Interval observers for linear systems with delay

Abstract

This paper focuses on the analysis of some classes of observers for linear systems with a point-wise delay. First, it is pointed out that, classical interval observers for systems without delays are not robust with respect to the presence of delays that appear in a specific structure location, no matter how small it is. Next, it is shown that, in general, for linear systems classical interval observers endowed with a point-wise delay are unstable. A new type of design enabling to circumvent this obstacle is proposed. It relies on framers that incorporate distributed delay terms.

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Interval Observers for Linear Systems with Delay
Fr´ ed´ eric Mazenc1, Silviu-Iulian Niculescu2, Olivier Bernard3,
Abstract—This paper focuses on the analysis of some classes
of observers for linear systems with a point-wise delay. First,
it is pointed out that, classical interval observers for systems
without delays are not robust with respect to the presence of
delays that appear in a specific structure location, no matter
how small it is. Next, it is shown that, in general, for linear
systems classical interval observers endowed with a point-wise
delay are unstable. A new type of design enabling to circumvent
this obstacle is proposed. It relies on framers that incorporate
distributed delay terms.
Keywords. Interval observers, exponential stability, delay.
I. INTRODUCTION
The interval observer method is a recent state estimation
technique, initiated in [4], and extended and applied in many
works, see, for instance, [22], [1], [19], [16]. It is used in
particular in biological contexts, where taking into account
the presence of uncertainties is essential.
Let us summarize this technique in the simple case where,
for the system input-free, there is no disturbance acting
on the output: Consider a nonlinear system Γ(Σ,Λ) whose
dynamics are defined by using a differential representation
Σ(f1,Rn,R?):
˙ x(t) = f1(x(t),w(t))
(1)
with x ∈ Rn, w ∈ R?, endowed with the output Λ(f2,Rq),
y(t) = f2(x(t)) ∈ Rq, where w is an unknown time-varying
function such that there are two known Lipschitz continuous
functions w−(t), w+(t) such that, for all t ≥ 0, component
by component, w−(t) ≤ w(t) ≤ w+(t). Then the dynamical
system Γo(Σo,Λo), of arbitrary dimension na, where Σo=
Σ(F,Rna,R2?,Rq) and Λo= Λ(h,R2n) is given by:
?
with w = (w−,w+) ∈ R2?and h : Rna×R2?×Rq?→ R2n,
h = (h−,h+) defines a framer if, for any initial conditions
x(0) ∈ Rn, z(0) ∈ Rna, such that the inequalities
x−
˙ z(t) = F(z(t),w(t),y(t)) ,
x+= h+(z,w,y) , x−= h−(z,w,y) ,
(2)
0≤ x(0) ≤ x+
0
(3)
with x±
solution of (1), (2) with x(0) and z(0) respectively as initial
condition satisfies, for all t ≥ 0, component by component,
the inequalities
0
= h±(z(0),w(0),y(0)), are satisfied, then the
x−(t) ≤ x(t) ≤ x+(t) .
(4)
1EPI MERE INRIA-INRA, UMR Analyse des Syst` emes et Biom´ etrie,
INRA 2, pl. Viala, 34060 Montpellier, France, mazenc@supagro.inra.fr.
2Laboratoire des Signaux et Syst` emes (LSS), CNRS-Sup´ elec, 3 rue Joliot
Curie, 91192 Gif-sur-Yvette, France, Silviu.Niculescu@lss.supelec.fr
3EPI COMORE INRIA Sophia-Antipolis, 2004 route des Lucioles, BP
93, 06902 Sophia-Antipolis, France, olivier.bernard@inria.fr.
Moreover, if the framer is stable, it is called an interval
observer.
The success of this state estimation method is mostly
due to two reasons: (i) First, the method applies and gives
satisfactory results when systems with large uncertainties are
studied; (ii) Second, this technique takes advantage of an
approximate knowledge of the initial condition and gives
information on the unknown state of the system studied at
any instant t ≥ 0 of time, whereas classical observers only
provide a useful information asymptotically, that is, from an
engineering point of view, for sufficiently large time-values.
Interval observers available in the literature are all made
for systems represented by ordinary differential equations.
