Observer-based Fault Detection for Piecewise Linear Systems:Discrete-time Cases

Jun Xu and Kai Yew Lum and Ai Poh Loh

Abstract— In this paper, we discuss the fault detection withunknown inputs for a class of discrete-time piecewise linearsystems. Piecewise linear systems are mostly partitioned basedon their state variables. Due to the system noise and estimationerrors, the transitions of actual state and its estimate may notbe synchronized, as well as the system modes. Motivated bythe recent works [31], [29], [32], [30], [27], we consider thefault detection problem using non-synchronized observer andpresents several less conservative design approaches.

Index Terms— Fault detection; piecewise linear systems;linear matrix inequality (LMI); Bilinear matrix inequality(BMI)

I. INTRODUCTION

Piecewise linear (PWL) systems have a wide range ofapplications in engineering. A large class of nonlinearcomponents, such as relay and saturation, are piecewiselinear [16]. Some special classes of hybrid systems andswitched systems are equivalent to PWL systems [14], [6].In fact, due to its universal approximation properties, smoothnonlinear systems can be analyzed in a PWL way [16].Meanwhile, a large class of intelligent systems, such as fuzzylogic (neural) systems, are modelled as PWL systems in [16].Thus PWL system is a powerful tool for analysis and designof nonlinear systems.

Although there are several different definitions for PWLsystems, a widely acceptable mathematical model of PWLsystems can be represented as follows [9], [16].

δx(t) = Aix(t) + B1iw(t) + B2if(t) + ai

y(t) = Cix(t) + D1iw(t) + D2if(t)for x(t) ∈ Si, i ∈ I

(1)

where x(t) ∈ Rn is the state vector, w(t) ∈ Rm

is the unknown input vector including modeling error,uncertain disturbance, process and measurement noises,y(t) ∈ R` is the measurement vector, and f(t) arethe fault vectors. δ denotes the derivative operator incontinuous time (i.e. δx(t) = (d/dt)x(t)) and the shiftforward operator in discrete time (i.e. δx(t) = x(t + 1)).Si = xt|Fixt + gi ≥ 0i∈I ⊆ Rn denotes partitions ofthe state space into a set of convex polyhedral subspaces

Jun Xu and Kai Yew Lum are with Temasek Laboratories, NationalUniversity of Singapore, 5 Sports Drive 2, Singapore 117508. Email:tslxuj,tsllumky@nus.edu.sg.

Ai Poh Loh is with Department of Electrical and Computer Engineering,National University of Singapore, 4 Engineering Drive 3 , Singapore117508. Email: elelohap@nus.edu.sg.

This work is supported by Defence Science & Technology Agency(DSTA) POD 513242.

1. We assume that all matrices mentioned are appropriatelydimensioned. In this paper, we do not consider the affineterm, i.e., we set ai = gi = 0, since the systems with affineterms can be easily handled similarly [9], [16].

It is no doubt that the fault of such complex systemsis inevitable, and fault detection is obviously an importantissue. When the physical model is known, the most popularapproaches are the mode-based fault detection methods,whose main idea is to use the redundancy in informationobtained from measurement in combination with the systemmodel [7]. Particularly, the observer-based design approachhas received much attentions [15], [7]. However, there havebeen very few existing works for PWL systems [26], nomatter what method is used.

In fact, even for normal estimation and filtering problemfor PWL systems, not many works have been reported yet,due to the complexity nature of these systems. Some resultsof the similar systems, such as slow-switching systems [1],[2] and Markovian systems [8] are presented. However, inthese works, the modes of the systems are known a priori orbased on probability properties. Feng proposes an observerdesign approach for discrete-time piecewise linear systems(DPS) in [10] and presents two filter design methods forthe same class systems in [11], where he assumes that thepartitions are defined in terms of the system output so that theplant and state estimator always switch to the same partitionat the same time. However, many PWL systems are morelikely partitioned based on the state space. Furthermore, thereinevitably exists measurement noise in the output. In suchsituations, there is no guarantee that the system state and theestimated state always stay in the same partition at the sametime. In other words, it is likely that the system state mightoperate in a different region as the estimated state from timeto time. This type of state estimation is referred to as non-synchronized state estimation.

