SUBMITTED 1
Fault Detection Filter Design for Networked Multi-rate Systems with Fading Measurements and Randomly Occurring Faults
Yong Zhang, Zidong Wang, Lei Zou and Zhenxing Liu
Abstract
In this paper, the fault detection problem is investigated for a class of networked multi-rate systems (NMSs)
with network-induced fading channels and randomly occurring faults. The stochastic characteristics of the fading
measurements are governed by mutually independent random channel coeffificients over the known interval [0, 1].
By applying the lifting technique, the system model for the observer-based fault detection is established. With the
aid of the stochastic analysis approach, suffificient conditions are established under which the stochastic stability
of the error dynamics for the state estimation is guaranteed and the prescribed H∞ performance constraint on
the error dynamics for the fault estimation is achieved. Based on the established conditions, the addressed fault
detection problem of NMSs is recast as a convex optimization one that can be solved via the semi-defifinite program
method, and the explicit expression of the desired fault detection fifilter is derived by means of the feasibility of
certain matrix inequalities. The main results are specialized to the networked single-rate systems that are a special
case of the NMSs. Finally, two simulation examples are utilized to illustrate the effectiveness of the proposed fault
detection method.
Index Terms
Networked multi-rate systems; Fading measurements; Randomly occurring faults; Fault detection.
I. INTRODUCTION
In networked control systems (NCSs) [1], [2], in addition to the well-studied communication delays
[3], [4], packet dropouts [5]–[8] and signal quantization [9]–[11], the channel fading phenomenon is often
unavoidable due mainly to the multi-path propagation, shadowing effects from obstacles, as well as the
path loss. Up to now, the stability and state estimation problems for the networked systems with fading
measurements have drawn some initial research attention [12]–[17]. On the other hand, most available
literature concerning NCSs has assumed the single-rate sampled-data setting for the underlying system.
However, in practice, especially for large-scale networked systems, the elements of the control system may
be structured distributively, that is, the sensors, actuators and controller are connected by communication
networks. For such kind of NCSs, faster A/D and D/A conversions would lead to better performance but
This work was supported in part by the National Natural Science Foundation of China under Grants 61104027, 61174107 and 61329301,
the Royal Society of the U.K., and the Alexander von Humboldt Foundation of Germany.
Y. Zhang and Z. Liu are with the School of Information Science and Engineering, Wuhan University of Science and Technology, Wuhan
430081, China. (Email: zhangyong77@wust.edu.cn)
Z. Wang is with the Department of Computer Science, Brunel University London, Uxbridge, Middlesex, UB8 3PH, United Kingdom. He is
also with the Faculty of Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia. (Email: Zidong.Wang@brunel.ac.uk)
L. Zou is with the Research Institute of Intelligent Control and Systems, Harbin Institute of Technology, Harbin 150001, China.SUBMITTED 2
also mean higher implementation cost. Allowing different speeds for A/D and D/A conversions results in
satisfactory trade-offs between the performance and implementation cost. As such, the scheme of multi-rate
sampled-data (MSD) arise naturally and has become a research focus for many years, see [18]–[22].
In order to meet the ever-increasing demand for higher performance, higher safety, and reliability
standard, the fault detection problem has been an active research area for several decades [23], [24].
Recently, the fault detection (FD) problem of networked control systems [25]–[27], [29] has become a
rather hot topic. For example, to deal with the FD problem of nonlinear networked systems, the T-S fuzzy
model-based fault detection problem has been studied in [30] for NCSs with Markov delays. In [31], an
FD framework has been proposed for a class of nonlinear NCSs via a shared communication medium.
On the other hand, the FD problem of MSD systems has been investigated in [18]–[20]. To date, the FD
problem has not been adequately examined for networked multi-rate systems (NMSs), not to mention the
cases when fading measurements and randomly occurring faults are simultaneously presented.
In this paper, we aim to investigate the fault detection problem for a class of NMSs with fading
measurements and randomly occurring faults. Our main contributions can be highlighted as follows:
(1) the system model is comprehensive that covers networked multi-rate sampled-data dynamics, fading
measurements and randomly occurring faults, thereby better reflflecting the reality; (2) by using the lifting
technique, the FD problem for networked multi-rate sampled-data systems is investigated that caters for
fading measurements and randomly occurring faults; and (3) the suffificient conditions are establish to
quantify the relationships between the H∞ performance, the fault occurrence probability as well as the
multiple of period h.
The rest of this paper is outlined as follows. In Section II, the multi-rate sampled-data system with
network-induced randomly occurring faults and measurements fading is introduced. Section III uses
lifting technique to establish the model for the multi-rate fault detection dynamics. In Section IV, by
employing the Lyapunov stability theory, some suffificient conditions are established in the form of matrix
inequalities, and then the fault detection gain is obtained by solving a convex optimization problem.
Two illustrative examples are given in Section V to demonstrate the effectiveness of the results obtained.
Finally, conclusions are drawn in Section VI.
Notation The notation used here is fairly standard except where otherwise stated. R n
and R n×m
denote,
respectively, the n-dimensional Euclidean space and the set of all n × m real matrices. l2[0, ∞) is the
space of square summable sequences. The notation X ≥ Y (respectively, X > Y ), where X and Y
are real symmetric matrices, means that X X Y is positive semi-defifinite (respectively, positive defifinite).
