# Fault Detection for Nonlinear Impulsive Switched Systems

Authors: Jian Li; Qingyu Su; Lingfang Sun; Bo Li
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper is concerned with the fault detection problem for nonlinear impulsive switched systems. Fault detection filters are designed such that the augmented systems are stable, and the residual signal
generated by the filters achieves theH∞-gain for disturbances and guarantees the H- performance for faults. Sufficient conditions for the solvability of this problem are formulated in terms of linear matrix inequalities; furthermore, the filter gains are characterized by a convex optimization problem. A simulation on a continuous stirred tank reactor control system is given to demonstrate the effectiveness of the proposed methods.


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## Body

## 1. Introduction

Switched systems are an important class of hybrid systems, which consist of a family of continuous-time or discrete-time subsystems and a switching law that specifies the switching between them. Study on this class of systems has attracted much attention in recent years for its theoretical significance [1–6] and engineering applications [7–9]. However, in a wide range of actual systems such as engineering, economics, and biology, there usually exist some impulses when the switched system is switching among subsystems. These special switched systems are called impulsive switched systems. As typical switched impulsive system, the circuit switching usually causes system state to change abruptly in some circuit systems [10]. The abrupt state at the time instant of switching could often lead to oscillations and instability and the lead poor performances. Recently, numbers of papers have focused on stability problem of switched impulsive systems [11–13].On the other hand, fault detection (FD) is an important topic in system engineering from the viewpoint of the higher demands for safety and reliability of control systems [14–17]. Among these model-based approaches, the class of procedures are that observers or filters are firstly designed to construct a residual signal which is used to generate an alarm when the residual evaluation function has larger than the threshold. During the past decades, Many results, using robust H∞ technique [18–21] and H∞/H- technique, investigate this important issue [22, 23].However, the problem of FD design in switched systems schemes is still in the early stage of development and a few results have been reported in the literatures [24–28]. It is worth noticing that all the aforementioned FD approaches for switched systems do not include impulsive switched systems. Moreover, since the FD approaches for switched systems without impulsive increments are not appropriate for switched systems with impulsive increments, the new FD technique need to solve the impulsive case. Because these impulsive switched systems exist widely, the FD problem for a general class of nonlinear impulsive switched systems is significant, both theoretically and practically.In this paper, the problem of FD filters for a general class of nonlinear impulsive switched systems is presented. Different from [10, 29] which only involve linear impulsive switched systems with linear impulsive increments, nonlinear impulsive switched systems with nonlinear impulsive increments have been considered in this paper. The main contributions of this paper include (i) the H- performance of nonlinear impulsive switched systems to directly reflect the effect on the residual signal from faults and obtain the better detection results and (ii) the smaller conservatism of sufficient conditions for the H- performance and the H∞ performance. Finally, the filters gains are characterized in terms of the solution of a convex optimization problem.The paper is organized as follows. Section2 introduces the problem under consideration and presents the design objectives. Section 3 illustrates the FD filter design approach in detail. The thresholds are given in Section 4. Two examples are given in Section 5 to demonstrate the proposed method. Conclusions of this paper are given in the last section.Notation. For a matrix A, AT denotes its transpose. For a symmetric matrix, A>0(A≥0) and A<0(A≤0) denote positive definiteness (positive semidefinite matrix) and negative definiteness (negative semidefinite matrix), respectively. The Hermitian part of a square matrix M is denoted by He(M):=M+MT. The symbol * within a matrix represents the symmetric entries.