For non-synchronized state estimation, Liu and Zhang[23] give a new filtering scheme using the most probabletrajectory estimated and a shift transformation. Their workcan be used to estimate the PWL systems, while the group ofcandidate distribution actually is not easy to be determined.Trecate et al. in [12] propose a state-smoothing algorithmfor hybrid systems based on moving-horizon estimation(MHE) by exploiting the equivalence between mixed logical

1For the further properties of these partitions, the reader may refer to [9],[16]. We assume that there is no Zeno behavior.

16th IEEE International Conference on Control ApplicationsPart of IEEE Multi-conference on Systems and ControlSingapore, 1-3 October 2007

MoC02.3

1-4244-0443-6/07/$20.00 ©2007 IEEE. 373

dynamic (MLD) systems and PWL systems. However, thisapproach actually requires a huge computational power.Birouche et al. present a solution for detecting the switchingtime instant and the corresponding active mode by using aseparated observer structure with very restrictive conditions[4]. Recently, Juloski et al. [18], [17] introduce a designprocedure for the Luenberger type of observer, which doesnot require information on currently active dynamics of abi-modal PWL system.

It is noted that some fault detection designs for fuzzysystems actually are similar to these of PWL systems, dueto the close relationship between them [16]. For example,Nguang et al. [24] consider the properties of the localregions and incorporate the partition information into design.However, no mode switching is considered and only a singleLyapunov function is employed there.

Motivated by recent works by Xu and Xie [31], [29],[32], where the H∞ and generalized H2 performance forboth continuous-time and discrete-time general PWL systemsare studied with more feasible form, and Wang et al. [27],where a L2-gain based fault detection performance index isused, we will study how to design the non-synchronized faultdetection observer with required performance for discrete-time PWL systems in an efficient way in this paper. Thereader may refer to [28] for the continuous-time counterpartand threshold design.

The continuation of this paper begins with the observerstructure and fault detection objective in Section II. InSection III, the main result is presented. We discuss the faultdetection observer design problem in two parts: one is toassume that the Lyapunov matrices are positive definite, andthe other removes this assumption. Finally, we draw someconclusion in Section IV. For convenience, we introducethe following notations: A > 0 (A < 0) means that A ispositive definite (negative definite). A ≯ 0 means A is notnecessary positive definite.A 0 implies that A has non-negative entries. ‖y‖`2[0,N ] =

∑Nt=0yT y 1

2 . ’*’ denotes anentry that can be deduced from the symmetry of the matrix,say, for example,

A B∗ C

=

A B

BT C

.

II. ESTIMATOR STRUCTURE AND PROBLEM STATEMENT

To begin processing the design problem, we make thefollowing assumptions.

A1: The system (1) is asymptotically stable;A2: w(t) and f(t) are `2-norm bounded;A3: D2i(x), i ∈ I is with full column rank.A4: The system dynamic is governed by the dynamic

of the local mode of Si when the dynamic jumpsfrom Si to Sj .

We introduce the following Luenberger-type observer

x(t + 1) = Aj x(t) + Lj(y(t) − Cj x(t))y(t) = Cj x(t) x(t) ∈ Sj , j ∈ I

(2)where Lj is the estimator gain to be designed.

In view of the system dynamics x(t) ∈ Si and theestimator dynamics x(t) ∈ Sj , the state estimation error

dynamics can be obtained by combining the system (1) andthe state estimator (2):

(Σ) :ξ(t + 1) = Aijξ(t) + B1ijw(t) + B2ijf(t)

rwf (t) = y(t) − y(t) = Cijξ(t) + D1iw(t) + D2if(3)

where

e(t) ∆= x(t) − x(t), ξ(t) ∆=[

x(t)e(t)

],

Aij =[

Ai 0Ai − Aj − LjCi + LjCj Aj − LjCj

],

B1ij =[

B1i

B1i − LjD1i

], B2ij =

[B2i

B2i − LjD2i

]Cij =

[Ci − Cj Cj

](4)

For ease of reference, we let

rw = rwf |f≡0

rf = rwf |w≡0

Note that the transition of the estimator state x(t) isbased on its estimated value. If x(t) and x(t) are closeenough such that e(t) can be ignored, i.e., x(t) and x(t)are synchronized in transition from one partition to another,then the method in [10], [25] may be applied here. However,in practice, x(t) and x(t) may not always be located in thesame partition, especially in the initial period. Thus they arenon-synchronized.

