Tracy–Widom distribution based fault detection approach: Application to aircraft sensor/actuator fault detection

ISA Transactions 51 (2012) 189–197
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ISA Transactions
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Tracy–Widom distribution based fault detection approach: Application to
aircraft sensor/actuator fault detection
Ch. Hajiyev∗
Istanbul Technical University, Faculty of Aeronautics and Astronautics, Maslak, 34469 Istanbul, Turkey
a r t i c l e i n f o
Article history:
Received 30 March 2011
Received in revised form
8 July 2011
Accepted 29 July 2011
Available online 19 August 2011
Keywords:
Fault detection
Kalman filtering
Innovation covariance
Largest eigenvalue
Aircraft
Flight control system
a b s t r a c t
The fault detection approach based on the Tracy–Widom distribution is presented and applied to the
aircraft flight control system. An operative method of testing the innovation covariance of the Kalman
filter is proposed. The maximal eigenvalue of the random Wishart matrix is used as the monitoring
statistic, and the testing problem is reduced to determine the asymptotics for the largest eigenvalue of the
Wishart matrix. As a result, an algorithm for testing the innovation covariance based on the Tracy–Widom
distribution is proposed. In the simulations, the longitudinal and lateral dynamics of the F-16 aircraft
model is considered, and detection of sensor and control surface faults in the flight control system which
affect the innovation covariance, are examined.
© 2011 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
The problem of sensor/actuator fault detection, in real time, ap-
pears in many problems of navigation and control [1–3]. Abnormal
measurements, sudden shifts appearing in the measuring channel,
faultiness of measuring devices, changes in statistical characteris-
tics of noises of an object or of measurements, malfunctions in the
computer, reduction in the control surface/actuator effectiveness,
friction between the moving parts of the control surfaces, partial
loss of a control surface (break off of a part of control surface) and
also a sharp change in the trajectory of a control process, etc. should
be enumerated among these faults.
Many fault detection filters have been developed to detect
and identify sensor and actuator faults by using analytical redun-
dancy [4–12]. In [4,5], algorithms for the detection and diagno-
sis of multiple faults in a dynamic system are described. They are
based on the Interacting Multiple-Model (IMM) estimation algo-
rithm, which is one of the most cost-effective adaptive estima-
tion techniques for systems involving structural and parametric
Abbreviations: ADC: Analog-to-Digital Converter; EKF: Extended Kalman Filter;
FDI: Fault Detection and Isolation; FTC: Fault Tolerant Control; GLR: Generalized
Likelihood Ratio; IMM: Interacting Multiple-Model; KF: Kalman Filter; MMAE:
Multiple model adaptive estimation; UAV: Unmanned Aerial Vehicle; UKF:
Unscented Kalman Filter; VISTA: Variable In-Flight Stability Test Aircraft.
∗ Tel.: +90 212 285 3105; fax: +90 212 285 3139.
E-mail address: cingiz@itu.edu.tr.
changes. The algorithms proposed provide an integrated frame-
work for the fault detection, diagnosis, and state estimation. In [9],
a nonlinear model of the aircraft’s dynamics replace the traditional
inertial navigation equations and is used in conjunction with the
IMM and the unscented Kalman filter (UKF) for improving state
estimation in the presence of inertial sensor faults. It is shown
that a UKF matched to the faulty conditions can maintain state
estimates with enough accuracy for the controller to continue
tracking the desired flight path, whereas a mismatched UKF often
results in estimate and control failures. Multiple model adaptive
estimation (MMAE) methods have been incorporated [6] into the
design of a flight control system for the variable in-flight stability
test aircraft (VISTA) F-16, providing it the capability to detect and
compensate sensor/actuator failures. The algorithm consists of a
‘‘front end’’ estimator for the control system, composed of a bank
of parallel Kalman filters, each matched to a specific hypothesis
about the failure status of the system (fully functional or a failure in
any one of the sensors or actuators), and a means of blending the
filter outputs through a probability-weighted average. In [10,11],
MMAE technique is used for the actuator fault detection and isola-
tion (FDI) purpose. In methods described in [4–6,9–11], the faults
are assumed to be known, and the Kalman filters are designed for
the known sensor/actuator faults. As the approach requires several
parallel Kalman filters, and the faults should be known, it can be
used in limited applications.
In [7], an analytical redundancy-based approach for detecting
and isolating sensor, actuator, and component (i.e., plant) faults
in complex dynamical systems, such as aircraft and spacecraft is
0019-0578/$ – see front matter © 2011 ISA. Published by Elsevier Ltd. All rights reserved.
doi:10.1016/j.isatra.2011.07.008

