A Survey on Model-Based Fault Detection Techniques for Linear Time-Invariant Systems with Numerical Analysis

Abstract

With the ongoing increase in complexity, less tolerance to performance degradation and safety requirements of practical systems has increased the necessity of fault detection (FD) as early as possible. During the last few decades, many research findings have been developed in fault diagnosis that addresses the issue of fault detection and isolation in linear and nonlinear systems. The paper's objective is to present a survey on various state-of-art model-based FD techniques developed for linear time-invariant (LTI) systems for the interested readers to learn about recent development in this field. Model-based FD techniques for LTI systems are classified as parameter-estimation methods, parity-space-based methods, and observer-based methods. The background and recent progress, in context to fault detection, of each of these methods and their practical applications are discussed in this paper. Furthermore, two different FD techniques are compared via analytical equations and simulation results obtained from the DC motor model. In the end, possible future research directions in model-based FD, particularly for the LTI system, are highlighted for prosperous researchers. A comparison and emerging research topic make this contribution different from the existing survey papers on FD.

Full text

Pertanika J. Sci. & Technol. 30 (1): 53 - 78 (2022)
ISSN: 0128-7680
e-ISSN: 2231-8526
Journal homepage: http://www.pertanika.upm.edu.my/
© Universiti Putra Malaysia Press
SCIENCE & TECHNOLOGY
Article history:
Received: 03 May 2021
Accepted: 15 September 2021
Published: 06 December 2021
ARTICLE INFO
DOI: https://doi.org/10.47836/pjst.30.1.04
E-mail addresses:
masoodjaffar@student.usm.my (Masood Ahmad)
eerosmiwati@usm.my (Rosmiwati Mohd-Mokhtar)
* Corresponding author
Review article
A Survey on Model-based Fault Detection Techniques for Linear
Time-Invariant Systems with Numerical Analysis
Masood Ahmad1,2 and Rosmiwati Mohd-Mokhtar1*
1School of Electrical and Electronic Engineering, Universiti Sains Malaysia, Engineering Campus, 14300
Nibong Tebal, Pulau Pinang, Malaysia
2Department of Electrical and Computer Engineering, COMSATS University Islamabad (Lahore Campus)
Lahore, Pakistan
ABSTRACT
With the ongoing increase in complexity, less tolerance to performance degradation and
safety requirements of practical systems has increased the necessity of fault detection
(FD) as early as possible. During the last few decades, many research ndings have been
developed in fault diagnosis that addresses the issue of fault detection and isolation in
linear and nonlinear systems. The paper’s objective is to present a survey on various
state-of-art model-based FD techniques developed for linear time-invariant (LTI) systems
for the interested readers to learn about recent development in this eld. Model-based
FD techniques for LTI systems are classied as parameter-estimation methods, parity-
space-based methods, and observer-based methods. The background and recent progress,
in context to fault detection, of each of these methods and their practical applications
are discussed in this paper. Furthermore, two dierent FD techniques are compared via
analytical equations and simulation results obtained from the DC motor model. In the end,
possible future research directions in model-based FD, particularly for the LTI system, are
highlighted for prosperous researchers. A comparison and emerging research topic make
this contribution dierent from the existing survey papers on FD.
Keywords: Fault detection, Kalman lter, LTI system,
model-based techniques, residual generation
INTRODUCTION
In the era of science and technology, every
engineering system demands accuracy,
reliability, and safety during its operation.
However, to realize such systems, i.e., high-

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Pertanika J. Sci. & Technol. 30 (1): 53 - 78 (2022)
Masood Ahmad and Rosmiwati Mohd-Mokhtar
speed trains, power systems, aircraft, and chemical plants, increases the system complexity
and nancial cost. Moreover, any abnormal behavior in a safety-critical system causes
performance degradation and leads to a dangerous situation. Thus, detecting and locating
the fault early is important to ensure safety and reliability by taking necessary measures
(Li et al., 2016; Franklin et al., 2019).
A fault is an unexpected event and input to the system that can occur in any part of the
system. Concerning the location of occurrence, it is generally classied as an actuator fault
(loss of control), a sensor fault (improper functioning of measuring components), and a
component fault (variation in the system’s internal parameters). Actuator and sensor faults
can be considered additive faults, while component faults are multiplicative faults (Frank
et al., 2000). Faults can also be classied according to their time behavior, i.e., abrupt fault,
incipient fault, and intermittent fault. Any fault in a system causes poor performance and
leads the entire system to collapse if it is not timely handled (Jie & Patton, 2012; Liu et
al., 2018; Na & Ahmad, 2019).
The fault diagnosis system is composed of three sub-systems. Each subsystem is merged
with the capabilities of detection, isolation, and identication (estimation) of the fault. FD is
the rst step in the fault diagnosis process that indicates the fault and its time of occurrence
in the system. Fault isolation determines the location of a fault, and fault identication
nds the type and size of a fault (Gao et al., 2015). Generally, fault diagnosis methods
are model-based and data-driven methods depending on the system model information
(Isermann, 1997). Data-driven-based FD methods solve the FD problem directly from
online process data. Therefore, these techniques are more suitable for large-scale complex
systems (Ding, 2014). On the other hand, model-based FD techniques utilize the analytical
model of the process that reveals the physical meaning of process dynamics through the
mathematical description. Ample of research has been done on model-based fault diagnosis
and their applications on various linear and nonlinear systems during the last few decades
(Gertler, 2017; Jie & Patton, 2012; Isermann, 2006; Ding, 2013). Therefore, model-based
techniques are chosen in many practical scenarios, provided that the analytical model of
the process is well-established.
The core idea of the model-based FD technique is to reconstruct/estimate the output
of a practical system using the analytical model, and the reconstructed output is compared
with the actual output of the system measured from sensors. The dierence between the
two outputs is a residual signal, which indicates fault occurrence in the practical system.
The model-based FD system is depicted in Figure 1. Residual evaluation refers to the
process of extracting fault information from residuals to dierentiate between fault and
disturbance. Finally, a binary decision about the occurrence of a fault is made by comparing
the evaluated residual signal with the pre-dened threshold. Hence, a model-based fault
detection system consists of two subsystems: residual generation, residual evaluation, and