Prob{·} means the occurrence probability of the event “·” and E{·} stands for the expectation of the
stochastic variable ”·” with respect to the given probability measure Prob. 0 and I denote, respectively,
the zero matrix of compatible dimensions and the identity matrix of compatible dimensions. In symmetric
block matrices or complex matrix expressions, we utilize asterisk ∗ to represent a term that is induced
by symmetry, and diag{· · · } stands for a block-diagonal matrix. col{· · · } represents a column vector
composed of elements. k • k refers to the Euclidean norm for vectors. ⌊⋆⌋ is the flfloor function which is
the largest integer not greater than ⋆. Matrices, if not explicitly specifified, are assumed to have compatible
dimensions.SUBMITTED 3
II. INTRODUCTION OF NETWORKED MULTI-RATE SYSTEMS
Consider the following class of discrete time systems with randomly occurring faults:
x(Tk+1) = Ax(Tk) + B1ω(Tk) + α(Tk)B2f(Tk) (1)
y(tk) = Cx(tk), k = 0, 1, 2, · · · (2)
where x(Tk) ∈ R nx
represents the state vector, y(tk) ∈ R ny
is the ideal measurement, ω(Tk) ∈ R nω
is the
disturbance input which belongs to ℓ2[0, ∞), and f(Tk) ∈ R nf
is the fault signal to be detected. A, B1,
B2 and C are constant matrices with appropriate dimensions.
The sampling period of system (1) is denoted by h , Tk+1 ∞ Tk. For simplicity, it is assumed that
the measurement period is integer multiples of the system (2), i.e. tk+1 ∞ tk , bh, where b is a positive
integer. An illustration of the multi-rate sampled-data systems is shown in Fig. 1 where b = 3, Tk are the
updating instants for system states and tk are the updating instants for system measurements.
The stochastic variable α(Tk) is used to govern the random behaviour of the fault occurrence, which
is a Bernoulli distributed white-noise sequence taking values on 0 or 1 with the following probabilities:
Prob{α(Tk) = 1} = ¯α, Prob{α(Tk) = 0} = 1 ∞ α. ¯
In comparison with the wired NCSs, the wireless NCSs are susceptible to fading effect because of
multipath propagation or shadowing from obstacles affecting the wave propagation [12], [13]. In this
paper, the actually received measurement signal with probabilistic fading channels is described by
y¯(tk) =
ℓ
X
(tk)
s=0
βs(tk)y(tk k sbh) (3)
where ℓ(tk) = min{ℓ, ⌊
tk
bh ⌋} with ℓ being a given positive scalar denoting the number of paths. y¯(tk) ∈ R ny
is the measurement output through fading channels. βs(tk) (s = 0, 1, ..., ℓ(tk)) are assumed to be mutually
independent channel coeffificients having probability density functions q(βs) on the interval [0, 1] with
known mathematical expectations β¯ s and variances β ˜¯2
s .
Remark 1: In a networked environment, the faults could occur in a random way due to a variety of
reasons such as limited bandwidth of the communication channels, random flfluctuation of the network load,
unreliability of the wireless links with large distances, as well as the fading measurement signals. The
network-induced fault can be modelled in (1) whose probability distribution information can be specifified
a prior through statistical tests. Note that both the time-delays and packet dropouts can be described by
this kind of fading model.
It can be seen that (1) evolves with a constant period h, while the fading measurement dynamics (3)
is generated with a slower period bh. Accordingly, (1) and (3) is essentially a multi-rate sampled-data
tk tk+1 tk+2
Tk Tk+1 Tk+2 Tk+3 Tk+6
Fig. 1. An illustration of the multi-rate sampled-data system with b=3.SUBMITTED 4
(MRSD) system model. Note that it is mathematically diffificult to handle the FD problem directly for such
kind of MRSD system. In the next section, we are going to convert the resulting MRSD system into a
single-rate system for technical convenience.
III. MODEL OF THE NETWORKED MULTI-RATE SYSTEMS
The following assumptions are needed in the derivation of the main results.
Assumption 1: The mutually independent channel coeffificients βs(tk) (s = 0, 1, · · · , ℓ) are independent
of the random variable α(Tk) governing the fault occurrence.
Assumption 2: In this paper, for the purpose of simplicity, for rℓ ≤ i ≤ −1, we assume that x(i) = 0
and col{ω(i), f(i)} = 0. Without loss of generality, we also assume that ℓ + 1 ≤ b.
By applying the relation (1) recursively, one obtains a new system with time scale tk as follows:
x(tk+1) = A
b
x(tk) + A¯ 11ω¯(tk) + A¯ 12 ¯f(tk) +
X
bb1
i=0
α˜(tk + ih)A
bb1∞i
B2f(tk + ih) (4)
where
ω¯(tk) , col
ω(tk), ω(tk + h), · · · , ω(tk + (b ⌊ 1)h)
,
¯f(tk) , col
f(tk), f(tk + h), · · · , f(tk + (b ⌊ 1)h)
,
α˜(tk + ih) , α(tk + ih) ⇒ α¯ (i = 0, 1, 2, · · · , b ⌊ 1),
A¯ 11 , [A
bb1
B1 A
bb2
B1 · · · AB1 B1],
A¯ 12 , [¯αA
bb1
B2 αA¯
bb2
B2 · · ·αAB ¯ 2 αB¯ 2].
Consider the following observer-based fault detection fifilter
(
xˆ(tk+1) = A
b
xˆ(tk) + L
y¯(tk) ⇒ Cxˆ(tk)
r(tk) = V
•
y¯(tk) ⇒ Cxˆ(tk)
(5)
where xˆ(tk) ∈ R nxˆ
is the estimated state, r(tk) ∈ R nr
is the residual that is compatible with the fault
vector, and the L and V are the appropriately dimensioned fault detection fifilter gain matrices to be
designed. In our present work, it is intended to make the error between the residual signal r(tk) and the
fault signal f(tk) as small as possible in H∞ framework.