190 Ch. Hajiyev / ISA Transactions 51 (2012) 189–197
developed. A statistical change detection technique based on a
modification of the standard generalized likelihood ratio (GLR)
statistic is used to detect faults in real time. The GLR test requires
the statistical characteristics of the system to be known before and
after the fault occurs. As this information is usually not available
after the fault, the method has limited applications in practice.
An integrated robust fault detection and isolation (FDI) and fault
tolerant control (FTC) scheme for a fault in the actuators or sensors
of linear stochastic systems, which are subjected to unknown
inputs (disturbances), is presented in [8]. The FDI modules are
constructed using banks of robust two-stage Kalman filters, which
simultaneously estimate the state and the fault bias, and generate
residual sets decoupled from unknown disturbances. All elements
of residual sets are evaluated by using a hypothesis statistical test,
and the fault is declared according to the decision logic prepared.
In this work, it is assumed that a single fault occurs at a time and
the fault treated is of a random bias type. The diagnostic method
presented in the article is valid only for the control surface FDI.
The parity relations based detection approach is widely applied
to various FDI problems, as described in several papers [12–17].
These techniques are indeed particularly relevant for fault diagno-
sis because they lead to very short time windows, which reduce the
detection time. These techniques are well-suited for deterministic
models. Uncertain parity relations are an interesting alternative to
the state-estimation approaches that require integration with re-
spect to the time and therefore set the problem of wrapping effect,
which is often solved by degrading the guarantee property [15].
In Refs. [18–24], the neural network based methods to detect
sensor/actuator failures are developed and discussed. In Ref. [18],
a neural network is proposed as an approach to the task of failure
detection, following the damage to an aerodynamic surface of an
aircraft flight control system. This structure, used for the state
estimation purpose, can be designed and trained online during
flight and generates a residual signal that indicates the damage
as soon as it occurs. In [19], the problem of detecting control
surface failures for a high performance aircraft is considered. The
detection model is developed using the linear dynamic model
of an F/A-18 aircraft. Two parallel models detect the existence
of a surface failure, whereas the isolation is achieved and the
magnitude of any one of the possible failure modes is estimated
by a decision algorithm using either neural networks or fuzzy
logic. Ref. [20] describes a study related to the testing and
validation of a neural-network based approach for the problem of
actuator failure detection and identification after the occurrence
of battle damage to an aircraft control surface. Online learning of
neural architecture, trained with the extended backpropagation
algorithm, have been tested under nonlinear conditions in the
presence of sensor noise. In [21], an approach for the fault detection
and diagnosis of the actuators and sensors in nonlinear systems is
presented. First, a known nonlinear system is considered, and an
adaptive diagnostic model incorporating the estimate of the fault
is constructed. Further, unknown nonlinear systems are studied
and a feedforward neural network is trained to estimate the
system states under healthy conditions. Genetic algorithms are
proposed as a means of optimizing the weighting connections of
the neural network and used in order to assist the diagnosis of
the fault. In [22], a neural network based method to detect faults
in nonlinear systems is proposed. Fault diagnosis is accomplished
by means of a bank of estimators, which provide estimates of
the parameters that describe actuator, plant, and sensor faults.
The problem of designing such estimators for general nonlinear
systems is solved by searching for optimal estimation functions.
These functions are approximated by feedforward neural networks
and the problem is reduced to finding the optimal neural weights. A
