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Fault detection for discrete-time switched systems with sensor stuck faults and servo inputs Guang-Xin Zhong a, Guang-Hong Yang a,b,n a b
College of Information Science and Engineering, Northeastern University, Shenyang 110819, PR China State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang 110819, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 30 October 2013 Received in revised form 17 March 2015 Accepted 16 May 2015 This paper was recommended for publication by Dr. Q.-G. Wang
This paper addresses the fault detection problem of switched systems with servo inputs and sensor stuck faults. The attention is focused on designing a switching law and its associated fault detection filters (FDFs). The proposed switching law uses only the current states of FDFs, which guarantees the residuals are sensitive to the servo inputs with known frequency ranges in faulty cases and robust against them in fault-free case. Thus, the arbitrarily small sensor stuck faults, including outage faults can be detected in finite-frequency domain. The levels of sensitivity and robustness are measured in terms of the finitefrequency H index and l2-gain. Finally, the switching law and FDFs are obtained by the solution of a convex optimization problem. & 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Fault detection Switched systems Sensor stuck faults Linear matrix inequality (LMI)
1. Introduction In recent years, the issue of fault detection (FD) has attracted amount of attention due to the increasing demand of safety and reliability standards in engineering practice [1–4]. Among the existing FD approaches, the model-based FD can replace hardware redundancy based on traditional schemes, which is widely applied for vehicle control systems, robots, power systems, manufacturing processes, see e.g. [5–8]. The basic idea of the mentioned subject is to design a detection unit to generate a signal named residual, which reflects the difference of system between faulty and faultfree cases. To this end, various types of the residual generator are proposed. For example, in [9], observer-based generator is developed to ensure detection performance by minimizing the errors between faults and residuals. To make the residuals be sensitive to faults and robust against disturbances, a mixed H 1 =H fault detection filter (FDF) is constructed in [10]. When the faults have known frequency domains, an improved frequency-based FDF is designed to reduce the conservatism [11]. On the other hand, switched systems have attracted increasing interest in the past three decades [12–14]. The systems appear in variety of applications, such as dc/dc convertors, FET transistors n Corresponding author at: College of Information Science and Engineering, Northeastern University, Shenyang 110819, PR China. Tel.: þ86 24 83681939. E-mail addresses: [email protected] (G.-X. Zhong), [email protected] (G.-H. Yang).
and HiMAT vehicle [15–17]. A switched system consists of a finite number of subsystems and a switching law orchestrating switching between these subsystems [18]. Lots of techniques have been developed to study the problems of stability and control for switched systems. For instance, [19] investigates the H 1 control problem using piecewise quadratic Lyapunov function. In [20], the issue of quadratic stability is studied based on the multiple Lyapunov-like function. In the proposed technical frameworks, some typical switching laws which depend on time, state or both are constructed. For the FD problem for switched systems, main objective is to design a switched law and residual generators. As is well known, a switched system can have good performance by properly constructing switching law. However, most relevant results pertain to time-dependent [21,22] or arbitrary switching [23]. Under the proposed switching laws, some FD performance in full-frequency domain are guaranteed. Recently, frequency-based FD approach is presented, which guarantees better FD performance [24]. Thus, it is worthwhile to consider how to design a novel switching law to satisfy finite-frequency FD performance of switched systems. On the other hand, sensor stuck fault is very common for control systems, which directly acts on the process measurement [25–27]. However, the fault type is not adequately addressed for switched systems. In the mentioned technical frameworks, the effect of sensor faults on residuals is directly considered. Clearly, when stuck faults have very small magnitudes, the fault sensitivity will be degraded. Even when the outage faults occur (stuck value is
http://dx.doi.org/10.1016/j.isatra.2015.05.006 0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: Zhong G-X, Yang G-H. Fault detection for discrete-time switched systems with sensor stuck faults and servo inputs. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.006i
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2
zero), it may fail to detect faults. Hence, another motivation of this paper arises: to detect arbitrarily small stuck faults including outage faults, can we present a novel FD scheme to remove the requirement of considering the relation between residuals and small sensor faults directly? Noting that the presence of servo inputs improves the performance of systems [28], we will solve the considered problem via introducing servo inputs. This paper studies the FD problem of switched systems with servo inputs and sensor stuck faults. The contributions lie in the following aspects: (i) Different from the previous results, a statedependent switching law is constructed, which uses only the states of FDFs. (ii) Under the switching law, the finite-frequency H index and l2-gain are introduced to guarantee better fault sensitivity for the system. (iii) By constructing the difference between the residuals and servo inputs in fault-free and faulty cases, arbitrarily small sensor stuck faults including outage faults can be detected. This overcomes the drawback caused by considering the relation between the residuals and sensor stuck faults. Finally, the switching law and FDFs are obtained by solving linear matrix inequalities (LMIs). The rest of the paper is organized as follows. In Section 2, some fundamental definitions, lemmas and main objectives are provided. The main results are proposed in Section 3. Section 4 gives the FD scheme. In Section 5, simulation examples are given to illustrate the effectiveness of the proposed method. Finally, some conclusions end the paper in Section 6. The notation used in this paper is fairly standard. For a matrix A, AT denotes its transpose. The Hermitian part of a square matrix M is denoted by HeðMÞ ¼ M þ M T . The symbol n within a matrix represents the symmetric entries. 0 and I represent the zero and identity matrix with the appropriate dimensions, respectively.
free and when F pli ¼ 1; p ¼ 0; l ¼ 1; …; m, all the sensors are fault0 free, that is, yF σ ðkÞ ¼ yðkÞ. Remark 1. By revisiting the existing FD method in [25], the stuck value fpli is required to be known, which enhance the sensitivity between residuals and sensor stuck faults. Hence, the method cannot be applied when fpli is unknown. In this paper, the sensor faults with unknown stuck values can be detected by introducing servo inputs. This removes the restriction of the existing results. 2.3. Fault detection filters with servo inputs For each subsystem i, the following FDF with servo input is constructed: p
^ þ 1Þ ¼ Afi xðkÞ ^ þ Bfi yF i ðkÞ þ Bfvi vi ðkÞ xðk p
^ þ Dfi yF i ðkÞ r p ðkÞ ¼ C fi xðkÞ
^ A R is the state of the filter and r p ðkÞ A R is the where xðkÞ residual. Afi, Bfi, Bfvi, Cfi and Dfi are filter matrices with appropriate dimensions, which are to be determined. Remark 2. Letting Bfvi ¼ 0 or vi ðkÞ ¼ 0, the proposed FDF is converted into the existing one without servo input [25,26]. On the other hand, it has been proven that frequency-based filter leads to less conservatism [6,24]. Thus, by constructing vi(k) in finitefrequency domain, the switched system can have better FD performance. The frequency-based FDF design conditions will be provided in this paper. Combining Eqs. (1)–(4), the overall system is described by p
where
2.1. System model Consider the following switched system: xðk þ 1Þ ¼ Aσ xðkÞ þ Bdσ dðkÞ þ Bvσ vσ ðkÞ ð1Þ m
where xðkÞ A R is the state, yðkÞ A R is the measured output, dðkÞ A Rr is the unknown input, and vσ ðkÞ A Rv is the servo input with known frequency range. The unknown input and the servo input belong to l2 ½0; 1Þ. The piecewise constant function σ ðkÞ : ½0; 1Þ-L ¼ f1; …; Ng is a switching signal, that is, when σ ðkÞ ¼ i, the ith subsystem is activated, where L is a finite set and the positive integer N is the number of subsystems. The matrices Ai, Bdi, Bvi, Ci, Ddi and Dvi are known constant matrices with the appropriate dimensions.
p
p
p
p
p
p
ð5Þ
"
# " # " # Ai Bdi 0 xðkÞ p p ~ xðkÞ ¼ ; Ai ¼ ; B di ¼ ; ^ Bfi F pi Ddi Bfi F pi C i Afi xðkÞ " # " # Bvi 0 p p p B vi ¼ ; C i ¼ Dfi F pi C i C fi ; ; F 1i ¼ Bfi ðI F pi Þ Bfi F pi Dvi þ Bfvi p
D di ¼ Dfi F pi Ddi ;
p
D vi ¼ Dfi F pi Dvi ;
p
F 2i ¼ Dfi ðI F pi Þ
2.4. Preliminaries The following definitions are given, which are essential for our derivation. Definition 1. For each senor fault mode p, switched system (5) under switching law σ is said to have a finite-frequency H index γ1, if the following inequalities
2.2. Fault model In this paper, we consider the following sensor stuck fault models: p
p
~ þ 1Þ ¼ A i xðkÞ ~ þ B di dðkÞ þ B vi vi ðkÞ þ F 1i f pi xðk p
n
s
~ þ D di dðkÞ þD vi vi ðkÞ þ F 2i f i r p ðkÞ ¼ C i xðkÞ
2. Problem formulation and preliminaries
yðkÞ ¼ C σ xðkÞ þ Ddσ dðkÞ þ Dvσ vσ ðkÞ
ð4Þ
n
p
yF σ ðkÞ ¼ F pσ yðkÞ þ ðI F pσ Þf σ ; p ¼ 0; 1; …; q ð2Þ p p p p T p where f σ ¼ f 1σ f 2σ … f mσ , f lσ ðl ¼ 1; …; mÞ is the unknown constant, which represents the stuck value of the lth sensor. p denotes the pth fault mode, q is the number of the total possible fault modes and F pσ is the diagonal matrix with the following form: ð3Þ F pσ ¼ diag F p1σ F p2σ … F pmσ ; F plσ ¼ 0 or 1 Then, for σ ¼ i, we know that, when F pli ¼ 0, p ¼ 1; …; q, the lth sensor gets stuck, when F pli ¼ 1, p ¼ 1; …; q, the lth sensor is fault-
1 X
r Tp ðkÞr p ðkÞ Z γ 21
k¼0
1 X
vTi ðkÞvi ðkÞ;
p ¼ 1; …; q
ð6Þ
k¼0
hold for all solutions of (5) such that 1 1 X θ 2X T ~ x~ T ðkÞ ~ þ 1Þ xðkÞÞð ~ ~ þ 1Þ xðkÞÞ ~ xðkÞ ðxðk xðk r 2 sin li 2 k¼0 k¼0 ð7Þ are satisfied for the low-frequency case j θi j r θli , ehθwi
1 X
~ þ 1Þ ehθ1i xðkÞÞð ~ ~ þ 1Þ ehθ2i x~ T ðkÞÞT r 0 ðxðk xðk
ð8Þ
k¼0
Please cite this article as: Zhong G-X, Yang G-H. Fault detection for discrete-time switched systems with sensor stuck faults and servo inputs. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.006i
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are satisfied for the middle-frequency case θ1i r θi r θ2i and 2.5. Problem statement
1 θ 2X T ~ þ 1Þ xðkÞÞð ~ ~ þ 1Þ xðkÞÞ ~ ~ x~ T ðkÞ ðxðk xðk Z 2 sin hi xðkÞ 2 k¼0 k¼0 1 X
ð9Þ are satisfied for the high-frequency case j θi j Z θhi , where θi is the frequency of servo input vi(k), θli, θ1i, θ2i and θhi are known constants, which represent the frequency ranges of vi(k). Definition 2. For fault-free mode p ¼0, switched system (5) under switching law σ is said to have a finite-frequency l2-gain γ2, if the following inequalities 1 X
γ
r T0 ðkÞr 0 ðkÞ r 22
k¼0
1 X
vTi ðkÞvi ðkÞ
ð10Þ
For system (5), we will design a novel switching law and FDFs with servo inputs such that the asymptotical stability of system is guaranteed. Moreover, 1. In faulty cases, maximizing the effect of servo inputs on residuals rp(k) in finite-frequency domain, while minimizing the effects of unknown input on residuals rp(k) in fullfrequency domain. Thus, (6) and the following conditions will be satisfied: 1 X
r Tp ðkÞr p ðkÞ r γ 23
k¼0
1 X
T
d ðkÞdðkÞ;
p ¼ 1; …; q
ð13Þ
k¼0
k¼0
hold for all solutions of (5) such that (7), (8), and (9) are satisfied in low-, middle-, and high-frequency ranges, respectively. Remark 3. Motivated by [29], Definitions 1 and 2 are presented, which are used to characterize the FD performance for multimodel systems. The proposed performance indexes γ1 and γ2 represent the sensitivity and robustness between the residuals and servo inputs in faulty and fault-free cases. Thus, the relation between the residuals and sensor stuck faults is not considered directly. When the faults have very small magnitudes, good fault sensitivity is still guaranteed. Also, we introduce some useful lemmas for later development. Lemma 1 (Galbusera et al. [30]). Given a nonsingular matrix V and symmetric matrices Y and Xi for all iA L satisfying " # Y I 40 ð11Þ n Xi it is possible to determine nonsingular matrices Ui, symmetric matrices Y^ i and X^ i , such that " # " # Y V Xi Ui 1 Si ¼ S ¼ 4 0; 40 ð12Þ i n Y^ i n X^ i
Lemma 2 (Projection lemma). Given a symmetric matrix matrices Γ and Λ, there exists a decision matrix X satisfying
Φ and
2. In fault-free case, the effect of servo inputs on residuals r 0 ðkÞ is minimized in finite-frequency domain, while the effect of unknown input on residuals r 0 ðkÞ is minimized in fullfrequency domain. Then, (10) and the following condition are used to characterize the robustness: 1 X
r T0 ðkÞr 0 ðkÞ r γ 24
k¼0
1 X
T
d ðkÞdðkÞ
ð14Þ
k¼0
Remark 4. Since the proposed FD method is to detect sensor stuck faults by considering the sensitivity between servo inputs and residuals in different cases, even if the sensor outage faults occur p (F pli ¼ 0, f lσ ¼ 0, l ¼ 1; …; m), the faults are still detected. The examples in Section 5 illustrate the effectiveness of the proposed method. Remark 5. For the unknown input d(k), it is difficult to obtain its frequency information. Hence, the standard l2-gain properties (13) and (14) in full-frequency domain are proposed. These properties guarantee that the residuals are robust against disturbance in faulty and fault-free cases. 3. Main results Here, we construct the following switching law: T ^ ^ σ ðxðkÞÞ ¼ argminx^ ðkÞY^ i xðkÞ iAL
ð15Þ
Φ þ Γ X ΛT þ ΛX T Γ T o 0
where Y^ i are positive definite matrices, which are to be designed.
if and only if the following projection inequalities with respect to X hold:
Remark 6. The switching law (15) is a special state-dependent switching law due to its dependence only on the states of FDFs, which is easy to design and realize in practice. In the following theorems, we will prove that the switching law guarantees the stability of system (5). In addition, compared with the existing results under arbitrary or average dwell time switching [22,23], the proposed switching law improves the FD performance by introducing the finite-frequency H index and l2-gain.
