# H

Authors: Guangfu Deng; Huihong Zhao
Journal: Discrete Dynamics in Nature and Society
 (2015)

---

## Abstract

This paper deals with the problem ofH
∞ fault detection for a class of linear discrete time-varying descriptor systems with missing measurements, and the missing measurements are described by a Bernoulli random binary switching sequence. We first translate the H
∞ fault detection problem into an indefinite quadratic form problem. Then, a sufficient and necessary condition on the existence of the minimum is derived. Finally, an observer-based H
∞ fault detection filter is obtained such that the minimum is positive and its parameter matrices are calculated recursively by solving a matrix differential equation. A numerical example is given to demonstrate the efficiency of the proposed method.


---

## Body

## 1. Introduction

During the last four decades, the fault diagnosis theory has received considerable attention, and many remarkable achievements have been obtained [1–6]. As mentioned in [2], it is well recognized that the model-based fault diagnosis techniques can be classified into three classical approaches: observer-based methods, parity space methods, and parameter identification based methods. Most of the achievements on fault diagnosis always assume that the observations contain the signal to be detected. However, in practice, the observation may contain the signal in a random manner. In this paper, the H
∞ fault detection problem for a class of linear discrete time-varying descriptor systems with random missing measurements is investigated.In practice, time variability is the inherent characteristic of most systems. Recently, research on fault diagnosis of linear systems with time-varying parameters has attracted more and more attention; see, for example, [7–22]. By dividing the filter gain matrix into two sections, an unknown input decoupling optimal filter for linear stochastic systems has been designed in [7], and its application on fault diagnosis has also been addressed. By employing the invariant subspace method and game theory, a game fault detection filter has been proposed in [8]. Based on parity space approach and stochastic signal processing methods, the fault detection and isolation have been studied for a class of linear discrete systems with stochastic inputs and deterministic disturbances and faults in [15, 21]. By applying adaptive observer method, a residual generator has been provided for linear discrete time-varying systems in [19, 22], and the residual adaptive threshold is derived by set-membership computations based on zonotopes. By assuming that the mean and variance of the fault and disturbances are known, the optimal fault detection filter for a class of stochastic systems has been developed in [10, 11, 13, 16]; however, in general, the prior information of the fault cannot be obtained. By solving a min-max problem with a generalized least-squares cost criterion, a generalized least-squares fault detection filter has been designed in [9, 12]. By using an adaptive observer method, a fault diagnosis technique for linear time-varying systems has been developed in [14, 17, 18]. However, to the best of authors’ knowledge, few reports on fault diagnosis problem of linear discrete time-varying descriptor systems have been published, which motivates the present study.In classical fault diagnosis theories, all residual signals are obtained in the case that the measured output contains valid information. However, the data packet dropout is inevitable in navigation and guidance system, industrial control system, and network control system. Thus, in recent years, increasing attention has been paid to fault diagnosis problems for systems with missing measurements [23–25]. By employing the linear matrix inequality technique, both full-order and reduced-order fault detection filters have been considered for a class of linear discrete time-invariant systems with missing measurements and parameter uncertainty in [23]. In finite frequency domain, the fault detection problem has been concerned with systems with missing measurements in [24], and the fault detection scheme has been utilized to an aircraft model. In [25], the missing measurements are described by Markov random process, and the residual generator is presented as a discrete-time Markovian jump linear system. Note that the existing results mainly focus on the nondescriptor systems; there are few achievements on fault detection problem for descriptor systems with missing measurements. Therefore, in this paper, we aim to design the H
∞ fault detection filter and residual evaluation scheme for a class of linear discrete time-varying descriptor systems with random missing measurements.In this paper, based on the estimation method proposed in [26], a new fault detection filter design approach is developed for a class of linear discrete time-varying descriptor systems with random missing measurements. First, the H
∞ fault detection problem is converted to the problem in which a certain indefinite quadratic form has a minimum and the fault detection filter parameter matrices are such that the minimum is positive. Then, by applying matrix analysis method, a necessary and sufficient condition for the indefinite quadratic form is analyzed. And by guaranteeing the positivity of the minimum, the parameter matrices of the fault detection filter are obtained. Moreover, the residual evaluation function and threshold are designed for the fault detection. Finally, a numerical example is provided to illustrate the performance of the H
∞ fault detection filter and the residual evaluation scheme.


## 2. Problem Statement

Consider a discrete time-varying descriptor system described by the following model:(1)
M
i
+
1
x
i
+
1
=
A
i
x
i
+
B
f
i
f
i
+
w
i
,
y
i
=
λ
i
C
i
x
i
+
D
f
i
f
i
+
v
i
,where x
(
i
)
∈
R
n, w
(
i
)
∈
R
n, y
(
i
)
∈
R
m, v
(
i
)
∈
R
m, and f
(
i
)
∈
R
p are the state, external disturbance input, uncertain measurement output, measurement noise, and fault to be detected, respectively; f
(
i
), w
(
i
), and v
(
i
) are bounded signals belonging to l
2
[
0
,
N
]; N is a positive integer; M
(
i
), A
(
i
), B
f
(
i
), C
(
i
), and D
f
(
i
) are known real-time-varying matrices with appropriate dimensions; and M
(
i
) is a singular matrix with rank
[
M
(
i
)
]
=
n
1, 0
<
n
1
<
n; the random variable λ
(
i
)
∈
R is a Bernoulli distributed white sequence taking the values of 0 and 1 with(2)
Prob
λ
i
=
1
=
E
λ
(
i
)
=
ρ
,
Prob
λ
i
=
0
=
1
-
E
λ
i
=
1
-
ρ
,
E
λ
2
i
=
ρ
,
E
1
-
λ
i
2
=
1
-
ρ
,
E
1
-
λ
i
λ
i
=
0
,where E
(
·
) denotes the mathematical expectation, Prob
(
·
) denotes the probability distribution, and ρ
∈
R is a known positive scalar.Hypothesis 1.The initial matrices of system (1) satisfy the condition that M
(
0
)
=
A
(
-
1
)
=
B
f
(
-
1
)
=
I.TheH
∞ fault detection problem under investigation in this paper can be stated as follows. Given a disturbance attenuation level γ
>
0, based on the measurement output sequence {
y
(
i
)
}
i
=
1
k, find a residual signal r
(
k
). If it exists, the following inequality is satisfied: (3)
sup
⁡
x
(
0
)
,
d
(
i
)
i
=
0
k
E
∑
i
=
1
k
r
i
-
f
i
-
1
T
r
i
-
f
i
-
1
+
∑
i
=
0
k
d
T
i
d
i
-
1
kkkkkk
×
E
M
0
x
0
-
A
-
1
x
˘
0
T
P
0
-
1
∑
i
=
0
k
d
T
i
d
i
kkkkkkikk
kkkkkkkk
×
M
0
x
0
-
A
-
1
x
˘
0
kkkkkkki
k
k
k
k
k
i
k
k
i
+
∑
i
=
0
k
d
T
i
d
i
-
1
<
γ
2
,where(4)
d
(
i
)
=
f
T
(
i
)
w
T
(
i
)
v
T
(
i
)
T
,
i
=
0,1
,
…
,
k
-
1
,
f
T
(
i
)
0
v
T
(
i
)
T
,
i
=
k
.
P
0 is a given positive definite matrix function which reflects the relative uncertainty of the initial state x
(
0
) about the initial state estimate x
˘
(
0
). Without loss of generality, let x
˘
(
0
)
=
0.Even if the measurement data are fully available, the valid information of faultf
(
i
) is not contained in the measurement output {
y
(
j
)
}
j
=
0
i when D
f
(
i
)
=
0. Thus, an H
∞ one-step lag fault detection issue is defined as (3) to overcome this problem.Define the following new variables:(5)
x
-
i
=
x
i
f
i
x
i
-
1
f
i
-
1
,
w
-
i
=
w
i
f
i
+
1
0
0
.Thus, system (1) can be described as the following augmented model:(6)
M
-
i
+
1
x
-
i
+
1
=
A
-
i
x
-
i
+
w
-
i
,
y
i
=
λ
i
C
-
i
x
-
i
+
v
i
,where(7)
M
-
i
+
1
=
M
i
+
1
0
0
0
0
I
0
0
0
0
I
0
0
0
0
I
,
A
-
i
=
A
i
B
f
i
0
0
0
0
0
0
I
0
0
0
0
I
0
0
,
C
-
i
=
C
i
D
f
i
0
0
.For the purpose of fault detection, the following observer-based fault detection filter is proposed as a residual generator:(8)
x
-
^
k
+
1
=
V
1
k
x
-
^
k
+
V
2
k
y
k
,
r
k
=
L
k
x
-
^
k
,
x
-
^
0
=
0
,where L
(
i
)
=
[
0
0
0
I
]. Thus, the fault estimation problem can come down to the following: find the parameter matrices V
1
(
k
) and V
2
(
k
); if they exist, the performance index (3) is satisfied.According to system (6), the performance index (3) is reexpressed as(9)
sup
⁡
x
0
,
w
-
i
,
v
i
i
=
0
k
E
∑
i
=
1
k
r
i
-
L
i
x
-
i
T
+
∑
i
=
0
k
v
T
i
v
i
-
1
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
∑
i
=
1
k
r
i
-
L
i
x
-
i
T
×
r
i
-
L
i
x
-
i
kkkkkkkkiiik
×
E
M
-
0
x
-
0
-
A
-
-
1
x
-
˘
0
T
P
-
0
-
1
∑
i
=
0
k
v
T
i
v
i
i
kkkkkkkkkkk
×
M
-
0
x
-
0
-
A
-
-
1
x
-
˘
0
kkkkkkkkkkki
+
∑
i
=
0
k
-
1
w
-
T
i
w
-
i
kkkkkkkkkkkkkkkkkkkkk
+
∑
i
=
0
k
v
T
i
v
i
-
1
<
γ
2
,where P
-
0
=
diag
{
P
0
,
I
p
,
I
n
,
I
p
} and x
-
˘
(
0
)
=
[
x
˘
T
(
0
)
0
0
0
]
T, and assume that x
(
-
1
)
=
0 and f
(
-
1
)
=
0.Define(10)
J
k
=
E
M
-
0
x
-
0
-
A
-
-
1
x
-
˘
0
T
P
-
0
-
1
∑
i
=
0
k
-
1
w
-
T
i
w
-
i
+
∑
i
=
0
k
v
T
i
v
i
kkkkki
×
M
-
0
x
-
0
-
A
-
-
1
x
-
˘
0
kkkkkk
+
∑
i
=
0
k
-
1
w
-
T
i
w
-
i
+
∑
i
=
0
k
v
T
i
v
i
-
γ
-
2
E
∑
i
=
1
k
r
i
-
L
i
x
-
i
T
r
i
-
L
i
x
-
i
.Then, the H
∞ fault detection problem is equivalent to the following: (1)
J
k of (10) has minimum with respect to {
x
-
(
i
)
}
i
=
0
k and {
w
-
(
i
)
}
i
=
0
k;
(2)
{
r
(
i
)
}
i
=
0
k can be chosen such that the value of J
k at its minimum is positive.In the following, we will first discuss the mathematical expectation ofJ
k based on system (6). Then, the existence of the minimum J
k over {
x
-
(
i
)
}
i
=
0
k and {
w
-
(
i
)
}
i
=
0
k can be derived. Finally, a solution to parameter matrices V
1
(
k
) and V
2
(
k
) will be obtained such that J
k
>
0.