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Pertanika J. Sci. & Technol. 30 (1): 53 - 78 (2022)
A Survey on Model-Based Fault Detection Techniques
threshold computation (Ding, 2013). In case of fault occurrence, an alarm is generated
to intimate the operator, or some control action is taken to compensate the eect of fault
for the smooth operation of an entire system. The process of modifying the control action
according to fault nature is called fault-tolerant control (FTC). A detailed discussion on
FTC is available in Zhang and Jiang (2008).
There are three main types of model-based fault detection techniques according to
the way of residual generation. They are known as observer-based FD techniques, parity-
space-based FD techniques, and parameter-estimation-based FD techniques (Isermann,
1984). Observer-based fault detection techniques correspond to the design of an observer
for estimating system output and residual generation. In the parity-space-based approach,
residual is generated by eliminating the initial states of dynamic systems and utilizing only
system input and measurement data within a nite time window. The prime objective of
both techniques is to ensure the robustness of residual against the process and measurement
of unknown inputs. Finally, the parameter-estimation approach is used to detect the slight
change/drift in the system parameters by comparing the actual parameters of the nominal
process with the estimated parameters.
Considering the massive monetary losses caused by the faults, there is a need to nd
the solution for FD problems in safety-critical applications in the presence of unknown
inputs and system parameter variations. External unknown inputs and parameter variations
make the FD more intricated. In addition, FD becomes more complex in the closed-loop
conguration because a fault in the system gets buried rapidly by the control actions. Most
of the existing survey papers discussed the FD methods for nominal linear systems. This
survey paper aims to look at another perspective, in which it presents state-of-art model-
based fault detection techniques recently developed for the linear time-invariant (LTI)
systems in open and closed-loop congurations and subjected to unknown inputs and
uncertainties simultaneously. Also, two well-known FD techniques, Kalman lter (Blanke
Figure 1. Model-based fault detection system (Chen et al., 2011)
Input
u(t)
Faults
Actuators Plant Sensors
Residual
evaluation
Fault alarm
Model-based fault
detection
Residuals
Output
y(t)
r(t)
Residual
generator
Plant model

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Pertanika J. Sci. & Technol. 30 (1): 53 - 78 (2022)
Masood Ahmad and Rosmiwati Mohd-Mokhtar
et al., 2015) and optimal FD lter (Ding, 2013), are compared using an analogy in observer
gain matrices and by simulations in this survey paper to demonstrate the clear picture of
the FD process. Sensor and actuator fault in a linear model of DC motor is detected using
both FD techniques. Their performance is compared in terms of detectability, computational
burden, and design complexity.
Unlike other review papers that focused on the same subject, our paper also included a
numerical comparison of fault detection techniques, and the simulation results were shown
for verication. Furthermore, based on the review, future research direction in model-based
fault detection approach is added to give the readers an idea and way forward to extend
the research in this eld. It is signicant because the review outcome from this paper can
be used as a reference for the readers of interest.
LTI SYSTEM WITH DISTURBANCE AND FAULT
LTI system subject to disturbance and fault can be expressed in the form of state-space
representation as Equations 1 and 2:
(1)
(2)
are state, input, output vectors respectively.
is
is
l2-norm bounded unknown input vector and
is
is l2-norm bounded unknown fault
vector. (A, B, C, D, Ed, Ef, Fd, Ff) are known matrices with appropriate dimensions.
Furthermore, Ef and Ff represent the place where the fault occurs and its inuence on the
system dynamics (Ahmad et al., 2017). Component/process fault or modeling error in the
system may cause the change of parameters of process dynamics. Equations 1 and 2 can
be represented by incorporating the process fault (Equation 3).
(3)
Where ( ) represents the component faults/modeling errors. There are
normally four types of sensor faults: sensor drift fault, sensor oset fault, xed scaling
factor fault, and sensor stuck fault. In addition, the same type of fault could be categorized
for actuators (Franklin et al., 2019). It is important to mention here that this survey only
focuses on additive fault detection techniques (sensor/actuator fault).

