HAL Id: hal-00497786
https://hal.archives-ouvertes.fr/hal-00497786
Submitted on 9 Apr 2014
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entic research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diusion de documents
scientiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
Adaptive observer for fault estimation in nonlinear
systems described by a Takagi-Sugeno model
Atef Kheder, Kamel Benothman, Mohamed Benrejeb, Didier Maquin
To cite this version:
Atef Kheder, Kamel Benothman, Mohamed Benrejeb, Didier Maquin. Adaptive observer for fault
estimation in nonlinear systems described by a Takagi-Sugeno model. 18th Mediterranean Conference
on Control and Automation, MED’10, Jun 2010, Marrakech, Morocco. pp.CDROM. �hal-00497786�
Adaptive observer for fault estimation in nonlinear systems described
by a Takagi-Sugeno model
Atef Khedher, Kamel Benothman, Mohamed Benrejeb and Didier Maquin
Abstract—This paper deals with the problem of fault
estimation for linear and nonlinear systems. An adaptive
proportional integral observer is designed to estimate both
the system state and sensor and actuator faults which can
affect the system. The model of the system is first augmented
in such a manner that the original sensor faults appear
as actuator faults in this new model. The faults are then
considered as unknown inputs and are estimated using a
classical proportional-integral observer. The proposed method
is first developed for linear systems and is then extended to
nonlinear ones that can be represented by a Takagi-Sugeno
model. In the two cases, examples of low dimensions illustrate
the effectiveness of the proposed method.
Index Terms— fault diagnosis, fault estimation, adaptive ob-
server, proportional-integral observer, state estimation, Takagi-
Sugeno model
I. INTRODUCTION
State estimation is an important field of research with
numerous applications in control and diagnosis. Generally
the whole system state is not always measurable and the
recourse to its estimation is a necessity.
An observer is generally a dynamical system allowing
the state reconstruction from the system model and the
measurements of its inputs and outputs [15]. For linear
models, state estimation methods are very efficient [5],
[13], [14]. However for many real systems, the linearity
hypothesis cannot be assumed. In that case, the synthesis
of a nonlinear observer allows the reconstruction of the
system state. For example, let us cite sliding mode observers
[4], the Thau-Luenberger observers [21] and observer for
nonlinear systems described by Takagi-Sugeno models [2].
Approaches using Takagi-Sugeno model (also known
as multiple model [17]) are the object of many works
in different contexts including the taking into account of
unknown inputs or parameter uncertainties [1], [7], [8].
Various studies dealing with the presence of unknown inputs
acting on the system were published [1], [5], [20]. Some
of them tried to reconstruct the system state in spite of the
unknown input existence. This reconstruction is assured via
the elimination of unknown inputs [6], [20]. Other works
Atef Khedher, Kamel Benothman and Mohamed Benrejeb are
with LARA Automatique, ENIT, BP 37, le Belv´ed`ere, 1002 Tu-
nis, khedher
atef@yahoo.fr, kamel.benothman@enim.
rnu.tn, mohamed.benrejeb.enit.rnu.tn
Didier Maquin is with the CRAN, UMR 7039, Nancy-Universit´e,
CNRS, 2, avenue de la Forˆet de Haye, 54516 Vandœuvre-l`es-Nancy,
didier.maquin@ensem.inpl-nancy.fr
choose to estimate, simultaneously, the unknown inputs and
system state [1], [5], [18]. Among the techniques that do
not require the elimination of the unknown inputs, Wang
[23] proposes an observer able to entirely reconstruct the
state of a linear system in the presence of unknown inputs
and in [16], to estimate the state, a model inversion method
is used. Using the Walcott and Zak structure observer [22]
Edwards et al. [3], [4] have also designed a convergent
observer using the Lyapunov approach.
Observers with unknown inputs are used to estimate
actuator faults which can be considered as unknown inputs.