Since delays are frequently encountered in many dynamical
systems (see, e.g. [8], [18] and the references therein), arising
as an intrinsic part of the system, through input and measure-
ment channels, feedback and propagation of some signals,
through heredity in population dynamics, and as a result of a
simplification of partial differential equations, it is appealing
to investigate if interval observers can be used in such a case.
To the best of the authors’ knowledge, such problems have
not been addressed in the delay case although some results on
the applications of interval analysis techniques [13] to delay
systems have been already discussed in the literature (see,
for instance, [14]). Finally, although constructing observers
for delay systems represents a subject of recurring interest
since the 70s (see, e.g., [2], [23], [5], [21], [6], [3], [24] and
the references therein), the method we are proposing is quite
new and takes extensively advantage more on interpreting the
framers’ inequalities in an appropriate ”functional” setting.
Since the interval observers technique allows to cope with
poorly known systems, one may expect this technique to be
robust with respect to the presence of a small delay in the
equations of the observed system: indeed, in some cases, a
small delay can be considered as a small perturbation which
does not significantly modify the behavior of a system (see,
e.g. [18] and the references therein). In the present paper,
we will show that this is not the case for systems endowed
with exponentially stable linear interval observers (Section
III). In spite of what the intuition may suggest, the classical
interval observers without delay in general do not work when
applied to exponentially stable linear systems with a point-
wise delay, no matter how small this delay is. Overcoming
this obstacle, that is, determining interval observers which
give a satisfactory estimation result when a small delay is
present, is not a trivial task. In particular, we will show that
the classical interval observer constructions lead, in general,
to framers with point-wise delays which are unstable. This
is a serious drawback because unstable framers no longer
Joint 48th IEEE Conference on Decision and Control and
28th Chinese Control Conference
Shanghai, P.R. China, December 16-18, 2009
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provide valuable estimates when the time becomes large.
It is worth noting that the results of this first part and the
techniques of their proofs have some similarities with those
of [9, Section 3], [10, Section III].
In the second part of the paper (Section IV), for a
general family of exponentially stable linear systems with a
point-wise delay, we propose a new construction of interval
observers which converge exponentially to the origin when
no disturbance is acting. The key feature of the framers
we introduce is that their dynamics have distributed delay
terms, although they are framers for systems with “selective”
memory, i.e. point-wise delays. The idea is based on a
classical model transformation of the original system by
using an appropriate integration over one delay interval.
Even if the advantages and the drawbacks of using such
a technique were largely discussed in the literature [20],
[8], [7] (and the references therein), the distributed delay
based-construction allows us a better comprehension of the
system’s dynamics as “pieces of trajectories” for a better use
in constructing exponentially stable interval observers in such
a setting. Some illustrative examples end the presentation.
II. NOTATIONS, DEFINITIONS
1) Classical notations and definitions.
Let T be a positive real number. Then, for a given
t ≥ 0, xt(·) denotes the function xt(θ) = x(t + θ),
∀θ ∈ [−T,0]. A time-delay system described by a func-
tional differential equation (FDE): ˙ χ(t) = F (t,χt), will
be denoted as Σ(F,Cn([−T,0])). For an initial condition
φχ ∈ Cn([−T,0]), we denote by χ(t0,φχ,Σ) the solution
of Σ(F,Cn([−T,0])) through (t0,φχ). In the presence of a
disturbance w ∈ Rnwand of an output yχ∈ Rny, the system
rewrites as:
?
and it will be denoted Γ(Σ,Λ). Here Σ corresponds to the
differential representation Σ(F,Cn([−T,0]),Cnw([−T,0])),
(or Σ(F,Cn([−T,0]),Rnw) and Λ to the output Λ(H,Rny).
Next, a square matrix is said to be cooperative if its off-
diagonal terms are nonnegative. A system ˙ x = f(x,w) is
said to be cooperative if the Jacobian matrix of f with respect
to x, denoted
[25]).
We introduce now a new definition of cooperativity tai-
lored for specific case of systems with point-wise delays:
˙ χ(t) = F (t,χt,wt)
yχ(t) = H(χt)
or
?
˙ χ(t) = F (t,χt,w(t)),
yχ(t) = H(χt)
Df
Dx(x,w) (or simply
Df
Dx), is cooperative (see
Definition 1. Consider a system with a point-wise delay
Σ(F,Cn([−τ,0]),R?) defined by:
˙ χ(t) = F (χ(t),χ(t − τ),w(t)),
with τ ≥ 0 and where F : Rn×Rn×R??→ Rnis a function
of class C1and w : R+ ?→ R?is a Lipschitz continuous
function.