T
Γ ? ΦΓ ? o 0 T
Λ ? ΦΛ ? o0
Lemma 3 (Finsler's lemma). Let ξ A Cn , Q A Cnn , and H A Cnm . Let H ? be any matrix such that H ? H ¼ 0. Then the following conditions are equivalent: 1. ξ Q ξ o 0; 8 Hn ξ ¼ 0; ξ a 0. 2. ( Y A Rmn : Q þ HY þ Y n Hn o 0. n
3.1. Conditions for performance (6)
Theorem 1. Assume that system (5) is asymptotically stable under the switching law (15), for a given scalar γ 1 4 0 and an arbitrary nonsingular matrix V, condition (6) is satisfied, if there exist symp p metric matrices P i , Y, X i , Q i 4 0, Φ1i 4 0, and matrices Ti, T2i, Ni, Hi,
Please cite this article as: Zhong G-X, Yang G-H. Fault detection for discrete-time switched systems with sensor stuck faults and servo inputs. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.006i
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4
Dfi ð 8 iA LÞ, p ¼ 1; …; q, such that the following inequalities hold: 2 3 p p 0 Ψ 2i P i HefΦ1i g Ψ 1i 6 7 6 n Ψ p3i Ψ p4i Ψ p5i 7 6 7 ð16Þ 6 p 7o0 6 7 n n 2I Ψ 6i 4 5 n
where 2 p
Pi ¼ 4
n
p P 1i
p P 2i p P 3i
n
"
Ψ p1i ¼ "
Ψ p4i ¼
Ψ p7i
n
3
2
5;
Qi ¼4
p
#
p Q 1i
n
"
Ψ Ψ
p 35i p 45i
#
"
Ψ p5i ¼
;
Ψ Ψ
Ψ 16i ¼ YBvi þ T 1i Dvi þ T 2i ;
Ψ p23i ¼ α ¼
p i Q 2i X i þ X i Ai ;
p P 1i
β
"
5;
Ψ p3i ¼
#
p 36i p 46i
Y
Xi
n
Xi
# ;
Ψ p24i ¼ α
p i Q 3i X i þX i Ai p p 1i Dfi F i C i
μ
T
p p Ψ p44i ¼ P 3i βi Q 3i þ He X i Ai μp2i ðDfi F pi C i þ Hi Þ T
Ψ p45i ¼ 12 ðC Ti F pi DTfi þ HTi Þ þ μp2i T
7 0 5;
I
0
6 J3 ¼ 4 0 0
I
3
7 I5 0
ð24Þ
3
0
7 pT C i D vi 7 7 5 pT p 2 γ 1 I D vi D vi
T
ð25Þ
hold for given Λi, Λ i ¼ Λi Λi , 2 3 2 I I T p 6 7 p A i 7; J~ ¼ 6 0 J~ 1i ¼ 6 4 4 4 T5 p 0 B vi
and J~ 4 , where
T p J p1i J~ 1i
¼ 0. Then, letting
3 0 7 I5 0
Then, (25) is reformulated as 2 T p P pi þ HefΛi g Q pi Λi þ Λi A i 6 6 6 n Ξ pi 6 4
3
p
ΛTi B vi jT
p
p
7 7
ΛTi B vi C i D vi 7 7o0 5 T p γ 21 I D vi D vi
n
ð26Þ
where pT
pT
p
Ξ pi ¼ P pi 2 cos θli Q pi þ HefA i Λi g C i C i
pT
Ψ p46i ¼ X i Bvi ðC Ti F pi DTfi þ HTi Þμp3i μp2i Dfi F pi Dvi ;
2
n
n
Ψ p36i ¼ YBvi þ T 1i Dvi þ T 2i C Ti F i DTfi μ3i μp1i Dfi F pi Dvi T
6 J2 ¼ 4 I 0
3
Using Lemma 2 and letting 2 P pi Q pi 6 pT p P pi 2 cos θli Q pi C i C i Υ pi ¼ 6 6 n 4
p
Ψ p7i ¼ γ 21 I He μp3i Dfi F pi Dvi
T
0
Υ i þ HefJ~1i Λ i J~4 g o 0
pT
Ψ 26i ¼ X i Bvi ;
pT
I
0
we known that (23) is satisfied if
Ψ p33i Ψ p34i ; n Ψ p44i
Ψ p34i ¼ P 2i βi Q 2i þ N i þ T 1i F pi C i þ ATi X i C Ti F pi DTfi μp2i μp1i ðDfi F pi C i þ Hi Þ
T
0
2
n
#
Ψ p14i ¼ αi Q 2i X i þN i þ T 1i F pi C i
p p i Q 1i þ He YAi þ T 1i F i C i
Ψ p35i ¼ 12 C Ti F pi DTfi þ μp1i ;
p 3 B vi 7 0 5;
p
Ai 6 J p1i ¼ 4 I
p J~ 1i
p
p
Φ1i ¼ "
Ψ p13i Ψ p14i Ψ 16i ; Ψ 2i ¼ Ψ 26i Ψ p23i Ψ p24i
p
Ψ
3
#
Ψ p13i ¼ αi Q 1i Y þ YAi þ T 1i F pi C i ;
p 33i
p Q 2i p Q 3i
where 2
T
Ψ p6i ¼ 12 Dfi F pi Dvi þ μp3i
where μpmi ; m ¼ 1; 2; 3 should be given beforehand. αi ¼ 1, βi ¼ 2 cos ðθli Þ in the low-frequency range j θvi j r θli , αi ¼ ehθci , βi ¼ 2 cos ðθwi Þ, θci ¼ ðθ1i þ θ2i Þ=2 and θwi ¼ ðθ2i θ1i Þ=2 in the middle-frequency range θ1i r θvi r θ2i , and αi ¼ 1; β i ¼ 2 cos ðθhi Þ in the high-frequency range θvi Z θhi . Moreover, the switching law and FDFs can be obtained by
Note that Φ1i 4 0 implies X i and Y X i are nonsingular. Suppose 1 X i ¼ X i , we know that Φ1i 4 0 is equivalent to (11). Then, based on Lemma 1, we have " # " # Y V Xi Ui Λi ¼ Γ ¼ ; ð27Þ i n Y^ i n X^ i 1
Letting Λi
¼ Γ i , we have
U i ¼ ðI X i YÞV T Y^ i ¼ V T ðY X 1 Þ 1 V
ð28Þ
i
Y^ i ¼ V T ðY X i Þ 1 V
ð17Þ
Afi ¼ V 1 ðN i YAi ÞðX i YÞ 1 V
ð18Þ
Bfi ¼ V 1 T 1i
ð19Þ
Bfvi ¼ V 1 T 2i
ð20Þ
C fi ¼ H i ðX i YÞ 1 V
ð21Þ
Clearly, (17) is obtained. Further, denote
Δ1i ¼
Y
I
VT
0
"
;
Δ2i ¼
I
0
0
Xi
#
Ωi ¼ Γ i Δ1i Δ2i
;
ð29Þ
Δ1i and Δ2i are full rank of column. Then, preh i T T multiplying diag Ωi Ωi I and post-multiplying diag Ωi Ωi I to
Obviously,
(26) and letting " P pi ¼ Proof. First, we consider the performance condition (6) when the servo input is in low-frequency domain. For p ¼ 1; …; q, denote " # " # Q pi P pi I 0 Σ pi ¼ Π ¼ ; n γ 21 I n P pi 2 cos θli Q pi Suppose the following conditions hold for j θvi j r θli : # " p # " p # " p # " p p T p p T p C i D vi C i D vi A i B vi B vi p Ai Σi Π o0 þ 0 I 0 I I 0 I 0
ð22Þ
J p1i ½J 3 Σ i J T3 þJ 2 Π J T2 J p1i o0 T
p
ð23Þ
P p2i
n
P p3i
#
" ;
Q pi ¼
Q p1i
Q p2i
n
Q p3i
# 40
we get 2 6 6 6 4
P i þ HefΦ1i g
p p Q i Φ1i þ A~ i
n
Ξ~ i
n
n
p
p
3
p B~ vi
7 pT p 7 B~ vi C~ i D vi 7 o 0 5 T p p γ 21 I D vi D vi
ð30Þ
where pT
Ξ~ i ¼ P i 2 cos θli Q i þ HefA~ i g C~ i C~ i ; P i ¼ ΩTi P pi Ωi p p p p p T T p T T p T A~ i ¼ Δ2i Δ1i A i Ωi ; B~ vi ¼ Δ2i Δ1i B vi ; C~ i ¼ C i Ωi ; Q i ¼ Ωi Q pi Ωi p
Clearly, (22) is equivalent to the following conditions:
P p1i
p
p
p
p
p
Please cite this article as: Zhong G-X, Yang G-H. Fault detection for discrete-time switched systems with sensor stuck faults and servo inputs. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.006i
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conditions in Theorem 1 guarantee the finite-frequency H index
Noting that (30) is equivalent to T
Γ p1i Φpi Γ p1i o 0
ð31Þ
I 60 6 p Γ 1i ¼ 6 40
0 I p C~
0
0
2
3 0 0 7 7 p 7; D vi 5
i
6 6 6 p Φi ¼ 6 6 6 4
I
p
P i HefΦ1i g
p p Q i Φ1i þ A~ i
0
n
Ξ^ i
pT 12C~ i
n
n
0
n
n
n
p
p B~ vi
3
7 7 7 7 7 p 12D vi 7 5 2 γ1I
p
p
p
Letting h i p p Rpi ¼ 0 C~ i I D vi
ð32Þ
we have Rpi Γ 1i ¼ 0. Using Lemma 3, we see (31) is guaranteed if the inequities p
Φpi þ HefLpi Rpi g o 0
ð33Þ
hold for some matrices p
Lji.
Next, setting
p
p
Q 1i ¼ Q p1i ;
P 3i ¼ P p1i þ HefP p2i U Ti X i g þ X i U i P p3i U Ti X i
p
N i ¼ ðYAi X i Þ þ VAci U Ti X i ;
Q 3i ¼ Q j1i þ HefQ p2i U Ti X i g þ X i U i Q p3i U Ti X i
H i ¼ C fi U Ti X i ;
and constructing h i T T T Lpi ¼ 0 μ p1i I μp3i ;
T
h
T 1i ¼ VBfi ;
T
T
μ p1i ¼ μp1i μp2i
T 2i ¼ VBfvi
i
ð35Þ Using zero initial condition and stability property of system (5), by accumulating both sides of (35) for k from 0 to 1, we get ð r Tp ðkÞr p ðkÞ þ γ 21 vTi ðkÞvi ðkÞÞ þ tr½Q ji Si o 0
ð36Þ
k¼0
where ~ þ 1Þ þ x~ T ðk þ 1ÞxðkÞ ~ 2 cos θli x~ T ðkÞxðkÞÞ ~ ðx~ T ðkÞxðk
k¼0
According to Definition 1, we have Si Z 0, which implies 1 X k¼0
r Tp ðkÞr p ðkÞ Z γ 21
vTi ðkÞvi ðkÞ;
Σ p24i ¼ N Ti þ C Ti F i T T1i ; pT
p 34i
T
¼ BTdi Y þ DTdi F pi T T1i ;
Σ 15i ¼ ATi X i ;
Σ p16i ¼ C Ti F pi DTfi T
Σ 25i ¼ ATi X i ;
Σ p26i ¼ C Ti F i DTfi þ HTi pT
Σ 35i ¼ BTdi X i ;
T
Σ p36i ¼ DTdi F pi DTfi
Proof. We choose the following Lyapunov functional candidate: ~ ~ ~ VðxðkÞÞ ¼ V σ ðxðkÞÞ ðxðkÞÞ ¼ x~ T ðkÞΛσ ðxðkÞÞ xðkÞ ~ ~
ð38Þ
where Λi is introduced via (27). Then, a state-dependent switching law is constructed: ~ ~ σ ðxðkÞÞ ¼ arg minx~ T ðkÞΛi xðkÞ
ð39Þ
iAL
Obviously, the switching law (39) is equivalent to (15). For σ ðkÞ ¼ σ ðk þ 1Þ ¼ i, we have
~ þ 1Þ x~ T ðkÞΛi xðkÞ ~ ¼ x~ T ðk þ 1ÞΛi xðk pT
pT
p
p
~ þ 2x~ T ðkÞA i Λi B di dðkÞ ¼ x~ T ðkÞA i Λi A i xðkÞ T
p
p
þ d ðkÞB di Λi B di dðkÞ T
By Schur complement, if the following inequities hold: 2 3 pT pT 0 A i Λi C i Λi 6 7 6 pT pT 7 2 6 n 7 γ I B Λ D i i di 7 o 0 3 6 6 7 n Λi 0 5 4 n n
n
n
ð40Þ
I
we get
Hence, we have 1 X
I
n
~ þ 1ÞÞ V i ðxðkÞÞ ~ ¼ V i ðxðk
~ þ 1Þ þ x~ T ðk þ 1ÞxðkÞ ~ 2 cos θli x~ T ðkÞxðkÞÞ ~ þ tr½Q pi ðx~ T ðkÞxðk o0
tr½Q ji Si Z 0.
n
~ þ 1ÞÞ V σ ðkÞ ðxðkÞÞ ~ ΔVðkÞ ¼ V σðk þ 1Þ ðxðk
~ x~ T ðk þ 1ÞP pi x~ Ti ðk þ1Þ r Tp ðkÞr p ðkÞ þ γ 21 vTi ðkÞvi ðkÞ x~ T ðkÞP pi xðkÞ
1 X
n
Moreover, Y^ i , Afi, Bfi and Cfi can be constructed by (17), (18), (19) and (21), respectively.
