V : Rn → R definedbyV (x) =12(M(x)−Mmin)2 if M(x) < Mmin12(M(x)−Mmax)2 if M(x) > Mmax0 otherwise.???????????????????????? ??????????????????????????????????????????????????????????????????????????????????????????Note that V is a Lyapunov function of the system on Rn+because it is continuous and V˙ (x) ≤ 0, ∀x ∈ Rn+ (see[5], pg.30). In fact, if x ∈ Rn+,V˙ (x) =(M(x)−Mmin) dM(x)dt if M(x) < Mmin(M(x)−Mmax) dM(x)dt if M(x) > Mmax0 otherwiseis nonpositive, as we next show.• Suppose that M(x) < Mmin. Since∑ni=1 (qi + ∆qi)xi + λ (M∗ −M(x)) ≥≥∑ni=1 qixi + ∆qminM(x) + λ (M∗ −M(x))=∑ni=1 qixi + (λ−∆qmin)(Mmin −M(x))> 0,it follows that u˜(x) > 0. Thus,u(x) = u˜(x)= (∑ni=1 bi)−1 (∑ni=1 (qi + ∆qi) xi++λ (M∗ −M(x))) .In this case,dM(x)dt =∑ni=1 ∆qixi + λ (M∗ −M(x))≥ ∆qminM(x) + λ (M∗ −M(x))= (λ−∆qmin)(Mmin −M(x))> 0andV˙ (x) =(M(x)−Mmin) dM(x)dt< 0.• Suppose that M(x) > Mmax. If u˜(x) < 0, u(x) = 0anddM(x)dt = −∑ni=1 qixi ≤ 0.Thus, in this case,V˙ (x) =(M(x)−Mmax) dM(x)dt≤ 0.If u˜(x) ≥ 0, since u(x) = u˜(x), it follows thatdM(x)dt =∑ni=1 ∆qixi + λ (M∗ −M(x)) .≤ ∆qmaxM(x) + λ (M∗ −M(x))= (λ−∆qmax)(Mmax −M(x))< 0andV˙ (x) =(M(x)−Mmax) dM(x)dt< 0.Applying LaSalle’s invariance principle (see [5], pg.30),it turns out that x(t) converges to the largest invariant setcontained in{x ∈ Rn+ : V˙ (x) = 0}= I1 ∪ I2,whereI1 ={x ∈ Rn+ : M(x) ∈ I(M∗)}andI2 ={x ∈ Rn+ : u(x) = 0,∑ni=1 qixi = 0 andM(x) > Mmax}.It can be shown that the set I2 has no invariant subset.Indeed, if this would be the case, u would be zero andthe assumption of full outflow connectedness would implythat M(x) → 0, meaning that x would leave the subset(see [1]). On the other hand, I1 is an invariant set, whichconcludes the proof of the theorem.Remark 3:i) If ∆q1 = ∆q2 = . . . = ∆qn = ∆q, since∆qmin = ∆qmax = ∆q, the state trajectoryx(t) converges to the iso-mass ΩM , with M =λλ−∆q M∗.ii) When ∆qi = 0, i = 1, . . . , n, we recover theresult from Theorem 1.Using the same kind of techniques as in Theorem 2, itis possible to show that:Proposition 4: Let (6) be a fully outflow connectedcompartmental system. Then, the state trajectories x(t)of the closed loop system (6)-(7), with arbitrary initialconditions x(0) ∈ Rn+, converge to the forward invariantsetΩ˜ ={x ∈ Rn+ : M(x) ∈ I˜(M∗)},where I˜(M∗) =[λλ+∆qM∗, λλ−∆qM∗]and ∆q =max {|∆qi|}, if the design parameter λ is taken to belarger than ∆q.Note that, contrary to what happens with the set I(M∗)in Theorem 2 (which may even not contain the value M∗),the set I˜(M∗) in Proposition 4 is a neighborhood of M∗.This allows to bound the absolute mass offset bymax{M∗ − λλ+∆qM∗, λλ−∆qM∗ −M∗}== ∆qλ−∆qM∗,leading the bound ∆qλ−∆q for the relative mass offset.Clearly, increasing the parameter λ contributes to increasethe robustness of the control law.Remark 5: Note that other bounds for the relativemass offset can be derived from set I(M∗), namelymax{ |∆qmin|λ−∆qmin ,|∆qmax|λ−∆qmax}.IV. SIMULATIONSIn this section, some simulation examples are presentedfor the neuromuscular blockade control. We consider thatthe patient’s real model is given by (4), with the followingvalues for the parameters (units = min−1): k12 =0.1928, k13 = 0.0017, k21 = 0.1556, q1 = 0.1047, q2 =0, q3 = 0.0836.Our aim is to stabilise the system mass on the valueM∗ = 1.1169 (corresponding to a 10% level of neuro-muscular blockade), using the control law (7). We