Abstract This paper solves the problem of fault detection for discrete-time hybrid sys-tems via a descriptor system method. These systems include control inputs, unknownbounded disturbances, and sensor faults. Based on the descriptor system method, afault detection observer is designed to obtain the estimation of the states, outputs andfaults, simultaneously. By means of the observer gain, the state error system is stochas-tically stable, and the disturbances effect on the residuals meets the H∞ performanceindex. The satisfactory sensitivity problem of the residuals to the faults is solved.A numerical example illustrates the effectiveness and applicability of the developedmethod.
Keywords Descriptor system method · Fault detection · Fault detection observerdesign · Discrete-time hybrid system · H∞ control
X. Zhu · Y. Xia (B) · M. WangSchool of Automation, Key Laboratory of Intelligent Control and Decision of Complex Systems,Beijing Institute of Technology, Beijing 100081, Chinae-mail: [email protected]
S. MaInstitute of Ordnance Technology, Ordnance Engineering College, Shijiazhuang 050000, Chinae-mail: [email protected]
Circuits Syst Signal Process
1 Introduction
It is well known that in modern engineering systems, several kinds of dynamic behav-ior in different parts of the system are shown, such as continuous dynamics, discretedynamics, jump phenomena, and logic commands. Hybrid systems model appropri-ately modern engineering systems. Among this kind of systems, jumping linear sys-tems have been a great practical important subject and attracted a lot of interest. Injumping linear systems, the dynamics of the discrete and continuous states are mod-eled, respectively, by a finite state Markov chain and linear difference equations sub-ject to the discrete chain. Markov jump linear system is by now a well-known class ofsystems with a wide potential of applicability, which includes applications in safety-critical and high-integrity systems. Recently, many results have been published onMarkov jump linear systems. For jump Markovian systems, stability analysis [15,32],adaptive backstepping controller design [28], reliable H∞ guaranteed cost controllerdesign [20], comparison principle and stability [17], robust state estimation, and faultdiagnosis [23] have been reported, respectively. The control problem for discrete sin-gular hybrid systems [30] has solved. H∞ filtering for nonlinear singular Markovianjumping systems has considered in [29].
Descriptor systems are a natural mathematical representation for many practicalsystems, because they provide a description of the dynamic as well as the algebraicrelationships between the chosen descriptor variables simultaneously. Descriptor sys-tems are also referred to as singular systems, implicit systems, generalized state-spacesystems, differential-algebraic systems, or semi-state systems. They exhibit a largerclass of systems than the normal linear system model and have extensive applications,for example, electrical circuits, power systems, economics, and other fields. In recentyears, many results on descriptor systems have been published. There are many meth-ods which dealt with the problem of singular control systems in [6]. The problemsof observability analysis and sensor location study [3], observer design [24], optimalfiltering [21], and output feedback controller [1] for descriptor systems, respectively,have been presented. Singular hybrid systems have been considered. Sliding modecontrol problem of singular stochastic hybrid systems [25] has been studied. The sta-bility problem of continuous singular hybrid systems [27,38] is considered. H∞ filterfor singular Markovian jump systems has been considered in [26].
Fault detection and isolation algorithm and their application in a wide range ofindustrial processes have been intensive research in the past three decades. Someresults on fault detection have been published. Among these model-based approaches,a residual signal is built and compared with a predefined threshold by means of thestate observer or filter. When the residual evaluation function has a value larger thanthe threshold, an alarm is produced. A source of the alarm comes from many industrialsystems’ unknown inputs, uncertainties, fault, and so on. Fault detection systems aresensitive to faults and robust to disturbances. There are also some recently publishedpapers on fault detection filter in the literature. The problems of fault detection for T-Sfuzzy discrete systems [31] in finite frequency domain, interior permanent-magnetmotor drive system for electric vehicle [14], linear delta operator system [33], andlinear systems [13,18] over networks are solved, respectively. A method to deal withfault detection of Markovian jump systems with sensor saturations and randomly
Circuits Syst Signal Process
varying nonlinearities has been given in [8]. Fuzzy model-based robust fault detectionwith stochastic mixed time-delays and successive packet dropouts has been publishedin [9]. Fault detection [5,12] for linear singular systems has been studied. For discreteMarkovian jump systems in [19,22,39], detection filter is designed. Fault detectionfilter for Markovian jump singular systems with intermittent measurements [37] hasbeen designed. Fault detection for discrete-time switched singular time-delay systemshas been studied in [16]. Based on a descriptor approach, robust observer has designedto deal with the problem of fault detection Takagi-Sugeno systems [2]. The problemsof fault diagnosis or fault-tolerant control are proposed in [34–36]. To the best of theauthors’ knowledge, the problem of fault detection for discrete-time hybrid systemsvia a descriptor system method has not been fully investigated yet. This motivates us tosolve this interesting and challenging problem, which has great potential in practicalapplications.