The system
Σ(F,Cn([−τ,0]),R?)
delayed cooperative if, for all (a,b,c) ∈ Rn× Rn× R?,
the matrix
DF
Db(a,b,c) are nonnegative.
(5)
issaidto be
DF
Da(a,b,c) is cooperative and the entries of
In the sequel, all the inequalities must be understood
component by component i.e. ?a = (?a1,...,?ar)?∈ Rr
and ?b= (?b1,...,?br)?∈ Rrare such that ?a≤ ?b if
and only if, for all i ∈ {1,...,r}, ?ai≤ ?bi. If M is a matrix
with the entries mi,j, then we denote by M+(resp. M−) the
matrix whose entries are max{0,mi,j} (resp. min{0,mi,j}).
2) Definitions of framers and interval observers.
As seen from the Introduction, two ingredients are essen-
tial in defining a framer: (i) the existence of the functions
w+and w−that “bound” the disturbance w on the whole
time-domain interval, and (ii) an appropriate trajectory-based
interpretation of the dynamics. Based on this observation,
there exist several possibilities to extend such definitions to
time-delay systems, and these extensions will finally depend
on the way the notion of state is interpreted and used in the
FDE setting. With the notations above, we have:
Definition 2. Consider a time-delay system of the form
Σ(F,Cn([−T,0]),C?([−T,0])), and described by:
˙ χ(t) = F (t,χt,wt),
(6)
with F of class C1, such that there are two known Lipschitz
continuous functions w−(t), w+(t) such that, for all t ≥
−T, w−(t) ≤ w(t) ≤ w+(t) and such that for all initial
conditions, the solutions of Σ are defined over [−T,+∞).
Then the dynamical system Γo(Σo,Λo) with Σo
Σ(ϕ,Cnz([−Tz,0]),C2?([−Tz,0])) (with ϕ a C1function and
Tz≥ T) and Λo= Λ(H,R2n) defined by
?
with w = (w−,w+) ∈ R2?, and H : Rnz ?→ R2n,
H = (H−,H+) is called a general framer if, for any
function φχ ∈ Cn([−T,0]), φZ ∈ Cnz([−Tz,0]) such that
the inequalities
=
˙Z(t)
χ+(t)
=
=
ϕ(t,Zt,wt) ,
H+(Z(t)),χ−(t) = H−(Z(t)) ,(7)
φχ−(t) ≤ φχ(t) ≤ φχ+(t)
with φχ−(t) = H−(φZ(t)), φχ+(t) = H+(φZ(t)) are
satisfied, then the solutions of (6), (7) with respectively φχ,
φZas initial condition exist and satisfy, for all t ≥ 0,
χ−(t) ≤ χ(t) ≤ χ+(t) .
It is important to point out that in the case when Tz> T,
there is no explicit constraint on the ”piece-of-trajectory” of
the initial condition φZof the general framer corresponding
to [−Tz,−T). However, the conditions imposed on the
corresponding solution on [0,Tz− T) can define “back-
wardly” appropriate constraints on such initial condition.
Some refinement of this notion is proposed in the next
paragraph by taking into account the systems’ dynamics from
one delay interval to the another. Finally, the definition above
can be easily adapted to the case when the disturbance w acts
“instantaneously” (no memory effect) on the dynamics of Σ
by replacing C?([−Tz,0]) with R?.
Consider now the case of a system with point-wise delay
and with the disturbance acting “instantaneously” on the
dynamics. Then, we have the following:
∀t ∈ [−T,0],
(8)
(9)
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Definition 3. Consider a system with a point-wise delay
Σ(F,Cn([−τ,0]),R?) defined by:
˙ χ(t) = F (χ(t),χ(t − τ),w(t)),
with τ ≥ 0 and F a function of class C1, such that there
are two known Lipschitz continuous functions w−(t), w+(t)
such that, for all t ≥ −τ, w−(t) ≤ w(t) ≤ w+(t) and such
that for all initial conditions, the solutions are defined over
[−τ,+∞).
