# On Fault Detection in Linear Discrete-Time, Periodic, and Sampled-Data Systems

Authors: P. Zhang; S. X. Ding
Journal: Journal of Control Science and Engineering
 (2008)

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## Abstract

This paper gives a review of some standard fault-detection (FD) problem
formulations in discrete linear time-invariant systems and the available solutions. Based on it, recent development of FD in periodic systems and sampled-data systems is reviewed and presented. The focus in this paper is on the robustness and sensitivity issues in designing model-based FD systems.


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## Body

## 1. Introduction

With the increasing requirements of modern complex control
systems on safety and reliability, model-based fault detection and isolation
(FDI) technology has attracted remarkable attention during the last three
decades [1–6]. In major industrial sectors, it has become an
important supporting technology and is replacing the traditional hardware
redundancy technique in part or totally. As a standard functional module, FDI
systems are increasingly integrated in modern technical systems and provide
valuable information for condition-based predictive maintenance, higher-level
fault tolerant control, and plant-wide production optimization.Though closely related to the development of control
and filtering theory, there are several distinct features of the model-based
FDI problems that justify the efforts made in this field. To evaluate the
performance of an FDI system in practice, miss alarm rate, false alarm rate, and
detection delay are the most important criteria that decide the acceptance of the
methods. It is widely accepted that these functional requirements can be
reformulated as a multi-objective problem. Enhancing the robustness of the FDI
system to unknown disturbances and modeling errors is an essential objective.
However, alone the robustness does not guarantee a good FDI performance. The
sensitivity of the FDI system to faults should be simultaneously improved. To
find the best compromise between the robustness and the sensitivity is thus the
central problem in model-based FDI. This is the first difference of FDI
problems from control and standard filtering problems, where the focus is put
on disturbance attenuation. Bearing this in mind, full-decoupling problem and
optimal design of FDI systems have been studied [3–6] and different types of indices have been introduced
to describe the sensitivity to the faults. Secondly, for the purpose of FDI, a
fault indicating signal, called residual, needs not only to be generated, but
also to be evaluated and, based on it, a decision for the existence, location,
and size of the faults needs to be made. Therefore, an FDI procedure includes
residual generation and residual evaluation. An integrated design of these two
parts is needed to guarantee the optimal FDI performance [7].In this paper, we will first give a review of some
standard fault detection (FD) problem formulations in discrete-time systems and
the available solutions. There are two types of discrete-time model-based FD
systems: the parity space and the observer-based ones. The former is, in its
original form, specially dedicated to the discrete-time systems [8], while the latter is analogous to the continuous-time
systems and its development shares the same essentials with the continuous-time
systems. Perhaps for this reason, besides the early research activity on
the parity space approaches, only few studies have been specifically devoted to
the FD problems in discrete-time systems. Recently, the intensive research on
networked control systems (NCS) and embedded systems considerably stimulates
the study on periodic, sampled-data systems [9]. The integration of data communication networks into
control systems introduces natural periodic behavior in the system dynamics and
the sampling effect is understood not only in view of the behavior of A/D and
D/A converters but also in the context of data transmission among the
subsystems. It can be observed that the recent studies on FD in periodic and
sampled-data systems are mainly based on the discrete-time model-based FD
methods. It is this fact that motivates us to give an overview of some standard
FD methods for discrete-time systems and, based on it, to review and present
some recent results on FD in periodic and sampled-data systems. Bearing in mind
that fault isolation problems can be principally reformulated as a robust fault detection problem [4, 5], our focus in this paper is on the robustness issues
in designing model-based FD systems.The paper is organized as follows. In Section2, we
review FD methods for discrete-time systems and address some important
relations between different methods. Section 3 is devoted to FD in
discrete-time periodic systems. In Section 4, FD in sampled-data systems is
addressed.Throughout this paper, standard notations of robust
control theory, for instance those used in [10], are adopted. We will use σ̲(X),σ¯(X) to denote the
minimum and maximum singular values of matrix X, respectively
and σi(X) to denote any
singular value of X that satisfies σ̲(X)≤σi(X)≤σ¯(X). ∥x∥E denotes the
Euclidean norm of vector x, ∥ξ∥2 the l2-norm of
discrete-time signal ξ or the ℒ2-norm of
continuous-time signal ξ, ∥ξ∥2,[a,b) the 2 norm of ξ over the
interval [a,b),  and ∥G∥∞ the H∞-norm of
transfer function matrix G. The superscript T denotes the
transpose of matrices and the superscript * denotes the
adjoint of operators. ℛℋ∞ stands for the
subspace that consists of all proper and real rational stable transfer function
matrices. In this paper, we call a state-space model (A,B,C,D) regular, if it
is detectable and has no invariant zeros on the unit circle and no unobservable
modes at the origin.


## 2. FD of Discrete LTI Systems

Linear time-invariant (LTI) systems are the simplest
class of systems. Although the handling of FD problems in discrete LTI systems
can often be done along the well-established framework of FD schemes for
continuos LTI systems, study on FD in discrete LTI systems is of primary
importance from the following three aspects:(i)
it gives
insight and often motivates extensions to more complex systems like periodic
and sampled-data systems addressed in the subsequent sections;(ii)
there are some
methods that have been developed specially for discrete LTI systems;(iii)
due to its
practical form for the direct online implementation, the discrete-time system
form is often favored in the applications.
In this section, basic ideas and solution procedures
of advanced FD methods for discrete LTI systems, divided into three groups,
will be reviewed:(i)
parity space
approaches, which are specific for discrete LTI systems and will be dealt to
some details;(ii)
the
parametrization of observer-based FD systems and post-filter design schemes;(iii)
fault detection
filter schemes, which are mostly studied and closely related to robust control
theory.
Thanks to the well-known relationships between the
technical features of the discrete- and continuous-time systems, many
well-established FD schemes for continuous LTI systems can be directly applied
to the latter two FD schemes. For this reason, we will restrict ourselves to
some representative methods and give a brief view of the analog application of
the methods for continuous-time systems to the discrete FD systems. Another
focus in this section is on the comparison and interpretation of the FD
methods.


### 2.1. System Models and Problem Formulation

Suppose that the discrete LTI systems are described by(1)ΣLTI:{x(k+1)=Ax(k)+Bu(k)+Edd(k)+Eff(k),y(k)=Cx(k)+Du(k)+Fdd(k)+Fff(k),
where x∈ℛn is the state
vector, u∈ℛp the vector of
control inputs, y∈ℛm the vector of
process outputs, d∈ℛnd the vector of
unknown disturbances, and f∈ℛnf the vector of
faults to be detected, A,B,Ed,Ef,C,D,Fd, and Ff are known
constant matrices of appropriate dimensions. In the frequency domain, system ΣLTI can be
equivalently described by
(2)y(z)=Gu(z)u(z)+Gd(z)d(z)+Gf(z)f(z),
where Gu(z),Gd(z), and Gf(z) denote,
respectively, the transfer function matrices from u,d, and f to y.Although the design of a model-based FD systems mainly
consists of three tasks: (a) residual generation, (b) residual evaluation, (c)
threshold determination, major research attention has been focused on the
residual generation with the following issues.(i)
Full decoupling problem, which deals with the design of a residual generator, so
that the residual signal r satisfies
(3)∀u,d,limk→∞r(k)=0iff=0,r(k)≠0iffi(k)≠0,i=1,…,nf.
If a full decoupling is realized, then the residual
evaluation reduces to detect the nonzeroness of the residual signal.(ii)
Optimal FD problem, which is to design the residual generator so that the residual
signal r is as small as
possible if f=0 and deviates
from 0 as much as
possible if fi(k)≠0, i=1,…,nf.
Considering that in the fault-free case the residual
signal r would, due to
the existence of d, differ from
zero, evaluation of the size of r is necessary in
order to distinguish the influence of the faults from that of the disturbances.
In this paper, the norm-based evaluation of the residual signal, denoted
by J=∥r∥ and, based on
it, threshold determination satisfying
(4)Jth=supf=0,d∥r∥
will be briefly reviewed.


### 2.2. Parity Space Approach

The parity space approach is based on the so-called
parity relation. Lets be an integer
denoting the length of a moving time window. The output of system (1) over the moving window [k−s,k] can be
expressed by the initial state x(k−s), the stacked control input vector uk,s, the stacked disturbance vector dk,s, and the stacked
fault vector fk,s as
(5)yk,s=Ho,sx(k−s)+Hu,suk,s+Hd,sdk,s+Hf,sfk,s,
where
(6)ξk,s=[ξ(k−s)ξ(k−s+1)⋮ξ(k)]withξstandingfory,u,d,fHo,s=[CCA⋮CAs],Hu,s=[DO⋯OCBD⋱⋮⋮⋱⋱OCAs−1B⋯CBD],Hd,s and Hf,s are constructed
similarly as Hu,s and can be
achieved by replacing B,D, respectively, by Ed,Fd and Ef,Ff. To satisfy the requirement on the residual signal, a
residual generator can be constructed as
(7)r(k)=vs(yk,s−Hu,suk,s),
where a design parameter vs called parity
vector is introduced to modulate the residual dynamics and improve the
sensitivity of the residual to the faults and the robustness to the
disturbances and the initial state. Usually, vsHo,s=0 is required to
eliminate the influence of the initial state and the past input signals (before
the time instant k−s).If the existence condition(8)rank[Ho,sHd,sHf,s]>rank[Ho,sHd,s]
is satisfied, then a full decoupling from both the
initial state and the disturbances can be achieved by solving
(9)vs[Ho,sHd,s]=0,vsHf,s≠0,
for vs, that is, vs lies in the
intersection between the left null space of [Ho,s□Hd,s] and the image
space of Hf,s. If a full decoupling is not achievable or not
desired, the FD problem is often formulated as to solve the optimization
problem
(10)maxvs,vsHo,s=0JPS(vs)=maxvs,vsHo,s=0supdk,s=0,fk,s≠0(rT(k)r(k)/fk,sTfk,s)supfk,s=0,dk,s≠0(rT(k)r(k)/dk,sTdk,s)=maxvs,vsHo,s=0vsHf,sHf,sTvsTvsHd,sHd,sTvsT
whose solution can be obtained by solving a
generalized eigenvalue-eigenvector problem [11].Solution to Optimization Problem (10)
LetNbasis denote the
basis of the left null space of Ho,s. Assume that λmax and ps,max are the maximal
generalized eigenvalue and the corresponding eigenvector to the generalized
eigenvalue-eigenvector problem
(11)ps,max(NbasisHf,sHf,sTNbasisT−λmaxNbasisHd,sHd,sTNbasisT)=0,
then optimization problem (10) is solved by
(12)vs=ps,maxNbasis.
It is pointed out in [12] that the solutions of a full decoupling or (10) are achieved at the cost of (considerably) reduced
fault detectability. This can be immediately seen with a look at the dynamics
of the residual signal
(13)r(k)=vs(yk,s−Hu,suk,s)=vs(Hf,sfk,s+Hd,sdk,s)
which shows that the influence of the fault expressed
by vsHf,s is structurally
reduced to a minimum, that is, rank(vsHf,s)=1. Reference [12] proposed  the use of a parity matrix Vs,(14)r(k)=Vs(yk,s−Hu,suk,s)=Vs(Hf,sfk,s+Hd,sdk,s)
instead of a parity vector aiming at enhancing the
influence of the faults on the residual signal. To this end, the following
optimization problems are formulated as:
(15)maxVs,VsHo,s=0JPS,∞/∞(Vs)=maxVs,VsHo,s=0supdk,s=0,fk,s≠0(rT(k)r(k)/fk,sTfk,s)supfk,s=0,dk,s≠0(rT(k)r(k)/dk,sTdk,s)=maxVs,VsHo,s=0σ¯2(VsHf,s)σ¯2(VsHd,s),(16)maxVs,VsHo,s=0JPS,−/∞(Vs)=maxVs,VsHo,s=0infdk,s=0,fk,s≠0(rT(k)r(k)/fk,sTfk,s)supfk,s=0,dk,s≠0(rT(k)r(k)/dk,sTdk,s)=maxVs,VsHo,s=0σ̲2(VsHf,s)σ¯2(VsHd,s),(17)maxVs,VsHo,s=0JPS,i/∞(Vs)=maxVs,VsHo,s=0σi2(VsHf,s)σ¯2(VsHd,s).
The difference in optimization problems (15)-(16) consists in that the former considers the maximal
influence of the faults on the residual amplitude, while the latter considers
the minimal influence. Optimization problem (17) is a generalization of (15)-(16) and takes into account the fault sensitivity in
different directions. The achievable optimal performance index of optimization
problem (15) is the same with that of (10). At the end of this subsection, we will show that
the solution of (17) would lead to maximizing the fault detectability in
the context of a tradeoff between false alarm rate and fault detectability.Solution to Optimization Problems (15), (16), and (17)
A solution to optimization problem (17) that also solves (15)-(16) simultaneously is given in [12] as follows, which is derived based on the observation
that for any matrices X1 and X2 of compatible
dimensions,
(18)σ¯(X1X2)≤σ¯(X1)σ¯(X2),σ̲(X1X2)≤σ¯(X1)σ̲(X2),σi(X1X2)≤σ¯(X1)σi(X2).
Assume that there is the following singular value
decomposition (SVD):(19)NbasisHd,s=U[SO]VT,
where U and V are unitary
matrices, S=diag{σ1,…,σγ}, then
optimization problems (15), (16), and (17) are solved by
(20)Vs=P¯sS−1UTNbasis,
where P¯s is any unitary
matrix of compatible dimensions.
Note that the solutions to the above optimization
problems are not unique. For instance, an alternative optimal solution for
problem (16) is
(21)Vs=P¯sPNbasis,
where P is the left
inverse of NbasisHf,s. On the other side, only solution (20) solves (15), (16), and (17) simultaneously. For this reason, (20) is called unified parity space solution.
To detect the faults successfully, the generated
residual signal should be further evaluated. For a residual signal generated by
means of the parity space approach, the Euclidean norm defined by(22)J=∥r(k)∥E=rT(k)r(k)
is a reasonable evaluation function. It follows from (14) and (4) that the corresponding threshold is determined by
(23)Jth=supf=0,d∥r(k)∥E=σ¯(VsHd,s)max∥dk,s∥E.
Based on the decision logic
(24)∥r(k)∥E≤Jth⇒fault-free,otherwisefaulty,
a decision for the occurrence of a fault can be
finally made.
In practice, false alarm rate and miss detection rate
are two important technical features for the performance evaluation of a fault detection system. Below, we will introduce these two concepts in the context
of the norm-based residual evaluation (22) and briefly compare the above-presented parity space
solutions.
Jthsetting under a
given false alarm rate.  Consider (23) and denote the upper bound of ∥dk,s∥E by δd. In the context of norm-based evaluation, the
objective of Jth setting is to
ensure that any disturbance whose size is not larger than the tolerant limit
should not cause an alarm. To express the strongest disturbance that is allowed
without causing a false alarm in relation to δd, we define false alarm rate (FAR) as
(25)FAR=1−αδd,0≤α≤δd⇒0≤1−αδd≤1,
that is, those disturbances whose size is not larger than α=(1−FAR)δd should not cause an alarm. Suppose that the
allowable FAR is now given. It is straightforward that the threshold should be
set as
(26)Jth=ασ¯(VsHd,s)=(1−FAR)δdσ¯(VsHd,s).
Note that in the norm-based residual evaluation, Jth is often set as δdσ¯(VsHd,s) which leads to
a zero FAR but may result
in a very conservative Jth setting.
To express the miss detection rate (MDR), we introduce
the set of detectable faults. Note that a fault can be detected if and only if(27)∥r∥E>Jth⟺∥Vs(Hd,sdk,s+Hf,sfk,s)∥E>Jth.
Hence, the set of detectable faults (SDF) ΩDE(Vs,Jth,d) is defined as
follows: given Vs and Jth(28)ΩDE(Vs,Jth,d)={fk,s∣(27)issatisfied}.
Given Jth, a parity space
matrix Vs,opt delivers a
residual signal with the lowest MDR if
(29)∀Vs,ΩDE(Vs,Jth,d)⊆ΩDE(Vs,opt,Jth,d),
that is, ΩDE(Vs,opt,Jth,d) includes the
largest number of detectable faults, which is equivalent with the lowest MDR.
The subsequent comparison study is done in the context
ofmaximizing SDF (i.e., minimizing MDR) under a given FAR.
Note that (27) can be, according to (19), rewritten into
(30)∥Vs(Hd,sdk,s+Hf,sfk,s)∥E>(1−FAR)δdσ¯(VsHd,s)⟺∥Q[IO]VTdk,s+QS−1UTNbasisHf,sfk,s∥E>(1−FAR)δdσ¯(Q[IO]VT)bysettingVs=QS−1UTNbasis,forsomeQ.
It turns out that ∀Q≠0, (30) holds only if
(31)∥[IO]VTdk,s+S−1UTNbasisHf,sfk,s∥E>(1−FAR)δd.
It means that  a parity matrix that ensures (31) would provide a maximal SDF. Note that the unified
parity space solution (20) delivers exactly (31). Thus, the unified parity space solution
maximizes SDF (i.e., minimizes MDR) under a given FAR.
For comparison, denote the SVD ofNbasisHf,s by
(32)NbasisHf,s=Uf[SfO]VfT.
Then the vector-valued solution (12) to optimization problem (10) and the matrix-valued solution (21) to optimization problem (16) can be, respectively, rewritten into
(33)vs=psS−1UTNbasis,ps(S−1UTUfSf2UfTUS−1−λmaxI)=0,Vs=(P¯sSf−1UfTUS)S−1UTNbasis.
Since, generally, ps,P¯sSf−1UfTUS are not unitary
matrices, we have finally
(34)ΩDE(Vs,Jth,d)⊂ΩDE(Vs,opt,Jth,d),Vs=ps,maxNbasis,Vs,opt=P¯sS−1UTNbasis,ΩDE(Vs,Jth,d)⊂ΩDE(Vs,opt,Jth,d),Vs=P¯sPNbasis,Vs,opt=P¯sS−1UTNbasis.
With the following remarks we would like to conclude
this subsection.(i)
Parity-space-based FD system design is characterized by the simple mathematical handling. It
only deals with matrix- and vector-valued operations. This fact attracts
attention from the industry for the application of parity-space-based methods.(ii)
There is a
one-to-one relationship between the parity-space approach and the observer-based approach that allows the design of an observer-based residual generator
based on a given parity vector [13, 14]. Based on this result, a strategy called parity-space design, observer-based implementation has been developed, which makes
use of the computational advantage of parity-space approaches for the FD system
design (selection of a parity vector or matrix) and then realizes the solution
in the observer form to ensure a numerically stable and less consuming online
computation. This strategy has been successfully used in the sensor-fault
detection in vehicles [15].(iii)
In the parity-space approaches, a high orders will improve
the optimal performance index JPS but, on the
other side, increase the online computational effort [16]. By introducing a low-order IIR (infinite impulse
response) filter, the performance of the parity-relation-based residual
generator can be much improved without significant increase of the order of the
parity relation [17]. Similar effect can be achieved by the closed-loop-observer-based implementation, as pointed out by [18].(iv)
The algebraic
form of the parity-space-based FD system allows a statistic test and norm-based
residual evaluation and threshold determination [19]. It may well bridge the statistical methods [1] and the observer-based methods.(v)
In the
framework of parity-space-based FD system design, system dynamic features like
transmission zeros, zeros in the right half plane (RHP), and so forth are not taken into
account. This may cause trouble at the online implementation. Also for this
reason, we are of the opinion that the strategy ofparity space design,
observer-based implementation would be helpful to solve this problem.