which follows
Si ¼
n
Σ p14i ¼ ATi Y þ C Ti F pi T T1i ; Σ
where μp1i ; μp2i and μp3i ; are turning parameters. Then we obtain (16), which guarantees # " # #" p " #T 8 " p p T Pp p < ~ Q pi xðkÞ A i B vi A i B vi i vi ðkÞ : I Q pi P pi 2 cos θli Q pi 0 I 0 9 2 3 T T " # p p p p = xðkÞ ~ C i D vi C i C i 5 þ4 o0 ð34Þ pT p pT p v ; 2 i ðkÞ D vi C i D vi D vi þ γ 1 I
1 X
n
T
p
Q 2i ¼ Q j1i þ Q p2i U Ti X i ;
Theorem 2. For a given scalar γ 3 4 0 and an arbitrary nonsingular matrix V, system (5) is asymptotically stable and satisfies (13) under the switching law (15), if there exist symmetric matrices Y, X i and matrices T1i, Ni, Hi, Dfi ð 8 i A LÞ, p ¼ 1; …; q, such that the following inequalities hold: 2 3 Y X i 0 Σ p14i Σ 15i Σ p16i 6 p p 7 6 n Xi 0 Σ 24i Σ 25i Σ 26i 7 6 7 p p 6 n 2 n γ 3 I Σ 34i Σ 35i Σ 36i 7 6 7 ð37Þ 6 7 o0 6 n n n Y Xi 0 7 6 7 6 7 4 n n n n Xi 0 5 where
p
P 2i ¼ P p1i þ P p2i U Ti X i ;
3.2. Conditions for stability and performance (13)
p B~ vi
Ξ^ i ¼ P i 2 cos θl Q i þ HefA~ i g
P 1i ¼ P p1i ;
γ1. Moreover, using the proposed multi-model approach, we known that γ1 characterizes the level of sensitivity for all the faulty systems in finite-frequency domain.
where 2
5
j θvi j r θli
k¼0
which means (6) is guaranteed in low-frequency case. Further, according to the similar lines in the proof above, we can obtain the sufficient conditions for middle- and high-frequency cases, which complete the proof.□ Remark 7. Based on the generalized Kalman–Yakubovich–Popov (GKYP) Lemma in [31,32], the proposed frequency-based
ΔVðkÞ þ γ 23 rTp ðkÞrp ðkÞ r dT ðkÞdðkÞ
ð41Þ
By the similar approach in Theorem 1, we know (37) is sufficient for (41). For σ ðkÞ ¼ i; σ ðk þ 1Þ ¼ jð 8 i ajÞ, the switching law (15) guarantees ~ þ 1ÞÞ V σ ðkÞ ðxðkÞÞ ~ ΔVðkÞ ¼ V σðk þ 1Þ ðxðk ~ þ 1ÞÞ V i ðxðkÞ ~ ¼ V j ðxðk ~ þ1ÞÞ V i ðxðkÞÞ ~ r V i ðxðk
ð42Þ
Please cite this article as: Zhong G-X, Yang G-H. Fault detection for discrete-time switched systems with sensor stuck faults and servo inputs. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.006i
G.-X. Zhong, G.-H. Yang / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
6
~ ~ which means VðxðkÞÞ ¼ V σ ðxðkÞÞ is the overall Lyapunov function of system (5). Clearly, (41) can be obtained again. Therefore, it follows 1 X
ΔVðkÞ o
k¼0
1 X
γ
1 X
2 T 3 r p ðkÞr p ðkÞ þ
k¼0
T
d ðkÞdðkÞ
ð43Þ
k¼0
In addition, system (5) is asymptotically stable under switching law (15) in the case of dðkÞ ¼ 0. Next, using zero initial condition and Vðxð1ÞÞ Z 0, we have 1 X
r Tp ðkÞr p ðkÞ r γ 23
k¼0
1 X
T
d ðkÞdðkÞ
k¼0
which completes the proof.□ 3.3. Conditions for performance (10) In order to attenuate the effect of servo inputs on residuals in fault-free case, the following theorem is presented. Theorem 3. Assume that the system (5) is asymptotically stable 0 under the switching law (15), for given scalars γ 2 4 0; li and an arbitrary nonsingular matrix V, condition (10) is satisfied in low-, middle- and high-frequency ranges, respectively, if there exist sym0 0 metric matrices P~ i , Y, X i , Q~ i 4 0, Φ1i 40 and matrices T1i, T2i, Ni Dfi ð 8 i A LÞ, such that the following inequities hold: 2 3 0 0 0 0 P~ i Ψ~ 1i 6 7 0 0 0 6 7 6 n Ψ~ 2i Ψ~ 3i Ψ~ 4i 7 ð44Þ 6 7o0 T 6 7 4 n n γ 22 I DTvi F 0i DTfi 5 n
n
I
n
where Φ1i, αi, and βi are defined in Theorem 1: 2 0 3 2 0 3 2 0 3 0 0 ~ ~0 Ψ Ψ P~ 1i P~ 2i Q~ 1i Q~ 2i 0 0 0 33i 34i ~ 5; Q~ ¼ 4 5; Ψ ¼ 4 5 P~ i ¼ 4 i 2i 0 0 0 n P~ 3i n Q~ 3i n Ψ~ 44i T h i T T 0T 0 0T 0T Ψ~ ¼ Ψ~ Ψ~ ; Ψ~ ¼ C T F 0 DT C T F 0 DT þ H T 3i
35i
45i
Ψ~ 1i ¼ αi Q~ i l0i Φ1i ; 0
0
i
4i
i
fi
i
i
fi
0
0
Ψ~ 44i ¼ P~ 3i βi Q~ 3i þHefl0i X i Ai g; 0
0
0
Ψ 045i ¼ l0i X i Bvi
Ψ~ 35i ¼ l0i ðYBvi þ T 1i F 0i Dvi þ T 2i Þ 0
Proof. For fault-free case (q ¼0), letting 2 T 3 2 T 3 " 0 " # 0 0 I 7 0 A0 B0 Ci I7 Ci 6 6 Ai ~ ~ vi i Σ Π þ 4 0T 5 i 4 0T 5 i I I 0 D vi 0 B vi 0
0
D vi 0
# o0
ð45Þ
In low-frequency range j θvi j o θli , suppose 2 3 " # 0 0 I 0 P^ i Q^ i 0 ~ ¼ 5; Π Σ~ i ¼ 4 0 0 0 n γ 22 I Q^ i P^ i 2 cos θli Q^ i
J 02i ½J 3 Σ~
Ψ
hold, where Λi is introduced via (27) and 2 3 0 0 0 P^ i Q^ i 6 7 T T 6 0 7 0 ^ ^ 0 2 cos θ Q^ 0 þ C 0 C 0 C 0 D 0 7; Υ~ i ¼ 6 P Q l i vi 7 i i i i 6 i 4 5 0T 0 0 D vi C i γ 22 I 2 3 2 3 I 0 0T 7 6 T 6 0 7 0 0 A l i 7; J ¼ J 3i ¼ 6 4 iI5 4i 4 T5 0 0 B vi where l0i are scalars should be known beforehand, which follows 2 3 0 0 0 0 0 P^ i Q^ i li Λi 6 7 6 0T 7 0 ^0 6 n 7 Ξ Λ B C i i vi 1i 6 7o0 ð49Þ 6 T 7 0 7 6 n γ 22 I D vi 5 4 n n n n I where 0
0T
Ξ^ 1i ¼ P^ i 2 cos θli Q^ i þ HefA i Λi g 0
0
Using the same approaches in Theorem 1, we obtain (44), which means 3" # 2 # " #T 8" 0 0 0 0 T 0 ^0 < ~ Q^ i xðkÞ A i B vi 4 P i 5 A i B vi 0 0 vi ðkÞ : I 0 I 0 Q^ i P^ i 2 cos θli Q 0i 9 2 T 3 T " # 0 0 0 0 > ~ C i D vi 7= xðkÞ 6 Ci Ci o0 ð50Þ þ 4 0T 0 5 T 0 0 ; vi ðkÞ D vi C i D vi D vi γ 2 I >
0 T 0 fi J 2 J 2i o 0
where J2 and J3 are obtained from (24) and 2 0 2 T 3 0 3 0 0 0T A i B vi C C C D vi i 6 7 6 i i 7 0 J 02i ¼ 4 I 5 0 5; Ψ fi ¼ 4 0T 0 0T 0 Dvi C i D vi D vi γ 22 I 0 I
3.4. Conditions for performance (14) Based on Theorem 2, the following theorem is provided to guarantee (14) for fault-free system. Theorem 4. For a given scalar γ 4 40 and an arbitrary nonsingular matrix V, system (5) is asymptotically stable and satisfies (14) under the switching law (15), if there exist symmetric matrices Y, X i and matrices Y, X i , T1i, Ni, Hi, Dfi ð 8 iA LÞ, such that the following inequalities hold: 2 3 0 Σ 014i Σ 15i Σ 016i Y X i 6 7 6 n Xi 0 Σ 024i Σ 15i Σ 026i 7 6 7 6 7 0 0 7 2 6 n n γ I Σ Σ Σ 35i 34i 36i 7 o 0 6 4 ð51Þ 6 7 6 n n n Y Xi 0 7 6 7 6 n n n n Xi 0 7 4 5 n n n n n I where
we see that (45) is satisfied if the following inequities hold: 0 T i J 3 þJ 2
ð48Þ
Then, according to the proof in Theorem 1, it is easy to complete this proof.□
i
Moreover, Y^ i , Afi, Bfi, Bfvi and Cfi can be constructed by (17), (18), (19), (20) and (21), respectively.
T
0
0
0
Ψ~ 34i ¼ P~ 2i βi Q~ 2i þl0i N i þ l0i ATi X i þ l0i T 1i F 0i C i ; 0
Υ~ i þ HefJ 03i Λi J 04i g o 0
2
Ψ~ 33i ¼ P~ 1i βi Q~ 1i þ Hefl0i ðYAi þ T 1i F 0i C i Þg 0
By the similar approach, (46) can be satisfied if
T
Σ 014i ¼ ATi Y þ C Ti F 0i T T1i ; ð46Þ
T
Σ 024i ¼ N Ti þ C Ti F 0i T T1i ; Σ 034i ¼ BTdi Y þ DTdi F T0i T T1i ;
ð47Þ
T
Σ 15i ¼ ATi X i ;
Σ 016i ¼ C Ti F 0i DTfi T
Σ 026i ¼ C Ti F 0i DTfi þ HTi Σ 35i ¼ BTdi X i ;
T
Σ 036i ¼ DTdi F 0i DTfi
Moreover, Y^ i , Afi, Bfi and Cfi can be constructed by (17), (18), (19) and (21), respectively.
Please cite this article as: Zhong G-X, Yang G-H. Fault detection for discrete-time switched systems with sensor stuck faults and servo inputs. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.006i
G.-X. Zhong, G.-H. Yang / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Remark 8. Based on Theorems 1–4, we know that FDFs (4) are valid for multiple switched systems, that is, faulty switched systems ðp ¼ 1; …; qÞ and fault-free switched system ðp ¼ 0Þ. Moreover, the internal stability of these systems can be guaranteed by Theorems 2 and 4. Remark 9. Letting Y i ¼ Y, N i ¼ N, T 1i ¼ T 1 , T 2i ¼ T 2 , X i ¼ X i , H i ¼ H and Dfi ¼ Df , the results in Theorems 1–4 can be used to design detector for linear time-invariant (LTI) systems with finitefrequency servo inputs.
7
where p ¼ 0; …; q. Further, denote following detection threshold as J th ¼ sup J 0 ðkÞ
ð53Þ
dðkÞ A l2
Then, the sensor stuck faults can be detected by the following test: ( J J p ðkÞ J rJ th Fault free ð54Þ J J p ðkÞ J 4J th Faulty
5. Examples In this section, two examples are given to illustrate the effectiveness of our method.
4. Fault detection scheme Summarily, the following theorem is given to solve the FD problem. Theorem 5. Under the switching law (15), system (5) is asymptotically stable and satisfies (6), (10), (13) and (14), if there exist 0 0 p p symmetric matrices P~ , P , Q~ 4 0, Q 4 0, Y, X i and matrices T1i, i
i
i
i
T2i, Ni, Hi, Dfi ð 8 i A LÞ, p ¼ 1; …; q, such that LMIs (16), (37), (44) and (51) hold. Moreover, Y^ i , Afi, Bfi, Bfvi and Cfi can be obtained from (17), (18), (19), (20) and (21), respectively. Finally, the FD problem is solved via the following optimization: max γ 1
Example 1. Consider the switched system (1), which consists of the two subsystems: 0:8 0 0:2 0:7 A1 ¼ ; Bd1 ¼ ; Bv1 ¼ 0:2 0:7 0:3 0:8 0:3 0:1 0:1 0:6 C1 ¼ ; Dd1 ¼ ; Dv1 ¼ 0:2 0:3 0:2 0:9 0:7 0:1 0:3 0:5 ; Bd2 ¼ ; Bv2 ¼ A2 ¼ 0:1 0:8 0:2 0:9 0:2 0:1 0:2 0:7 ; Dd2 ¼ ; Dv2 ¼ C2 ¼ 0:3 0:3 0:1 0:7
s:t: (16), (37), (44) and (51) for given scalars
γ2, γ3 and γ4.
To reduce the conservatism in LMI computation, the following algorithm is given. Algorithm 1. Let ϵ be a given large enough constant specifying a stop criterion of the algorithm. Step 1: Maximize γ1 subject to the LMIs (16), (37), (44) and (51). ~ 0ð0Þ , P pð0Þ , Q~ 0ð0Þ , Q pð0Þ , The optimal solutions are denoted as γ ð0Þ i i i 1 , Pi ð0Þ
ð0Þ ð0Þ ð0Þ ð0Þ Y ð0Þ , X i , T ð0Þ 1i , T 2i , N i , H i , and Dfi . 0ð0Þ 0ð0Þ pð0Þ pð0Þ ð0Þ Y ð0Þ , X i and Dð0Þ into Step 2: Substitute P~ i , P i , Q~ i , Q i fi
γ 2 ¼ γ 4 ¼ 1, μ1i ¼ ½0:9 0:8T , Chosen vi ðkÞ ¼ 0:5 ðθvi ¼ 0Þ, T μ2i ¼ ½0:8 0:7 , μ3i ¼ 4:1 and li ¼1 for 8 i A f1; 2g, we consider the following cases: Case 1 (sensor outage (stuck values are zero) case): Suppose the sensor 2 of the 1st subsystem and sensor 1 of the 2nd subsystem get outage at 20 time step. Then, γ1 for different γ3 are shown in Table 1. Obviously, there is a trade-off between sensitivity and robustness. Based on Theorem 5, we obtain the following FDFs: 0:5723 0:3016 15:1583 0:2149 Af 1 ¼ ; Bf 1 ¼ ; 0:3060 0:2105 32:1568 2:6434 11:9692 Bfv1 ¼ ; C f 1 ¼ ½0:1571 0:0855; 15:9948
LMIs (16), (37), (44) and (51), wherein γ1, T1i, T2i, Ni and Hi are seen as unknown matrices to be determined. Further, maximize
2.5
γ1
ð1Þ subject to the above LMIs and denote their solutions as γ ð1Þ 1 , T 1i , ð1Þ and H ð1Þ T ð1Þ 2i , N i i . Next, substitute them into the above LMIs again 0ð1Þ 0ð1Þ pð1Þ pð1Þ ð1Þ and denote P~ i , P i , Q~ i , Q i , Y ð1Þ , X i and Dfið1Þ as unknown ðnÞ ðnÞ T 1i , T ðnÞ and H ðnÞ are 2i , N i i ðnÞ If 1 o , repeat the above
the solutions of the nth optimization. γ optimization, else continue. Step 3: When γ ðnÞ 1 Z ϵ, for any n, stop.