This paper solves the problem of fault detection for discrete-time hybrid systems.The main contributions of this paper are: (I) two performance indexes are introducedto measure the disturbance robustness and the fault sensitivity for control inputs,unknown bounded disturbances, and sensor faults. (II) Based on the descriptor systemmethod, this paper’s aim is that a fault detection observer to obtain the estimation ofthe states, outputs, and faults, simultaneously. The fault detection observer not onlyguarantees that the state error system is stochastically stable but also that the distur-bance effect on the residuals satisfies the prescribed H∞ performance. The achievingsatisfactory sensitivity problem of the residuals to faults is solved. (III) A numericalexample demonstrates the effectiveness and applicability of the developed theoreticalresults.
Notation: The notation used throughout the paper is standard. The superscript Tstands for matrix transposition; R
n denotes the n-dimensional Euclidean space; Irepresents the unit matrix; diag{·} stands for a block-diagonal matrix; ‖ · ‖ denotesthe Euclidean norm of a vector and its induced norm of a matrix; and the signals thatare square integrable over [0,∞) are denoted by L2[0,∞) with the norm ‖ · ‖; E{·}denotes the expectation operator with respect to probability measure P; ∗ representsa term that is induced by symmetry.
2 System Description and Problem Formulation
In this paper, suppose that the system mode switching is governed by rk . The discrete-time hybrid systems can be described by the following equation:
xk+1 = A1rk xk + B1rk uk + C1rk ωk,
yk = A2rk xk + C2rk ωk + D2rk fk, (1)
where xk ∈ Rn is the state, yk ∈ R
r is the measured output. fk ∈ Rr denotes the
sensor fault. uk ∈ Rs denotes the control input. ωk ∈ R
p denotes the unknownbounded disturbance. uk , fk , and ωk belong to L2[0,∞). A1rk , B1rk , C1rk , A2rk ,and C2rk are known matrices with appropriate dimensions, and D2rk is assumed fullcolumn rank. rk, k ∈ Z is a time homogeneous Markov chain taking values in a finite
Circuits Syst Signal Process
set S = {1, 2, . . . , S}, with stationary transition probability matrix � = [πi j ]i, j∈S
with πi j = Pr{rk+1 = j |rk = i} ≥ 0, for i, j ∈ S and∑S
j=1 πi j = 1.Define
hk = D2i fk, xk =[
xk
hk
]
, (2)
then we can have the following system:
E xk+1 = A1i xk + B1i uk + C1iωk + D1i hk,
yk = A2i xk + C2iωk = A2i xk + hk + C2iωk, (3)
where
E =[
I 00 0
]
, A1i =[
A1i 00 −I
]
, B1i =[
B1i
0
]
, C1i =[
C1i
0
]
,
D1i =[
0I
]
, A2i = [A2i I
], A2i = [
A2i 0].
We design a fault detection observer to obtain the estimation of states xk , outputs yk ,and faults fk , simultaneously.