## 2. Problem Formulation

### 2.1. System Model

Consider the following nonlinear impulsive switched systems:(1)x˙(t)=Aσ(t)x(t)+Bσ(t)1d(t)+Bσ(t)2f(t)+Υσ(t)(t,x(t)),t≠tk,Δx(t)=Hσ(t)x(t)+Ωσ(t)x(t)(t,x(t)),t=tk,y(t)=Cσ(t)x(t)+Dσ(t)1d(t)+Dσ(t)2f(t),x(t0+)=x0,
where x(t)∈Rn is the state, y(t)∈Rm is the measured output, and d(t)∈Rnd and f(t)∈Rnf are the disturbance input and the fault input, respectively. It is assumed that both the disturbance input and the fault input are energy bounded; then it is demanded that they belong to l2[0,∞). tk, k=0,1,2,…, are impulsive switching time points satisfying 0=t0<t1<⋯<tk<⋯<t∞=∞. The piecewise constant function σ(t) is a switching rule which takes its values in the finite set 𝒩={1⋯N} (N>1 is the number of subsystems). The index σ(t) denotes the sequence of the active subsystem; that is, when σ(t)=j, the jth subsystems is activated at the time point t=tk. Υσ(t)(t,x(t)):[t0,∞)×Rn→Rn, which is globally the Lipschitz continuous, and Υσ(t)(t,0)≡0 for all t∈[t0,∞)Δx(tk)=x(tk+)-x(tk-)=x(tk+)-x(tk), with x(tk+)=limt→tk+x(t) and x(tk)=x(tk-)=limt→tk-x(t); that is, the solution x(t) is left continuous. Ωσ(t)(t,x(t)):[t0,∞)×Rn→Rn is nonlinear function, and Ωσ(t)(t,0)≡0 for all t∈[t0,∞). The jth subsystem is denoted by the matrices Aj, Bj1,  Bj2, Cj,  Dj1, and Dj2 with appropriate dimensions.We introduce the following assumptions for nonlinear impulsive switched systems (1).Assumption 1.
There exist nonnegative scalarsgj>0, j∈𝒩, such that
(2)Υσ(t)T(t,x(t))Υσ(t)(t,x(t))≤gjxT(t)x(t).Assumption 2.
Denoteρσ(t)(·) as the spectral radius for each subsystem. ∥Ωσ(t)x(t)(t,x(t))∥≤ρσ(t)(Hσ(t)+1)∥x(t)∥ for t∈[t0,∞).For the purpose of the fault detection, the following FD filters are designed:(3)x˙f(t)=Afσ(t)xf(t)+Bfσ(t)y(t),r(t)=Cfσ(t)xf(t)+Dfσ(t)y(t),
where xf(t) is the state of the filter and r(t) is the residual signal. The matrices Afσ(t),Bfσ(t),Cfσ(t), and Dfσ(t) with appropriate dimensions are to be determined.Combining (1) and (3), the nonlinear impulsive switched system can be written as
(4)x~˙(t)=𝒜σ(t)x~(t)+ℬσ(t)1d(t)+ℬσ(t)2f(t)+𝒬σ(t)(t,x~(t)),t≠tk,Δx~(t)=ℋσ(t)x~(t)+𝒪σ(t)(t,x~(t)),t=tk,r(t)=𝒞σx~(t)+𝒟σ1d(t)+𝒟σ2f(t),
where
(5)x~(t)=[xT(t)xfT(t)]T,Δx~(tk)=x~(tk+)-x~(tk-)=x~(tk+)-x~(tk),ℋσ(t)=[Hσ(t)000],𝒪j(t,x~(t))=[Ωj(t,x(t))0]
and when σ(t)=j,
(6)[𝒜jℬj1ℬj2𝒬j(t,x~(t))𝒞j𝒟j1𝒟j20]=[Aj0Bj1Bj2Υj(t,x(t))BfjCjAfjBfjDj1BfjDj20DfjCjCfjDfjDj1DfjDj20].
To present the purpose of this paper more precisely, the following definition is introduced.Definition 3.
Let Assumptions1 and 2 be satisfied and d(t)=0. Nonlinear impulsive switched system (4) under zero initial conditions is said to be stable with the H--gain β, if the condition holds that
(7)∫t0∞‍∥r(s)∥2ds≥β2∫t0∞‍∥f(s)∥2ds.


