Robust fault detection for switched positive linear systems with time varying delays

Research Article
Robust fault detection for switched positive linear systems
with time-varying delays
Mei Xiang, Zhengrong Xiang n
School of Automation, Nanjing University of Science and Technology, Nanjing 210094, People′s Republic of China
a r t i c l e i n f o
Article history:
Received 4 December 2012
Received in revised form
9 June 2013
Accepted 27 July 2013
Available online 14 September 2013
This paper was recommended for
publication by Dr. Q.-G. Wang.
Keywords:
Fault detection
Switched positive systems
Time-varying delays
Positive L1 filtering
Average dwell time
a b s t r a c t
This paper investigates the problem of robust fault detection for a class of switched positive linear
systems with time-varying delays. The fault detection filter is used as the residual generator, in which the
filter parameters are dependent on the system mode. Attention is focused on designing the positive filter
such that, for model uncertainties, unknown inputs and the control inputs, the error between the
residual and fault is minimized. The problem of robust fault detection is converted into a positive L1
filtering problem. Subsequently, by constructing an appropriate multiple co-positive type Lyapunov–
Krasovskii functional, as well as using the average dwell time approach, sufficient conditions for the
solvability of this problem are established in terms of linear matrix inequalities (LMIs). Two illustrative
examples are provided to show the effectiveness and applicability of the proposed results.
& 2013 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
A switched positive linear system (SPLS) is a type of hybrid
dynamical system that consists of a number of positive subsystems
[1,2] and a switching signal, which defines a specific positive
subsystem being activated during a certain interval. SPLS deserves
investigation both for practical applications and for theoretical
reasons. Especially, the positivity constraint is pervasive in engi-
neering practice as well as in chemical, biological and economic
modeling, etc. See for instance, communication networks [3], the
viral mutation dynamics [4], formation flying [5], and system
theory [6–10]. It should be pointed out that studying SPLS is more
challenging than that of general switched system or positive
system because, in order to obtain some results, one has to
combine the features of positive systems and switched systems.
Recently, some results on the stability analysis of SPLSs have been
obtained [11–13].
In practice, time-delay phenomena in dynamic systems widely
exist. Although many results have been reported for time-delay
systems [14–25], only recently has the SPLS with time delay
become a topic of major interest [26–28], which is theoretically
challenging and of fundamental importance to numerous
applications.
On the other hand, fault detection and isolation (FDI) in
dynamic systems has been an active field of research during the
past decades, and some model-based fault detection approaches
have been proposed in [29–38]. The basic idea of the model based
FDI is to use state observers or filters to generate a residual signal
and, based on this, to determine the residual evaluation function
compared with a predefined threshold. When the residual evalua-
tion function has a value larger than the threshold, an alarm is
generated. It is well known that unknown inputs, control inputs,
and model uncertainties are coupled in many industrial systems,
which are the sources of false alarms and can corrupt the
performance of the FDI system. This means that FDI systems have
to be sensitive to faults and simultaneously robust to the unknown
inputs and the model uncertainties. Therefore it is of great
significance to design a robust FDI system that provides both
sensitivity to faults and robustness to the unknown inputs and the
model uncertainties, that is, the robust FDI issue, see for example,
[39,40]. Recently, an H1-filtering formulation has been presented
to solve the robust FDI and robust fault detection filter (RFDF)
design problems for switched systems (see, e.g., [41–47] and the
references therein). However, to the best of our knowledge, the
RFDF problem of SPLSs has not been fully investigated. Moreover,
the method given in [41–47] cannot be applied to SPLSs, and this
constitutes the main motivation of the present study.
In this paper, we are interested in dealing with the problem of
RFDF by constructing an appropriate multiple co-positive type
Lyapunov–Krasovskii functional as well as applying the average
dwell time approach for SPLSs with time-varying delays. The main
contributions of this paper can be summarized as follows: (i) the
residual generator is constructed based on the filter, and the
design of RFDF is formulated as positive L1 filtering problem.
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ISA Transactions
0019-0578/$ - see front matter & 2013 ISA. Published by Elsevier Ltd. All rights reserved.
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n
Corresponding author. Tel.: +86 1395 1012 297; fax: +86 25 84313 809.
E-mail addresses: xiangzr@mail.njust.edu.cn, xiangzr@tom.com (Z. Xiang).
ISA Transactions 53 (2014) 10–16

The objective is to make the error between the fault and the
residual as small as possible, and increase robustness of the
residual to the unknown inputs and the model uncertainties; (ii)
by using the average dwell time approach, sufficient conditions for
the existence of such filter are established in terms of linear matrix
inequalities (LMIs). The parameterized matrices of this filter are
constructed by solving the corresponding LMIs; and (iii) two
simulation examples are presented to demonstrate the effective-
ness of the proposed methods.