For system (3), consider the following fault detection observer
ˆxk = ˜xk + Li A2i xk,
Ei ˜xk+1 = Fi ˜xk + B1i uk,
ˆyk = A2i ˆxk = A2i xk,
εk = yk − ˆyk,
(4)
where ˜xk is the auxiliary state, ˆxk = [x T
k hTk
]Tis the state estimation of states xk , and
εk denotes residual signal that carries information on time and location of the occur-rence of the faults, respectively. Matrices Ei , Fi , and Li are the observer’s parametersto be obtained. Note that the estimation of the faults fk is (DT
2i D2i )−1 DT
2i hk .
Remark 1 In this paper, we design a full-order observer to solve the problem of faultdetection for discrete-time hybrid systems. However, there exists incomplete informa-tion in the filtering issues, such as [10,11]. The distributed filter problem of systemsover sensor networks with successive packet dropouts [10] has solved. The robust H∞filter problem of nonlinear networked systems with packet dropouts has provided in[11].
Define the state estimation error ek = xk − ˆxk , then we have
(E + Ei Li A2i )xk+1 − Ei ˆxk+1
= ( A1i + Fi Li A2i )xk − Fi ˆxk + (D1i + Fi Li )hk + C1iωk . (5)
Circuits Syst Signal Process
Set
E + Ei Li A2i = Ei , Fi = A1i + Fi , Li A2i , D1i + Fi Li = 0,
and then we can have
Fi =[
A1i 0−A2i −I
]
, Li =[
0I
]
, Ei =[
I + Ri A2i Ri
Qi A2i Qi
]
,
Ei xk+1 − Ei ˆxk+1 = Fi xk − Fi ˆxk + C1iωk .
(6)
Let
E−1i =
[I −Ri Q−1
i−A2i Q−1
i + A2i Ri Q−1i
]
,
E−1i C1i =
[C1i
−A2i C1i
]
, (7)
E−1i Fi =
[A1i + Ri Q−1
i A2i Ri Q−1i
−A2i A1i − (Q−1i + A2i Ri Q−1
i )A2i −(Q−1i + A2i Ri Q−1
i )
]
,
hence the following state error system is obtained
ek+1 = E−1i Fi ek + E−1
i C1iωk,
εk = yk − ˆyk = A2i ek + D2i fk + C2iωk, (8)
where Ri and Qi guarantee matrix Ei ’s nonsingularity.
Remark 2 We can convert the proposed fault detection observer design into matricesRi and Qi design such that the state error system is stochastically stable, residuals εk
are as robust as possible to disturbances ωk and as sensitive as possible to faults fk .
Based on the aforementioned analysis, the problem of fault detection for system (1)and observer (4) can be viewed as designing the appropriate fault detection observerto make the system stochastically stable, minimize the effect of the unknown boundeddisturbances, and enhance the effect of the sensor faults. In order to detect the sensorfaults, the widely adopted approach is to choose an appropriate threshold Jth andto determine the evaluation function JN (εk). That is, a threshold Jth and a residualevaluation function JN (εk) can be chosen
JN (εk) =√√√√ 1
N
k0+N∑
k=k0
εTk εk,
Jth = supuk∈L2,ωk∈L2, fk=0
JN (εk),
(9)
Circuits Syst Signal Process
where k0 denotes the initial evaluation time instant, and N is the evaluation timewindow and limited. It is shown that the computation of Jth involves the determinationof unknown bounded disturbances ωk and control inputs uk on residuals εk .
Based on this, the occurrence of the sensor faults can be detected via the followinglogic rule:
JN (εk) > Jth ⇒ Fault ⇒ Alarm,
JN (εk) ≤ Jth ⇒ NoAlarm. (10)
It is shown that the faults are detected by comparing the residual evolution functionand the predefined threshold. When the level of the residual evaluation function islarger than the threshold level, the alarm is generated, which shows that the faults aredetected.
The paper designs the fault detection observer (4) to get the minimum of γ and themaximum of β. The observer has the best robustness to disturbances ωk and the bestsensitivity to faults fk .
Now, we can present a useful definition.