Then, the dynamical system Γo(Σo,Λo) with Σo
Σ(ϕ,Cnz([−kτ,0]),C?([−kτ,0])) (ϕ a C1function and k ≥
1 a positive integer) and Λo= Λ(H,R2n) given by:
?
with w = (w−,w+) ∈ R2?, and H : Rnz?→ R2n, H =
(H−,H+) is called a specific framer if, for any functions
φχ∈ Cn([−τ,0]), and any function φZ∈ Cnz([−τ,(k−1)τ])
such that the inequalities
(10)
=
˙Z(t)
χ+(t)
=
=
ϕ(Zt,wt) ,
H+(Z(t)),χ−(t) = H−(Z(t)) ,(11)
φχ−(t) ≤ χ(t) ≤ φχ+(t) , ∀t ∈ [−τ,(k − 1)τ]
with φχ−(t) = H−(φZ(t)), φχ+(t) = H+(φZ(t)) are
satisfied, then, for all t ≥ (k − 1)τ, the solution of (11)
exists and satisfies, for all t ≥ 0,
χ−(t) ≤ χ(t) ≤ χ+(t) .
The refinement proposed in the definition of specific
framers is represented by the way to treat the initial condition
of the corresponding interval observer with respect to the
dynamics of the original system. Indeed, the solution of (10)
through (0,φχ) can be constructed by using the so-called
“step-by-step” procedure, i.e. for all integers i ≥ 0, the
solution on the time-interval [iτ,(i + 1)τ) is derived by a
simple integration of an ordinary differential equation (ODE)
including the solution on [(i − 1)τ,iτ) (see, e.g., [11], [18]
for further discussions). This particular point is reflected by
the inequalities (12) above.
(12)
(13)
Definition 4. A general or a specific framer for re-
spectively the system (6) or the system (11) is an
exponentially stable interval observer if all its solutions con-
verge exponentially to the origin when the time goes to the
infinity when w is identically equal to zero.
III. LACK OF ROBUSTNESS OF INTERVAL OBSERVERS
WITH RESPECT TO DELAYS
In general, interval observers currently used for expo-
nentially stable linear systems in the absence of delay are
not robust when an arbitrarily small delay is present in the
dynamics. Indeed, when the system is cooperative with
respect to χ(t) but not delayed cooperative, the properties of
state enclosing are not guaranteed. However, if the system
is delayed cooperative, then the classical construction of
framers yields exponentially stable interval observers.
To exhibit the lack of robustness mentioned above, we
consider the simple delay system:
˙ x(t) = −x(t − τ)
(14)
with x ∈ R, τ ≥ 0 and no output, which is not delayed
cooperative. One can easily prove that the restriction τ ∈
[0,π/2) implies that the system (14) admits the origin as an
exponentially stable equilibrium point.
A. A non-robust interval observer
When τ = 0, the system (14) is cooperative, and it is a
well-known fact that the two copies of the system:
?
compose a framer. Moreover it is exponentially stable: all the
solutions of (15) converge exponentially to the origin when
the time goes to the infinity.
When τ > 0, the behavior of the equations is tremendously
different: the system (15) is no longer a framer for the system
(14) in any reasonable sense. To understand why, let us
choose for the z+-subsystem of (15) and the system (14)
the initial conditions: (i) φz+(t) = 0, for all t ∈ [−τz,0],
where τzis any real number larger than τ, and (ii) φx(t) = t,
for all t ∈ [−τ,0]. Then the corresponding solutions satisfy
z+(t) = x(t)+= 0, ∀t ≥ −τ and
x(t)=
˙ z+(t)
x+(t)
=
=
−z+(t) , ˙ z−(t) = −z−(t) ,
z+(t) , x−(t) = z−(t)
(15)
−?t
0φx(m − τ)dm , ∀t ∈ [0,τ].
(16)
Consequently, for all t ∈ [0,τ], x(t) = −t2
x(t)+. Hence, the inequality x(t) ≤ x(t)+does not hold for
all t ≥ 0. It follows that the system (15) is not a general
framer to the system (14).
One can prove that it is not a specific framer to (14)
when the assumption τ ≤1
is imposed by exhibiting families of solutions which are in
contradiction with the definition of specific observers.