The rest of this paper is organized as follows. In Section 2,
system formulation and some necessary lemmas are given. In
Section 3, a sufficient condition for the existence of L1-gain
performance for SPLS with time-varying delay is established. Then
based on the above result, the RFDF design problem is solved. Two
numerical examples are provided to illustrate the design results in
Section 4. Concluding remarks are given in Section 5.
1.1. Notations
In this paper, A⪰0ð⪯0Þ means that all entries of matrix A are
non-negative (non-positive); A≻0ð≺0Þ means that all entries of A
are positive (negative); A≻BðA⪰BÞ means that AÀB≻0ðAÀB⪰0Þ. AT
means the transpose of the matrix A; RðRþÞ is the set of all real
(positive real) numbers; Rn
ðRn
þÞ is the n-dimensional real (positive
real) vector space; RnÂk
is the set of all real matrices of
ðn Â kÞ-dimension. ‖x‖ ¼ ∑n
k ¼ 1jxkj, where xk is the k-th element
of xARn
. L1½t0; 1Þ is the space of absolute integrable vector-valued
functions on ½t0; 1Þ, i.e., we say z : ½t0; 1Þ-Rk
is in L1½t0; 1Þ if
R 1
t0
‖zðtÞ‖ dt o1.
2. Problem statements and preliminaries
Consider the following switched linear systems with time-
varying delays:
_xðtÞ ¼ AsðtÞxðtÞ þ AdsðtÞxðtÀdðtÞÞ þ BsðtÞuðtÞ þ EsðtÞwðtÞ þ GsðtÞf ðtÞ;
yðtÞ ¼ CsðtÞxðtÞ þ CdsðtÞxðtÀdðtÞÞ þ DsðtÞuðtÞ þ FsðtÞwðtÞ þ HsðtÞf ðtÞ;
xðt0 þ θÞ ¼ φðθÞ; θA Àh2; 0
Â Ã
;
8
><
>:
ð1Þ
where xðtÞARn
is the state, yðtÞARq
is the measured output; and
uðtÞARm
, wðtÞARp
, f ðtÞARr̄
are the control input, disturbance input
and the fault input, respectively, which belong to L1 ½0; 1Þ;
sðtÞ : ½0; 1Þ-N ¼ f1; 2; ⋯; Ng is the switching signal with N
being the number of subsystems; Αi, Αdi, Bi, Ei, Gi, Ci, Cdi, Di, Fi
and Hi, iAN, are constant matrices with appropriate dimensions;
φðθÞ is a vector-valued initial function defined on interval ½Àh2; 0Š,
h2 40; t0 ¼ 0 is the initial time, and tκ denotes the κ-th switching
instant; dðtÞ is assumed to be the interval time-varying delay
satisfying either of the following two cases:
0rh1 rdðtÞrh2; _dðtÞrτo1; ðC1Þ
0rh1 rdðtÞrh2 ðC2Þ
where h1 and h2 are positive scalars representing the upper and
the lower bounds of the time delay, respectively.
Remark 1. It is clear that (C2) contains (C1). The time-varying
delay dðtÞ is differentiable and bounded with a constant delay-
derivative bound in (C1), whereas it is continuous and bounded in
(C2). Then, in the sense, the criteria obtained under (C1) are less
conservative than those under (C2). However, if the information of
the derivation of time delay is unknown, only (C2) can be used to
deal with the situation.
Definition 1. System (1) is said to be positive if, for any initial
conditions φðθÞ⪰0; θA½Àh2; 0Š, uðtÞ⪰0, wðtÞ⪰0, f ðtÞ⪰0 and any
switching signals sðtÞ, the corresponding state trajectory xðtÞ⪰0
and output yðtÞ⪰0 hold for all tZt0.
Definition 2. [48] A is called a Metzler matrix, if the off-diagonal
entries of matrix A are non-negative.
The following lemma can be obtained from Lemma 3 in [26]
and Proposition 1 in [27].
Lemma 1. System (1) is positive if and only if Ai, iAN, are Metzler
matrices, Adi⪰0, Bi⪰0, Ei⪰0, Gi⪰0, Ci⪰0, Cdi⪰0, Di⪰0, Fi⪰0, and Hi⪰0,
iAN.
An FDI system consists of a residual generator and a residual
evaluation stage including an evaluation function and a threshold.
For the purpose of residual generation, the following fault detec-
tion filter is constructed as a residual generator
_^xðtÞ ¼ Af sðtÞ ^xðtÞ þ Bf sðtÞyðtÞ;
rðtÞ ¼ Cf sðtÞ ^xðtÞ þ Df sðtÞyðtÞ;
(
ð2Þ
where ^xðtÞARn
and rðtÞARl
are the state and the residual, respec-
tively. Αf i, Bf i, Cf i and Df i, iAN, are the parameterized filter
matrices to be determined.
