to unknown inputs has received considerable attention [4] and [11]. However, a very few works have been developed ... state space. ..... a symmetric positive definite matrix, unique solution of the Lyapunov equation (21). A11i. T. 11i. P. P. Qi. 1.
Proceedings of the 42nd IEEE Conference on Decision and Control Maui, Hawaii USA, December 2003
TuM14-2
Sliding mode multiple observer for fault detection and isolation Abdelkader Akhenak, Mohammed Chadli, Didier Maquin, José Ragot Centre de Recherche en Automatique de Nancy - CNRS UMR 7039 Institut National Polytechnique de Lorraine 2, avenue de la forêt de Haye - 54516 Vandoeuvre-lès-Nancy cedex {aakhenak, mchadli, dmaquin, jragot}@ensem.inpl-nancy.fr Abstract
was found that the system states could be forced to reach and subsequently remain on a pre-defined surface in the state space. Whilst constrained to this surface, the resulting reduced-order motion – referred to as the sliding motion – was shown to be insensitive to any uncertainty or external disturbance signals which were implicit in the input of the system. This inherent robustness property has resulted in world wide interest and research in the area of sliding mode control. These ideas have subsequently been employed in other situations including the problem of state estimation via an observer.
This paper deals with the design of a sliding mode multiple observer (an observer based on a multiple model) allowing to estimate the state vector of a non linear dynamical system. This latter is influenced by unknown inputs which act on it through a known transmission matrix. The state estimation and consequently the output estimation can therefore be classically used for detecting and isolating faults. Keywords: multiple model, multiple observer, sliding mode, state estimation, unknown inputs. 1
The earliest work of Utkin is based on a discontinuous structure for the observer as described in [5]. Walcott and Zak use a Lyapunov-based approach to formulate and synthesize an observer which, under appropriate assumptions, exhibits asymptotic state error decay in the presence of bounded nonlinearities and uncertainties on the input of the system [4]. Edwards and Spurgeon propose an observer strategy, similar in style to that of Walcott and Zak, which circumvents the use of a symbolic manipulation and offers an explicit design algorithm. Within the framework of the multiple model approach, the synthesis of regulators by using sliding mode was also considered [10].
Introduction
The general procedure for using an observer for fault detection and isolation consists of three main steps: 1. 2. 3.
Estimating the output measurement of the system by using an adapted structure of the observer. Comparing the estimated and the measured outputs, i.e. generating the so-called residuals. Analyzing the residuals and deciding if a fault occurred or not.
The decision process may be based on a simple threshold test applied on the instantaneous value or on a moving average of the residuals. However, when the system under consideration is subject to unknown disturbances or unknown inputs, to properly achieve fault detection needs the effect of the disturbance to be de-coupled from the residual signal; that allows to avoid false alarms in the detection procedure. This problem is known in the literature as the robust fault detection problem which is mainly solved by using unknown input observers [7].
The presented work consists in conceiving a sliding mode multiple observer, capable of reconstructing the state and the output vectors of a system when some inputs are unknown, such as each local observer is modeled in the same way of Walcott’s and Zak’s observer (1988). 2.1 Multiple model representation Let us consider a nonlinear system represented by the following multiple model (with r is the number of local models) with unknown inputs:
The problem of state estimation of linear systems subject to unknown inputs has received considerable attention [4] and [11]. However, a very few works have been developed for nonlinear systems [3] and [12]. The purpose of this work is to propose a methodology for the design of a nonlinear observer of this type of systems. 2
%K (1) &Kx1t6 = ∑ µ 2ξ1t672A x1t6 + B u1t6 + R u1t6 + D 7 y t = Cx t '16 16 %K µ 2ξ1t67 = 1 such that : &∑ K'0 ≤ µ 2ξ1t67 ≤ 1 ∀ i ∈ ;1,..., r@ where x1t 6 ∈ R is the state vector, u1t 6 ∈ R is the input vector, u1t 6 ∈ R the vector of unknown inputs and y1t 6 ∈ R the vector of measurable output. For the ith r
i
i
i
i
i
i=1
r
Sliding mode multiple observer
i=1
i
i
The concept of sliding mode emerged from the Soviet Union in the late sixties where the effects of introducing discontinuous control action into dynamical systems were explored. By the use of a judicious switched control law, it
0-7803-7924-1/03/$17.00 ©2003 IEEE
n
m
q
p
953
!