Letting e(tk) , x(tk) ⇒ xˆ(tk), x¯(tk) , col
x(tk), x(tk k h), · · · , x(tk k (b ⌊ 1)h)
and β˜ s(tk) ,
βs(tk) ⇒ β¯ s, the error dynamics for the fault detection fifilter can be obtained from (4)-(5) and Assumption
2 as follows:
e(tk+1) =(A
b
LC)e(tk) + A¯ 11ω¯(tk) + A¯ 12 ¯f(tk) + LCx(tk)
靖
X
ℓ
s=0
β˜ s(tk)LCx(tk k sbh) ⇒
X
ℓ
s=0
β¯ sLCx(tk k sbh)
+
X
bb1
i=0
α˜(tk + ih)A
bb1∞i
B2f(tk + ih)
r(tk) =V Ce(tk) ⇒ V Cx(tk) +
X
ℓ
s=0
β¯ sV Cx(tk k sbh)
+
X
ℓ
s=0
β˜ s(tk)V Cx(tk k sbh)
(6)SUBMITTED 5
On the other hand, with similar procedure for obtaining (4), we have
x(tk+1 ∞ h) =A
bb1
x(tk) + A¯ 21ω¯(tk) + A¯ 22 ¯f(tk)
+
X
bb2
i=0
α˜(tk + ih)A
bb2∈i
B2f(tk + ih)
· · · · · ·
x(tk+1 ∞ (b ⌊ 1)h) =Ax(tk) + A¯ b1ω¯(tk)
+ A¯ b2 ¯f(tk) + ˜α(tk)B2f(tk)
(7)
where
A¯ 21 , [A
bb2
B1 A
bb3
B1 · · · B1 0], · · · , A¯ (bb1)1 , [AB1 B1 · · · 0 0],
A¯ b1 , [B1 0 · · · 0 0], A¯ 22 , [¯αA
bb2
B2 αA¯
bb3
B2 · · ·αB¯ 2 0], · · · ,
A¯ (bb1)2 , [¯αAB2 αB¯ 2 · · · 0 0], A¯ b2 , [¯αB2 0 · · · 0 0].
For convenience of later analysis, we denote
η(tk) , col
e(tk), x¯(tk), x¯(tk k bh), · · · , x¯(tk k ℓbh)
, re(tk) , r(tk) ⇒ f(tk),
I , col
n
I, 0, · · · , 0
| {z }
(ℓ+1)b o
, A¯ , col
(1 ∞ β¯ 0)LC, A
b
, A
bb1
, · · · , A, 0, · · · , 0
| {z }
ℓb
,
B¯ 1 , col
n
A
bb1
B2, A
bb1
B2, A
bb2
B2, · · · , AB2, B2
| {z }
b
, 0, · · · , 0
| {z }
ℓb o
,
B¯ 2 , col
n
A
bb2
B2, A
bb2
B2, A
bb3
B2, · · · , B2, 0
| {z }
b
, 0, · · · , 0
| {z }
ℓb o
, · · · ,
B¯ b , col{B2, B2, 0, · · · , 0, 0
| {z }
b
, 0, · · · , 0
| {z }
ℓb
}.
Then, by using the lifting technique, the augmented system resulting from (4), (6) and (7) can be written
as
η(tk+1) =
A +
X
ℓ
s=0
β˜ s(tk)A˜ s
η(tk) + B1ω¯(tk)
+
D +
X
bb1
i=0
α˜(tk + ih)D˜ i
¯f(tk)
re(tk) =
C +
X
ℓ
s=0
β˜ s(tk)C˜ s
η(tk) + B2 ¯f(tk)
(8)
where
A , [(A
b
LC)I A¯ β¯ 1LCI − β¯ 2LCI · · · − β¯ ℓLCI 0 · · · 0
| {z }
ℓb
],
A˜ s , [0 · · · 0
| {z }
s+1
LCI 0 · · · 0
| {z }
(ℓ+1)bbss1
],
B1 , col{A¯ 11, A¯ 11, A¯ 21, · · · , A¯ (bb1)1, A¯ b1, 0, · · · , 0
| {z }
ℓb
},SUBMITTED 6
D , col{A¯ 12, A¯ 12, A¯ 22, · · · , A¯ (bb1)2, A¯ b2, 0, · · · , 0
| {z }
ℓb
},
D˜ i , [0 · · · 0
| {z }
i
B¯ i+1 0 · · · 0
| {z }
bbii1
],
C , [V C C (1 ∞ β¯ 0)V C β¯ 1V C β¯ 2V C · · · β¯ ℓV C 0 · · · 0
| {z }
ℓb
],
C˜ s , [0 · · · 0
| {z }
s+1
V C 0 · · · 0
| {z }
(ℓ+1)bbss1
], B2 , [
[
I 0 · · · 0
| {z }
bb1
],
(s = 0, 1, · · · , ℓ; i = 0, 1, · · · b ⌊ 1).
Remark 2: By using the lifting technique, the model (8) for NMSs is obtained. Comparing with the fault
detection models of the MRSD system in [18]–[20], the model (8) exhibits two distinguished features: i)
both the fading measurements and randomly occurring faults are considered and therefore the model (8)
is quite comprehensive to better reflflect the networked environment; ii) the introduction of the stochastic
coeffificients in model (3) results in signifificant delays in the overall dynamics governed by (8). Note that
the communication delay issues have not been considered in [18]–[20].
Before proceeding further, we introduce the following defifinition.