Remark 2. The residual generator is a positive switched system,
and the parameters of filter (2) depend on system modes. In our
paper, the filter (2) is assumed to be switched synchronously with
the switching signal in system (1). This means that the switching
signal in filter (2) is the same as that in system (1).
For the purpose of fault detection, it is not necessary to
estimate the fault f ðtÞ. Similar to [40,41], a suitable weighting
matrix Qf ðsÞ was introduced to limit the frequency interval, in
which the fault should be identified, and the system performance
could be improved. Considering that the FDI problem is a special
case of RFDF with Qf ðsÞ ¼ I, our attentions will be only focused on
the RFDF problem.
One minimal realization of ^f ðsÞ ¼ Qf ðsÞf ðsÞ is supposed to be
_xðtÞ ¼ AQ xðtÞ þ BQ f ðtÞ;
^f ðtÞ ¼ CQ xðtÞ þ DQ f ðtÞ;
8
<
:
ð3Þ
where xðtÞARnf
is the state of the weighted fault, f ðtÞ is the original
fault and ^f ðtÞARr̄
is the weighted fault. ΑQ , BQ , CQ , DQ are assumed
to be known real constant matrices with appropriate dimensions.
Denoting eðtÞ ¼ rðtÞÀ^f ðtÞ, and augmenting the model of system
(1) to include the states of (2) and (3), we can obtain the
augmented system as follows
_~xðtÞ ¼ ~AsðtÞ ~xðtÞ þ ~AdsðtÞ ~xðtÀdðtÞÞ þ ~EsðtÞ ~wðtÞ;
eðtÞ ¼ ~CsðtÞ ~xðtÞ þ ~CdsðtÞ ~xðtÀdðtÞÞ þ ~FsðtÞ ~wðtÞ;
8
<
:
ð4Þ
where
~xðtÞ ¼
xðtÞ
^xðtÞ
xðtÞ
2
6
4
3
7
5; ~wðtÞ ¼
uðtÞ
wðtÞ
f ðtÞ
2
6
4
3
7
5; ~Ai ¼
Ai 0 0
Bf iCi Af i 0
0 0 AQ
2
6
4
3
7
5; ~Adi ¼
Adi 0 0
Bf iCdi 0 0
0 0 0
2
6
4
3
7
5;
~Ei ¼
Bi Ei Gi
Bf iDi Bf iFi Bf iHi
0 0 BQ
2
6
4
3
7
5; ~Ci ¼ Df iCi Cf i ÀCQ
h i
; ~Cdi ¼ Df iCdi 0 0
h i
~Fi ¼ Df iDi Df iFi Df iHiÀDQ
h i
; iAN:
Definition 3. [28] System (4) is said to be exponentially stable
under switching signal sðtÞ, if there exist constants αZ1 and β40,
such that the solution ~xðtÞ of system (4) satisfies ‖~xðtÞ‖r
α‖~xðt0Þ‖CeÀβðtÀt0Þ
, 8tZt0, where ‖~xðt0Þ‖C ¼ supt0Àh2 rδ r t0
‖~xðδÞ‖
È É
.
M. Xiang, Z. Xiang / ISA Transactions 53 (2014) 10–16 11

Definition 4. [49] For any switching signal sðtÞ and any
T2 4T1 Z0, let NsðT1; T2Þ denote the switching number of sðtÞ
over the interval ½T1; T2Þ. For given Ta 40 and N0 Z0, if the
inequality NsðT1; T2ÞrN0 þ ðT2ÀT1Þ=Ta holds, then the positive
constant Ta is called an average dwell time and N0 is called a
chattering bound.
Remark 3. In our paper, we do not assume that N0 ¼ 0. This
assumption corresponds to that sðtÞ cannot switch at all on any
interval of length smaller than Ta, which degenerates into dwell
time case. In general, as pointed in [50], if we discard the first N0
switches, the average time between consecutive switches is at
least Ta. From this point of view, adopting N0 Z0 is more general
and natural than N0 ¼ 0.