"# $
local model, A i ∈ R n × n is the state matrix, Bi ∈ R n × m is
In − p ~ T= C1
of the unknown inputs and Di ∈ R n × l
~ where T is a non singular matrix. With respect to this new coordinate system, the new output distribution matrix can be written as: ~ ~ C = CT −1 = 0 I p (6)
the matrix of input, R i ∈ R n ×q is the distribution matrix
is a matrix
depending on the operating point. Moreover, C ∈ R p × n is the matrix of output. It is assumed that the matrices R i are perfectly known ; on the contrary the time evolution of u t is unknown. Finally, ξ t represents the vector of decision depending on the input and/or the measurable state variables: the value of ξ t allows to specify what are the active local models at time t. The procedure that allows to obtain this structure and to estimate its parameters is not developed here. Let us only state that one can either uses techniques of parametric estimation [8] or linearization techniques [9].
16
16 16
0 C2
(5)
The other system matrices are written as: ~ ~ ~ ~ ~ ~ ~ ~ B A A A i = TA i T −1 = ~ 11i ~ 12i , Bi = TBi = ~ 1i B2i A 21i A 22i ~ ~ ~ ~ ~ D1i ~ R 1i Di = TDi = ~ , R i = TR i = ~ D2i R 2i
!
"# #$
"# ! #$
2.2 Multiple observer structure
The Lyapunov matrices
The proposed observer for the multiple model (1), is a linear combination of local observers, each of them having the structure proposed by Walcott and /DN In this part, we consider that the inputs u t are
~ ~ T ~ P = T −1 PT −1
16
4 9 ~ ~ = 4T 9 Q T
~ −1 Qi ~ ~~ CT FiT = PR i T
⋅
represents the Euclidean norm. It is also assumed that there exist matrices G i ∈ R n × p such that A 0 i = A i − G iC have stable eigenvalues and that there exist Lyapunov pairs P, Q i such that the
1
CT FiT = PR i ,
6
;
@
i
~ where ~ x t = Tx t = ~ x1 t
(3)
i i
16
1 6 ~x 1t6
16
4
6
Proposition 1: let
where
C1 ∈ R
following change ~ ~ x t = Tx t ,
16
16
i
of
∈R
1 6
and det C2 ≠ 0 .
coordinates
is
then
i
−1
11i 21i
i
i
1i 2i
i
"# #$
!
16
"# #$
~ B A 12i , Bi = TBi = 1i B2i A 22i ~ ~ R 1i = TR i = et C = CT −1 R 2i
"# ! #$
4A~ , B~ , R~ , C~9 be a local model for ~ which there exists a pair 4P, F 9 defined by constraints (8-
(4) 2
@
16
C = C1 C2 p×p
; 9
4
%KA = TA~ T = A K& !A ~ D KKD = TD = "#, R !D #$ '
6, C
(10)
9
9
As the outputs of the system are to be considered for the design of the observer, it is logical to effect a coordinates change so that the outputs directly appear as components of the new state vector. Without loss of generality, the output distribution matrix can always be written as:
1
T
~t . singular similarity transformation T , where x t = Tx Then, the system matrices are written in the new base as follows [4]:
Let us suppose that all the pairs A i , C are observable.
p× n −p
(9)
2
4
2.3 First change of coordinates
1
i
Now the result concerning the conception of a robust observer in the presence of unknown inputs established by Walcott and Zak may be used. This result is then extended to the conception of a multiple observer. ~ ~ ~ ~ Let the local models A i , Bi , R i , C defined by equation (9) ~ where A i are stable matrices ∀ i ∈ 1,..., r , and ~ ~ ~ ~ A i , Bi , R i , C be related to A i , Bi , R i , C by a non-
variables νi t , with νi t ∈ R q , which guarantee the exponential convergence of x t towards x t . Let us note that equation (2) allows to isolate the unknown inputs. In order to estimate the state vector of the system (1), we are going to proceed to two successive coordinate changes of the state vector.