robust FDI scheme for a general class of nonlinear systems subject
to state and sensor unmodeled dynamics, uncertainties, and noise
disturbances is presented in [23]. The methodology is based on a
neural-network-based observer strategy that can detect, isolate,
and identify both actuator and sensor faults. In [24], a sensor fault
detection and accommodation scheme has been demonstrated
on a nonlinear unmanned aerial vehicle (UAV) model using
an extended minimum resource allocation network radial basis
function neural networks. The methods based on artificial neural
networks and genetic algorithms do not have physical bases.
Therefore, according to the different data corresponding to the
same event, the model gives different solutions. Thus, the model
should be continuously trained by using the new data.
Ref. [25] focuses on specific issues relative to real-time on-
line estimation of aircraft aerodynamic parameters at nominal
and post-actuator failure flight conditions. A specific parameter
identification method, based on the Fourier transform, has been
applied to the approximated mathematical model of the NASA
IFCS F-15 aircraft. The direct evaluation of the stability and control
derivatives has been considered versus the estimation of the
coefficients of the state space system matrices. This method may
not produce good results when the number of the stability and
control derivatives is high.
In [26], a novel scheme for autonomous component health
management with failed actuator detection and identification and
failed sensor detection, identification, and avoidance is presented.
This scheme has features that are very superior to those with triple
redundant sensing and voting, yet requires fewer sensors; it can
be applied to any system with redundant sensing. This method
requires the hardware redundancy and consequently equipment
expenditures. Moreover, an obvious defect of hardware methods
is the deterioration of the large-sized standard characteristics of
the equipment [27].
In this direction of studies, it is necessary to mention the
theory of diagnostics of a dynamic system by the innovation of
the Kalman filter [2,28–30]. The advantages of these methods are
as follows: they provide the monitoring of the correctness of the
result obtained by the current working input actions; they do not
require a priori information about the values of the changes in the
statistical characteristics of the innovation in case of fault; they
allow one to solve the fault detection problem in real time; they
require small computational expenditure for realization, since they
do not increase the dimension of the initial problem, in contrast to
the most algorithmic methods.
As is known [28], in the case where a system is normally
operated, the normalized innovation of the Kalman filter, which
is compatible with the model of dynamics, is the white Gaussian
noise with zero mean and identity covariance matrix. The faults
that appear in the system of estimations lead to the changes
in these statistical characteristics of the normalized innovation.
Therefore, in this case, the fault detection problem is reduced
to the problem of the fastest detection of the deviation of these
characteristics from nominal.
In [30], the sensor and control surface/actuator failures that
affect the mean of the innovation have been considered. The
methods of testing the correspondence between the innovation
and the white noise and revealing the change of its expectation are
based on the classical statistical methods and they are considered
in detail in the literature [2,28,30]; therefore, it shall not be
concentrated on testing these characteristics.
Testing, in real time, the Kalman filter innovation covariance
turns out to be a very complicated and not a well developed topic,
since there are difficulties in the determination of the confidence
domain for a random matrix. Moreover, existing methods of high-
dimensional statistical analysis [31,32] usually lead to asymptotic
distributions; this sharply diminishes the operativeness of these
methods.
Therefore, one makes use of a scalar measure of this matrix
in practice, such as the trace, sum of the matrix elements,