### 2.2. Problem Formulation

The design problem of the fault detection filters to be addressed in this paper can be expressed as follows.The frameworks of FD filter design: given nonlinear impulsive switched system (1), the FD filters (3) are designed such that nonlinear impulsive switched system (4) is stable and the fault effects on the residual signal are maximized, while the disturbance effects on the residual signals are minimized. Our design objective of the FD filters can now be formulated as the following performances:
(8)∫t0∞‍∥r(s)∥2ds≥β2∫t0∞‍∥f(s)∥2ds,(9)∫t0∞‍∥r(s)∥2ds≤γ2∫t0∞‍∥d(s)∥2ds.Remark 4.
Condition (8) is expressed to maximize the effects of the fault f(t) on the residual output r(t) for impulsive switched system (4). That is, the residual output r(t) is sensitive for the fault f(t). Condition (9) is used for the disturbance attenuation performance, which minimizes the disturbance effects on the all residual outputs and ensures that the disturbances are not disastrous.


## 2.1. System Model

Consider the following nonlinear impulsive switched systems:(1)x˙(t)=Aσ(t)x(t)+Bσ(t)1d(t)+Bσ(t)2f(t)+Υσ(t)(t,x(t)),t≠tk,Δx(t)=Hσ(t)x(t)+Ωσ(t)x(t)(t,x(t)),t=tk,y(t)=Cσ(t)x(t)+Dσ(t)1d(t)+Dσ(t)2f(t),x(t0+)=x0,
where x(t)∈Rn is the state, y(t)∈Rm is the measured output, and d(t)∈Rnd and f(t)∈Rnf are the disturbance input and the fault input, respectively. It is assumed that both the disturbance input and the fault input are energy bounded; then it is demanded that they belong to l2[0,∞). tk, k=0,1,2,…, are impulsive switching time points satisfying 0=t0<t1<⋯<tk<⋯<t∞=∞. The piecewise constant function σ(t) is a switching rule which takes its values in the finite set 𝒩={1⋯N} (N>1 is the number of subsystems). The index σ(t) denotes the sequence of the active subsystem; that is, when σ(t)=j, the jth subsystems is activated at the time point t=tk. Υσ(t)(t,x(t)):[t0,∞)×Rn→Rn, which is globally the Lipschitz continuous, and Υσ(t)(t,0)≡0 for all t∈[t0,∞)Δx(tk)=x(tk+)-x(tk-)=x(tk+)-x(tk), with x(tk+)=limt→tk+x(t) and x(tk)=x(tk-)=limt→tk-x(t); that is, the solution x(t) is left continuous. Ωσ(t)(t,x(t)):[t0,∞)×Rn→Rn is nonlinear function, and Ωσ(t)(t,0)≡0 for all t∈[t0,∞). The jth subsystem is denoted by the matrices Aj, Bj1,  Bj2, Cj,  Dj1, and Dj2 with appropriate dimensions.We introduce the following assumptions for nonlinear impulsive switched systems (1).Assumption 1.
There exist nonnegative scalarsgj>0, j∈𝒩, such that
(2)Υσ(t)T(t,x(t))Υσ(t)(t,x(t))≤gjxT(t)x(t).Assumption 2.
Denoteρσ(t)(·) as the spectral radius for each subsystem. ∥Ωσ(t)x(t)(t,x(t))∥≤ρσ(t)(Hσ(t)+1)∥x(t)∥ for t∈[t0,∞).For the purpose of the fault detection, the following FD filters are designed:(3)x˙f(t)=Afσ(t)xf(t)+Bfσ(t)y(t),r(t)=Cfσ(t)xf(t)+Dfσ(t)y(t),
where xf(t) is the state of the filter and r(t) is the residual signal. The matrices Afσ(t),Bfσ(t),Cfσ(t), and Dfσ(t) with appropriate dimensions are to be determined.Combining (1) and (3), the nonlinear impulsive switched system can be written as
(4)x~˙(t)=𝒜σ(t)x~(t)+ℬσ(t)1d(t)+ℬσ(t)2f(t)+𝒬σ(t)(t,x~(t)),t≠tk,Δx~(t)=ℋσ(t)x~(t)+𝒪σ(t)(t,x~(t)),t=tk,r(t)=𝒞σx~(t)+𝒟σ1d(t)+𝒟σ2f(t),
where
(5)x~(t)=[xT(t)xfT(t)]T,Δx~(tk)=x~(tk+)-x~(tk-)=x~(tk+)-x~(tk),ℋσ(t)=[Hσ(t)000],𝒪j(t,x~(t))=[Ωj(t,x(t))0]
and when σ(t)=j,
(6)[𝒜jℬj1ℬj2𝒬j(t,x~(t))𝒞j𝒟j1𝒟j20]=[Aj0Bj1Bj2Υj(t,x(t))BfjCjAfjBfjDj1BfjDj20DfjCjCfjDfjDj1DfjDj20].
To present the purpose of this paper more precisely, the following definition is introduced.Definition 3.
Let Assumptions1 and 2 be satisfied and d(t)=0. Nonlinear impulsive switched system (4) under zero initial conditions is said to be stable with the H--gain β, if the condition holds that
(7)∫t0∞‍∥r(s)∥2ds≥β2∫t0∞‍∥f(s)∥2ds.