Remark 1: For the case x(t) ∈ Si and x(t) ∈ Sj , we have

Fijξ(t)∆=

[Fi 0Fj −Fj

]ξ(t) ≥ 0 (5)

We define new partition Sij = ξ(t)|Fijξ(t) ≥ 0, anddenote the index pair set for the DPS as Φd. It is easy tocheck that Sij is still a convex polyhedron. We also denotethe index set of the system jump from Sij to Skl for theDPS as Ψd.

The system state and estimated state in DPS is much morecomplex than that of continuous-time cases. As mentionedbefore, the two consecutive system states x(t) and x(t + 1)may belong to any partitions in DPS. Thus we define aset Ω that represents all transitions from one partition toanother which happen in the system state and estimated state,that is, Ω ∆= i, j|x(t) ∈ Si, xt ∈ Sj , i 6= j, i, j ∈ I.As a consequence, there are four cases about the dynamicstransitions of the combined system (3) from ξ(t) to ξ(t + 1),where x(t) ∈ Si, x(t + 1) ∈ Sk, x(t) ∈ Sj and x(t + 1) ∈Sl [30]:

Ψd∆=

(i, j, k, l)|

Case 1 : i = j, k = l ∈ ICase 2 : i = j ∈ I, k, l ∈ ΩCase 3 : i, j ∈ Ω, k = l ∈ ICase 4 : i, j ∈ Ω, k, l ∈ Ω

(6)

It is well-known that the two main properties for faultdetection algorithms are robustness and sensitivity. A lotof analysis and synthesis based on H∞ optimization havebeen proposed. However, since it is rather difficult to obtain

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the transfer function of nonlinear systems, hence, a directconsideration on H∞ performance is not straightforward.Instead, we use the `2 gain performance. The problems underconsideration are stated as follows:

Robust Fault Detection Problem: Consider the PWLsystem (1). Design an observer of the form (2) such that

1) The system (3) is asymptotically stable;2) The fault detection “unknown input signal” gain ratio

J = γ1γ2

is made small where

‖rw‖`2 < γ1‖w‖`2 , w 6= 0 (7)‖rf‖`2 > γ2‖f‖`2 , f 6= 0 (8)

under zero initial conditions.

III. OBSERVER DESIGN

The following analysis results can be deduced based onthe stability results and definition of the piecewise Lyapunovfunction components. We first present some analyticalresults. And then we address the fault detection observerdesign problem in two situations depending on whether theLyapunov matrix Pij is positive definite.

Lemma 1: The system (3) is asymptotically stable andsatisfies the condition (7), if there exits a solution (Pij =PT

ij , U1ijkl 0) to the following inequalities:

Pij − FTij V1ijFij > 0, ∀(i, j) ∈ Φd (9)[

$(1)ijkl AT

ijPklBij + CTijD1i

BTijPklAij + DT

1iCij BTijPklBij − γ2

1I + DT1iD1i

]< 0,

(10)∀(i, j, k, l) ∈ Ψd

where $(1)ijkl = AT

ijPklAij − Pij + FTij U1ijklFij + CT

ijCij .Proof: We define a Lyapunov function candidate as

follows:vij(ξ) = ξT Pijξ, ξ ∈ Sij

Due to (9), the Lyapunov function is positive definite. Then,along the state trajectory of the system (3),

∆vij = ζT

[AT

ijPklAij − Pij PklB1ij

BT1ijPkl BT

1ijPklB1ij

]ζ (11)

where ζ = [ξT wT ]T .The existence of (Pij , Pkl, U1ijkl, (i, j, k, l) ∈ Ψd) to

(10) implies that

ATijPklAij − Pij < −CT

ijCij − FTij U1ijklFij < 0

which implies the stability of the system (3) [16].Next, (10) can be rewritten as:[

ATijPklAij − Pij PklB1ij

BT1ijPkl BT

1ijPklB1ij

]+

[CT

ijCij CTijD1i

∗∗ −γ21I + DT

1iD1i

]+

[FT

ij UijFij 00 0

]< 0

(12)

Multiplying (12) by ζT and ζ from the left and the right,respectively, we have:

∆vij +(‖ rw ‖2 −γ2

1 ‖ w ‖2)≤ −ξT FT

ij U1ijklFijξ < 0(13)

By piecewise summing (13) from 0 to ∞, the globalLyapunov function v(ξ(t)) satisfies

v(ξ(∞))− v(ξ(0))+∞∑

t=0

(‖ rw ‖2 −γ21 ‖ w ‖2)dt < 0 (14)

which implies that

‖ rw ‖2`2< γ2

1 ‖ w ‖2`2 (15)

due to the asymptotical stability of the system (1) and (3)and zero initial conditions. Thus we complete the proof.