## 3. Main Results

From (6) and (10), we have (11)
J
k
=
E
M
-
0
x
-
0
-
A
-
-
1
x
-
˘
0
T
P
-
0
-
1
M
-
0
x
-
0
-
A
-
-
1
x
-
˘
0
T
kki
×
M
-
0
x
-
0
-
A
-
-
1
x
-
˘
0
+
E
∑
i
=
0
k
-
1
M
-
i
+
1
x
-
i
+
1
-
A
-
i
x
-
i
T
∑
i
=
0
k
-
1
M
-
i
+
1
x
-
i
+
1
-
A
-
i
x
-
i
T
kki
×
M
-
i
+
1
x
-
i
+
1
-
A
-
i
x
-
i
+
E
∑
i
=
0
k
y
i
-
λ
i
C
-
i
x
-
i
T
∑
i
=
0
k
y
i
-
λ
i
C
-
i
x
-
i
T
kki
×
y
i
-
λ
i
C
-
i
x
-
i
-
γ
-
2
E
∑
i
=
1
k
r
i
-
L
i
x
-
i
T
r
i
-
L
i
x
-
i
.Notice that(12)
E
y
(
i
)
-
λ
(
i
)
C
-
(
i
)
x
-
(
i
)
T
y
(
i
)
-
λ
(
i
)
C
-
(
i
)
x
-
(
i
)
=
E
(
i
)
C
-
T
(
i
)
C
-
(
i
)
x
-
(
i
)
y
T
(
i
)
y
(
i
)
-
λ
(
i
)
y
T
(
i
)
C
-
(
i
)
x
-
(
i
)
-
λ
(
i
)
x
-
T
(
i
)
k
k
k
k
k
k
j
×
C
-
T
(
i
)
y
(
i
)
+
λ
2
(
i
)
x
-
T
(
i
)
C
-
T
(
i
)
C
-
(
i
)
x
-
(
i
)
=
y
T
(
i
)
y
(
i
)
-
ρ
y
T
(
i
)
C
-
(
i
)
x
-
(
i
)
-
ρ
x
-
T
(
i
)
C
-
T
(
i
)
y
(
i
)
+
ρ
x
-
T
(
i
)
C
-
T
(
i
)
C
-
(
i
)
x
-
(
i
)
=
y
i
-
ρ
C
-
i
x
-
i
T
y
i
-
ρ
C
-
i
x
-
i
+
ρ
-
ρ
2
x
-
T
(
i
)
C
-
T
(
i
)
C
-
(
i
)
x
-
(
i
)
=
y
(
i
)
-
C
~
(
i
)
x
-
(
i
)
T
y
(
i
)
-
C
~
(
i
)
x
-
(
i
)
+
F
i
x
-
i
T
F
i
x
-
i
,where(13)
C
~
i
=
ρ
C
-
i
,
F
i
=
ρ
-
ρ
2
C
-
i
.Then, we have (14)
J
k
=
M
-
0
x
-
0
-
A
-
-
1
x
-
˘
0
T
P
-
0
-
1
×
M
-
0
x
-
0
-
A
-
-
1
x
-
˘
0
+
∑
i
=
0
k
-
1
M
-
i
+
1
x
-
i
+
1
-
A
-
i
x
-
i
T
k
×
M
-
i
+
1
x
-
i
+
1
-
A
-
i
x
-
i
+
∑
i
=
0
k
y
i
-
C
~
i
x
-
i
T
y
(
i
)
-
C
~
(
i
)
x
-
(
i
)
+
∑
i
=
0
k
F
i
x
-
i
T
F
(
i
)
x
-
(
i
)
-
γ
-
2
∑
i
=
1
k
r
i
-
L
i
x
-
i
T
r
i
-
L
i
x
-
i
.Thus, J
k can be further expressed as(15)
J
k
=
Σ
k
X
k
-
Y
k
T
Π
k
Σ
k
X
k
-
Y
k
,where(16)
X
k
=
x
-
k
x
-
k
-
1
x
-
k
-
2
⋮
x
-
2
x
-
1
x
-
0
,
Y
k
=
y
-
k
y
-
k
-
1
y
-
k
-
2
⋮
y
-
2
y
-
1
y
-
0
,
Σ
k
=

Ω
k
Ψ
k
-
1
0
0
0
0
0
0
Ω
k
-
1
Ψ
k
-
2
0
0
0
0
0
0
Ω
k
-
2
⋱
0
0
0
0
0
0
⋱
Ψ
2
0
0
0
0
0
0
Ω
2
Ψ
1
0
0
0
0
0
0
Ω
1
Ψ
0
0
0
0
0
0
0
Ω
0