### 2.3. Parametrization of FD Systems and Post-Filter Design

Observer-based FD system design for continuous LTI
systems has been widely studied in the literature [3–6]. In this and the next subsections, the analog form of
those known results will be briefly reviewed. Attention will be paid to the
comparison study when it is special for discrete LTI systems.Let(M^u(z),N^u(z)) be a left
coprime factorization pair of Gu(z), that is, Gu(z)=M^u−1(z)N^u(z),M^u(z),N^u(z)∈ℛℋ∞ [10]. In [20], a parametrization of all LTI residual generators for
system (1) described by
(35)r(z)=R(z)(M^u(z)y(z)−N^u(z)u(z))
is presented, where R(z)∈ℛℋ∞ is the
so-called postfilter that is arbitrarily selectable. Suppose that Gu(z)=(A,B,C,D) is a detectable
state space realization of Gu(z). Then M^u(z),N^u(z) can be computed
as follows [10]:
(36)M^u(z)=I−C(zI−A+LC)−1L,N^u(z)=D+C(zI−A+LC)−1(B−LD).
It is now a well-known result that(i)
M^u(z)y(z)−N^u(z)u(z)=y(z)−y^(z), where y^(z) is the output
estimation delivered by a full-order observer;(ii)
givenL1 and L2, there exists an ℛℋ∞-invertible
postfilter Q(z)=I+C(zI−A+L1C)−1(L2−L1) so that
(37)M^u,L1(z)y(z)−N^u,L1(z)u(z)=Q(z)(M^u,L2(z)y(z)−N^u,L2(z)u(z)),
where (M^u,L1(z),N^u,L1(z)) and (M^u,L2(z),N^u,L2(z)) are the left
coprime factorization pair of Gu(z), which are computed according to (36) with L=L1 and L=L2, respectively.(iii)
all LTI
residual generators can be expressed by a series connection of a full-order
observer and a postfilter, and are therefore called observer-based residual
generators. Moreover, the selection of the postfilter can be done independent
of the observer design.
Due to the latter fact, we concentrate in this
subsection on the selection of R(z). The dynamics of a residual generator (35) is governed by
(38)r(z)=R(z)M^u(z)(Gd(z)d(z)+Gf(z)f(z)).
The full decoupling problem is to find the postfilter R(z) so that
(39)R(z)M^u(z)Gd(z)=0,R(z)M^u(z)Gf(z)≠0.
If the l2-norm of the
residual signal is used as evaluation function, then the optimal FD problem is
formulated as optimization problems
(40)supR(z)∈ℛℋ∞JFRE,∞/∞(R)=supR(z)∈ℛℋ∞∥R(z)M^u(z)Gf(z)∥∞∥R(z)M^u(z)Gd(z)∥∞,(41)supR(z)∈ℛℋ∞JFRE,−/∞(R)=supR(z)∈ℛℋ∞∥R(z)M^u(z)Gf(z)∥−∥R(z)M^u(z)Gd(z)∥∞,(42)supR(z)∈ℛℋ∞JFRE,i/∞(R)=supR(z)∈ℛℋ∞σi(R(ejω)M^u(ejω)Gf(ejω))∥R(z)M^u(z)Gd(z)∥∞(43)supR(z)∈ℛℋ∞1×mJFRE,2/2(R)=supR(z)∈ℛℋ∞1×m(∫02πR(ejω)M^u(ejω)Gf(ejω)Gf∗(ejω)×M^u∗(ejω)R∗(ejω)dω/∫02πR(ejω)×M^u(ejω)Gd(ejω)Gd∗(ejω)M^u∗(ejω)×R∗(ejω)dω),
where σi(R(ejω)M^u(ejω)Gf(ejω)) represents the fault sensitivity at different levels at the frequency ω,
(44)∥R(z)M^u(z)Gf(z)∥−=infω(σ̲(R(ejω)M^u(ejω)Gf(ejω)))=infω,i(σi(R(ejω)M^u(ejω)Gf(ejω)))
though not a norm, it is interpreted as the worst-case
fault sensitivity. Optimization problems (40), (41), and (43) are often called the H∞/H∞,H−/H∞, and H2/H2 optimization,
respectively.The ratio-type performance index given in (40) and (43) is the first one that was introduced for the
FD purpose [11, 21]. Currently, the index of the form
(45)∥R(z)M^u(z)Gd(z)∥<γ,∥R(z)M^u(z)Gf(z)∥>β
becomes more popular, where γ,β are some
constants. The FD system design is often formulated as maximizing β under a given γ. The third index
type is often met in the robust control theory and formulated as
(46)Jf−d=αf∥R(z)M^u(z)Gf(z)∥−αd∥R(z)M^u(z)Gd(z)∥,
where αf,αd>0 are some given
constants. The FD system design is then achieved by maximizing Jf−d. In [22], it has been demonstrated that the above three types
indices are equivalent in a certain sense. With this fact in mind, in this
paper we only consider optimizations under ratio-type indices (40)–(43).Solution to Optimization Problems (40)–(42)
LetM^u(z)Gd(z)=Gdo(z)Gdi(z) be a
co-inner-outer factorization of M^u(z)Gd(z) [10], where Gdo(z) is the ℛℋ∞-left-invertible
co-outer, Gdi(z) is the co-inner
containing all the right half complex plane zeros of M^u(z)Gd(z) and satisfying Gdi(z)Gdi∗(z)=I. Based on the relations
(47)∥G1(z)G2(z)∥∞≤∥G1(z)∥∞∥G2(z)∥∞,∥G1(z)G2(z)∥−≤∥G1(z)∥∞∥G2(z)∥−σi(G1(ejω)G2(ejω))≤∥G1(z)∥∞σi(G2(ejω)),∀i
it has been proven in [23, 24], similar to the results for continuous LTI systems
given in [22] and recently in [25], that optimization problems (40)–(42) are solved simultaneously by
(48)R(z)=Gdo−1(z).
For this reason, (48) is called unified solution.Another solution to optimization problem (41), if Gf(z) is ℛℒ∞-left-invertible,
is
(49)R(z)=Gfo−1(z),
where Gfo(z) is the co-outer
of M^u(z)Gf(z).The main purpose of the co-inner-outer factorization
is to separate the nonminimum phase zeros so that the rest part ofM^u(z)Gd(z) or M^u(z)Gf(z) is ℛℋ∞-left invertible.
Note that the co-inner-outer factorization is not unique. Therefore, the
optimal postfilter R(z) is also not
unique [26].Solution to Optimization Problem (43)
The optimal solution to optimization problem (43) is a frequency selector as follows [27]:(50)R(z)=fω0(z)p(z),JFRE,2/2,opt=supωλmax(ω),
where fω0(z) is an ideal
frequency-selective filter with the selective frequency at ω0, which satisfies
(51)fω0(ejω)q(ejω)=0,ω≠ω0∫02πfω0(ejω)q(ejω)q∗(ejω)fω0∗(ejω)dω=q(ejω0)q∗(ejω0),∀qT(z)∈ℛℋ2,λmax(ω),p(ejω) are,
respectively, the maximal generalized eigenvalue and corresponding eigenvector
of the following generalized eigenvalue-eigenvector problem:
(52)p(ejω)(λmax(ω)M^u(ejω)Gd(ejω)Gd∗(ejω)M^u∗(ejω)−M^u(ejω)Gf(ejω)Gf∗(ejω)M^u∗(ejω))=0
and
ω0 is the
frequency at which λmax(ω) achieves its
maximum, that is,
(53)λmax(ω0)=supωλmax(ω).
In practice, usually a narrow bandpass filter is
implemented as frequency selector. From the viewpoint of FAR and MDR, the
frequency selector may cause loss of fault sensitivity and restrict the
application of the H2/H2 optimal
residual generator.Recently, [27] reported a very interesting result on the
relationship between the parity-space vector and the solution to optimization
problem (43). It has been shown that
(54)lims→∞minvsvsHf,sHf,sTvsTvsHd,sHd,sTvsT=λmax(ωopt)
and lims→∞vs correspond to
a bandpass filter. This result not only reveals the physical interpretation of
the standard optimal selection of parity vectors but also provides us with an
efficient tool to approximate the optimal solution to optimization problem (43). Moreover, based on it, advanced parity-space
approaches using wavelet transform have been proposed [28, 29].As to the residual evaluation and threshold
determination, thel2-norm is the
mostly used evaluation function, which leads to
(55)J=∥r∥2,Jth=supf=0,d∥r∥2=∥R(z)M^u(z)Gd(z)∥∞max∥d∥2.
In case of applying the unified solution (48), we have
(56)Jth=supf=0,d∥r∥2=max∥d∥2.
Analog to the results in [30] and the discussion in the last subsection, it can be
proven that the unified solution (48) minimizes MDR under a given FAR, where MDR and FAR
are defined in the context of the norm-based residual evaluation.


### 2.4. Fault Detection Filter Design

fault detection filter (FDF) is a special kind of
observer-based FD systems (35) with a constant post-filter and constructed as
(57)x^(k+1)=Ax^(k)+Bu(k)+L(y(k)−y^(k)),r(k)=W(y(k)−y^(k)),y^(k)=Cx^(k)+Du(k),
where the observer gain matrix L and the
weighting matrix W are design
parameters. Due to its state space expression and close relation to the
observer design, FDF study receives most research attention. The dynamics of
residual generator (57) is governed by
(58)r(z)=Grd(z)d(z)+Grf(z)f(z),Grd(z)=W(Fd+C(zI−A+LC)−1(Ed−LFd)),Grf(z)=W(Ff+C(zI−A+LC)−1(Ef−LFf)).
In the FDF design, the full-decoupling problem is to
design L and W such that
(59)Grd(z)=0,Grf(z)≠0
while the optimal FD problems are formulated so as to
choose matrices L,W that solve the
optimization problem [5, 7, 31]
(60)supL,WJOBS,∞/∞(L,W)=supL,W∥Grf(z)∥∞∥Grd(z)∥∞,(61)supL,WJOBS,−/∞(L,W)=supL,W∥Grf(z)∥−∥Grd(z)∥∞,(62)supL,WJOBS,i/∞(L,W)=supL,Wσi(Grf(ejω))∥Grd(z)∥∞.Solution to Optimization Problems (60)–(62)
Because the FDF (57) is a special case of (35), the optimal solutions to optimization problems (60)–(62) can be derived based on the state space realization of
the optimal post-filter R(z) given by (48), as shown in [24]. A unified optimal solution to optimization problems
(60)–(62) is associated to a discrete-time algebraic Riccati
system (DTARS) [24, 32]. Assume that (A,Ed,C,Fd) is regular,
then
(63)L=−LdT,W=Wd
solve optimization problems (60)–(62) simultaneously, where Wd is the left
inverse of a full column rank matrix Hd satisfying HdHdT=CXdCT+FdFdT and (Xd,Ld) is the
stabilizing solution to the DTARS
(64)[AXdAT−Xd+EdEdTAXdCT+EdFdTCXdAT+FdEdTCXdCT+FdFdT][ILd]=0.
An alternative solution to optimization problem (61), if (A,Ef,C,Ff) is regular, is
given by
(65)L=−LfT,W=Wf,
where Wf is the left
inverse of a full column rank matrix Hf satisfying HfHfT=CXfCT+FfFfT and (Xf,Lf) is the
stabilizing solution to the DTARS
(66)[AXfAT−Xf+EfEfTAXfCT+EfFfTCXfAT+FfEfTCXfCT+FfFfT][ILf]=0.
Recently, application of LMI-technique (linear matrix
inequality) to solve (60) and (61) for continuous LTI systems has been reported [33–36]. The core of those approaches consists in formulating
(60) or (61) as a multiobjective optimization problem and solving
them based on an iterative computation of two LMIs. It can be proven that these
solutions can, in the ideal case, converge to the optimal solution (63). The extension of these results to the discrete FDF
is straightforward and will not be discussed in this paper.At the end of this subsection, we would like to
introduce a very interesting result achieved by the comparison study between
the well-known Kalman-filter-based residual generation [37] and the unified solution (63). Given system
(67)x(k+1)=Ax(k)+Bu(k)+Edζ(k),y(k)=Cx(k)+Du(k)+Fdη(k),
where ζ(k),η(k) are independent
zero-mean Gaussian white noise processes with cov(ζ(k)ζT(k))=I, cov(η(k)ηT(k))=I, then a Kalman filter with
(68)x^(k+1∣k)=Ax^(k∣k−1)+Bu(k)+L(y(k)−y^(k∣k−1)),(69)y^(k∣k−1)=Cx^(k∣k−1)+Du(k),(70)L=APCT(CPCT+R)−1,R=FdFdT(>0),(71)APAT−P−APCT(CPCT+R)−1CPAT+EdEdT=0,P>0,delivers an innovation as residual signal, where x^(k∣k−1) is a prediction
of x(k) based on the
data up to k−1. Comparing the above Kalman filter algorithm with the
unified solution (63) makes it clear that both solutions are quite
similar. Indeed, (67) can be brought into the general form of (1) by letting d=[ζTηT]T and, as a
result, (71) can be regarded as a special case of (64). Remember that the Kalman filter delivers, in the
context of statistic tests, a minimum MDR under a given FAR. In comparison, in
the context of norm-based definition of MDR and FAR, the unified solution (63) provides us with the same result.


## 2.1. System Models and Problem Formulation

Suppose that the discrete LTI systems are described by(1)ΣLTI:{x(k+1)=Ax(k)+Bu(k)+Edd(k)+Eff(k),y(k)=Cx(k)+Du(k)+Fdd(k)+Fff(k),
where x∈ℛn is the state
vector, u∈ℛp the vector of
control inputs, y∈ℛm the vector of
process outputs, d∈ℛnd the vector of
unknown disturbances, and f∈ℛnf the vector of
faults to be detected, A,B,Ed,Ef,C,D,Fd, and Ff are known
constant matrices of appropriate dimensions. In the frequency domain, system ΣLTI can be
equivalently described by
(2)y(z)=Gu(z)u(z)+Gd(z)d(z)+Gf(z)f(z),
where Gu(z),Gd(z), and Gf(z) denote,
respectively, the transfer function matrices from u,d, and f to y.Although the design of a model-based FD systems mainly
consists of three tasks: (a) residual generation, (b) residual evaluation, (c)
threshold determination, major research attention has been focused on the
residual generation with the following issues.(i)
Full decoupling problem, which deals with the design of a residual generator, so
that the residual signal r satisfies
(3)∀u,d,limk→∞r(k)=0iff=0,r(k)≠0iffi(k)≠0,i=1,…,nf.
If a full decoupling is realized, then the residual
evaluation reduces to detect the nonzeroness of the residual signal.(ii)
Optimal FD problem, which is to design the residual generator so that the residual
signal r is as small as
possible if f=0 and deviates
from 0 as much as
possible if fi(k)≠0, i=1,…,nf.
Considering that in the fault-free case the residual
signal r would, due to
the existence of d, differ from
zero, evaluation of the size of r is necessary in
order to distinguish the influence of the faults from that of the disturbances.
In this paper, the norm-based evaluation of the residual signal, denoted
by J=∥r∥ and, based on
it, threshold determination satisfying
(4)Jth=supf=0,d∥r∥
will be briefly reviewed.