Definition 1 Considering system (3), observer (4) is a H∞ fault detection observer ifthe following conditions hold
– For all i = 1, . . ., S, system (8) is stochastically stable when ωk = 0 and fk = 0.– For a given scalar γ > 0 and any nonzero disturbances ωk , the following inequality
holds
E
{ ∞∑
k=0
εTk εk
}
< γ 2∞∑
k=0
ωTk ωk, (11)
that is, ‖εk‖E2 < γ ‖ωk‖2, where
‖εk‖E2 = E
{ ∞∑
k=0
εTk εk
} 12
.
– For a given scalar β > 0 and any nonzero faults fk , the following inequality holds
E
{ ∞∑
k=0
εTk εk
}
> β2∞∑
k=0
f Tk fk, (12)
that is, ‖εk‖E2 > β‖ fk‖2.
Lemma 1 (Schur complement) [4] Given constant matrices �1, �2, and �3 withappropriate dimensions, where �T
1 = �1 and �T2 = �2 > 0, then
�1 + �T3 �−1
2 �3 < 0,
Circuits Syst Signal Process
if and only if
[�1 �T
3�3 −�2
]
< 0 or
[−�2 �3
�T3 �1
]
< 0.
3 Robustness Analysis
In no fault case, this paper presents that system (8) is stochastically stable, and the effectof disturbances ωk on residuals εk meets (11). Then gains Ri and Qi are designed,that is, the gains of observer (4) are determined. When fk = 0, system (8) is rewrittenas follows:
ek+1 = E−1i Fi ek + E−1
i C1iωk,
εk = A2i ek + C2iωk .(13)
Lemma 2 For all i = 1, . . ., S, system (13) is stochastically stable when ωk = 0 ifthere exist positive definite symmetric matrices Pi such that (14) holds.
(E−1i Fi )
T Pi (E−1i Fi ) − Pi < 0, (14)
where Pi = ∑Sj=1 πi j Pj .
Proof When ωk = 0, system (13) is rewritten
ek+1 = E−1i Fi ek,
εk = A2i ek .(15)
Construct the following Lyapunov functional
V (ek) = eTk Pi ek . (16)
By calculating the difference of V (k) along the trajectories (15), we can get the fol-lowing equality:
Vk = E{V (ek+1)} − V (ek)
= E
{eT
k+1 Pi ek+1
}− eT
k Pi ek
= eTk ((E−1
i Fi )T Pi (E−1
i Fi ) − Pi )ek . (17)
According to (14), we can obtain that
Vk = eTk ((E−1
i Fi )T Pi (E−1
i Fi ) − Pi )ek < 0
for ek = 0.Therefore, system (13) is stochastically stable when ωk = 0. That completes the
proof. �
Circuits Syst Signal Process
Theorem 1 For a given scalar γ > 0, system (13) is stochastically stable whenω(k) = 0 and satisfies (11) if there exist positive definite symmetric matrices Pi suchthat (18) holds.
⎡
⎢⎢⎣
−Pi 0 (E−1i Fi )
T AT2i∗ −γ 2 I (E−1C1i )
T CT2i
∗ ∗ −P−1i 0
∗ ∗ ∗ −I
⎤
⎥⎥⎦ < 0. (18)
Proof When ωk = 0, the form of system (13) is equivalent to system (15). Similar toLemma 2, it is easy to obtain that system (13) is stochastically stable when ωk = 0.
In the following, we can show that (11) is satisfied. Under the zero initial conditionand ωk = 0, define the following performance index:
J1 = E
{εT
k εk
}− γ 2ωT
k ωk + Vk, (19)
where γ > 0 is the prescribed attenuation level. Then we can have
J1 = E
{εT
k εk
}− γ 2ωT
k ωk + Vk
= E
{εT
k εk
}− γ 2ωT
k ωk + E
{eT
k+1 Pi ek+1
}− eT
k Pi ek
= ( A2i ek + C2iωk)T ( A2i ek + C2iωk) − γ 2ωT
k ωk
+[E−1i Fi ek + E−1
i C1iωk]T Pi [E−1i Fi ek + E−1
i C1iωk] − eTk Pi ek
= T �, (20)
where
� =[
(E−1i Fi )
T Pi (E−1i Fi ) − Pi + AT
2i A2i (E−1i Fi )
T Pi E−1i C1i + A2i C2i
∗ −γ 2 I + (E−1i C1i )
T Pi E−1i C1i + CT
2i C2i
]
,
=[
ek
ωk
]
.