,
Π
k
=
Φ
k
0
0
0
0
Φ
k
-
1
0
0
0
0
⋱
0
0
0
0
Φ
0
,
Ω
i
=
M
-
i
F
i
C
~
i
L
i
,
Ω
0
=
M
-
0
F
0
C
~
0
,
Ψ
i
-
1
=
-
A
-
i
-
1
0
0
0
,
Φ
i
=
I
0
0
0
0
I
0
0
0
0
I
0
0
0
0
-
γ
-
2
I
,
Φ
0
=
P
-
0
-
1
0
0
0
I
0
0
0
I
,
y
-
i
=
0
0
y
i
r
i
,
y
-
0
=
x
-
˘
0
0
y
0
,
i
=
1,2
,
…
,
k
.In virtue of the above variables, for all k
>
0, we have (17)
X
k
=
x
-
(
k
)
X
k
-
1
,
Y
k
=
y
-
(
k
)
Y
k
-
1
,
Σ
k
=
Ω
(
k
)
α
(
k
-
1
)
0
Σ
k
-
1
,
Π
k
=
Φ
(
k
)
0
0
Π
k
-
1
,
α
k
-
1
=
Ψ
k
-
1
0
⋯
0
.Lemma 1 (see [26]).
Consider matricesα, β, R, and x of appropriate dimensions, and R is symmetric. If and only if α
T
R
α
≥
0 and Ker
⁡
(
α
T
R
α
)
⊂
Ker
⁡
(
R
α
), for any β, we have (18)
inf
⁡
x
α
x
-
β
T
R
α
x
-
β
>
-
∞
.If the minimum is attained, it is unique if and only if α
T
R
α
>
0. Moreover, the optimal solution is derived by x
^
=
(
α
T
R
α
)
-
1
α
T
R
β.


### 3.1. Existence Conditions of the Minimum

According to Lemma1, J
k has the minimum if and only if Σ
k
T
Π
k
Σ
k
>
0. When k
=
0, let(19)
P
0
=
Σ
0
T
Π
0
Σ
0
=
M
-
0
F
0
C
~
0
T
P
-
0
-
1
0
0
0
I
0
0
0
I
M
-
0
F
0
C
~
0
=
M
-
T
0
P
-
0
-
1
M
-
0
+
C
~
T
0
C
~
0
+
F
T
0
F
0
.Therefore, when k
=
0, J
0 has the minimum if and only if P
(
0
)
>
0.Furthermore, for allk
>
0, we can obtain the following equation from (17):(20)
Σ
k
T
Π
k
Σ
k
=
Ω
T
k
Φ
k
Ω
k
Ω
T
k
Φ
k
α
k
-
1
α
T
k
-
1
Φ
k
Ω
k
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
+
α
T
k
-
1
Φ
k
α
k
-
1
.To ensure that (20) is positive definite, Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
+
α
T
(
k
-
1
)
Φ
(
k
)
α
(
k
-
1
) must be positive definite. Assume that Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
>
0 is satisfied, and note that(21)
α
T
(
k
-
1
)
Φ
(
k
)
α
(
k
-
1
)
=
Ψ
T
(
k
-
1
)
0
Φ
(
k
)
Ψ
(
k
-
1
)
0
=
-
A
-
(
k
-
1
)
0
0
0
T
I
0
0
0
0
I
0
0
0
0
I
0
0
0
0
-
γ
-
2
I
-
A
-
(
k
-
1
)
0
0
0
0
0
0
=
A
-
T
k
-
1
A
-
k
-
1
0
0
0
≥
0
.Therefore, Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
+
α
T
(
k
-
1
)
Φ
(
k
)
α
(
k
-
1
)
>
0. If and only if the Schur complement of Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
+
α
T
(
k
-
1
)
Φ
(
k
)
α
(
k
-
1
) in (20) is positive definite, we have Σ
k
T
Π
k
Σ
k
>
0.Define(22)
P
k
=
Ω
T
k
Φ
k
Ω
k
-
Ω
T
k
Φ
k
α
k
-
1
×
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
+
α
T
k
-
1
Φ
k
α
k
-
1
-
1
×
α
T
k
-
1
Φ
k
Ω
k
=
Ω
T
(
k
)
Φ
(
k
)
×
I
-
α
k
-
1
α
T
(
k
-
1
)
Φ
(
k
)
k
k
k
k
o
×
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
+
α
T
k
-
1
Φ
k
α
k
-
1
-
1
k
k
k
k
o
×
α
T
(
k
-
1
)
Φ
(
k
)
Ω
(
k
)
=
Ω
T
(
k
)
Φ
(
k
)
×
I
+
α
k
-
1
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
-
1
α
T
(
k
-
1
)
Φ
(
k
)
-
1
×
Ω
k
.And notice that (23)
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
=
Ω
T
k
-
1
Φ
k
-
1
Ω
k
-
1
Ω
T
k
-
1
Φ
k
-
1
α
k
-
2
α
T
k
-
2
Φ
k
-
1
Ω
k
-
1
Σ
k
-
2
T
Π
k
-
2
Σ
k
-
2
+
α
T
k
-
2
Φ
k
-
1
α
k
-
2
=
I
Ω
T
k
-
1
Φ
k
-
1
α
k
-
2
Σ
k
-
2
T
Π
k
-
2
Σ
k
-
2
+
α
T
k
-
2
Φ
k
-
1
α
k
-
2
-
1
0
I
×
P
k
-
1
0
0
Σ
k
-
2
T
Π
k
-
2
Σ
k
-
2
+
α
T
k
-
2
Φ
k
-
1
α
k
-
2
×
I
0
Σ
k
-
2
T
Π
k
-
2
Σ
k
-
2
+
α
T
(
k
-
2
)
Φ
(
k
-
1
)
α
(
k
-
2
)
-
1
α
T
(
k
-
2
)
Φ
(
k
-
1
)
Ω
(
k
-
1
)
I
.Then, (24)
α
k
-
1
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
-
1
α
T
(
k
-
1
)
=
Ψ
(
k
-
1
)
0
×
P
(
k
-
1
)
0
0
Σ
k
-
2
T
Π
k
-
2
Σ
k
-
2
+
α
T
(
k
-
2
)
Φ
(
k
-
1
)
α
(
k
-
2
)
-
1
×
Ψ
T
(
k
-
1
)
0
=
Ψ
k
-
1
P
-
1
k
-
1
Ψ
T
k
-
1
.Moreover, we obtain(25)
P
k
=
Ω
T
(
k
)
Φ
(
k
)
×
I
+
α
k
-
1
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
-
1
α
T
k
-
1
Φ
k
-
1
×
Ω
(
k
)
=
Ω
T
(
k
)
Φ
(
k
)
×
I
+
Ψ
k
-
1
P
-
1
k
-
1
Ψ
T
k
-
1
Φ
(
k
)
-
1
Ω
(
k
)
=
M
-
(
k
)
F
(
k
)
C
~
(
k
)
L
(
k
)
T
I
0
0
0
0
I
0
0
0
0
I
0
0
0
0
-
γ
-
2
I
×
I
+
-
A
-
(
k
-
1
)
0
0
0
P
-
1
(
k
-
1
)
-
A
-
(
k
-
1
)
0
0
0
T
k
k
k
k
k
k
k
×
I
0
0
0
0
I
0
0
0
0
I
0
0
0
0
-
γ
-
2
I
-
A
-
(
k
-
1
)
0
0
0
T
-
1
M
-
(
k
)
F
(
k
)
C
~
(
k
)
L
(
k
)
=
M
-
T
k
F
T
k
C
~
T
k
-
γ
-
2
L
T
k
×
I
+
A
-
k
-
1
P
-
1
k
-
1
A
-
T
k
-
1
-
1
0
0
0
0
I
0
0
0
0
I
0
0
0
0
I
×
M
-
k
F
k
C
~
k
L
k
=
M
-
T
(
k
)
I
+
A
-
k
-
1
P
-
1
k
-
1
A
-
T
k
-
1
-
1
M
-
(
k
)
+
C
~
T
k
C
~
k
+
F
T
k
F
k
-
γ
-
2
L
T
k
L
k
.Thus, it is readily known that J
k has a minimum if and only if P
(
k
)
>
0.In light of the above discussion, we have the following results.Theorem 2.
Consider the linear discrete time-varying descriptor system (1), given a scalar γ
>
0; then J
k has a minimum over {
x
-
(
i
)
}
i
=
0
k and {
w
-
(
i
)
}
i
=
0
k if and only if P
(
k
)
>
0
(
k
=
0,1
,
…
,
k
), where(26)
P
k
=
M
-
T
k
I
+
A
-
k
-
1
P
-
1
k
-
1
A
-
T
k
-
1
-
1
×
M
-
(
k
)
+
C
~
T
(
k
)
C
~
(
k
)
+
F
T
(
k
)
F
(
k
)
-
γ
-
2
L
T
k
L
k
,
P
0
=
M
-
T
0
P
-
0
-
1
M
-
0
+
C
~
T
0
C
~
0
+
F
T
0
F
0
.