## 2.2. Parity Space Approach

The parity space approach is based on the so-called
parity relation. Lets be an integer
denoting the length of a moving time window. The output of system (1) over the moving window [k−s,k] can be
expressed by the initial state x(k−s), the stacked control input vector uk,s, the stacked disturbance vector dk,s, and the stacked
fault vector fk,s as
(5)yk,s=Ho,sx(k−s)+Hu,suk,s+Hd,sdk,s+Hf,sfk,s,
where
(6)ξk,s=[ξ(k−s)ξ(k−s+1)⋮ξ(k)]withξstandingfory,u,d,fHo,s=[CCA⋮CAs],Hu,s=[DO⋯OCBD⋱⋮⋮⋱⋱OCAs−1B⋯CBD],Hd,s and Hf,s are constructed
similarly as Hu,s and can be
achieved by replacing B,D, respectively, by Ed,Fd and Ef,Ff. To satisfy the requirement on the residual signal, a
residual generator can be constructed as
(7)r(k)=vs(yk,s−Hu,suk,s),
where a design parameter vs called parity
vector is introduced to modulate the residual dynamics and improve the
sensitivity of the residual to the faults and the robustness to the
disturbances and the initial state. Usually, vsHo,s=0 is required to
eliminate the influence of the initial state and the past input signals (before
the time instant k−s).If the existence condition(8)rank[Ho,sHd,sHf,s]>rank[Ho,sHd,s]
is satisfied, then a full decoupling from both the
initial state and the disturbances can be achieved by solving
(9)vs[Ho,sHd,s]=0,vsHf,s≠0,
for vs, that is, vs lies in the
intersection between the left null space of [Ho,s□Hd,s] and the image
space of Hf,s. If a full decoupling is not achievable or not
desired, the FD problem is often formulated as to solve the optimization
problem
(10)maxvs,vsHo,s=0JPS(vs)=maxvs,vsHo,s=0supdk,s=0,fk,s≠0(rT(k)r(k)/fk,sTfk,s)supfk,s=0,dk,s≠0(rT(k)r(k)/dk,sTdk,s)=maxvs,vsHo,s=0vsHf,sHf,sTvsTvsHd,sHd,sTvsT
whose solution can be obtained by solving a
generalized eigenvalue-eigenvector problem [11].Solution to Optimization Problem (10)
LetNbasis denote the
basis of the left null space of Ho,s. Assume that λmax and ps,max are the maximal
generalized eigenvalue and the corresponding eigenvector to the generalized
eigenvalue-eigenvector problem
(11)ps,max(NbasisHf,sHf,sTNbasisT−λmaxNbasisHd,sHd,sTNbasisT)=0,
then optimization problem (10) is solved by
(12)vs=ps,maxNbasis.
It is pointed out in [12] that the solutions of a full decoupling or (10) are achieved at the cost of (considerably) reduced
fault detectability. This can be immediately seen with a look at the dynamics
of the residual signal
(13)r(k)=vs(yk,s−Hu,suk,s)=vs(Hf,sfk,s+Hd,sdk,s)
which shows that the influence of the fault expressed
by vsHf,s is structurally
reduced to a minimum, that is, rank(vsHf,s)=1. Reference [12] proposed  the use of a parity matrix Vs,(14)r(k)=Vs(yk,s−Hu,suk,s)=Vs(Hf,sfk,s+Hd,sdk,s)
instead of a parity vector aiming at enhancing the
influence of the faults on the residual signal. To this end, the following
optimization problems are formulated as:
(15)maxVs,VsHo,s=0JPS,∞/∞(Vs)=maxVs,VsHo,s=0supdk,s=0,fk,s≠0(rT(k)r(k)/fk,sTfk,s)supfk,s=0,dk,s≠0(rT(k)r(k)/dk,sTdk,s)=maxVs,VsHo,s=0σ¯2(VsHf,s)σ¯2(VsHd,s),(16)maxVs,VsHo,s=0JPS,−/∞(Vs)=maxVs,VsHo,s=0infdk,s=0,fk,s≠0(rT(k)r(k)/fk,sTfk,s)supfk,s=0,dk,s≠0(rT(k)r(k)/dk,sTdk,s)=maxVs,VsHo,s=0σ̲2(VsHf,s)σ¯2(VsHd,s),(17)maxVs,VsHo,s=0JPS,i/∞(Vs)=maxVs,VsHo,s=0σi2(VsHf,s)σ¯2(VsHd,s).
The difference in optimization problems (15)-(16) consists in that the former considers the maximal
influence of the faults on the residual amplitude, while the latter considers
the minimal influence. Optimization problem (17) is a generalization of (15)-(16) and takes into account the fault sensitivity in
different directions. The achievable optimal performance index of optimization
problem (15) is the same with that of (10). At the end of this subsection, we will show that
the solution of (17) would lead to maximizing the fault detectability in
the context of a tradeoff between false alarm rate and fault detectability.Solution to Optimization Problems (15), (16), and (17)
A solution to optimization problem (17) that also solves (15)-(16) simultaneously is given in [12] as follows, which is derived based on the observation
that for any matrices X1 and X2 of compatible
dimensions,
(18)σ¯(X1X2)≤σ¯(X1)σ¯(X2),σ̲(X1X2)≤σ¯(X1)σ̲(X2),σi(X1X2)≤σ¯(X1)σi(X2).
Assume that there is the following singular value
decomposition (SVD):(19)NbasisHd,s=U[SO]VT,
where U and V are unitary
matrices, S=diag{σ1,…,σγ}, then
optimization problems (15), (16), and (17) are solved by
(20)Vs=P¯sS−1UTNbasis,
where P¯s is any unitary
matrix of compatible dimensions.
Note that the solutions to the above optimization
problems are not unique. For instance, an alternative optimal solution for
problem (16) is
(21)Vs=P¯sPNbasis,
where P is the left
inverse of NbasisHf,s. On the other side, only solution (20) solves (15), (16), and (17) simultaneously. For this reason, (20) is called unified parity space solution.
To detect the faults successfully, the generated
residual signal should be further evaluated. For a residual signal generated by
means of the parity space approach, the Euclidean norm defined by(22)J=∥r(k)∥E=rT(k)r(k)
is a reasonable evaluation function. It follows from (14) and (4) that the corresponding threshold is determined by
(23)Jth=supf=0,d∥r(k)∥E=σ¯(VsHd,s)max∥dk,s∥E.
Based on the decision logic
(24)∥r(k)∥E≤Jth⇒fault-free,otherwisefaulty,
a decision for the occurrence of a fault can be
finally made.
In practice, false alarm rate and miss detection rate
are two important technical features for the performance evaluation of a fault detection system. Below, we will introduce these two concepts in the context
of the norm-based residual evaluation (22) and briefly compare the above-presented parity space
solutions.
Jthsetting under a
given false alarm rate.  Consider (23) and denote the upper bound of ∥dk,s∥E by δd. In the context of norm-based evaluation, the
objective of Jth setting is to
ensure that any disturbance whose size is not larger than the tolerant limit
should not cause an alarm. To express the strongest disturbance that is allowed
without causing a false alarm in relation to δd, we define false alarm rate (FAR) as
(25)FAR=1−αδd,0≤α≤δd⇒0≤1−αδd≤1,
that is, those disturbances whose size is not larger than α=(1−FAR)δd should not cause an alarm. Suppose that the
allowable FAR is now given. It is straightforward that the threshold should be
set as
(26)Jth=ασ¯(VsHd,s)=(1−FAR)δdσ¯(VsHd,s).
Note that in the norm-based residual evaluation, Jth is often set as δdσ¯(VsHd,s) which leads to
a zero FAR but may result
in a very conservative Jth setting.
To express the miss detection rate (MDR), we introduce
the set of detectable faults. Note that a fault can be detected if and only if(27)∥r∥E>Jth⟺∥Vs(Hd,sdk,s+Hf,sfk,s)∥E>Jth.
Hence, the set of detectable faults (SDF) ΩDE(Vs,Jth,d) is defined as
follows: given Vs and Jth(28)ΩDE(Vs,Jth,d)={fk,s∣(27)issatisfied}.
Given Jth, a parity space
matrix Vs,opt delivers a
residual signal with the lowest MDR if
(29)∀Vs,ΩDE(Vs,Jth,d)⊆ΩDE(Vs,opt,Jth,d),
that is, ΩDE(Vs,opt,Jth,d) includes the
largest number of detectable faults, which is equivalent with the lowest MDR.
The subsequent comparison study is done in the context
ofmaximizing SDF (i.e., minimizing MDR) under a given FAR.
Note that (27) can be, according to (19), rewritten into
(30)∥Vs(Hd,sdk,s+Hf,sfk,s)∥E>(1−FAR)δdσ¯(VsHd,s)⟺∥Q[IO]VTdk,s+QS−1UTNbasisHf,sfk,s∥E>(1−FAR)δdσ¯(Q[IO]VT)bysettingVs=QS−1UTNbasis,forsomeQ.
It turns out that ∀Q≠0, (30) holds only if
(31)∥[IO]VTdk,s+S−1UTNbasisHf,sfk,s∥E>(1−FAR)δd.
It means that  a parity matrix that ensures (31) would provide a maximal SDF. Note that the unified
parity space solution (20) delivers exactly (31). Thus, the unified parity space solution
maximizes SDF (i.e., minimizes MDR) under a given FAR.
For comparison, denote the SVD ofNbasisHf,s by
(32)NbasisHf,s=Uf[SfO]VfT.
Then the vector-valued solution (12) to optimization problem (10) and the matrix-valued solution (21) to optimization problem (16) can be, respectively, rewritten into
(33)vs=psS−1UTNbasis,ps(S−1UTUfSf2UfTUS−1−λmaxI)=0,Vs=(P¯sSf−1UfTUS)S−1UTNbasis.
Since, generally, ps,P¯sSf−1UfTUS are not unitary
matrices, we have finally
(34)ΩDE(Vs,Jth,d)⊂ΩDE(Vs,opt,Jth,d),Vs=ps,maxNbasis,Vs,opt=P¯sS−1UTNbasis,ΩDE(Vs,Jth,d)⊂ΩDE(Vs,opt,Jth,d),Vs=P¯sPNbasis,Vs,opt=P¯sS−1UTNbasis.
With the following remarks we would like to conclude
this subsection.(i)
Parity-space-based FD system design is characterized by the simple mathematical handling. It
only deals with matrix- and vector-valued operations. This fact attracts
attention from the industry for the application of parity-space-based methods.(ii)
There is a
one-to-one relationship between the parity-space approach and the observer-based approach that allows the design of an observer-based residual generator
based on a given parity vector [13, 14]. Based on this result, a strategy called parity-space design, observer-based implementation has been developed, which makes
use of the computational advantage of parity-space approaches for the FD system
design (selection of a parity vector or matrix) and then realizes the solution
in the observer form to ensure a numerically stable and less consuming online
computation. This strategy has been successfully used in the sensor-fault
detection in vehicles [15].(iii)
In the parity-space approaches, a high orders will improve
the optimal performance index JPS but, on the
other side, increase the online computational effort [16]. By introducing a low-order IIR (infinite impulse
response) filter, the performance of the parity-relation-based residual
generator can be much improved without significant increase of the order of the
parity relation [17]. Similar effect can be achieved by the closed-loop-observer-based implementation, as pointed out by [18].(iv)
The algebraic
form of the parity-space-based FD system allows a statistic test and norm-based
residual evaluation and threshold determination [19]. It may well bridge the statistical methods [1] and the observer-based methods.(v)
In the
framework of parity-space-based FD system design, system dynamic features like
transmission zeros, zeros in the right half plane (RHP), and so forth are not taken into
account. This may cause trouble at the online implementation. Also for this
reason, we are of the opinion that the strategy ofparity space design,
observer-based implementation would be helpful to solve this problem.


## 2.3. Parametrization of FD Systems and Post-Filter Design

Observer-based FD system design for continuous LTI
systems has been widely studied in the literature [3–6]. In this and the next subsections, the analog form of
those known results will be briefly reviewed. Attention will be paid to the
comparison study when it is special for discrete LTI systems.Let(M^u(z),N^u(z)) be a left
coprime factorization pair of Gu(z), that is, Gu(z)=M^u−1(z)N^u(z),M^u(z),N^u(z)∈ℛℋ∞ [10]. In [20], a parametrization of all LTI residual generators for
system (1) described by
(35)r(z)=R(z)(M^u(z)y(z)−N^u(z)u(z))
is presented, where R(z)∈ℛℋ∞ is the
so-called postfilter that is arbitrarily selectable. Suppose that Gu(z)=(A,B,C,D) is a detectable
state space realization of Gu(z). Then M^u(z),N^u(z) can be computed
as follows [10]:
(36)M^u(z)=I−C(zI−A+LC)−1L,N^u(z)=D+C(zI−A+LC)−1(B−LD).
It is now a well-known result that(i)
M^u(z)y(z)−N^u(z)u(z)=y(z)−y^(z), where y^(z) is the output
estimation delivered by a full-order observer;(ii)
givenL1 and L2, there exists an ℛℋ∞-invertible
postfilter Q(z)=I+C(zI−A+L1C)−1(L2−L1) so that
(37)M^u,L1(z)y(z)−N^u,L1(z)u(z)=Q(z)(M^u,L2(z)y(z)−N^u,L2(z)u(z)),
where (M^u,L1(z),N^u,L1(z)) and (M^u,L2(z),N^u,L2(z)) are the left
coprime factorization pair of Gu(z), which are computed according to (36) with L=L1 and L=L2, respectively.(iii)
all LTI
residual generators can be expressed by a series connection of a full-order
observer and a postfilter, and are therefore called observer-based residual
generators. Moreover, the selection of the postfilter can be done independent
of the observer design.
Due to the latter fact, we concentrate in this
subsection on the selection of R(z). The dynamics of a residual generator (35) is governed by
(38)r(z)=R(z)M^u(z)(Gd(z)d(z)+Gf(z)f(z)).
The full decoupling problem is to find the postfilter R(z) so that
(39)R(z)M^u(z)Gd(z)=0,R(z)M^u(z)Gf(z)≠0.
If the l2-norm of the
residual signal is used as evaluation function, then the optimal FD problem is
formulated as optimization problems
(40)supR(z)∈ℛℋ∞JFRE,∞/∞(R)=supR(z)∈ℛℋ∞∥R(z)M^u(z)Gf(z)∥∞∥R(z)M^u(z)Gd(z)∥∞,(41)supR(z)∈ℛℋ∞JFRE,−/∞(R)=supR(z)∈ℛℋ∞∥R(z)M^u(z)Gf(z)∥−∥R(z)M^u(z)Gd(z)∥∞,(42)supR(z)∈ℛℋ∞JFRE,i/∞(R)=supR(z)∈ℛℋ∞σi(R(ejω)M^u(ejω)Gf(ejω))∥R(z)M^u(z)Gd(z)∥∞(43)supR(z)∈ℛℋ∞1×mJFRE,2/2(R)=supR(z)∈ℛℋ∞1×m(∫02πR(ejω)M^u(ejω)Gf(ejω)Gf∗(ejω)×M^u∗(ejω)R∗(ejω)dω/∫02πR(ejω)×M^u(ejω)Gd(ejω)Gd∗(ejω)M^u∗(ejω)×R∗(ejω)dω),
where σi(R(ejω)M^u(ejω)Gf(ejω)) represents the fault sensitivity at different levels at the frequency ω,
(44)∥R(z)M^u(z)Gf(z)∥−=infω(σ̲(R(ejω)M^u(ejω)Gf(ejω)))=infω,i(σi(R(ejω)M^u(ejω)Gf(ejω)))
though not a norm, it is interpreted as the worst-case
fault sensitivity. Optimization problems (40), (41), and (43) are often called the H∞/H∞,H−/H∞, and H2/H2 optimization,
respectively.The ratio-type performance index given in (40) and (43) is the first one that was introduced for the
FD purpose [11, 21]. Currently, the index of the form
(45)∥R(z)M^u(z)Gd(z)∥<γ,∥R(z)M^u(z)Gf(z)∥>β
becomes more popular, where γ,β are some
constants. The FD system design is often formulated as maximizing β under a given γ. The third index
type is often met in the robust control theory and formulated as
(46)Jf−d=αf∥R(z)M^u(z)Gf(z)∥−αd∥R(z)M^u(z)Gd(z)∥,
where αf,αd>0 are some given
constants. The FD system design is then achieved by maximizing Jf−d. In [22], it has been demonstrated that the above three types
indices are equivalent in a certain sense. With this fact in mind, in this
paper we only consider optimizations under ratio-type indices (40)–(43).Solution to Optimization Problems (40)–(42)
LetM^u(z)Gd(z)=Gdo(z)Gdi(z) be a
co-inner-outer factorization of M^u(z)Gd(z) [10], where Gdo(z) is the ℛℋ∞-left-invertible
co-outer, Gdi(z) is the co-inner
containing all the right half complex plane zeros of M^u(z)Gd(z) and satisfying Gdi(z)Gdi∗(z)=I. Based on the relations
(47)∥G1(z)G2(z)∥∞≤∥G1(z)∥∞∥G2(z)∥∞,∥G1(z)G2(z)∥−≤∥G1(z)∥∞∥G2(z)∥−σi(G1(ejω)G2(ejω))≤∥G1(z)∥∞σi(G2(ejω)),∀i
it has been proven in [23, 24], similar to the results for continuous LTI systems
given in [22] and recently in [25], that optimization problems (40)–(42) are solved simultaneously by
(48)R(z)=Gdo−1(z).
For this reason, (48) is called unified solution.Another solution to optimization problem (41), if Gf(z) is ℛℒ∞-left-invertible,
is
(49)R(z)=Gfo−1(z),
where Gfo(z) is the co-outer
of M^u(z)Gf(z).The main purpose of the co-inner-outer factorization
is to separate the nonminimum phase zeros so that the rest part ofM^u(z)Gd(z) or M^u(z)Gf(z) is ℛℋ∞-left invertible.
Note that the co-inner-outer factorization is not unique. Therefore, the
optimal postfilter R(z) is also not
unique [26].Solution to Optimization Problem (43)
The optimal solution to optimization problem (43) is a frequency selector as follows [27]:(50)R(z)=fω0(z)p(z),JFRE,2/2,opt=supωλmax(ω),
where fω0(z) is an ideal
frequency-selective filter with the selective frequency at ω0, which satisfies
(51)fω0(ejω)q(ejω)=0,ω≠ω0∫02πfω0(ejω)q(ejω)q∗(ejω)fω0∗(ejω)dω=q(ejω0)q∗(ejω0),∀qT(z)∈ℛℋ2,λmax(ω),p(ejω) are,
respectively, the maximal generalized eigenvalue and corresponding eigenvector
of the following generalized eigenvalue-eigenvector problem:
(52)p(ejω)(λmax(ω)M^u(ejω)Gd(ejω)Gd∗(ejω)M^u∗(ejω)−M^u(ejω)Gf(ejω)Gf∗(ejω)M^u∗(ejω))=0
and
ω0 is the
frequency at which λmax(ω) achieves its
maximum, that is,
(53)λmax(ω0)=supωλmax(ω).
In practice, usually a narrow bandpass filter is
implemented as frequency selector. From the viewpoint of FAR and MDR, the
frequency selector may cause loss of fault sensitivity and restrict the
application of the H2/H2 optimal
residual generator.Recently, [27] reported a very interesting result on the
relationship between the parity-space vector and the solution to optimization
problem (43). It has been shown that
(54)lims→∞minvsvsHf,sHf,sTvsTvsHd,sHd,sTvsT=λmax(ωopt)
and lims→∞vs correspond to
a bandpass filter. This result not only reveals the physical interpretation of
the standard optimal selection of parity vectors but also provides us with an
efficient tool to approximate the optimal solution to optimization problem (43). Moreover, based on it, advanced parity-space
approaches using wavelet transform have been proposed [28, 29].As to the residual evaluation and threshold
determination, thel2-norm is the
mostly used evaluation function, which leads to
(55)J=∥r∥2,Jth=supf=0,d∥r∥2=∥R(z)M^u(z)Gd(z)∥∞max∥d∥2.
In case of applying the unified solution (48), we have
(56)Jth=supf=0,d∥r∥2=max∥d∥2.
Analog to the results in [30] and the discussion in the last subsection, it can be
proven that the unified solution (48) minimizes MDR under a given FAR, where MDR and FAR
are defined in the context of the norm-based residual evaluation.


## 2.4. Fault Detection Filter Design

fault detection filter (FDF) is a special kind of
observer-based FD systems (35) with a constant post-filter and constructed as
(57)x^(k+1)=Ax^(k)+Bu(k)+L(y(k)−y^(k)),r(k)=W(y(k)−y^(k)),y^(k)=Cx^(k)+Du(k),
where the observer gain matrix L and the
weighting matrix W are design
parameters. Due to its state space expression and close relation to the
observer design, FDF study receives most research attention. The dynamics of
residual generator (57) is governed by
(58)r(z)=Grd(z)d(z)+Grf(z)f(z),Grd(z)=W(Fd+C(zI−A+LC)−1(Ed−LFd)),Grf(z)=W(Ff+C(zI−A+LC)−1(Ef−LFf)).
In the FDF design, the full-decoupling problem is to
design L and W such that
(59)Grd(z)=0,Grf(z)≠0
while the optimal FD problems are formulated so as to
choose matrices L,W that solve the
optimization problem [5, 7, 31]
(60)supL,WJOBS,∞/∞(L,W)=supL,W∥Grf(z)∥∞∥Grd(z)∥∞,(61)supL,WJOBS,−/∞(L,W)=supL,W∥Grf(z)∥−∥Grd(z)∥∞,(62)supL,WJOBS,i/∞(L,W)=supL,Wσi(Grf(ejω))∥Grd(z)∥∞.Solution to Optimization Problems (60)–(62)
Because the FDF (57) is a special case of (35), the optimal solutions to optimization problems (60)–(62) can be derived based on the state space realization of
the optimal post-filter R(z) given by (48), as shown in [24]. A unified optimal solution to optimization problems
(60)–(62) is associated to a discrete-time algebraic Riccati
system (DTARS) [24, 32]. Assume that (A,Ed,C,Fd) is regular,
then
(63)L=−LdT,W=Wd
solve optimization problems (60)–(62) simultaneously, where Wd is the left
inverse of a full column rank matrix Hd satisfying HdHdT=CXdCT+FdFdT and (Xd,Ld) is the
stabilizing solution to the DTARS
(64)[AXdAT−Xd+EdEdTAXdCT+EdFdTCXdAT+FdEdTCXdCT+FdFdT][ILd]=0.
An alternative solution to optimization problem (61), if (A,Ef,C,Ff) is regular, is
given by
(65)L=−LfT,W=Wf,
where Wf is the left
inverse of a full column rank matrix Hf satisfying HfHfT=CXfCT+FfFfT and (Xf,Lf) is the
stabilizing solution to the DTARS
(66)[AXfAT−Xf+EfEfTAXfCT+EfFfTCXfAT+FfEfTCXfCT+FfFfT][ILf]=0.
Recently, application of LMI-technique (linear matrix
inequality) to solve (60) and (61) for continuous LTI systems has been reported [33–36]. The core of those approaches consists in formulating
(60) or (61) as a multiobjective optimization problem and solving
them based on an iterative computation of two LMIs. It can be proven that these
solutions can, in the ideal case, converge to the optimal solution (63). The extension of these results to the discrete FDF
is straightforward and will not be discussed in this paper.At the end of this subsection, we would like to
introduce a very interesting result achieved by the comparison study between
the well-known Kalman-filter-based residual generation [37] and the unified solution (63). Given system
(67)x(k+1)=Ax(k)+Bu(k)+Edζ(k),y(k)=Cx(k)+Du(k)+Fdη(k),
where ζ(k),η(k) are independent
zero-mean Gaussian white noise processes with cov(ζ(k)ζT(k))=I, cov(η(k)ηT(k))=I, then a Kalman filter with
(68)x^(k+1∣k)=Ax^(k∣k−1)+Bu(k)+L(y(k)−y^(k∣k−1)),(69)y^(k∣k−1)=Cx^(k∣k−1)+Du(k),(70)L=APCT(CPCT+R)−1,R=FdFdT(>0),(71)APAT−P−APCT(CPCT+R)−1CPAT+EdEdT=0,P>0,delivers an innovation as residual signal, where x^(k∣k−1) is a prediction
of x(k) based on the
data up to k−1. Comparing the above Kalman filter algorithm with the
unified solution (63) makes it clear that both solutions are quite
similar. Indeed, (67) can be brought into the general form of (1) by letting d=[ζTηT]T and, as a
result, (71) can be regarded as a special case of (64). Remember that the Kalman filter delivers, in the
context of statistic tests, a minimum MDR under a given FAR. In comparison, in
the context of norm-based definition of MDR and FAR, the unified solution (63) provides us with the same result.