ϵ
According to above scheme, the residual evaluation function Jp(k) is chosen as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u k uX ð52Þ r Tp ðkÞr p ðkÞ J p ðkÞ ¼ t
2
Switching signal
matrices. Solve these LMIs and let γ
ðnÞ 1 ,
1.5
1
k ¼ k0
Table 1 Comparison of the finite frequency H index γ1 for different γ3. γ3 γ1
0.8 3.0258
1 3.3825
1.2 3.5986
1.4 3.6387
0.5 0 1.6 3.7120
5
10
15
20
25
30
35
40
Time in samples Fig. 1. Switching signal.
Please cite this article as: Zhong G-X, Yang G-H. Fault detection for discrete-time switched systems with sensor stuck faults and servo inputs. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.006i
G.-X. Zhong, G.-H. Yang / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
8
2.5
1.8
10
7
Faulty case 9 2
Faulty case
Faulty case 1.6
Fault−free case
Fault−free case
6
8
1.4
7
1.2
Faulty case 5
5 4
0.8 0.6
Fault−free case
0.5
Res. eva. function
1
1
6
Residual signal
Res. eva. function
Residual signal
1.5
0.4
Fault−free case
3
4
3
2 Threshold
Threshold
0.2
2 0
0
1
1 −0.2 −0.5
0
0 0
10
20
30
0
40
10
20
30
0
40
10
30
0
40
Time in samples
Time in samples
Time in samples
20
Fig. 2. Residual and evaluation function of the system with servo inputs in outage case.
10
1.8 1.6
6
0.3 0.2
4
1 0.8 0.6
3
0.4
2
1
1 −0.2
0
20
40
0
0
20
40
Time in samples
Fig. 3. Residual and evaluation function of the system without servo inputs in outage case.
Df 1 ¼ ½5:2290 2:5149;
Af 2 ¼
5:7842
24:0894
0:5506
12:1864
C f 2 ¼ ½0:1820 0:1773;
Moreover, we have 0:1112 0:0448 Y^ 1 ¼ ; 0:0448 0:1096
;
0:3300
0:4031
0:1004 0:4563 20:2003 Bfv2 ¼ 14:9077
;
Df 2 ¼ ½ 3:3921 4:5038
0:1177 Y^ 2 ¼ 0:0463
0 0
Time in samples
3
0
0
Bf 2 ¼
4
0.2
2
0.1
−0.1
stuck at 0.2 5 Res. eva. function
0.4
Residual signal
Res. eva. function
Residual signal
0.5
stuck at 0.15
stuck at 0.2
1.2
5
stuck at 0.1
6
stuck at 0.15
1.4
0.8
stuck at 0.05
stuck at 0.1
7
0.6
40
7 stuck at 0.05
0.7
30
Fig. 4. Residual and evaluation function of the system with servo inputs when the sensors stuck at 0.05.
8 0.9
20
Time in samples
0:0463
0:1089
^ is shown in Fig. 1. Then, according to switching law (15), σ ðxðkÞÞ Under the switching law, the residual and evaluation output in faulty and fault-free cases are shown in Fig. 2, which means the sensor outage faults are detected. Further, we compare it with the existing one without introducing servo inputs [33]. Fig. 3 shows that the previous method fails to detect the sensor outage faults.
20
40
0
20
40
Time in samples
Time in samples
Fig. 5. Residual and evaluation function of the system without servo inputs for different sensor stuck faults.
Case 2 (sensor stuck faults with small amplitudes): Assume sensor 2 of the 1st subsystem and sensor 1 of the 2nd subsystem is stuck at 0.05 when k¼ 20. Letting vi ðkÞ ¼ 0:4, our residual and evaluation output are shown in Fig. 4. Meanwhile, using the method in [33], the residual and evaluation output for different small stuck faults are shown in Fig. 5, which illustrates that our approach is more effective.
Example 2. Consider a model of the PWM (Pulse-Width-Modulation)-driven boost converter in [34]. It can be converted into a switched system with two subsystems, where the nominal system matrices are given by 2
0:94 6 0:3 A1 ¼ 4 0:25
0:1 0:95 0:06
0:06
3
0:3 7 5; 0:63
2
0:93 6 0:14 A2 ¼ 4 0:16
0:08 0:66 0:4
0:07
3
0:20 7 5 0:66
Please cite this article as: Zhong G-X, Yang G-H. Fault detection for discrete-time switched systems with sensor stuck faults and servo inputs. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.006i
G.-X. Zhong, G.-H. Yang / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 3
2
12
191:5019 6 Bf 1 ¼ 4 43:7046 60:8278
Faulty case Fault−free case 2.5
10
Res. eva. function
Residual signal
8
1
2
6
2
Threshold
2
−0.5
0 0
10
20
30
0
40
Time in samples
10
20
30
40
Time in samples
Fig. 6. Residual and evaluation function when the sensors stuck at 0.03 when k ¼ 15. 3
2
3 115:0460 6 26:7914 7 Bfv1 ¼ 4 5 36:4248
0:5898
1:0755
63:3837
3
0:5381
7 12:8387 5; 29:0708
2
12:0416
3
6 7 Bfv2 ¼ 4 15:3132 5 4:0493
C f 2 ¼ ½0:0013 0:0008 0:0001; Df 2 ¼ ½ 3:0106 3:1308 2 3 0:0068 0:0004 0:0049 6 7 0:0191 0:0044 5; Y^ 1 ¼ 4 0:0004
Fault−free case
0
3
0:2513 7 5; 0:1211
0:0133
6 Bf 2 ¼ 4 2:5579 1:1725
4 0.5
0:5358
C f 1 ¼ ½ 0:0004 0:0072 0:0109; Df 1 ¼ ½3:1960 2:9683 2 3 0:8091 0:2600 0:0368 6 0:5722 0:2466 7 Af 2 ¼ 4 0:0044 5;
2
1.5
9
12 Faulty case
0:0049
0:0074 6 Y^ 2 ¼ 4 0:0001 0:0059
0:0044 0:0001 0:0250 0:0075
0:0218 3 0:0059 0:0075 7 5 0:0230
Then our residual and evaluation function in stuck and outage cases are shown in Figs. 6 and 7, respectively, which further demonstrate the effectiveness of the proposed method.
Fault−free case
2.5
6. Conclusions
10
In this paper, the FD problem for discrete-time switched systems with sensor stuck faults and servo inputs is studied. A novel FD scheme is proposed, where the states of residual generators are used to construct the switching signals. The switching strategy guarantees the finite-frequency l2-gain and H index, which means the residuals sensitive to servo inputs in faulty cases and robust against them in fault-free case. According to the difference of the residuals in these cases, the sensor stuck and outage faults are both detected.
2
Res. eva. function
Residual signal
8 1.5
1
6
4 Fault−free case
0.5
Threshold
2
0
−0.5
Acknowledgments
0 0
10
20
30
0
40
Time in samples
10
20
30
40
Time in samples
Fig. 7. Residual and evaluation function when the outage faults occur when k ¼ 20.
Suppose other matrices to be 2 3 2 3 2 3 2 3 0:2 0:05 0:2 0:1 6 7 6 7 6 7 6 7 Bd1 ¼ 4 0:2 5; Bd2 ¼ 4 0:08 5; Bv1 ¼ 4 0:2 5; Bv2 ¼ 4 0:1 5 0:1 0:08 0:1 0:2 2 3 0:3 0:2 0:8 0:8 6 7 Dd1 ¼ ; Dd2 ¼ 4 0:2 5; Dv1 ¼ ; Dv2 ¼ 0:2 0:6 0:9 C1 ¼
0:1
0:1
0:1
0:2
0:08
0:09
;
C2 ¼
0:3
0:4
0:1
0:2
0:1
0:08
Denote vi ðkÞ ¼ 0:8 and suppose sensor 1 of the 1st subsystem and sensor 2 of the 2nd subsystem get stuck at 0.03 when k ¼15 or outage when k ¼20. Using Theorem 5, we have 2 3 0:8491 0:1661 0:3336 6 0:0380 0:7937 0:2929; 7 Af 1 ¼ 4 5; 0:0212 0:1946 0:5376
This work was supported in part by the Funds of National Science of China (Grant nos. 61273148, 61420106016), the Foundation for the Author of National Excellent Doctoral Dissertation of PR China (Grant no. 201157), and the Research Fund of State Key Laboratory of Synthetical Automation for Process Industries (Grant no. 2013ZCX01). References [1] Frank PM, Ding SX. Survey of robust residual generation and evaluation methods in observer-based fault detection systems. J Process Control 1997;7 (6):403–24. [2] Frank PM, Ding SX. Frequency domain approach to optimally robust residual generation and evaluation for model-based fault diagnosis. Automatica 1994;30(5):789–804. [3] Ding SX. Model based fault diagnosis techniques—design schemes, algorithms and tools. Berlin: Springer-Verlag; 2008. [4] Zhong MY, Ding SX, Lam J, Wang HB. An LMI approach to design robust fault detection filter for uncertain LTI systems. Automatica 2003;39(3):543–50. [5] Giovanini L, Dondo R. A fault detection and isolation filter for discrete linear systems. ISA Trans 2003;42(4):643–9. [6] Yang GH, Wang H. Fault detection for a class of uncertain state-feedback control systems. IEEE Trans Control Syst Technol 2010;18(1):201–12. [7] Li XJ, Yang GH. Fault detection for linear stochastic systems with sensor stuck faults. Optim Control Appl Methods 2010;33(1):61–80. [8] Liu J, Wang JL, Yang GHA. LMI approach to minimum sensitivity analysis with application to fault detection. Automatica 2005;41(11):1995–2004. [9] Zhang P, Ding X. An integrated trade-off design of observer-based fault detection systems. Automatica 2008;44(7):1886–94.
Please cite this article as: Zhong G-X, Yang G-H. Fault detection for discrete-time switched systems with sensor stuck faults and servo inputs. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.006i
10
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[10] Ding SX. Integrated design of feedback controllers and fault detectors. Annu Rev Control 2003;33(2):124–35. [11] Wang H, Yang GH. A finite frequency approach to filter design for uncertain discrete-time system. Int J Adapt Control Signal Process 2008;22(6):533–50. [12] Liberzon D, Morse AS. Basic problems in stability and design of switched systems. IEEE Trans Control Syst 1999;19(5):59–70. [13] Zhao J, Hill DJ. On the stability, l2-gain and H 1 control for switched systems. Automatica 2008;44(2):126–32. [14] Zhang D, Yu L, Wang QG, Ong CJ. Estimator design for discrete-time switched neural networks with asynchronous switching and time-varying delay. IEEE Trans Neural Netw Learn Syst 2012;23(5):827–34. [15] Ma Y, Kawakami H, Tse CK. Bifurcation analysis of switched dynamical systems with periodically moving borders. IEEE Trans Circuits Syst I 2004;51 (6):1184–93. [16] Allerhand LI, Shaked U. Robust state-dependent switching of linear systems with dwell time. IEEE Trans Autom Control 2013;58(4):994–1001. [17] Li J, Yang GH. Fault detection for switched linear parameter-varying systems: an average dwell-time approach. IET Control Theory Appl 2013;7(8):1120–30. [18] Lin H, Antsaklis PJ. Stability and stabilizability of switched linear systems: a survey of recent results. IEEE Trans Autom Control 2009;54(2):308–22. [19] Deaecto GS, Geromel JC, Daafouz J. Dynamic output feedback H 1 control of switched linear systems. Automatica 2012;47(8):1713–20. [20] Zhao J, Dimirovski GM. Quadratic stability of a class of switched nonlinear systems. IEEE Trans Autom Control 2004;49(4):574–8. [21] Li J, Yang GH. Asynchronous fault detection filter design approach for discretetime switched linear systems. Int J Robust Nonlinear Control 2014;24 (1):70–96. [22] Zhang D, Yu L, Zhang WA. Delay-dependent fault detection for switched linear systems with time-varying delays: the average dwell time approach. Signal Process 2011;91(4):831–40.
[23] Wang D, Wang W, Shi P. Robust fault detection for switched linear systems with state delays. IEEE Trans Syst Man Cybern Part B: Cybern 2009;39 (3):800–5. [24] Yang HJ, Xia YQ, Liu B. Fault detection for T–S fuzzy discrete systems in finitefrequency domain. IEEE Trans Syst Man Cybern Part B: Cybern 2011;41 (4):911–20. [25] Wang H, Yang GH. Fault detection for linear uncertain systems with sensor faults. IET Control Theory Appl 2010;4(6):923–35. [26] Wu LG, Ho D. Fuzzy filter design for Itô stochastic systems with application to sensor fault detection. IEEE Trans Fuzzy Syst 2009;17(1):233–42. [27] Tan CP, Edwards C. Sliding mode observers for detection and reconstruction of sensor faults. Automatica 2002;38(10):1815–21. [28] Andoh F. Moment of inertia identification using the time average of the product of torque reference input and motor position. IEEE Trans Power Electron 2007;22(6):2534–42. [29] Ding DW, Yang GH. Fuzzy filter design for nonlinear systems in finitefrequency domain. IEEE Trans Fuzzy Syst 2010;18(5):935–45. [30] Galbusera L, Bolzern P, Deaecto GS, Geromel JC. State and output feedback H 1 control of time-delay switched linear systems. Int J Robust Nonlinear Control 2012;22(15):1674–90. [31] Iwasaki T, Hara S. A generalized KYP lemma: unified frequency domain inequalities with design applications. IEEE Trans Autom Control 2005;50 (1):41–59. [32] Ding DW, Yang GH. Finite frequency H1 filter for uncertain discrete-time switched linear systems. Prog Nat Sci 2009;19(11):1625–33. [33] Wang H, Yang GH. A finite frequency domain approach to fault detection for linear discrete-time systems. Int J Control 2008;81(71):1162–71. [34] Zhang LX, Cui NG, Liu M, Zhao Y. Asychronous filtering of discrete-time switched linear systems with average dwell time. IEEE Trans Circuits Syst 2011;58(5):1109–18.