Ch. Hajiyev / ISA Transactions 51 (2012) 189–197 191
generalized variance (determinant), eigenvalues of the matrix, etc.,
where each characterizes one or another geometrical parameter
of the correlation ellipsoid. The algorithm for testing the trace
of the innovation covariance matrix is presented in [30]. But the
trace testing algorithm ignores the off-diagonal elements of the
covariance matrix. Therefore, this algorithm cannot detect very
small changes in the system [2].
Most of the fault detection tests are based on the statistical
properties of the eigenvalues of the sample covariance matrix [33,
34]. In [35], an algorithm based on the geometrical location of these
eigenvalues has been proposed. In [36], a new kind of test based on
an analytic expression of the ordered eigenvalues profile, obtained
under noise only hypothesis. Strategy in this work consists of
looking for a break in the profile by comparing observed profile
and the noise only one. The decision is taken by comparing the
error of prediction with the threshold, which is obtained by solving
the integral equation. Unfortunately, the distributions that enter
into this equation are not known analytically, hence it is difficult
to determine the threshold and perform the algorithm proposed.
There exist some interesting results on the distribution of
eigenvalues, the characteristic function of eigenvalues, and the
distribution and moments of the smallest eigenvalue of Wishart
distributed matrices [37–40]. But the application of the works
mentioned to the fault detection problem of multidimensional
dynamic systems turns out to be very complicated, since there are
difficulties in determining the confidence domain (or intervals) for
the eigenvalues of the random matrix.
In the present paper, the maximal eigenvalue of the matrix is
used for the scalar measure of the Wishart matrix tested and the
operative algorithm for testing the innovation covariance using the
Tracy–Widom distribution is proposed.
The structure of this paper is as follows. In Section 2, the fault
detection problem in multidimensional dynamic systems based
on the innovation of the Kalman filter is formulated. New results
for testing the innovation covariance based on the Tracy–Widom
distribution are given in Section 3. The AFTI/F-16 aircraft model
description and extended Kalman filter for the F-16 nonlinear
dynamic model estimation are given in Section 4. In Section 5,
some simulations are carried out for the sensor/actuator fault
detection problem in the AFTI/F-16 aircraft flight control system.
The changes that affect the innovation covariance have been
considered. Section 6 gives a brief summary of the results obtained
and the conclusion.
2. Statement of the problem
Let us consider a class of systems described by differential equa-
tions of the form
x(k + 1) = Φ(k + 1, k)x(k) + G(k + 1, k)w(k)
z(k) = H(k)x(k) + υ(k), (1)
where x(k) is the n-dimensional state vector of the system, Φ(k +
1, k) is the transition matrix of order n × n of the system, w(k) is
the random n-dimensional vector of system noises, G(k + 1, k) is
the transition matrix of system noises of order n × n, z(k) is the
s-dimensional measurement vector, H(k) is the measurement ma-
trix of the system of order s × n, υ(k) is the random s-dimensional
vector of measurement noises. It is assumed that the random vec-
tors w(k), υ(k), and x(0) are mutually independent white Gaussian
processes with zero expectations and covariance matrices defined
by the relations:
E[w(k)wT
(j)] = Q (k)δ(kj),
E[υ(k)υT
(j)] = R(k)δ(kj),
E[x(0)xT
(0)] = P(0),
where δ(kj) is the Kronecker symbol. Under the above-mentioned
a priori information, the estimation of the state vector ˆx(k/k) and
the estimation error covariance P(k/k) are found with the help of
the optimal Kalman filter [41]. Moreover, if the optimal filter is nor-
mally operating, then the normalized innovation
˜v(k) = [H(k)P(k/k − 1)HT
(k) + R(k)]−1/2
× [z(k) − H(k)ˆx(k/k − 1)] (2)
where ˆx(k/k − 1) is the extrapolation value by one step,
P(k/k − 1) = Φ(k, k − 1)P(k − 1/k − 1)ΦT
× (k, k − 1) + G(k, k − 1)Q (k − 1)GT
(k, k − 1)
is the extrapolation error covariance and P(k−1/k−1) is the esti-
mation error covariance in the preceding step, is a white Gaussian
noise with zero mean and identity covariance matrix [28]:
E[˜v(k)] = 0, E[˜v(k)˜vT
(j)] = P˜v = Iδ(kj).
Here I is the identity matrix. Note that the index k/k − 1 thus indi-
cates one step predicted values, whereas k/k denotes the estimate
at step k using all measurements including z(k).
The changes in the properties of the system or characteristics
of perturbations (faults of measuring devices, abnormal measure-
ments, changes in statistical characteristics of noises of the object
or of measurements, reduction in the control surface/actuator ef-
fectiveness, etc.) leading to a change in the innovation covariance
are considered in this study.
Consider the dynamical system (1) which is subjected to sud-
den changes. Our purpose is to detect on-line whether a change
occurred or not by using the largest eigenvalue of the covariance
matrix of the normalized innovation.
3. Algorithm for solution via the Tracy–Widom distribution
Two hypotheses are introduced:
γ0: no sensor fault occurs;
γ1: a sensor fault occurs.
Let us write the expression for the sample covariance matrix of the
sequence ˜ν(k):
ˆS(k) =
1
M − 1
k−
j=k−M+1
[˜ν(j) − ¯˜ν(k)][˜ν(j) − ¯˜ν(k)]T
(3)
where
¯˜ν(k) =
1
M
k−
j=k−M+1
˜ν(j) (4)
is the sample mean; M is the number of realizations used (the
width of the sliding window).
As is known, [31], under the validity of the hypotheses γo, the
random matrix
A(k) = (M − 1)ˆS(k) (5)
has the Wishart distribution with M degrees of freedom and is
denoted by Ws(M, P˜ν):
A ∼ Ws(M, P˜ν), (6)
where s and P˜ν are the dimension and covariance of the normalized
innovation ˜ν respectively. In testing statistical hypotheses, the
testing of the Wishart statistics (6) is complicated and not
well developed in view of the difficulty of constructing the
confidence domain for a random matrix. In practice, one of the
scalar measures of the above-mentioned matrix is usually applied
for testing random matrices. In this paper, the construction of
confidence intervals for the maximal eigenvalue of the matrix A(k)
is considered.