Lemma 2: The system (3) is asymptotically stable andsatisfies the condition (8), if there exits a solution (Qij =QT

ij , U2ijkl 0) to the following inequalities:

Qij − FTij V2ijFij > 0, ∀(i, j) ∈ Φd (16)

[−$

(2)ijkl CT

ijD2i − ATijQklBij

DT2iCij − BT

ijQklAij DT2iD2i − BT

ijQklBij − γ22I

]> 0, ∀(i, j, k, l) ∈ Ψd

(17)where $

(2)ijkl = AT

ijQklAij − Qij + FTij U2ijklFij − CT

ijCij .Proof: We define a Lyapunov function candidate as

follows:

vij(ξ) = ξT Qijξ, ξ ∈ Sij

The rest is similar to that of Lemma 1. In fact, based on(17), we have

−v(ξ(∞))+v(ξ(0))+∞∑0

(‖ rf ‖2 −γ2 ‖ f ‖2)dt > 0 (18)

which implies that

‖ rf ‖2`2> γ2

2 ‖ f ‖2`2 (19)

for zero initial conditions. Thus we complete the proof.Theorem 1: Consider the discrete-time PWL system

defined by (1). The observer (2) that solves the robust faultdetection problem if for some ε1, ε2 > 0, there exists asolution (Pij = PT

ij , Qij = QTij , V1ij 0, V2ij

0, U1ijkl 0, U2ijkl 0, (i, j, k, l) ∈ Ψd) to (9)-(10)and (16)-(17).

Note that in Theorem 1, Pij and Qij are not necessarilypositive definite, which makes the design more difficult. Infact, in this setting, the well-known Schur complement [5]cannot be applied here. In the following, based on whetherthe Lyapunov matrices are positive definite, we divide ourdiscussion into two cases: Vij ≡ 0 and Vij 0.

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A. Vij ≡ 0To enable an estimator design, we can apply the Schur

complement to (10) and (17) and obtain −Pkl PklAij PklBij

∗ ϑ(1)ijkl CT

ijD1i

∗ ∗ −γ21I + DT

1iD1i

< 0 (20)

where ϑ(1)ijkl = −Pij + FT

ij U1ijklFij + CTijCij . −Qkl QklAij QklBij

∗ ϑ(2)ijkl −CT

ijD2i

∗ ∗ γ22I − DT

2iD2i

< 0 (21)

where ϑ(2)ijkl = −Qij + FT

ij U2ijklFij − CTijCij .

Remark 2: Based on the inequality (20), a direct designcan be obtained by letting Qij = Pij and Pij =[

P(1)ij 00 P (3)

], (i, j, k, l) ∈ Ψd. However, this approach

will be very conservative in general due to the structureconstraints on Pij .

In the following, we shall focus on how to alleviatethe conservatism. Some technical lemmas derived from theresults in [30], [31] will be presented first.

Lemma 3: [30], [31] The inequality (10) of Theorem 1,can be implied by Pkl − 2Pij PijAij PijBij

∗ ϑ(1)ijkl CT

ijD1i

∗ ∗ −γ2I + DT1iD1i

< 0 (22)

Remark 3: Note that (Pij − σPkl)P−1kl (Pij − σPkl) ≥ 0,

where σ is a real scalar, thus we have PijP−1kl Pij ≥ 2σPij−

σ2Pkl. So the (1,1)-block of inequality (22), in fact, can bereplaced by σ2Pkl−2σPij . The additional σ may bring someflexibilities.

Further, to remove the structural constraint on Pij , weresort to the following lemma.

Lemma 4: [30], [31] There exists a solution (Pij >0, U1ijkl 0) to inequality (22) of Lemma 3, if and onlyif there exists a solution (Pij > 0, U1ijkl 0, Gij) to thefollowing inequality for ∀i, j, k, l ∈ Ψd:

Pkl − 2Pij GTijAij GT

ijBij GTij − Pij

∗ ϑ(1)ijkl CT

ijD1i −ε1ATijGij

∗ ∗ −γ21I + DT

1iD1i −ε1BTijGij

∗ ∗ ∗ −ε1(Gij + GTij)

< 0

(23)where ε1 is a positive scalar.Similarly, for (17) of Theorem 1, we have the followingresult.