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A Survey on Model-Based Fault Detection Techniques
PARAMETER-ESTIMATION-BASED RESIDUAL GENERATION
The core of the parameter-estimation technique is based on system identication by utilizing
the system’s measured input and output data. In this technique, system parameters of a
practical system are identied either oine or online under the normal operating condition
while assuming that the fault is reected in the system’s physical parameters. In the context
of FD, residual is dened as a comparison between nominal parameters of the system in
a fault-free case and estimated parameters. The estimated parameters should match with
the system parameters in a fault-free situation. Any discrepancy in process parameters
indicates the change/fault in the system. Parameters are estimated using parameter-
estimation algorithms, i.e., least squares (LS), recursive least squares (RLS), regularized
LS, or extended least squares (ELS). These methods can be applied to any engineering
system, provide the inherent information of system dynamics. The exploitation of these
methods leads to an ecient fault detection and control system (Ding, 2013). Jesica
and Poznyak (2018) developed a new technique using the Kalman lter and instrument
variable method for parameter estimation in the stochastic system. The proposed technique
minimized the inuence of Gaussian noise and removed the biases in estimation, which
remains available in standard least square methods. The designed scheme also improved
the convergence speed.
Bachir et al. (2006) used the oine parameter-estimation technique for stator inter-turn
short circuit fault and broken rotor bar detection. In this study, a new model of an induction
machine for stator and rotor has been developed for fault detection. They introduced the
new parameters in the original model of the induction machine for stator inter-turn fault
detection and the design of a new faulty model for broken rotor bars detection.
Generally, the parameter-estimation technique requires one input and output signal,
and it provides a more detailed picture of internal process quantities. Therefore, this
technique is more suitable for component fault detection. However, it can be used for
sensor/actuator fault detection as well. The major disadvantage is that it always needs
an excitation signal for initiating the parameter-estimation process, which may not be
suitable for the process, operating at a stationary point (Isermann, 2006). In addition,
the parameter-estimation technique is less robust to unknown inputs that may aect the
estimation process. Nevertheless, the performance of parameter-estimation-based- FD
systems has been demonstrated by many successful applications in industrial processes
and automatic control systems (Belmokhtar et al., 2015; Ye et al., 2015; Herrera & Yao,
2018; Khang et al., 2018).
PARITY-SPACE-BASED RESIDUAL GENERATION
In a parity-space-based approach, a residual is generated by eliminating the eect of initial
states of a dynamic system and utilizing only the system input and measurement data within

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Pertanika J. Sci. & Technol. 30 (1): 53 - 78 (2022)
Masood Ahmad and Rosmiwati Mohd-Mokhtar
a nite time window. The inconsistency arises in the residual in case of abnormal behavior
evolving in the system dynamics. For example, the parity-space-based fault detection
system has been studied (Sun et al., 2019; Zhang et al., 2006; Zhong et al., 2018). This
section discusses parity-space-based FD for a linear discrete-time system in terms of design
and implementation issues.
Consider Equation 1. The following parity relation can be established (Equation 4).
(4)
Where , while and s is
the order of parity-space and
,
Hd,s, Hf,s can be obtained by substituting (Ed, Fd) and (Ef, Ff) in place of (B,D) in Hu,s.
Residual generator based on parity relation vector can be written as Equation 5.
(5)
is a residual signal and M is donated as a parity-space matrix, which contains a set
of parity vectors and is dened as parity space, . Let us denote vs
is the parity vector and holds the condition then residual in Equation 5 can be
represented as Equation 6.
(6)
Equation 6 clearly shows that parity relation-based residual design only requires the
computation of parity vector vs. In a fault-free case, (f(k) = 0), ds(k) can be perfectly decoupled
from the residual if the rank condition is satised (M[Ho,s Hd,s] = 0). However, the condition
looks stringent and very hard to satisfy for practical systems. In such cases, the preferred
solution for residual design is to apply the optimization technique to make an appropriate
trade-o between robustness and sensitivity. For this purpose, several objective functions,
with the prime aim to achieve the trade-o between robustness against disturbances and
sensitivity to faults, have been dened. The following performance index is widely adopted
for a parity-space-based residual generation (Equation 7) (Ding, 2013; Gertler, 2017).

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Pertanika J. Sci. & Technol. 30 (1): 53 - 78 (2022)
A Survey on Model-Based Fault Detection Techniques
(7)
For successful fault detection, the l2-norm of a residual signal in Equation 6 is the most
commonly used evaluation function and is dened as Equation 8 (Ding, 2013).
(8)
A threshold can be set as ( is an upper bound of disturbance energy)
in a fault-free case. In the last step, the decision logic is used to declare the fault alarm.
An optimized parity-spaced-based fault detection algorithm was developed in
(Odendaal & Jones, 2014) for actuator fault detection in Meraka Modular UAV. The study
optimized the parity relation, obtained from the standard parity-space approach, using the
transformation matrix that forms the residual more sensitive to the fault. The said approach
improved the computational burden compared to the online computation of covariance
matrices at every instant in the Extended Kalman lter. As compared to FD techniques in
an open-loop, fault detection in a closed-loop control system is much complicated because
a closed-loop is more robust against exogenous inputs. External inputs surround a fault
signal with low amplitude, and closed-loop control input makes the fault signal smaller. The
phenomena reduce the fault detection performance and fault detection rate signicantly.
Sun et al. (2019) proposed a parity-space transformation-based fault detection system for
the closed-loop control system. A stable kernel matrix for parity-space transformation was
designed to obtain the more accurate parity-space in a closed-loop system that improves
the fault detection performance. Furthermore, the fault detection rate has been improved
by accumulating the residual in a time window.
Zhong et al. (2018) demonstrated an integrated design of residual generation and
residual evaluation for fault detection in a linear discrete-time system subject to unknown
input without complete knowledge of probability distribution. The study focuses on parity-
space-based FD design to achieve an optimal trade-o between false alarm rate (FAR)
and missed detection rate (MDR). Determination of parity-space vector and optimization
of FAR and MDR are formulated in the minimum error minimax probability machine
(MEMPM) framework. The proposed algorithm delivers an optimal trade-o between
MDR and FAR in the worst-case scenario of unknown inputs without the information of
stochastic distribution.