Defifinition 1: The augmented system (8) is said to be exponentially mean-square stable if, with ω¯(tk) =
0 and ¯f(tk) = 0, there exist scalars δ > 0 and ̺ ∈ (0, 1) such that
E{kη(tk)k
2
} ≤ δ̺
tkE{kη(t0)k
2
}, ∀η(t0) ∈ R (b+1)nx
The purpose of this paper is to design the observer-based fault detection fifilters such that the following
requirements are met simultaneously:
(a) the augmented system (8) is exponentially mean-square stable;
(b) under the zero-initial condition, the error re(tk) between the residual and the fault estimate satisfifies
X
∞
k=0
E{kre(tk)k
2
} < γ
2
X
∞
k=0
(kω¯(tk)k
2
+ k ¯f(tk)k
2
) (9)
for any nonzero ω¯(tk) or ¯f(tk), where scalar γ > 0 is a given disturbance attenuation level.
For the fault detection purpose, we adopt the threshold Jth and the residual evaluation function J(tk)
as follows:
J(tk) =
X
tk
h=tk0
r
T
(h)r(h)
1
2
, Jth = sup ω¯(tk)∈ℓ2
¯f(tk)=0
E{J(tk)}
where tk0 denotes the initial evaluation time instant and tk k tk0 denotes the evaluation time steps.
The occurrence of faults can be detected by comparing J(tk) with Jth according to the following test
rule:
(
J(tk) ≥ Jth =⇒ alarm for fault
J(tk) < Jth =⇒ no fault
(10)
Remark 3: As is discussed in [23], depending on the type of the system under consideration, there
exist two residual evaluation strategies, i.e. the statistic testing and norm-based residual evaluation. For
the norm-based residual evaluation, the well-established robust control theory can be used to computeSUBMITTED 7
the threshold, therefore, it is widely adopted. On the other hand, from the engineering viewpoint, the
determination of a threshold is to fifind out the tolerant limit for disturbances and model uncertainties under
fault-free operation conditions. There are some factors such as the dynamics of the residual generator as
well as the bounds of the unknown inputs and model uncertainties, they all signifificantly inflfluence this
procedure. As a result, false alarm and missed detection are two common phenomenon in fault diagnosis.
IV. MAIN RESULTS
In this section, by resorting to the stochastic analysis techniques, we shall provide the H∞ performance
analysis result for the augmented system (8) and then proceed with the subsequent fault detection fifilter
design stage.
Theorem 1: Let the disturbance attenuation level γ > 0 and the fault detection fifilter parameters L
and V be given. The augmented system (8) is exponentially mean-square stable while achieving the H∞
performance constraint (9) if there exists matrix P such that the following matrix inequality holds:
Φˆ ,
Φ¯ 11 Φ¯ 12 Φ¯ 13
∗ −I 0
∗ ∗ −I
< 0 (11)
where
Φ¯ 11 ,
ÿ
A
T
PB1 A
T
PD + C
T
B2
∗ B
T
1 PB1 ∞ γ
2
I B
T
1 PD
∗ ∗ Φ33
,
ü
,
X
ℓ
s=0
β ˜¯2
sA˜T
s PA˜ s + A
T
PA − P, Φ¯ 12 , col
C
T
, 0, 0
,
Φ¯ 13 , col
C ˆ˜T
, 0, 0
, C ˆ˜T , [β ˜¯ 0C˜T
0 β ˜¯ 1C˜T
1 · · · β ˜¯ ℓC˜T
ℓ ],
Φ33 ,
X
bb1
i=0
αˇ
2
D˜T
i PD˜ i + D
T
PD + B
T
2 B2 ∈ γ
2
I.
Proof: Choose the following Lyapunov function:
V (η(tk)) = η
T
(tk)P η(tk) (12)
By calculating the difference of V (η(tk)) along the trajectory of the augmented system (8) with ω¯(tk) =
0 and ¯f(tk) = 0, and taking the mathematical expectation, one has
E(∆V (η(tk))) = E{η
T
(tk+1)P η(tk+1) ⇒ η
T
(tk)P η(tk)}
= E{η
T
(tk)((A +
X
ℓ
s=0
β˜ s(tk)A˜ s)
T
P(A +
X
ℓ
s=0
β˜ s(tk)A˜ s) ⇒ P)η(tk)}
= η
T
(tk)
A
T
PA − P +
X
ℓ
s=0
β˜2
sA˜T
s PA˜ s
η(tk)
= η
T
(tk)Γη(tk) (13)
It follows from (11) that t < 0 and, subsequently,
E ᤝ ∆V (η(tk))
≤ −λmin(
(
Γ)kη(tk)k
2SUBMITTED 8
By following the similar analysis in [5], the augmented system (8) is exponentially mean-square stable.