Definition 5. [28] For λ40 and γ40, system (4) is said to have
L1-gain performance under switching signal sðtÞ, if the following
conditions are satisfied:
(a) System (4) is exponentially stable when ~wðtÞ ¼ 0;
(b) under zero initial condition, i.e., φðθÞ ¼ 0; θA½Àh2; 0Š, it
holds that
Z 1
t0
eÀλðtÀt0Þ
‖eðtÞ‖dt rγ
Z 1
t0
‖ ~wðtÞ‖dt; ~wðtÞa0:
Now, the RFDF design problem to be addressed in this paper
can be transformed into an L1 filtering problem. What we need to
do here is to develop the fault detection filter (2) for positive
switched delay system (1) such that the resulting augmented
system (4) is exponentially stable under the switching signals with
average dwell time when ~wðtÞ ¼ 0, and under zero-initial condi-
tion, the infimum of γ is made small in the feasibility of
sup
~wðtÞa 0; ~wðtÞ AL1½0;1Þ
R1
0 eÀλt
‖eðtÞ‖dt
R 1
0 ‖ ~wðtÞ‖dt
oγ; γ40 ð5Þ
Remark 4. It is noted from (4) and (5) that the residual rðtÞ
generated by filter (2) provides an estimate of the weighted fault
^f ðtÞ. The stable weighting matrix Qf ðsÞ is given. Thus, detection and
isolation of the fault f ðtÞ can be achieved by examining the values
of the residual rðtÞ. That is, the designed filter not only detects the
fault, but also can isolate it. This method to detect the fault has
been developed in [41–47], but it has not been extended to
positive switched systems.
After designing the residual generator, the remaining impor-
tant task is to evaluate the generated residual. One of the widely
adopted approaches is to select a threshold and a residual
evaluation function. In this paper, the residual evaluation function
is chosen as
JrðTÞ ¼
Z T
0
eÀλτ
‖rðτÞ‖dτ ð6Þ
where T is the evaluation time window.
Once the evaluation function has been selected, we are able
to determine the threshold. It is reasonable to choose the thresh-
old as
Jth ¼ sup
w A L1½0;1Þ;uAL1½0;1Þ;f ¼ 0
JrðTÞ ð7Þ
Based on this, the faults can be detected by using the following
logical relationships
JrðTÞ4Jth ) With Faults ) Alarm ð8Þ
JrðTÞrJth ) No Faults ð9Þ
3. Main results
In this section, sufficient conditions for the existence of RFDF
can be given. To obtain the main result, we firstly consider the
exponential stability of the following positive switched system
_~xðtÞ ¼ ~AsðtÞ ~xðtÞ þ ~AdsðtÞ ~xðtÀdðtÞÞ ð10Þ
where ~xðtÞARM
þ , M ¼ 2n þ nf ; ~Ai, iAN, are Metzler constant
matrices, and ~Adi⪰0, iAN, are constant matrices; dðtÞ satisfies
(C1) or (C2).
Lemma 2. [28] Given a positive constant λ, if there exist
vi; υi,ϑi ARM
þ and ςi ARM
, iAN, such that
Ψi ¼ diag ψi1; ψi2; ⋯; ψiM; ψ′i1; ψ′i2; ⋯; ψ′iM
È É
⪯0; ð11Þ
Πij ¼ diagfπij1; πij2; ⋯; πijM; π′ij1; π′ij2; ⋯; π′ijMg⪯0; 8ði; jÞAN Â N;
ð12Þ
hold, then under the following average dwell time scheme
Ta 4Tn
a ¼ ln μ=λ; ð13Þ
with μZ1 satisfies
vi⪯ μvj; υi⪯ μυj; ϑi⪯ μϑj; 8ði; jÞAN Â N ð14Þ
system (10) with (C1) is exponentially stable, and the solution of
the system satisfies
‖~xðtÞ‖r
ðε2 þ h2ε3 þ h
2
2ε4Þ
ε1
eÀðλÀln μ
Ta
ÞðtÀt0Þ
‖~xðt0Þ‖C ð15Þ
where ε1 ¼ min
ði;rÞA NÂM
virf g, ε2 ¼ max
ði;rÞA NÂM
virf g, ε3 ¼ max
ði;rÞ ANÂM
virf g,
ε4 ¼ max
ði;rÞ ANÂM
ϑirf g,
ψir ¼ ~a
T
irvi þ λvir þ υir þ h2ϑir þ ςir; ψ′ir ¼ ~a
T
dirviÀð1ÀτÞeÀλh2
υirÀςir;
πijr ¼ À~a
T
jrςiÀeÀλh2
ϑir; π′ijr ¼ À~a
T
djrςi; rAM ¼ 1; 2; ⋯; Mf g;
~airð~adirÞ represents the r-th column vector of matrix ~Aið ~AdiÞ;
vi ¼ vi1; vi2; ⋯; viM½ ŠT
, υi ¼ υi1; υi2; ⋯; υiM½ ŠT
, ϑi ¼ ϑi1; ϑi2; ⋯; ϑiM½ ŠT
,
ςi ¼ ςi1; ςi2; ⋯; ςiM½ ŠT
.