Please cite this article as: Zhong G-X, Yang G-H. Fault detection for discrete-time switched systems with sensor stuck faults and servo inputs. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.006i
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Fault detection for discrete-time switched systems with sensor stuck faults and servo inputs Guang-Xin Zhong a, Guang-Hong Yang a,b,n a b
College of Information Science and Engineering, Northeastern University, Shenyang 110819, PR China State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang 110819, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 30 October 2013 Received in revised form 17 March 2015 Accepted 16 May 2015 This paper was recommended for publication by Dr. Q.-G. Wang
This paper addresses the fault detection problem of switched systems with servo inputs and sensor stuck faults. The attention is focused on designing a switching law and its associated fault detection filters (FDFs). The proposed switching law uses only the current states of FDFs, which guarantees the residuals are sensitive to the servo inputs with known frequency ranges in faulty cases and robust against them in fault-free case. Thus, the arbitrarily small sensor stuck faults, including outage faults can be detected in finite-frequency domain. The levels of sensitivity and robustness are measured in terms of the finitefrequency H index and l2-gain. Finally, the switching law and FDFs are obtained by the solution of a convex optimization problem. & 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Fault detection Switched systems Sensor stuck faults Linear matrix inequality (LMI)
1. Introduction In recent years, the issue of fault detection (FD) has attracted amount of attention due to the increasing demand of safety and reliability standards in engineering practice [1–4]. Among the existing FD approaches, the model-based FD can replace hardware redundancy based on traditional schemes, which is widely applied for vehicle control systems, robots, power systems, manufacturing processes, see e.g. [5–8]. The basic idea of the mentioned subject is to design a detection unit to generate a signal named residual, which reflects the difference of system between faulty and faultfree cases. To this end, various types of the residual generator are proposed. For example, in [9], observer-based generator is developed to ensure detection performance by minimizing the errors between faults and residuals. To make the residuals be sensitive to faults and robust against disturbances, a mixed H 1 =H fault detection filter (FDF) is constructed in [10]. When the faults have known frequency domains, an improved frequency-based FDF is designed to reduce the conservatism [11]. On the other hand, switched systems have attracted increasing interest in the past three decades [12–14]. The systems appear in variety of applications, such as dc/dc convertors, FET transistors n Corresponding author at: College of Information Science and Engineering, Northeastern University, Shenyang 110819, PR China. Tel.: þ86 24 83681939. E-mail addresses: [email protected] (G.-X. Zhong), [email protected] (G.-H. Yang).
and HiMAT vehicle [15–17]. A switched system consists of a finite number of subsystems and a switching law orchestrating switching between these subsystems [18]. Lots of techniques have been developed to study the problems of stability and control for switched systems. For instance, [19] investigates the H 1 control problem using piecewise quadratic Lyapunov function. In [20], the issue of quadratic stability is studied based on the multiple Lyapunov-like function. In the proposed technical frameworks, some typical switching laws which depend on time, state or both are constructed. For the FD problem for switched systems, main objective is to design a switched law and residual generators. As is well known, a switched system can have good performance by properly constructing switching law. However, most relevant results pertain to time-dependent [21,22] or arbitrary switching [23]. Under the proposed switching laws, some FD performance in full-frequency domain are guaranteed. Recently, frequency-based FD approach is presented, which guarantees better FD performance [24]. Thus, it is worthwhile to consider how to design a novel switching law to satisfy finite-frequency FD performance of switched systems. On the other hand, sensor stuck fault is very common for control systems, which directly acts on the process measurement [25–27]. However, the fault type is not adequately addressed for switched systems. In the mentioned technical frameworks, the effect of sensor faults on residuals is directly considered. Clearly, when stuck faults have very small magnitudes, the fault sensitivity will be degraded. Even when the outage faults occur (stuck value is
http://dx.doi.org/10.1016/j.isatra.2015.05.006 0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: Zhong G-X, Yang G-H. Fault detection for discrete-time switched systems with sensor stuck faults and servo inputs. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.006i
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zero), it may fail to detect faults. Hence, another motivation of this paper arises: to detect arbitrarily small stuck faults including outage faults, can we present a novel FD scheme to remove the requirement of considering the relation between residuals and small sensor faults directly? Noting that the presence of servo inputs improves the performance of systems [28], we will solve the considered problem via introducing servo inputs. This paper studies the FD problem of switched systems with servo inputs and sensor stuck faults. The contributions lie in the following aspects: (i) Different from the previous results, a statedependent switching law is constructed, which uses only the states of FDFs. (ii) Under the switching law, the finite-frequency H index and l2-gain are introduced to guarantee better fault sensitivity for the system. (iii) By constructing the difference between the residuals and servo inputs in fault-free and faulty cases, arbitrarily small sensor stuck faults including outage faults can be detected. This overcomes the drawback caused by considering the relation between the residuals and sensor stuck faults. Finally, the switching law and FDFs are obtained by solving linear matrix inequalities (LMIs). The rest of the paper is organized as follows. In Section 2, some fundamental definitions, lemmas and main objectives are provided. The main results are proposed in Section 3. Section 4 gives the FD scheme. In Section 5, simulation examples are given to illustrate the effectiveness of the proposed method. Finally, some conclusions end the paper in Section 6. The notation used in this paper is fairly standard. For a matrix A, AT denotes its transpose. The Hermitian part of a square matrix M is denoted by HeðMÞ ¼ M þ M T . The symbol n within a matrix represents the symmetric entries. 0 and I represent the zero and identity matrix with the appropriate dimensions, respectively.
free and when F pli ¼ 1; p ¼ 0; l ¼ 1; …; m, all the sensors are fault0 free, that is, yF σ ðkÞ ¼ yðkÞ. Remark 1. By revisiting the existing FD method in [25], the stuck value fpli is required to be known, which enhance the sensitivity between residuals and sensor stuck faults. Hence, the method cannot be applied when fpli is unknown. In this paper, the sensor faults with unknown stuck values can be detected by introducing servo inputs. This removes the restriction of the existing results. 2.3. Fault detection filters with servo inputs For each subsystem i, the following FDF with servo input is constructed: p
^ þ 1Þ ¼ Afi xðkÞ ^ þ Bfi yF i ðkÞ þ Bfvi vi ðkÞ xðk p
^ þ Dfi yF i ðkÞ r p ðkÞ ¼ C fi xðkÞ
^ A R is the state of the filter and r p ðkÞ A R is the where xðkÞ residual. Afi, Bfi, Bfvi, Cfi and Dfi are filter matrices with appropriate dimensions, which are to be determined. Remark 2. Letting Bfvi ¼ 0 or vi ðkÞ ¼ 0, the proposed FDF is converted into the existing one without servo input [25,26]. On the other hand, it has been proven that frequency-based filter leads to less conservatism [6,24]. Thus, by constructing vi(k) in finitefrequency domain, the switched system can have better FD performance. The frequency-based FDF design conditions will be provided in this paper. Combining Eqs. (1)–(4), the overall system is described by p
where
2.1. System model Consider the following switched system: xðk þ 1Þ ¼ Aσ xðkÞ þ Bdσ dðkÞ þ Bvσ vσ ðkÞ ð1Þ m
where xðkÞ A R is the state, yðkÞ A R is the measured output, dðkÞ A Rr is the unknown input, and vσ ðkÞ A Rv is the servo input with known frequency range. The unknown input and the servo input belong to l2 ½0; 1Þ. The piecewise constant function σ ðkÞ : ½0; 1Þ-L ¼ f1; …; Ng is a switching signal, that is, when σ ðkÞ ¼ i, the ith subsystem is activated, where L is a finite set and the positive integer N is the number of subsystems. The matrices Ai, Bdi, Bvi, Ci, Ddi and Dvi are known constant matrices with the appropriate dimensions.
p
p
p
p
p
p
ð5Þ
"
# " # " # Ai Bdi 0 xðkÞ p p ~ xðkÞ ¼ ; Ai ¼ ; B di ¼ ; ^ Bfi F pi Ddi Bfi F pi C i Afi xðkÞ " # " # Bvi 0 p p p B vi ¼ ; C i ¼ Dfi F pi C i C fi ; ; F 1i ¼ Bfi ðI F pi Þ Bfi F pi Dvi þ Bfvi p
D di ¼ Dfi F pi Ddi ;
p
D vi ¼ Dfi F pi Dvi ;
p
F 2i ¼ Dfi ðI F pi Þ
2.4. Preliminaries The following definitions are given, which are essential for our derivation. Definition 1. For each senor fault mode p, switched system (5) under switching law σ is said to have a finite-frequency H index γ1, if the following inequalities
2.2. Fault model In this paper, we consider the following sensor stuck fault models: p
p
~ þ 1Þ ¼ A i xðkÞ ~ þ B di dðkÞ þ B vi vi ðkÞ þ F 1i f pi xðk p
n
s
~ þ D di dðkÞ þD vi vi ðkÞ þ F 2i f i r p ðkÞ ¼ C i xðkÞ
2. Problem formulation and preliminaries
yðkÞ ¼ C σ xðkÞ þ Ddσ dðkÞ þ Dvσ vσ ðkÞ
ð4Þ
n
p
yF σ ðkÞ ¼ F pσ yðkÞ þ ðI F pσ Þf σ ; p ¼ 0; 1; …; q ð2Þ p p p p T p where f σ ¼ f 1σ f 2σ … f mσ , f lσ ðl ¼ 1; …; mÞ is the unknown constant, which represents the stuck value of the lth sensor. p denotes the pth fault mode, q is the number of the total possible fault modes and F pσ is the diagonal matrix with the following form: ð3Þ F pσ ¼ diag F p1σ F p2σ … F pmσ ; F plσ ¼ 0 or 1 Then, for σ ¼ i, we know that, when F pli ¼ 0, p ¼ 1; …; q, the lth sensor gets stuck, when F pli ¼ 1, p ¼ 1; …; q, the lth sensor is fault-
1 X
r Tp ðkÞr p ðkÞ Z γ 21
k¼0
1 X
vTi ðkÞvi ðkÞ;
p ¼ 1; …; q
ð6Þ
k¼0
hold for all solutions of (5) such that 1 1 X θ 2X T ~ x~ T ðkÞ ~ þ 1Þ xðkÞÞð ~ ~ þ 1Þ xðkÞÞ ~ xðkÞ ðxðk xðk r 2 sin li 2 k¼0 k¼0 ð7Þ are satisfied for the low-frequency case j θi j r θli , ehθwi
1 X
~ þ 1Þ ehθ1i xðkÞÞð ~ ~ þ 1Þ ehθ2i x~ T ðkÞÞT r 0 ðxðk xðk
ð8Þ
k¼0
Please cite this article as: Zhong G-X, Yang G-H. Fault detection for discrete-time switched systems with sensor stuck faults and servo inputs. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.006i
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are satisfied for the middle-frequency case θ1i r θi r θ2i and 2.5. Problem statement
1 θ 2X T ~ þ 1Þ xðkÞÞð ~ ~ þ 1Þ xðkÞÞ ~ ~ x~ T ðkÞ ðxðk xðk Z 2 sin hi xðkÞ 2 k¼0 k¼0 1 X
ð9Þ are satisfied for the high-frequency case j θi j Z θhi , where θi is the frequency of servo input vi(k), θli, θ1i, θ2i and θhi are known constants, which represent the frequency ranges of vi(k). Definition 2. For fault-free mode p ¼0, switched system (5) under switching law σ is said to have a finite-frequency l2-gain γ2, if the following inequalities 1 X
γ
r T0 ðkÞr 0 ðkÞ r 22
k¼0
1 X
vTi ðkÞvi ðkÞ
ð10Þ
For system (5), we will design a novel switching law and FDFs with servo inputs such that the asymptotical stability of system is guaranteed. Moreover, 1. In faulty cases, maximizing the effect of servo inputs on residuals rp(k) in finite-frequency domain, while minimizing the effects of unknown input on residuals rp(k) in fullfrequency domain. Thus, (6) and the following conditions will be satisfied: 1 X
r Tp ðkÞr p ðkÞ r γ 23
k¼0
1 X
T
d ðkÞdðkÞ;
p ¼ 1; …; q
ð13Þ
k¼0
k¼0
hold for all solutions of (5) such that (7), (8), and (9) are satisfied in low-, middle-, and high-frequency ranges, respectively. Remark 3. Motivated by [29], Definitions 1 and 2 are presented, which are used to characterize the FD performance for multimodel systems. The proposed performance indexes γ1 and γ2 represent the sensitivity and robustness between the residuals and servo inputs in faulty and fault-free cases. Thus, the relation between the residuals and sensor stuck faults is not considered directly. When the faults have very small magnitudes, good fault sensitivity is still guaranteed. Also, we introduce some useful lemmas for later development. Lemma 1 (Galbusera et al. [30]). Given a nonsingular matrix V and symmetric matrices Y and Xi for all iA L satisfying " # Y I 40 ð11Þ n Xi it is possible to determine nonsingular matrices Ui, symmetric matrices Y^ i and X^ i , such that " # " # Y V Xi Ui 1 Si ¼ S ¼ 4 0; 40 ð12Þ i n Y^ i n X^ i
Lemma 2 (Projection lemma). Given a symmetric matrix matrices Γ and Λ, there exists a decision matrix X satisfying
Φ and
2. In fault-free case, the effect of servo inputs on residuals r 0 ðkÞ is minimized in finite-frequency domain, while the effect of unknown input on residuals r 0 ðkÞ is minimized in fullfrequency domain. Then, (10) and the following condition are used to characterize the robustness: 1 X
r T0 ðkÞr 0 ðkÞ r γ 24
k¼0
1 X
T
d ðkÞdðkÞ
ð14Þ
k¼0
Remark 4. Since the proposed FD method is to detect sensor stuck faults by considering the sensitivity between servo inputs and residuals in different cases, even if the sensor outage faults occur p (F pli ¼ 0, f lσ ¼ 0, l ¼ 1; …; m), the faults are still detected. The examples in Section 5 illustrate the effectiveness of the proposed method. Remark 5. For the unknown input d(k), it is difficult to obtain its frequency information. Hence, the standard l2-gain properties (13) and (14) in full-frequency domain are proposed. These properties guarantee that the residuals are robust against disturbance in faulty and fault-free cases. 3. Main results Here, we construct the following switching law: T ^ ^ σ ðxðkÞÞ ¼ argminx^ ðkÞY^ i xðkÞ iAL
ð15Þ
Φ þ Γ X ΛT þ ΛX T Γ T o 0
where Y^ i are positive definite matrices, which are to be designed.
if and only if the following projection inequalities with respect to X hold:
Remark 6. The switching law (15) is a special state-dependent switching law due to its dependence only on the states of FDFs, which is easy to design and realize in practice. In the following theorems, we will prove that the switching law guarantees the stability of system (5). In addition, compared with the existing results under arbitrary or average dwell time switching [22,23], the proposed switching law improves the FD performance by introducing the finite-frequency H index and l2-gain.