This estimation can be obtained using a proportional integral
observer [12], [19]. In most cases, a physical process can
be subjected to disturbances which have as origin the noises
due to its environment, uncertainty of measurements, sensor
and/or actuator faults. These disturbances have harmful
effects on the normal behavior of the process and their
estimation can be used to conceive a control strategy able to
minimize their effects. In the case of sensor faults, Edwards
[5] proposes, for linear systems, to use a new state which is
a filtered version of the output, to conceive an augmented
system in which the sensor faults appear as unknown inputs.
This formulation was also used by [9]–[11], [24] to be able
to estimate the faults.
In many cases, systems are affected by faults of different
nature such as sensor or actuator faults, so, in this paper,
a proportional integral observer is conceived to estimate,
simultaneously, the state and theses two kind of faults. The
extension of this method to nonlinear systems described by
Takagi-Sugeno models is proposed thereafter.
The paper is organised as follows. Section II presents
the proposed method of faults estimation for linear systems.
In section III the extension of the proposed method for
nonlinear systems described by Takagi-Sugeno models is
made. Two simulations examples are proposed to validate
the method for linear and nonlinear systems.
II. LINEAR SYSTEM CASE
The objective of this part is to estimate a fault affecting
a linear system via an adaptive proportional integral state
observer.
A. Problem formulation
Consider the linear model affected by a sensor fault, an
actuator fault and a measurement noise described by:
˙x(t) = Ax(t) + Bu(t) + Ef
a
(t) (1a)
y(t) = Cx(t) + F f
s
(t) + Dw(t) (1b)
where x(t) ∈ IR
n
represents the system state, y(t) ∈ IR
m
is
the measured output, u(t) ∈ IR
r
is the known system input,
f
a
(t) and f
s
(t) represent respectively actuator and sensor
faults and w(t) is the measurement noise. A, B and C are
known constant matrices with appropriate dimensions. E, F
and D are respectively the actuator fault, the sensor fault
and the noise distribution matrices which are assumed to be
known. Consider also the state z(t) ∈ IR
p
that is a filtered
version of the output y(t) [5]. This state is given by:
˙z(t) = −
¯
Az(t) +
¯
ACx(t) +
¯
AF f
s
(t) +
¯
ADw(t) (2)
where −
¯
A ∈ IR
p×p
is a stable matrix. Let us introduce
the augmented state X(t) =
x
T
(t) z
T
(t)
T
and the cor-
responding augmented system given by:
˙
X(t) = A
a
X(t) + B
a
u(t) + E
a
f(t) + F
a
w(t) (3a)
Y (t) = C
a
X(t) (3b)
with:
A
a
=
A 0
¯
AC −
¯
A
, B
a
=
B
0
, E
a
=
E 0
0
¯
AF
F
a
=
0
¯
AD
, C
a
=
0 I
, f (t) =
f
a
(t)
f
s
(t)
(4)
The structure of the chosen observer is as follows:
˙
ˆ
X(t) = A
a
ˆ
X(t) + B
a
u(t) + E
a
ˆ
f(t) + K
˜
Y (t)
˙
ˆ
f(t) = L
˜
Y (t)
ˆ
Y (t) = C
a
ˆ
X(t)
(5)
where
ˆ
X(t) is the estimated augmented state,
ˆ
f(t) represents
the estimated fault,
ˆ
Y (t) is the estimated output, K is the
proportional observer gain and L is the integral gain to be
computed.
˜
Y (t) = Y (t) −
ˆ
Y (t). Let us define the state
estimation error ˜x(t) and the fault estimation error
˜
f(t):
˜x(t) = X(t) −
ˆ
X(t) and
˜
f(t) = f(t) −
ˆ
f(t) (6)
The dynamics of the state estimation error is given by the
computation of
˙
˜x(t) which can be written:
˙
˜x(t) =
˙
X(t) −
˙
ˆ
X(t)
= (A
a
− KC
a
)˜x(t) + E
a
˜
f(t) + F
a
w(t) (7)
The dynamics of the fault estimation error is:
˙
˜
f(t) =
˙
f(t) −
˙
ˆ
f(t)
=
˙
f(t) −LC
a
˜x(t) (8)
Let us introduce:
ϕ(t) =
˜x(t)
˜
f(t)
and ε(t) =
w(t)
˙
f(t)
(9)
From (7) and (8), one can obtain:
˙ϕ(t) = A
0
ϕ(t) + B
0
ε(t) (10)
with:
A
0
=
A
a
− KC
a
E
a
−LC
a
0
and B
0
=
F
a
0
0 I
(11)
In order to analyse the convergence of the generalized
estimation error ϕ(t), let us consider the following quadratic
Lyapunov candidate function V (t):
V (t) = ϕ
T
(t)P ϕ(t) (12)
where P denotes a positive definite matrix.