The result above can be easily extended to more general
linear systems as follows1:
2+ τt > 0 =
e, which simplifies the analysis,
Proposition 1. Let A = (ai,j) be a matrix of dimension
n × n. Assume that there exists j ∈ {1,...,n} such that
aj,j< 0. Let τ > 0. Then the system
?
is not a general interval observer for the system
˙ z+(t) = Az+(t) , ˙ z−(t) = Az−(t) ,
x+= z+, x−= z−,
(17)
˙ x(t) = Aξ(t)
(18)
where ξ(t) is the vector with xj(t − τ) as j-th component
and xi(t) as i-th component when i ?= j.
B. Interval observers with delay
The next natural question which arises is whether expo-
nentially stable interval observers with the point-wise delay
τ exist for the system (14), that is whether there is an
exponentially stable framer (possibly non-linear) of the form
?
˙ z(t)
x+(t)
=
=
F(z(t),z(t − τ)) ,
h+(z(t)) , x−(t) = h−(z(t)) ,
(19)
1For the proof, see the full version of the paper [17].
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for the system (14). We have no answer to this general
question. However, we show below that the straightforward
adaptation of the framer (15) is not stable.
The classical strategy of framer design leads us to consider
the following system:
?
If we choose initial conditions satisfying φ−
φ+
x+− x, ˜ x−= x − x−, we obtain
˙˜ x+(t) = ˜ x−(t − τ) ,˙˜ x−(t) = ˜ x+(t − τ) .
One can easily prove that any solution of (21) with non-
negative initial conditions is nonnegative for all t ≥ −τ. It
follows that the inequalities
˙ z+(t) = −z−(t − τ) , ˙ z−(t) = −z+(t − τ) ,
x+(t) = z+(t) , x−(t) = z−(t) .
(20)
x(t) ≤ φx(t) ≤
x(t) for all t ∈ [−τ,0], then, with the notations ˜ x+=
(21)
x−(t) ≤ x(t) ≤ x+(t)
(22)
are satisfied for all t ≥ 0, which implies that (20) is a general
framer for (14). But the system (21) is unstable: almost all
its solutions go to the infinity when t → ∞. Therefore the
system (20) is not an exponentially stable interval observer.
Furthermore, we have the following result2:
Proposition 2. The system (14) does not possess a general
exponentially stable framer of the form
?
where ζ ∈ Rnζ, nζ ∈ N, nζ ≥ 1 is arbitrary and Fi, Hi
(i = 1,2) are matrices of appropriate dimensions.
˙ζ(t) = F1ζ(t) + F2ζ(t − τ) ,
x+(t) = H1ζ(t),x−(t) = H2ζ(t) ,
(23)
In contrast to Proposition 2, we give a result below,
which shows that in particular cases, the presence of a
point-wise delay does not preclude from constructing of a
general asymptotically stable framer with a point-wise delay.
This result emphasizes the importance of the notion delayed
cooperative systems3:
Proposition 3. Consider the delayed cooperative system with
a point-wise delay Σ(F,Cn([−τ,0]))
˙ χ(t) = F(χ(t),χ(t − τ)) .
Assume that this system is a delayed cooperative system and
is globally asymptotically stable. Then the dynamical system
?
is globally asymptotically stable and is a general framer for
Σ(F,Cn([−τ,0])).
Remark 1. For the sake of clarity, Proposition 3 is devoted
to systems having no disturbance. Immediate extensions to
cases where disturbances are present can be deduced.
(24)
˙ χ+(t) = F (χ+(t),χ+(t − τ)),
˙ χ−(t) = F (χ−(t),χ−(t − τ)),
(25)
2For the proof, see the full version of the paper [17].
3For the proof, see the full version of the paper [17].
IV. LINEAR SYSTEMS WITH A POINT-WISE DELAY
In the previous section, we have listed 3 difficulties arising
when one aims at constructing interval observers for linear
systems with delay. In the sequel, we show how these ob-
stacles can be overcome for a large family of linear systems
with a point-wise delay (which contains the system (14)) by
exhibiting a particular family of framers. Their key features
is that they have a distributed term in their equations and
are exponentially stable interval observers, when the delay
is smaller than some upper bound. Besides, these interval
observers are specific framers, but not general framers.
We consider a linear system with a single delay:
˙ x(t)=
Ax(t) + Bx(t − τ) + w(t)
(26)
with x ∈ Rn, w ∈ Rnand where A and B are constant
square matrices, w a Lipschitz continuous function and
where there is no output. Let ai,j (resp. bi,j) denote the
entries of A (resp. B). Introduce now the quantity:
?