## 2.2. Problem Formulation

The design problem of the fault detection filters to be addressed in this paper can be expressed as follows.The frameworks of FD filter design: given nonlinear impulsive switched system (1), the FD filters (3) are designed such that nonlinear impulsive switched system (4) is stable and the fault effects on the residual signal are maximized, while the disturbance effects on the residual signals are minimized. Our design objective of the FD filters can now be formulated as the following performances:
(8)∫t0∞‍∥r(s)∥2ds≥β2∫t0∞‍∥f(s)∥2ds,(9)∫t0∞‍∥r(s)∥2ds≤γ2∫t0∞‍∥d(s)∥2ds.Remark 4.
Condition (8) is expressed to maximize the effects of the fault f(t) on the residual output r(t) for impulsive switched system (4). That is, the residual output r(t) is sensitive for the fault f(t). Condition (9) is used for the disturbance attenuation performance, which minimizes the disturbance effects on the all residual outputs and ensures that the disturbances are not disastrous.


## 3. The Fault Detection Filter Design

Before beginning this section, the following lemmas are needed to present our main results.Lemma 5 (see [30]).
Letϵ>0 be a given scalar and Ξ∈Rp×q be a matrix such that ΞTΞ≤I, where I is an identity matrix with appropriate dimension. Then 2xTΞy≤ϵxTx+ϵ-1yTy for all x∈Rp and y∈Rq.Lemma 6 (see [31]).
LetP∈Rn×n be a given symmetric positive definite matrix and let Q∈Rn×n be a given symmetric matrix. Then
(10)λmin{P-1Q}x(t)TPx(t)≤x(t)TQx(t)≤λmax{P-1Q}x(t)TPx(t)
for all x(t)∈Rn, while λmax{·} and λmin{·} denote, respectively, the largest and the smallest eigenvalues of the matrix inside the brackets.In this section, sufficient conditions on the existence of the FD filters for nonlinear impulsive switched systems (4) would be given, and the desired filters can be obtained.