16
i
2.4 Isolating the unknown inputs
One can determine the matrices G i and the control
16
i
Summarizing, the change of coordinates allows to express directly the output vector as a part of the state vector.
i
16
i
2
r
i=1
(8-b) (8-c)
16
%K A x 1t6 + B u1t6 − x1 t 6 = ∑ µ 2ξ1 t 67 ν G Cx t − y t + K t &K 2 1 6 1 67 1 6 'y 1t6 = Cx 1t6 i
−1
i=1
(2-b)
i
i
(8-a)
r
are satisfied for some Fi ∈ R q × p . The proposed observer has the form: i
and the structural
i
%K~ ~ ~ ~ ~ x1 t 6 = ∑ µ 2ξ1 t 674 A ~ x 1 t 6 + B u1 t 6 + R u 1 t 6 + D 9 &K 'y1t6 = ~x 1t6
(2-a)
∀ i ∈ 1,..., r
(7)
According to definitions (7), the system (1) can be rewritten under the following form:
structural constraints: T A 0i P + PA 0 i = −Q i
"# ! #$ 1P, Q 6
constraints (2) became, in the new coordinates, as follows:
16
bounded, such as u t ≤ ρ, where ρ is scalar and
"# ! #$
The
i
i
i
i
operated:
c).
954
Then,
there
exists
a
non-singular
similarity
4
9
transformation T so that the quadruple A i , Bi , R i , C in
P=
the new coordinates exhibits the following properties: 1. A 0 i = A i − G i C = A 011i ∈ R 1 A 11i
011i
A 012 i
021i
A 022 i
"# where #$
n−p × n−p
−1
4 9G 0 "# where P = !P F $ i
* * ∈ R p × p with P22 = P22
4 8
* 22
* T 22 i
T
>0
~ 3. C = CT −1 = 0 Ip
! 61
T~ P 0 4. P = T −1 PT −1 = 1 0 P2
4 9
where P1 ∈ R 1
n−p × n−p
6
"# #$
Ip
A 012 i
021i
A 022 i
⇒
%&A 'A
T
"# P #$ ! 0
1
"# A P #$ ! A
"# + P P #$ ! 0 0
1
2
0
011i
A 012 i
2
021i
A 022 i
"# < 0 #$
T 011i P1 + P1A 011i < 0 T 022 i P2 + P2 A 022 i < 0
11
= A 11i
4G C 9
and since
i
11
= 0,
9
~ ~ ~ ~ model described by A i , Bi , R i , C . Then, there is a non-
4
9
singular transformation T from which the multiple model with unknown inputs can be written in the following form:
%K &Kx1t6 = ∑ µ 2ξ1t674A x1t6 + B u1t6 + R u1t6 + D 9 'y 1 t 6 = x 1 t 6 r
"# #$
i
(11-a)
49
! ~ P " P # I #$ !P
~ ~ P R i = TR i = 11 0
!