Lemma 3. Given a positive constant λ, if there exist vi, ϑi ARM
þ and
ςi ARM
, iAN, such that
Ψi ¼ diag ψi1; ψi2; ⋯; ψiM; ψ′i1; ψ′i2; ⋯; ψ′iM
È É
⪯0 ð16Þ
Πij ¼ diagfπij1; πij2; ⋯; πijM; π′ij1; π′ij2; ⋯; π′ijMg⪯0; 8ði; jÞAN Â N;
ð17Þ
hold, then system (10) with (C2) is exponentially stable
for any switching signal sðtÞ with average dwell time satisfy-
ing (13), and the solution of the system satisfies
‖~xðtÞ‖r
ðε2þh
2
2ε4Þ
ε1
eÀðλÀ1nm
Ta
ÞðtÀt0Þ
‖~xðt0Þ‖, where ψir ¼ ~a
T
irviþ
λvir þ h2ϑir þ ςir, ψ′ir ¼ ~a
T
dirviÀςir, πijr and π′ijr are defined in
Lemma 2.
Proof. By removing the terms containing υi in Lemma 2, we can
easily get Lemma 3, and the detailed proof is omitted here.
3.1. L1-gain performance analysis
Based on Lemma 2, the following theorem presents sufficient
conditions for the existence of L1-gain performance for system
(4) under (C1).
M. Xiang, Z. Xiang / ISA Transactions 53 (2014) 10–1612

Theorem 1. Given positive constants λ and γ, if there exist
vi; υi,ϑi ARM
þ and ςi ARM
, iAN, such that, for 8ði; jÞAN Â N,
Χi ¼ diag χi1; χi2; ⋯; χiM; χ′i1; χ′i2; ⋯; χ′iM; χ″i1; χ″i2; ⋯; χ″iM1
È É
⪯0 ð18Þ
Θij ¼ diag θij1; θij2; ⋯; θijM; θ′ij1; θ′ij2; ⋯; θ′ijM; θ″ij1; θ″ij2; ⋯; θ″
ijM1
n o
⪯0
ð19Þ
then system (4) under (C1) is exponentially stable with the L1-gain
performance for any switching signal sðtÞ satisfying the average
dwell time (13), where
χir ¼ ~a
T
irvi þ λvir þ υir þ h2ϑir þ ςir þ ‖~cir‖; χ′ir ¼ ~a
T
dirviÀð1ÀτÞeÀλh2
υirÀςir þ ‖~cdir‖;
χ″ir1
¼ ~e
T
ir1
vi þ ‖~f ir1
‖Àγ; θijr ¼ À~a
T
jrςiÀeÀλh2
ϑir; θ′ijr ¼ À~a
T
djrςi; θ″ijr1
¼ À~e
T
jr1
ςi; rAM;
r1 Af1; 2; ⋯; M1g; M1 ¼ p þ m þ r
À
:
~airð~adir; ~cir; ~cdirÞ represents the r1-th column vector of matrix
~Að ~Adi; ~Ci; ~CdiÞ, and ~eir1
ð~f ir1
Þ represents the r-th column vector of
matrix ~Eið~FiÞ.
Proof. Choose the multiple co-positive type Lyapunov–Krasovskii
functional for system (4) as follows
Viðt; ~xðtÞÞ ¼ Vi1ðt; ~xðtÞÞ þ Vi2ðt; ~xðtÞÞ þ Vi3ðt; ~xðtÞÞ ð20Þ
where
Vi1ðt; ~xðtÞÞ ¼ ~xT
ðtÞvi
Vi2ðt; ~xðtÞÞ ¼
Z t
tÀdðtÞ
eλðsÀtÞ ~xT
ðsÞυids
Vi3ðt; ~xðtÞÞ ¼
Z 0
Àh2
Z t
tþθ
eλðsÀtÞ ~xT
ðsÞϑi ds dθ
and vi; υi,ϑi ARM
þ , iAN, λ40.
For the sake of simplicity, Viðt; ~xðtÞÞ is written as ViðtÞ in
this paper.
By Lemma 2, the exponential stability of system (4) with
~wðtÞ ¼ 0 is ensured if (18)-(19) hold. To show the L1-gain perfor-
mance, for any T 40, denote t1; t2; ⋯; tκ; tκþ1; ⋯; tNsðt0;TÞ the switch-
ing instants on the interval ½t0; TÞ. For any tA½tκ; tκþ1Þ, we have
VsðtÞðtÞreÀλðtÀtκ Þ
VsðtκÞðtκÞÀ
Z t
tκ
eÀλðtÀsÞ
ΛðsÞ ds ð21Þ
where ΛðsÞ ¼ ‖eðsÞ‖Àγ‖ ~wðsÞ‖.