### 3.1. The Fault Sensitiveness Performance (8)

Considering nonlinear impulsive switched system (4) with d(t)=0, we have
(11)x~˙(t)=𝒜σ(t)x~(t)+ℬσ(t)2f(t)+𝒬σ(t)(t,x~(t)),t≠tk,Δx~=ℋσ(t)x~(t)+𝒪σ(t)(t,x~(t)),t=tk,r(t)=𝒞σ(t)x~(t)+𝒟σ(t)2f(t);H- performance for nonlinear impulsive switched system (11) is given.Lemma 7.
Letαj, ε be constants satisfying 0<αj<1, ε>0, j∈𝒩, and Assumptions 1 and 2 hold. Furthermore, suppose that nonlinear impulsive switched system (11) switches from ith subsystem to jth subsystem as switched time point tk. If there exists the Lyapunov functions candidate Vσ(t)(t)=Vj(t)=x~(t)T𝒫jx~(t),j∈𝒩, satisfying the following inequalities
(12)[𝒜jT𝒫j+𝒫j𝒜j+ε-1𝒫j2+αj𝒫j+εgjI-𝒞jT𝒞j𝒫jℬj2-𝒞jT𝒟j2ℬj2T𝒫j-𝒟j2T𝒞jβ2I-𝒟j2T𝒟j2]<0,j∈𝒩,
then nonlinear impulsive switched system (11) is stable with H--gain β for any switching signal satisfying
(13)ln(μij|k)-αi(tk-tk-1)<0,i∈𝒩,j∈𝒩,i≠j,k=1,2,…,
where μij|k=((δ+1)λmax{(ℋj+I)T𝒫j(ℋj+I)}+(δ-1+1)ρj2λmax{𝒫j})/λmin{𝒫i},δ>0.Proof.
Whent∈(tk,tk+1] and jth subsystem is activated, evaluating the time derivative of Vj(t) along the trajectory of nonlinear impulsive switched system (11) gives
(14)V˙j(t)=x~T(t)(𝒜jT𝒫j+𝒫j𝒜j)x~(t)+x~T(t)𝒫jℬj2f(t)+f(t)Tℬj2T𝒫jx~(t)+𝒬j(t,x~(t))T𝒫jx~(t)+x~T(t)𝒫𝒬j(t,x~(t)).
By Lemma 5, it is clear that
(15)2𝒬j(t,x~(t))T𝒫jx~(t)≤ε𝒬j(t,x~(t))T𝒬j(t,x~(t))+ε-1x~T(t)𝒫j2x~(t).
Therefore, when assuming the zero input (i.e., f(t)=0) and using Assumptions 1, we have the following condition from (14) and (15):
(16)V˙j∥f(t)=0(t)≤x~T(t)Qjx~(t),
where Qj=𝒜jT𝒫j+𝒫j𝒜j+εgjI+ε-1𝒫j2. If (12) holds, then Qj+αj𝒫j≤0, which implies that Qj≤-αj𝒫j≤0. Thus, from (16), we have V˙j(t)|f(t)=0(t)≤-αjVj(t)≤0. Integrating this inequality gives
(17)Vj(t)≤e-αj(t-tk)Vj(tk+),t∈(tk,tk+1].
At the impulsive switching time point tk, it has
(18)Vj(tk+)=(Δx~(tk)+x~(tk))T𝒫j(Δx~(tk)+x~(tk))=[(ℋj+I)x~(tk)]T𝒫j[(ℋj+I)x~(tk)]+2𝒪jT(t,x~(t))𝒫j[(ℋj+I)x~(tk)]+𝒪jT(t,x~(t))𝒪j(t,x~(t)).
By Lemma 6, it has
(19)Vj(tk+)≤(δ+1)[(ℋj+I)x~(tk)]T𝒫j(ℋj+I)x~(tk)+(δ-1+1)𝒪jT(t,x~(t))𝒫j𝒪j(t,x~(t))≤{(δ+1)λmax{(ℋj+I)T𝒫j(ℋj+I)}+(δ-1+1)ρj2λmax{𝒫j}}x~T(tk)x~(tk)≤μij|kVi(tk),
where μij|k=((δ+1)λmax{(ℋj+I)T𝒫j(ℋj+I)}+(δ-1+1)ρj2λmax{𝒫j})/λmin{𝒫i}, δ>0. Therefore, from (17) and (19), we have