## 3. FD of Discrete-Time Linear Periodic Systems

In this section, we review some recent results on FD
in discrete linear time periodic (LTP) systems. LTP is a special kind of linear
time-varying systems described by(72)ΣLTP:{x(k+1)=A(k)x(k)+B(k)u(k)+Ed(k)d(k)+Ef(k)f(k),y(k)=C(k)x(k)+D(k)u(k)+Fd(k)d(k)+Ff(k)f(k),
where A(k),B(k),C(k),D(k),Ed(k),Ef(k),Fd(k), and Ff(k) are known
bounded and real periodic matrices of period θ, that is, ∀k, A(k+θ)=A(k), and so forth. There is not only continuous interest and
development of periodic control and filtering theory [38–42], but also increasing applications of periodic control
in practice like helicopter vibration control, satellite attitude control as
well as wind turbine. The FD problem of periodic systems has been considered in [24, 32, 43–45]. Basically, there are two ways to handle the FD
problem of LTP systems, as shown below.


### 3.1. FD Schemes Based on Lifted LTI Reformulation

It is well known that there is a strong correspondence
between discrete LTP systems and discrete LTI systems [42]. Therefore, FD system design for the LTP system (72) can be carried out as follows:(i)
lift the LTP
system (72) into a discrete LTI system,(ii)
design residual
generator(s) based on the lifted LTI reformulation,(iii)
using either
parallel residual generators or select the parameters of the residual generator
to satisfy the causality condition.LetΨA(k)(k1,k0) ( k1≥k0) denote the state transition matrix of LTP system (72)
(73)ΨA(k)(k1,k0)={Iifk1=k0,A(k1−1)A(k1−2)⋯A(k0)ifk1>k0.
Periodic system (72) can be lifted into a discrete LTI system described
by [42]
(74)xτ((k+1)θ+τ)=Aτxτ(kθ+τ)+Bτuτ(kθ+τ)+Ed,τdτ(kθ+τ)+Ef,τfτ(kθ+τ),yτ(kθ+τ)=Cτxτ(kθ+τ)+Dτuτ(kθ+τ)+Fd,τdτ(kθ+τ)+Ff,τfτ(kθ+τ),
where τ is an integer
between 0 and θ−1 denoting the
initial time, the state vector of the lifted system is xτ(kθ+τ)=x(kθ+τ), ητ with η standing for u,y,d,f is the
augmented signal defined by
(75)ητ(kθ+τ)=[η(kθ+τ)η(kθ+τ+1)⋮η(kθ+τ+θ−1)],Aτ=ΨA(k)(τ+θ,τ),Bτ=[X′Y′⋯Z′],Cτ=[C(τ)C(τ+1)ΨA(k)(τ+1,τ)⋮C(τ+θ−1)ΨA(k)(τ+θ−1,τ)],Dτ=[Dτ,1,1O⋯ODτ,2,1Dτ,2,2⋱⋮⋮⋱⋱ODτ,θ,1⋯Dτ,θ,θ−1Dτ,θ,θ],Dτ,i,j={D(τ+i−1)fori=j,C(τ+i−1)ΨA(k)(τ+i−1,τ+j)B(τ+j−1)fori>j,
where  Bτ,1=ΨA(k)(τ+θ,τ+1)B(τ), Bτ,2=ΨA(k)(τ+θ,τ+2)B(τ+1), Bτ,θ=B(τ+θ−1), and Ed,τ,Fd,τ and Ef,τ,Ff,τ are defined in
a way similar to Cτ,Dτ.


#### 3.1.1. Observer-Based FD System Design and Implementation

Assume that(A(k),C(k)) is detectable.
Then (Aτ,Cτ) is detectable
and an observer-based LTI residual generator can be designed based on lifted
reformulation (74) as
(76)x^τ((k+1)θ+τ)=Aτx^τ(kθ+τ)+Bτuτ(kθ+τ)+Lτ(yτ(kθ+τ)−y^τ(kθ+τ)),rτ(kθ+τ)=Wτ(yτ(kθ+τ)−y^τ(kθ+τ)),y^τ(k)=Cτx^τ(kθ+τ)+Dτuτ(kθ+τ),
where Lτ and Wτ are constant
matrices and can be designed with FD approaches for the discrete LTI systems
introduced in the last section to realize full decoupling or optimal FD.
Observer (76) reconstructs the outputs over one period y(kθ+τ),…,y(kθ+τ+θ−1) based on the
estimation of state vector x(kθ+τ). Both the state vector of observer (76) and the residual signal are updated every θ time instants.In fault detection, the detection delay should be as
small as possible. Therefore, it is advantageous if a residual signal can be
obtained at each time instant. To this aim,(i)
a bank of LTI
residual generators (76) can be used, each of which is designed for τ=0,1,…,θ−1, respectively [43]. This scheme is characterized by a simple design but
needs much online computational efforts.(ii)
If the
weighting matrixWτ is designed to
satisfy the causality constraint, that is, Wτ is a lower
block triangular matrix in the form of
(77)Wτ=[Wτ,1,1O⋯OWτ,2,1Wτ,2,2⋱⋮⋮⋱⋱OWτ,θ,1⋯Wτ,θ,θ−1Wτ,θ,θ],
then, for a given integer τ, the residual generator (76) can be implemented as
(78)x^τ((k+1)θ+τ)=Aτx^τ(kθ+τ)+Bτuτ(kθ+τ)+Lτ([y(kθ+τ)⋮y(kθ+τ+θ−1)]−[y^(kθ+τ)⋮y^(kθ+τ+θ−1)]),r(kθ+τ+j)=[Wτ,j+1,1⋯Wτ,j+1,j+1]×([y(kθ+τ)⋮y(kθ+τ+j)]−[y^(kθ+τ)⋮y^(kθ+τ+j)]),y^(kθ+τ+j)=C(τ+j)ΨA(k)(τ+j,τ)x^τ(kθ+τ)+[Dτ,j+1,1⋯Dτ,j+1,j+1][u(kθ+τ)⋮u(kθ+τ+j)],j=0,1,…,θ−1.
In this case, the state estimation is still updated at
every θ time instants,
but at each time instant kθ+τ+j, j=0,1,…,θ−1, a residual signal r(kθ+τ+j) is calculated
from control input and measured outputs available up to the time instant kθ+τ+j.


#### 3.1.2. Parity-Relation-Based FD System Design and Implementation

Similarly, a parity-relation-based LTI residual
generator can be built as follows:(79)rτ(kθ+τ)=Vτ,s(yτ,k,s−Hu,suτ,k,s),yτ,k,s=[yτ((k−s)θ+τ)yτ((k−s+1)θ+τ)⋮yτ(kθ+τ)],uτ,k,s=[uτ((k−s)θ+τ)uτ((k−s+1)θ+τ)⋮uτ(kθ+τ)],Hu,s=[DτO⋯OCτBτDτ⋱⋮⋮⋱⋱OCτAτs−1Bτ⋯CτBτDτ],
where Vτ,s is a constant
parity matrix,
(80)Vτ,s[CτCτAτ⋮CτAτs]=0.
To get a residual signal at each time instant, we can
use a bank of parity-relation-based residual generators, each one for τ=0,1,…,θ−1, respectively. Alternatively, we can also impose a
structural constant on parity matrix Vτ,s as
(81)Vτ,s=[Vτ,s,0Vτ,s,1⋯Vτ,s,s−1Vτ,s,s],Vτ,s,s=[(Vτ,s,s)1,1O⋯O(Vτ,s,s)2,1(Vτ,s,s)2,2⋱⋮⋮⋱⋱O(Vτ,s,s)θ,1⋯(Vτ,s,s)θ,θ−1(Vτ,s,s)θ,θ],
that is, the last block in Vτ,s is a lower
triangular matrix, then for a fixed τ, residual generator (79) can be implemented in such a way that only control
inputs and measured outputs available up to the time instant kθ+τ+j are needed for
the calculation of r(kθ+τ+j), j=0,1,…,θ−1.


### 3.2. FD Schemes Based on Periodic Model

In this subsection, we will show that the parity-space approach and the observer-based FD approach can be directly extended to
periodic systems, which do not need the temporary step of lifted LTI
reformulation and lead to a simplified design and implementation.


#### 3.2.1. Periodic Parity-Space Approach

The extension of the parity-space approach to periodic
systems is straightforward, because the parity-space approach can handle each
time instant independently [45]. The input-output relation of periodic system (72) during the moving window [k−s,k] can be
expressed by
(82)yk,s=Ho,s,kx(k−s)+Hu,s,kuk,s+Hd,s,kdk,s+Hf,s,kfk,s.
While the vectors yk,s,uk,s,dk,s, and fk,s in (82) are built in exactly the same way as in the LTI case
according to (6), the matrices Ho,s,k,Hu,s,k,Hd,s,k, and Hf,s,k in (82) are not constant matrices but the following periodic
matrices:
(83)Ho,s,k=[C(k−s)C(k−s+1)A(k−s)⋮C(k)A(k−1)⋯A(k−s+1)A(k−s)],Hu,s,k=[D(k−s)O⋯OC(k−s+1)B(k−s)D(k−s+1)⋱⋮⋮⋱⋱OC(k)A(k−1)⋯⋯D(k)A(k−s+1)B(k−s)],Hd,s,k, and Hf,s,k are similar as Hu,s,k with B(j),D(j) replaced,
respectively, by Ed(j),Fd(j), and Ef(j),Ff(j), j=k−s,…,k.A periodic residual generator can be built as(84)r(k)=Vk(yk,s−Hu,s,kuk,s),(85)=(a)Vk(Hd,s,kdk,s+Hf,s,kfk,s),
where Vk is a θ-periodic
parity matrix (or vector) that satisfies VkHo,s,k=0, equation (85) represents the
residual dynamics.If the rank condition(86)rank[Ho,s,kHd,s,kHf,s,k]>rank[Ho,s,kHd,s,k]
holds for any k, then a full decoupling can be achieved by solving
(87)Vk[Ho,s,kHd,s,k]=0,VkHf,s,k≠0
for Vk over one period
at k,k+1,…,k+θ−1. The residual evaluation consists in detecting the
deviation of residual r(k) from 0. Especially, if
(88)rank[Ho,s,0⋯Ho,s,θ−1Hd,s,0⋯Hd,s,θ−1Hf,s,0⋯Hf,s,θ−1]>rank[Ho,s,0⋯Ho,s,θ−1Hd,s,0⋯Hd,s,θ−1],
then the full decoupling can be achieved by a constant
parity matrix (or vector) Vk. However, condition (88) is rather restrictive in practice.In case that a full decoupling is not achievable,
optimization problems similar to (15)–(17) are formulated as
(89)maxVk,VkHo,s,k=0JLTP,PS,∞/∞(Vk)=maxVk,VkHo,s,k=0σ¯2(VkHf,s,k)σ¯2(VkHd,s,k),(90)maxVk,VkHo,s,k=0JLTP,PS,−/∞(Vk)=maxVk,VkHo,s,k=0σ̲2(VkHf,s,k)σ¯2(VkHd,s,k),(91)maxVk,VkHo,s,k=0JLTP,PS,i/∞(Vk)=maxVk,VkHo,s,k=0σi2(VkHf,s,k)σ¯2(VkHd,s,k),
which are solved over one period to get the optimal
periodic parity matrix Vk. Because the parity-space approach handles each time
instant independently and there is no stability problem, the solutions of
problems (87)–(91) at each time instant are independent of each other
and can be obtained following the procedures introduced in Section 2.2. The
threshold Jth can be
calculated by (23) or (26) according to the requirement on FAR, while the
residual evaluation function is selected as the amplitude (22) of the residual signal.If the orders of the parity
relation (82) is an integer multiple of the period θ, then the periodic parity-space approach is
equivalent with a bank of residual generators (79). In comparison, the periodic parity space approach
provides more flexibility. The order of the parity relation s needs not to be
related to the period θ. Moreover, s may take
different values at different time instants. In this case, the threshold for
the residual evaluation may need to be chosen differently at different time.


#### 3.2.2. Periodic Observer-Based Approach

Assume that(A(k),C(k)) is detectable.
A periodic observer-based residual generator can be constructed as
(92)x^(k+1)=A(k)x^(k)+B(k)u(k)+L(k)(y(k)−y^(k)),r(k)=W(k)(y(k)−y^(k)),y^(k)=C(k)x^(k)+D(k)u(k),
where L(k) and W(k) are θ-periodic
observer gain matrix and weighting matrix, respectively. The residual dynamics
is governed by
(93)e(k+1)=(A(k)−L(k)C(k))e(k)+(Ed(k)−L(k)Fd(k))d(k)+(Ef(k)−L(k)Ff(k))f(k),r(k)=W(k)(C(k)e(k)+Fd(k)d(k)+Ff(k)f(k)),
where e(k)=x(k)−x^(k).To enhance the robustness of the FD system to the
unknown disturbances without loss of the sensitivity to the faults, the optimal
design problem is formulated as(94)supL(k),W(k)JLTP,OBS,∞/∞(L(k),W(k))=supL(k),W(k)supd=0,f∈l2−{0}(∥r∥2/∥f∥2)supf=0,d∈l2−{0}(∥r∥2/∥d∥2),(95)supL(k),W(k)JLTP,OBS,−/∞(L(k),W(k))=supL(k),W(k)infd=0,f∈l2−{0}(∥r∥2/∥f∥2)supf=0,d∈l2−{0}(∥r∥2/∥d∥2). The solutions of optimization problems (94)-(95) are derived by solving an equivalent optimization
problem for the cyclically lifted LTI systems first and then recover the periodic
matrices L(k) and W(k) [24].Solution to Optimization Problems (94)-(95)
Assume that(A(k),Ed(k),C(k),Fd(k)) is regular.
Then
(96)L(k)=−LdT(k),W(k)=Wd(k)
solve optimization problems (94)-(95) simultaneously, where Wd(k) is the left
inverse of a full column rank matrix Hd(k) satisfying Hd(k)HdT(k)=C(k)Xd(k)CT(k)+Fd(k)FdT(k), and (Xd(k),Ld(k)) is the
stabilizing solution to the difference periodic Riccati system (DPRS)
(97)[Ωd,11Ωd,12Ωd,12TΩd,22]×[ILd(k)]=0,
where Ωd,11=A(k)Xd(k)AT(k)−Xd(k+1)+Ed(k)EdT(k), Ωd,12=A(k)Xd(k)CT(k)+Ed(k)FdT(k), and Ωd,22=C(k)Xd(k)CT(k)+Fd(k)FdT(k). An alternative solution to problem (95), if (A(k),Ef(k),C(k),Ff(k)) is regular, is
given by
(98)L(k)=−LfT(k),W(k)=Wf(k),
where Wf(k) is the left
inverse of a full column rank matrix Hf(k) satisfying Hf(k)HfT(k)=C(k)Xf(k)CT(k)+Ff(k)FfT(k), and (Xf(k),Lf(k)) is the
stabilizing solution to the DPRS
(99)[Ωf,11Ωf,12Ωf,12TΩf,22]×[ILf(k)]=0,
where Ωf,11=A(k)Xf(k)AT(k)−Xf(k+1)+Ef(k)EfT(k), Ωf,12=A(k)Xf(k)CT(k)+Ef(k)FfT(k),  and Ωf,22=C(k)Xf(k)CT(k)+Ff(k)FfT(k). It is interesting to note the following connections
between different approaches.
(i)
Periodic
observer-based residual generator (92) can be rewritten into the form of lifted
reformulation-based LTI observer (76) with
(100)Lτ=[Lτ,1Lτ,2⋯Lτ,θ],Lτ,i=ΨA(k)−L(k)C(k)(τ+θ,τ+i)L(τ+i−1),Wτ=[Wτ,1,1O⋯OWτ,2,1Wτ,2,2⋱⋮⋮⋱⋱OWτ,θ,1⋯Wτ,θ,θ−1Wτ,θ,θ],Wτ,i,j=W(τ+i−1),ifi=j,Wτ,i,j=−W(τ+i−1)C(τ+i−1)×ΨA(k)−L(k)C(k)(τ+i−1,τ+j)L(τ+j−1)ifi>j,i=1,2,…,θ,j=1,2,…,θ.
Recalling the discussion in Section 3.1.1, the
physical meaning is that the periodic observer-based residual generator
naturally satisfies the causality condition. It is further proven that, if the
parameters (L(k),W(k)) of the periodic
observer-based residual generator (92) solve optimization problems (94) or (95), then the parameters (Lτ,Wτ) of the LTI observer (76) got by (100) will solve optimization problems in the form of (60)-(61).(ii)
Similar as in
LTI systems, the periodic parity-space approach and periodic observer-based
approach are closely related. Assume that the periodic vector(101)vk=[vk,0vk,1⋯vk,s]
satisfies vkHo,s,k=0. Then a periodic functional observer-based residual
generator in the form of
(102)z(k+1)=G(k)z(k)+H(k)u(k)+L(k)y(k),r(k)=w(k)z(k)+q(k)u(k)+p(k)y(k)
with z∈ℛs, r∈ℛ, can be readily obtained as [45]
(103)G(k)=[00⋯0gk,110⋱⋮gk,2⋮⋱⋱0⋮0⋯10gk,s−10⋯01gk,s],L(k)=−[vk+s,0vk+s−1,1⋮vk+1,s−1]−[gk,1gk,2⋮gk,s]vk,s,w(k)=[00⋯0−1],p(k)=vk,s,H(k)=T(k+1)B(k)−L(k)D(k),q(k)=−p(k)D(k),
where periodic scalars gk,1,…,gk,s appearing in
matrices G(k),L(k) are free
parameters and should be selected to guarantee the stability of G(k). Moreover, if vk realizes a full
decoupling from the unknown disturbances, that is, vk[Ho,s,kHd,s,k]=0, then the functional observer-based residual
generator (102) also achieves a full decoupling, that is, r(k)=0, ∀u(k),d(k). This provides an approach to design full decoupling
observer-based residual generator.(iii)
We would like
to point out that, for the LTI system (1), a residual generator with periodic gain matrix L(k) and periodic weighting
matrix W(k) will not
improve the FD performance under performance index (94).


## 3.1. FD Schemes Based on Lifted LTI Reformulation

It is well known that there is a strong correspondence
between discrete LTP systems and discrete LTI systems [42]. Therefore, FD system design for the LTP system (72) can be carried out as follows:(i)
lift the LTP
system (72) into a discrete LTI system,(ii)
design residual
generator(s) based on the lifted LTI reformulation,(iii)
using either
parallel residual generators or select the parameters of the residual generator
to satisfy the causality condition.LetΨA(k)(k1,k0) ( k1≥k0) denote the state transition matrix of LTP system (72)
(73)ΨA(k)(k1,k0)={Iifk1=k0,A(k1−1)A(k1−2)⋯A(k0)ifk1>k0.
Periodic system (72) can be lifted into a discrete LTI system described
by [42]
(74)xτ((k+1)θ+τ)=Aτxτ(kθ+τ)+Bτuτ(kθ+τ)+Ed,τdτ(kθ+τ)+Ef,τfτ(kθ+τ),yτ(kθ+τ)=Cτxτ(kθ+τ)+Dτuτ(kθ+τ)+Fd,τdτ(kθ+τ)+Ff,τfτ(kθ+τ),
where τ is an integer
between 0 and θ−1 denoting the
initial time, the state vector of the lifted system is xτ(kθ+τ)=x(kθ+τ), ητ with η standing for u,y,d,f is the
augmented signal defined by
(75)ητ(kθ+τ)=[η(kθ+τ)η(kθ+τ+1)⋮η(kθ+τ+θ−1)],Aτ=ΨA(k)(τ+θ,τ),Bτ=[X′Y′⋯Z′],Cτ=[C(τ)C(τ+1)ΨA(k)(τ+1,τ)⋮C(τ+θ−1)ΨA(k)(τ+θ−1,τ)],Dτ=[Dτ,1,1O⋯ODτ,2,1Dτ,2,2⋱⋮⋮⋱⋱ODτ,θ,1⋯Dτ,θ,θ−1Dτ,θ,θ],Dτ,i,j={D(τ+i−1)fori=j,C(τ+i−1)ΨA(k)(τ+i−1,τ+j)B(τ+j−1)fori>j,
where  Bτ,1=ΨA(k)(τ+θ,τ+1)B(τ), Bτ,2=ΨA(k)(τ+θ,τ+2)B(τ+1), Bτ,θ=B(τ+θ−1), and Ed,τ,Fd,τ and Ef,τ,Ff,τ are defined in
a way similar to Cτ,Dτ.