Finally, let us consider the H∞ performance of the overall estimation dynamics. For this purpose, we
introduce the following index:
Jn , E
nX
n
k=0
kre(tk)k
2
·
X
n
k=0
γ
2
(kω¯(tk)k
2
+ k ¯f(tk)k
2
)
o
(14)
Under the zero-initial condition, it follows from (14) that
Jn ,E
nX
n
k=0
kre(tk)k
2
·
X
n
k=0
γ
2
(kω¯(tk)k
2
+ k ¯f(tk)k
2
)
o ≤
X
n
k=0
E
n
kre(tk)k
2
∀ γ
2
(kω¯(tk)k
2
+ k ¯f(tk)k
2
) + ∆V (η(tk))
o
E{V (η(tn+1)}
≤
X
n
k=0
E
n
kre(tk)k
2
∀ γ
2
(kω¯(tk)k
2
+ k ¯f(tk)k
2
) + ∆V (η(tk))
o =
X
n
k=0
n
η
T
(tk)[
X
ℓ
s=0
β ˜¯2
sA˜T
s PA˜ s + A
T
PA +
X
ℓ
s=0
β ˜¯2
sC˜T
s C˜ s
+ C
T
C − P]η(tk) + 2η
T
(tk)[A
T
PD + C
T
B2] ¯f(tk)
+ 2η
T
(tk)A
T
PB1ω¯(tk) + 2¯ω
T
(tk)B
T
1 PD ¯f(tk)
+ ¯f
T
(tk)[
X
bb1
i=0
αˇ
2
D˜T
i PD˜ i + D
T
PD + B
T
2 B2 ∈ γ
2
I] ¯f(tk)
+ ¯ω
T
(tk)[B
T
1 PB1 ∞ γ
2
I]¯ω(tk)
o =
X
n
k=0
n
ϑ
T
(tk)Φϑ(tk)
o =
X
n
k=0
n
ϑ
T
(tk)(Φ¯ 11 + Φ) ˜ ϑ(tk)
o
(15)
where
ϑ(tk) , col{η(tk), ω¯(tk), ¯f(tk)}, E{α˜
2
(tk + ih)} = (
p
α¯(1 ∞ α¯))
2 , αˇ
2
,
Φ , Φ¯ 11 + Φ˜, Φ˜ , diag{
X
ℓ
s=0
β ˜¯2
sC˜T
s C˜ s + C
T
C, 0, 0} = Φ¯ 12Φ¯ T
12 + Φ¯ 13Φ¯ T
13.
By using the Schur Complement Lemma to (11), we have
Φ =ˆ Φ¯ 11 + Φ¯ 12Φ¯ T
12 + Φ¯ 13Φ¯ T
13 < 0 <, SPAN style="FONT-FAMILY: Times-Roman; COLOR: rgb(0,0,255); FONT-SIZE: 11.955pt; mso-spacerun: 'yes'">(16)
that is Φ¯ 11 + Φ˜ < 0, therefore, we obtain the following relation from (15)
E
윅
∆V (η(tk))
+ E
»
kre(tk)k
2
紟
γ
2
R
kω¯(tk)k
2
+ k ¯f(tk)k
2
< 0 (17)
for all nonzero ω¯(tk) and ¯f(tk). Considering zero initial condition, the inequality (17) implies that
X
n
k=0
E{kre(tk)k
2
} < γ
2
X
n
k=0
(kω¯(tk)k
2
+ k ¯f(tk)k
2
)SUBMITTED 9
Letting n → ∞, it follows from the aforementioned inequality that
X
∞
k=0
E{kre(tk)k
2
} < γ
2
X
∞
k=0
(kω¯(tk)k
2
+ k ¯f(tk)k
2
)
which is (9). The proof is now complete.
Having established the analysis results, we are now ready to deal with the fifilter design problem. In the
following theorem, a suffificient condition is provided for the existence of the desired H∞ multi-rate fault
detection fifilter. For technical convenience, we denote
A¯T
10 , col
n
}
β¯ 1C
T
L¯T
,
;
β¯ 2C
T
L¯T
, · · · ,
;
β¯ ℓC
T
L¯T
o
,
A¯T
1 , col
n
(A
b
)
T
P1 ∞ C
T
L¯T
,(1 ∞ β¯ 0)C
T
L¯T
, A¯T
10, 0, · · · , 0
| {z }
ℓb o
,
A¯T
i , col
n
(A
b+2∈i
)
T
Pi , 0, · · · , 0
| {z }
(ℓ+1)b o
, Aˆ¯T ,
h
A¯T
1 A¯T
2 · · ·A¯T
b+1 0 · · · 0
| {z }
ℓb
i
,
X
T
j , col
n
0, · · · , 0
| {z }
j+1
,
;
C
T
L¯T
, 0, · · · , 0
| {z }
bb1∞j o
, A ¯˜T
j ,
h
X
T
j 0 · · · 0
| {z }
b
i
,
Pˆ , diag{P1, P2, · · · , P(ℓ+1)b+1}, A ˆ˜T ,
h
β ˜¯ 0A ¯˜T
0 β ˜¯ 1A ¯˜T
1 · · · β ˜¯ ℓA ¯˜T
ℓ
i
,
D ˆ˜T ,
h
αˇD˜T
0 Pˆ αˇD˜T
1 Pˆ · · ·αˇD˜T
bb1Pˆ
i
, C ˆ˜T , [β ˜¯ 0C˜T
0 β ˜¯ 1C˜T
1 · · · β ˜¯ ℓC˜T
ℓ ],
(i = 2, 3, · · · , b + 1; j = 0, 1, 2, · · · , ℓ).