Let D denote the diagonal matrix such that all the diagonal
terms of the matrix −D + A + B are equal to zero and
E = A + B − D
We introduce now the following assumptions:
Assumption 1. The matrix
?
is Hurwitz.
Assumption 2. The Lipschitz continuous functions w+,w−
satisfying the property
η = max max
i,j{|ai,j|},max
i,j{|bi,j|}
?
.
(27)
and
EP= D + E+.
(28)
M =
EP
−E−
−E−
EP
?
(29)
w−(t) ≤ w(t) ≤ w+(t),t ≥ −τ,
(30)
are supposed known.
Assumption 1 guarantees that there exists a matrix Q =
Q?> 0 such that
QM + M?Q = −In,
(31)
where Indenotes the n × n identity matrix.
We are ready to state and prove the main result:
Theorem 1. Assume that the system (26) satisfies Assump-
tions 1 and 2. Then one can construct a linear specific framer
for this system. This specific framer is an exponentially stable
interval observer when the delay is such that
?
Discussion of Theorem 1
• We have chosen to study linear systems with an additive
disturbance w(t): our motivation is the importance of sys-
tems of this type. Observe that other types of disturbances
can be considered. For instance the case where the entries
of A or B are not well known may be interesting, but this
is beyond the scope of the present work.
• When A+B is cooperative and Hurwitz, then Assumption
1863
τ
∈
0,
1
16n2η2||Q||
?
.
(32)
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1 is automatically satisfied since EP= A+B and E−= 0.
• One can easily deduce that Assumptions 1-2 ensure that


is an exponentially stable interval observer for the system
(26) when τ = 0 and w(t) = 0 for all t ≥ 0. Observe that
the fact that (33) is an exponentially stable interval observer
for the system (26) when τ = 0 and w(t) = 0 for all t ≥ 0
implies that A + B is a Hurwitz matrix.
Proof. Let us introduce the notations
?t−τ
νj(t)=
?t
˙ x(t) = (D + E)x(t) − B[x(t) − x(t − τ)] + w(t) . (36)
Then, for all t ≥ τ, the solutions of the system (26) satisfy
˙ x(t) = (D+E)x(t)+Gν(t)+Hξ(t)+w(t)−BWI(t). (37)
We rewrite this system

˙ z+(t)
˙ z−(t)
x+(t)
=
=
=
EPz+(t) + E−z−(t) ,
EPz−(t) + E−z+(t) ,
z+(t) , x−(t) = z−(t) ,
(33)
ξj(t)=
t−2τxj(r)dr , ξ = (ξ1....ξn)?,
?t
t−τw(r)dr and G = −BA, H = −B2. Then
∀t ≥ 0, (26) rewrites as
(34)
t−τxj(r)dr , ν = (ν1....νn)?,
(35)
WI(t) =
˙ x(t) = (EP+ E−)x(t) + G+ν(t) + +G−ν(t) + H+ξ(t)
+H−ξ(t) − B+WI(t) − B−WI(t) + w(t) .
This system leads us to consider, for the system (26), the
candidate interval observer




with ξ+
W+
symmetrically deduced from z−
To show that the system (39) is a specific interval observer
for the system (26), we consider for (26) an initial condition
φx ∈ Cn([−τ,0]) which, due to the linearity of (26),
generates for all t ≥ −τ a solution χ(t) and, for (39), an
initial condition φz= (φz+,φz−)?∈ C2n([−τ,τ]) such that,
for all t ∈ [−τ,τ],
φz−(t) ≤ x(t) ≤ φz+(t) .
Next, we introduce the notations: ˜ x+= x+−x, ˜ x−= x−x−,
˜ξ+
=
˜ ν+
=
˜ξ−
=
˜ ν−
=
(38)














I(t) =?t

˙ z+(t)=
EPz+(t) + E−z−(t) + H+ξ+(t)
+H−ξ−(t) + G+ν+(t) + G−ν−(t)
+w+(t) − B+W−
EPz−(t) + E−z+(t) + H+ξ−(t)
+H−ξ+(t) + G+ν−(t) + G−ν+(t)
+w−(t) − B+W+
z+(t) ,x−(t) = z−(t) ,
?t−τ
j(t) and w−(t).
