INTRODUCTION
Extensive class of FDI methods are developed analytically on the basis
of mathematical models describing systems. In the last 30 years, a few
researches have been performed in this region (Patton and Chen, 1995).
A modelbased FDI produces residual so that one can find fault through
it. Therefore this method does not detect the fault directly and this
can be mentioned as a main problem. For suitable residual production,
different suggestions have been proposed. For example, one can refer to
parity space (Ding et al., 2002) and residual production on the
basis of observer (Commault et al., 2003). Observer based residual
generation estimates the fault through output or state estimation of the
system using a proper observer. The residual here means fault estimation.
When the residual is evaluated, fault signal is constructed and separated
from the residual (Emami Naeini et al., 1988). For this purpose,
fixed and adaptive threshold level is used. In presence of uncertainty
or noise and in order to prevent from false alarms, selected FDI scheme
must be robust. It means it has to distinguish the fault and unknown input
impacts (Chen and Patton, 1999; Ibaraki et al., 2005; Chen and
Seif, 2006).
The fault estimation approaches imposed by use of observers are commonly
direct ways providing fault information and isolation. This type method
provides an estimate of the size and severity of the fault, which are
important in many applications. For example, one can refer to observer
design by use of sliding mode method. This method attracted attention
of many researchers due to its more robustness in comparison with design
based observer of Luenberger (Edwards and Spurgeon, 1994; Edwards et
al., 2000; Chen et al., 2005). For a class of linear systems,
a canonical form is suggested and then input and output faulty signals
are estimated using sliding mode observer (Edwards et al., 2000).
Problem associated with sliding mode observers in separation of actuator
faults is solved in (Chen et al., 2005) using a bank of sliding
mode observers.
In this study a canonical form together with adaptive technique are employed
to design an adaptive observer for FDI applications (Edwards et al.,
2000). Simulation results presented in this study show that the adaptive
observer is robust against parameters uncertainties. An adaptive observer
is designed for a model of an aircraft and its simulation results are
compared with those of the sliding mode given in Edwards et al.
(2000). This comparison illustrates robustness and eligibility of the
proposed method and it can be suggested as an alternative method for FDI
design with some confidences.
PROBLEM FORMULATION
Consider the following system:
where, Aε R^{nxn}, Bε R^{nxm}, CεR^{pxn},
Dε R^{nxq}, with q < p < n, X(t)εR^{n}
is the state vector, u(t)ε R^{m} is the input vector and
y(t)ε R^{p} is the output vector. Also f_{a}(t) and
f_{s}(t) represent the actuator and sensor faults, respectively
with limited value. Here, u (t) and y (t) are available. The matrices
C and D are assumed to be of full rank and we have the following relation
for system:
• 
Rank (CD) = q 
• 
Invariant zeros of (A,D,C) must lie in the open LHP 
In this case on the basis of (Edwards et al., 2000), one can find
linear transformation like T so that after its application in (1), the
following equation can be obtained (Z = TX):
where, z_{1}(t)ε R^{(np)}, z_{2}(t)ε
R^{p} and A_{11}(t)ε R^{(np)x(np)} is stable
matrix. There are the following relations between system 1 and 2:
OBSERVER DESIGN WITH SLIDING MODE
For system (2), consider observer equation as follows:
where,
is a stable design matrix that should be determined by designer. The discontinuous
vector v and error vector e_{y} are defined as follows:
where, P_{2}ε R^{pxp} Lyapunov Matrix for
and the scalar ρ is selected so that:
If we want to write observer relation for Eq. 1, it
is enough to apply linear transformation
on relation 4:
In Edwards et al. (2000), it is shown that one can estimate f_{a}(t)
and f_{s}(t) in term of v approximately:
where, scalar δ is small positive and it has been assume that f_{s}(t)
has slow changes.
OBSERVER DESIGN WITH ADAPTIVE TECHNIQUES
Theorem: For system (2) and observer (4) and by assuming stability
of matrix ,
by use of vector:
In finite time, one can make error estimation is quadratically stable
if fault is fixed or limited and has slow varying. In relation (11) P_{2}ε
R^{pxp} Lyapunov Matrix for
and α is design constant.
Prove: At first, by assuming f_{s}(t) = 0 and defining
and
we have:
With regard to stability of matrix A_{11}ε R^{(np)x(np)},
stability of e_{1}(t) is proved easily. By defining and
considering stability e_{1}(t), for proving stability of its output
estimation error, we rewrite:
By defining Lyapunov function as:
And defining
which Q is positive definite matrix and using (11), we have for its derivation
of V:
Therefore e_{y}(t) is stable and tends to zero.
With regard to the fact that estimation error e_{1}(t) and e_{y}
(t) become zero and use of relation (13) in steady state, one can reach
the following result:
Now, for state which f_{a} = 0 and f_{s} is available,
we rewrite the relations:
By use of the above relation, one can rewrite relations (12) and (13)
as follow:
It is found that f_{s} and
appear as disturbance in equation. With regard to the fact that value
of fault is zero or it has been assumed to have slow change, therefore
by selecting big value of α, estimation error can tend to zero. With
this assuming in steady state, one can reach the following relation like
(10):
It is clear that if matrix (A_{22}A_{21}A_{11}^{1}A_{12})
is nonsingular, one can calculate fault through the above relation; otherwise
due to defect of rank in matrix, one can not use directly the above said
relation. This occurs in aircraft model which is mentioned further.
SIMULATION RESULTS
Here, in order to show ability of method, linear model of longitudinal
movement of aircraft has been studied:
Where:
By application of Eigenstructure assignment method of Wang et al.
(2005) and considering closed loop roots according to below, state feedback
gain U(t) KX(t) is calculated as follows:
By use of algorithm (Edwards et al., 2000), one can find linear
transformation so that equation can be converted to canonical form (2):
By selecting general
form of observer is obtained. For two schemes mentioned for observer,
P_{2} = eye(5) has been considered which holds in relation
For the observer designed on the basis of sliding mode, value of ρ
= 75 and for observer designed on basis of adaptive method, value of α
= 50 has been considered. Now, if there is any change in equilibrium state
to the extent of 1rad sec^{1} in pitch rate, simulation will
have been performed in different states. (fs_{i} is indicator
of fault in ith of sensor and fa_{j} is indicator of fault in
jth of actuator):
• 
By assuming fa_{2} (t) = 2u(t2) (Fig.
1) and fs_{1} = u(t2) (Fig. 2) for
two observers designed with sliding mode and adaptive methods, simulations
are done and reconstructed fault signals have been drawn (Fig.
3, 4) 
• 
With above assumptions of (A) and 20% of changes in parameters of
system, simulations have been repeated (Fig. 5,
6) 
• 
By assuming fa_{2} (t) = r(t5)2r(t10)+r(t15) (Fig.
7) and fs_{1} = 0.1 [(r(t5)2r(t10)+r(t15)] (Fig.
8) for two designed with sliding mode and adaptive methods, simulations
are done and reconstructed fault signals have been drawn (Fig.
9, 10) 