Let λmax be the largest principal component variance of the
covariance matrix (3), or the largest eigenvalue of an s-variate
Wishart distribution (6) on M degrees of freedom with identity
covariance.
Consider the limit of large s and M with M/s = γ ≥ 1. When
centered by
µs =
√
M − 1 +
√
s
2
(7)
and scaled by
σs =
√
M − 1 +
√
s
 
1
√
M − 1
+
1
√
s
1/3
, (8)
the distribution of λmax approaches the Tracy–Widom law of order
1, which is defined in terms of the Painleve II differential equation
and can be numerically evaluated and tabulated in software [42].
The next theorem is proved in [42].
Theorem. Under the above conditions, if, M
s
→ γ ≥ 1 then
˜λmax =
λmax − µs
σs
∼ F1 (9)
where ˜λmax is centered and scaled as in (7) and (8) respectively, largest
sample eigenvalue of the random Wishart matrix (5).
The theorem is stated for situations in which M ≥ s. From the
numerical work [43], Tracy and Widom report that the F1 distri-
bution, plotted in Fig. 1, has a mean which is equal to −1.21, and
a standard deviation which is equal to 1.27. The density is asym-
metric (See Fig. 1). Numerical table look-up for this distribution is
analogous to using the traditional statistical distributions.
By selecting lower β1 = 1 − α1 and upper β2 = 1 − α2
confidence limits (α1 and α2 are the levels of significance)
P{F1 < F1(M, s, β1)} = β1; 0 < β1 < 1 (10)
P{F1 < F1(M, s, β2)} = β2; 0 < β2 < 1 (11)
from the equations above, the threshold values F1(M, s, β1) and
F1(M, s, β2) will be determined [44]. Under the validity of the
hypotheses γ1, the left hand side of expression (9) tends to exceed
the threshold value F1(M, s, β1) or F1(M, s, β2). Then the decision
rule on the current state of the system of estimation with respect
to the hypotheses introduced will be written in the form
γo : ˜λmax(k) ≥ F1(M, s, β1) and
˜λmax(k) ≤ F1(M, s, β2) fault free
γ1 : ˜λmax(k) < F1(M, s, β1) or
˜λmax(k) > F1(M, s, β2) with fault.
(12)
Consequently, by comparing the above-defined centered and
scaled largest sample eigenvalue of the Wishart matrix A(k)
with the confidence limits obtained for the corresponding Tracy–
Widom distribution F1, it is possible to detect sensor/actuator
faults using decision rule (12).
Fig. 1 shows a graph of the probability density function of the
Tracy–Widom distribution F1 and the computed confidence limits
for β1 = 1 − α1 = 0.01 and β2 = 1 − α2 = 0.99, where the
numerals indicate: 1 and 3—the sensor fault detection zone; 2—the
fault free zone.
The expressions for the areas shown in Fig. 1 are written in the
following form:
β1 =
∫ F1(β1)
−∞
f (F1) dF1;
β2 =
∫ F1(β2)
−∞
f (F1) dF1; (13)
β2 − β1 =
∫ F1(β2)
F1(β1)
f (F1) dF1.
Fig. 1. Density of the Tracy–Widom distribution F1 with the fault free and fault
detection domains.
4. EKF for the F-16 aircraft state estimation
The technique for fault detection is applied to an unstable multi-
input multi-output model of an AFTI/F-16 fighter. The fighter is
stabilized by means of a linear quadratic optimal controller. The
control gain brings all the eigenvalues that are outside the unit
circle, inside the unit circle. It also keeps the mechanical limits on
the deflections of control surfaces. The model of the fighter is as
follows [45]:
x(k + 1) = Ax(k) + Bu(k) + F(x(k)) + w(k). (14)
The state variables are: x = [v, α, q, θ, β, p, r, φ, ψ]T
, where, v is
the forward velocity, α is the angle of attack, q is the pitch rate,
θ is the pitch angle, β is the side-slip angle, p is the roll rate, r
is the yaw rate, φ is the roll angle, and ψ is the yaw angle, w(k)
is the system noise with zero mean and the correlation matrix
E[w(k)wT
(j)] = Q (k)δ(kj). The fighter has six control surfaces
and hence the six control inputs are: u = [δHR, δHL, δFR, δFL, δC , δR],
where δHR and δHL are the deflections of the right and left horizontal
stabilizers, δFR and δFL are the deflections of the right and left flaps,
δC and δR are the canard and rudder deflections. The following hard
bounds (mechanical limits) on the deflections of control surfaces
are assumed: |δHR, δHL| ≤ 0.44 rad, |δFR, δFL| ≤ 0.35 rad, |δC | ≤
0.47 rad and |δR| ≤ 0.52 rad. A, B, and F(x) are calculated for the
sampling period of 0.03 s.
Let us define the estimated vector as:
xT
(k) = [ν(k), α(k), q(k), θ(k), β(k), p(k), r(k), φ(k), ψ(k)]
and apply the Kalman filter to estimate this vector. The measure-
ment equations can be written as:
z(k) = Hx(k) + υ(k), (15)
where H is the measurement matrix, which is 9 × 9 unit matrix,
υ(k) is the random vector of Gaussian noise of measurements with
zero mean and the correlation matrix E[υ(k)υT
(j)] = R(k)δ(kj). By
using the quasi-linearization method, let us linearize the equation
(14):
x(k) = Aˆx(k − 1) + Bû(k − 1) + F(ˆx(k − 1)) + A[x(k − 1)
− ˆx(k − 1)] + Fx(k − 1)

x(k − 1) − ˆx(k − 1)

+ B

u(k − 1) − û(k − 1)

+ w(k − 1) (16)
where Fx =
∂F
∂x

ˆx(k−1)
is the Jacobian of F(x(k − 1)) evaluated at
x(k − 1) = ˆx(k − 1).
The following recursive EKF algorithm for the state vector esti-
mation of the F-16 fighter motion is obtained in [45]:
Equation of the estimation value
ˆx(k) = Aˆx(k − 1) + Bû(k − 1) + F(ˆx(k − 1)) + K(k)ν(k). (17)

