Fault Diagnosis for a Class of Nonlinear Systems by means of a
Polynomial Observer
J. L. Mata-Machuca, R. Mart´ ınez-Guerra, D. Hern´ andez-S´ anchez
Department of Automatic Control, CINVESTAV,IPN, Mexico D.F., Mexico
Phone (+52)555-747-3800 ext.4222 Fax (+52)555-747-3982 E-mail:jmata@ctrl.cinvestav.mx
Abstract— This paper deals about the problem of fault
diagnosis of a class of nonlinear systems using the theory of state
observers. In order to reconstruct the faults of the system, a
polynomial observer is proposed, which includes in its structure
correction terms of high order. The methodology consists of
add to the original system the faults as new states, which
increase the order of the observer. This scheme reconstructs
simultaneously faults and state variables. In addition, as a
comparative study, it has been designed an observer of reduced
order. Both techniques are applied to fault diagnosis of a three-
tank system.
Index Terms— fault diagnosis, observers, nonlinear system.
I. INTRODUCTION
A fault is a not allowed deviation of at least one charac-
teristic property or parameter of the system with respect to
its usual, normal or acceptable condition.
In the last three decades a considerable number of papers
related to the problem of fault diagnosis have been reported
[1]–[4]. Among the publications around the diagnosis of
faults have different approaches, for example, in [5]–[7] used
differential geometric tools; papers [8]–[10] are based on an
alternative method using differential algebraic techniques.
This work considers the problem of fault diagnosis of a
class of nonlinear systems. The outputs are mainly obtained
by sensors measurements. The outputs and the number of
faults determine if the system is diagnosable or not [11],
[12]. The fault diagnosis problem is considered as a problem
of fault signals observation. In this sense, the diagnosability
of a system is given by the algebraic observability condition
of the fault [8]. The main contribution of this article consists
of the solution of the fault diagnosis problem by means
of a polynomial observer. This work combines the ideas
of [13] on the design of observers for nonlinear Lipschitz
systems with the method introduced in [11], adding high
order correction terms
1
, such that the asymptotic convergence
of the observer is guaranteed. In addition, is designed an
observer of reduced order based on the free model approach,
which converges asymptotically. The proposed schemes to
the fault diagnosis and reconstructions of states are applied
to a three-tank system.
1
As it is well known, an Extended Luenberger observer can be seen as
a first order Taylor series around the estimated state, therefore to improve
the estimation performance high order terms are included in the observer
structure.
II. PROBLEM STATEMENT
Let us consider the following nonlinear system:
˙ x(t)= g(x, u, f )
y(t)= h(x, u)
(1)
where x ∈ R
n
is the state vector, u ∈ R
l
is the known
input vector, f =(f
1
,f
2
, ..., f
μ
) ∈ R
μ
is the faults vector
(unknown inputs), and y ∈ R
p
is the output vector. With g,h
analytic functions.
The fault vector f is unknown, which can be interpreted as
a state with uncertainties. The faults estimation is obtained by
an extended state. Consider the following extended system:
˙ x(t)= g(x, u, f )
˙
f
j
(t)= Ω
j
(x, u, f ) , 1 ≤ j ≤ μ
y(t)= h(x, u)
(2)
where Ω = (Ω
1
, Ω
2
,..., Ω
μ
) ∈ R
μ
is an unknown bounded
function, that is to say,
Ω(x, u, f )≤ N< ∞ (3)
The system (2) can be expressed in the following form,
˙ x(t)= Ax +Ψ(x, u, f )
˙
f
j
(t)= Ω
j
(x, u, f ) , 1 ≤ j ≤ μ
y(t)= Cx
(4)
where Ψ(x, u, f ) is a nonlinear function that satisfies the
Lipschitz condition, i.e.,
Ψ(x, u, f ) - Ψ(ˆ x, u, f )≤ Lx - ˆ x (5)
III. OBSERVABILITY AND DIAGNOSABILITY:
DIFFERENTIAL ALGEBRAIC APPROACH
Before proposing an observer for the extended system
(4), in this section are presented some definitions about
observability and diagnosability.
III-A. Definitions
The observability notion (diagosability) of a system, linear
or nonlinear, consists of the possibility of reconstructing the
state x (fault f ), having the knowledge of the output y and,
the input u, and possibly a finite number of their derivatives,
(y
(k)
, k ≥ 0 and u
(l)
(t), l ≥ 0). Next some definitions
concerning differential algebra [10], [11], [14] appear.
Definition 1: Let L and K be differentials fields. A dif-
ferential field extension L/K is given by two differentials
fields K, L, such that:
2010 7th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE 2010)
Tuxtla Gutiérrez, Chiapas, México. September 8-10, 2010.
IEEE Catalog Number: CFP10827-ART
ISBN: 978-1-4244-7314-4
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158