### 3.2. Design of theH
∞ Fault Detection Filter

According to Lemma1, it is known that if J
k has a minimum over {
x
-
(
i
)
}
i
=
0
k and {
w
-
(
i
)
}
i
=
0
k, the optimal solution is(27)
X
^
k
∣
k
=
Σ
k
T
Π
k
Σ
k
-
1
Σ
k
T
Π
k
Y
k
.When k
=
0, we have(28)
x
-
^
0
∣
0
=
Σ
0
T
Π
0
Σ
0
-
1
Σ
0
T
Π
0
Y
0
=
Ω
T
0
Φ
0
Ω
0
-
1
Ω
T
0
Φ
0
x
-
˘
0
0
y
0
=
P
-
1
0
M
-
0
F
0
C
~
0
T
P
-
0
-
1
0
0
0
I
0
0
0
I
x
-
˘
0
0
y
0
=
P
-
1
0
M
-
T
0
P
-
0
-
1
x
-
˘
0
+
P
-
1
0
C
~
T
0
y
0
.Furthermore, notice that(29)
x
-
^
k
∣
k
X
^
k
-
1
∣
k
=
Ω
T
(
k
)
Φ
(
k
)
Ω
(
k
)
Ω
T
(
k
)
Φ
(
k
)
α
(
k
-
1
)
α
T
(
k
-
1
)
Φ
(
k
)
Ω
(
k
)
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
+
α
T
(
k
-
1
)
Φ
(
k
)
α
(
k
-
1
)
-
1
×
Ω
(
k
)
α
(
k
-
1
)
0
Σ
k
-
1
T
Φ
(
k
)
0
0
Π
k
-
1
y
-
(
k
)
Y
k
-
1
=
Ω
T
(
k
)
Φ
(
k
)
Ω
(
k
)
Ω
T
(
k
)
Φ
(
k
)
α
(
k
-
1
)
α
T
(
k
-
1
)
Φ
(
k
)
Ω
(
k
)
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
+
α
T
(
k
-
1
)
Φ
(
k
)
α
(
k
-
1
)
-
1
×
Ω
T
k
Φ
k
y
-
k
α
T
k
-
1
Φ
k
y
-
k
+
Σ
k
-
1
T
Π
k
-
1
Y
k
-
1
=
H
11
,
k
H
12
,
k
H
21
,
k
H
22
,
k
×
Ω
T
k
α
T
k
-
1
Φ
k
y
-
k
+
0
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
X
^
k
-
1
k
-
1
.Then, (30)
x
-
^
k
∣
k
=
H
11
,
k
Ω
T
k
+
H
12
,
k
α
T
k
-
1
Φ
(
k
)
y
-
(
k
)
+
H
12
,
k
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
X
^
k
-
1
∣
k
-
1
,
Σ
k
T
Π
k
Σ
k
-
1
=
Ω
T
(
k
)
Φ
(
k
)
Ω
(
k
)
Ω
T
k
Φ
k
α
k
-
1
α
T
k
-
1
Φ
(
k
)
Ω
(
k
)
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
+
α
T
k
-
1
Φ
k
α
k
-
1
-
1
=
Φ
k
α
k
-
1
-
1
I
Ω
T
k
Φ
k
α
k
-
1
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
+
α
T
k
-
1
Φ
k
α
k
-
1
-
1
0
I
k
k
×
P
(
k
)
0
0
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
+
α
T
k
-
1
Φ
k
α
k
-
1
k
k
×
I
0
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
+
α
T
k
-
1
Φ
k
α
k
-
1
-
1
α
T
k
-
1
Φ
(
k
)
Ω
(
k
)
I
-
1
=
I
0
-
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
+
α
T
k
-
1
Φ
k
α
k
-
1
-
1
α
T
k
-
1
Φ
(
k
)
Ω
(
k
)
I
×
P
-
1
(
k
)
0
0
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
+
α
T
k
-
1
Φ
k
α
k
-
1
-
1
×
I
-
Ω
T
(
k
)
Φ
(
k
)
α
(
k
-
1
)
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
+
α
T
k
-
1
Φ
k
α
k
-
1
-
1
0
I
=
H
11
,
k
H
12
,
k
H
21
,
k
H
22
,
k
.Hence,(31)
H
11
,
k
=
P
-
1
k
,
H
12
,
k
=
-
P
-
1
k
Ω
T
k
Φ
k
α
k
-
1
×
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
+
α
T
k
-
1
Φ
k
α
k
-
1
-
1
=
-
P
-
1
(
k
)
Ω
T
(
k
)
×
Φ
(
k
)
-
Φ
(
k
)
α
(
k
-
1
)
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
-
1
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
-
1
α
T
(
k
-
1
)
-
1
k
k
k
k
k
×
α
T
k
-
1
k
k
k
k
k
i
×
Φ
-
1
k
+
α
k
-
1
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
-
1
k
k
k
k
k
k
k
k
k
×
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
-
1
α
T
(
k
-
1
)
-
1
×
α
(
k
-
1
)
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
-
1
=
-
P
-
1
(
k
)
Ω
T
(
k
)
×
Φ
-
1
(
k
)
+
α
(
k
-
1
)
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
-
1
k
k
k
k
k
k
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
-
1
×
α
T
k
-
1
-
1
×
α
(
k
-
1
)
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
-
1
=
-
P
-
1
(
k
)
Ω
T
(
k
)
×
Φ
-
1
k
+
Ψ
k
-
1
P
-
1
k
-
1
Ψ
T
k
-
1
-
1
×
α
k
-
1
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
-
1
.From the above analysis, we obtain(32)
H
11
,
k
Ω
T
(
k
)
Φ
(
k
)
+
H
12
,
k
α
T
k
-
1
Φ
(
k
)
=
P
-
1
(
k
)
Ω
T
(
k
)
Φ
(
k
)
-
P
-
1
k
Ω
T
k
×
Φ
-
1
(
k
)
+
Ψ
k
-
1
P
-
1
k
-
1
Ψ
T
k
-
1
-
1
×
α
k
-
1
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
-
1
α
T
k
-
1
Φ
k
=
P
-
1
(
k
)
Ω
T
(
k
)
×
Φ
k
Ψ
k
-
1
P
-
1
k
-
1
Ψ
T
k
-
1
Φ
k
k
k
k
k
i
k
k
-
Φ
-
1
(
k
)
+
Ψ
k
-
1
P
-
1
k
-
1
Ψ
T
k
-
1
-
1
k
k
k
k
k
k
k
×
Ψ
k
-
1
P
-
1
k
-
1
Ψ
T
k
-
1
Φ
k
=
P
-
1
k
Ω
T
k
×
Φ
-
1
k
+
Ψ
k
-
1
P
-
1
k
-
1
Ψ
T
k
-
1
-
1
.As such,(33)
x
-
^
(
k
∣
k
)
=
P
-
1
(
k
)
Ω
T
(
k
)
×
Φ
-
1
(
k
)
+
Ψ
k
-
1
P
-
1
k
-
1
Ψ
T
k
-
1
-
1
×
y
-
(
k
)
-
Ψ
(
k
-
1
)
x
-
^
(
k
-
1
∣
k
-
1
)
.Thus,(34)
x
-
^
k
∣
k
=
P
-
1
k
Ω
T
k
×
Φ
-
1
(
k
)
+
Ψ
k
-
1
P
-
1
k
-
1
Ψ
T
k
-
1
-
1
×
y
-
(
k
)
-
Ψ
k
-
1
x
-
^
k
-
1
∣
k
-
1
=
P
-
1
(
k
)
M
-
(
k
)
F
(
k
)
C
~
(
k
)
L
(
k
)
T
×
I
0
0
0
0
I
0
0
0
0
I
0
0
0
0
-
γ
-
2
I
-
1
+
-
A
-
(
k
-
1
)
0
0
0
-
A
-
(
k
-
1
)
0
0
0
T
k
k
k
k
k
k
k
k
×
P
-
1
(
k
-
1
)
-
A
-
(
k
-
1
)
0
0
0
T
-
1
×
0
0
y
k
r
k
-
-
A
-
k
-
1
0
0
0
x
-
^
k
-
1
∣
k
-
1
=
P
-
1
(
k
)
M
-
(
k
)
F
(
k
)
C
~
(
k
)
L
(
k
)
T
×
I
+
A
-
k
-
1
P
-
1
k
-
1
A
-
T
k
-
1
-
1
0
0
0
0
I
0
0
0
0
I
0
0
0
0
-
γ
-
2
I
×
A
-
k
-
1
x
-
^
k
-
1
∣
k
-
1
0
y
k
r
k
=
P
-
1
(
k
)
M
-
T
(
k
)
×
I
+
A
-
k
-
1
P
-
1
k
-
1
A
-
T
k
-
1
-
1
×
A
-
(
k
-
1
)
x
-
^
k
-
1
∣
k
-
1
+
P
-
1
(
k
)
C
~
T
(
k
)
y
(
k
)
-
γ
-
2
P
-
1
(
k
)
L
T
(
k
)
r
(
k
)
.Based on the above discussion, we present the main results of this paper.Theorem 3.
Consider system (1), given a scalar γ
>
0; then the H
∞ fault detection filter (8) that achieves (3) exists if, and only if, P
(
k
)
>
0
(
k
=
0,1
,
…
,
k
) and(35)
V
1
k
=
P
(
k
)
+
γ
-
2
L
T
(
k
)
L
(
k
)
-
1
M
-
T
(
k
)
×
I
+
A
-
k
-
1
P
-
1
k
-
1
A
-
T
k
-
1
-
1
A
-
k
-
1
V
2
k
=
P
k
+
γ
-
2
L
T
k
L
k
-
1
C
~
T
k
,where P
(
k
) is calculated by (26).Proof.
Note thatJ
k has a minimum over {
x
-
(
i
)
}
i
=
0
k and {
w
-
(
i
)
}
i
=
0
k if and only if P
(
k
)
>
0
(
k
=
0,1
,
…
,
k
), and the minimum is (36)
min
⁡
x
i
i
=
0
k
J
k
=
M
-
0
x
-
^
0
∣
k
-
A
-
-
1
x
-
˘
0
T
P
-
0
-
1
×
M
-
(
0
)
x
-
^
0
∣
k
-
A
-
(
-
1
)
x
-
˘
0
+
∑
i
=
0
k
-
1
M
-
i
+
1
x
-
^
i
+
1
∣
k
-
A
-
i
x
-
^
i
∣
k
T
×
M
-
(
i
+
1
)
x
-
^
i
+
1
∣
k
-
A
-
(
i
)
x
-
^
i
∣
k
+
∑
i
=
0
k
y
i
-
C
~
i
x
-
^
i
∣
k
T
×
y
(
i
)
-
C
~
(
i
)
x
-
^
i
∣
k
+
∑
i
=
0
k
F
i
x
-
^
i
∣
k
T
F
i
x
-
^
i
∣
k
-
γ
-
2
∑
i
=
1
k
r
i
-
L
i
x
-
^
i
∣
k
T
×
r
i
-
L
i
x
-
^
i
∣
k
.It is readily seen that a positive minimum of J
k is guaranteed by setting r
(
i
)
=
L
(
i
)
x
-
^
(
i
∣
k
), i
=
0,1
,
…
,
k. Furthermore, substituting r
(
k
)
=
L
(
k
)
x
-
^
(
k
∣
k
) into (34), we obtain(37)
x
-
^
k
∣
k
=
P
(
k
)
+
γ
-
2
L
T
(
k
)
L
(
k
)
-
1
×
M
-
T
(
k
)
I
+
A
-
k
-
1
P
-
1
k
-
1
A
-
T
k
-
1
-
1
k
k
k
k
k
i
I
+
A
-
k
-
1
P
-
1
k
-
1
A
-
T
k
-
1
-
1
×
A
-
k
-
1
x
-
^
k
-
1
∣
k
-
1
+
C
~
T
(
k
)
y
(
k
)
.In the light of the above equation, the parameter matrices of (8) can be given by (35). Hence, the theorem is proven.From Theorem3 in this section, the H
∞ fault detection filter r
(
k
) can be computed in the following steps.Step 1.
Setγ
>
0, P
0
>
0, x
˘
(
0
)
=
0, and k
=
0; calculate P
-
0 and x
-
˘
(
0
) in (9).Step 2.
CalculateP
(
0
) and x
-
^
(
0
∣
0
) using (26) and (28).Step 3.
IfP
(
0
)
>
0, let r
(
0
)
=
L
(
0
)
x
-
^
(
0
∣
0
), and go to Step 4; otherwise, exit.Step 4.
Letk
=
k
+
1; compute P
(
k
) using (26).Step 5.
IfP
(
k
)
>
0, compute V
1
(
k
), V
2
(
k
), and r
(
k
) using (35) and (8), and go to Step 4; otherwise, exit.Step 6.
Repeat Steps4 to 5 till k
=
N.Remark 4.
Note that the system augmentation has been applied to design theH
∞ fault detection filter (8); it may lead to more expensive computational cost. Fortunately, an H
∞ simultaneous state and unknown input estimator for descriptor system have been proposed in [27], and an H
∞ fixed-lag smoother for missing measurements system has been given in [28], so it is possible, in the future, to design a new H
∞ fault detection filter with lower computational cost by using the algorithm given in [27, 28].