### 3.1.1. Observer-Based FD System Design and Implementation

Assume that(A(k),C(k)) is detectable.
Then (Aτ,Cτ) is detectable
and an observer-based LTI residual generator can be designed based on lifted
reformulation (74) as
(76)x^τ((k+1)θ+τ)=Aτx^τ(kθ+τ)+Bτuτ(kθ+τ)+Lτ(yτ(kθ+τ)−y^τ(kθ+τ)),rτ(kθ+τ)=Wτ(yτ(kθ+τ)−y^τ(kθ+τ)),y^τ(k)=Cτx^τ(kθ+τ)+Dτuτ(kθ+τ),
where Lτ and Wτ are constant
matrices and can be designed with FD approaches for the discrete LTI systems
introduced in the last section to realize full decoupling or optimal FD.
Observer (76) reconstructs the outputs over one period y(kθ+τ),…,y(kθ+τ+θ−1) based on the
estimation of state vector x(kθ+τ). Both the state vector of observer (76) and the residual signal are updated every θ time instants.In fault detection, the detection delay should be as
small as possible. Therefore, it is advantageous if a residual signal can be
obtained at each time instant. To this aim,(i)
a bank of LTI
residual generators (76) can be used, each of which is designed for τ=0,1,…,θ−1, respectively [43]. This scheme is characterized by a simple design but
needs much online computational efforts.(ii)
If the
weighting matrixWτ is designed to
satisfy the causality constraint, that is, Wτ is a lower
block triangular matrix in the form of
(77)Wτ=[Wτ,1,1O⋯OWτ,2,1Wτ,2,2⋱⋮⋮⋱⋱OWτ,θ,1⋯Wτ,θ,θ−1Wτ,θ,θ],
then, for a given integer τ, the residual generator (76) can be implemented as
(78)x^τ((k+1)θ+τ)=Aτx^τ(kθ+τ)+Bτuτ(kθ+τ)+Lτ([y(kθ+τ)⋮y(kθ+τ+θ−1)]−[y^(kθ+τ)⋮y^(kθ+τ+θ−1)]),r(kθ+τ+j)=[Wτ,j+1,1⋯Wτ,j+1,j+1]×([y(kθ+τ)⋮y(kθ+τ+j)]−[y^(kθ+τ)⋮y^(kθ+τ+j)]),y^(kθ+τ+j)=C(τ+j)ΨA(k)(τ+j,τ)x^τ(kθ+τ)+[Dτ,j+1,1⋯Dτ,j+1,j+1][u(kθ+τ)⋮u(kθ+τ+j)],j=0,1,…,θ−1.
In this case, the state estimation is still updated at
every θ time instants,
but at each time instant kθ+τ+j, j=0,1,…,θ−1, a residual signal r(kθ+τ+j) is calculated
from control input and measured outputs available up to the time instant kθ+τ+j.


### 3.1.2. Parity-Relation-Based FD System Design and Implementation

Similarly, a parity-relation-based LTI residual
generator can be built as follows:(79)rτ(kθ+τ)=Vτ,s(yτ,k,s−Hu,suτ,k,s),yτ,k,s=[yτ((k−s)θ+τ)yτ((k−s+1)θ+τ)⋮yτ(kθ+τ)],uτ,k,s=[uτ((k−s)θ+τ)uτ((k−s+1)θ+τ)⋮uτ(kθ+τ)],Hu,s=[DτO⋯OCτBτDτ⋱⋮⋮⋱⋱OCτAτs−1Bτ⋯CτBτDτ],
where Vτ,s is a constant
parity matrix,
(80)Vτ,s[CτCτAτ⋮CτAτs]=0.
To get a residual signal at each time instant, we can
use a bank of parity-relation-based residual generators, each one for τ=0,1,…,θ−1, respectively. Alternatively, we can also impose a
structural constant on parity matrix Vτ,s as
(81)Vτ,s=[Vτ,s,0Vτ,s,1⋯Vτ,s,s−1Vτ,s,s],Vτ,s,s=[(Vτ,s,s)1,1O⋯O(Vτ,s,s)2,1(Vτ,s,s)2,2⋱⋮⋮⋱⋱O(Vτ,s,s)θ,1⋯(Vτ,s,s)θ,θ−1(Vτ,s,s)θ,θ],
that is, the last block in Vτ,s is a lower
triangular matrix, then for a fixed τ, residual generator (79) can be implemented in such a way that only control
inputs and measured outputs available up to the time instant kθ+τ+j are needed for
the calculation of r(kθ+τ+j), j=0,1,…,θ−1.


## 3.1.1. Observer-Based FD System Design and Implementation

Assume that(A(k),C(k)) is detectable.
Then (Aτ,Cτ) is detectable
and an observer-based LTI residual generator can be designed based on lifted
reformulation (74) as
(76)x^τ((k+1)θ+τ)=Aτx^τ(kθ+τ)+Bτuτ(kθ+τ)+Lτ(yτ(kθ+τ)−y^τ(kθ+τ)),rτ(kθ+τ)=Wτ(yτ(kθ+τ)−y^τ(kθ+τ)),y^τ(k)=Cτx^τ(kθ+τ)+Dτuτ(kθ+τ),
where Lτ and Wτ are constant
matrices and can be designed with FD approaches for the discrete LTI systems
introduced in the last section to realize full decoupling or optimal FD.
Observer (76) reconstructs the outputs over one period y(kθ+τ),…,y(kθ+τ+θ−1) based on the
estimation of state vector x(kθ+τ). Both the state vector of observer (76) and the residual signal are updated every θ time instants.In fault detection, the detection delay should be as
small as possible. Therefore, it is advantageous if a residual signal can be
obtained at each time instant. To this aim,(i)
a bank of LTI
residual generators (76) can be used, each of which is designed for τ=0,1,…,θ−1, respectively [43]. This scheme is characterized by a simple design but
needs much online computational efforts.(ii)
If the
weighting matrixWτ is designed to
satisfy the causality constraint, that is, Wτ is a lower
block triangular matrix in the form of
(77)Wτ=[Wτ,1,1O⋯OWτ,2,1Wτ,2,2⋱⋮⋮⋱⋱OWτ,θ,1⋯Wτ,θ,θ−1Wτ,θ,θ],
then, for a given integer τ, the residual generator (76) can be implemented as
(78)x^τ((k+1)θ+τ)=Aτx^τ(kθ+τ)+Bτuτ(kθ+τ)+Lτ([y(kθ+τ)⋮y(kθ+τ+θ−1)]−[y^(kθ+τ)⋮y^(kθ+τ+θ−1)]),r(kθ+τ+j)=[Wτ,j+1,1⋯Wτ,j+1,j+1]×([y(kθ+τ)⋮y(kθ+τ+j)]−[y^(kθ+τ)⋮y^(kθ+τ+j)]),y^(kθ+τ+j)=C(τ+j)ΨA(k)(τ+j,τ)x^τ(kθ+τ)+[Dτ,j+1,1⋯Dτ,j+1,j+1][u(kθ+τ)⋮u(kθ+τ+j)],j=0,1,…,θ−1.
In this case, the state estimation is still updated at
every θ time instants,
but at each time instant kθ+τ+j, j=0,1,…,θ−1, a residual signal r(kθ+τ+j) is calculated
from control input and measured outputs available up to the time instant kθ+τ+j.


## 3.1.2. Parity-Relation-Based FD System Design and Implementation

Similarly, a parity-relation-based LTI residual
generator can be built as follows:(79)rτ(kθ+τ)=Vτ,s(yτ,k,s−Hu,suτ,k,s),yτ,k,s=[yτ((k−s)θ+τ)yτ((k−s+1)θ+τ)⋮yτ(kθ+τ)],uτ,k,s=[uτ((k−s)θ+τ)uτ((k−s+1)θ+τ)⋮uτ(kθ+τ)],Hu,s=[DτO⋯OCτBτDτ⋱⋮⋮⋱⋱OCτAτs−1Bτ⋯CτBτDτ],
where Vτ,s is a constant
parity matrix,
(80)Vτ,s[CτCτAτ⋮CτAτs]=0.
To get a residual signal at each time instant, we can
use a bank of parity-relation-based residual generators, each one for τ=0,1,…,θ−1, respectively. Alternatively, we can also impose a
structural constant on parity matrix Vτ,s as
(81)Vτ,s=[Vτ,s,0Vτ,s,1⋯Vτ,s,s−1Vτ,s,s],Vτ,s,s=[(Vτ,s,s)1,1O⋯O(Vτ,s,s)2,1(Vτ,s,s)2,2⋱⋮⋮⋱⋱O(Vτ,s,s)θ,1⋯(Vτ,s,s)θ,θ−1(Vτ,s,s)θ,θ],
that is, the last block in Vτ,s is a lower
triangular matrix, then for a fixed τ, residual generator (79) can be implemented in such a way that only control
inputs and measured outputs available up to the time instant kθ+τ+j are needed for
the calculation of r(kθ+τ+j), j=0,1,…,θ−1.


## 3.2. FD Schemes Based on Periodic Model

In this subsection, we will show that the parity-space approach and the observer-based FD approach can be directly extended to
periodic systems, which do not need the temporary step of lifted LTI
reformulation and lead to a simplified design and implementation.


### 3.2.1. Periodic Parity-Space Approach

The extension of the parity-space approach to periodic
systems is straightforward, because the parity-space approach can handle each
time instant independently [45]. The input-output relation of periodic system (72) during the moving window [k−s,k] can be
expressed by
(82)yk,s=Ho,s,kx(k−s)+Hu,s,kuk,s+Hd,s,kdk,s+Hf,s,kfk,s.
While the vectors yk,s,uk,s,dk,s, and fk,s in (82) are built in exactly the same way as in the LTI case
according to (6), the matrices Ho,s,k,Hu,s,k,Hd,s,k, and Hf,s,k in (82) are not constant matrices but the following periodic
matrices:
(83)Ho,s,k=[C(k−s)C(k−s+1)A(k−s)⋮C(k)A(k−1)⋯A(k−s+1)A(k−s)],Hu,s,k=[D(k−s)O⋯OC(k−s+1)B(k−s)D(k−s+1)⋱⋮⋮⋱⋱OC(k)A(k−1)⋯⋯D(k)A(k−s+1)B(k−s)],Hd,s,k, and Hf,s,k are similar as Hu,s,k with B(j),D(j) replaced,
respectively, by Ed(j),Fd(j), and Ef(j),Ff(j), j=k−s,…,k.A periodic residual generator can be built as(84)r(k)=Vk(yk,s−Hu,s,kuk,s),(85)=(a)Vk(Hd,s,kdk,s+Hf,s,kfk,s),
where Vk is a θ-periodic
parity matrix (or vector) that satisfies VkHo,s,k=0, equation (85) represents the
residual dynamics.If the rank condition(86)rank[Ho,s,kHd,s,kHf,s,k]>rank[Ho,s,kHd,s,k]
holds for any k, then a full decoupling can be achieved by solving
(87)Vk[Ho,s,kHd,s,k]=0,VkHf,s,k≠0
for Vk over one period
at k,k+1,…,k+θ−1. The residual evaluation consists in detecting the
deviation of residual r(k) from 0. Especially, if
(88)rank[Ho,s,0⋯Ho,s,θ−1Hd,s,0⋯Hd,s,θ−1Hf,s,0⋯Hf,s,θ−1]>rank[Ho,s,0⋯Ho,s,θ−1Hd,s,0⋯Hd,s,θ−1],
then the full decoupling can be achieved by a constant
parity matrix (or vector) Vk. However, condition (88) is rather restrictive in practice.In case that a full decoupling is not achievable,
optimization problems similar to (15)–(17) are formulated as
(89)maxVk,VkHo,s,k=0JLTP,PS,∞/∞(Vk)=maxVk,VkHo,s,k=0σ¯2(VkHf,s,k)σ¯2(VkHd,s,k),(90)maxVk,VkHo,s,k=0JLTP,PS,−/∞(Vk)=maxVk,VkHo,s,k=0σ̲2(VkHf,s,k)σ¯2(VkHd,s,k),(91)maxVk,VkHo,s,k=0JLTP,PS,i/∞(Vk)=maxVk,VkHo,s,k=0σi2(VkHf,s,k)σ¯2(VkHd,s,k),
which are solved over one period to get the optimal
periodic parity matrix Vk. Because the parity-space approach handles each time
instant independently and there is no stability problem, the solutions of
problems (87)–(91) at each time instant are independent of each other
and can be obtained following the procedures introduced in Section 2.2. The
threshold Jth can be
calculated by (23) or (26) according to the requirement on FAR, while the
residual evaluation function is selected as the amplitude (22) of the residual signal.If the orders of the parity
relation (82) is an integer multiple of the period θ, then the periodic parity-space approach is
equivalent with a bank of residual generators (79). In comparison, the periodic parity space approach
provides more flexibility. The order of the parity relation s needs not to be
related to the period θ. Moreover, s may take
different values at different time instants. In this case, the threshold for
the residual evaluation may need to be chosen differently at different time.


### 3.2.2. Periodic Observer-Based Approach

Assume that(A(k),C(k)) is detectable.
A periodic observer-based residual generator can be constructed as
(92)x^(k+1)=A(k)x^(k)+B(k)u(k)+L(k)(y(k)−y^(k)),r(k)=W(k)(y(k)−y^(k)),y^(k)=C(k)x^(k)+D(k)u(k),
where L(k) and W(k) are θ-periodic
observer gain matrix and weighting matrix, respectively. The residual dynamics
is governed by
(93)e(k+1)=(A(k)−L(k)C(k))e(k)+(Ed(k)−L(k)Fd(k))d(k)+(Ef(k)−L(k)Ff(k))f(k),r(k)=W(k)(C(k)e(k)+Fd(k)d(k)+Ff(k)f(k)),
where e(k)=x(k)−x^(k).To enhance the robustness of the FD system to the
unknown disturbances without loss of the sensitivity to the faults, the optimal
design problem is formulated as(94)supL(k),W(k)JLTP,OBS,∞/∞(L(k),W(k))=supL(k),W(k)supd=0,f∈l2−{0}(∥r∥2/∥f∥2)supf=0,d∈l2−{0}(∥r∥2/∥d∥2),(95)supL(k),W(k)JLTP,OBS,−/∞(L(k),W(k))=supL(k),W(k)infd=0,f∈l2−{0}(∥r∥2/∥f∥2)supf=0,d∈l2−{0}(∥r∥2/∥d∥2). The solutions of optimization problems (94)-(95) are derived by solving an equivalent optimization
problem for the cyclically lifted LTI systems first and then recover the periodic
matrices L(k) and W(k) [24].Solution to Optimization Problems (94)-(95)
Assume that(A(k),Ed(k),C(k),Fd(k)) is regular.
Then
(96)L(k)=−LdT(k),W(k)=Wd(k)
solve optimization problems (94)-(95) simultaneously, where Wd(k) is the left
inverse of a full column rank matrix Hd(k) satisfying Hd(k)HdT(k)=C(k)Xd(k)CT(k)+Fd(k)FdT(k), and (Xd(k),Ld(k)) is the
stabilizing solution to the difference periodic Riccati system (DPRS)
(97)[Ωd,11Ωd,12Ωd,12TΩd,22]×[ILd(k)]=0,
where Ωd,11=A(k)Xd(k)AT(k)−Xd(k+1)+Ed(k)EdT(k), Ωd,12=A(k)Xd(k)CT(k)+Ed(k)FdT(k), and Ωd,22=C(k)Xd(k)CT(k)+Fd(k)FdT(k). An alternative solution to problem (95), if (A(k),Ef(k),C(k),Ff(k)) is regular, is
given by
(98)L(k)=−LfT(k),W(k)=Wf(k),
where Wf(k) is the left
inverse of a full column rank matrix Hf(k) satisfying Hf(k)HfT(k)=C(k)Xf(k)CT(k)+Ff(k)FfT(k), and (Xf(k),Lf(k)) is the
stabilizing solution to the DPRS
(99)[Ωf,11Ωf,12Ωf,12TΩf,22]×[ILf(k)]=0,
where Ωf,11=A(k)Xf(k)AT(k)−Xf(k+1)+Ef(k)EfT(k), Ωf,12=A(k)Xf(k)CT(k)+Ef(k)FfT(k),  and Ωf,22=C(k)Xf(k)CT(k)+Ff(k)FfT(k). It is interesting to note the following connections
between different approaches.
(i)
Periodic
observer-based residual generator (92) can be rewritten into the form of lifted
reformulation-based LTI observer (76) with
(100)Lτ=[Lτ,1Lτ,2⋯Lτ,θ],Lτ,i=ΨA(k)−L(k)C(k)(τ+θ,τ+i)L(τ+i−1),Wτ=[Wτ,1,1O⋯OWτ,2,1Wτ,2,2⋱⋮⋮⋱⋱OWτ,θ,1⋯Wτ,θ,θ−1Wτ,θ,θ],Wτ,i,j=W(τ+i−1),ifi=j,Wτ,i,j=−W(τ+i−1)C(τ+i−1)×ΨA(k)−L(k)C(k)(τ+i−1,τ+j)L(τ+j−1)ifi>j,i=1,2,…,θ,j=1,2,…,θ.
Recalling the discussion in Section 3.1.1, the
physical meaning is that the periodic observer-based residual generator
naturally satisfies the causality condition. It is further proven that, if the
parameters (L(k),W(k)) of the periodic
observer-based residual generator (92) solve optimization problems (94) or (95), then the parameters (Lτ,Wτ) of the LTI observer (76) got by (100) will solve optimization problems in the form of (60)-(61).(ii)
Similar as in
LTI systems, the periodic parity-space approach and periodic observer-based
approach are closely related. Assume that the periodic vector(101)vk=[vk,0vk,1⋯vk,s]
satisfies vkHo,s,k=0. Then a periodic functional observer-based residual
generator in the form of
(102)z(k+1)=G(k)z(k)+H(k)u(k)+L(k)y(k),r(k)=w(k)z(k)+q(k)u(k)+p(k)y(k)
with z∈ℛs, r∈ℛ, can be readily obtained as [45]
(103)G(k)=[00⋯0gk,110⋱⋮gk,2⋮⋱⋱0⋮0⋯10gk,s−10⋯01gk,s],L(k)=−[vk+s,0vk+s−1,1⋮vk+1,s−1]−[gk,1gk,2⋮gk,s]vk,s,w(k)=[00⋯0−1],p(k)=vk,s,H(k)=T(k+1)B(k)−L(k)D(k),q(k)=−p(k)D(k),
where periodic scalars gk,1,…,gk,s appearing in
matrices G(k),L(k) are free
parameters and should be selected to guarantee the stability of G(k). Moreover, if vk realizes a full
decoupling from the unknown disturbances, that is, vk[Ho,s,kHd,s,k]=0, then the functional observer-based residual
generator (102) also achieves a full decoupling, that is, r(k)=0, ∀u(k),d(k). This provides an approach to design full decoupling
observer-based residual generator.(iii)
We would like
to point out that, for the LTI system (1), a residual generator with periodic gain matrix L(k) and periodic weighting
matrix W(k) will not
improve the FD performance under performance index (94).