T
Γ ? ΦΓ ? o 0 T
Λ ? ΦΛ ? o0
Lemma 3 (Finsler's lemma). Let ξ A Cn , Q A Cnn , and H A Cnm . Let H ? be any matrix such that H ? H ¼ 0. Then the following conditions are equivalent: 1. ξ Q ξ o 0; 8 Hn ξ ¼ 0; ξ a 0. 2. ( Y A Rmn : Q þ HY þ Y n Hn o 0. n
3.1. Conditions for performance (6)
Theorem 1. Assume that system (5) is asymptotically stable under the switching law (15), for a given scalar γ 1 4 0 and an arbitrary nonsingular matrix V, condition (6) is satisfied, if there exist symp p metric matrices P i , Y, X i , Q i 4 0, Φ1i 4 0, and matrices Ti, T2i, Ni, Hi,
Please cite this article as: Zhong G-X, Yang G-H. Fault detection for discrete-time switched systems with sensor stuck faults and servo inputs. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.006i
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4
Dfi ð 8 iA LÞ, p ¼ 1; …; q, such that the following inequalities hold: 2 3 p p 0 Ψ 2i P i HefΦ1i g Ψ 1i 6 7 6 n Ψ p3i Ψ p4i Ψ p5i 7 6 7 ð16Þ 6 p 7o0 6 7 n n 2I Ψ 6i 4 5 n
where 2 p
Pi ¼ 4
n
p P 1i
p P 2i p P 3i
n
"
Ψ p1i ¼ "
Ψ p4i ¼
Ψ p7i
n
3
2
5;
Qi ¼4
p
#
p Q 1i
n
"
Ψ Ψ
p 35i p 45i
#
"
Ψ p5i ¼
;
Ψ Ψ
Ψ 16i ¼ YBvi þ T 1i Dvi þ T 2i ;
Ψ p23i ¼ α ¼
p i Q 2i X i þ X i Ai ;
p P 1i
β
"
5;
Ψ p3i ¼
#
p 36i p 46i
Y
Xi
n
Xi
# ;
Ψ p24i ¼ α
p i Q 3i X i þX i Ai p p 1i Dfi F i C i
μ
T
p p Ψ p44i ¼ P 3i βi Q 3i þ He X i Ai μp2i ðDfi F pi C i þ Hi Þ T
Ψ p45i ¼ 12 ðC Ti F pi DTfi þ HTi Þ þ μp2i T
7 0 5;
I
0
6 J3 ¼ 4 0 0
I
3
7 I5 0
ð24Þ
3
0
7 pT C i D vi 7 7 5 pT p 2 γ 1 I D vi D vi
T
ð25Þ
hold for given Λi, Λ i ¼ Λi Λi , 2 3 2 I I T p 6 7 p A i 7; J~ ¼ 6 0 J~ 1i ¼ 6 4 4 4 T5 p 0 B vi
and J~ 4 , where
T p J p1i J~ 1i
¼ 0. Then, letting
3 0 7 I5 0
Then, (25) is reformulated as 2 T p P pi þ HefΛi g Q pi Λi þ Λi A i 6 6 6 n Ξ pi 6 4
3
p
ΛTi B vi jT
p
p
7 7
ΛTi B vi C i D vi 7 7o0 5 T p γ 21 I D vi D vi
n
ð26Þ
where pT
pT
p
Ξ pi ¼ P pi 2 cos θli Q pi þ HefA i Λi g C i C i
pT
Ψ p46i ¼ X i Bvi ðC Ti F pi DTfi þ HTi Þμp3i μp2i Dfi F pi Dvi ;
2
n
n
Ψ p36i ¼ YBvi þ T 1i Dvi þ T 2i C Ti F i DTfi μ3i μp1i Dfi F pi Dvi T
6 J2 ¼ 4 I 0
3
Using Lemma 2 and letting 2 P pi Q pi 6 pT p P pi 2 cos θli Q pi C i C i Υ pi ¼ 6 6 n 4
p
Ψ p7i ¼ γ 21 I He μp3i Dfi F pi Dvi
T
0
Υ i þ HefJ~1i Λ i J~4 g o 0
pT
Ψ 26i ¼ X i Bvi ;
pT
I
0
we known that (23) is satisfied if
Ψ p33i Ψ p34i ; n Ψ p44i
Ψ p34i ¼ P 2i βi Q 2i þ N i þ T 1i F pi C i þ ATi X i C Ti F pi DTfi μp2i μp1i ðDfi F pi C i þ Hi Þ
T
0
2
n
#
Ψ p14i ¼ αi Q 2i X i þN i þ T 1i F pi C i
p p i Q 1i þ He YAi þ T 1i F i C i
Ψ p35i ¼ 12 C Ti F pi DTfi þ μp1i ;
p 3 B vi 7 0 5;
p
Ai 6 J p1i ¼ 4 I
p J~ 1i
p
p
Φ1i ¼ "
Ψ p13i Ψ p14i Ψ 16i ; Ψ 2i ¼ Ψ 26i Ψ p23i Ψ p24i
p
Ψ
3
#
Ψ p13i ¼ αi Q 1i Y þ YAi þ T 1i F pi C i ;
p 33i
p Q 2i p Q 3i
where 2
T
Ψ p6i ¼ 12 Dfi F pi Dvi þ μp3i
where μpmi ; m ¼ 1; 2; 3 should be given beforehand. αi ¼ 1, βi ¼ 2 cos ðθli Þ in the low-frequency range j θvi j r θli , αi ¼ ehθci , βi ¼ 2 cos ðθwi Þ, θci ¼ ðθ1i þ θ2i Þ=2 and θwi ¼ ðθ2i θ1i Þ=2 in the middle-frequency range θ1i r θvi r θ2i , and αi ¼ 1; β i ¼ 2 cos ðθhi Þ in the high-frequency range θvi Z θhi . Moreover, the switching law and FDFs can be obtained by
Note that Φ1i 4 0 implies X i and Y X i are nonsingular. Suppose 1 X i ¼ X i , we know that Φ1i 4 0 is equivalent to (11). Then, based on Lemma 1, we have " # " # Y V Xi Ui Λi ¼ Γ ¼ ; ð27Þ i n Y^ i n X^ i 1
Letting Λi
¼ Γ i , we have
U i ¼ ðI X i YÞV T Y^ i ¼ V T ðY X 1 Þ 1 V
ð28Þ
i
Y^ i ¼ V T ðY X i Þ 1 V
ð17Þ
Afi ¼ V 1 ðN i YAi ÞðX i YÞ 1 V
ð18Þ
Bfi ¼ V 1 T 1i
ð19Þ
Bfvi ¼ V 1 T 2i
ð20Þ
C fi ¼ H i ðX i YÞ 1 V
ð21Þ
Clearly, (17) is obtained. Further, denote
Δ1i ¼
Y
I
VT
0
"
;
Δ2i ¼
I
0
0
Xi
#
Ωi ¼ Γ i Δ1i Δ2i
;
ð29Þ
Δ1i and Δ2i are full rank of column. Then, preh i T T multiplying diag Ωi Ωi I and post-multiplying diag Ωi Ωi I to
Obviously,
(26) and letting " P pi ¼ Proof. First, we consider the performance condition (6) when the servo input is in low-frequency domain. For p ¼ 1; …; q, denote " # " # Q pi P pi I 0 Σ pi ¼ Π ¼ ; n γ 21 I n P pi 2 cos θli Q pi Suppose the following conditions hold for j θvi j r θli : # " p # " p # " p # " p p T p p T p C i D vi C i D vi A i B vi B vi p Ai Σi Π o0 þ 0 I 0 I I 0 I 0
ð22Þ
J p1i ½J 3 Σ i J T3 þJ 2 Π J T2 J p1i o0 T
p
ð23Þ
P p2i
n
P p3i
#
" ;
Q pi ¼
Q p1i
Q p2i
n
Q p3i
# 40
we get 2 6 6 6 4
P i þ HefΦ1i g
p p Q i Φ1i þ A~ i
n
Ξ~ i
n
n
p
p
3
p B~ vi
7 pT p 7 B~ vi C~ i D vi 7 o 0 5 T p p γ 21 I D vi D vi
ð30Þ
where pT
Ξ~ i ¼ P i 2 cos θli Q i þ HefA~ i g C~ i C~ i ; P i ¼ ΩTi P pi Ωi p p p p p T T p T T p T A~ i ¼ Δ2i Δ1i A i Ωi ; B~ vi ¼ Δ2i Δ1i B vi ; C~ i ¼ C i Ωi ; Q i ¼ Ωi Q pi Ωi p
Clearly, (22) is equivalent to the following conditions:
P p1i
p
p
p
p
p
Please cite this article as: Zhong G-X, Yang G-H. Fault detection for discrete-time switched systems with sensor stuck faults and servo inputs. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.006i
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conditions in Theorem 1 guarantee the finite-frequency H index
Noting that (30) is equivalent to T
Γ p1i Φpi Γ p1i o 0
ð31Þ
I 60 6 p Γ 1i ¼ 6 40
0 I p C~
0
0
2
3 0 0 7 7 p 7; D vi 5
i
6 6 6 p Φi ¼ 6 6 6 4
I
p
P i HefΦ1i g
p p Q i Φ1i þ A~ i
0
n
Ξ^ i
pT 12C~ i
n
n
0
n
n
n
p
p B~ vi
3
7 7 7 7 7 p 12D vi 7 5 2 γ1I
p
p
p
Letting h i p p Rpi ¼ 0 C~ i I D vi
ð32Þ
we have Rpi Γ 1i ¼ 0. Using Lemma 3, we see (31) is guaranteed if the inequities p
Φpi þ HefLpi Rpi g o 0
ð33Þ
hold for some matrices p
Lji.
Next, setting
p
p
Q 1i ¼ Q p1i ;
P 3i ¼ P p1i þ HefP p2i U Ti X i g þ X i U i P p3i U Ti X i
p
N i ¼ ðYAi X i Þ þ VAci U Ti X i ;
Q 3i ¼ Q j1i þ HefQ p2i U Ti X i g þ X i U i Q p3i U Ti X i
H i ¼ C fi U Ti X i ;
and constructing h i T T T Lpi ¼ 0 μ p1i I μp3i ;
T
h
T 1i ¼ VBfi ;
T
T
μ p1i ¼ μp1i μp2i
T 2i ¼ VBfvi
i
ð35Þ Using zero initial condition and stability property of system (5), by accumulating both sides of (35) for k from 0 to 1, we get ð r Tp ðkÞr p ðkÞ þ γ 21 vTi ðkÞvi ðkÞÞ þ tr½Q ji Si o 0
ð36Þ
k¼0
where ~ þ 1Þ þ x~ T ðk þ 1ÞxðkÞ ~ 2 cos θli x~ T ðkÞxðkÞÞ ~ ðx~ T ðkÞxðk
k¼0
According to Definition 1, we have Si Z 0, which implies 1 X k¼0
r Tp ðkÞr p ðkÞ Z γ 21
vTi ðkÞvi ðkÞ;
Σ p24i ¼ N Ti þ C Ti F i T T1i ; pT
p 34i
T
¼ BTdi Y þ DTdi F pi T T1i ;
Σ 15i ¼ ATi X i ;
Σ p16i ¼ C Ti F pi DTfi T
Σ 25i ¼ ATi X i ;
Σ p26i ¼ C Ti F i DTfi þ HTi pT
Σ 35i ¼ BTdi X i ;
T
Σ p36i ¼ DTdi F pi DTfi
Proof. We choose the following Lyapunov functional candidate: ~ ~ ~ VðxðkÞÞ ¼ V σ ðxðkÞÞ ðxðkÞÞ ¼ x~ T ðkÞΛσ ðxðkÞÞ xðkÞ ~ ~
ð38Þ
where Λi is introduced via (27). Then, a state-dependent switching law is constructed: ~ ~ σ ðxðkÞÞ ¼ arg minx~ T ðkÞΛi xðkÞ
ð39Þ
iAL
Obviously, the switching law (39) is equivalent to (15). For σ ðkÞ ¼ σ ðk þ 1Þ ¼ i, we have
~ þ 1Þ x~ T ðkÞΛi xðkÞ ~ ¼ x~ T ðk þ 1ÞΛi xðk pT
pT
p
p
~ þ 2x~ T ðkÞA i Λi B di dðkÞ ¼ x~ T ðkÞA i Λi A i xðkÞ T
p
p
þ d ðkÞB di Λi B di dðkÞ T
By Schur complement, if the following inequities hold: 2 3 pT pT 0 A i Λi C i Λi 6 7 6 pT pT 7 2 6 n 7 γ I B Λ D i i di 7 o 0 3 6 6 7 n Λi 0 5 4 n n
n
n
ð40Þ
I
we get
Hence, we have 1 X
I
n
~ þ 1ÞÞ V i ðxðkÞÞ ~ ¼ V i ðxðk
~ þ 1Þ þ x~ T ðk þ 1ÞxðkÞ ~ 2 cos θli x~ T ðkÞxðkÞÞ ~ þ tr½Q pi ðx~ T ðkÞxðk o0
tr½Q ji Si Z 0.
n
~ þ 1ÞÞ V σ ðkÞ ðxðkÞÞ ~ ΔVðkÞ ¼ V σðk þ 1Þ ðxðk
~ x~ T ðk þ 1ÞP pi x~ Ti ðk þ1Þ r Tp ðkÞr p ðkÞ þ γ 21 vTi ðkÞvi ðkÞ x~ T ðkÞP pi xðkÞ
1 X
n
Moreover, Y^ i , Afi, Bfi and Cfi can be constructed by (17), (18), (19) and (21), respectively.