The problem of robust state and fault estimation reduces
to finding the gains K and L of the observer to ensure an
asymptotic convergence of ϕ(t) toward zero if ε(t) = 0 and
to ensure a bounded error when ε(t) 6= 0, i.e.:
lim
t→∞
ϕ(t) = 0 for ε(t) = 0
kϕ(t)k
Q
ϕ
≤ λkε(t)k
Q
ε
for ε(t) 6= 0
(13)
where λ > 0 is the attenuation level. To satisfy the con-
straints (13), it is sufficient to find a Lyapunov function V (t)
such that:
˙
V (t) + ϕ
T
(t)Q
ϕ
ϕ(t) −λ
2
ε
T
(t)Q
ε
ε(t) < 0 (14)
where Q
ϕ
and Q
ε
are two positive definite matrices.
The inequality (14) can also be written as:
ψ(t)
T
Ωψ(t) < 0 (15)
with:
ψ(t) =
ϕ(t)
ε(t)
, Ω =
A
T
0
P + P A
0
+ Q
ϕ
P B
0
B
T
0
P −λ
2
Q
ε
(16)
The quadratic form in (15) is negative if Ω < 0. The
matrix A
0
can be expressed as:
A
0
=
˜
A −
˜
K
˜
C (17)
with:
˜
A =
A
a
E
a
0 0
,
˜
K =
K
L
,
˜
C =
C
a
0
(18)
The presence of the terms P
˜
K and λ
2
let the inequality
Ω < 0 nonlinear, to linearize it, let us define the following
changes of variables G = P
˜
K and m = λ
2
. The matrix Ω
can then be written as:
Ω =
P
˜
A +
˜
A
T
P − G
˜
C −
˜
C
T
G
T
+ Q
ϕ
P B
0
B
T
0
P −mQ
ε
(19)
The resolution of the inequality Ω < 0, that is now linear
with regard to the different unknowns, leads to find the
matrices P and G and the scalar m. The gain matrix
˜
K
is determined via the resolution of
˜
K = P
−1
G and the
attenuation level λ is given by λ =
√
m.
B. Example
Let us consider the linear system described by the follow-
ing matrices:
A =
−0.3 −3 −0.5 0.1
−0.7 −5 2 4
2 −0.5 −5 −0.9
−0.7 −2 1 −0.9
, B =
1 2
5 1
4 −3
1 2
,
D =
0.5 0.5
0.2 0.2
0.1 0.1
0 0.1
, F =
4 6
0 0
−4 2
7 6
C = I and E = B. The system input u(t) is defined by
u(t) =
u
T
1
(t) u
T
2
(t)
T
, where u
1
(t) is a telegraph type
signal varying between zero and one and u
2
(t) is defined by
u
2
(t) = 0.4 + 0.25 sin(πt). The actuator fault f
a
(t) is made
up of two components
f
a
(t) =
f
T
a1
(t) f
T
a2
(t)
T
(20)
with:
f
a1
(t) =
0.4 sin(πt), 15 s < t < 75 s
0, otherwise
,
f
a2
(t) =
0, t < 20 s
0.3, 20 s < t < 80 s
0.5, t > 80 s
and the sensor fault f
s
(t) is defined as follows:
f
s
(t) =
f
T
s1
(t) f
T
s2
(t)
T
(21)
with:
f
s1
(t) =
0, t ≤ 35 s
0.6, t > 35 s
, f
s2
(t) =
0, t ≤ 25 s
sin(0.6π t), t > 25 s
To define the state z, one choose
¯
A = 25 × I, where I is
the identity matrix.