12 p
* 11 * 21
* 12 * 22
T i
p
* T 22 i
2
1i
1i
22 i 2
2i
2i
2i
1
According to equation (13), the proposed multiple observer has the following form:
(11-b)
%K A x 1t6 + B u1t6 + D − x1t 6 = ∑ µ 2ξ1t 67 &K G e 1t6 + K ν 1t6 K'y 1t6 = Cx 1t6 A "# and K = 0 " with G = !P R #$ !A − A #$ where x 1t 6 represents the estimated state vector. A r
i
i
12 i
i
22 i
@
C ∀ i ∈ 1,..., r . Using the = PR i ~ partitioning of P and T , a direct computation leads to:
i
i y
i=1
4 9
;
12 i 2
21i 1
i
i=1
Lyapunov matrix for A 0 i and satisfies the structural FiT
11i 1
2
~ If there exists a Lyapunov matrix P which satisfies T~ constraints (8), then the matrix P = T −1 PT −1 is a T
i
i=1 r
♦
and so property 2 is proved
(13)
r
~ ~ ~ ~ obtains R i = P −1CT FiT . If P −1 is −1 ~ * P * P12 P12 = 11* , one obtains: ~ * P21 P22 P22
"# "# #$ ! $ 0 "# P " 0 " F = # # P #$ !I $ !P F $
i
1 6 ∑ µ 2ξ1t674A x 1t6 + A x 1t6 + B u1t6 + D 9 A x 1 t 6 + A x 1 t 6 + B u1 t 6 + x 1t 6 = ∑ µ 2ξ1t 67 R u1t6 + D y1 t 6 = x 1 t 6 Notice that x 1t 6 does not depend explicitly upon the unknown inputs u1t 6 . x 1 t =
♦
From equation (8), one ~ ~ P expressed as P −1 = ~11 P21
i
or in a developed form:
the expression considering the
equation (6) and (11-a).
i
2
4 9
In the new coordinate system, ~ C = CT −1 = 0 Ip is obtained by
i
i=1
~ which is non-singular since det T = det P11 ≠ 0 because ~ ~T P11 = P11 > 0.
constraints
011i
4
%K& K'
Let us define the change of coordinates using the following transformation: 11
A !A
Let us suppose that there exists a pair of Lyapunov ~ ~ matrices P, Q i checking the constraint (8) for each local
~ ~ ~ n−p × n−p ~ P P12 P11 ∈ R 1 6 1 6 is regular P = ~11 where ~ ~ ~ n − p6× p 1 T P12 ∈ R and P22 ∈ R p × p P12 P22
~ P12
"# #$
2.5 Synthesis of a multiple observer
written as follows:
P~ T= !0
p
"# #$ !
∀ G i ∈ R n × p (see property 3), the matrices A 11i are also stable, so property 1 is proved.
9
"# #$
"# #$!
11 T 12
4 9
4 9
!
0
As A 011i = A 11i − G i C
and P2 ∈ R p × p .
~ Proof: Let us consider the pairs P, Fi associated to the ~ ~ ~ ~ ~ local model A i , Bi , R i , C and the Lyapunov matrix P
4
~ −1 ~ −1 ~ ~ ~ P12 P11 − P11 P12 P −1 0 = 11 (12) ~ Ip P22 0 0 P2
"# P~ ~ I #$ !P
−1 11 T −1 12 11
~ ~ T ~ −1~ where P2 = P22 − P12 P11 P12 and thus P has the required block diagonal structure of property 4. ♦ Finally, as the Lyapunov matrix P related to A 0 i has been demonstrated to be block diagonal, the matrices A 011i and A 022 i are stable. Indeed, from equations (2-a) and (12), one obtains:
6 1 6 are stable that implies that n−p × n−p ∈ R 1 6 1 6 are stable.
~ G i = TT 2. R i
A !A
P~ ~ ~ !− P P
s 22
i
i
i
(14)
i
−1 2
2i
s 22
is a stable matrix and the discontinuous vector functions νi t are defined as follows:
955
16
% 1 6 K&K '
T T νi t = −ρEi Ei Ei
4
8
−1
16
16
e Ty t P2 R 2 i
if e y t ≠ 0 elsewhere
0
16 16 16
16
;
@
e y t = y t − y t , Ei = e Ty t R 2 i , ∀ i ∈ 1,..., r
with:
becomes: Taking into account (22), the expression of V
(15) and
T
4 8P
= −Q2
2
16
r
∑ µ 1ξ64−e Q e − ~e Q ~e
= V
16
i
2
i
i=1
(17)
s 22 2
21i 1 −1 2 2i i
1
6
+
e2T P2e2
i
T 1
T 11i 1
T 2
s 22 i
T 1
T 21i 2 2
T 2
1
11i
1
2
s 22
T
2
2i i
T 2 2
2 2i
+ 2ρ e2T P2 R 2 i
2i
9
T 2i
1i 1
2 2i
2
;
7
@
2. Let us suppose now that the output error e2 is zero; the is then written as: function V r
∑ µ 1ξ64−e Q e − ~e Q ~e 8 < 0
2
i=1
T 2i
1i 1
2 2i
;
7
@
Thus, we have demonstrated that the errors e1 t and e2 t tighten towards zero in an exponential way.