which follows
Si ¼
n
Σ p14i ¼ ATi Y þ C Ti F pi T T1i ; Σ
where μp1i ; μp2i and μp3i ; are turning parameters. Then we obtain (16), which guarantees # " # #" p " #T 8 " p p T Pp p < ~ Q pi xðkÞ A i B vi A i B vi i vi ðkÞ : I Q pi P pi 2 cos θli Q pi 0 I 0 9 2 3 T T " # p p p p = xðkÞ ~ C i D vi C i C i 5 þ4 o0 ð34Þ pT p pT p v ; 2 i ðkÞ D vi C i D vi D vi þ γ 1 I
1 X
n
T
p
Q 2i ¼ Q j1i þ Q p2i U Ti X i ;
Theorem 2. For a given scalar γ 3 4 0 and an arbitrary nonsingular matrix V, system (5) is asymptotically stable and satisfies (13) under the switching law (15), if there exist symmetric matrices Y, X i and matrices T1i, Ni, Hi, Dfi ð 8 i A LÞ, p ¼ 1; …; q, such that the following inequalities hold: 2 3 Y X i 0 Σ p14i Σ 15i Σ p16i 6 p p 7 6 n Xi 0 Σ 24i Σ 25i Σ 26i 7 6 7 p p 6 n 2 n γ 3 I Σ 34i Σ 35i Σ 36i 7 6 7 ð37Þ 6 7 o0 6 n n n Y Xi 0 7 6 7 6 7 4 n n n n Xi 0 5 where
p
P 2i ¼ P p1i þ P p2i U Ti X i ;
3.2. Conditions for stability and performance (13)
p B~ vi
Ξ^ i ¼ P i 2 cos θl Q i þ HefA~ i g
P 1i ¼ P p1i ;
γ1. Moreover, using the proposed multi-model approach, we known that γ1 characterizes the level of sensitivity for all the faulty systems in finite-frequency domain.
where 2
5
j θvi j r θli
k¼0
which means (6) is guaranteed in low-frequency case. Further, according to the similar lines in the proof above, we can obtain the sufficient conditions for middle- and high-frequency cases, which complete the proof.□ Remark 7. Based on the generalized Kalman–Yakubovich–Popov (GKYP) Lemma in [31,32], the proposed frequency-based
ΔVðkÞ þ γ 23 rTp ðkÞrp ðkÞ r dT ðkÞdðkÞ
ð41Þ
By the similar approach in Theorem 1, we know (37) is sufficient for (41). For σ ðkÞ ¼ i; σ ðk þ 1Þ ¼ jð 8 i ajÞ, the switching law (15) guarantees ~ þ 1ÞÞ V σ ðkÞ ðxðkÞÞ ~ ΔVðkÞ ¼ V σðk þ 1Þ ðxðk ~ þ 1ÞÞ V i ðxðkÞ ~ ¼ V j ðxðk ~ þ1ÞÞ V i ðxðkÞÞ ~ r V i ðxðk
ð42Þ
Please cite this article as: Zhong G-X, Yang G-H. Fault detection for discrete-time switched systems with sensor stuck faults and servo inputs. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.006i
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6
~ ~ which means VðxðkÞÞ ¼ V σ ðxðkÞÞ is the overall Lyapunov function of system (5). Clearly, (41) can be obtained again. Therefore, it follows 1 X
ΔVðkÞ o
k¼0
1 X
γ
1 X
2 T 3 r p ðkÞr p ðkÞ þ
k¼0
T
d ðkÞdðkÞ
ð43Þ
k¼0
In addition, system (5) is asymptotically stable under switching law (15) in the case of dðkÞ ¼ 0. Next, using zero initial condition and Vðxð1ÞÞ Z 0, we have 1 X
r Tp ðkÞr p ðkÞ r γ 23
k¼0
1 X
T
d ðkÞdðkÞ
k¼0
which completes the proof.□ 3.3. Conditions for performance (10) In order to attenuate the effect of servo inputs on residuals in fault-free case, the following theorem is presented. Theorem 3. Assume that the system (5) is asymptotically stable 0 under the switching law (15), for given scalars γ 2 4 0; li and an arbitrary nonsingular matrix V, condition (10) is satisfied in low-, middle- and high-frequency ranges, respectively, if there exist sym0 0 metric matrices P~ i , Y, X i , Q~ i 4 0, Φ1i 40 and matrices T1i, T2i, Ni Dfi ð 8 i A LÞ, such that the following inequities hold: 2 3 0 0 0 0 P~ i Ψ~ 1i 6 7 0 0 0 6 7 6 n Ψ~ 2i Ψ~ 3i Ψ~ 4i 7 ð44Þ 6 7o0 T 6 7 4 n n γ 22 I DTvi F 0i DTfi 5 n
n
I
n
where Φ1i, αi, and βi are defined in Theorem 1: 2 0 3 2 0 3 2 0 3 0 0 ~ ~0 Ψ Ψ P~ 1i P~ 2i Q~ 1i Q~ 2i 0 0 0 33i 34i ~ 5; Q~ ¼ 4 5; Ψ ¼ 4 5 P~ i ¼ 4 i 2i 0 0 0 n P~ 3i n Q~ 3i n Ψ~ 44i T h i T T 0T 0 0T 0T Ψ~ ¼ Ψ~ Ψ~ ; Ψ~ ¼ C T F 0 DT C T F 0 DT þ H T 3i
35i
45i
Ψ~ 1i ¼ αi Q~ i l0i Φ1i ; 0
0
i
4i
i
fi
i
i
fi
0
0
Ψ~ 44i ¼ P~ 3i βi Q~ 3i þHefl0i X i Ai g; 0
0
0
Ψ 045i ¼ l0i X i Bvi
Ψ~ 35i ¼ l0i ðYBvi þ T 1i F 0i Dvi þ T 2i Þ 0
Proof. For fault-free case (q ¼0), letting 2 T 3 2 T 3 " 0 " # 0 0 I 7 0 A0 B0 Ci I7 Ci 6 6 Ai ~ ~ vi i Σ Π þ 4 0T 5 i 4 0T 5 i I I 0 D vi 0 B vi 0
0
D vi 0
# o0
ð45Þ
In low-frequency range j θvi j o θli , suppose 2 3 " # 0 0 I 0 P^ i Q^ i 0 ~ ¼ 5; Π Σ~ i ¼ 4 0 0 0 n γ 22 I Q^ i P^ i 2 cos θli Q^ i
J 02i ½J 3 Σ~
Ψ
hold, where Λi is introduced via (27) and 2 3 0 0 0 P^ i Q^ i 6 7 T T 6 0 7 0 ^ ^ 0 2 cos θ Q^ 0 þ C 0 C 0 C 0 D 0 7; Υ~ i ¼ 6 P Q l i vi 7 i i i i 6 i 4 5 0T 0 0 D vi C i γ 22 I 2 3 2 3 I 0 0T 7 6 T 6 0 7 0 0 A l i 7; J ¼ J 3i ¼ 6 4 iI5 4i 4 T5 0 0 B vi where l0i are scalars should be known beforehand, which follows 2 3 0 0 0 0 0 P^ i Q^ i li Λi 6 7 6 0T 7 0 ^0 6 n 7 Ξ Λ B C i i vi 1i 6 7o0 ð49Þ 6 T 7 0 7 6 n γ 22 I D vi 5 4 n n n n I where 0
0T
Ξ^ 1i ¼ P^ i 2 cos θli Q^ i þ HefA i Λi g 0
0
Using the same approaches in Theorem 1, we obtain (44), which means 3" # 2 # " #T 8" 0 0 0 0 T 0 ^0 < ~ Q^ i xðkÞ A i B vi 4 P i 5 A i B vi 0 0 vi ðkÞ : I 0 I 0 Q^ i P^ i 2 cos θli Q 0i 9 2 T 3 T " # 0 0 0 0 > ~ C i D vi 7= xðkÞ 6 Ci Ci o0 ð50Þ þ 4 0T 0 5 T 0 0 ; vi ðkÞ D vi C i D vi D vi γ 2 I >
0 T 0 fi J 2 J 2i o 0
where J2 and J3 are obtained from (24) and 2 0 2 T 3 0 3 0 0 0T A i B vi C C C D vi i 6 7 6 i i 7 0 J 02i ¼ 4 I 5 0 5; Ψ fi ¼ 4 0T 0 0T 0 Dvi C i D vi D vi γ 22 I 0 I
3.4. Conditions for performance (14) Based on Theorem 2, the following theorem is provided to guarantee (14) for fault-free system. Theorem 4. For a given scalar γ 4 40 and an arbitrary nonsingular matrix V, system (5) is asymptotically stable and satisfies (14) under the switching law (15), if there exist symmetric matrices Y, X i and matrices Y, X i , T1i, Ni, Hi, Dfi ð 8 iA LÞ, such that the following inequalities hold: 2 3 0 Σ 014i Σ 15i Σ 016i Y X i 6 7 6 n Xi 0 Σ 024i Σ 15i Σ 026i 7 6 7 6 7 0 0 7 2 6 n n γ I Σ Σ Σ 35i 34i 36i 7 o 0 6 4 ð51Þ 6 7 6 n n n Y Xi 0 7 6 7 6 n n n n Xi 0 7 4 5 n n n n n I where
we see that (45) is satisfied if the following inequities hold: 0 T i J 3 þJ 2
ð48Þ
Then, according to the proof in Theorem 1, it is easy to complete this proof.□
i
Moreover, Y^ i , Afi, Bfi, Bfvi and Cfi can be constructed by (17), (18), (19), (20) and (21), respectively.
T
0
0
0
Ψ~ 34i ¼ P~ 2i βi Q~ 2i þl0i N i þ l0i ATi X i þ l0i T 1i F 0i C i ; 0
Υ~ i þ HefJ 03i Λi J 04i g o 0
2
Ψ~ 33i ¼ P~ 1i βi Q~ 1i þ Hefl0i ðYAi þ T 1i F 0i C i Þg 0
By the similar approach, (46) can be satisfied if
T
Σ 014i ¼ ATi Y þ C Ti F 0i T T1i ; ð46Þ
T
Σ 024i ¼ N Ti þ C Ti F 0i T T1i ; Σ 034i ¼ BTdi Y þ DTdi F T0i T T1i ;
ð47Þ
T
Σ 15i ¼ ATi X i ;
Σ 016i ¼ C Ti F 0i DTfi T
Σ 026i ¼ C Ti F 0i DTfi þ HTi Σ 35i ¼ BTdi X i ;
T
Σ 036i ¼ DTdi F 0i DTfi
Moreover, Y^ i , Afi, Bfi and Cfi can be constructed by (17), (18), (19) and (21), respectively.
Please cite this article as: Zhong G-X, Yang G-H. Fault detection for discrete-time switched systems with sensor stuck faults and servo inputs. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.006i
G.-X. Zhong, G.-H. Yang / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Remark 8. Based on Theorems 1–4, we know that FDFs (4) are valid for multiple switched systems, that is, faulty switched systems ðp ¼ 1; …; qÞ and fault-free switched system ðp ¼ 0Þ. Moreover, the internal stability of these systems can be guaranteed by Theorems 2 and 4. Remark 9. Letting Y i ¼ Y, N i ¼ N, T 1i ¼ T 1 , T 2i ¼ T 2 , X i ¼ X i , H i ¼ H and Dfi ¼ Df , the results in Theorems 1–4 can be used to design detector for linear time-invariant (LTI) systems with finitefrequency servo inputs.
7
where p ¼ 0; …; q. Further, denote following detection threshold as J th ¼ sup J 0 ðkÞ
ð53Þ
dðkÞ A l2
Then, the sensor stuck faults can be detected by the following test: ( J J p ðkÞ J rJ th Fault free ð54Þ J J p ðkÞ J 4J th Faulty
5. Examples In this section, two examples are given to illustrate the effectiveness of our method.
4. Fault detection scheme Summarily, the following theorem is given to solve the FD problem. Theorem 5. Under the switching law (15), system (5) is asymptotically stable and satisfies (6), (10), (13) and (14), if there exist 0 0 p p symmetric matrices P~ , P , Q~ 4 0, Q 4 0, Y, X i and matrices T1i, i
i
i
i
T2i, Ni, Hi, Dfi ð 8 i A LÞ, p ¼ 1; …; q, such that LMIs (16), (37), (44) and (51) hold. Moreover, Y^ i , Afi, Bfi, Bfvi and Cfi can be obtained from (17), (18), (19), (20) and (21), respectively. Finally, the FD problem is solved via the following optimization: max γ 1
Example 1. Consider the switched system (1), which consists of the two subsystems: 0:8 0 0:2 0:7 A1 ¼ ; Bd1 ¼ ; Bv1 ¼ 0:2 0:7 0:3 0:8 0:3 0:1 0:1 0:6 C1 ¼ ; Dd1 ¼ ; Dv1 ¼ 0:2 0:3 0:2 0:9 0:7 0:1 0:3 0:5 ; Bd2 ¼ ; Bv2 ¼ A2 ¼ 0:1 0:8 0:2 0:9 0:2 0:1 0:2 0:7 ; Dd2 ¼ ; Dv2 ¼ C2 ¼ 0:3 0:3 0:1 0:7
s:t: (16), (37), (44) and (51) for given scalars
γ2, γ3 and γ4.
To reduce the conservatism in LMI computation, the following algorithm is given. Algorithm 1. Let ϵ be a given large enough constant specifying a stop criterion of the algorithm. Step 1: Maximize γ1 subject to the LMIs (16), (37), (44) and (51). ~ 0ð0Þ , P pð0Þ , Q~ 0ð0Þ , Q pð0Þ , The optimal solutions are denoted as γ ð0Þ i i i 1 , Pi ð0Þ
ð0Þ ð0Þ ð0Þ ð0Þ Y ð0Þ , X i , T ð0Þ 1i , T 2i , N i , H i , and Dfi . 0ð0Þ 0ð0Þ pð0Þ pð0Þ ð0Þ Y ð0Þ , X i and Dð0Þ into Step 2: Substitute P~ i , P i , Q~ i , Q i fi
γ 2 ¼ γ 4 ¼ 1, μ1i ¼ ½0:9 0:8T , Chosen vi ðkÞ ¼ 0:5 ðθvi ¼ 0Þ, T μ2i ¼ ½0:8 0:7 , μ3i ¼ 4:1 and li ¼1 for 8 i A f1; 2g, we consider the following cases: Case 1 (sensor outage (stuck values are zero) case): Suppose the sensor 2 of the 1st subsystem and sensor 1 of the 2nd subsystem get outage at 20 time step. Then, γ1 for different γ3 are shown in Table 1. Obviously, there is a trade-off between sensitivity and robustness. Based on Theorem 5, we obtain the following FDFs: 0:5723 0:3016 15:1583 0:2149 Af 1 ¼ ; Bf 1 ¼ ; 0:3060 0:2105 32:1568 2:6434 11:9692 Bfv1 ¼ ; C f 1 ¼ ½0:1571 0:0855; 15:9948
LMIs (16), (37), (44) and (51), wherein γ1, T1i, T2i, Ni and Hi are seen as unknown matrices to be determined. Further, maximize
2.5
γ1
ð1Þ subject to the above LMIs and denote their solutions as γ ð1Þ 1 , T 1i , ð1Þ and H ð1Þ T ð1Þ 2i , N i i . Next, substitute them into the above LMIs again 0ð1Þ 0ð1Þ pð1Þ pð1Þ ð1Þ and denote P~ i , P i , Q~ i , Q i , Y ð1Þ , X i and Dfið1Þ as unknown ðnÞ ðnÞ T 1i , T ðnÞ and H ðnÞ are 2i , N i i ðnÞ If 1 o , repeat the above
the solutions of the nth optimization. γ optimization, else continue. Step 3: When γ ðnÞ 1 Z ϵ, for any n, stop.