The innovation sequence
ν(k) = z(k) − H

Aˆx(k − 1) + Bˆu(k − 1) + F(ˆx(k − 1))

. (18)
The gain matrix of Kalman filter
K(k) = P(k)HT
R−1
(k). (19)
The covariance matrices of estimation errors
P(k) = M(k) − M(k)HT

R(k) + HM(k)HT
−1
HM(k). (20)
The covariance matrices of extrapolation errors
M(k) = AP(k − 1)AT
+ BDu(k − 1)BT
+ Fx(k − 1)P(k − 1)FT
x (k − 1) + GQ (k − 1)GT
(21)
where Du is the error covariance of the control input.
When the Kalman filter is used, the convergence of estimated
values to the actual values of parameters depends on the stability
condition of the filter. The stability of conventional digital filters
is easily analyzed with z-transform methods [2,46]. The problems
of the stability analysis of KF, divergence in the KF and the
methods of numerical stabilization of KF are widely discussed and
described in the monograph of the author [2]. Indeed, the extended
Kalman filter proposed is not so different from the Kalman filters
investigated in [2] in the point of view of the structure.
In [47], a sufficient condition has been presented to ensure the
stability of the state estimator with a Kalman filter that is subject
to finite word length effects. If the tile sufficiency condition is not
satisfied, it does not necessarily imply the system instability, but
instability may really occur if the analog-to-digital converter (ADC)
length and the tile mantissa length are too short in application. This
criterion shows that a state estimator, which may be unstable with
respect to round off errors of quantization, can be stabilized by
choosing the appropriate ADC and mantissa lengths. In fact, since
the ADC noise and the computational round off errors will appear
continuously, it is clear that the estimation error will not approach
zero.
Using the robust stability criterion proposed in [47], it is sug-
gested that the word length should be greater than or equal to 9
bits; otherwise, it may lead to an unstable response. The F-16 air-
craft estimator system that uses the Kalman filter is operated on
a computer system with a sufficiently long word length, and the
ADC for the output measurement has more than l0 bits. Therefore,
the instability of the Kalman filter with noise in F-16 applications
is unlikely.
5. Sensor/actuator fault detection simulation results
Let us show that, on the basis of the algorithm for testing the
innovation covariance proposed in this paper, one can, in a timely
manner, detect the faults appearing in the measuring channel.
Measurements were processed using the EKF (17)–(21). The ex-
pression for the normalized innovation of EKF is:
˜ν(k) =

R(k) + HM(k)HT
−1/2
ν(k). (22)
To detect faults changing the innovation covariance, the above cen-
tered and scaled largest sample eigenvalue of the Wishart matrix
A(k) is used. In the simulations, M = 20, s = 9, β1 = 0.01 and
β2 = 0.99 are taken, and the threshold values F1(M, s, β1) and
F1(M, s, β2) are found as −3.9 and 2.02 respectively. The follow-
ing cases are examined:
(i) There is no fault in the system. Results obtained are presented
in Figs. 2–3. Fig. 2 shows admissible bounds of the statistic ˜λmax(k)
and the plot of its behavior in the case of normal functioning of the
system. As is expected, at all points, −3.9 < ˜λmax(k) < 2.02. The
Fig. 2. Graph of the statistic ˜λmax(k) for normal operating of the system.
Fig. 3. Behavior of the normalized innovation ˜νq(k) in the case of normal operating
of the system.
corresponding normalized innovation in the third measurement
channel (pitch rate gyroscope channel) ˜νq(k) is shown in Fig. 3.
The threshold values corresponding to the level of significance
of α = 0.01 are ±2.58. The graphs of the normalized innovation
in the other measurement channels are very similar to the ones in
Fig. 3.
(ii) A sensor fault in the system. To verify the efficiency of the
algorithm proposed, beginning from the step k = 30, a fault
in the third measurement channel (pitch rate gyroscope fault)
is simulated; the noise variance in the pitch rate gyroscope is
changed as follows:
zq(k) = zq(k) + 2vq(k), (k ≥ 30).
The simulation results corresponding to this case are presented in
Figs. 4–5.
Fig. 4 shows that the value of ˜λmax(k) increases after the 30th
step and exceeds its threshold at the step k = 60 (0.9 s after the
fault occurs). As a result, based on the decision rule (12), the sensor
fault in the system is noted.
The behavior of the appropriate normalized innovation ˜νq(k) is
presented in the Fig. 5. Similar simulation results for the air speed
indicator and angle of attack sensor are presented in Figs. 6–9.
In simulations, the noise variances in the aircraft sensors in-
vestigated (pitch rate gyroscope, air speed indicator and angle of
attack sensor) are changed by multiplying them with 2, i.d. the