Fig. 1: 
Fault signal occurred in second actuator (constant fault) 

Fig. 2: 
Fault signal occurred in first sensor (constant fault) 

Fig. 3: 
Reconstructed fault signal for state that constant fault has
occurred in second actuator 
• 
With above assumptions of (C) and 20% of changes in
parameters of system, simulations have been repeated (Fig.
11, 12) 
• 
By assuming 

for two observers designed with sliding mode and adaptive
methods, simulations are done and reconstructed fault signals have
been drawn (Fig. 15, 16) 

Fig. 4: 
Reconstructed fault signal for state that constant fault has
occurred in first sensor 

Fig. 5: 
Reconstructed fault signal for state that 20% of changes have
been made in parameters of system and constant fault occurred in second
actuator 

Fig. 6: 
Reconstructed fault signal for state that 20% of changes
have been made in parameters of system and constant fault occurred
in first sensor 

Fig. 7: 
Fault signal occurred in second actuator (triangular
fault) 

Fig. 8: 
Fault signal occurred in first sensor (triangular fault) 

Fig. 9: 
Reconstructed fault signal for state that triangular fault
has occurred in second actuator 
• 
With above assumptions of (E) and 20% of changes in
parameters of system, simulations have been repeated (Fig.
17, 18) 
For reconstruction of fault signal, relation (9) and (10) have been used for
design of observer on the basis of sliding mode and relation (17) and (21) have
been used for design of observer on the basis of adaptive method.

Fig. 10: 
Reconstructed fault signal for state that triangular fault
has occurred in first sensor 

Fig. 11: 
Reconstructed fault signal for state that 20% of changes have
been made in parameters of system and triangular fault occurred in second
actuator 

Fig. 12: 
Reconstructed fault signal for state that 20% of changes have
been made in parameters of system and triangular fault occurred in first
sensor 

Fig. 13: 
Fault signal occurred in second actuator (incipient fault) 

Fig. 14: 
Fault signal occurred in first sensor (incipient fault) 

Fig. 15: 
Reconstructed fault signal for state that incipient fault
has occurred in second actuator 
According
to (10) and (21) , it is clear that none of the two methods can recognize fault
in second sensor due to defect of rank in matrix (A_{22}A_{21}A_{11}^{1}A_{12}).
Result of simulation show simplicity and ability of the mentioned method.

Fig. 16: 
Reconstructed fault signal for state that incipient
fault has occurred in first sensor 

Fig. 17: 
Reconstructed fault signal for state that 20% of changes have
been made in parameters of system and incipient fault occurred in second
actuator 

Fig. 18: 
Reconstructed fault signal for state that 20% of changes
have been made in parameters of system and incipient fault occurred
in first sensor 
CONCLUSION
In this study, by use of adaptive technique a novel FDI is imposed. This
method can reconstruct a class of actuator and sensor faults. In order
to show capability of the recommended method, simulation results have
been compared with those obtained by SMO method on a model of an aircraft.
Simulation results confirm the capability of the mentioned method to reconstruct
fault signal, so SMO can be replaced by adaptive observer with some advantages
for fault detection and isolation purposes.