INTRODUCTION
The failures which occur in the system usually cause enormous loss
of productivity, expensive equipment and human lives ultimately, so the
design and analysis of fault detection scheme has attached great importance
and has received considerable attention. During the last two decades,
various approaches to fault detection (FD) using modelbased analytical
redundancy have been represented. The studies based on these methods mainly
focused on the robustness with respect to unknown factors (the disturbance,
noise) and the sensitivity to early fault. Several state estimationbased
FD methods are proposed, such as Luenberger robust observer (Ibaraki
et al., 2005), linear matrix inequalities (LMI) (Wang et al.,
2007), Hinfinite estimator (Collins and Song, 2000{Collins, 2000 #2})
and multiplemodel estimation method (Hofbaur and Williams, 2004).
Unlike the fault detection problem, which has been extensively investigated
in the literature, the fault isolation (FI) problem has received less
attention, especially in the case of nonlinear uncertain systems. In this
study, a key design issue of the fault isolation scheme is the adaptive
residual threshold associated with each isolation estimator. That is,
the residual of each fault isolation estimator is associated with an adaptive
threshold, which can be implemented online by using linear filtering methods
(Basin et al., 2005). The occurrence of a particular fault is excluded
if at least one of the residual components of the corresponding isolation
estimator exceeds its threshold in a finite time. Fault isolation is achieved
when all faults but one is excluded.
Meanwhile, the application of FD (without the consideration of FI) scheme
aims at LTI and openloop system. Such as the H_{–} estimation
algorithms to faulty aircraft (Melody et al., 2001). The problem
of fault detection was addressed through H_{–} robust estimator
using PopovTsypkin multipliers in (Collins and Song, 2000). Their analysis
were based on openloop linear timeinvariant (LTI) system response where
the filters were designed for a nominal condition and tested for a second
offnominal LTI plant for robustness analysis. In (Felicio et al.,
2002), the H_{–}FD Ricattibased approach is used to design
FD filters for an inverted pendulum. But the application was not fully
carried out in this case (i.e., no simulations). In fact, most practical
conditions of the system are nonlinear in nature and most failures are
more accurately modeled as nonlinear functions about the state/output
and input variables.
So, in this study, a FDI scheme using some nonlinear estimators to detect
and isolate the incipient faults in an aerocraft’s nonlinear uncertain
closedloop system is significative. In the presence of a failure, the
function exported by the online approximator which is the important component
of the nonlinear estimator can be used as an estimate of the possible
nonlinear fault function. Once the fault is detected, the isolation estimators
are activated for the purpose of fault isolation. A type of fault that
has occurred can be isolated if the residual associated with the matched
isolation estimator remains below its corresponding adaptive threshold
and at least one of the components of the residuals associated with all
the other estimators exceeds its threshold at some finite time. The simulation
results show the effectiveness of the application.
PROBLEM FORMULATION
Here, the nonlinear system in presence of the fault is formulated
as below:
The differential equation presents
the nominal model, η = [η,...η_{n}]^{T}
describe the model uncertainty, f = [f_{1},...f_{n}] presents
the fault occurs in the system. Many fault diagnosis schemes simplify
the fault function to the time function, actually, most fault is the nonlinear
function of the state variables x and input vector u.
Each time component β_{i}(i = 1,...n) can be described as
the form below (Trunov and Polycarpou, 2000):
where, γ_{i} is an unknown constant and γ_{i}>0,
it presents the rate at which the fault changes in state variable x evolves.
Assumption 1:The state variable x(t) is available for measurement
Assumption 2:Each modeling uncertainty function η_{i}
is bounded by a known constant
NONLINEAR FAULT DETECTION ISOLATION SCHEME
A class of N+1 nonlinear adaptive estimators are used in the proposed
FDI scheme, where, N is the number of nonlinear faults. One of the nonlinear
adaptive estimators is the fault detection and approximation estimator
used to detect faults. The remaining ones are fault isolation estimators
that are used for isolation purposes only after a fault has been detected.