Lemma 5: [30], [31] There exists a solution (Qij >0, U2ijkl 0) to inequality (17) of Theorem 1, if thereexists a solution (Qij > 0, U2ijkl 0, Gij) to the followinginequality for ∀i, j, k, l ∈ Ψd:

Qkl − 2Qij GTijAij GT

ijBij GTij − Qij

∗ ϑ(1)ijkl −CT

ijD2i −ε1ATijGij

∗ ∗ γ21I − DT

2iD2i −ε1BTijGij

∗ ∗ ∗ −ε1(Gij + GTij)

< 0

(24)

where ε2 is a positive scalar.Remark 4: The key idea of Lemmas 4 and 5 is to

eliminate the coupling between the Lyapunov matrices Pkl

and the system matrices with the aid of slack variable Gij .Now we can choose

Pij =

[P

(1)ij P

(2)ij

P(2)T

ij P(3)ij

], Qij =

[Q

(1)ij Q

(2)ij

Q(2)T

ij Q(3)ij

](i, j) ∈ Φd. In order to obtain a design method based onLMIs, we let Glij =

G(1)lij

G(2)lij

0 G(3)j

, l = 1, 2, for (i, j) ∈

Φd.We note that Gij is invertible, so is G(3)j . Based on the

above lemmas, we obtain the following theorem.Theorem 2: Consider the system defined by (1). Given

scalars γ1, γ2 > 0, there exists an observer (2) that solvesthe robust fault detection problem if for some ε1, ε2 > 0,there exists a solution (Pij > 0, Qij > 0, Gij , U1ijkl 0, U2ijkl 0, Wj) to the following LMIs for ∀(i, j, k, l) ∈Ψd:

P(1)kl

− 2P(1)ij

P(2)kl

− 2P(2)ij

G(1)T

1ijAi 0

∗ P(3)kl

− 2P(3)ij

χ(1)ij

G(3)T

jAj − WjCj

∗ ∗ χ(4)ij

U(2)1ijkl

− P(2)ij

+ (Ci − Cj)T Cj

∗ ∗ ∗ U(3)1ijkl

− P(3)ij

+ CTj Cj

∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗

G(1)T

1ijBi G

(1)T

1ij− P

(1)ij

−P(2)ij

χ(3)Tij

G(2)T

1ij− P

(2)T

ijG

(3)T

j− P

(3)ij

(Ci − Cj)T D1i −ε1ATi G

(1)1ij

−ε1χ(2)ij

CTj D1i 0 −ε1(AT

j G(3)j

− CTj W T

j )

−γ21I + DT

1iD1i −ε1BTi G

(1)1ij

−ε1χ(3)ij

∗ −ε1(G(1)T

1ij+ G

(1)1ij

) −ε1G(2)1ij

∗ ∗ −ε1(G(3)T

j+ G

(3)j

)

< 0

(25)

Q(1)kl

− 2Q(1)ij

Q(2)kl

− 2Q(2)ij

G(1)T

2ijAi 0

∗ Q(3)kl

− 2Q(3)ij

χ(5)ij

G(3)T

jAj − WjCj

∗ ∗ χ(8)ij

U(2)2ijkl

− Q(2)2ij

− (Ci − Cj)T Cj

∗ ∗ ∗ −Q(3)2ij

+ U(3)2ijkl

− CTj Cj

∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗

G(1)T

2ijBi G

(1)T

2ij− Q

(1)ij

−Q(2)ij

χ(7)Tij

G(2)T

2ij− Q

(2)T

ijG

(3)T

j− Q

(3)ij

−(Ci − Cj)T D2i −ε2ATi G

(1)2ij

−ε2χ(6)ij

−CTj D2i 0 −ε2(AT

j G(3)j

− CTj W T

j )

γ22I − DT

2iD2i −ε2BTi G

(1)2ij

−ε2χ(7)ij

∗ −ε2(G(1)T

2ij+ G

(1)2ij

) −ε2G(2)2ij

∗ ∗ −ε2(G(3)T

j+ G

(3)j

)

< 0

(26)

where χ(1)ij

= G(2)T

1ijAi + G

(3)T

j(Ai − Aj) − Wj(Ci − Cj), χ

(2)ij

= ATi G

(2)1ij

+

(Ai − Aj)T G(3)j

− (Ci − Cj)T W Tj , χ

(3)ij

= BTi G

(2)1ij

+ BTi G

(3)j

− DTi W T

j , χ(4)ij

=

−P(1)ij

+ U(1)1ijkl

+ (Ci − Cj)T (Ci − Cj)), χ(5)ij

= G(2)T

2ijAi + G

(3)T

j(Ai − Aj) −

Wj(Ci − Cj), χ(6)ij

= ATi G

(2)2ij

+ (Ai − Aj)T G(3)j

− (Ci − Cj)T W Tj , χ

(7)ij

=

BTi G

(2)2ij

+BTi G

(3)j

−DTi W T

j , and χ(8)ij

= −Q(1)ij

+ U(1)2ijkl

−(Ci −Cj)T (Ci −Cj)).