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Masood Ahmad and Rosmiwati Mohd-Mokhtar
Equation 6 demonstrates that the residual generator requires the past data of input and
output measurements; hence, this residual generator is preferred for discrete-time dynamic
systems only. Furthermore, it is important to mention that parity-space-based residual
generators are more sensitive to unknown inputs because of their open-loop structure
than the observer-based residual generator, which has a closed-loop conguration and is
less sensitive to unknown input and system uncertainties. Despite the advantage of design
simplicity, knowledge of previous data and constraints on parity-space order makes the
parity-space-based residual generator non-ideal for online implementation. The solution
to these two problems is a one-to-one mapping between the design parameters of the
observer-based technique and the parity-space-based technique. This scheme is known
as parity-space design, observer-based implementation (Ding, 2013; Isermann, 2006).
By taking advantage of both designs, less design eort of parity-space vector, and online
realization of the observer, this scheme can also be used for continuous-time systems.
The parity-space-based technique is well applied for FD purposes in the induction
motor drive system (Dybkowski & Klimkowski, 2017), electromechanical brake systems
(Hwang & Huh, 2015), vehicles (Wang et al., 2019), and power systems (Rasoolzadeh &
Salmasi, 2020).
OBSERVER-BASED RESIDUAL GENERATION
A survey on observer-based FD techniques for the LTI system is presented in this section.
FD system based on Kalman lter is presented rst for stochastic LTI systems with
Gaussian noise. Next, unknown input observer (UIO) is discussed for perfectly decoupling
the unknown input, followed by a discussion on optimal observer design for deterministic
LTI systems subjected to norm-bounded unknown inputs and uncertainties.
Kalman Filter-based Fault Detection
A stochastic system is a dynamic system subjected to a stochastic/random noise, i.e.,
Gaussian noise with mean value and variance. The rst stochastic fault detection system
was developed using the innovations (residuals) generated by the Kalman lter (Mehra &
Peschon, 1971). A well-known Kalman lter, which looks like an observer in a deterministic
environment, is used to estimate the system’s state based on a series of measurements taken
in time, having system inaccuracies and statistical noise. The residual signal contains the
information of fault and the mutual eect of uncertainties, process, and measurement noise.
Stochastic residual evaluation techniques are used to eliminate these undesired eects
and recover the fault information from the residual. These techniques use the residual’s
statistical properties, i.e., mean, variance, and covariance, and try to detect statistical
parameters change. Several statistical methods are available in the literature to evaluate

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Pertanika J. Sci. & Technol. 30 (1): 53 - 78 (2022)
A Survey on Model-Based Fault Detection Techniques
the residual generated by the Kalman lter for fault detection purposes. A few of these
methods are the chi-square test (Da & Lin, 1996), multiple hypothesis tests (Bøgh, 1995),
generalized likelihood ratio (GLR) test (Willsky & Jones, 1976), and cumulative sum
algorithms (Nikiforov et al., 1993).
Kalman lter provides an optimal estimate of system states, i.e., minimum covariance
of error between estimated and actual states of the system. The unied approach is presented
in Doraiswami and Cheded (2013) to detect and isolate the fault in a linear discrete-time
system with measurement and system noise using the Kalman lter. Switched Kalman lter
is designed for sensor fault detection and isolation in power converters (Kleilat et al., 2018).
The proposed lter is an extended version of the standard Kalman lter combined with a
disturbance decoupling observer and has conrmed satisfactory results disturbance. The
incipient sensor fault detection in the continuous stirred-tank reactor benchmark process has
been addressed in Gautam et al. (2017) using Kalman lter and GLR test. In this technique,
signal to noise ratio (SNR) index is used to determine the threshold for successful fault
detection with minimum false alarm and missed detection rate.
As physical and technological constraints arose in the industrial system due to
complexity and robustness, many researchers developed modied and enhanced versions
of the Kalman lter to cope with the advanced requirements of the system. As a result
of dedicated research towards the development of stochastic fault detection techniques,
in parallel with ongoing research on deterministic fault detection techniques, improved
versions of Kalman lter is obtained, such as extended Kalman lters (EKF), unscented
Kalman lters (UKF), adaptive Kalman lters, and augmented state Kalman lters.
An Extended Kalman lter is commonly used for estimating the non-measurable states
of a nonlinear dynamic system (Jokic et al., 2018). Extended Kalman lter has shown good
results random disturbances; however, owing to nonlinearity in states and measurements of
the system, it is imperative to get the linearized and Jacobian matrix of the system model.
Moreover, the linearization process reduces the estimation performance of EKF that might
lead to instability of the ltering process for fault detection purposes.
Unscented Kalman lter overcomes the drawbacks of the EKF approach. UKF uses
the unscented transform, i.e., a good approximation of the stochastic distribution of state
rather than nonlinear function. Therefore, this method is straightforward and ecient for
estimating the system states for nonlinear dynamic systems leading to better fault detection
performance (Khazraj et al., 2016). However, in some cases, EKF requires a precise value
of measurement and process noise covariance matrix. This condition is not often practical,
leading to another version of the Kalman lter, i.e., adaptive Kalman lter.
An adaptive Kalman lter is used to tune the measurement noise covariance matrix and
system noise covariance matrix according to noise conditions to obtain a satisfactory fault
diagnosis (Hajiyev & Soken, 2013). According to the literature, researchers classify the