ϵ
According to above scheme, the residual evaluation function Jp(k) is chosen as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u k uX ð52Þ r Tp ðkÞr p ðkÞ J p ðkÞ ¼ t
2
Switching signal
matrices. Solve these LMIs and let γ
ðnÞ 1 ,
1.5
1
k ¼ k0
Table 1 Comparison of the finite frequency H index γ1 for different γ3. γ3 γ1
0.8 3.0258
1 3.3825
1.2 3.5986
1.4 3.6387
0.5 0 1.6 3.7120
5
10
15
20
25
30
35
40
Time in samples Fig. 1. Switching signal.
Please cite this article as: Zhong G-X, Yang G-H. Fault detection for discrete-time switched systems with sensor stuck faults and servo inputs. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.006i
G.-X. Zhong, G.-H. Yang / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
8
2.5
1.8
10
7
Faulty case 9 2
Faulty case
Faulty case 1.6
Fault−free case
Fault−free case
6
8
1.4
7
1.2
Faulty case 5
5 4
0.8 0.6
Fault−free case
0.5
Res. eva. function
1
1
6
Residual signal
Res. eva. function
Residual signal
1.5
0.4
Fault−free case
3
4
3
2 Threshold
Threshold
0.2
2 0
0
1
1 −0.2 −0.5
0
0 0
10
20
30
0
40
10
20
30
0
40
10
30
0
40
Time in samples
Time in samples
Time in samples
20
Fig. 2. Residual and evaluation function of the system with servo inputs in outage case.
10
1.8 1.6
6
0.3 0.2
4
1 0.8 0.6
3
0.4
2
1
1 −0.2
0
20
40
0
0
20
40
Time in samples
Fig. 3. Residual and evaluation function of the system without servo inputs in outage case.
Df 1 ¼ ½5:2290 2:5149;
Af 2 ¼
5:7842
24:0894
0:5506
12:1864
C f 2 ¼ ½0:1820 0:1773;
Moreover, we have 0:1112 0:0448 Y^ 1 ¼ ; 0:0448 0:1096
;
0:3300
0:4031
0:1004 0:4563 20:2003 Bfv2 ¼ 14:9077
;
Df 2 ¼ ½ 3:3921 4:5038
0:1177 Y^ 2 ¼ 0:0463
0 0
Time in samples
3
0
0
Bf 2 ¼
4
0.2
2
0.1
−0.1
stuck at 0.2 5 Res. eva. function
0.4
Residual signal
Res. eva. function
Residual signal
0.5
stuck at 0.15
stuck at 0.2
1.2
5
stuck at 0.1
6
stuck at 0.15
1.4
0.8
stuck at 0.05
stuck at 0.1
7
0.6
40
7 stuck at 0.05
0.7
30
Fig. 4. Residual and evaluation function of the system with servo inputs when the sensors stuck at 0.05.
8 0.9
20
Time in samples
0:0463
0:1089
^ is shown in Fig. 1. Then, according to switching law (15), σ ðxðkÞÞ Under the switching law, the residual and evaluation output in faulty and fault-free cases are shown in Fig. 2, which means the sensor outage faults are detected. Further, we compare it with the existing one without introducing servo inputs [33]. Fig. 3 shows that the previous method fails to detect the sensor outage faults.
20
40
0
20
40
Time in samples
Time in samples
Fig. 5. Residual and evaluation function of the system without servo inputs for different sensor stuck faults.
Case 2 (sensor stuck faults with small amplitudes): Assume sensor 2 of the 1st subsystem and sensor 1 of the 2nd subsystem is stuck at 0.05 when k¼ 20. Letting vi ðkÞ ¼ 0:4, our residual and evaluation output are shown in Fig. 4. Meanwhile, using the method in [33], the residual and evaluation output for different small stuck faults are shown in Fig. 5, which illustrates that our approach is more effective.
Example 2. Consider a model of the PWM (Pulse-Width-Modulation)-driven boost converter in [34]. It can be converted into a switched system with two subsystems, where the nominal system matrices are given by 2
0:94 6 0:3 A1 ¼ 4 0:25
0:1 0:95 0:06
0:06
3
0:3 7 5; 0:63
2
0:93 6 0:14 A2 ¼ 4 0:16
0:08 0:66 0:4
0:07
3
0:20 7 5 0:66
Please cite this article as: Zhong G-X, Yang G-H. Fault detection for discrete-time switched systems with sensor stuck faults and servo inputs. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.006i
G.-X. Zhong, G.-H. Yang / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 3
2
12
191:5019 6 Bf 1 ¼ 4 43:7046 60:8278
Faulty case Fault−free case 2.5
10
Res. eva. function
Residual signal
8
1
2
6
2
Threshold
2
−0.5
0 0
10
20
30
0
40
Time in samples
10
20
30
40
Time in samples
Fig. 6. Residual and evaluation function when the sensors stuck at 0.03 when k ¼ 15. 3
2
3 115:0460 6 26:7914 7 Bfv1 ¼ 4 5 36:4248
0:5898
1:0755
63:3837
3
0:5381
7 12:8387 5; 29:0708
2
12:0416
3
6 7 Bfv2 ¼ 4 15:3132 5 4:0493
C f 2 ¼ ½0:0013 0:0008 0:0001; Df 2 ¼ ½ 3:0106 3:1308 2 3 0:0068 0:0004 0:0049 6 7 0:0191 0:0044 5; Y^ 1 ¼ 4 0:0004
Fault−free case
0
3
0:2513 7 5; 0:1211
0:0133
6 Bf 2 ¼ 4 2:5579 1:1725
4 0.5
0:5358
C f 1 ¼ ½ 0:0004 0:0072 0:0109; Df 1 ¼ ½3:1960 2:9683 2 3 0:8091 0:2600 0:0368 6 0:5722 0:2466 7 Af 2 ¼ 4 0:0044 5;
2
1.5
9
12 Faulty case
0:0049
0:0074 6 Y^ 2 ¼ 4 0:0001 0:0059
0:0044 0:0001 0:0250 0:0075
0:0218 3 0:0059 0:0075 7 5 0:0230
Then our residual and evaluation function in stuck and outage cases are shown in Figs. 6 and 7, respectively, which further demonstrate the effectiveness of the proposed method.
Fault−free case
2.5
6. Conclusions
10
In this paper, the FD problem for discrete-time switched systems with sensor stuck faults and servo inputs is studied. A novel FD scheme is proposed, where the states of residual generators are used to construct the switching signals. The switching strategy guarantees the finite-frequency l2-gain and H index, which means the residuals sensitive to servo inputs in faulty cases and robust against them in fault-free case. According to the difference of the residuals in these cases, the sensor stuck and outage faults are both detected.
2
Res. eva. function
Residual signal
8 1.5
1
6
4 Fault−free case
0.5
Threshold
2
0
−0.5
Acknowledgments
0 0
10
20
30
0
40
Time in samples
10
20
30
40
Time in samples
Fig. 7. Residual and evaluation function when the outage faults occur when k ¼ 20.
Suppose other matrices to be 2 3 2 3 2 3 2 3 0:2 0:05 0:2 0:1 6 7 6 7 6 7 6 7 Bd1 ¼ 4 0:2 5; Bd2 ¼ 4 0:08 5; Bv1 ¼ 4 0:2 5; Bv2 ¼ 4 0:1 5 0:1 0:08 0:1 0:2 2 3 0:3 0:2 0:8 0:8 6 7 Dd1 ¼ ; Dd2 ¼ 4 0:2 5; Dv1 ¼ ; Dv2 ¼ 0:2 0:6 0:9 C1 ¼
0:1
0:1
0:1
0:2
0:08
0:09
;
C2 ¼
0:3
0:4
0:1
0:2
0:1
0:08
Denote vi ðkÞ ¼ 0:8 and suppose sensor 1 of the 1st subsystem and sensor 2 of the 2nd subsystem get stuck at 0.03 when k ¼15 or outage when k ¼20. Using Theorem 5, we have 2 3 0:8491 0:1661 0:3336 6 0:0380 0:7937 0:2929; 7 Af 1 ¼ 4 5; 0:0212 0:1946 0:5376
This work was supported in part by the Funds of National Science of China (Grant nos. 61273148, 61420106016), the Foundation for the Author of National Excellent Doctoral Dissertation of PR China (Grant no. 201157), and the Research Fund of State Key Laboratory of Synthetical Automation for Process Industries (Grant no. 2013ZCX01). References [1] Frank PM, Ding SX. Survey of robust residual generation and evaluation methods in observer-based fault detection systems. J Process Control 1997;7 (6):403–24. [2] Frank PM, Ding SX. Frequency domain approach to optimally robust residual generation and evaluation for model-based fault diagnosis. Automatica 1994;30(5):789–804. [3] Ding SX. Model based fault diagnosis techniques—design schemes, algorithms and tools. Berlin: Springer-Verlag; 2008. [4] Zhong MY, Ding SX, Lam J, Wang HB. An LMI approach to design robust fault detection filter for uncertain LTI systems. Automatica 2003;39(3):543–50. [5] Giovanini L, Dondo R. A fault detection and isolation filter for discrete linear systems. ISA Trans 2003;42(4):643–9. [6] Yang GH, Wang H. Fault detection for a class of uncertain state-feedback control systems. IEEE Trans Control Syst Technol 2010;18(1):201–12. [7] Li XJ, Yang GH. Fault detection for linear stochastic systems with sensor stuck faults. Optim Control Appl Methods 2010;33(1):61–80. [8] Liu J, Wang JL, Yang GHA. LMI approach to minimum sensitivity analysis with application to fault detection. Automatica 2005;41(11):1995–2004. [9] Zhang P, Ding X. An integrated trade-off design of observer-based fault detection systems. Automatica 2008;44(7):1886–94.
Please cite this article as: Zhong G-X, Yang G-H. Fault detection for discrete-time switched systems with sensor stuck faults and servo inputs. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.006i
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[10] Ding SX. Integrated design of feedback controllers and fault detectors. Annu Rev Control 2003;33(2):124–35. [11] Wang H, Yang GH. A finite frequency approach to filter design for uncertain discrete-time system. Int J Adapt Control Signal Process 2008;22(6):533–50. [12] Liberzon D, Morse AS. Basic problems in stability and design of switched systems. IEEE Trans Control Syst 1999;19(5):59–70. [13] Zhao J, Hill DJ. On the stability, l2-gain and H 1 control for switched systems. Automatica 2008;44(2):126–32. [14] Zhang D, Yu L, Wang QG, Ong CJ. Estimator design for discrete-time switched neural networks with asynchronous switching and time-varying delay. IEEE Trans Neural Netw Learn Syst 2012;23(5):827–34. [15] Ma Y, Kawakami H, Tse CK. Bifurcation analysis of switched dynamical systems with periodically moving borders. IEEE Trans Circuits Syst I 2004;51 (6):1184–93. [16] Allerhand LI, Shaked U. Robust state-dependent switching of linear systems with dwell time. IEEE Trans Autom Control 2013;58(4):994–1001. [17] Li J, Yang GH. Fault detection for switched linear parameter-varying systems: an average dwell-time approach. IET Control Theory Appl 2013;7(8):1120–30. [18] Lin H, Antsaklis PJ. Stability and stabilizability of switched linear systems: a survey of recent results. IEEE Trans Autom Control 2009;54(2):308–22. [19] Deaecto GS, Geromel JC, Daafouz J. Dynamic output feedback H 1 control of switched linear systems. Automatica 2012;47(8):1713–20. [20] Zhao J, Dimirovski GM. Quadratic stability of a class of switched nonlinear systems. IEEE Trans Autom Control 2004;49(4):574–8. [21] Li J, Yang GH. Asynchronous fault detection filter design approach for discretetime switched linear systems. Int J Robust Nonlinear Control 2014;24 (1):70–96. [22] Zhang D, Yu L, Zhang WA. Delay-dependent fault detection for switched linear systems with time-varying delays: the average dwell time approach. Signal Process 2011;91(4):831–40.
[23] Wang D, Wang W, Shi P. Robust fault detection for switched linear systems with state delays. IEEE Trans Syst Man Cybern Part B: Cybern 2009;39 (3):800–5. [24] Yang HJ, Xia YQ, Liu B. Fault detection for T–S fuzzy discrete systems in finitefrequency domain. IEEE Trans Syst Man Cybern Part B: Cybern 2011;41 (4):911–20. [25] Wang H, Yang GH. Fault detection for linear uncertain systems with sensor faults. IET Control Theory Appl 2010;4(6):923–35. [26] Wu LG, Ho D. Fuzzy filter design for Itô stochastic systems with application to sensor fault detection. IEEE Trans Fuzzy Syst 2009;17(1):233–42. [27] Tan CP, Edwards C. Sliding mode observers for detection and reconstruction of sensor faults. Automatica 2002;38(10):1815–21. [28] Andoh F. Moment of inertia identification using the time average of the product of torque reference input and motor position. IEEE Trans Power Electron 2007;22(6):2534–42. [29] Ding DW, Yang GH. Fuzzy filter design for nonlinear systems in finitefrequency domain. IEEE Trans Fuzzy Syst 2010;18(5):935–45. [30] Galbusera L, Bolzern P, Deaecto GS, Geromel JC. State and output feedback H 1 control of time-delay switched linear systems. Int J Robust Nonlinear Control 2012;22(15):1674–90. [31] Iwasaki T, Hara S. A generalized KYP lemma: unified frequency domain inequalities with design applications. IEEE Trans Autom Control 2005;50 (1):41–59. [32] Ding DW, Yang GH. Finite frequency H1 filter for uncertain discrete-time switched linear systems. Prog Nat Sci 2009;19(11):1625–33. [33] Wang H, Yang GH. A finite frequency domain approach to fault detection for linear discrete-time systems. Int J Control 2008;81(71):1162–71. [34] Zhang LX, Cui NG, Liu M, Zhao Y. Asychronous filtering of discrete-time switched linear systems with average dwell time. IEEE Trans Circuits Syst 2011;58(5):1109–18.
Please cite this article as: Zhong G-X, Yang G-H. Fault detection for discrete-time switched systems with sensor stuck faults and servo inputs. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.006i