The structure of the fault detection estimator and the fault isolation
estimators are given below, respectively.
Fault detection and approximation estimator: Based on (1), a nonlinear
FD estimator of the form (4) is considered:
where, is
the estimated state vector, presents
an online approximation model, is
an adjustable weight vector, G^{d} = (g^{d}_{l},...,g^{d}_{n})
is a positive value. The initial value is
chosed such that ,
which corresponds to the case where the system is normal (no fault).
A key component of the nonlinear adaptive estimator described by (4)
is the online approximator, denoted by ,
whose ith component has the structure as (5):
The next step in the construction of the fault detection estimator is
the design of the learning algorithm for updating the weights .
Let be
the state estimation error. Using techniques from adaptive control (Lyapunov
synthesis method) (Trunov and Polycarpou, 2000), the learning algorithm
of the online approximator is chosen as follows:
The projection operator P restricts the weights estimate to
a predefined region .
is
a symmetric positive definite learning rate matrix and Z denotes the gradient
matrix of the online approximator with respect to its adjustable weights,
i.e., .
The deadzone operator is
defined as follows:
The deadzone operator which
acts on the variable ε^{d}_{i} such that the output
of the operator is zero while no fault in system, that is .
The detection of the fault occurs if at least one state ε^{d}_{i}
of the state estimation error exceeds its deadzone boundary .
More precisely, the fault detection time is defined as the first instant
of time such that ,
that is:
By Eq. 1 and 4 it can be easily verified
that each component of the state estimation error satisfies:
Because of ,
then using the Eq. (3), the FD threshold can
be get as follow:
Next, the sensitivity of the fault detection scheme will be considered.
The faults occur at some unknown time T_{z} and develop with the
rate γ_{i}.
Theory 1: Consider the nonlinear fault diagnosis scheme described
by Eq. 4 and 6. If there exists an interval
of time [T_{z}+t_{1}, T_{z}+t_{2}], such
that at least one component f_{i}(x(t), u(t)) of the fault vector
f(x(t), u(t)) satisfies:
then the fault can be detected, that is (the
proof can be found in Frank and Seliger (1991).
Fault isolation estimators: Once a fault has been detected, the
isolation scheme is activated. Then construct the following nonlinear
adaptive estimators as isolation estimators:
where, for
i = 1,...,n, l...,N is the estimate of the fault parameter vector
in the ith state variable. For notational simplicity and without loss
of generality, assume that g^{l}_{i} = g_{i},
for all l = 1,...,N.
The design of fault isolation estimators is similar to the design of
the fault detection estimator. Each isolation estimator corresponds to
one of the possible types of nonlinear faults. Specifically,
if let be the ith component of the state estimation error vector of the
l th estimator, then the learning algorithm is chosen as (Zhang
et al., 2002):
where, is
a symmetric positive definite learning rate matrix.
The faultisolation decision scheme is based on the generalized observer
scheme (GOS) principle: if the lth fault occurs at some time T_{z}
and is detected at time T_{d}, then a set of adaptive thresholds
λ^{l}_{i} (t), 1,...,n can be designed such
that the ith component of the state estimation error associated with the
lth estimator satisfies ,
for all t>=T_{d}. If for each rε{1,...,N}\{l}, there
exist some finite time t^{r}>=T_{d} and some iε{1,...,n}
such that ,
then the possibility that the fault l may have occurred can be deduced.
The absolute fault isolation time is defined as T^{l}_{s}
= t^{r}.
Theorem 1: If the incipient fault l occurs, then for all
t>=T_{d} and for all iε{1,...,n}, the ith component of the
state estimation error of the lth isolation estimator satisfies the following
inequality:
Where:
represents the fault function estimation error in the case of a matched
fault.