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Pertanika J. Sci. & Technol. 30 (1): 53 - 78 (2022)
Masood Ahmad and Rosmiwati Mohd-Mokhtar
adaptive lters into four types: Bayesian-based, maximum likelihood-based, correlation-
based, and covariance matching techniques (Tripathi et al., 2016). In addition, the augmented
Kalman lter is often used to estimate the system states and fault and disturbance signals
simultaneously (Gannouni & Hmida, 2017). Various applications of Kalman lter-based
fault detection system can be found in the gas turbine engine (Pourbabaee et al. 2016),
synchronous generator (Nadarajan et al., 2016), power systems (Liu & He, 2017), wind
turbine system (Cho et al., 2018), and aircraft (Marzat et al., 2012).
UIO-based Fault Detection
Fault detection system should be robust against all undesired inputs such as process and
measurement disturbance. Initially, it was proposed to decouple the unknown input from
the state estimation process using the disturbance distribution matrix. If system states are
decoupled from unknown disturbance, then residual is, obviously, also independent (Ding
& Frank, 1990; Wünnenberg & Frank, 1987). The decoupling observer is known as the
unknown input observer (UIO), a type of Luenberger observer, principally to estimate
state variables.
In the Eigen structure assignment method, left eigenvectors of the observer gain matrix
are assigned so that gain’s left eigenvectors are orthogonal to the disturbance distribution
matrix to make residual signal robust (Patton & Chen, 2000). In this approach, instead of
decoupling unknown input from the state estimation process, the residual signal is made
independent of unknown input. The geometric approach is used by (Hur & Ahn, 2014) to
decouple the eect of disturbance from the residual. All the above approaches (UIO, Eigen
structure assignment approach, and geometric approach) address the problem of disturbance
decoupling from the residual. However, these approaches are not capable of handling the
model uncertainties. One possible solution to tackle the model uncertainties is to model
them as an unknown input and then apply the disturbance decoupling techniques further.
The inverter incipient sensor fault is successfully detected and accommodated in
three-phase PWM inverters in the traction system (Zhang et al., 2017). UIO technique has
been widely used in many applications of FD, aircraft systems (Hur & Ahn, 2014), gas
turbine engines (Dai et al., 2009), cyber-physical systems, and wind energy systems (Zhu
& Gao, 2014). However, as stated in Ding (2013), the existing condition of the disturbance
decoupling technique is stringent and hard to achieve for many practical systems. Moreover,
the decoupling technique is also not suitable for cases where fault vector lies in the same
space as disturbance vector that may lead to a decoupling of fault signal from residual as
well, just like disturbance. An alternate strategy, widely adopted, is to design an observer
to make a suitable compromise between robustness to unknown input and sensitivity to
fault. It makes the fault detection problem a multi-objective design problem.

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A Survey on Model-Based Fault Detection Techniques
Optimized Observer/FDF-based Fault Detection
Fault Detection Filter (FDF), a well-known realization of full order state observer, generates
the residual for fault detection purposes. FDF structure for Equations 1 and 2 can be
represented by the following Equation 9:
(9)
, is the state estimation vector and measurement estimation vector,
respectively. is a so-called residual signal. L and V are two design parameters of FDF.
Observer gain L is determined in such a way that estimation error asymptotically goes to
zero. The residual in Equation 9 can be rewritten in the frequency domain as Equation 10:
(10)
Where
It can be observed from Equation 9 that residual is dependent on fault and unknown
input signal. norm represents the maximum inuence of disturbance on residual and
is widely used to improve the robustness of residual against the unknown inputs (Zhou &
Zhang, 2019). Furthermore, to analytically represent the inuence of fault on the residual,
norm, H2 norm, and H_ index are successfully adopted for FDF design. Robustness to
unknown input/disturbance while, at the same time, sensitivity to fault makes the design
of FDF a multi-objective optimization problem. This way, optimal observer gain can be
obtained by solving the following optimization problem over some specied frequency
range (Equation 11).
(11)
There are several forms of performance indices (i.e., ) are available for
solving Equation 11. In addition, L*,V* are optimal parameters of FDF in Equation 9
that deliver a residual which has maximum robustness to unknown inputs and maximum
sensitivity to faults (Ding, 2013).