According to (18), we can obtain that � < 0, which shows that J1 < 0.Summing up both sides of inequality J1 < 0, and considering the zero initial
condition and
E{eTk+1 Pi ek+1}|k→∞ > 0,
we can have
E
{ ∞∑
k=0
{εT
k εk
}}
< γ 2∞∑
k=0
{ωT
k ωk
}
Circuits Syst Signal Process
Therefore, ‖εk‖E2 < γ ‖ωk‖2 holds for any nonzero disturbances ωk ∈ L2[0,∞). That
completes the proof. �Based on the above discussion, the condition of system’s stability with the disturbancesattenuation γ is obtained, but gains Qi and Ri are not given. The following theoremis introduced to design gains Qi and Ri .
Theorem 2 Consider system (1) with observer (4). For a given scalar γ > 0, system(13) is stochastically stable when ωk = 0 and satisfies (11), if there exist symmetricmatrices Pi11, Pi22, matrices Mi , Ni , nonsingular matrices �i11 and �i22 such that(21) holds.
⎡
⎢⎢⎢⎢⎢⎢⎣
−Pi11 0 0 AT1i�i11 + AT
2i MTi∗ −Pi22 0 MT
i∗ ∗ −γ 2 I CT1i�i11
∗ ∗ ∗ Pi11 − �i11 − �Ti11∗ ∗ ∗ ∗
∗ ∗ ∗ ∗−AT
1i AT2i�i22 − AT
2i N Ti AT
2i−N Ti 0
−CT1i AT
2i�i22 CT2i
0 0Pi22 − �i22 − �T
i22 0∗ −I
⎤
⎥⎥⎥⎥⎥⎥⎦
< 0.
(21)
Qi and Ri satisfy
Qi = (�−Ti22 Ni − A2i�
−Ti11 Mi )
−1, Ri = �−1i11 Mi Qi (22)
such that the gains of observer (4) are determined.
Proof Applying Schur complement to (18) and pre- and post-multiplying bydiag{I, I,�T
i , I } and diag{I, I,�i , I }, respectively, we have
⎡
⎢⎢⎣
−Pi 0 (E−1i Fi )
T �i AT2i∗ −γ 2 I (E−1C1i )
T �i CT2i
∗ ∗ −�Ti P−1
i �i 0∗ ∗ ∗ −I
⎤
⎥⎥⎦ < 0. (23)
Note that
(Pi − �i )T P−1
i (Pi − �i ) ≥ 0
implies
− �Ti P−1
i �i ≤ Pi − (�i + �Ti ), (24)
Circuits Syst Signal Process
we can have
⎡
⎢⎢⎣
−Pi 0 (E−1i Fi )
T �i AT2i∗ −γ 2 I (E−1C1i )
T �i CT2i∗ ∗ Pi − (�i + �T
i ) 0∗ ∗ ∗ −I
⎤
⎥⎥⎦ < 0. (25)
Defining Pi = diag{Pi11, Pi22}, �i = diag{�i11,�i22}, and considering (7), (25)is rewritten as follows:
⎡
⎢⎢⎢⎢⎢⎢⎣
−Pi11 0 0 �14 �15 AT2i∗ −Pi22 0 �24 �25 0
∗ ∗ −γ 2 I �34 �35 CT2i∗ ∗ ∗ �44 0 0
∗ ∗ ∗ ∗ �55 0∗ ∗ ∗ ∗ ∗ −I
⎤
⎥⎥⎥⎥⎥⎥⎦
< 0, (26)
where
�14 = (A1i + Ri Q−1i A2i )
T �i11,�24 = Q−Ti RT
i �i11,�34 = CT1i�i11,
�44 = Pi11 − �i11 − �Ti11,�15 = (−A2i A1i − (Q−1
i + A2i Ri Q−1i )A2i )
T �i22,
�25 = −(Q−1i + A2i Ri Q−1
i )T �i22,�35 = −CT1i AT
2i�i22,
�55 = Pi22 − �i22 − �Ti22.