## 3.2.1. Periodic Parity-Space Approach

The extension of the parity-space approach to periodic
systems is straightforward, because the parity-space approach can handle each
time instant independently [45]. The input-output relation of periodic system (72) during the moving window [k−s,k] can be
expressed by
(82)yk,s=Ho,s,kx(k−s)+Hu,s,kuk,s+Hd,s,kdk,s+Hf,s,kfk,s.
While the vectors yk,s,uk,s,dk,s, and fk,s in (82) are built in exactly the same way as in the LTI case
according to (6), the matrices Ho,s,k,Hu,s,k,Hd,s,k, and Hf,s,k in (82) are not constant matrices but the following periodic
matrices:
(83)Ho,s,k=[C(k−s)C(k−s+1)A(k−s)⋮C(k)A(k−1)⋯A(k−s+1)A(k−s)],Hu,s,k=[D(k−s)O⋯OC(k−s+1)B(k−s)D(k−s+1)⋱⋮⋮⋱⋱OC(k)A(k−1)⋯⋯D(k)A(k−s+1)B(k−s)],Hd,s,k, and Hf,s,k are similar as Hu,s,k with B(j),D(j) replaced,
respectively, by Ed(j),Fd(j), and Ef(j),Ff(j), j=k−s,…,k.A periodic residual generator can be built as(84)r(k)=Vk(yk,s−Hu,s,kuk,s),(85)=(a)Vk(Hd,s,kdk,s+Hf,s,kfk,s),
where Vk is a θ-periodic
parity matrix (or vector) that satisfies VkHo,s,k=0, equation (85) represents the
residual dynamics.If the rank condition(86)rank[Ho,s,kHd,s,kHf,s,k]>rank[Ho,s,kHd,s,k]
holds for any k, then a full decoupling can be achieved by solving
(87)Vk[Ho,s,kHd,s,k]=0,VkHf,s,k≠0
for Vk over one period
at k,k+1,…,k+θ−1. The residual evaluation consists in detecting the
deviation of residual r(k) from 0. Especially, if
(88)rank[Ho,s,0⋯Ho,s,θ−1Hd,s,0⋯Hd,s,θ−1Hf,s,0⋯Hf,s,θ−1]>rank[Ho,s,0⋯Ho,s,θ−1Hd,s,0⋯Hd,s,θ−1],
then the full decoupling can be achieved by a constant
parity matrix (or vector) Vk. However, condition (88) is rather restrictive in practice.In case that a full decoupling is not achievable,
optimization problems similar to (15)–(17) are formulated as
(89)maxVk,VkHo,s,k=0JLTP,PS,∞/∞(Vk)=maxVk,VkHo,s,k=0σ¯2(VkHf,s,k)σ¯2(VkHd,s,k),(90)maxVk,VkHo,s,k=0JLTP,PS,−/∞(Vk)=maxVk,VkHo,s,k=0σ̲2(VkHf,s,k)σ¯2(VkHd,s,k),(91)maxVk,VkHo,s,k=0JLTP,PS,i/∞(Vk)=maxVk,VkHo,s,k=0σi2(VkHf,s,k)σ¯2(VkHd,s,k),
which are solved over one period to get the optimal
periodic parity matrix Vk. Because the parity-space approach handles each time
instant independently and there is no stability problem, the solutions of
problems (87)–(91) at each time instant are independent of each other
and can be obtained following the procedures introduced in Section 2.2. The
threshold Jth can be
calculated by (23) or (26) according to the requirement on FAR, while the
residual evaluation function is selected as the amplitude (22) of the residual signal.If the orders of the parity
relation (82) is an integer multiple of the period θ, then the periodic parity-space approach is
equivalent with a bank of residual generators (79). In comparison, the periodic parity space approach
provides more flexibility. The order of the parity relation s needs not to be
related to the period θ. Moreover, s may take
different values at different time instants. In this case, the threshold for
the residual evaluation may need to be chosen differently at different time.


## 3.2.2. Periodic Observer-Based Approach

Assume that(A(k),C(k)) is detectable.
A periodic observer-based residual generator can be constructed as
(92)x^(k+1)=A(k)x^(k)+B(k)u(k)+L(k)(y(k)−y^(k)),r(k)=W(k)(y(k)−y^(k)),y^(k)=C(k)x^(k)+D(k)u(k),
where L(k) and W(k) are θ-periodic
observer gain matrix and weighting matrix, respectively. The residual dynamics
is governed by
(93)e(k+1)=(A(k)−L(k)C(k))e(k)+(Ed(k)−L(k)Fd(k))d(k)+(Ef(k)−L(k)Ff(k))f(k),r(k)=W(k)(C(k)e(k)+Fd(k)d(k)+Ff(k)f(k)),
where e(k)=x(k)−x^(k).To enhance the robustness of the FD system to the
unknown disturbances without loss of the sensitivity to the faults, the optimal
design problem is formulated as(94)supL(k),W(k)JLTP,OBS,∞/∞(L(k),W(k))=supL(k),W(k)supd=0,f∈l2−{0}(∥r∥2/∥f∥2)supf=0,d∈l2−{0}(∥r∥2/∥d∥2),(95)supL(k),W(k)JLTP,OBS,−/∞(L(k),W(k))=supL(k),W(k)infd=0,f∈l2−{0}(∥r∥2/∥f∥2)supf=0,d∈l2−{0}(∥r∥2/∥d∥2). The solutions of optimization problems (94)-(95) are derived by solving an equivalent optimization
problem for the cyclically lifted LTI systems first and then recover the periodic
matrices L(k) and W(k) [24].Solution to Optimization Problems (94)-(95)
Assume that(A(k),Ed(k),C(k),Fd(k)) is regular.
Then
(96)L(k)=−LdT(k),W(k)=Wd(k)
solve optimization problems (94)-(95) simultaneously, where Wd(k) is the left
inverse of a full column rank matrix Hd(k) satisfying Hd(k)HdT(k)=C(k)Xd(k)CT(k)+Fd(k)FdT(k), and (Xd(k),Ld(k)) is the
stabilizing solution to the difference periodic Riccati system (DPRS)
(97)[Ωd,11Ωd,12Ωd,12TΩd,22]×[ILd(k)]=0,
where Ωd,11=A(k)Xd(k)AT(k)−Xd(k+1)+Ed(k)EdT(k), Ωd,12=A(k)Xd(k)CT(k)+Ed(k)FdT(k), and Ωd,22=C(k)Xd(k)CT(k)+Fd(k)FdT(k). An alternative solution to problem (95), if (A(k),Ef(k),C(k),Ff(k)) is regular, is
given by
(98)L(k)=−LfT(k),W(k)=Wf(k),
where Wf(k) is the left
inverse of a full column rank matrix Hf(k) satisfying Hf(k)HfT(k)=C(k)Xf(k)CT(k)+Ff(k)FfT(k), and (Xf(k),Lf(k)) is the
stabilizing solution to the DPRS
(99)[Ωf,11Ωf,12Ωf,12TΩf,22]×[ILf(k)]=0,
where Ωf,11=A(k)Xf(k)AT(k)−Xf(k+1)+Ef(k)EfT(k), Ωf,12=A(k)Xf(k)CT(k)+Ef(k)FfT(k),  and Ωf,22=C(k)Xf(k)CT(k)+Ff(k)FfT(k). It is interesting to note the following connections
between different approaches.
(i)
Periodic
observer-based residual generator (92) can be rewritten into the form of lifted
reformulation-based LTI observer (76) with
(100)Lτ=[Lτ,1Lτ,2⋯Lτ,θ],Lτ,i=ΨA(k)−L(k)C(k)(τ+θ,τ+i)L(τ+i−1),Wτ=[Wτ,1,1O⋯OWτ,2,1Wτ,2,2⋱⋮⋮⋱⋱OWτ,θ,1⋯Wτ,θ,θ−1Wτ,θ,θ],Wτ,i,j=W(τ+i−1),ifi=j,Wτ,i,j=−W(τ+i−1)C(τ+i−1)×ΨA(k)−L(k)C(k)(τ+i−1,τ+j)L(τ+j−1)ifi>j,i=1,2,…,θ,j=1,2,…,θ.
Recalling the discussion in Section 3.1.1, the
physical meaning is that the periodic observer-based residual generator
naturally satisfies the causality condition. It is further proven that, if the
parameters (L(k),W(k)) of the periodic
observer-based residual generator (92) solve optimization problems (94) or (95), then the parameters (Lτ,Wτ) of the LTI observer (76) got by (100) will solve optimization problems in the form of (60)-(61).(ii)
Similar as in
LTI systems, the periodic parity-space approach and periodic observer-based
approach are closely related. Assume that the periodic vector(101)vk=[vk,0vk,1⋯vk,s]
satisfies vkHo,s,k=0. Then a periodic functional observer-based residual
generator in the form of
(102)z(k+1)=G(k)z(k)+H(k)u(k)+L(k)y(k),r(k)=w(k)z(k)+q(k)u(k)+p(k)y(k)
with z∈ℛs, r∈ℛ, can be readily obtained as [45]
(103)G(k)=[00⋯0gk,110⋱⋮gk,2⋮⋱⋱0⋮0⋯10gk,s−10⋯01gk,s],L(k)=−[vk+s,0vk+s−1,1⋮vk+1,s−1]−[gk,1gk,2⋮gk,s]vk,s,w(k)=[00⋯0−1],p(k)=vk,s,H(k)=T(k+1)B(k)−L(k)D(k),q(k)=−p(k)D(k),
where periodic scalars gk,1,…,gk,s appearing in
matrices G(k),L(k) are free
parameters and should be selected to guarantee the stability of G(k). Moreover, if vk realizes a full
decoupling from the unknown disturbances, that is, vk[Ho,s,kHd,s,k]=0, then the functional observer-based residual
generator (102) also achieves a full decoupling, that is, r(k)=0, ∀u(k),d(k). This provides an approach to design full decoupling
observer-based residual generator.(iii)
We would like
to point out that, for the LTI system (1), a residual generator with periodic gain matrix L(k) and periodic weighting
matrix W(k) will not
improve the FD performance under performance index (94).


## 4. FD of Sampled-Data Systems

The study on FD problems of sampled-data (SD) systems
has been motivated by the digital implementation of controllers and monitoring
systems. Figure1 sketches a typical application of an FD system in a process
control system. The process under consideration is a continuous-time process.
Both the controller and the FD system are discrete-time systems which are
implemented on a computer system or on an embedded microprocessor. The sensor
output signals are discretized by the A/D converters and then fed to the
controller as well as to the FD system. The D/A converters convert the
discrete-time control input signals into continuous-time signals. Since both
continuous-time and discrete-time signals exist in the system, the system
design should be indeed considered from the viewpoint of an SD system [46, 47]. The intersample behavior is the main factor that
should be considered in developing FD algorithms for the SD systems. In
practice, it happens often that the A/D and D/A converters are working at
different sampling rates [48–53]. In this section, we will review the FD methods for
the SD systems with various sampling mechanisms.Figure 1
Schematic description of the application of an FDI system in a process control system.


### 4.1. System Description

Assume that, in the SD systems, the process is a
continuous LTI process represented by(104)x˙(t)=Acx(t)+Bcu(t)+Edcd(t)+Efcf(t),y(t)=Cx(t),
where Ac,Bc,Edc,Efc, and C are known
constant matrices of appropriate dimensions. In single-rate sampled-data
(SSD) systems, the A/D converter and the D/A converter are, respectively,
described by
(105)ψ(k)=y(kh),(106)u(t)=υ(k),kh⩽t<(k+1)h,
where h is the sampling
period, ψ∈ℛm is the sampled
process output signal, υ∈ℛp is the
discrete-time control input sequence delivered by the controller program. In multirate
sampled-data (MSD) systems, the A/D converters and the D/A converters may
work with different sampling rates and thus modeled, respectively, by
(107)ψl(kl)=yl(klTy,l),l=1,2,…,m;kl=0,1,2,…,(108)uj(t)=υj(kj),kjTu,j≤t<(kj+1)Tu,j,j=1,2,…,p;kj=0,1,2,…,
where Ty,l and Tu,j denote,
respectively, the sampling periods of the A/D converter in the lth output
channel and the D/A converter in the jth input
channel. A more general class of systems are nonuniformly sampled-data
(NSD) systems, where the sampling instants may be multirate, asynchronous,
and nonequidistantly distributed, that is,
(109)ψl(kl)=yl(ty,kl),l=1,2,…,m;kl=0,1,2,…,uj(t)=υj(kj),tu,kj≤t<tu,kj+1,j=1,2,…,p;kj=0,1,2,…,
where ty,kl represent the
sampling instants in the lth output
channel and tu,kj the time
instants at which the jth control
input is updated. It is worth mentioning that a special kind of NSD systems,
where the sampling instants are nonequidistant spaced but periodic, has been
studied in the literature rather intensively [54–57].


### 4.2. FD of SSD Systems

Conventionally, an FD system can be designed for the
SSD system by indirect approaches, that is,(i)
analog design
and SD implementation, or(ii)
discrete-time
design based on the discretization of the process model.
Motivated by the development of sampled-data control [46, 47], in the last years the FD problem of the SSD systems
have been studied from the viewpoint of direct design to take into
account the intersample behavior and eliminate the approximation made during
the design [26, 58–60].The dynamics of the SSD system at the sampling
instants can be described by(110)x(k+1)=Ax(k)+Bυ(k)+d¯(k)+f¯(k),ψ(k)=Cx(k),
where
(111)A=eAch,B=∫0heActBcdt,d¯(k)=∫0heAc(h−τ)Edcd(kh+τ)dτ,f¯(k)=∫0heAc(h−τ)Efcf(kh+τ)dτ.
It is worth noticing that in SD systems there is
a significant difference between u(t) and d(t),f(t). Due to the D/A converter (106), u(t) is a piecewise constant signal. The influence of u(t) on y(t) is exactly
known from the information of υ(k) and can thus be
completely compensated in residual generation. In comparison, d(t) and f(t) are unknown
signals. Hence, the key is to study the influence of continuous-time signals d(t) and f(t) on the
discrete-time sampled output signals ψ(k) and residual
signals r(k).


#### 4.2.1. Parity-Relation-Based FD Scheme for SSD Systems

A parity-relation-based residual generator(112)r(k)=Vs(ψk,s−Hu,sυk,s)
can be used for residual generation, where VsHo,s=0, ψk,s,υk,s,Ho,s, and Hu,s are constructed
according to (6). To describe the intersample behavior, for a
continuous-time signal δ(t) with δ standing for d and f, an operator is defined as follows:
(113)Ψδδk,s(t)=[δ¯(k−s)δ¯(k−s+1)⋮δ¯(k)]=[∫0heAc(h−τ)Eδcδ((k−s)h+τ)dτ∫0heAc(h−τ)Eδcδ((k−s+1)h+τ)dτ⋮∫0heAc(h−τ)Eδcδ(kh+τ)dτ].
The residual dynamics can be expressed with the help
of operators as
(114)r(k)=VsH(Ψddk,s(t)+Ψffk,s(t)),H=[OO⋯OCO⋱⋮⋮⋱⋱OCAs−1⋯CO].
The influence of the continuous-time signal δ(t) over the time
interval [(k−s)h,(k+1)h) on the
discrete-time signal Ψδδk,s(t) is measured by
(115)σ¯(VsHΨδ(Ψδ)∗HTVsT)=supδ∈ℒ2,[(k−s)h,(k+1)h)rT(k)r(k)∥δ∥2,[(k−s)h,(k+1)h)2,σ̲(VsHΨδ(Ψδ)∗HTVsT)=infδ∈ℒ2,[(k−s)h,(k+1)h)rT(k)r(k)∥δ∥2,[(k−s)h,(k+1)h)2,
where (Ψδ)∗ denotes the
adjoint of the operator Ψδ which uniquely
satisfies
(116)〈Ψδδk,s(t),βs(k)〉=〈δk,s(t),(Ψδ)∗βs(k)〉
for any vector βs(k) of compatible
dimensions. The optimization problems are thus formulated as
(117)maxVs,VsHo,s=0JSSD,PS,∞/∞(Vs)=maxVs,VsHo,s=0σ¯(VsHΨf(Ψf)∗HTVsT)σ¯(VsHΨd(Ψd)∗HTVsT),maxVs,VsHo,s=0JSSD,PS,−/∞(Vs)=maxVs,VsHo,s=0σ̲(VsHΨf(Ψf)∗HTVsT)σ¯(VsHΨd(Ψd)∗HTVsT),maxVs,VsHo,s=0JSSD,PS,i/∞(Vs)=maxVs,VsHo,s=0σi(VsHΨf(Ψf)∗HTVsT)σ¯(VsHΨd(Ψd)∗HTVsT).
An analytical expression can be obtained for Ψδ(Ψδ)∗ as
(118)Ψδ(Ψδ)∗=diag{E¯δE¯δT,E¯δE¯δT,…,E¯δE¯δT},E¯δE¯δT=∫0heAcτEδcEδcTeAcTτdτ.
As a result, optimization problems (117) are transformed into some equivalent optimization
problems:
(119)maxVs,VsHo,s=0JSSD,PS,∞/∞(Vs)=maxVs,VsHo,s=0σ¯2(VsH¯f,s)σ¯2(VsH¯d,s),maxVs,VsHo,s=0JSSD,PS,−/∞(Vs)=maxVs,VsHo,s=0σ̲2(VsH¯f,s)σ¯2(VsH¯d,s),maxVs,VsHo,s=0JSSD,PS,i/∞(Vs)=maxVs,VsHo,s=0σi2(VsH¯f,s)σ¯2(VsH¯d,s),
where H¯f,s and H¯d,s are built based
on (A,E¯f,C,O) and (A,E¯d,C,O) in a way
similar to Hu,s in (6). The equivalent optimization problems (119) are of the standard form and can be solved as
introduced in Section 2.2.


#### 4.2.2. Observer-Based FD Scheme for SSD Systems

An observer-based residual generator is constructed as(120)x^(k+1)=Ax^(k)+Bυ(k)+L(ψ(k)−ψ^(k)),ψ^(k)=Cx^(k),r(k)=W(ψ(k)−ψ^(k)).
To describe the influence of continuous-time signals d(t) and f(t) on the
discrete-time residual signal r(k) in the
frequency domain, operator Γδ (δ standing for d or f) is introduced
as
(121)Γδδ(jω)=1h∑k=−∞∞Gδ(jω+jkωs)δ(jω+jkωs),Gδ(s)=C(sI−Ac)−1Eδc,ωs=2πh.
Based on it, the residual dynamics can be expressed as
(122)r(ejωh)=WM^u(ejωh)(Γdd(jω)+Γff(jω)),M^u(ejωh)=I−C(ejwhI−A+LC)−1L.
Let E¯δ be given by (118). Then
(123)ΓδΓδ∗=C(ejωhI−A)−1E¯δE¯δT(e−jωhI−AT)−1CT.
Based on it, the H2/H2 optimal design
problem is solved [59]. Further, it was shown that the H∞/H∞,H−/H∞ and Hi/H∞ design problems
of the SSD system are equivalent to that of a discrete LTI system and can be
obtained by solving equivalent optimization problems [26, 60]:
(124)supL,WJSSD,OBS,∞/∞(L,W)=supL,Wsupd=0,f∈ℒ2−{0}(∥r∥2/∥f∥2)supf=0,d∈ℒ2−{0}(∥r∥2/∥d∥2)=supL,W∥G¯rf(z)∥∞∥G¯rd(z)∥∞,supL,WJSSD,OBS,−/∞(L,W)=supL,Winfd=0,f∈ℒ2−{0}(∥r∥2/∥f∥2)supf=0,d∈ℒ2−{0}(∥r∥2/∥d∥2)=supL,W∥G¯rf(z)∥−∥G¯rd(z)∥∞,supL,WJSSD,OBS,i/∞(L,W)=supL,Wσi(G¯rf(ejωh))∥G¯rd(z)∥∞G¯rδ(z)=WC(zI−A+LC)−1E¯δ,δ=d,f.


#### 4.2.3. Influence of Sampling on Full Decoupling

No matter which residual generation approach is adopted,
due to the sampling effect, the full decoupling becomes more difficult in SD
systems than in the original continuous-time systems, because after sampling
the dimension of the influence space of the unknown disturbances becomes(125)rankE¯d=rankE¯dE¯dT=rank∫0heAcτEdcEdcTeAcTτdτ≥rankEdc,
that is, the equivalent number of the unknown
disturbances may increase [58].