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Masood Ahmad and Rosmiwati Mohd-Mokhtar
Optimal FDF design for linear system formulated as a multi-objective optimization
problem (Ding et al., 2000) and unied solution for the optimization problem is obtained
using factorization technique which is realized by solving the Riccati equation. Aguilera
et al. (2016) designed an observer-based fault detection system to detect the current sensor
fault in the induction motor drive. A dierential geometric approach is used in this study
to detect and isolate the single and multiple faults, i.e., disconnection, oset, and constant
gain faults.
The prime objective of all techniques developed for the solution of Equation 11 is
to obtain an optimal trade-o between robustness and sensitivity. These techniques are
well-suited for fault detection in linear systems subjected to unknown disturbances only.
However, in model-based FD techniques, another issue often encounters in the residual
generation process is model mismatching. A perfect mathematical model of a practical
system is never available because of modeling error, process linearization, and component
aging issues. Hence, optimization techniques that solve Equation 11 cannot be applied to
uncertain systems.
System uncertainty severely aects the output estimation that leads to poor performance
of fault detection. Robust FDF has been designed for continuous LTI systems subjected
to disturbance and norm-bounded uncertainty (Zhong et al., 2003). Robust FD problem
is formulated as model matching problem, and solution of the optimization problem
is presented in linear matrix inequality (LMI) form in the said paper. Li et al. (2013)
extended the same work discussed in Ding et al. (2000) for continuous-time linear uncertain
systems subjected to polytopic uncertainty utilizing the iterative LMI approach. Farhat
and Koenig (2015) formulated the proportional integral observer (PIO) design problem as
a multi-objective optimization problem for the continuous-time linear uncertain system.
The robustness to disturbance and uncertainty has been ensured minimizing the norm
of Grd in the LMI framework.
Although the designed robust FDF, with and without system uncertainty, somehow
minimizes the eect of disturbances and uncertainties, these are not completely decoupled
from the residual. These unknown inputs still inuence the residual. In such cases, the
appropriate residual evaluation function and threshold computation selection are integral
in successful fault detection. The nal decision on the occurrence of a fault is made using
a simple comparison between residual evaluation function and threshold.
There are two widely accepted ways to generate evaluation function and threshold
depending on the system’s dynamics under consideration. Generally, the norm-based
residual evaluation function is used for a deterministic system in which the energy of
unknown input is bounded under a certain limit . On the other hand,
a statistical-based residual evaluation function is adopted for stochastic systems. For
deterministic systems, l2-norm is a mostly used evaluation function and is dened in
[8]. Peak value, root mean square value, and moving average of residual is also used

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A Survey on Model-Based Fault Detection Techniques
for residual evolution function. We refer our esteemed readers to Ding et al. (2003) for
more knowledge and computational skills. In case, residual is dependent on disturbance,
uncertainty, and fault signal, then residual, generated by any of the methods discussed
above, can be represented as Equation 12:
(12)
In a fault-free case, , the threshold can be dened as Equation 13:
(13)
Other Observer-based Fault Detection Techniques
Besides the previously discussed Kalman lter, UIO, and optimized FDF schemes, other
types of fault detection observers in the literature have been investigated for LTI systems.
Those are a proportional-integral observer (Do et al., 2018; Yang et al., 2020), sliding
mode observer ( Zhirabok et al., 2018; Zhang et al., 2019), interval observer (Pourasghar
et al., 2020; Zammali et al. 2020), and adaptive observer (Lijia et al., 2019; Perrin et al.,
2004). These observers are designed to estimate the system output with minimum output
estimation error and fast convergence speed. Then output estimation error is used as a
residual to indicate the fault occurrence.
DISCUSSION
In this section, a comparison among various fault detection techniques discussed so far
is given in terms of robustness, complexity, and performance. The robustness of the fault
detection method is checked by the measure of sensitivity to noise, disturbance, and
uncertainty. Likewise, robustness and the performance of the fault detection method are
determined in terms of FAR and MDR. Based on the above survey, the following points
are highlighted:
1. An observer generates the residual for fault detection, and it is synthesized to zero
in fault-free cases. Observer-based and parity-space-based FD methods produce
the alike residual in terms of residual characteristics. However, the observer-based
method shows more robustness to uncertainty as compared to the parity-space-
based residual generator.
2. FDF is a complete state observer, while an unknown input observer is a reduced-
order observer and might be considered where full state estimation is not required.
UIO has more robustness to unknown inputs with increased complexity and
computational eort.
3. Observer-based and parity-space-based fault detection methods are designed for
additive faults and perform well when the plant model is perfect.