In this situation, the observer gains can be given by:

Lj = G(3)−Tj Wj , j ∈ I

The above result applies partition-dependent Lyapunovfunctions and will be less conservative than the method statedin Remark 4. However, there is still structural constraint

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on Gij . In the following, we will show how to removethe structural constraint completely by an iterative LMIapproach. In fact, we can easily see the following conditionsare equivalent to the inequality (10) and (17) of the theorem1. −Pkl Aij B1ij

∗ ϑ(1)ijkl CT

ijD1i

∗ ∗ −γ21I + DT

1iD1i

< 0, ∀(i, j, k, l) ∈ Ψd

(27) −Qkl Aij B2ij

∗ ϑ(2)ijkl −CT

ijD2i

∗ ∗ γ22I − DT

2iD2i

< 0, ∀(i, j, k, l) ∈ Ψd

(28)

PklPkl = I, k, l ∈ Φd (29)

QklQkl = I, k, l ∈ Φd (30)

Note that (29) can be weakened to the following well-knownsemi-definite programming relaxation:[

−Pkl I

I −Pkl

]≤ 0, k, l ∈ Φd (31)

Observe that the condition PklPkl = I is equivalent totrace(PklPkl) = 2n, thus we can solve the equalityconstraint (29) by solving the following optimizationproblem

min∑

k,l∈Ω

trace(PklPkl), subject to (31) (32)

Similarly, (30) can be weakened as[−Qkl I

I −Qkl

]≤ 0, k, l ∈ Φd (33)

The above problem is not convex since the costfunction in (32) is bilinear. This bilinear problem has beeninvestigated by many researchers for static output feedbackcontrol of continuous-time systems. In fact, some efficientcomputational algorithms, such as the cone complementaritylinearization methods[13] and sequential linear programmingmatrix method (SLPMM)[22], have been known. In thispaper, we borrow the main idea of SLPMM because SLPMMalways generates a strictly decreasing sequence of theobjective function value which is bounded below by someinteger, and thus it is convergent.

Now we extend the SLPMM to solve the state estimationproblem and have the following steps:

Algorithm 1: SLPMM FOR ESTIMATOR DESGIN

Step 1 Obtain an initial set (P 0kl, Q

0kl, P

0kl, Q

0kl) by solving

(31), (33), (28) and (27) for ∀(i, j, k, l) ∈ Ψ.Step 2 Given P t

kl, P tkl, Qt

kl and Qtkl, where t is a counter,

solve the following optimization problem for somePkl > 0, Qkl > 0, Pkl > 0, Qkl > 0:

min∑

k,l∈Φd

trace(PklPtkl + P t

klPkl + QklQtkl + Qt

klQkl)

s.t. (28), (27), (31), (33), ∀(i, j, k, l) ∈ Ψ(34)

Step 3 If∑

k,l∈Φd

trace(PklPtkl + P t

klPkl − 2P tklP

tkl) +

trace(QklQtkl + Qt

klQkl − 2QtklQ

tkl) ≤ ε, where

ε is a pre-defined sufficiently small positive scalar,substitute Pkl into (20) and Qkl into (21) . If (20)and (21) are feasible, we obtain a proper observergain. Stop. Otherwise let ε = ε/κ, where κ > 1is a given scalar. If ε < ε0, where ε0 is a givensufficiently small positive valve value, stop. We failto find an observer gain.

Step 4 Compute α, β ∈ [0 1] by solvingmin

∑k,l∈Φd

trace((1 − α)Pkl + αPtkl)((1 − α)Pkl + αP

tkl))

+trace((1 − α)Qkl + αQtkl)((1 − α)Qkl + αQt

kl))

Set P t+1kl = (1 − α)Pkl + αP t

kl, Qt+1kl = (1 −

β)Qkl+βQtkl. t = t+1, P t+1

kl = (1−α)Pkl+αP tkl,

Qt+1kl = (1− β)Qkl + βQt

kl. t = t + 1. Go to Step2.