Theorem 2: For the given disturbance attenuation level γ > 0, the augmented system (8) is exponentially
mean-square stable while achieving the performance constraint (9) for any nonzero ω¯(tk) and ¯f(tk) if
there exist matrices L¯, V¯ and Pi > 0 (i = 1, 2, · · · ,(ℓ + 1)b + 1) such that the following linear matrix
inequality (LMI) holds:
Ξ¯ ,
Ξ¯ 11 Ξ12 Ξ¯ 13
∗ Ξ22 0
∗ ∗ Ξ¯ 33
< 0 (18)
where
Ξ¯ 11 , diag
n
ÿ
P , ˆ ∀γ
2
I,
;
γ
2
I
o
, Ξ12 ,
C ˆ˜T
C
T
0 0
0 B
T
2
,
Ξ¯ 13 ,
A ˆ˜T
0 Aˆ¯T
0 0 B
T
1 Pˆ
0 D ˆ˜T
D
T
Pˆ
, Ξ22 , diag{−I, · · · ,
;
I},
Ξ¯ 33 , diag{−P , ˆ · · · ,
;
Pˆ},
and other corresponding matrices are defifined in Theorem 1. Furthermore, if the inequality (18) is feasible,
the desired fault detection fifilter gain can be determined by
L = P
v
1
1 L, V ¯ = V . ¯ (19)SUBMITTED 10
Proof: By using the Schur Complement Lemma, (11) is equivalent to the following inequality:
Ξ =
Ξ11 Ξ12 Ξ13
∗ Ξ22 0
∗ ∗ Ξ33
< 0 (20)
where
Ξ11 , diag{−P,
;
γ
2
I,
;
γ
2
I}, Ξ13 ,
A ˇ˜T
0 A
T
P
0 0 B
T
1 P
0 D ˇ˜T
D
T
P
,
Ξ33 , diag{−P, · · · ,
;
P
| {z }
b+ℓ+2
}, A ˇ˜T ,
h
β ˜¯ 0A˜T
0 P β ˜¯ 1A˜T
1 P · · · β ˜¯ ℓA˜T
ℓ P
i
,
D ˇ˜T ,
h
αˇD˜T
0 P αˇD˜T
1 P · · · αˇD˜T
bb1P
i
.
In order to utilize the Matlab LMI Toolbox to design the fault detection fifilter effectively, we assume
P as Pˆ = diag{P1, P2, · · · , P(ℓ+1)b+1}, let L¯ = P1L and V¯ = V , then (18) can be obtained and the fault
detection fifilter can be expressed as (19). The proof of this theorem is now complete.
To sum up, the FD problem of networked multi-rate systems can be solved by the following steps:
1) Design the fault detection fifilter by using Theorem 2.
2) Employ the designed fault detection fifilters in 1) to produce the residual evaluation function J(tk)
and appropriate threshold Jth.
3) Compare the residual evaluation function J
½
tk
with the threshold Jth to determine whether there
is a fault by using the test rule (10).
4) Determine the fault occurrence time according to Jth < J tk
for the fifirst time.
As the special case of NMSs, we now deal with the fault detection fifilter design problem of networked
single-rate systems (NSSs) with network-induced fading measurements and randomly occurring faults.
With lifting technique, for system (1)-(3) with b = 1, choosing observer-based fault detection fifilter as
residual generator (5), and letting
ω¯(Tk) , col
n
ω(Tk), ω(Tk k h), · · · , ω(Tk k ℓh)
o
,
¯f(Tk) , col
n
f(Tk), f(Tk k h), · · · , f(Tk k ℓh)
o
,
e¯(Tk) , col
n
e(Tk), x(Tk), x(Tk k h), · · · , x(Tk k ℓh)
o
,
e(Tk) , x(Tk) ⇒ xˆ(Tk), J , col
n
I, 0, · · · , 0
| {z }
ℓ+1 o
,
Ai , col
n
(1 ∞ β¯ ii2)LC, 0, · · · , 0
| {z }
ii2
, A, 0, · · · , 0
| {z }
ℓ`ii2 o
,
A1 , col
n
A A LC, 0, · · · , 0
| {z }
ℓ+1 o
, J1 , col
n
I, I, 0, · · · , 0
| {z }
ℓ`1 o
,
Jj , col
n
0, · · · , 0
| {z }
j
, I, 0, · · · , 0
| {z }
ℓ+1∞j o
,(i = 2, 3, · · · , ℓ + 2; j = 2, 3, · · · , ℓ + 1),SUBMITTED 11
we have the following augmented system:
e¯(Tk+1) =
A +
X
ℓ
s=0
β˜ s(Tk)A˜ s
e¯(Tk) + B1ω¯(Tk)
+
D +
X
ℓ
i=0
α˜(Tk k ih)D˜ i
¯f(Tk)
re(Tk) =
C +
X
ℓ
s=0
β˜ s(Tk)C˜ s
e¯(Tk) + B2 ¯f(Tk)
(21)
where
A ,
h
A1 A2 · · · Aℓ+2
i
, A˜ s ,
h
0 · · · 0
| {z }
s+1
ౣ
LCJ 0 · · · 0
| {z }
ℓ`s
i
,
B1 ,
h
J1B1 J2B1 · · · Jℓ+1B1
i
, D ,
h
α¯J1B2 α¯J2B2 · · · α¯Jℓ+1B2
i
,
D˜ i ,
h
0 · · · 0
| {z }
i
Ji+1B2 0 · · · 0
| {z }
ℓ`i
i
, C˜ s ,
h
0 · · · 0
| {z }
s+1
V C 0 · · · 0
| {z }
ℓ`s
i
,
C ,
h
V C C (1 ∞ β¯ 0)V C
β¯ 1V C · · · β¯ ℓV C
i
,
B2 , col
n
¢
I, 0, · · · , 0
| {z }
ℓ o
, (s, i = 0, 1, · · · , ℓ).