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Pertanika J. Sci. & Technol. 30 (1): 53 - 78 (2022)
Masood Ahmad and Rosmiwati Mohd-Mokhtar
4. Observer-based and parity-space-based FD system design requires the knowledge
of robust control theory when there is uncertainty and unmodeled disturbance
in the linear systems. These unwanted inputs are handled by selecting a higher
detection threshold in the second step of the fault detection process. However, a
higher detection threshold causes to increase in the FAR, and a lower detection
threshold cause to increase in the MDR. Therefore, a feasible trade-o is required
between FAR and MDR for successful fault detection.
5. Optimization-based FD methods are formulated so that the sensitivity of residual
to unknown inputs is minimized along with improved sensitivity to faults. Such
methods provide a solution in terms of mathematical multi-objective functions.
There could be one disadvantage: it might not guarantee the usefulness and
performance of the solution in some applications. Hence, special care is required
while implementing optimized FD methods in the underlying application, which
may also increase complexity.
6. Kalman lter-based FD system is used for stochastic systems, and much knowledge
of statistical analysis and probability is required. Dierent versions of the Kalman
lter can be applied to nonlinear systems and time-varying systems as well.
7. Fault detection methods relying on system identication are useful for linear and
nonlinear systems, but the performance of such FD systems entirely depends
on detecting the variation in system parameters. Moreover, these fault detection
methods are benecial for detecting small and incipient faults.
8. The major disadvantage of the model-based FD method is to get the precise
mathematical model of the system. This reason restricts the application of the
model-based FD methods to industrial systems. In such cases, data-driven
techniques are advantageous when the system model is unavailable (Denkena et
al., 2020).
9. The parameter-estimation method produced good results in detecting structural
damage, while all other model-based methods are well suited for detecting the
actuator and sensor faults.
10. The selection of an appropriate detection technique depends on the reliability of
the available knowledge of the system.
COMPARISON OF FAULT DETECTION TECHNIQUES
This section compares two well-known fault detection techniques in terms of detectability,
design complexity, and computational load. For this purpose, a linear DC motor model is
simulated in MATLAB for sensor and actuator fault detection. First, a unied solution of
an optimal FDF developed in Ding (2013) for Equations 1 and 2 subjected to deterministic
disturbance is compared with Kalman lter-based optimal estimator for stochastic LTI

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A Survey on Model-Based Fault Detection Techniques
system (Blanke et al., 2015). Both FD techniques conrm good performance in terms
of fault detection if an unknown disturbance lies within bound and there is no system
uncertainty. Second, the state-space model of DC motor in Equations 1 and 2 has the
following matrices:
, , ,
, ,
Kalman Filter Design
Residual generation for stochastic LTI system (Equation 14):
(14)
Let and , be zero-mean process and measurement noise with following
covariance matrix, . The dynamics of the
Kalman lter-based residual generator is represented by Equation 15:
(15)
K is Kalman lter gain and can be determined as ,
While P can be obtained by solving the following Riccati Equation 16:
(16)
The residual evaluation and threshold are given as Equation 17:
(17)
Where .Threshold could be defined as
, where is given false alarm rate (FAR) and n is the number of output
signals. The threshold is computed using chi-square (X2) test.
Unied Solution of Optimal FDF
Residual generation for deterministic LTI system in Equations 1 and 2, d(k) is l2 norm
bounded unknown disturbance, holds . Optimal FDF of the form in
Equation 11, observer gain L and post-lter V is obtained using the unied solution

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Pertanika J. Sci. & Technol. 30 (1): 53 - 78 (2022)
Masood Ahmad and Rosmiwati Mohd-Mokhtar
Equation 18:
(18)
Where, X is a solution of the above Riccati equation. Residual obtained from unied
solution is an optimal solution because, in fault-free case, the eect of d(k) on the residual
is uniform in the entire subspace spanned by the disturbance d(k). Thus Equation 9 serves
as a residual evaluation function, and the threshold is taken as the upper limit of disturbance
energy (Equation 19).
(19)
Table 1 describes the comparison in gain matrices of the Kalman lter and optimal FDF. It
is shown that the gain of both lters depends on their noise characteristics. The following
comparison reveals no dierence between these two lters, and they deliver optimal residual
in terms of robustness if there is no system uncertainty.
Table 1
Comparison between FD techniques via gain equations
is zero-mean white noise with the
covariance matrix
is an all-pass filter in fault-free case, i.e., the
beauty of a unified solution
is zero-mean white noise with the
covariance matrix
is an all-pass filter in fault-free case, i.e., the
beauty of a unified solution
is zero-mean white noise with the
covariance matrix
is an all-pass filter in fault-free case, i.e., the
beauty of a unified solution
is zero-mean white noise with the
covariance matrix
is an all-pass filter in fault-free case, i.e., the
beauty of a unified solution
r (k) is zero-mean white noise
with the covariance matrix σ2v
r (k) is an all-pass lter in fault-free case,
i.e., the beauty of a unied solution
Comparison by Simulation
For simulation purposes, the following assumptions are considered,
,
,
, the covariance of Gaussian noise:
and .

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Pertanika J. Sci. & Technol. 30 (1): 53 - 78 (2022)
A Survey on Model-Based Fault Detection Techniques
The following simulation results have been observed: Figure 2 illustrates that system
output estimates are close to actual measurement in fault-free cases. Thus, residual obtained
from both schemes prove robustness to unknown disturbance/noise. Figure 3 demonstrates
a speed sensor fault case in which ramp input as an incipient fault is injected at 2.2 sec.
It is observed that the detection time of the Kalman lter is high as compared to optimal
FDF. In this case, the Kalman lter has shown less robustness to a time-varying fault in
the DC motor speed sensor. Finally, Figure 4 conrms the eectiveness of both schemes
in terms of robustness and detection time in actuator stuck fault conditions.
Simulation results conrm that both approaches have shown better performance under
unknown inputs. However, they have limited capability of FD in the presence of system
uncertainty and may deliver large FAR and MDR. Therefore, a fault-sensitive lter has
Figure 2. Residual, f(k) = 0 from (a) Kalman lter (left); and (b) optimal FDF
Figure 3. Evaluation function and threshold in incipient fault at t =2.2sec: (a) Kalman lter; and (b) optimal FDF
(a) (b)
(a) (b)
Time (sec) Time (sec)
Error magnitude
Error magnitude
Magnitude
Time (sec)
Magnitude
Time (sec)