(20)Vj(t)≤e-αj(t-tk)Vj(tk+)≤μij|ke-αj(t-tk)Vi(tk)≤μij|ke-αj(t-tk)e-αi(tk-tk-1)Vi(tk-1+)≤⋯≤≤(∏l=1,q≠pq∈𝒩,p∈𝒩kμqp|le-αq(tl-tl-1))e-αj(t-tk)×Vσ(t0)(t0),
where μqp|l means the lth switching impulsive value from qth subsystem to pth subsystems, and αq denotes the decay rate of the Lyapunov function for qth subsystem. Since (13) holds; that is, there exists μqp|l>0 such that ln(μqp|l)-αq(tl-tl-1)<0, then
(21)g(t-t0)=-αj(t-tk)+∑l=1,q≠pq∈𝒩,p∈𝒩kln(μqp|l)-αq(tl-tl-1)<0.
It follows that
(22)Vσ(t)(t)≤eg(t-t0)Vσ(t0)(t0).
Therefore, we conclude that Vσ(t)(t) converges to zero as t→∞ then nonlinear impulsive switched system (11) with f(t)=0 is stable.
Secondly, establish theH- performance defined in (7) for nonlinear impulsive switched system (11). Consider the following performance index:
(23)J=∫t0∞‍(β2∥f(s)∥2-∥r(s)∥2)ds.
For any nonzerof(t)∈l2[t0,∞) and zero initial condition x~(t0), let Γ1(t)=β2fT(t)f(t)-rT(t)r(t) and consider the Lyapunov functions as Vj(t)=x~(t)T𝒫jx~(t). For jth subsystem, it has
(24)Vj(x~(t))+αjVj(x~(t))+Γ1(t)≤𝒳1TRj1𝒳1,j∈𝒩,
where
(25)𝒳1=[x~(t)Tf(t)T]T,Rj1=[𝒜jT𝒫j+𝒫j𝒜j+ε-1𝒫j2+αj𝒫j+εgjI-𝒞jT𝒞j𝒫jℬj2-𝒞jT𝒟j2ℬj2T𝒫j-𝒟j2T𝒞jβ2I-𝒟j2T𝒟j2].
Equation (12) is equivalent to Rj1<0. Then, it has Vj(x~(t))+αjVj(x~(t))+Γ1(t)≤0. By iteration operation on the above inequality for t∈(tk,tk+1], we have
(26)Vj(t)≤e-αj(t-tk)Vj(tk+)-∫tkt‍Γ1(s)ds.
Along the same lines as the impulsive switching time point tk, it has from (19) and (26) that
(27)Vj(t)≤e-αj(t-tk)Vj(tk+)-∫tkt‍Γ1(s)ds≤μij|ke-αj(t-tk)Vi(tk)-∫tkt‍Γ1(s)ds≤μij|ke-αj(t-tk-1)Vi(tk-1+)-μij|ke-αj(t-tk)×∫tk-1tk‍Γ1(s)ds-∫tkt‍Γ1(s)ds≤⋯≤≤eg(t-t0)Vσ(t0)(t0)-∫t0t‍eg(t-s)Γ1(s)ds,
where g(t-t0) and g(t-s) are defined as (21).
Under the zero initial condition, (27) implies ∫t0t‍eg(t-s)Γ1(s)ds<0. Because 0<g(t-s)<1,  s∈[t0,t), it has ∫t0t‍Γ1(s)ds<∫t0t‍eg(t-s)Γ1(s)ds<0; that is,
(28)β2∫t0t‍fT(s)f(s)ds≤∫t0t‍rT(s)r(s)ds.
When t→∞, nonlinear impulsive switched system (11) has H- performance, which completes the proof.Remark 8.