startby taking the design parameter λ = 0.2.In the first simulation, depicted in Fig. 3, it is assumedthat the nominal patient model coincides with the real???????????????????????? ??????????????????????????????????????????????????????????????????????????????????????????one, i.e., ∆qi = 0, i = 1, 2, 3. As expected, the systemmass converges to M∗.Fig. 4 shows the result of a simulation scenario, wherethe ∆qi’s are taken to be all equal, namely ∆q1 = ∆q2 =∆q3 = 0.03. In this case, illustrating Remark 3i), thesystem mass reaches the set-point M = λλ−∆q M∗ =1.3141.The simulations in Fig. 5 correspond to the case wherethe ∆qi’s are different. The three illustrated cases corre-spond to the following situations: M∗ lies outside I(M∗),M∗ lies in the interval I(M∗) and M∗ coincides with oneof the bounds of this interval.Fig. 6 illustrates the case where not only the qi’s aresubject to uncertainties, but also the other parameters,showing that the controller performance only depends onthe ∆qi’s.Finally, Fig. 7 illustrates the behavior of the mass ofthe controlled system for different values of the parameterλ, under a fixed uncertainty for the system parameters.According to the definition of I(M∗), one observes thatthe increasing of λ corresponds to the decrease of thefinal mass offset.0 20 40 60 80 100 12000.20.40.60.811.21.41.61.82Time (min)Mass (µg/kg)  system massdesired massbounds of I(M*)Fig. 3. Simulation for the neuromuscular blockade control, considering∆qi = 0, i = 1, 2, 3.0 20 40 60 80 100 12000.20.40.60.811.21.41.61.82Time (min)Mass (µg/kg)  system massdesired massbounds of I(M*)Fig. 4. Simulation for the neuromuscular blockade control, considering∆qi = 0.03, i = 1, 2, 3.0 20 40 60 80 100 12000.20.40.60.811.21.41.61.82Time (min)Mass (µg/kg)  system massdesired massbounds of I(M*)(a)0 20 40 60 80 100 12000.20.40.60.811.21.41.61.82Time (min)Mass (µg/kg)  system massdesired massbounds of I(M*)(b)0 20 40 60 80 100 12000.20.40.60.811.21.41.61.82Time (min)Mass (µg/kg)  system massdesired massbounds of I(M*)(c)Fig. 5. Simulations for the neuromuscular blockade control. (a) Simula-tion with ∆q1 = 0.08,∆q2 = 0.07 and ∆q3 = 0.05; the system masslays asymptotically in the interval I(M∗) = [1.4893, 1.8616]. (b) Sim-ulation with ∆q1 = −0.09, ∆q2 = 0.02 and ∆q3 = 0.01; the systemmass lays asymptotically in the interval I(M∗) = [0.7703, 1.2411] .(c) Simulation with ∆q1 = −0.08,∆q2 = 0 and ∆q3 = −0.01;the system mass lays asymptotically in the interval I(M∗) =[0.7978, 1.1169] .???????????????????????? ??????????????????????????????????????????????????????????????????????????????????????????0 20 40 60 80 100 12000.20.40.60.811.21.41.61.82Time (min)Mass (µg/kg)  system massdesired massbounds of I(M*)(a)0 20 40 60 80 100 12000.20.40.60.811.21.41.61.82Time (min)Mass (µg/kg)  system massdesired massbounds of I(M*)(b)0 20 40 60 80 100 12000.20.40.60.811.21.41.61.82Time (min)Mass (µg/kg)  system massdesired massbounds of I(M*)(c)Fig. 6. Simulations for the neuromuscular blockade control when allthe parameters are subject to uncertainties. These simulations whereobtained considering the same ∆qi’s as in the simulations in Fig. 5and other parameters uncertainties, namely ∆k13 = 0.02, ∆k12 =0.04 and ∆k21 = −0.03. Notice that the simulation results are exactlythe same as the corresponding ones in Fig. 5.0 20 40 60 80 100 