16
some definite the matrices
16
In conclusion, the multiple observer of the system (1) can be written as follows:
%K A x 1t6 + B u1t6 + D − &Kx1t6 = ∑ µ 2ξ1t67 G 2Cx 1t6 − y1t67 + K ν 1t6 'y 1t6 = Cx 1t6 %KG = TT~ A " , K = TT~ P TT~ R 4 9 4 9 K& 4 9 !A − A ##$ KK ν 1t6 = %K&−ρE 4E E 8 e 1t6P R if e 1t6 ≠ 0 K' 0 elsewhere ' r
i
i
−1
22 i
i
∑ µ 1ξ6 e−ePQAe e− e+ Q2eeR+ eν A− 2ePPe R+ u
where
16
i
T i
i
(23)
i i
−1
12 i
i
The derivative (19) can be shown to be:
i
i
i=1
(21)
T 21i 2 2 T 2 2 2i
T 1
i
for e1, ~ e2i ≠ 0, ∀ i ∈ 1,..., r
p×p
T P1 + P1Α11i = −Q A11i i
i=1
9
for e1, ~ e2 i ≠ 0, ∀ i ∈ 1,..., r
(20)
T 2 2 1 T 2 2i i
T 2i
1i 1
T 1
i=1
which are symmetric and definite positive too. n−p × n−p Let P1 ∈ R 1 6 1 6 a symmetric positive definite matrix, unique solution of the Lyapunov equation (21).
T T 1 i 1 2 T 2 2 21i 1
T 1
i
i
= V
= A T P Q −1P A + Q Q i 21i 2 2 2 21i 1i
− 2ρ e2T P2 R 2i − 2e2T P2 R 2i u
i=1 r
T 2 2 21i 1 T 2 2 2i
and Q 2 ∈ R Proof: let Q 1i positive matrices, and consider ∈ R 1 n − p 6× 1 n − p 6 defined by: Q i
i
2 2i
∑ µ 1ξ64−e Q e − ~e Q ~e − 2ρ e P R ~ ~ = ∑ µ 2ξ1t 674 − e Q e − e Q e 8 < 0
(19)
2
n−p × n−p ∈ R1 6 1 6
r
T 2i
1i 1
r
Proposition 2: there exists a symmetric positive definite matrix P2 checking (16), such that the dynamical errors (17) are asymptotically stable.
= V
T 1
i
≤ V
(18)
e 4A P + P A 9e + e A P P A e + + 4 9 V = ∑ µ 1ξ6 e A P e + e P A e + 2e R ν − 2e P R u i=1
∑ µ 1ξ64−e Q e − ~e Q ~e
As the unknown inputs are bounded, then:
Its derivative in respect to time, evaluated along the trajectory of the system by using equations (2) and (16), may be expressed as: r
9
i=1
In order to show the exponential convergence of this observer, let us consider the following Lyapunov function: V e 1, e 2 =
+ 2e 2T R 2 i ν i − 2e 2T P2 R 2 i u
r
= V
2i
Lyapunov equation
e1T P1e1
2 2i
1. Let us suppose that the output error e2 is different from zero. By using the expression (15) of νi , the derivative of the function V becomes:
11i 1
i=1 r
T 2i
1i 1
i=1
r
1
T 1
i
derivative, their dynamic evolutions check:
%K K&e 1t6 = ∑ µ 2ξ1t67A e 1t6 KKe 1t6 = ∑ µ 2ξ1t67 A e 1t6 + A e 1t6 + P R ν − R u1t6 K'
2 2i
21i 1 T 2 2 2i
By using the equation (20), the derivative of the Lyapunov function becomes:
(16)
16 16
T 2i
T −1 21i 2 2 2 T 2 2i i
i
with ~ e2i = e2 − Q 2−1P2 A 21ie1 .