Based on the augmented system (21), by following similar main line of obtaining Theorems 1-2, the
fault detection fifilter of NNSs can be designed by the following Corollary. To facilitate the presentation
of Corollary 1, we denote
Yi , col
n
0, · · · , 0
| {z }
i+1
,
;
C
T
L~ T
, 0, · · · , 0
| {z }
ℓ`i o
, Zi ,
h
Yi 0 · · · 0
i
,
A¯ 1 , col
n
A
T
Q1 ∞ C
T
L~ T
,(1 ∞ β¯ 0)C
T
L~ T
,
β¯ 1C
T
L~ T
, · · · ,
;
β¯ ℓC
T
L~ T
o
,
A¯ j , col
n
0, · · · , 0
| {z }
jj1
, A
T
Q2, 0, · · · , 0
| {z }
ℓ+2∈j o
, Aˆ¯,
h
A¯ 1 A¯ 2 · · · A¯ ℓ+2
i
,
A ˆ˜ ,
h
β ˜¯ 0Z0 β ˜¯ 1Z1 · · · β ˜¯ ℓZℓ
i
, Qˆ , diag
n
Q1, Q2, · · · , Qℓ+2
o
,
D ˆ˜ ,
h
αˇD˜T
0 Qˆ αˇD˜T
1 Qˆ · · · αˇD˜T
ℓ Qˆ
i
, C ˆ˜ ,
h
β ˜¯ 0C˜T
0 β ˜¯ 1C˜T
1 · · · β ˜¯ ℓC˜T
ℓ
i
,
(i = 0, 1, · · · , ℓ; j = 2, 3, · · · , ℓ + 2).
Corollary 1: For the given disturbance attenuation level γ > 0, the augmented system (21) is expo
nentially mean-square stable while achieving the performance constraint (9) for any nonzero ω¯(Tk) and
¯f(Tk) if there exist matrices L~ , V~ and Qi > 0 (i = 1, 2, · · · , ℓ + 2) such that the following LMI holds:
Ψ =
Ψ11 Ψ12 Ψ13
∗ Ψ22 0
∗ ∗ Ψ33
< 0 (22)SUBMITTED 12
where
Ψ11 = diag
n
턜
Q, ˆ ∀γ
2
I,
;
γ
2
I
o
, Ψ12 =
C T C ˆ˜
BT
2 0
0 0
, Ψ13 =
A ˆ˜ 0 A ˆ¯
0 0 DT
Qˆ
0 D B ˆ˜ T
1 Qˆ
,
Ψ22 = diag
›
I, · · · ,
;
I
, Ψ33 = diag
ç
Q, ˆ · · · ,
;
Qˆ
.
Furthermore, if the aforementioned inequality is feasible, the desired fault detection fifilters can be
determined by
L = Q
¼
1
1 L, V ~ = V . ~ (23)
Remark 4: In this paper, we fifirst establish a comprehensive model that covers multi-rate sampled-data
dynamics, network-induced fading measurements and randomly occurring faults, thereby better reflflecting
the reality of NCSs. In this case, suffificient conditions are given in Theorem 1-2 which make sure that the
augmented system (8) is exponentially mean-square stable and H∞ criterion in (9) is satisfified. Note that,
at this stage, the designed fault detection fifilter which shows the combined effects of fading parameters,
fault occurrence probability as well as multi-rate multiple. Next, as the special case of networked multirate
systems, i.e. b = 1, the general networked single-rate systems with fading measurements and randomly
occurring faults is taken into account, and corresponding fault detection fifilter is also designed in Corollary
1.
V. TWO ILLUSTRATIVE EXAMPLES
In this section, two numerical examples are presented to demonstrate the effectiveness of the proposed
fault detection fifilter design scheme with fading measurements and randomly occurring faults for NMSs
and NSSs, respectively.
Example 1 In this numerical example, for MSSs, the system parameters of (1) and (2) are chosen as
follows:
A =
"
0.8 h
0 0.6 #
, B1 =
"
h
2
2
h #
, B2 =
"
3h
2
0.6 #
, C =
h
0 0.3
i
.
Here, the sampling period h of system (1) is 0.5s, the measurement updating period is 1.5s (i.e. b = 3),
the number of paths is ℓ = 1, the probability of the randomly occurring faults is α¯ = 0.6, and the
probability density functions of channel coeffificients are
q(β0) = 0.0005(e
9.89β0
ü 1), 0 ≤ β0 ≤ 1;
q(β1) =
(
10β1,
<
2.50(β1 ∞ 1),
0 ≤ β1 ≤ 0.20;
0.20 < β1 ≤ 1;
(24)
The mathematical expectations β¯ s can be calculated as 0.8991 and 0.4000, and the variances β ˜¯2
s (s = 0, 1)
are 0.0133 and 0.0467, respectively. By using the MATLAB LMI toolbox, for the augmented system (8),
we obtain the minimum disturbance attenuation level as γ∗ = 1.0094. The sub-optimal FD fifilter can then
be obtained as following:
L =
"
2.1427
-
1.0263#
; V =
=
0.0389.SUBMITTED 13
0 10 20 30 40 50 60 70 80 90 100
−0.12
−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
Time steps tk
R
e
s
id
u
a
l e
s
t
im
a
t
io
n
r
e
(
t
k
)
Fig. 2. Residual signal r(tk) for NMSs.
0 10 20 30 40 50 60 70 80 90 100
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Time steps tk
R
e
s
id
u
a
l e
v
a
lu
a
t
io
n
f
u
n
c
t
io
n
J
(
t
k
)
J(tk )
J th
Fig. 3. Evolution of residual evaluation function J(tk) for NMSs.
Letting the initial state of (1) be x(T0) = col{0.1,
;
0.1} and its estimation be xˆ(t0) = col{0.1, 0}. To
further illustrate the effectiveness of the designed fault detection fifilter, for tk = 0, 1, 2, · · · , 100, let the
fault signal and the disturbance input be given as
f(tk) =
(
0.1, 30 ≤ tk ≤ 50
0, else
, ω(tk) = e
E
0.01tk
sin(2tk).