Set Mi = �Ti11 Ri Q−1
i , Ni = �Ti22(Q−1
i + A2i Ri Q−1i ), then (21) and (22) have
been obtained. That completes the proof. �
4 Fault Sensitivity Analysis without Disturbances Case
In no disturbances case, we will prove that system (8) is stochastically stable whenfk = 0 and the faults fk effect on residuals εk satisfies (12). Next, gains Ri and Qi
are designed, that is, the gains of observer (4) are determined. When ωk = 0, we canhave the following system:
ek+1 = E−1i Fi ek,
εk = A2i ek + D2i fk .(27)
Remark 3 Similar to the definition in [7], β is used to measure the sensitivity of theresidual to the sensor fault under no disturbance case. The larger β is, the more highthe sensitivity of the fault detection observer to the sensor faults becomes.
Theorem 3 For a given scalar β > 0, system (27) is stochastically stable when fk = 0and satisfies (12) if there exist positive definite symmetric matrices Pi such that the
Circuits Syst Signal Process
following inequality holds
[(E−1
i Fi )T Pi (E−1
i Fi ) − Pi − AT2i A2i
∗AT
2i D2i
β2 I − DT2i D2i
]
< 0. (28)
Proof When fk = 0, the form of system (27) is equivalent to system (15). Similar toLemma 2, it is obvious to obtain that system (27) is stochastically stable when fk = 0.
In the following, we can show that (12) is satisfied. Under the zero initial conditionand fk = 0, define the following performance index
J2 = E
{εT
k εk
}− γ 2ωT
k ωk − Vk (29)
where β > 0 is the prescribed attenuation level. Then we can have
J2 ≥ E
{εT
k εk
}− β2 f T
k fk − Vk
= E
{εT
k εk
}− β2 f T
k fk − E{V (ek+1)} + V (ek)
= ( A2i ek + D2i fk)T ( A2i ek + D2i fk) − β2 f T
k fk
−(E−1i Fi ek)
T Pi (E−1i Fi ek) + eT
k Pi ek
= ϒT �ϒ, (30)
where
� =[−(E−1
i Fi )T Pi (E−1
i Fi ) + Pi + AT2i A2i
∗AT
2i D2i
−β2 I + DT2i D2i
]
, ϒ =[
ek
fk
]
.
According to (28), we can obtain that � > 0, which shows that J2 > 0.Summing up both sides of inequality J2 > 0, and considering the zero initial
condition and
E
{eT
k+1 Pi ek+1
}|k→∞ > 0,
we can have
E
{ ∞∑
k=0
{εT
k εk
}}
> β2∞∑
k=0
{f Tk fk
}
Therefore, ‖εk‖E2 > β‖ fk‖2 holds for any nonzero faults fk ∈ L2[0,∞). That
completes the proof. �Based on the above discussion, the condition of system’s stability with the fault
attenuation β is obtained, but gains Qi and Ri are not shown. The following theoremis introduced to design gains Qi and Ri .
Circuits Syst Signal Process
Theorem 4 Consider system (1) with observer (4). For a given scalar β > 0, system(27) is stochastically stable when fk = 0 and satisfies (12), if there exist symmetricmatrices Pi11, Pi22, matrices Mi , Ni , nonsingular matrices �i11 and �i22 such that(31) holds.
⎡
⎢⎢⎢⎢⎣
−Pi11 − AT2i A2i 0 AT
2i D2i
∗ −Pi22 0∗ ∗ β2 I − DT
2i D2i
∗ ∗ ∗∗ ∗ ∗
AT1i�i11 + AT
2i MTi −AT
1i AT2i�i22 − AT
2i N Ti
MTi −N T
i0 0
Pi11 − �i11 − �Ti11 0
∗ Pi22 − �i22 − �Ti22
⎤
⎥⎥⎥⎥⎦
< 0. (31)
Qi and Ri satisfy (22) such that the gains of observer (4) are determined.