Using the previously described method with Q
ϕ
= Q
ǫ
= I
leads to the obtention of the observer gain K and L. The
resulting attenuation level is λ = 0.3162 and:
K =
116.0253 −40.0901 −23.0977 −37.0989
−16.9977 78.3129 30.9322 56.9434
−161.4062 149.2055 4.9692 135.7089
112.4707 −74.5075 34.8011 −91.8975
12.0452 18.2418 −3.6436 25.1209
−2.9528 22.0100 6.1366 22.9357
6.0123 −2.7081 5.2831 −1.9449
0.5412 −3.6433 2.3397 8.0328
L =
−76.7558 122.3517 22.1039 125.0592
185.6710 −104.5700 −37.2061 −17.9442
6.9387 −25.9424 −107.0508 152.6953
49.4870 −71.5776 56.8719 108.1356
The simulation results are shown in the figures 1 and 2. This
method allows to estimate well the faults affecting the system
even in the case of time-varying faults.
0 10 20 30 40 50 60 70 80 90 100
−1
−0.5
0
0.5
1
0 10 20 30 40 50 60 70 80 90 100
−1
−0.5
0
0.5
1
Fig. 1. Actuator faults and their estimation
0 10 20 30 40 50 60 70 80 90 100
−1
−0.5
0
0.5
1
0 10 20 30 40 50 60 70 80 90 100
−1
−0.5
0
0.5
1
Fig. 2. Sensor faults and their estimation
III. EXTENSION TO MULTIPLE MODEL REPRESENTATION
The objective of this part is to extend the previous pro-
posed method to nonlinear systems represented by a Takagi-
Sugeno model.
A. Problem formulation
Consider the following nonlinear Takagi-Sugeno system
affected by sensor faults, actuator faults and a measurement
noise described by:
˙x(t) =
M
X
i=1
µ
i
(ξ(t))(A
i
x(t) + B
i
u(t) + E
i
f
a
(t))(22a)
y(t) = Cx(t) + F f
s
(t) + Dw(t) (22b)
where x(t) ∈ IR
n
represents the system state, y(t) ∈ R
m
is the measured output, u(t) ∈ IR
r
is the system input,
f
a
(t) and f
s
(t) represents respectively actuator and sensor
faults and w(t) is the measurement noise. A
i
, B
i
and C
are known constant matrices with appropriate dimensions.
E
i
, F and D are respectively the actuator faults, the sensor
faults and the noise distribution matrices which are assumed
to be known. The scalar M represents the number of local
models. The weighting functions µ
i
are nonlinear and depend
on the decision variable ξ(t) which must be measurable.
The weighting functions satisfy the convex sum property
expressed in the following equations:
0 ≤ µ
i
(ξ(t)) ≤ 1,
M
X
i=1
µ
i
(ξ(t)) = 1 (23)
Let us consider the state z ∈ IR
p
given by:
˙z(t) =
M
X
i=1
µ
i
(ξ(t))(−
¯
A
i
z(t)+
¯
A
i
Cx(t ) +
¯
A
i
F f
s
(t) +
¯
A
i
Dw(t))
(24)
where −
¯
A
i
, i ∈ 1, .., M are stable matrices. The dynamics
of the augmented state X(t) =
x
T
(t) z
T
(t)
T
is
governed by:
˙
X(t) =
M
X
i=1
µ
i
(ξ(t))(A
ai
X(t) + B
ai
u(t) + E
ai
f(t) + F
ai
w(t))
(25a)
Y (t) = C
a
X(t) (25b)
with:
A
ai
=
A
i
0
¯
A
i
C −
¯
A
i
, B
ai
=
B
i
0
, (26)
E
ai
=
E 0
0
¯
A
i
F
, F
ai
=
0
¯
A
i
D
(27)
The matrices C
a
and f are given by the equation (4). The
structure of the proportional integral observer is chosen as
follows:
˙
ˆ
X(t) =
M
X
i=1
µ
i
(ξ(t))(A
ai
ˆ
X(t) + B
ai
u(t) + E
ai
ˆ
f ( t) + K
i
˜
Y (t))
(28)
ˆ
f ( t) =
M
X
i=1
µ
i
(ξ(t))L
i
˜
Y (t) (29)
ˆ
Y (t) = C
a
ˆ
X(t) (30)
where
ˆ
X(t) is the estimated system state,
ˆ
f(t) represents
the estimated fault,
ˆ
Y (t) is the estimated output, K
i
are the
local model proportional observer gains and L
i
are the local
model integral gains to be computed and
˜
Y (t) = Y (t)−
ˆ
Y (t).