## 3.1. Existence Conditions of the Minimum

According to Lemma1, J
k has the minimum if and only if Σ
k
T
Π
k
Σ
k
>
0. When k
=
0, let(19)
P
0
=
Σ
0
T
Π
0
Σ
0
=
M
-
0
F
0
C
~
0
T
P
-
0
-
1
0
0
0
I
0
0
0
I
M
-
0
F
0
C
~
0
=
M
-
T
0
P
-
0
-
1
M
-
0
+
C
~
T
0
C
~
0
+
F
T
0
F
0
.Therefore, when k
=
0, J
0 has the minimum if and only if P
(
0
)
>
0.Furthermore, for allk
>
0, we can obtain the following equation from (17):(20)
Σ
k
T
Π
k
Σ
k
=
Ω
T
k
Φ
k
Ω
k
Ω
T
k
Φ
k
α
k
-
1
α
T
k
-
1
Φ
k
Ω
k
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
+
α
T
k
-
1
Φ
k
α
k
-
1
.To ensure that (20) is positive definite, Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
+
α
T
(
k
-
1
)
Φ
(
k
)
α
(
k
-
1
) must be positive definite. Assume that Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
>
0 is satisfied, and note that(21)
α
T
(
k
-
1
)
Φ
(
k
)
α
(
k
-
1
)
=
Ψ
T
(
k
-
1
)
0
Φ
(
k
)
Ψ
(
k
-
1
)
0
=
-
A
-
(
k
-
1
)
0
0
0
T
I
0
0
0
0
I
0
0
0
0
I
0
0
0
0
-
γ
-
2
I
-
A
-
(
k
-
1
)
0
0
0
0
0
0
=
A
-
T
k
-
1
A
-
k
-
1
0
0
0
≥
0
.Therefore, Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
+
α
T
(
k
-
1
)
Φ
(
k
)
α
(
k
-
1
)
>
0. If and only if the Schur complement of Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
+
α
T
(
k
-
1
)
Φ
(
k
)
α
(
k
-
1
) in (20) is positive definite, we have Σ
k
T
Π
k
Σ
k
>
0.Define(22)
P
k
=
Ω
T
k
Φ
k
Ω
k
-
Ω
T
k
Φ
k
α
k
-
1
×
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
+
α
T
k
-
1
Φ
k
α
k
-
1
-
1
×
α
T
k
-
1
Φ
k
Ω
k
=
Ω
T
(
k
)
Φ
(
k
)
×
I
-
α
k
-
1
α
T
(
k
-
1
)
Φ
(
k
)
k
k
k
k
o
×
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
+
α
T
k
-
1
Φ
k
α
k
-
1
-
1
k
k
k
k
o
×
α
T
(
k
-
1
)
Φ
(
k
)
Ω
(
k
)
=
Ω
T
(
k
)
Φ
(
k
)
×
I
+
α
k
-
1
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
-
1
α
T
(
k
-
1
)
Φ
(
k
)
-
1
×
Ω
k
.And notice that (23)
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
=
Ω
T
k
-
1
Φ
k
-
1
Ω
k
-
1
Ω
T
k
-
1
Φ
k
-
1
α
k
-
2
α
T
k
-
2
Φ
k
-
1
Ω
k
-
1
Σ
k
-
2
T
Π
k
-
2
Σ
k
-
2
+
α
T
k
-
2
Φ
k
-
1
α
k
-
2
=
I
Ω
T
k
-
1
Φ
k
-
1
α
k
-
2
Σ
k
-
2
T
Π
k
-
2
Σ
k
-
2
+
α
T
k
-
2
Φ
k
-
1
α
k
-
2
-
1
0
I
×
P
k
-
1
0
0
Σ
k
-
2
T
Π
k
-
2
Σ
k
-
2
+
α
T
k
-
2
Φ
k
-
1
α
k
-
2
×
I
0
Σ
k
-
2
T
Π
k
-
2
Σ
k
-
2
+
α
T
(
k
-
2
)
Φ
(
k
-
1
)
α
(
k
-
2
)
-
1
α
T
(
k
-
2
)
Φ
(
k
-
1
)
Ω
(
k
-
1
)
I
.Then, (24)
α
k
-
1
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
-
1
α
T
(
k
-
1
)
=
Ψ
(
k
-
1
)
0
×
P
(
k
-
1
)
0
0
Σ
k
-
2
T
Π
k
-
2
Σ
k
-
2
+
α
T
(
k
-
2
)
Φ
(
k
-
1
)
α
(
k
-
2
)
-
1
×
Ψ
T
(
k
-
1
)
0
=
Ψ
k
-
1
P
-
1
k
-
1
Ψ
T
k
-
1
.Moreover, we obtain(25)
P
k
=
Ω
T
(
k
)
Φ
(
k
)
×
I
+
α
k
-
1
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
-
1
α
T
k
-
1
Φ
k
-
1
×
Ω
(
k
)
=
Ω
T
(
k
)
Φ
(
k
)
×
I
+
Ψ
k
-
1
P
-
1
k
-
1
Ψ
T
k
-
1
Φ
(
k
)
-
1
Ω
(
k
)
=
M
-
(
k
)
F
(
k
)
C
~
(
k
)
L
(
k
)
T
I
0
0
0
0
I
0
0
0
0
I
0
0
0
0
-
γ
-
2
I
×
I
+
-
A
-
(
k
-
1
)
0
0
0
P
-
1
(
k
-
1
)
-
A
-
(
k
-
1
)
0
0
0
T
k
k
k
k
k
k
k
×
I
0
0
0
0
I
0
0
0
0
I
0
0
0
0
-
γ
-
2
I
-
A
-
(
k
-
1
)
0
0
0
T
-
1
M
-
(
k
)
F
(
k
)
C
~
(
k
)
L
(
k
)
=
M
-
T
k
F
T
k
C
~
T
k
-
γ
-
2
L
T
k
×
I
+
A
-
k
-
1
P
-
1
k
-
1
A
-
T
k
-
1
-
1
0
0
0
0
I
0
0
0
0
I
0
0
0
0
I
×
M
-
k
F
k
C
~
k
L
k
=
M
-
T
(
k
)
I
+
A
-
k
-
1
P
-
1
k
-
1
A
-
T
k
-
1
-
1
M
-
(
k
)
+
C
~
T
k
C
~
k
+
F
T
k
F
k
-
γ
-
2
L
T
k
L
k
.Thus, it is readily known that J
k has a minimum if and only if P
(
k
)
>
0.In light of the above discussion, we have the following results.Theorem 2.
Consider the linear discrete time-varying descriptor system (1), given a scalar γ
>
0; then J
k has a minimum over {
x
-
(
i
)
}
i
=
0
k and {
w
-
(
i
)
}
i
=
0
k if and only if P
(
k
)
>
0
(
k
=
0,1
,
…
,
k
), where(26)
P
k
=
M
-
T
k
I
+
A
-
k
-
1
P
-
1
k
-
1
A
-
T
k
-
1
-
1
×
M
-
(
k
)
+
C
~
T
(
k
)
C
~
(
k
)
+
F
T
(
k
)
F
(
k
)
-
γ
-
2
L
T
k
L
k
,
P
0
=
M
-
T
0
P
-
0
-
1
M
-
0
+
C
~
T
0
C
~
0
+
F
T
0
F
0
.