Let us denote state estimation and output errors as e1 t = x 1 t − x 1 t and e2 t = e y t . By direct time
16
T 1
µi ξ
i=1
P2 ∈ R p × p is the unique symmetric positive definite solution of the Lyapunov equation: s s P2 A 22 + A 22
−e 4Q − A P Q P A 9e − 1 6 ∑ ~e Q ~e + 2e R ν − 2e P R u r
= V
s 22 − T 1 i
16 16
i
T y
2
2i
16
−1 2
i
y
;
@
e y t = Cx t − y t and Ei = eTy t R2 i , ∀ i ∈ 1,..., r
It is easy to verify that: T −1 −1 2 P2 A 21i e 1 Q 2 e 2 − Q 2 P2 A 21i e 1 = T T T T T -1 2 2 − e1 A 21i P2e 2 − e 2 P2 A 21i e1 + e 1 A 21i P2Q 2 P2 A 21i e1
4e − Q 4e Q e 2 T 2
8 4
3
9
8
(22)
Example
The selected nonlinear system is represented on figure 1. It results from a traditional benchmark [6] and
956
Simulation results
schematizes a hydraulic process made up of three tanks. These three tanks T1, T2 and T3 , with identical sections A, are connected one to each other by cylindrical pipes with identical sections Sn . The output valve is located at the
The simulation results are represented on the following figures. The convergence of the state vector of the multiple observer towards those of the multiple model is quite good. At the vicinity of t=0, the disparity between estimated and actual state is due to the choice of initial conditions.
output of tank T2 ( T2 it ensures to empty the tank filled by the pump flows 1 and 2 with respectively rates Q 1 t
16
16
and Q 2 t ). Two combinations of the three water levels are measured. The communication pipes between the tanks are equipped with manually adjustable ball valves, which allow the corresponding pump to be closed or open. The three levels x 1, x 2 and x 3 are governed by the constraint x 1 > x 3 > x 2 ; the process model is given by the equation (24).
x 10
10
Indeed, taking into account the fundamental laws of conservation of the fluid, one can describe the operating mode of each tank; one then obtains a nonlinear model expressed by the following state equations [6]:
%KA dx 1t6 = Q 1t6 − α S 42g2x 1t6 − x 1t679 KK dxdt1t6 KKA dt = Q 1t6 + α S 42g2x 1t6 − x 1t679 − α S 42g2 x 1t 679 &K dx 1t 6 KKA dt = α S 42g2x 1t6 − x 1t679 − KK α S 42g 2 x 1t 6 − x 1t 679 + Qf 1t 6 ' where α , α and α are constants. Qf 1t 6
Q 1(t)
-5
15
5 0 x 10
500
1000
1500
1000
1500
1000
1500
Q 2(t)
-5
2 .5
1/ 2
1
1
1 n
1
3
2
1/ 2
2
2
3 n
3
2
1 .5
1/ 2
2 n
(24)
2
1/ 2
3
1 n
1
0
3
x 10
1/ 2
3 n
3
2
4 3 2
3
1
16 16
0 0
4
i
i
i
500
Figure 1: Multiple model inputs
%K &Kx1t6 = ∑ µ 2ξ1t672A x1t6 + B u1t6 + R u1t6 + D 7 'y1t6 = Cx1t6 i
16
u t
2
denotes an additional mass flow caused by a leak that constitutes the unknown input and g is the gravity constant. The multiple model (1), with ξ t = u t , which approximates the nonlinear system (24), is described by: 1
500 -5
a c tu a l a n d e s tim a te d x 1 1 .1
i
i=1
1 0 .9
The matrices A i , Bi , C, and Di are calculated by linearizing the initial system (24) around different points chosen in the operating range of the system. Four local models have been selected in an heuristic way. That number guarantees a good approximation of the state of the real system by the multiple model.