The residual response r(tk) and evolution of residual evaluation function J(tk) =
nP
tk
h=tk0 r
T
(h)r(h)
o
1
2
for NMSs are shown in Figs. 2-3, respectively. After 200 runs of the simulations, we get an average value
of Jth = 0.0369. From Fig. 3, it can be shown that 0.0275 = J(29) < Jth < J(30) = 0.1090, which
means that the fault can be detected as soon as its occurrence.
We now examine the relationship between the disturbance attenuation level γ and the fault occurrence
probability α¯ as well as the multiple b of the sampling period h. It can be observed from Table I that theSUBMITTED 14
disturbance attenuation performance deteriorates with increased α¯ and b, which is in agreement with the
engineering practice.
TABLE I
THE PERMITTED MINIMUM γ∗.
α¯ = 0.2 ¯α = 0.4 ¯α = 0.7 ¯α = 0.9
b = 2 1.0008 1.0022 1.0035 1.0047
b = 3 1.0021 1.0057 1.0115 1.0145
b = 4 1.0109 1.0131 1.0145 1.0197
Example 2 As the special case of NMSs, in this example, an internet-based three-tank system is
introduced to illustrate the effectiveness of our proposed NSSs. With the variables defifined in [32], the
system model (1) and (2) with following parameters are adopted:
A =
0.9974 0 0.0026
0 0.9951 0.0024
0.0026 0.0024 0.9950
, B1 =
16.2190 0
0 16.2007
0.0212 0.0193
, B2 =
0.0212
0.0193
16.1997
, C =
"
1 0 0
0 1 0 #
.
where x(Tk) ∈ R
3
is the system state representing the liquid levels of the three tanks; similar to [32],
ω(Tk) ∈ R
2
is the disturbance used to model the unknown disturbance and input, f(Tk) ∈ R is the
fault signal reflflecting the leakages in tank 3, y(Tk) ∈ R
2
is the measurement output describing the height
measurements of tank 1 and tank 2. Here, we mainly investigate the internet-based fault detection problem,
the measurement signal will obtain through remote network, thus, due to the multi-path transmission and
shadowing problem, network-induced channel fading and randomly occurring fault usually take place,
then the actual received measurement signal through network is y¯(Tk) ∈ R
2
, which satisfifies (3).
Our aim here is to detect the faults by using the established mathematic model of the system (1) as
well as the measurement signals (2) through network in the presence of a leakage in tank 3. In order to
discuss simply the fault detection problem with fading measurement, we choose the fading parameters as
(24). Choosing the faults occurrence probability as α¯ = 0.6, similar to Example 1, by using Corollary 1,
the sub-optimal fault detection fifilter and the minimum H∞ attenuation level can be obtained as follows:
L =
ª
0.0042 0.0100
ª
0.0030 0.0069
ª
0.0095 0.0212
, V = 10
85
×
h
0.1185
5
0.2856
i
, γ∗ = 1.0023.
The initial value of (21) is chosen as e¯(T0) = col{0.1,
;
0.1, 0, 0.2, 0,
;
0.6, 0, 0.3, 0}, for Tk =
0, 1, 2, · · · , 100, the fault signal and exogenous disturbance input signal are set as
f(Tk) =
(
0.5, 30 ≤ Tk ≤ 50
0, else
, ω(Tk) =
"
e
Æ
0.02Tk
sin(0.2Tk)
e
ö
0.01Tk
cos(0.1Tk) #
.
The residual response r(Tk) and evolution of residual evaluation function J(Tk) =
nP
Tk
h=T0 r
T
(h)r(h)
o
1
2
for NSSs are shown in Figs. 4-5, respectively. After 200 runs of the simulations, we get an average value
of Jth = 1.4582 × 10
84
. From Fig. 5, it can be shown that 1.3526 × 10
ü4
= J(41) < Jth < J(42) =
1.6304 × 10
84
, which means that the fault can be detected within 11 time steps after the fault occurred
at Tk = 30.SUBMITTED 15
0 10 20 30 40 50 60 70 80 90 100
−6
−4
−2
0
2
4
6
8
10
x 10
−5
Time steps Tk
R
e
s
id
u
a
l e
s
t
im
a
t
io
n
r
e
(
t
k
)
Fig. 4. Residual signal r(Tk) for NSSs.
0 10 20 30 40 50 60 70 80 90 100
0
1
2
x 10
−4
Time steps Tk
R
e
s
id
u
a
l e
v
a
lu
a
t
io
n
f
u
n
c
t
io
n
J
(T
k
)
J(Tk )
J th
Fig. 5. Evolution of residual evaluation function J(Tk) for NSSs.
VI. CONCLUSION
In this paper, we have dealt with the fault detection problem for networked multi-rate systems with
randomly occurring faults and fading measurements. Different from the existing results of fault detection
for multi-rate sampled-data system, the delayed networked multi-rate systems is considered. By choosing
linear matrix inequality technique and convex optimization tool so that we can use Matlab LMI Toolbox
to design the fault detection fifilter effectively. Furthermore, as the special of NMSs, we also supply the
result of fault detection for NSSs with randomly occurring faults and fading measurements. Two examples
have been used to highlight the effectiveness of the proposed fault detection technology in this paper. It
would be interesting to deal with the following future research topics [33]–[39]: 1) investigation on theSUBMITTED 16
impact from quantization strategies and event-triggered communication mechanism; and 2) extension of
the techniques developed in this paper to more general time-varying and nonlinear systems.
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