From (14), one obtains
Vsðtκ ÞðtκÞrμVsðtκÀ1ÞðtÀ
κ Þ; κ ¼ 1; 2; ⋯; Nsðt0;TÞÀ1; ð22Þ
Combining (21) and (22) leads to
VsðTÞðTÞrμeÀλðTÀtNsðt0;TÞÞ
VsðtÀ
Nsðt0;TÞ
Þðt-
Nsðt0;TÞÞÀ
Z T
tNsðt0;TÞ
eÀλðTÀsÞ
ΛðsÞ ds
r⋯
rμNsðt0;TÞ
Vsðt0Þðt0ÞeÀλðTÀt0Þ
ÀμNsðt0;TÞ
Z t1
t0
eÀλðTÀsÞ
ΛðsÞ ds
ÀμNsðt0;TÞÀ1
Z t2
t1
eÀλðTÀsÞ
ΛðsÞdsÀ⋯À
Z T
tNsðt0;TÞ
eÀλðTÀsÞ
ΛðsÞ ds
¼ eÀλðTÀt0ÞþNsðt0;TÞln μ
Vsðt0Þðt0ÞÀ
Z T
t0
eÀλðTÀsÞþNsðs;TÞln μ
ΛðsÞ ds ð23Þ
Under zero initial condition, it can be obtained from (23) that
0rÀ
Z T
t0
ΛðsÞ ds ð24Þ
Considering that t0 ¼ 0, it follows from (24) that
Z T
0
ΛðsÞ dsr0 ð25Þ
Multiplying both sides of (25) by eÀNsð0;TÞln μ
yields
Z T
0
eÀλðTÀsÞÀNsð0;sÞln μ
‖eðsÞ‖ dsrγ
Z T
0
eÀλðTÀsÞÀNsð0;sÞln μ
‖ ~wðsÞ‖ ds
Then following the proof line of Theorem 2 presented in [28], we
obtain from (13) that system (4) satisfies (5).
This completes the proof.
3.2. Design of the RFDF
Now we are in a position to present a solution to the RFDF
design problem.
Denote
Ki1 ¼ AT
f i
viðnþ1Þ
⋮
viðnþnÞ
2
6
4
3
7
5; Ki2 ¼ BT
f i
viðnþ1Þ
⋮
viðnþnÞ
2
6
4
3
7
5; Ki3 ¼ Cf i; Ki4 ¼ Df i; ð26Þ
then based on Theorem 1, a sufficient condition for the existence
of RFDF of form (2) is presented in the following theorem.
Theorem 2. Consider system (1) with (C1), for given positive
constants λ and γ, the RFDF design problem is solvable if there
exist vi; υi,ϑi ARM
þ , ςi ARM
, Ki1 ARn
, Ki2 ARm
, and matrices Ki3 ARlÂn
,
Ki4 ARlÂq
, iAN, such that (13), (14) (18) and (19) hold. Moreover, if
(18)–(19) have a feasible solution, the RFDF parameter matrices
can be constructed by (26).
Proof. By Theorem 1, system (4) is exponentially stable and
satisfies (5) for any switching signals sðtÞ with the average dwell
time satisfying (13) if (18)–(19) hold.
We partition vi, υi, ϑi and ςi as
vi ¼ vi1; vi2; ⋯; vin; viðnþ1Þ; ⋯; við2nÞ; við2nþ1Þ; ⋯; viM
Â ÃT
;
υi ¼ υi1; υi2; ⋯; υin; υiðnþ1Þ; ⋯; υið2nÞ; υið2nþ1Þ; ⋯; υiM
Â ÃT
;
ϑi ¼ ϑi1; ϑi2; ⋯; ϑin; ϑiðnþ1Þ; ⋯; ϑið2nÞ; ϑið2nþ1Þ; ⋯; ϑiM
Â ÃT
;
ςi ¼ ςi1; ςi2; ⋯; ςin; ςiðnþ1Þ; ⋯; ςið2nÞ; ςið2nþ1Þ; ⋯; ςiM
Â ÃT
Substituting (26) into (18) and (19), it can be obtained that the
claim of the theorem is true.
The proof is completed.
Remark 5. It should be noted that an H1-filtering formulation has
been presented in [41–47] to solve the robust FDI and RFDF design
problems for non-positive switched systems, but these proposed
methods cannot be applied to positive switched systems. In
Theorem 2, an L1 filtering formulation, which is different from
the H1-filtering one, is proposed to solve the RFDF design problem
of positive switched systems, and this is the major contribution of
the paper.
Remark 6. When μ ¼ 1 in (14), which leads to vi ¼ vj; υi ¼ υj;
ϑi ¼ ϑj; 8ði; jÞAN Â N and Tn
a ¼ 0, the augmented system (4) pos-
sesses a common Lyapunov functional and the switching signals
can be arbitrary. This implies that the problem of RFDF design is
solvable under the arbitrary switching signals.
Remark 7. In the above criterion, the derivative of the interval
time-varying delay is known, that is the system satisfies (C1).