### 4.3. FD of MSD Systems

Under some assumptions, the FD problem of MSD systems
has been considered in [61–64]. The MSD system is in nature a periodic system. The
system period, denoted by T, is the least common multiple of the sampling periods [52]. The maximal common multiplier of the sampling
periods is often called base period. From the FD viewpoint, in MSD systems only
those time instants with sampled outputs are of interest. In [65], the basic idea of the proposed FD approach is to get
the input-output relations of the MSD at the base periods at first and then
downsample them according to different sampling periods to get the parity
relation of the MSD system. In [66], a so-called fast rate residual generator is
proposed, which generates a residual signal as soon as a new measurement is
available. The basic idea is to reformulate the MSD system as a system with
periodic output equations. The problem of fast rate residual generation is
further pursued by [67, 68], where the basic ideas are, respectively, to compute
the parity matrix Vs and the
post-filter R(z) for lifted
system model. To satisfy the causality constraint, the freedoms in matrix P¯s and R(z) are used. Most
recently, a unified and simplified approach to the FD of the MSD systems is
proposed by [69], which considers the special problem formulation of
fault detection and shows clearly the difference between control problem and FD
problem. The basic idea of [69] is to remodel the MSD system as a nonuniformly
sampled-data system and then use periodic or time varying system theory to
design the FD system.Denote with{tk} the sequence of
time instants at which one or more sampled outputs are available, t0<t1<⋯<tk<tk+1<⋯. Let ψ¯(k) represent the
vector of sampled output signals at time instant tk. The dimension of ψ¯(k) is time-varying
and upper bounded by m. Let x(k)=x(tk). For the purpose of FD, the MSD system described by (104), (107), and (108) can be equivalently remodeled as
(126)x(k+1)=A(k)x(k)+u¯(k)+d¯(k)+f¯(k),ψ¯(k)=C(k)x(k),
where
(127)A(k)=eAc(tk+1−tk),u¯(k)=∫tktk+1eAc(tk+1−t)Bcu(t)dt,d¯(k)=∫tktk+1eAc(tk+1−t)Edcd(t)dt,f¯(k)=∫tktk+1eAc(tk+1−t)Efcf(t)dt.
The new description is different from other
descriptions available in the literature [66–68] and considers the transition of system dynamics only
at the time instants with sampled outputs. The terms d¯(k) and f¯(k) characterize,
respectively, the influence of the disturbances and the faults on the sampled
outputs. As υ1,…,υp are available
and the models of the D/A converters (108) are known, u¯(k) and thus the
influence of the control input vector on the sampled outputs can be completely
reconstructed and easily compensated. The matrices A(k) and C(k) are
time varying matrices, as tk+1−tk is time-varying
with respect to time k. In the MSD system, due to the periodicity of the
sampling time sequence, A(k),C(k) are periodic
matrices. FD systems can be designed for the MSD system based on the
time-varying model (126).


#### 4.3.1. Parity-Relation-Based FD Scheme for MSD Systems

The input-output relationship of (126) over the moving horizon [tk−s,tk], where s denotes the
length of the moving horizon, is
(128)ψ(k)=Ho,s,kx(k−s)+Hs,k(u(k)+d(k)+f(k)),
where ψ(k),u(k),d(k), and f(k) are stacked
vectors based on ψ¯(j),u¯(j),d¯(j), and f¯(j), j=k−s,…,k, respectively, Ho,s,k is constructed
according to (83),
(129)Hs,k=[OO⋯C(k−s+1)⋱⋱⋮⋱OC(k)A(k−1)⋯A(k−s+1)⋯C(k)].
Build a parity-relation-based residual generator with
(130)r(k)=Vk(ψ(k)−Hs,ku(k)),VkHo,s,k=0,
where Vk is a
periodically time-varying parity matrix (or vector). To describe the influence
of continuous-time signals d(t),f(t) on the
multirate-sampled outputs, linear time-varying operators Ψkδ (δ standing for d or f),
(131)δ(k)=Ψkδδk(t)=[∫tk−stk−s+1eAc(tk−s+1−t)Eδcδ(t)dt∫tk−s+1tk−s+2eAc(tk−s+2−t)Eδcδ(t)dt⋮∫tk−1tkeAc(tk−t)Eδcδ(t)dt],
are introduced and the residual dynamics is described
by
(132)r(k)=VkHs,k(d(k)+f(k))=VkHs,k(Ψkddk(t)+Ψkffk(t)).
The optimal selection of Vk can be
formulated similar to (117) with Ψd(Ψd)∗ and Ψf(Ψf)∗ substituted by Ψkd(Ψkd)∗ and Ψkf(Ψkf)∗, respectively. Using the same technique, Ψkδ(Ψkδ)∗ are derived to be
(133)Ψkδ(Ψkδ)∗=diag{Ψk,sδ,…,Ψk,1δ},Ψk,jδ=∫0tk−j+1−tk−jeActEδcEδcTeAcTtdt,j=s,…,1.
Due to the periodicity of Ψkδ(Ψkδ)∗, the optimization problem needs to be solved over one
period.


#### 4.3.2. Observer-Based FD Scheme for MSD Systems

For the aim of fault detection, a fast rate
time-varying observer-based residual generator can be constructed as(134)x^(k+1)=A(k)x^(k)+u¯(k)+L(k)(ψ¯(k)−ψ^(k)),r(k)=W(k)(ψ¯(k)−ψ^(k)),ψ^(k)=C(k)x^(k),
where the gain matrix L(k) and the
weighting matrix W(k) are
time-varying matrices to be determined. The dimensions of L(k) and W(k) may change with
the number of available sampled output signals. Considering the periodicity of
the matrices A(k),C(k), (134) can be designed as a periodic observer. Define the
state estimation error as e(k)=x(k)−x^(k). The dynamics of residual generator (134) is governed by
(135)e(k+1)=(A(k)−L(k)C(k))e(k)+d¯(k)+f¯(k),r(k)=W(k)C(k)e(k).
Introduce linear time-varying operators Γkδ (δ standing for d or f),
(136)Γkδδk(t)=∫tktk+1eAc(tk+1−t)Eδcδ(t)dt,
to rewrite the residual dynamics as
(137)e(k+1)=(A(k)−L(k)C(k))e(k)+Γkddk(t)+Γkffk(t),r(k)=W(k)C(k)e(k).
To enhance the robustness of the FD system to the
unknown disturbances without loss of the sensitivity to the faults, the design
problem is formulated as
(138)supL(k),W(k)JMSD,OBS,∞/∞(L(k),W(k))=supL(k),W(k)supd=0,f∈ℒ2−{0}(∥r∥2/∥f∥2)supf=0,d∈ℒ2−{0}(∥r∥2/∥d∥2),supL(k),W(k)JMSD,OBS,−/∞(L(k),W(k))=supL(k),W(k)infd=0,f∈ℒ2−{0}(∥r∥2/∥f∥2)supf=0,d∈ℒ2−{0}(∥r∥2/∥d∥2).
By analyzing Γkδ(Γkδ)∗, optimization problems (138) are transformed into equivalent optimization
problems of discrete LTP system
(139)e(k+1)=(A(k)−L(k)C(k))e(k)+E¯d(k)deq(k)+E¯f(k)feq(k),r(k)=W(k)C(k)e(k),
where the l2-norms of deq(k) and feq(k) in (139) have the same upper bounds, respectively, with the ℒ2-norms of d(t) and f(t) in (104), the matrices E¯d(k) and E¯f(k) are
time-varying matrices reflecting the sampling effect and satisfy
(140)E¯δ(k)E¯δT(k)=∫0tk+1−tkeActEδcEδcTeAcTtdt.
Then, optimization problems (138) can be solved with the approaches introduced in
Section 3.2.2.


### 4.4. FD of NSD Systems

The same design procedures introduced in the last
subsection can be applied to the FD of NSD systems by reordering the sampling
instants. The main difference lies in that in general NSD systems,A(k),C(k),E¯d(k),E¯f(k) are
time-varying matrices but not periodic matrices. In consequence, for the NSD
systems(i)
if the parity
space approach is used, then the time-varying parity matrixV(k) needs to be
calculated at each time instant,(ii)
if the observer-based approach is adopted, then the observer gain matrixL(k) needs to
guarantee the stability of the resulting linear time-varying system.


### 4.5. Influence of Sampling Period on Optimal FD Performance

Sampling period is an important parameter in SD
systems. Recently, the influence of the sampling period on the optimal FD
performance has been investigated in [70, 71]. Suppose that for a given continuous-time process (104) three different sampling schemes are considered:
single-rate sampling with sampling period h, single-rate sampling with sampling period ρh, multirate sampling with base period h and system
period ρh, where ρ≥1 is an integer.
It is proven that the optimal performance indeces are related by
(141)JSSD,OBS,∞/∞,h≥JMSD,OBS,∞/∞≥JSSD,OBS,∞/∞,ρh,JSSD,PS,∞/∞,h≥JMSD,PS,∞/∞≥JSSD,PS,∞/∞,ρh.
That means that, with the increase of the sampling period,
the FD performance will become worse. It can be intuitively interpreted as the
consequence of information reduction caused by the increase of the sampling
period. However, we would like to emphasize that such a conclusion does not hold
for all performance indices, for instance, the H−/H∞ index.


## 4.1. System Description

Assume that, in the SD systems, the process is a
continuous LTI process represented by(104)x˙(t)=Acx(t)+Bcu(t)+Edcd(t)+Efcf(t),y(t)=Cx(t),
where Ac,Bc,Edc,Efc, and C are known
constant matrices of appropriate dimensions. In single-rate sampled-data
(SSD) systems, the A/D converter and the D/A converter are, respectively,
described by
(105)ψ(k)=y(kh),(106)u(t)=υ(k),kh⩽t<(k+1)h,
where h is the sampling
period, ψ∈ℛm is the sampled
process output signal, υ∈ℛp is the
discrete-time control input sequence delivered by the controller program. In multirate
sampled-data (MSD) systems, the A/D converters and the D/A converters may
work with different sampling rates and thus modeled, respectively, by
(107)ψl(kl)=yl(klTy,l),l=1,2,…,m;kl=0,1,2,…,(108)uj(t)=υj(kj),kjTu,j≤t<(kj+1)Tu,j,j=1,2,…,p;kj=0,1,2,…,
where Ty,l and Tu,j denote,
respectively, the sampling periods of the A/D converter in the lth output
channel and the D/A converter in the jth input
channel. A more general class of systems are nonuniformly sampled-data
(NSD) systems, where the sampling instants may be multirate, asynchronous,
and nonequidistantly distributed, that is,
(109)ψl(kl)=yl(ty,kl),l=1,2,…,m;kl=0,1,2,…,uj(t)=υj(kj),tu,kj≤t<tu,kj+1,j=1,2,…,p;kj=0,1,2,…,
where ty,kl represent the
sampling instants in the lth output
channel and tu,kj the time
instants at which the jth control
input is updated. It is worth mentioning that a special kind of NSD systems,
where the sampling instants are nonequidistant spaced but periodic, has been
studied in the literature rather intensively [54–57].


## 4.2. FD of SSD Systems

Conventionally, an FD system can be designed for the
SSD system by indirect approaches, that is,(i)
analog design
and SD implementation, or(ii)
discrete-time
design based on the discretization of the process model.
Motivated by the development of sampled-data control [46, 47], in the last years the FD problem of the SSD systems
have been studied from the viewpoint of direct design to take into
account the intersample behavior and eliminate the approximation made during
the design [26, 58–60].The dynamics of the SSD system at the sampling
instants can be described by(110)x(k+1)=Ax(k)+Bυ(k)+d¯(k)+f¯(k),ψ(k)=Cx(k),
where
(111)A=eAch,B=∫0heActBcdt,d¯(k)=∫0heAc(h−τ)Edcd(kh+τ)dτ,f¯(k)=∫0heAc(h−τ)Efcf(kh+τ)dτ.
It is worth noticing that in SD systems there is
a significant difference between u(t) and d(t),f(t). Due to the D/A converter (106), u(t) is a piecewise constant signal. The influence of u(t) on y(t) is exactly
known from the information of υ(k) and can thus be
completely compensated in residual generation. In comparison, d(t) and f(t) are unknown
signals. Hence, the key is to study the influence of continuous-time signals d(t) and f(t) on the
discrete-time sampled output signals ψ(k) and residual
signals r(k).


### 4.2.1. Parity-Relation-Based FD Scheme for SSD Systems

A parity-relation-based residual generator(112)r(k)=Vs(ψk,s−Hu,sυk,s)
can be used for residual generation, where VsHo,s=0, ψk,s,υk,s,Ho,s, and Hu,s are constructed
according to (6). To describe the intersample behavior, for a
continuous-time signal δ(t) with δ standing for d and f, an operator is defined as follows:
(113)Ψδδk,s(t)=[δ¯(k−s)δ¯(k−s+1)⋮δ¯(k)]=[∫0heAc(h−τ)Eδcδ((k−s)h+τ)dτ∫0heAc(h−τ)Eδcδ((k−s+1)h+τ)dτ⋮∫0heAc(h−τ)Eδcδ(kh+τ)dτ].
The residual dynamics can be expressed with the help
of operators as
(114)r(k)=VsH(Ψddk,s(t)+Ψffk,s(t)),H=[OO⋯OCO⋱⋮⋮⋱⋱OCAs−1⋯CO].
The influence of the continuous-time signal δ(t) over the time
interval [(k−s)h,(k+1)h) on the
discrete-time signal Ψδδk,s(t) is measured by
(115)σ¯(VsHΨδ(Ψδ)∗HTVsT)=supδ∈ℒ2,[(k−s)h,(k+1)h)rT(k)r(k)∥δ∥2,[(k−s)h,(k+1)h)2,σ̲(VsHΨδ(Ψδ)∗HTVsT)=infδ∈ℒ2,[(k−s)h,(k+1)h)rT(k)r(k)∥δ∥2,[(k−s)h,(k+1)h)2,
where (Ψδ)∗ denotes the
adjoint of the operator Ψδ which uniquely
satisfies
(116)〈Ψδδk,s(t),βs(k)〉=〈δk,s(t),(Ψδ)∗βs(k)〉
for any vector βs(k) of compatible
dimensions. The optimization problems are thus formulated as
(117)maxVs,VsHo,s=0JSSD,PS,∞/∞(Vs)=maxVs,VsHo,s=0σ¯(VsHΨf(Ψf)∗HTVsT)σ¯(VsHΨd(Ψd)∗HTVsT),maxVs,VsHo,s=0JSSD,PS,−/∞(Vs)=maxVs,VsHo,s=0σ̲(VsHΨf(Ψf)∗HTVsT)σ¯(VsHΨd(Ψd)∗HTVsT),maxVs,VsHo,s=0JSSD,PS,i/∞(Vs)=maxVs,VsHo,s=0σi(VsHΨf(Ψf)∗HTVsT)σ¯(VsHΨd(Ψd)∗HTVsT).
An analytical expression can be obtained for Ψδ(Ψδ)∗ as
(118)Ψδ(Ψδ)∗=diag{E¯δE¯δT,E¯δE¯δT,…,E¯δE¯δT},E¯δE¯δT=∫0heAcτEδcEδcTeAcTτdτ.
As a result, optimization problems (117) are transformed into some equivalent optimization
problems:
(119)maxVs,VsHo,s=0JSSD,PS,∞/∞(Vs)=maxVs,VsHo,s=0σ¯2(VsH¯f,s)σ¯2(VsH¯d,s),maxVs,VsHo,s=0JSSD,PS,−/∞(Vs)=maxVs,VsHo,s=0σ̲2(VsH¯f,s)σ¯2(VsH¯d,s),maxVs,VsHo,s=0JSSD,PS,i/∞(Vs)=maxVs,VsHo,s=0σi2(VsH¯f,s)σ¯2(VsH¯d,s),
where H¯f,s and H¯d,s are built based
on (A,E¯f,C,O) and (A,E¯d,C,O) in a way
similar to Hu,s in (6). The equivalent optimization problems (119) are of the standard form and can be solved as
introduced in Section 2.2.


### 4.2.2. Observer-Based FD Scheme for SSD Systems

An observer-based residual generator is constructed as(120)x^(k+1)=Ax^(k)+Bυ(k)+L(ψ(k)−ψ^(k)),ψ^(k)=Cx^(k),r(k)=W(ψ(k)−ψ^(k)).
To describe the influence of continuous-time signals d(t) and f(t) on the
discrete-time residual signal r(k) in the
frequency domain, operator Γδ (δ standing for d or f) is introduced
as
(121)Γδδ(jω)=1h∑k=−∞∞Gδ(jω+jkωs)δ(jω+jkωs),Gδ(s)=C(sI−Ac)−1Eδc,ωs=2πh.
Based on it, the residual dynamics can be expressed as
(122)r(ejωh)=WM^u(ejωh)(Γdd(jω)+Γff(jω)),M^u(ejωh)=I−C(ejwhI−A+LC)−1L.
Let E¯δ be given by (118). Then
(123)ΓδΓδ∗=C(ejωhI−A)−1E¯δE¯δT(e−jωhI−AT)−1CT.
Based on it, the H2/H2 optimal design
problem is solved [59]. Further, it was shown that the H∞/H∞,H−/H∞ and Hi/H∞ design problems
of the SSD system are equivalent to that of a discrete LTI system and can be
obtained by solving equivalent optimization problems [26, 60]:
(124)supL,WJSSD,OBS,∞/∞(L,W)=supL,Wsupd=0,f∈ℒ2−{0}(∥r∥2/∥f∥2)supf=0,d∈ℒ2−{0}(∥r∥2/∥d∥2)=supL,W∥G¯rf(z)∥∞∥G¯rd(z)∥∞,supL,WJSSD,OBS,−/∞(L,W)=supL,Winfd=0,f∈ℒ2−{0}(∥r∥2/∥f∥2)supf=0,d∈ℒ2−{0}(∥r∥2/∥d∥2)=supL,W∥G¯rf(z)∥−∥G¯rd(z)∥∞,supL,WJSSD,OBS,i/∞(L,W)=supL,Wσi(G¯rf(ejωh))∥G¯rd(z)∥∞G¯rδ(z)=WC(zI−A+LC)−1E¯δ,δ=d,f.


### 4.2.3. Influence of Sampling on Full Decoupling

No matter which residual generation approach is adopted,
due to the sampling effect, the full decoupling becomes more difficult in SD
systems than in the original continuous-time systems, because after sampling
the dimension of the influence space of the unknown disturbances becomes(125)rankE¯d=rankE¯dE¯dT=rank∫0heAcτEdcEdcTeAcTτdτ≥rankEdc,
that is, the equivalent number of the unknown
disturbances may increase [58].