Proof Applying Schur complement to (28), pre- and post-multiplying bydiag{I, I,�T
i } and diag{I, I,�i }, respectively, we have
⎡
⎣−Pi − AT
2i A2i AT2i D2i
∗ β2 I − DT2i D2i
∗ ∗
(E−1i Fi )
T �i
0−�T
i P−1i �i
⎤
⎦ < 0.
According to (24), we can have
⎡
⎣−Pi − AT
2i A2i AT2i D2i
∗ β2 I − DT2i D2i
∗ ∗
(E−1i Fi )
T �i
0Pi − (�i + �T
i )
⎤
⎦ < 0.
Defining Pi = diag{Pi11, Pi22}, �i = diag{�i11,�i22}, and considering (7), wecan have
2i D2i and �14, �15, �24, �25, �44,�55 have been defined above.
Set Mi = �Ti11 Ri Q−1
i and Ni = �Ti22(Q−1
i + A2i Ri Q−1i ), then (31) and (22) have
been obtained. That completes the proof. �
Circuits Syst Signal Process
5 Fault Sensitivity Analysis with Disturbances Case
In disturbances case, this paper gives the worst case fault sensitivity measure for thefault detection observer. γ and β can measure the worst case fault sensitivity.
Theorem 5 Consider system (8). For given scalars γ > 0 and β > 0, system (8) isstochastically stable when ωk = 0 and fk = 0 and satisfies (11) and (12) if there existpositive definite symmetric matrices Pi such that (18) and (28) hold.
Theorem 6 Consider system (1) with observer (4). For given scalars γ > 0 andβ > 0, system (8) is stochastically stable when ωk = 0 and fk = 0 and satisfies (11)and (12), if there exist symmetric matrices Pi11, Pi22, matrices Mi , Ni , nonsingularmatrices �i11, �i22 such that (21) and (31) hold, respectively. Then Qi and Ri satisfy(22), that is, the gains of observer (4) are obtained.
Remark 4 Based on Theorem 1, 2, 3, and 4, it is easy to summarize Theorem 5 and 6.
6 Illustrative Example
In this section, we will provide a numerical example to show the effectiveness andapplicability of the developed method in this paper.
We assume that the hybrid discrete-system involves two modes:Mode 1:
xk+1 = A11xk + B11uk + C11ωk,
yk = A21xk + C21ωk + D21 fk,
where
A11 =[
0.3 0.4−0.1 0.4
]
, B11 =[
00
]
, C11 =[
0.01−0.01
]
,
A21 =[
1 00 1
]
, C21 =[
0.1−0.2
]
, D21 =[
1.2 00 1.2
]
.
Mode 2:
xk+1 = A12xk + B12uk + C12ωk,
yk = A22xk + C22ωk + D22 fk,
where
A12 =[ −0.5 −0.2
0.1 1.1
]
, B12 =[
00
]
, C12 =[
0.70.1
]
,
A22 =[
1.2 00 1.2
]
, C22 =[
0.01−0.03
]
, D22 =[
1.3 00 1.3
]
.
Circuits Syst Signal Process
The transition probabilities are given as follows
π =[
0.6 0.40.7 0.3
]
.