Fig. 4. Behavior of the statistic ˜λmax(k) in case of changes in noise variance in the
pitch rate gyroscope.
Fig. 5. Behavior of the normalized innovation ˜νq(k) in case of changes in noise
variance in the pitch rate gyroscope.
air speed indicator (fault occurs at iteration 40, detection time: 10 iterations (0.3 s)).
standard deviations of the measurement noises increase 2 times.
Consequently, the range of the random measurement noises in-
creases 2 times too. Because of the randomness of measurement
noises, the simulated measurements of the appropriate sensor may
Fig. 7. Behavior of the normalized innovation ˜νu(k) in case of changes in noise
variance in the air speed indicator.
angle of attack sensor (fault occurs at iteration 73, detection time: 43 iterations
(1.29 s)).
Fig. 9. Behavior of the normalized innovation ˜να(k) in case of changes in noise
variance in the angle of attack sensor.
be from the ‘‘good’’ sample range (for example, corresponding to a
normal standard deviation) or ‘‘bad’’ one (corresponding to an in-
creased standard deviation). Consequently, the realizations of the

Fig. 10. Behavior of the statistic ˜λmax(k) in case of control surface fault.
innovation sequences entering into the sample covariance matrix
(3) may be from the ‘‘good’’ or ‘‘bad’’ samples. As a result, the values
of the monitoring statistics ˜λmax(k) increase when the ‘‘bad’’ sam-
ple is used in the expression (3), and decrease when the ‘‘good’’
sample is used in the (3). This effect is natural when the covari-
ance matrix is tested. Therefore, the monitoring statistics may ex-
ceed their admissible bound and after certain iterations, may drop
significantly below the admissible bound. This movement can be
observed from Figs. 4, 6, 8. The decision about fault detection can
be made after the first exceeded admissible bound (if it is not a
false alarm). If the number of successive faults (exceeds admissi-
ble bound) is more than n (n is the previously determined number,
usually n > 3), then the decision ‘‘fault occurs’’ is made. Other-
wise the fault is assumed as the false alarm, and is not taken into
consideration.
Sensor faults considered may be caused by abnormal measure-
ments, friction between the moving parts of the sensors and other
difficulties such as the decrease of the instrument accuracy, and
the increase of the background noise etc.
(iii) An actuator fault in the system. Two kinds of faults can occur
in an actuator:
(a) actuator surface fault;
(b) actuator motor fault.
The fault detection algorithms proposed are used below to
detect the actuator surface faults. The control derivatives that
correspond to the first control surface (right horizontal stabilizer)
have been changed at iteration 30 as follows;
B (i, 1) = B (i, 1) × 0.2; i = 1, 9. (23)
The behavior of the statistic ˜λmax(k) when the right horizontal
stabilizer is failed is given in Fig. 10. As it is presented in Fig. 10,
˜λmax(k) is lower than the threshold until the change of the control
derivatives. In contrast, when the control derivatives are changed,
˜λmax(k) increases, and at the step k = 43 (0.39 s after fault occurs)
it exceeds its admissible bound. As a result, the fault in the system
is detected via the decision rule (12). This fault causes a change
in the mean and the covariance of the innovation sequence. The
innovation ˜νq(k) in case of control surface fault is shown in Fig. 11.
Fault detection results for the fourth (left flap) and fifth (canard)
surfaces via the approach proposed are presented in Figs. 12 and 13
respectively. The control derivatives that correspond to the left flap
have been changed at iteration 30 as follows;
B (i, 4) = B (i, 4) × 0.6; i = 1, 9. (24)
Fig. 11. Behavior of the normalized innovation ˜νq(k) in case of control surface fault.
Fig. 12. Behavior of the statistic ˜λmax(k) in case of left flap fault.
The behavior of the statistic ˜λmax(k) in case of left flap fault is given
in Fig. 12.
Fig. 12 shows that the value of ˜λmax(k) exceeds its threshold at
the step k = 186 (4.68 s after the fault occurs). As a result, based
on the decision rule (12), a fault in the system is noted. Similar
simulation results for the canard fault are presented in Fig. 13.
The change of the control derivatives (elements of the control
distribution matrix B) may be caused by the faults such as the
reduction in the control surface/actuator effectiveness, the friction
between the moving parts of the control surfaces, partial loss of a
control surface (break off of a part of control surface) etc.
Note that the inertia (the delay) of fault detection depends on
the number of the samples (the width of the sliding window) M in
this case and with an increase in this number, this characteristic
worsens. On the other hand, a very small value of M leads to
frequent false faults. Furthermore, the estimates of the sample
mean, the sample covariance matrix and consequently, the values
of the statistic ˜λmax(k), in general, will be biased for small sample
sizes. When the estimation window is larger, then the biasness of
the estimate is less unlikely. However, larger estimation window
reduces the ability of the algorithm to correctly trace the high-
frequency changes of the trajectory, e.g. turns [48]. Therefore,
the trade-off between the bias, the frequent false faults on the
one hand and the tractability of the estimates, the bad inertia
characteristic of the fault detection on the other hand should be
taken into account according to the application at hand. In addition,