## 3.2. Design of theH
∞ Fault Detection Filter

According to Lemma1, it is known that if J
k has a minimum over {
x
-
(
i
)
}
i
=
0
k and {
w
-
(
i
)
}
i
=
0
k, the optimal solution is(27)
X
^
k
∣
k
=
Σ
k
T
Π
k
Σ
k
-
1
Σ
k
T
Π
k
Y
k
.When k
=
0, we have(28)
x
-
^
0
∣
0
=
Σ
0
T
Π
0
Σ
0
-
1
Σ
0
T
Π
0
Y
0
=
Ω
T
0
Φ
0
Ω
0
-
1
Ω
T
0
Φ
0
x
-
˘
0
0
y
0
=
P
-
1
0
M
-
0
F
0
C
~
0
T
P
-
0
-
1
0
0
0
I
0
0
0
I
x
-
˘
0
0
y
0
=
P
-
1
0
M
-
T
0
P
-
0
-
1
x
-
˘
0
+
P
-
1
0
C
~
T
0
y
0
.Furthermore, notice that(29)
x
-
^
k
∣
k
X
^
k
-
1
∣
k
=
Ω
T
(
k
)
Φ
(
k
)
Ω
(
k
)
Ω
T
(
k
)
Φ
(
k
)
α
(
k
-
1
)
α
T
(
k
-
1
)
Φ
(
k
)
Ω
(
k
)
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
+
α
T
(
k
-
1
)
Φ
(
k
)
α
(
k
-
1
)
-
1
×
Ω
(
k
)
α
(
k
-
1
)
0
Σ
k
-
1
T
Φ
(
k
)
0
0
Π
k
-
1
y
-
(
k
)
Y
k
-
1
=
Ω
T
(
k
)
Φ
(
k
)
Ω
(
k
)
Ω
T
(
k
)
Φ
(
k
)
α
(
k
-
1
)
α
T
(
k
-
1
)
Φ
(
k
)
Ω
(
k
)
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
+
α
T
(
k
-
1
)
Φ
(
k
)
α
(
k
-
1
)
-
1
×
Ω
T
k
Φ
k
y
-
k
α
T
k
-
1
Φ
k
y
-
k
+
Σ
k
-
1
T
Π
k
-
1
Y
k
-
1
=
H
11
,
k
H
12
,
k
H
21
,
k
H
22
,
k
×
Ω
T
k
α
T
k
-
1
Φ
k
y
-
k
+
0
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
X
^
k
-
1
k
-
1
.Then, (30)
x
-
^
k
∣
k
=
H
11
,
k
Ω
T
k
+
H
12
,
k
α
T
k
-
1
Φ
(
k
)
y
-
(
k
)
+
H
12
,
k
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
X
^
k
-
1
∣
k
-
1
,
Σ
k
T
Π
k
Σ
k
-
1
=
Ω
T
(
k
)
Φ
(
k
)
Ω
(
k
)
Ω
T
k
Φ
k
α
k
-
1
α
T
k
-
1
Φ
(
k
)
Ω
(
k
)
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
+
α
T
k
-
1
Φ
k
α
k
-
1
-
1
=
Φ
k
α
k
-
1
-
1
I
Ω
T
k
Φ
k
α
k
-
1
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
+
α
T
k
-
1
Φ
k
α
k
-
1
-
1
0
I
k
k
×
P
(
k
)
0
0
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
+
α
T
k
-
1
Φ
k
α
k
-
1
k
k
×
I
0
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
+
α
T
k
-
1
Φ
k
α
k
-
1
-
1
α
T
k
-
1
Φ
(
k
)
Ω
(
k
)
I
-
1
=
I
0
-
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
+
α
T
k
-
1
Φ
k
α
k
-
1
-
1
α
T
k
-
1
Φ
(
k
)
Ω
(
k
)
I
×
P
-
1
(
k
)
0
0
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
+
α
T
k
-
1
Φ
k
α
k
-
1
-
1
×
I
-
Ω
T
(
k
)
Φ
(
k
)
α
(
k
-
1
)
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
+
α
T
k
-
1
Φ
k
α
k
-
1
-
1
0
I
=
H
11
,
k
H
12
,
k
H
21
,
k
H
22
,
k
.Hence,(31)
H
11
,
k
=
P
-
1
k
,
H
12
,
k
=
-
P
-
1
k
Ω
T
k
Φ
k
α
k
-
1
×
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
+
α
T
k
-
1
Φ
k
α
k
-
1
-
1
=
-
P
-
1
(
k
)
Ω
T
(
k
)
×
Φ
(
k
)
-
Φ
(
k
)
α
(
k
-
1
)
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
-
1
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
-
1
α
T
(
k
-
1
)
-
1
k
k
k
k
k
×
α
T
k
-
1
k
k
k
k
k
i
×
Φ
-
1
k
+
α
k
-
1
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
-
1
k
k
k
k
k
k
k
k
k
×
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
-
1
α
T
(
k
-
1
)
-
1
×
α
(
k
-
1
)
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
-
1
=
-
P
-
1
(
k
)
Ω
T
(
k
)
×
Φ
-
1
(
k
)
+
α
(
k
-
1
)
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
-
1
k
k
k
k
k
k
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
-
1
×
α
T
k
-
1
-
1
×
α
(
k
-
1
)
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
-
1
=
-
P
-
1
(
k
)
Ω
T
(
k
)
×
Φ
-
1
k
+
Ψ
k
-
1
P
-
1
k
-
1
Ψ
T
k
-
1
-
1
×
α
k
-
1
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
-
1
.From the above analysis, we obtain(32)
H
11
,
k
Ω
T
(
k
)
Φ
(
k
)
+
H
12
,
k
α
T
k
-
1
Φ
(
k
)
=
P
-
1
(
k
)
Ω
T
(
k
)
Φ
(
k
)
-
P
-
1
k
Ω
T
k
×
Φ
-
1
(
k
)
+
Ψ
k
-
1
P
-
1
k
-
1
Ψ
T
k
-
1
-
1
×
α
k
-
1
Σ
k
-
1
T
Π
k
-
1
Σ
k
-
1
-
1
α
T
k
-
1
Φ
k
=
P
-
1
(
k
)
Ω
T
(
k
)
×
Φ
k
Ψ
k
-
1
P
-
1
k
-
1
Ψ
T
k
-
1
Φ
k
k
k
k
k
i
k
k
-
Φ
-
1
(
k
)
+
Ψ
k
-
1
P
-
1
k
-
1
Ψ
T
k
-
1
-
1
k
k
k