Using the expressions of ˜x(t) and
˜
f(t) given by the
equation (6), the dynamics of the state reconstruction error
is given by:
˙
˜x(t) =
M
X
i=1
µ
i
(ξ(t))((A
ai
− K
i
C
a
)˜x(t) + E
ai
˜
f(t) + F
ai
w(t))
(31)
The fault estimation error can be expressed as:
˙
˜
f(t) =
˙
f(t) −
M
X
i=1
µ
i
(ξ(t))L
i
C
a
˜x(t) (32)
Using the definitions of ϕ and ε given in (9) and omitting
to denote the dependance with regard to the time t, the
equations (31) and (32) can be written:
˙ϕ = A
m
ϕ + B
m
ε (33)
with:
A
m
=
M
X
i=1
µ
i
(ξ)
˜
A
0i
and B
m
=
M
X
i=1
µ
i
(ξ)
˜
B
0i
(34)
where:
˜
A
0i
=
A
ai
− K
i
C
a
E
ai
−L
i
C
a
0
,
˜
B
0i
=
F
ai
0
0 I
(35)
By considering the Lyapunov function V (t) given in (12),
and following the same reasoning as for linear systems,
convergence of state and fault estimation errors as well as
attenuation level are guaranteed if:
ψ(t)
T
Ω
m
ψ(t) < 0 (36)
with:
ψ =
ϕ
ε
, Ω
m
=
A
T
m
P + P A
m
+ Q
ϕ
P B
m
B
T
m
P −λ
2
Q
ε
(37)
The inequality (36) holds if Ω
m
< 0. Following the same
steps as for the linear case, let us define:
A
0i
=
˜
A
i
−
˜
K
i
˜
C (38)
with:
˜
A
i
=
A
ai
E
ai
0 0
,
˜
K
i
=
K
i
L
i
,
˜
C =
C
a
0
(39)
Using the changes of variables G
i
= P
˜
K
i
and m = λ
2
,
the matrix Ω
m
can be written as:
Ω
m
=
M
X
i=1
µ
i
(ξ(t))Ω
i
(40)
with:
Ω
i
=
P
˜
A
i
+
˜
A
T
i
P − G
i
˜
C −
˜
C
T
G
T
i
+ Q
ϕ
P B
m
B
T
m
P −mQ
ε
(41)
Sufficient conditions ensuring the negativity of Ω
m
can be
expressed as:
Ω
i
< 0, ∀i ∈ {1, . . . , M } (42)
Solving LMI’s (42) leads to the determination of the
matrices P and G
i
and the scalar m. The gain matrices are
then deduced:
˜
K
i
= P
−1
G
i
.
B. Example
Consider the nonlinear system described by a the Takagi-
Sugeno model given by the equation (22) with:
A
1
=
−0.3 −3 −0.5 0.1
−0.7 −5 2 4
2 −0.5 −5 −0.9
−0.7 −2 1 −0.9
, B
1
=
1 2
5 1
4 −3
1 0
A
2
=
−0.7 −7 −1.5 −7
−0.2 −2 0.6 1.3
5 −1.5 −9 −3.9
−0.4 −1 −0.3 −1
, B
2
=
1 1
2 1
0 2
−1 −2
D =
0.5 0.5
0.2 0.2
0.1 0.1
0 0.1
, F =
3.25 5
0 0.5
−3.25 1.75
5.75 5
E
1
= B
1
, E
2
= B
2
, M = 2, ξ(t) = u(t), C = I