12000.511.522.5Time (min)Mass (µg/kg)  system massdesired massbounds of I(M*)(a)0 20 40 60 80 100 12000.511.522.5Time (min)Mass (µg/kg)  system massdesired massbounds of I(M*)(b)0 20 40 60 80 100 12000.511.522.5Time (min)Mass (µg/kg)  system massdesired massbounds of I(M*)(c)Fig. 7. Simulations for the neuromuscular blockade control. Thesesimulations where obtained considering ∆q1 = −0.02, ∆q2 =0.01 and ∆q3 = −0.03, and different values of λ. (a) Simulation forλ = 0.02. (b) Simulation for λ = 0.2. (c) Simulation for λ = 2.???????????????????????? ??????????????????????????????????????????????????????????????????????????????????????????CONCLUSIONThis paper presents a study of the performance ofthe control law (3) proposed in [1] when applied to themass control of compartmental systems with parametersuncertainties.It turns out that the asymptotical mass values lay inan interval whose limits can be expressed in terms of theuncertainty bounds. This allows to derive bounds for theasymptotical mass offset.In order to illustrate the obtained results, simulationsfor the case of neuromuscular blockade automatic controlby atracurium infusion were carried out. These simula-tions suggest that, in this case, the asymptotical valuesof the drug mass reach a constant value in the afore-mentioned interval. This issue is currently under furtherinvestigation.ACKNOWLEDGMENTThis work was partially supported by FCT throughthe Unidade de Investigação Matemática e Aplicações(UIMA), Universidade de Aveiro, Portugal.REFERENCES[1] Bastin, G. and A. Provost (2002). “Feedback stabilisation withpositive control of dissipative compartmental systems”. In: Pro-ceedings of the 15th International Symposium on MathematicalTheory of Networks and Systems. Notre Dame, Indiana, USA.[2] Godfrey, K. (1983). Compartmental Models and Their Application.Academic Press.[3] Haddad, W., T. Hayakawa and J. Bayley (2003). “ Adaptivecontrol for non-negative and compartmental dynamical systemswith applications to general anesthesia”. International Journal ofAdaptive Control and Signal Processing, vol. 17, pp. 209-235.[4] Jacquez, J. and C. Simon (1993). “Qualitative theory of compart-mental systems”. SIAM Review, vol. 35, no. 1, pp. 43-79.[5] LaSalle, J. P. (1976). The Stability of Dynamical Systems. SIAM,Bristol, England.[6] Lemos, J., T. Mendonça and E. Mosca (1991). “Long-rangeadaptive control with input constraints”. International Journal ofControl, vol. 54, pp. 289-306.[7] Linkens, D. (1994). Intelligent control in biomedicine. Taylor andFrancis, London.[8] Magalhães, H. (2005). “Identification and control of positiveand compartmental systems applied to neuromuscular blockade”.In: Preprints of the 16th IFAC World Congress. Prague, CzechRepublic.[9] Mendonça, T. and P. Lago (1998). “PID control strategies for theautomatic control of neuromuscular blockade”. Control Engineer-ing Practice, vol. 6, pp. 1225-1231.[10] Sandberg, W. (1978). “On the mathematical foundations of com-partmental analysis in biology, medicine and ecology”, IEEETransactions on Circuits and Systems, vol. 25, pp. 273-279.???????????????????????? ??????????????????????????????????????????????????????????????????????????????????????????

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Control of uncertain compartmental systems

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