k
k
k
k
×
Ψ
k
-
1
P
-
1
k
-
1
Ψ
T
k
-
1
Φ
k
=
P
-
1
k
Ω
T
k
×
Φ
-
1
k
+
Ψ
k
-
1
P
-
1
k
-
1
Ψ
T
k
-
1
-
1
.As such,(33)
x
-
^
(
k
∣
k
)
=
P
-
1
(
k
)
Ω
T
(
k
)
×
Φ
-
1
(
k
)
+
Ψ
k
-
1
P
-
1
k
-
1
Ψ
T
k
-
1
-
1
×
y
-
(
k
)
-
Ψ
(
k
-
1
)
x
-
^
(
k
-
1
∣
k
-
1
)
.Thus,(34)
x
-
^
k
∣
k
=
P
-
1
k
Ω
T
k
×
Φ
-
1
(
k
)
+
Ψ
k
-
1
P
-
1
k
-
1
Ψ
T
k
-
1
-
1
×
y
-
(
k
)
-
Ψ
k
-
1
x
-
^
k
-
1
∣
k
-
1
=
P
-
1
(
k
)
M
-
(
k
)
F
(
k
)
C
~
(
k
)
L
(
k
)
T
×
I
0
0
0
0
I
0
0
0
0
I
0
0
0
0
-
γ
-
2
I
-
1
+
-
A
-
(
k
-
1
)
0
0
0
-
A
-
(
k
-
1
)
0
0
0
T
k
k
k
k
k
k
k
k
×
P
-
1
(
k
-
1
)
-
A
-
(
k
-
1
)
0
0
0
T
-
1
×
0
0
y
k
r
k
-
-
A
-
k
-
1
0
0
0
x
-
^
k
-
1
∣
k
-
1
=
P
-
1
(
k
)
M
-
(
k
)
F
(
k
)
C
~
(
k
)
L
(
k
)
T
×
I
+
A
-
k
-
1
P
-
1
k
-
1
A
-
T
k
-
1
-
1
0
0
0
0
I
0
0
0
0
I
0
0
0
0
-
γ
-
2
I
×
A
-
k
-
1
x
-
^
k
-
1
∣
k
-
1
0
y
k
r
k
=
P
-
1
(
k
)
M
-
T
(
k
)
×
I
+
A
-
k
-
1
P
-
1
k
-
1
A
-
T
k
-
1
-
1
×
A
-
(
k
-
1
)
x
-
^
k
-
1
∣
k
-
1
+
P
-
1
(
k
)
C
~
T
(
k
)
y
(
k
)
-
γ
-
2
P
-
1
(
k
)
L
T
(
k
)
r
(
k
)
.Based on the above discussion, we present the main results of this paper.Theorem 3.
Consider system (1), given a scalar γ
>
0; then the H
∞ fault detection filter (8) that achieves (3) exists if, and only if, P
(
k
)
>
0
(
k
=
0,1
,
…
,
k
) and(35)
V
1
k
=
P
(
k
)
+
γ
-
2
L
T
(
k
)
L
(
k
)
-
1
M
-
T
(
k
)
×
I
+
A
-
k
-
1
P
-
1
k
-
1
A
-
T
k
-
1
-
1
A
-
k
-
1
V
2
k
=
P
k
+
γ
-
2
L
T
k
L
k
-
1
C
~
T
k
,where P
(
k
) is calculated by (26).Proof.
Note thatJ
k has a minimum over {
x
-
(
i
)
}
i
=
0
k and {
w
-
(
i
)
}
i
=
0
k if and only if P
(
k
)
>
0
(
k
=
0,1
,
…
,
k
), and the minimum is (36)
min
⁡
x
i
i
=
0
k
J
k
=
M
-
0
x
-
^
0
∣
k
-
A
-
-
1
x
-
˘
0
T
P
-
0
-
1
×
M
-
(
0
)
x
-
^
0
∣
k
-
A
-
(
-
1
)
x
-
˘
0
+
∑
i
=
0
k
-
1
M
-
i
+
1
x
-
^
i
+
1
∣
k
-
A
-
i
x
-
^
i
∣
k
T
×
M
-
(
i
+
1
)
x
-
^
i
+
1
∣
k
-
A
-
(
i
)
x
-
^
i
∣
k
+
∑
i
=
0
k
y
i
-
C
~
i
x
-
^
i
∣
k
T
×
y
(
i
)
-
C
~
(
i
)
x
-
^
i
∣
k
+
∑
i
=
0
k
F
i
x
-
^
i
∣
k
T
F
i
x
-
^
i
∣
k
-
γ
-
2
∑
i
=
1
k
r
i
-
L
i
x
-
^
i
∣
k
T
×
r
i
-
L
i
x
-
^
i
∣
k
.It is readily seen that a positive minimum of J
k is guaranteed by setting r
(
i
)
=
L
(
i
)
x
-
^
(
i
∣
k
), i
=
0,1
,
…
,
k. Furthermore, substituting r
(
k
)
=
L
(
k
)
x
-
^
(
k
∣
k
) into (34), we obtain(37)
x
-
^
k
∣
k
=
P
(
k
)
+
γ
-
2
L
T
(
k
)
L
(
k
)
-
1
×
M
-
T
(
k
)
I
+
A
-
k
-
1
P
-
1
k
-
1
A
-
T
k
-
1
-
1
k
k
k
k
k
i
I
+
A
-
k
-
1
P
-
1
k
-
1
A
-
T
k
-
1
-
1
×
A
-
k
-
1
x
-
^
k
-
1
∣
k
-
1
+
C
~
T
(
k
)
y
(
k
)
.In the light of the above equation, the parameter matrices of (8) can be given by (35). Hence, the theorem is proven.From Theorem3 in this section, the H
∞ fault detection filter r
(
k
) can be computed in the following steps.Step 1.
Setγ
>
0, P
0
>
0, x
˘
(
0
)
=
0, and k
=
0; calculate P
-
0 and x
-
˘
(
0
) in (9).Step 2.
CalculateP
(
0
) and x
-
^
(
0
∣
0
) using (26) and (28).Step 3.
IfP
(
0
)
>
0, let r
(
0
)
=
L
(
0
)
x
-
^
(
0
∣
0
), and go to Step 4; otherwise, exit.Step 4.
Letk
=
k
+
1; compute P
(
k
) using (26).Step 5.
IfP
(
k
)
>
0, compute V
1
(
k
), V
2
(
k
), and r
(
k
) using (35) and (8), and go to Step 4; otherwise, exit.Step 6.
Repeat Steps4 to 5 till k
=
N.Remark 4.
Note that the system augmentation has been applied to design theH
∞ fault detection filter (8); it may lead to more expensive computational cost. Fortunately, an H
∞ simultaneous state and unknown input estimator for descriptor system have been proposed in [27], and an H
∞ fixed-lag smoother for missing measurements system has been given in [28], so it is possible, in the future, to design a new H
∞ fault detection filter with lower computational cost by using the algorithm given in [27, 28].