B. Vij 0In existing synthesis methods for discrete-time PWL

systems, the assumption that Lyapunov matrices are positivedefinite prevails. However, based on Theorem 1, it is nota necessary condition. A sufficient condition for a properpiecewise quadratic Lyapunov function v(ξ) only requiresthat its components vij(ξ) be positive in the local region.Since (10) and (17) are neither LMI nor BMI, whilst wecannot apply the Schur complement to them in this case, wehave to find other solutions. The following technical lemmassimilar to these of [31] suggest one possible approach.

Lemma 6: Inequality (10) of Theorem 1, isequivalent to the following inequality for some(Pij = PT

ij , Υ1ijkl, Γ1ij , V1ij 0, U1ijkl 0) for∀(i, j, k, l) ∈ Ψd:[

%(1)ijkl + AT

1ijΥ1ijkl + ΥT1ijklA1ij −ΥT

1ijkl + AT1ijΓ1kl

∗ Pkl − Γ1kl − ΓT1kl

]< 0

(35)where %

(1)ijkl =

−Pij + F T

ij U1ijklFij + CTijCij CT

ijD1i

∗ DT1iD1i − γ2

1I

,

A1ij =

Aij B1ij

.

Lemma 7: Inequality (17) of Theorem 1, isequivalent to the following inequality for some(Qij = QT

ij , Υ2ijkl,Γ2ij , V2ij 0, U2ijkl 0) for∀(i, j, k, l) ∈ Ψd:[

%(2)2ijkl + AT

2ijΥ2ijkl + ΥT2ijklA2ij −ΥT

ijkl + AT2ijΓ2kl

∗ Qkl − Γ2kl − ΓT2kl

]< 0

(36)where %

(2)ijkl =

−Qij + F T

ij U2ijklFij − CTijCij −CT

ijD2i

∗ γ22I − DT

2iD2i

and A2ij =

Aij B2ij

.

Remark 5: Note that the proofs of Lemmas 6 and 7 arebased on Finsler’s lemma [5]. Using the technique introducedin [31], we can see that if we let Υ1ijkl = Υ1kl and Υ2ijkl =Υ2kl, the equivalence still holds. Thus the following resultfollows.

Theorem 3: Consider the system defined by (1). Givenscalars γ1, γ2 > 0, there exists an observer (2) that solvesthe robust fault detection problem if for ∀(i, j, k, l) ∈

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Ψd, there exists a solution (Lj , Pij = PTij , Qij =

QTij , Υ1ijkl, Γ1ij , Υ2ijkl, Γ2ij , V1ij 0, U1ijkl 0, V2ij

0, U2ijkl 0) to the LMIs (9), (16) and BMIs (35)-(36).Remark 6: There are several existing (iterative)

algorithms to BMI problems, such as the branch andbound algorithm, V-K iterative algorithm, path-followingalgorithm, and method-of-centers-like algorithm for localregion, branch and bound algorithm and trust region strategyfor global optimization (see [19], [3] and references therein).We can also apply the commercial software: PENBMI tosolve this problem [21]. We omit the detail steps here.

IV. DISCUSSING CONCLUSION

In this paper, we have considered the non-synchronizedfault detection problem using Luenberger-type observer fordiscrete-time PWL systems. We have proposed severaltechniques to achieve less conservative design methods froma simple setting (Lyapunov matrices are positive definite) to amore complex setting (Lyapunov matrices are not necessarilypositive definite).

However, there are many issues to be explored further,such as the effect of switching sequence and threshold. Aswe know, when a “constant” fault occurs, such as “lock-in-place” and “outage” of actuators, the system comes intoan unknown mode. To identify whether the system switchesto such unknown mode or another normal mode, a properthreshold must be carefully chosen to decrease the falsedetection and missing detection rate. In the case that thesefault modes are known, different thresholds can be used. Asimple setting is with two thresholds: one for mode switchingand the other for the normal case (non-switching).

A two-step design for refinement can be carried out. Afterdetermining the switch sequence of the original system (1),we may require that the estimated state switching sequenceshall follow the original system state switch sequence. Suchcondition is not involved in our current design. Meanwhile,if the switch sequence is determined, the number of LMIswill be reduced. A possible solution might be based on theone-step reachable set [20]. Hence a BMI approach with theinitial solution from current design can be obtained.

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