0 .8 0 .7 0 .6 0
500
1000
1500
2000
a c tu a l a n d e s tim a te d x 2 0 .4
0 .3
Q1
Q2 x1
T1
T3
0 .2
0 .1
x3
0
T2
x2
500
1000
1500
2000
a c tu a l a n d e s tim a te d x 3
0 .8 0 .7 0 .6
Sn
0 .5 0 .4
16
Qf t
0 .3 0
Schema 1 : Three tank system
500
1000
1500
Figure 2: State estimation
957
2000
4
Conclusion
Takagi-Sugeno fuzzy models. IEEE Trans. on Fuzzy Systems, 8 (3), pp. 297-313, 2000.
In that paper, the design of a sliding mode non linear observer based on a multiple model has been proposed. The design of such observer relies on the existence of some matrices, namely P, Q i , Fi , i = 1,..., r , ensuring, on one hand, the stability of the observer and, on second hand, satisfying a structural constraint allowing to isolate the unknown but bounded inputs in a particular part of the state vector.
;
10. Y. Blanco, F. Gouaisbaut, W. Perruquetti, P. Borne. Sliding mode controller design using polytopic formulation. IEEE Conference on Decision and Control, CDC'2001, Orlando USA, December 2001.
@
11. Y. Guan, M. Saif. A novel approach to the design of unknown input observers. IEEE Trans. on Automatic Control, 36 (5), pp. 632-635, 1991.
Of course, the existence of such matrices depends on the number of unknown inputs with regards to the number of the measurements and the rank of the different associated matrices ; this point has not precisely been discussed in this paper because of space lacking. A first attempt of using this type of observer for fault detection and isolation has been presented on a well known three tank system. The quality of the obtained results seems to be sufficient to allow faults to be detected despite the presence of unknown inputs. Future works will deal with magnitude estimation of the unknown inputs.
12. D. Koenig, S. Mammar. Design of the class of reduced order unknown inputs nonlinear observer for fault diagnosis. American control conference, ACC’2001, Arlington, USA, June 2001.
References 1.
C. Edwards and S. K. Spurgeon. On the development of discontinuous observers. Int. J. Control, 59 (5), pp. 1211-1229, 1994.
2.
C. Edwards and S. K. Spurgeon. Sliding mode observers for fault detection and isolation. Automatica, 36 (4), pp. 541-553, 2000.
3.
A. Akhenak, M. Chadli, D. Maquin, J. Ragot. Multiple observer with unknown inputs. Application to a three tank system. IAR Annual Meeting, ICD, 2002.
4. B. L. Walcott and S. H. Zak. Observation of dynamical systems in the presence of bounded nonlinearities/uncertainties. IEEE Conference on Decision and Control, CDC’1988, pp. 961-966 , 1988. 5. V. I. Utkin. Principles of identification using sliding regimes. Soviet Physic: Doklady, Sliding mode in control optimisation (Berlin: Springer-Verlag) 26, pp. 271-272, 1992. 6. A. Zolghadri, D. Henry, M. Monsion. Design of nonlinear observers for fault diagnosis: a case study. Control Engineering Practice, 4 (11), pp. 1535-1544, 1996. 7.
R. J. Patton and J. Chen. Observer-based fault detection and isolation robustness and applications. Control Engineering Practice. 5 (5), pp. 671-682. 1997.
8.
K. Gasso, G. Mourot and J. Ragot. Structure identification in multiple model representation: elimination and merging of local models. IEEE Conference on Decision and Control, CDC'2001, Orlando, USA, December 2001.
9.
T. A. Johansen, R. Shorten, R. Murray-Smith. On the interpretation and identification dynamic
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