I(t) − B−W+
I(t) ,
˙ z−(t)=
I(t) − B−W−
I(t) ,
x+(t)=
(39)
j(t) =
t−2τz+
j(r)dr, ν+
j(t) =
j(t), ν−
?t
t−τz+
j(r)dr,
I(t) are
t−τw+(r)dr and ξ−
j(t) and W−
(40)
j(t)
j(t)
j(t)
j(t)
?t−τ
?t−τ
t−2τ[x+
?t
?t
j(r) − xj(r)]dr,
t−τ[x+
t−2τ[xj(r) − x−
t−τ[xj(r) − x−
j(r) − xj(r)]dr,
j(r)]dr,
j(r)]dr.
(41)
Then, using the fact that t ≥ τ, the solutions of the system
(26) satisfy (37), we deduce that, for all t ≥ τ,


with
P1(t)
−B−(W+
P2(t)
+B−(W−
Notice that Assumption 2 implies that, for all t ≥ 0, P1(t) ≥
0, P2(t) ≥ 0. Next, observe that Epis cooperative and all
the entries of the matrices −E−, G+, H+, −G−, −H−are
nonnegative. It follows that Lemma 1 in the appendix applies
to (42). On the other hand, (40) implies that (˜ x+(t), ˜ x−(t)) is
a solution of the system (42) such that ˜ x+(t) ≥ 0, ˜ x−(t) ≥ 0
for all t ∈ [−τ,τ]. It follows that ˜ x+(t) ≥ 0, ˜ x−(t) ≥ 0 for
all t ≥ −τ. This allows us to conclude that the system (39)
is a specific interval observer for the system (26).
Fnally, one can establish that Assumption 1 implies that
the system (42) is exponentially stable when the disturbance
w is identically equal to zero and and when τ satisfies
(32) through the construction of a Lyapunov-Krasovskii
functional4.





˙˜ x+(t)=
EP˜ x+(t) − E−˜ x−(t) + G+˜ ν+(t)
−G−˜ ν−(t) + H+˜ξ+(t) − H−˜ξ−(t) + P1(t),
˙˜ x−(t)=
EP˜ x−(t) − E−˜ x+(t) + G+˜ ν−(t)
−G−˜ ν+(t) + H+˜ξ−(t) − H−˜ξ+(t) + P2(t),
(42)
=
w+(t) − w−(t) − B+(W−
I(t) − WI(t)) ,
w(t) − w−(t) + B+(W+
I(t) − WI(t)) .
I(t) − WI(t))
=
I(t) − WI(t))
(43)
V. ILLUSTRATION
An anaerobic digestion process used to produce methane
from an organic pollutant s ∈ R using a bacterial species
X ∈ R can be represented as follows [15]:
?
with D > 0, the dilution rate, Y > 0, the yield coefficient
and Sin(t) > 0 the substrate input concentration, possibly
time-varying. The growth rate µ(s) is represented by a
non-monotonic function of Haldane type. Without any loss
of generality, we can assume that D = 1 and Y = 1.
Indeed, these constants can be removed through a change
of variable and a time-rescaling. The methane flow rate
y2(t) = µ(s(t))X(t) can be online measured. Moreover, the
substrate is off-line measured. This involves a delay τ > 0
in the measurement, so that y1(t) = s(t − τ).
We assume that Sin(t) = ψ + v(t), where ψ is an
unknown positive constant and v(t) is an unknown Lipschitz
continuous function. Then (44) becomes
?
We assume that there are known Lipschitz continuous func-
tions v−, v+such that, for all t ≥ −τ, the inequalities
−ψ
The problem of estimating the influent substrate concen-
tration, and especially ψ, is of key importance, together with
˙ s(t)
˙X(t)
=
=
D[Sin(t) − s(t)] −1
[µ(s(t)) − D]X(t) ,
Yµ(s(t))X(t) ,
(44)
˙ s(t)
˙X(t)
=
=
ψ + v(t) − s(t) − µ(s(t))X(t) ,
[µ(s(t)) − 1]X(t) .
(45)
2≤ v−(t) ≤ v(t) ≤ v+(t) are satisfied.
4For the proof, see the full version of the paper [17].
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