Unfortunately, in many engineering applications, it is difficult to
obtain the information, and only the case (C2) is satisfied. For (C2),
Theorem 2 is still applicable provided that the terms containing υi
in Theorem 2 are removed. Next we will extend Theorem 2 to the
case (C2) as follows.

Corollary 1. Consider system (1) with (C2), for given positive
constants λ and γ, the RFDF problem is solvable if there exist
vi,ϑi ARM
þ , ςi ARM
, Ki1 ARn
, Ki2 ARm
, and matrices Ki3 ARlÂn
,
Ki4 ARlÂq
, iAN, such that (13)–(14) and
Ξi ¼ diag ξi1; ξi2; ⋯; ξiM; ξ′i1; ξ′i2; ⋯; ξ′iM; ξ″i1; ξ″i2; ⋯; ξ″
iM1
n o
⪯0 ð27Þ
Θij ¼ diag θij1; θij2; ⋯; θijM; θ′ij1; θ′ij2; ⋯; θ′ijM; θ″ij1; θ″ij2; ⋯; θ″
ijM1
n o
⪯0
ð28Þ
where
ξir ¼ ~a
T
irvi þ λvir þ h2ϑir þ ςir þ ‖~cir‖; ξ′ir ¼ ~a
T
dirviÀςir
þ‖~cdir‖; ξ″
ir1
¼ ~eT
ir1
vi þ ‖~f ir1
‖Àγ;
and Θij are defined in Theorem 1. Moreover, if (27)–(28) have a
feasible solution, the RFDF parameter matrices can be constructed
by (26).
Proof. According to Lemma 3 and Theorem 2, the result can be
easily obtained.
4. Illustrative examples
In this section, two examples are presented to check the
validity of the proposed results.
Example 1. Consider system (1) with parameters as follows:
A1 ¼
À5 1
2 À4
!
; Ad1 ¼
0:1 0:2
0:1 0:2
!
; B1 ¼
0:3
0:4
!
; E1 ¼
0:2
0:1
!
;
G1 ¼
0:6
0:5
!
; C1 ¼ 0:2 0:1
Â Ã
; Cd1 ¼ 0:2 0:1
Â Ã
; D1 ¼ 0:1; F1 ¼ 0:4; H1 ¼ 0:6:
A2 ¼
À7 2
3 À6
!
; Ad2 ¼
0:1 0:1
0:2 0:1
!
; B2 ¼
0:2
0:1
!
; E2 ¼
0:5
0:3
!
; G2 ¼
0:3
0:2
!
C2 ¼ 0:3 0:2
Â Ã
; Cd2 ¼ 0:1 0:2
Â Ã
; D2 ¼ 0:5; F2 ¼ 0:2; H2 ¼ 0:2;
h1 ¼ 0:2; h2 ¼ 0:4; τ ¼ 0:4:
By Lemma 1, the trajectories of such system remain positive if
φðtÞ⪰0, tA Àh2; 0
Â Ã
. Our purpose here is to develop the fault
detection filter (2) for positive switched delay system (1) such
that the resulting augmented system (4) is exponentially stable
with the L1-gain performance.
In this paper, the weighted matrix of the fault is supposed to be
Qf ðsÞ ¼ 5=ðs þ 5Þ with the minimal realization: AQ ¼ À5, BQ ¼ 5,
CQ ¼ 1 and DQ ¼ 0. In this example, the disturbance is
wðtÞ ¼ 0:1eÀ0:04t
cos ð0:3πtÞ, the control input uðtÞ is the unit step
function, and the fault signal f ðtÞ is set up as
f ðtÞ ¼
0:4t; 3rtr10
0; others
(
Take λ ¼ 0:8 and γ ¼ 1, then solving (18) in Theorem 2 gives
rise to
v1 ¼
0:5350
0:5375
0:5052
0:5003
0:0135
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
; v2 ¼
0:5116
0:5092
0:4838
0:4682
0:0730
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
; υ1 ¼
0:5369
0:5765
0:4958
0:4926
0:3983
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
; υ2 ¼
0:7653
0:7375
0:4814
0:4702
0:5151
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
;
ϑ1 ¼
0:6662
0:7401
0:7928
0:9127
0:7563
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
; ϑ2 ¼
0:7353
0:8384
0:8247
0:9548
0:8201
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
; ς1 ¼
0:0539
0:0655
0:2975
0:2923
0:0584
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
; ς2 ¼
0:0611
0:0735
0:2759
0:2615
0:0622
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
;
K11 ¼
À2:5416
À2:5639
!
; K21 ¼
À2:4455
À2:4209
!