## 4. Residual Evaluation

When the design of theH
∞ fault detection filter has been completed, the next task is residual evaluation. First, the following residual evaluation function and the threshold are introduced to facilitate fault detection:(38)
J
r
,
k
=
∑
i
=
0
k
r
T
i
r
i
,
J
th
k
=
sup
⁡
w
i
,
v
i
∈
l
2
,
f
i
=
0
E
J
r
,
k
.The following strategy is applied for fault detection:(39)
J
r
,
k
>
J
th
k
⟹
There
is
fault
⟹
alarm
J
r
,
k
≤
J
th
k
⟹
There
is
no
fault
.If, for all 0
≤
i
≤
N, f
(
i
)
≡
0, the system (1) can be redescribed as follows:(40)
M
i
+
1
x
i
+
1
=
A
i
x
i
+
w
i
,
y
i
=
λ
i
C
i
x
i
+
v
i
.Employing a similar technical line of Section 3 in this paper, we can obtain the residual signal of system (40). For a given scalar γ
c
>
0, by employing performance index (3) and Theorem 3, we can judge whether the residual evaluation function with zero initial conditions achieves the following inequality: (41)
J
r
,
k
<
γ
c
2
E
∑
i
=
0
k
w
T
i
w
i
+
∑
i
=
0
k
v
T
i
v
i
.Note that w
(
i
) and v
(
i
) are bounded signals, so there exist m
1
(
k
) and m
2
(
k
) such that(42)
∑
i
=
0
k
w
T
(
i
)
w
(
i
)
≤
m
1
k
,
∑
i
=
0
k
v
T
i
v
i
≤
m
2
k
.Suppose that the minimum γ
c achieving (41) is γ
c
min
⁡; then we have (43)
J
r
,
k
<
γ
c
min
⁡
2
m
1
k
+
m
2
k
.Thus, the residual threshold can be defined as(44)
J
th
k
=
γ
c
min
⁡
2
m
1
k
+
m
2
k
.Finally, it can be judged based on the strategy given in (39).


## 5. A Numerical Example

Consider the discrete system (1) with the following parameters:(45)
M
k
=
2.9
0
0
0
2.5
+
0.3
sin
⁡
k
0
0
0
0
,
A
k
=
2.1
e
-
k
0
0
1.6
0
0
0
-
1.9
,
B
f
k
=
3.2
2.7
4.5
,
C
k
=
2.6
0
1.8
.Set γ
=
0.25 and P
0
=
I. The unknown signals w
(
k
), v
(
k
), and f
(
k
) are supposed to be(46)
w
k
=
0.3
sin
⁡
k
0.2
cos
⁡
⁡
3
k
0.4
cos
⁡
⁡
2
k
,
f
(
k
)
=
0
0
≤
k
<
40
,
1
,
40
≤
k
≤
60
,
v
k
=
0.5
sin
4
k
,
0
,
k
>
60
.By applying Theorem 3, the H
∞ fault detection filter is designed. Figure 1 shows the residual signal r
(
k
) when ρ
=
1 and D
f
(
k
)
=
1.8, and Figure 2 shows the residual evaluation function J
(
r
,
k
) and threshold J
th
(
k
) when ρ
=
1 and D
f
(
k
)
=
1.8. Figure 3 shows the residual signal r
(
k
) when ρ
=
1 and D
f
(
k
)
=
0, and Figure 4 shows the residual evaluation function J
(
r
,
k
) and threshold J
th
(
k
) when ρ
=
1 and D
f
(
k
)
=
0. It is shown that the tracking performance of H
∞ fault detection filtering is good in the above two cases. Figure 5 shows the variation law of random parameter λ
(
k
) when ρ
=
0.9. Figure 6 shows the residual signal r
(
k
) when ρ
=
0.9 and D
f
(
k
)
=
1.8, and Figure 7 shows the residual evaluation function J
(
r
,
k
) and threshold J
th
(
k
) when ρ
=
0.9 and D
f
(
k
)
=
1.8. Figure 8 shows the residual signal r
(
k
) when ρ
=
0.9 and D
f
(
k
)
=
0, and Figure 9 shows the residual evaluation function J
(
r
,
k
) and threshold J
th
(
k
) when ρ
=
0.9 and D
f
(
k
)
=
0. It is shown that the tracking performance of H
∞ fault detection filtering is weakening when the system has missing measurements, but the threshold J
th
(
k
) in Figures 7 and 9 has good performance.Figure 1
Residual signalr
(
k
) when ρ
=
1 and D
f
(
k
)
=
1.8.Figure 2
Residual evaluation functionJ
(
r
,
k
) and threshold J
th
(
k
) when ρ
=
1 and D
f
(
k
)
=
1.8.Figure 3
Residual signalr
(
k
) when ρ
=
1 and D
f
(
k
)
=
0.Figure 4
Residual evaluation functionJ
(
r
,
k
) and threshold J
th
(
k
) when ρ
=
1 and D
f
(
k
)
=
0.Figure 5
Random parametersλ
(
k
) when ρ
=
0.9.Figure 6
Residual signalr
(
k
) when ρ
=
0.9 and D
f
(
k
)
=
1.8.Figure 7
Residual evaluation functionJ
(
r
,
k
) and threshold J
th
(
k
) when ρ
=
0.9 and D
f
(
k
)
=
1.8.Figure 8
Residual signalr
(
k
) when ρ
=
0.9 and D
f
(
k
)
=
0.Figure 9
Residual evaluation functionJ
(
r
,
k
) and threshold J
th
(
k
) when ρ
=
0.9 and D
f
(
k
)
=
0.


## 6. Conclusions

In this paper, the Bernoulli random binary switching sequence has been applied to describe missing measurements, and it has been used as a basis on the study of theH
∞ fault detection problem for linear discrete time-varying descriptor systems. The main contribution of this paper is to develop a novel approach on solving the H
∞ fault detection problem for time-varying descriptor systems with missing measurements. An augmented system has been first obtained, and a relationship has been established between the H
∞ fault detection problem and a certain indefinite quadratic form problem. By applying matrix analysis method, a necessary and sufficient existence condition and the explicit formula of the H
∞ fault detection filter have been derived. The residual evaluation function and the threshold have been designed to facilitate fault detection. Finally, the proposed algorithm has been proved to be effective by a numerical example.



---

*Source: 867206-2015-10-05.xml*