## 4.2.1. Parity-Relation-Based FD Scheme for SSD Systems

A parity-relation-based residual generator(112)r(k)=Vs(ψk,s−Hu,sυk,s)
can be used for residual generation, where VsHo,s=0, ψk,s,υk,s,Ho,s, and Hu,s are constructed
according to (6). To describe the intersample behavior, for a
continuous-time signal δ(t) with δ standing for d and f, an operator is defined as follows:
(113)Ψδδk,s(t)=[δ¯(k−s)δ¯(k−s+1)⋮δ¯(k)]=[∫0heAc(h−τ)Eδcδ((k−s)h+τ)dτ∫0heAc(h−τ)Eδcδ((k−s+1)h+τ)dτ⋮∫0heAc(h−τ)Eδcδ(kh+τ)dτ].
The residual dynamics can be expressed with the help
of operators as
(114)r(k)=VsH(Ψddk,s(t)+Ψffk,s(t)),H=[OO⋯OCO⋱⋮⋮⋱⋱OCAs−1⋯CO].
The influence of the continuous-time signal δ(t) over the time
interval [(k−s)h,(k+1)h) on the
discrete-time signal Ψδδk,s(t) is measured by
(115)σ¯(VsHΨδ(Ψδ)∗HTVsT)=supδ∈ℒ2,[(k−s)h,(k+1)h)rT(k)r(k)∥δ∥2,[(k−s)h,(k+1)h)2,σ̲(VsHΨδ(Ψδ)∗HTVsT)=infδ∈ℒ2,[(k−s)h,(k+1)h)rT(k)r(k)∥δ∥2,[(k−s)h,(k+1)h)2,
where (Ψδ)∗ denotes the
adjoint of the operator Ψδ which uniquely
satisfies
(116)〈Ψδδk,s(t),βs(k)〉=〈δk,s(t),(Ψδ)∗βs(k)〉
for any vector βs(k) of compatible
dimensions. The optimization problems are thus formulated as
(117)maxVs,VsHo,s=0JSSD,PS,∞/∞(Vs)=maxVs,VsHo,s=0σ¯(VsHΨf(Ψf)∗HTVsT)σ¯(VsHΨd(Ψd)∗HTVsT),maxVs,VsHo,s=0JSSD,PS,−/∞(Vs)=maxVs,VsHo,s=0σ̲(VsHΨf(Ψf)∗HTVsT)σ¯(VsHΨd(Ψd)∗HTVsT),maxVs,VsHo,s=0JSSD,PS,i/∞(Vs)=maxVs,VsHo,s=0σi(VsHΨf(Ψf)∗HTVsT)σ¯(VsHΨd(Ψd)∗HTVsT).
An analytical expression can be obtained for Ψδ(Ψδ)∗ as
(118)Ψδ(Ψδ)∗=diag{E¯δE¯δT,E¯δE¯δT,…,E¯δE¯δT},E¯δE¯δT=∫0heAcτEδcEδcTeAcTτdτ.
As a result, optimization problems (117) are transformed into some equivalent optimization
problems:
(119)maxVs,VsHo,s=0JSSD,PS,∞/∞(Vs)=maxVs,VsHo,s=0σ¯2(VsH¯f,s)σ¯2(VsH¯d,s),maxVs,VsHo,s=0JSSD,PS,−/∞(Vs)=maxVs,VsHo,s=0σ̲2(VsH¯f,s)σ¯2(VsH¯d,s),maxVs,VsHo,s=0JSSD,PS,i/∞(Vs)=maxVs,VsHo,s=0σi2(VsH¯f,s)σ¯2(VsH¯d,s),
where H¯f,s and H¯d,s are built based
on (A,E¯f,C,O) and (A,E¯d,C,O) in a way
similar to Hu,s in (6). The equivalent optimization problems (119) are of the standard form and can be solved as
introduced in Section 2.2.


## 4.2.2. Observer-Based FD Scheme for SSD Systems

An observer-based residual generator is constructed as(120)x^(k+1)=Ax^(k)+Bυ(k)+L(ψ(k)−ψ^(k)),ψ^(k)=Cx^(k),r(k)=W(ψ(k)−ψ^(k)).
To describe the influence of continuous-time signals d(t) and f(t) on the
discrete-time residual signal r(k) in the
frequency domain, operator Γδ (δ standing for d or f) is introduced
as
(121)Γδδ(jω)=1h∑k=−∞∞Gδ(jω+jkωs)δ(jω+jkωs),Gδ(s)=C(sI−Ac)−1Eδc,ωs=2πh.
Based on it, the residual dynamics can be expressed as
(122)r(ejωh)=WM^u(ejωh)(Γdd(jω)+Γff(jω)),M^u(ejωh)=I−C(ejwhI−A+LC)−1L.
Let E¯δ be given by (118). Then
(123)ΓδΓδ∗=C(ejωhI−A)−1E¯δE¯δT(e−jωhI−AT)−1CT.
Based on it, the H2/H2 optimal design
problem is solved [59]. Further, it was shown that the H∞/H∞,H−/H∞ and Hi/H∞ design problems
of the SSD system are equivalent to that of a discrete LTI system and can be
obtained by solving equivalent optimization problems [26, 60]:
(124)supL,WJSSD,OBS,∞/∞(L,W)=supL,Wsupd=0,f∈ℒ2−{0}(∥r∥2/∥f∥2)supf=0,d∈ℒ2−{0}(∥r∥2/∥d∥2)=supL,W∥G¯rf(z)∥∞∥G¯rd(z)∥∞,supL,WJSSD,OBS,−/∞(L,W)=supL,Winfd=0,f∈ℒ2−{0}(∥r∥2/∥f∥2)supf=0,d∈ℒ2−{0}(∥r∥2/∥d∥2)=supL,W∥G¯rf(z)∥−∥G¯rd(z)∥∞,supL,WJSSD,OBS,i/∞(L,W)=supL,Wσi(G¯rf(ejωh))∥G¯rd(z)∥∞G¯rδ(z)=WC(zI−A+LC)−1E¯δ,δ=d,f.


## 4.2.3. Influence of Sampling on Full Decoupling

No matter which residual generation approach is adopted,
due to the sampling effect, the full decoupling becomes more difficult in SD
systems than in the original continuous-time systems, because after sampling
the dimension of the influence space of the unknown disturbances becomes(125)rankE¯d=rankE¯dE¯dT=rank∫0heAcτEdcEdcTeAcTτdτ≥rankEdc,
that is, the equivalent number of the unknown
disturbances may increase [58].


## 4.3. FD of MSD Systems

Under some assumptions, the FD problem of MSD systems
has been considered in [61–64]. The MSD system is in nature a periodic system. The
system period, denoted by T, is the least common multiple of the sampling periods [52]. The maximal common multiplier of the sampling
periods is often called base period. From the FD viewpoint, in MSD systems only
those time instants with sampled outputs are of interest. In [65], the basic idea of the proposed FD approach is to get
the input-output relations of the MSD at the base periods at first and then
downsample them according to different sampling periods to get the parity
relation of the MSD system. In [66], a so-called fast rate residual generator is
proposed, which generates a residual signal as soon as a new measurement is
available. The basic idea is to reformulate the MSD system as a system with
periodic output equations. The problem of fast rate residual generation is
further pursued by [67, 68], where the basic ideas are, respectively, to compute
the parity matrix Vs and the
post-filter R(z) for lifted
system model. To satisfy the causality constraint, the freedoms in matrix P¯s and R(z) are used. Most
recently, a unified and simplified approach to the FD of the MSD systems is
proposed by [69], which considers the special problem formulation of
fault detection and shows clearly the difference between control problem and FD
problem. The basic idea of [69] is to remodel the MSD system as a nonuniformly
sampled-data system and then use periodic or time varying system theory to
design the FD system.Denote with{tk} the sequence of
time instants at which one or more sampled outputs are available, t0<t1<⋯<tk<tk+1<⋯. Let ψ¯(k) represent the
vector of sampled output signals at time instant tk. The dimension of ψ¯(k) is time-varying
and upper bounded by m. Let x(k)=x(tk). For the purpose of FD, the MSD system described by (104), (107), and (108) can be equivalently remodeled as
(126)x(k+1)=A(k)x(k)+u¯(k)+d¯(k)+f¯(k),ψ¯(k)=C(k)x(k),
where
(127)A(k)=eAc(tk+1−tk),u¯(k)=∫tktk+1eAc(tk+1−t)Bcu(t)dt,d¯(k)=∫tktk+1eAc(tk+1−t)Edcd(t)dt,f¯(k)=∫tktk+1eAc(tk+1−t)Efcf(t)dt.
The new description is different from other
descriptions available in the literature [66–68] and considers the transition of system dynamics only
at the time instants with sampled outputs. The terms d¯(k) and f¯(k) characterize,
respectively, the influence of the disturbances and the faults on the sampled
outputs. As υ1,…,υp are available
and the models of the D/A converters (108) are known, u¯(k) and thus the
influence of the control input vector on the sampled outputs can be completely
reconstructed and easily compensated. The matrices A(k) and C(k) are
time varying matrices, as tk+1−tk is time-varying
with respect to time k. In the MSD system, due to the periodicity of the
sampling time sequence, A(k),C(k) are periodic
matrices. FD systems can be designed for the MSD system based on the
time-varying model (126).


### 4.3.1. Parity-Relation-Based FD Scheme for MSD Systems

The input-output relationship of (126) over the moving horizon [tk−s,tk], where s denotes the
length of the moving horizon, is
(128)ψ(k)=Ho,s,kx(k−s)+Hs,k(u(k)+d(k)+f(k)),
where ψ(k),u(k),d(k), and f(k) are stacked
vectors based on ψ¯(j),u¯(j),d¯(j), and f¯(j), j=k−s,…,k, respectively, Ho,s,k is constructed
according to (83),
(129)Hs,k=[OO⋯C(k−s+1)⋱⋱⋮⋱OC(k)A(k−1)⋯A(k−s+1)⋯C(k)].
Build a parity-relation-based residual generator with
(130)r(k)=Vk(ψ(k)−Hs,ku(k)),VkHo,s,k=0,
where Vk is a
periodically time-varying parity matrix (or vector). To describe the influence
of continuous-time signals d(t),f(t) on the
multirate-sampled outputs, linear time-varying operators Ψkδ (δ standing for d or f),
(131)δ(k)=Ψkδδk(t)=[∫tk−stk−s+1eAc(tk−s+1−t)Eδcδ(t)dt∫tk−s+1tk−s+2eAc(tk−s+2−t)Eδcδ(t)dt⋮∫tk−1tkeAc(tk−t)Eδcδ(t)dt],
are introduced and the residual dynamics is described
by
(132)r(k)=VkHs,k(d(k)+f(k))=VkHs,k(Ψkddk(t)+Ψkffk(t)).
The optimal selection of Vk can be
formulated similar to (117) with Ψd(Ψd)∗ and Ψf(Ψf)∗ substituted by Ψkd(Ψkd)∗ and Ψkf(Ψkf)∗, respectively. Using the same technique, Ψkδ(Ψkδ)∗ are derived to be
(133)Ψkδ(Ψkδ)∗=diag{Ψk,sδ,…,Ψk,1δ},Ψk,jδ=∫0tk−j+1−tk−jeActEδcEδcTeAcTtdt,j=s,…,1.
Due to the periodicity of Ψkδ(Ψkδ)∗, the optimization problem needs to be solved over one
period.


### 4.3.2. Observer-Based FD Scheme for MSD Systems

For the aim of fault detection, a fast rate
time-varying observer-based residual generator can be constructed as(134)x^(k+1)=A(k)x^(k)+u¯(k)+L(k)(ψ¯(k)−ψ^(k)),r(k)=W(k)(ψ¯(k)−ψ^(k)),ψ^(k)=C(k)x^(k),
where the gain matrix L(k) and the
weighting matrix W(k) are
time-varying matrices to be determined. The dimensions of L(k) and W(k) may change with
the number of available sampled output signals. Considering the periodicity of
the matrices A(k),C(k), (134) can be designed as a periodic observer. Define the
state estimation error as e(k)=x(k)−x^(k). The dynamics of residual generator (134) is governed by
(135)e(k+1)=(A(k)−L(k)C(k))e(k)+d¯(k)+f¯(k),r(k)=W(k)C(k)e(k).
Introduce linear time-varying operators Γkδ (δ standing for d or f),
(136)Γkδδk(t)=∫tktk+1eAc(tk+1−t)Eδcδ(t)dt,
to rewrite the residual dynamics as
(137)e(k+1)=(A(k)−L(k)C(k))e(k)+Γkddk(t)+Γkffk(t),r(k)=W(k)C(k)e(k).
To enhance the robustness of the FD system to the
unknown disturbances without loss of the sensitivity to the faults, the design
problem is formulated as
(138)supL(k),W(k)JMSD,OBS,∞/∞(L(k),W(k))=supL(k),W(k)supd=0,f∈ℒ2−{0}(∥r∥2/∥f∥2)supf=0,d∈ℒ2−{0}(∥r∥2/∥d∥2),supL(k),W(k)JMSD,OBS,−/∞(L(k),W(k))=supL(k),W(k)infd=0,f∈ℒ2−{0}(∥r∥2/∥f∥2)supf=0,d∈ℒ2−{0}(∥r∥2/∥d∥2).
By analyzing Γkδ(Γkδ)∗, optimization problems (138) are transformed into equivalent optimization
problems of discrete LTP system
(139)e(k+1)=(A(k)−L(k)C(k))e(k)+E¯d(k)deq(k)+E¯f(k)feq(k),r(k)=W(k)C(k)e(k),
where the l2-norms of deq(k) and feq(k) in (139) have the same upper bounds, respectively, with the ℒ2-norms of d(t) and f(t) in (104), the matrices E¯d(k) and E¯f(k) are
time-varying matrices reflecting the sampling effect and satisfy
(140)E¯δ(k)E¯δT(k)=∫0tk+1−tkeActEδcEδcTeAcTtdt.
Then, optimization problems (138) can be solved with the approaches introduced in
Section 3.2.2.


## 4.3.1. Parity-Relation-Based FD Scheme for MSD Systems

The input-output relationship of (126) over the moving horizon [tk−s,tk], where s denotes the
length of the moving horizon, is
(128)ψ(k)=Ho,s,kx(k−s)+Hs,k(u(k)+d(k)+f(k)),
where ψ(k),u(k),d(k), and f(k) are stacked
vectors based on ψ¯(j),u¯(j),d¯(j), and f¯(j), j=k−s,…,k, respectively, Ho,s,k is constructed
according to (83),
(129)Hs,k=[OO⋯C(k−s+1)⋱⋱⋮⋱OC(k)A(k−1)⋯A(k−s+1)⋯C(k)].
Build a parity-relation-based residual generator with
(130)r(k)=Vk(ψ(k)−Hs,ku(k)),VkHo,s,k=0,
where Vk is a
periodically time-varying parity matrix (or vector). To describe the influence
of continuous-time signals d(t),f(t) on the
multirate-sampled outputs, linear time-varying operators Ψkδ (δ standing for d or f),
(131)δ(k)=Ψkδδk(t)=[∫tk−stk−s+1eAc(tk−s+1−t)Eδcδ(t)dt∫tk−s+1tk−s+2eAc(tk−s+2−t)Eδcδ(t)dt⋮∫tk−1tkeAc(tk−t)Eδcδ(t)dt],
are introduced and the residual dynamics is described
by
(132)r(k)=VkHs,k(d(k)+f(k))=VkHs,k(Ψkddk(t)+Ψkffk(t)).
The optimal selection of Vk can be
formulated similar to (117) with Ψd(Ψd)∗ and Ψf(Ψf)∗ substituted by Ψkd(Ψkd)∗ and Ψkf(Ψkf)∗, respectively. Using the same technique, Ψkδ(Ψkδ)∗ are derived to be
(133)Ψkδ(Ψkδ)∗=diag{Ψk,sδ,…,Ψk,1δ},Ψk,jδ=∫0tk−j+1−tk−jeActEδcEδcTeAcTtdt,j=s,…,1.
Due to the periodicity of Ψkδ(Ψkδ)∗, the optimization problem needs to be solved over one
period.


## 4.3.2. Observer-Based FD Scheme for MSD Systems

For the aim of fault detection, a fast rate
time-varying observer-based residual generator can be constructed as(134)x^(k+1)=A(k)x^(k)+u¯(k)+L(k)(ψ¯(k)−ψ^(k)),r(k)=W(k)(ψ¯(k)−ψ^(k)),ψ^(k)=C(k)x^(k),
where the gain matrix L(k) and the
weighting matrix W(k) are
time-varying matrices to be determined. The dimensions of L(k) and W(k) may change with
the number of available sampled output signals. Considering the periodicity of
the matrices A(k),C(k), (134) can be designed as a periodic observer. Define the
state estimation error as e(k)=x(k)−x^(k). The dynamics of residual generator (134) is governed by
(135)e(k+1)=(A(k)−L(k)C(k))e(k)+d¯(k)+f¯(k),r(k)=W(k)C(k)e(k).
Introduce linear time-varying operators Γkδ (δ standing for d or f),
(136)Γkδδk(t)=∫tktk+1eAc(tk+1−t)Eδcδ(t)dt,
to rewrite the residual dynamics as
(137)e(k+1)=(A(k)−L(k)C(k))e(k)+Γkddk(t)+Γkffk(t),r(k)=W(k)C(k)e(k).
To enhance the robustness of the FD system to the
unknown disturbances without loss of the sensitivity to the faults, the design
problem is formulated as
(138)supL(k),W(k)JMSD,OBS,∞/∞(L(k),W(k))=supL(k),W(k)supd=0,f∈ℒ2−{0}(∥r∥2/∥f∥2)supf=0,d∈ℒ2−{0}(∥r∥2/∥d∥2),supL(k),W(k)JMSD,OBS,−/∞(L(k),W(k))=supL(k),W(k)infd=0,f∈ℒ2−{0}(∥r∥2/∥f∥2)supf=0,d∈ℒ2−{0}(∥r∥2/∥d∥2).
By analyzing Γkδ(Γkδ)∗, optimization problems (138) are transformed into equivalent optimization
problems of discrete LTP system
(139)e(k+1)=(A(k)−L(k)C(k))e(k)+E¯d(k)deq(k)+E¯f(k)feq(k),r(k)=W(k)C(k)e(k),
where the l2-norms of deq(k) and feq(k) in (139) have the same upper bounds, respectively, with the ℒ2-norms of d(t) and f(t) in (104), the matrices E¯d(k) and E¯f(k) are
time-varying matrices reflecting the sampling effect and satisfy
(140)E¯δ(k)E¯δT(k)=∫0tk+1−tkeActEδcEδcTeAcTtdt.
Then, optimization problems (138) can be solved with the approaches introduced in
Section 3.2.2.


## 4.4. FD of NSD Systems

The same design procedures introduced in the last
subsection can be applied to the FD of NSD systems by reordering the sampling
instants. The main difference lies in that in general NSD systems,A(k),C(k),E¯d(k),E¯f(k) are
time-varying matrices but not periodic matrices. In consequence, for the NSD
systems(i)
if the parity
space approach is used, then the time-varying parity matrixV(k) needs to be
calculated at each time instant,(ii)
if the observer-based approach is adopted, then the observer gain matrixL(k) needs to
guarantee the stability of the resulting linear time-varying system.


## 4.5. Influence of Sampling Period on Optimal FD Performance

Sampling period is an important parameter in SD
systems. Recently, the influence of the sampling period on the optimal FD
performance has been investigated in [70, 71]. Suppose that for a given continuous-time process (104) three different sampling schemes are considered:
single-rate sampling with sampling period h, single-rate sampling with sampling period ρh, multirate sampling with base period h and system
period ρh, where ρ≥1 is an integer.
It is proven that the optimal performance indeces are related by
(141)JSSD,OBS,∞/∞,h≥JMSD,OBS,∞/∞≥JSSD,OBS,∞/∞,ρh,JSSD,PS,∞/∞,h≥JMSD,PS,∞/∞≥JSSD,PS,∞/∞,ρh.
That means that, with the increase of the sampling period,
the FD performance will become worse. It can be intuitively interpreted as the
consequence of information reduction caused by the increase of the sampling
period. However, we would like to emphasize that such a conclusion does not hold
for all performance indices, for instance, the H−/H∞ index.


## 5. Concluding Remarks

In this paper, standard methods for FD in discrete LTI
systems have been reviewed and recent development in FD for discrete LTP and SD
systems has been summarized. In case of discrete LTI systems, the basic idea,
full decoupling and optimization problems, and the corresponding solutions are
introduced. It can be seen that different FD approaches are closely related to
each other. The FD problem of discrete LTP systems can be handled either
indirectly by lifting or directly by considering the periodicity of the system
matrices. In SD systems the main problem is to take into account the
intersample behavior and to develop direct FD approaches. With the aid of
operators, the FD problem of SSD, MSD, and NSD systems can be transformed,
respectively, into the FD problem of discrete LTI, LTP, and linear time-varying
systems. The methods introduced in this paper have found several interesting
applications in the emerging research area of embedded networked control
systems (emNCS) [72, 73]. Because of the limited data rate, the sampling
mechanisms become an important design parameter in emNCS and have decisive
influence on the real-time network and computing performance and FD
performance.



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*Source: 849546-2008-01-03.xml*