Proof: On the basis of (1) and (12),
in the presence of the fault l, the ith component of the error
dynamics of the lth isolation estimator for is given by:
Then the solution to the previous differential equation is
where, ψ^{l}_{i}(t) is defined as (15).
By taking norms, the Eq. 14 can be get
So, the proof is complete.
Because in Eq. 14 the fault approximation error ψ^{l}_{i}(t),
the fault evolution rate γ_{i} and the fault occurrence time
T_{z} are unknown, it cannot be directly used as a threshold function
for fault isolation. However, as the estimate
belongs to the known compact parameter set Θ^{l}_{i},
letting the parameter set Θ^{l}_{i} be a hypersphere
with center o^{l}_{i} and radius can
be get. Then
Meanwhile, for the incipient fault time profile given by Eq.
2, assume that the unknown fault evolution rate satisfies
, where, denotes
a known lower bound on the unknown fault evolution rate γ_{i}
. So, the Eq. 7 can be get:
Hence, based on (14), (16) and (17),
the following threshold functions for fault isolation are chosen as below
Next, the fault isolability condition of the proposed FDI scheme will
be analyzed. Intuitively, faults are easier to isolate if they are sufficiently
mutually different in terms of a suitable measure. In the following analysis,
a fault mismatch function in the form is introduced:
which can be interpreted as the difference between the actual lth
fault function in the ith state equation, represented by ,
u(t)) and the estimated fault function associated
with any other isolation estimator whose structure does not match the
actual fault. The following theorem characterizes the class of incipient
nonlinear faults that are isolable by the proposed FDI scheme.
Theorem 2: Consider the fault isolation scheme described by Eq.
12, (13) and (18). The incipient
fault is isolable if for each rε{1,...,N}\{l} there exist
some time t^{r}> T_{d} and some iε{1,...,n} such
that the ith component m^{l}r_{i}(t) of the fault
mismatch function satisfies the following inequality:
Proof: Based on (1) and (14),
in the presence of the fault, the ith component of the error dynamics
associated with the estimator r is given by:
the solution of the above differential equation is
By using the triangle inequality, Eq. 21 is obtained
So, the threshold for the state estimation error of the rth estimator
is:
Therefore, if (20) is fulfilled, the occurrence of
the fault r is excluded at time t^{r}, i.e., .
If this is satisfied for each rε{1,...,N}\{l}, then the ith
fault can be isolated, thus concluding the proof.
AEROCRAFT’S NONLINEAR CLOSELOOP SYSTEM
Aerocraft’s nonlinear model: Consider the tailcontrolled
pitchaxis missile airframe (Kim et al., 2004) depicted in (Fig.
1).
The aircraft nonlinear model for the longitudinal axis are presented.

Fig. 1: 
Aerocraft airframe 
Table 1: 
Model parameters 

where, α denotes the angle of attack, q is the pitch rotational
rate, δ is the tail fin deflection, C_{n} and C_{m}
are the aerodynamic coefficients, the expressions of which depend on α,
δ and the Mach number M (1.5~3). The parameters can be find in Table
1.
The output of the system is the normal acceleration.
The aerodynamic surface actuator and reaction jet actuator dynamics are
modeled as:
Nonlinear control law design: Since the present missile configuration
is tail controlled, it has a strong nonminimum phase behavior with respect
to the normal acceleration. Consequently, a straightforward application
of the feedback linearization technique will not be successful, because
the unstable zerodynamics would not assure the internal stability of the
system (Devaud et al., 2000).
Two approaches have been proposed to overcome this difficulty. First
of these is the timescale separation of the system dynamics into slow
and fast modes. The second approach is to redefine the system outputs
to suppress the zerodynamics. The first method will be adopt in the following
sections. The first step in the two timescale design process is to split
the system dynamics into time scales based upon the notion of slow and
fast dynamic modes. Note that even if a clear separation between the modes
are not present in the openloop dynamics, mode separation can be enforced
during control system design. For the present case, the actuator dynamics,
together with the pitch rate dynamics are included in the fast timescale.