According to Theorem 6, we can design the observer. When γ is reduced to 0.5 andβ is increased to 1, we can obtain R1, R2, Q1, and Q2
Q1 =[−222.0280 −237.0832
702.2211 −759.9567
]
, R1 =[ −90.2786 90.6397
−124.8776 123.7723
]
,
Q2 =[−8.4589 5.2822
0.1201 −12.8784
]
, R2 =[−0.0811 −0.0739
−0.5391 6.9261
]
,
that is, the observer’s parameters are shown as follows:
We can consider the initial condition that x(0) = [1 −0.5
]T , the control input
uk = sin(k), the sensor fault signal fk =[
fk1fk2
]
as follows:
fk1 =⎧⎨
⎩
0, 0 < k ≤ 1000.5, 100 < k ≤ 200
0, 200 < k, fk2 =
⎧⎨
⎩
0, 0 < k ≤ 1001, 100 < k ≤ 200
0, 200 < k,
and the unknown bounded disturbance ωk is shown by Fig. 1.For the assumption of r0 = 1, the switching signal rk is plotted in Fig. 2, where
“1” and “2” represent, respectively, the first and the second subsystem.Figure 3 shows the response of generated residual εk1 for the fault fk1, where sk1
represents generated residual εk1 with fault fk1, and sk10 represents generated residual
Circuits Syst Signal Process
0 50 100 150 200 250 300
−0.1
−0.05
0
0.05
0.1
0.15
Time step k
Am
plitu
de
wk
Fig. 1 Unknown disturbance ωk
0 50 100 150 200 250 3000.5
1
1.5
2
2.5
Time step k
Am
plitu
de
rk
Fig. 2 Switching signal
εk1 without the fault fk1, respectively. Figure 4 shows the response of generatedresidual εk2 for the fault fk2, where sk2 represents generated residual εk2 with fault fk2,and sk20 represents generated residual εk2 without the fault fk2, respectively. Despitethe disturbance ωk in the system, the sensitivity of residuals to faults is shown.
To detect the fault, we choose the residual evaluation function as stated in (9), andthe residual evaluation function is shown in Figs. 5 and 6, respectively. By calculation,
Circuits Syst Signal Process
0 50 100 150 200 250 300−0.5
0
0.5
1
1.5
2
Time step k
Am
plitu
de
sk1
sk10
Fig. 3 Generated residual εk1 for fk1
0 50 100 150 200 250 300−0.5
0
0.5
1
1.5
2
Time step k
Am
plitu
de
sk2
sk20
Fig. 4 Generated residual εk2 for fk2
one obtains that J103(εk1) =√
1103
∑103k=0 εT
k1εk1 = 0.0581 > Jth1 = 0.0423 andJ102(εk2) = 0.0783 > Jth2 = 0.0417, the faults fk1 and fk2 can be detected ask = 103 and k = 102, respectively. This means that the fault can be detected threetime steps and two time steps, respectively, after its occurrence.
The numerical example shows the sensitivity of residual signals to faults and therobustness against the disturbances. However, in fact, fault detection has extensive
Circuits Syst Signal Process
0 50 100 150 200 250 3000
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Time step k
Am
plitu
default f
1
no fault f1
Fig. 5 Residual evaluation for fk1
0 50 100 150 200 250 3000
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Time step k
Am
plitu
de
fault f2
no fault f2
Fig. 6 Residual evaluation for fk2
applications, such as flight control systems, attitude control systems, and so on in themodel of the aircraft engineering. That is very significant.
Remark 5 In this paper, we make a bridge between the fault detection and discrete-timehybrid systems. By using the descriptor system method, the fault detection problemcan be dealt with for the discrete-time hybrid systems. Some comparison results aregiven in Table 1.
Circuits Syst Signal Process
Table 1 Comparisons of the existing results
Reference Contributions
[5], [12] Singular systems
[19], [39] Markovian jump systems
[16], [37] Singular Markovian jump systems
[2] Continuous-time systems and the descriptor system method
This paper Discrete-time hybrid systems and the descriptor system method
7 Conclusion
We have solved the problem of fault detection for discrete-time hybrid systems via adescriptor system method. This paper designs a fault detection observer, which doesnot only guarantee the error dynamics is stochastically stable, but also make that thedisturbances effect on the residuals satisfy the prescribed H∞ performance and thatthe expected sensitivity of the residuals to the faults is achieved. Finally, a numericalexample demonstrates the effectiveness and applicability of the developed results.Further research work is that this class of fault detection technology is applied to theactual systems.
Acknowledgments The authors would like to thank the referees for their valuable and helpful commentswhich have improved the presentation. The work was supported by the National Basic Research Programof China (973 Program) (2012CB720000), the National Natural Science Foundation of China (61225015),Foundation for Innovative Research Groups of the National Natural Science Foundation of China (GrantNo. 61321002 ), the Ph.D. Programs Foundation of Ministry of Education of China (20111101110012),and CAST Foundation (CAST201210).
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