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Masood Ahmad and Rosmiwati Mohd-Mokhtar
Figure 5. Residual in actuator stuck case with system uncertainty
Figure 4. Evaluation function and threshold in actuator stuck fault at t = 2.5-3.5sec: (a) Kalman lter; and
(b) optimal FDF
(a) (b)
been designed for LTI uncertain system to handle the uncertainty problem (Ahmad &
Mohd-Mokhtar, 2020). Figure 5 shows residual obtained from H-index FDF for uncertain
DC motor system and residual from optimal FDF of nominal DC motor system. It has been
shown that H-index fault-sensitive FDF can minimize the eect of system uncertainties and
enhance the fault sensitivity (Ahmad & Mohd-Mokhtar, 2020) and shows the approximate
performance of optimal FDF designed for nominal DC motor. Detail comparison of the
three FD schemes is given in Table 2.
Magnitude
Time (sec)
Magnitude
Time (sec)
Error Magnitude
Time (sec)
0.06
0.04
0.02
0
-0.02
-0.04
-0.06
-0.08
-0.1
-0.12
-0.14
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
FDF zero uncertainty
H-index FDF

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Pertanika J. Sci. & Technol. 30 (1): 53 - 78 (2022)
A Survey on Model-Based Fault Detection Techniques
Table 2
General comparison of three FD schemes
Kalman Filter Optimal FDF H-index FDF
Detectability Can detect all faults but may
increase FAR when the fault has
a low fault to noise ratio
Can detect all faults
simulated in this paper
Can detect all faults
simulated in this paper
Detection time Fast Fast Fast
Computation load Moderate Low Moderate
Robustness to
uncertainty
FAR high in the simulation case MDR high in the
simulation case
Acceptable robustness
Robustness to initial
states
Less robust than optimal FDF More robust than other Least robust
MODERN TRENDS IN MODEL-BASED FD FOR LTI SYSTEMS
1. As discussed above, a model-based FD system consists of two stages: residual
generation and residual evaluation. The prime objective of the residual generation
stage is to generate the optimal residual. The objective of the residual evaluation and
threshold stage is to ensure maximum fault detectability. Thus, the overall objective
of an optimal fault detection system is to achieve the maximum fault detectability
and maintain zero false alarm for a deterministic type of system or achieve the fault
detectability under some allowable false alarm rate under stochastic noise. Based on the
survey, most of the results mentioned above focus on the eort of residual generation.
However, little attention is given to the residual evaluation and corresponding threshold
computation. Hence, there is an indispensable need to design the residual generation
and residual evaluation in an integrated way, rather than dealing with them separately,
to optimize certain criteria for better performance of the fault detection system.
2. Even though very nice results are available for model-based fault detection, there is
still a scarcity of research for uncertain linear systems. The standalone design of the
residual generator and residual evaluator and an optimally integrated fault detection
system, as discussed in 1, for the continuous and discrete-time linear uncertain system,
is still an open and challenging topic for researchers.
3. The fault should be estimated after it has been detected so that its eect can be
compensated to maintain the reliability of a practical system in a faulty situation. An
optimal fault estimator and fault-tolerant controller for nominal and uncertain systems
is also an ongoing open research topic.
4. It is also observed that minimal eort has been devoted to developing fault detection
systems in a closed-loop environment. Open-loop fault detection techniques become
ineective in a closed-loop environment. Thus, it is necessary to work on this topic
which is also an emerging topic.

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Pertanika J. Sci. & Technol. 30 (1): 53 - 78 (2022)
Masood Ahmad and Rosmiwati Mohd-Mokhtar
5. Besides the process and measurement noise, there is also multiplicative noise in the
system. Due to this, traditional methods could not produce the optimal performance
for fault detection. Hence, there is a need to address the fault detection problem for
LTI systems involving multiplicative noise.
In some instances, recorded or online data of system inputs and measurements are
only available. In such scenarios, model-based fault detection methods cannot be directly
applied. Therefore, model-based and data-driven or signal processing-based fault detection
methods can be combined to design optimal FD systems for complex systems where the
mathematical model of the system is not possible. As technology advances and more
techniques are developed regarding fault detection in LTI systems, the combination of
data-driven and model-based fault detection methods are the future research directions. The
integration of the two methods provides new opportunities and challenges. Furthermore,
the integration of robust control and machine learning techniques for observer-based FD
and estimation scheme for linear systems open new research directions in the model-based
FD framework.
CONCLUSION
The safety, reliability and desired performance of a practical system must be maintained at
all the time during its operation. Fault detection plays a paramount role in accomplishing
these objectives. This paper discusses various model-based residual generation techniques,
including FDF, UIO, parity-space, optimization-based, Kalman filter, and system
identication approach. Three dierent FD approaches were illustrated to address the FD
problem in the DC motor system, and their performance is compared. The arising issues and
emerging research topics on fault detection for LTI systems were explained. The outcome
from this paper may assist in further research in the future.
ACKNOWLEDGMENT
The work is supported under the HRDI-UESTPS scholarship program funded by the Higher
Education Commission (HEC) of Pakistan and partially supported by the USM RUI Grant:
1001/PELECT/8014093.
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