; K12 ¼ 0:0476; K22 ¼ 0:1130;
K13 ¼ 0:5135 0:5069
Â Ã
; K14 ¼ 0:1724; K23 ¼ 0:4856 0:4664
Â Ã
; K24 ¼ 0:3288:
It is not hard to find that (19) is satisfied. By utilizing the LMI Toolbox,
it follows from (14) that μ ¼ 5:4223. From (26), the desired filter can
be obtained with the parameterized matrices as follows:
and Tn
a ¼ 2:1132 is derived by (13), this means that a desired filter
under the switching signals with average dwell time is developed
by the proposed method. The switching signal shown in Fig. 1 is
generated by choosing Ta ¼ 2:2. Fig. 2 depicts the fault signal f ðtÞ.
The generated residual rðtÞ is shown in Fig. 3. The threshold can be
determined as Jth ¼ 0:2174 for t ¼ 20 s. Fig. 4 shows the evolution
of residual evaluation function JrðtÞ, in which the solid line is fault-
free case, the dashed line is the case with fault f ðtÞ. The simulation
results show that JrðtÞ ¼ 0:220340:2174 when t ¼ 4:5 s, which
means that the fault f ðtÞ can be detected 1.5 s after its occurrence.
0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
Time(s)
Systemmode
Fig. 1. Switching signal for (C1).
0 2 4 6 8 10 12 14 16 18 20
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Time(s)
Faultsignalf(t)
Fig. 2. Fault signal for (C1).

Example 2. As for (C2), consider system (1) with the parameters
which are the same as those in Example 1. Take λ ¼ 0:8 and γ ¼ 1,
then solving (27) in Corollary 1 gives rise to
v1 ¼
0:5420
0:4978
0:3680
0:2702
0:0378
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
; v2 ¼
0:8151
1:0154
0:4659
0:6346
0:0445
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
; ϑ1 ¼
2:1023
1:6523
3:0730
2:5621
1:8244
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
;
ϑ2 ¼
1:8225
5:7836
2:7172
8:0219
1:8510
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
; ς1 ¼
0:2919
0:2794
0:3039
0:2238
0:1678
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
; ς2 ¼
0:4683
0:7668
0:3884
0:5310
0:1582
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
K11 ¼
À2:4352
À1:9124
!
; K21 ¼
À2:6248
À5:3095
!
; K12 ¼ 0:2539; K22 ¼ 0:8469;
K13 ¼ 0:3039 0:2238
Â Ã
; K14 ¼ 0:0559; K23 ¼ 0:3884 0:5310
Â Ã
; K24 ¼ 0:1306:
It is not hard to find that (28) is satisfied. By utilizing the LMI
Toolbox, it follows from (14) that μ ¼ 3:5003. From (26), the
desired filter can be obtained with the parameterized matrices
as follows
0 2 4 6 8 10 12 14 16 18 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Time(s)
Residualsignalr(t)
Fig. 3. Residual signal for (C1).
0 2 4 6 8 10 12 14 16 18 20
0
0.05
0.1
0.15
0.2
0.25
Time(s)
Res.eve.functionJr(t)
Fault-free case
Fault case
Fig. 4. Evolution of residual evaluation function for (C1).
0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
Time(s)
Systemmode
Fig. 5. Switching signal for (C2).
0 2 4 6 8 10 12 14 16 18 20
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Time(s)
Residualsignalr(t)
Fig. 6. Residual signal for (C2).
0 2 4 6 8 10 12 14 16 18 20
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Time(s)
Res.eva.functionJr(t)
Fault-free case
Fault case
Fig. 7. Evolution of residual evaluation function for (C2).

and Tn
a ¼ 1:5661 is given by (13), this means that a desired filter
under the switching signals with average dwell time is developed
by the proposed method. The switching signal depicted in Fig. 5 is
generated by choosing Ta ¼ 1:6. wðtÞ, f ðtÞ and uðtÞ are the same as
those in Example 1. The generated residual rðtÞ is shown in Fig. 6.
The threshold can be determined as Jth ¼ 0:1004 for t ¼ 20 s. Fig. 7
shows the evolution of residual evaluation function JrðtÞ, in which
the solid line is fault-free case, the dashed line is the case with the
fault f ðtÞ. The simulation results show that JrðtÞ ¼ 0:101540:1004
when t ¼ 5 s, which means that the fault f(t) can be detected 2 s
after its occurrence.
5. Conclusions
In this paper, we have studied the robust fault detection
problem for a class of switched positive linear systems with
time-varying delays. By converting the problem of robust fault
detection into positive L1 filtering problem, sufficient conditions
for the existence of such filter are established, and the desired
filter matrices can be constructed easily through the solution of
LMIs. Finally, two illustrative examples are provided to show the
effectiveness and applicability of the proposed results. Our future
work will focus on the design of robust fault detection for discrete
time switched positive systems.
Acknowledgment
This work was supported by the National Natural Science
Foundation of China under Grant no. 61273120.
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