The normal acceleration dynamics is considered to be the slow time scale
mode.
Control of slow subsystem: The state is attack angle α and
the output is the pitch angle rate q. The subsystem is described as follow:
The desired closedloop aerodynamic of the slow subsystem can be described
by first order module as follow:
Then the desired input of the slow subsystem can be computed by Eq.
26 and Eq. 27
Control of fast subsystem: The state and output variants are identical
q and the input is the rudder deflect angle δ. The system is described
as follow:
The desired aerodynamic of the fast subsystem can be denoted by a second
order modul as follow:
Proceeding as before, the nonlinear control law for the aerodynamic surface
actuator in the fast timescale is given by:
SIMULATION RESULTS
Here, the proposed FDI scheme is applied to detect and isolate incipient
faults in a nonlinear aerocraft’s control system described as Eq.
22, 24, 25 and 31.
The faults which occur in the aerocraft’s system are described as (two
faults):
Where, θ^{1}εΘ^{1} = [1.5, 1.5] and θ^{2}εΘ^{2}
= [1.5, 1.5]. Assume that the unknown incipientfault evolution rate
γ defined in (2) satisfies:
Fault detection: Based on Eq. 22, the nonlinear
FD estimator is constructed as follow:
The online approximator is
implemented as a continuous radial basis function (RBF) neural network
(Trunov and Polycarpou, 2000), which are described by:
where, the weight can
be tuned by Eq. 6 and choose a uniform width σ =
0.6 for the basis functions and 11 fixed centers c_{j} (i.e., m = 11), which are evenly distributed
in the interval [2,2]. The stability and faultdetectability properties
of the fault detection estimator have been investigated in (Trunov and
Polycarpou, 2000).
Fault isolation: By using the methodology described earlier, a
bank of two isolation estimators is designed
The control input is set as Eq. 31. The modeling uncertainty
is assumed to arise out of a 5% inaccuracy in the value of a_{i},
b_{i}, c_{i}, (i = m, n). The bounding function.
Moreover, set g_{i} = 5(i = 1, 2).
The experiment results: Figure 2 shows the simulation
results when an incipient fault of type 1, with θ^{1} = 0.75
and the fault evolution rate γ = 0.2, occurs at t = 10 sec.

Fig. 2: 
(a) Timebehaviors of the fault
function (solid line) and the RBF neuralnetwork output (dashdotted
line) associated with the fault detection estimator (b) Time behaviors
of the state estimation error (solid line) associated with the FDAE
and the deadzone threshold (dashdotted line) (the fault detection
time instant is shown by an arrow) (c) and (d) Timebehaviors of
the state estimation errors (solid lines) and the thresholds (dashdotted
lines) associated with the two isolation estimators (the fault isolation
time instant is shown by an arrow) 
The evolution of the actual fault function f (solid line) and the output
of the neural network approximator (dashdotted
line) associated with the fault detection estimator are shown in
Fig. 2a. The state estimation error (solid line) of the fault detection
estimator and its corresponding deadzone threshold (dashdotted line)
are shown in Fig. 2b. As shown, the fault is detected
at approximately T_{d} = 11.2 sec. Moreover, in Fig.
2c and d, the residuals ε^{l}(t) (solid
lines) and their corresponding thresholds λ^{l}(t) (dashdotted
lines), associated with each isolation estimator, are shown. It can be
seen that the residual of estimator 1 always remains below its threshold,
whereas the residual of estimator 2 exceeds its threshold at approximately
T^{l}_{s} = 15.5 sec, thus allowing the isolation of fault
1.
CONCLUSION
In this study, a fault detection and isolation scheme using some
nonlinear estimators to detect and isolate the incipient faults is presented
and an aerocraft’s nonlinear closedloop system using dynamic inversion
method is constructed. Then the FDI scheme is applied in the aerocraft’s
faulty system constructed before. The simulation results show the effectiveness
of the application.