Fig. 13. Behavior of the statistic ˜λmax(k) in case of canard fault (fault occurs at
iteration 30, detection time: 20 iterations (0.6 s)).
the proper choice of the sliding window size depends significantly
on the motional dynamics.
The window size M is chosen empirically since there is no
theoretically justified choice for the number M at present. Since the
method proposed is stated for situations in which M ≥ s (s is the
innovation dimension), as a result of the performed simulations,
the values between s and 20 are recommended for the width of
the sliding window M in the aircraft sensor/actuator fault detection
implementations.
The introduction of the developed sensor/actuator fault detec-
tion algorithms does not distort the estimation results of the fil-
ter and has no influence on their accuracy. The simulation results
justify the theoretical calculation obtained and show the practical
applicability of the fault detection algorithms proposed.
A Robust Kalman Filter (RKF) [30] can be used in order to isolate
the detected sensor and control surface faults which occurred
in the aircraft control system. A Kalman filter that satisfies the
Doyle–Stein condition is referred to as RKF. The use of the RKF is
very useful in the isolation of sensor and control surface faults as it
is insensitive to the latter faults.
The aircraft sensor/actuator fault isolation problem may be
solved also via adaptive robust Kalman filters (ARKF) with single
and multiple measurement noise scale factors which are proposed
in [49,50]. These ARKF algorithms are insensitive to the sensor
faults and can be used to isolate the detected sensor and control
surface faults.
6. Conclusion
In this paper, the fault detection approach based on the
Tracy–Widom distribution is presented and applied to the aircraft
flight control system. An operative method for sensor/actuator
fault detection based on testing the innovation covariance of
the Kalman filter is proposed. The maximal eigenvalue of the
covariance matrix is used in this process as the monitoring statistic,
and the testing problem is reduced to determine the asymptotics
for the largest eigenvalue of the random Wishart matrix. As a
result, an algorithm for testing the innovation covariance based
on the Tracy–Widom distribution is proposed. The sensor/actuator
fault detection algorithm is based on the computation of the
centered and scaled maximal eigenvalue of the Wishart matrix and
its comparison with the confidence bounds of the Tracy–Widom
distribution.
The Tracy–Widom distribution proposed based fault detection
approach is applied to F-16 aircraft flight control system with
sensor and actuator failures. An extended Kalman filter has been
developed for nonlinear flight dynamics estimation of an F-16
fighter. Faults in the sensors and actuators affect the characteristics
of the innovation of the EKF. The faults that affect the covariance
of the innovation have been considered.
In the simulations, the longitudinal and lateral dynamics of
the F-16 aircraft model is considered, and detection of faults in
the different type of sensors and control surfaces are examined.
The performed simulations confirm the theoretical results and
show the effectiveness of the fault detection approach proposed
in aircraft flight control system applications.
The advantages of the method proposed are as follows:
• provides the monitoring of the correctness of the result ob-
tained by the current working input actions;
• does not require a priori information about the values of the
changes in the statistical characteristics of the innovation in
case of fault;
• allows one to solve the fault detection problem in real time;
• requires small computational expenditure for realization since
they do not increase the dimension of the initial problem, in
contrast to the most algorithmic methods.
• the introduction of developed failure detection algorithm does
not distort the estimation results of the filter and has no influ-
ence on their accuracy.
The failure detection method presented has the following dis-
advantages:
• this method is of a statistical approach and a particular statistics
must be accumulated,
• the method has no ability to determine the value of the fault
(fault identification).
It is shown that the inertia of fault detection depends on the
number of samples M and with an increase in this number, this
characteristic worsens. On the other hand, a very small value of M
leads to frequent false faults. Some recommendations for choosing
the number of samples M in practical implementations are given
in this study.
The algorithm proposed in this paper can be applied in the prob-
lems of fault detection, fault diagnosis and model validation of
multidimensional dynamic systems. The algorithm can be success-
fully used in flight control systems, navigation and guidance sys-
tems of aircrafts, satellites, ships, robotics, etc.
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