Simultaneous Sensor and Actuator Fault Reconstruction and Diagnosis
using Generalized Sliding Mode Observers
R. Raoufi†∗and H. J. Marquez‡
Abstract—A new filter for state and fault estimation in a class
of nonlinear systems is presented in this paper. The observer
benefits from both sliding mode control and singular systems
theory. The novelty of this approach is based upon dealing with
systems prone to faults at sensors and actuators during the
course of the system’s operation coincidentally. Conditions and
proofs of conversion for the proposed observer are presented.
A noticeable feature of the proposed approach is that the state
trajectories do not leave the sliding manifold even in presence
of sensor/actuator faults. This allows for actuator faults to
be reconstructed based upon information retrieved from the
equivalent output error injection signal. Due to employing a
generalized state space form (singular system theory), the sensor
faults are also estimated.
It is often the case when dealing with complex systems
requiring safe operation, that some form of supervisory
function is needed to indicate undesired process states or
“faults”. Faulty signals can exist in actuators, sensors and
process components that can deteriorate normal operation or
even lead to instability. Taking immediate and appropriate
actions in order to preserve safe operation while avoiding
possibly catastrophic damages is crucial. Thus, fault
detection and isolation (FDI) is of significant technical
importance. Model-based FDI schemes employ measured
variables and use mathematical models of the system
to detect abnormal conditions. Once a fault is detected,
diagnosis and isolation is performed, and decisions and
counteractions are then taken. Model-based FDI schemes
have proved effective in many successful implementations;
(see  and the references therein). In particular, observer-
based techniques for FDI have drawn significant attention
(see for example , , , ,  and references
therein). This paper deals with observer design for fault
detection and estimation using sliding mode observers
(SMOs) in a generalized state space form inspired by
singular system theory. Due to their ability to cope with
model uncertainties, SMOs offer great potential in fault
detection applications .
 introduced the use of the sliding mode approach in
observer design and used the Lyapunov theory to prove
stability.  proposed an alternative approach to the design
of a sliding mode observer using a discontinuous sliding
∗Corresponding author. Email: Raoufi@ece.ualberta.ca
†Department of Electrical and Computer Engineering, The University of
Alberta, Alberta, Canada.
‡Department of Electrical and Computer Engineering, The University of
Alberta, Alberta, Canada.
term fed back through a suitable gain. SMOs for linear
unknown input systems were studied in .  proposed
a canonical sliding observer form design for linear system
where a sufficient condition for stability based on linear
matrix inequalities (LMIs) was derived. Observation of
linear systems with unknown inputs via high-order sliding-
mode was addressed in  and . More recently,
development of sliding mode observers for unknown input
systems was proposed in . A robust fault detection
method for nonlinear systems with disturbances was studied
in  where strict geometric conditions where exploited.
Applications of SMO for fault tolerant control of linear
systems were addressed in  and .
The main focus of this paper is to explain the design
of sliding mode observers for the problem of fault
reconstruction and FDI. Recently,  and  proposed
that using the variable structure control law of the SMO
and the concept of equivalent output injection, a fault
can be reconstructed to any required accuracy for linear
systems. Sensor fault reconstruction using SMO was studied
in , further extending the results for linear systems
with disturbance and uncertainty. For nonlinear Lipschitz
systems,  addressed SMO based fault reconstruction by
assuming that disturbances are matched and can be lumped
into the so-called matching condition. Inspired by the
theory of singular systems,  proposed a new generalized
state-space observer design to estimate unknown signals
for a class of nonlinear systems. In  an interesting
method to design descriptor observers for systems with
measurement noise and application to sensor fault diagnosis
was proposed. State/noise observer for descriptor systems
with application to sensor fault diagnosis was also studied in
. The approaches of these references play an important
role in inspiring our observer design.
Fault reconstruction is excellent for directly isolating the
flaws within a system by revealing which sensor or actuator
is faulty and is useful for diagnosing incipient and small
faults. The detailed knowledge of the fault’s shape, obtained
from fault reconstruction, can highly facilitate the fault
tolerant control design. However, in practical systems, it is
often the case where actuators and also sensors suffer from
faults during the course of the system’s operation. Both
Actuators and Sensors can suffer from faults either alone,
at separate times or simultaneously. In this case, detection
and reconstruction of all faults is highly important. The
co-existence of unknown fault at both some sensor(s) and
2010 American Control Conference
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June 30-July 02, 2010
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matrix. Let L4= Λ−1L4, we obtain the stable matrix ΛA
From (10) and (20), we obtain
˙ e1= (A1−L1C1)e1+A2e3+B1f−B1ν(t)+BΦ1eΦ (61)
Let L1= (A1− As
˙ e1= As
1e1+ A2e3+ B1(f − ν(t)) + BΦ1eΦ
Consider V1= eT
1P1e1and V3= eT
3P3e3. We define
V = V1+ V3.
Then the derivative of V1and V3are given by:
1P1B1(f − ν) + (f − ν)TBT
From (12) it follows that e1= C−1
the switching gain (21) and (2) we obtain
1S1y − ˆ x1. Then using
1P1B1(fa(t) − ν(t)) =
1P1B1fa(t) − (ρ + ρ0)?eT
1P1B1? − (ρ + ρ0)?eT
Consequently, by choosing L4= Λ−1P−1
to (63), (64) and (65), the stability criteria ˙V
equivalent to the following inequality
1P1B1? = −ρ0?eT
1P1B1? < 0
3 V and with regard
< 0 is
1 P1+ P1As
?T. Thus if −Q < 0,
3P3ΛFsfs(t) + fs(t)TFT
sΛTP3e3 < 0
where ˜ eT=?
˙V ≤ −?˜ e?(λmin(Q)?˜ e? − 2?P3ΛFs?ω0),
?˜ e? ≥ 2λmin(Q)−1?P3ΛFs?ω0
it follows that˙V < 0 which guarantees that the magnitude
of the error is ultimately bounded with respect to the set:
˜ e : ?˜ e? <2?P3ΛFs?ω0
+ ε0,ε0> 0
where ε0 is an arbitrarily small positive scalar. This
completes the proof.
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