For switched systems, the Lyapunov function is discontinuous at the time instant of switching; that is to say, this Lyapunov function does not have its derivative in the whole time domain. Then the piecewise Lyapunov function was utilized in Lemma7. Each subsystem has one corresponding Lyapunov function, and each Lyapunov function has its derivative in the corresponding subsystem time domain.Based on Lemma7, the following theorem is given to obtain the fault sensitiveness performance conditions in terms of linear matrix inequalities.Theorem 9.
Letαj, β, and ε be constants satisfying 0<αj<1, β>0, ε>0 and Assumptions 1 and 2 hold. If there exist matrix variables Yaj, Xaj, Maj, Saj, Zaj, Nj, Rj, A^fj, B^fj, C^fj, D^fj, and symmetric positive-definite matrices
(29)𝒫aj=[Paj1Paj2*Paj3]>0,j∈𝒩,
satisfying the following inequalities
(30)[Θa1Θa2Θa300*Θa4Θa5𝒫aj0**Θa60RjT***-ε-1I0****-I]<0,j∈𝒩,
where(31)Θa1=[-He(Xaj)-Yaj-a1NjT*-He(a2Nj)],Θa3=[-Zaj+XajTBj2+a1B^fjDj2-a5Njvj+YajTBj2+a2B^fjDj2],Θa2=[Paj1-Maj+XajTAj+a1B^fjCjPaj2-Saj+a1A^fjPaj2T-a3Nj+YajTAj+a2B^fjCjPaj3-a4Nj+a2A^fj],Θa4=[αjPaj1+εgjI+He(MajTAj+a3B^fjCj)αjPaj2+a4A^fj+AjTSaj+a4CjTB^fjT*αjPaj3+εgjI+He(a4A^fj)],Θa5=[MajTBj2+a3B^fjDj2+AjTZaj+a5CjTB^fjTvj-CjTD^fjTSajTBj2+a4B^fjDj2+a5A^fjTvj-C^fjT],Θa6=β2I+He(Bj2TZaj+a5Dj2TB^fjTvj-D^fjDj2),then, for any switching signal satisfying (13), nonlinear impulsive switched system (11) is stable and guarantees the H- performance (8). Moreover, if (30) is feasible, then the FD filter gains in form of (3) can be given by
(32)[AfjBfjCfjDfj]=[Nj00Rj]-T[A^fjB^fjC^fjD^fj].Proof.
Now, to establish the convex condition, (12) can be rewritten as
(33)𝒰Tℳaj𝒰<0,
where
(34)𝒰=[𝒜jℬj2I00I],ℳaj=[0𝒫aj0𝒫ajαj𝒫aj+ε-1𝒫aj2+εgjI-𝒞jT𝒞j-𝒞jT𝒟j20-𝒟j2T𝒞jβ2I-𝒟j2T𝒟j2].
On the other hand, we also can have
(35)[-I𝒜jℬj2]𝒰=0.
Based on Finsler’s Lemma, it has(36)ℳaj+He([-IA~jTB~j2T]𝒲aj)<0,
where 𝒲aj introduced by Finsler’s Lemma is the matrix variable of appropriate dimensions. Partition 𝒲aj as 𝒲aj=[𝒲aj1𝒲aj2𝒲aj3], and
(37)𝒲aj1=[XajYaja1Nja2Nj],𝒲aj2=[MajSaja3Nja4Nj],𝒲aj3=[Zaja5Njvj].
Due to -𝒞jT𝒞j, -𝒞jT𝒟j2 and -𝒟j2T𝒟j2 in ℳaj, then the condition (36) is non-convex. To establish the convex condition, (36) can be rewritten as
(38)𝒰-Tℳ-aj𝒰-<0,
where(39)𝒰-=[I000I000I0𝒞j𝒟j2],ℳ-aj=[-He(𝒲aj1)𝒫aj-𝒲aj2+𝒲aj1T𝒜j-𝒲aj3+𝒲aj1Tℬj20*αj𝒫aj+εgjI+ε-1𝒫aj2+He(𝒜jT𝒲aj2)𝒜jT𝒲aj3+𝒲aj2Tℬj20**β2I+He(ℬj2T𝒲aj3)0***0***-I].Using Finsler’s Lemma again, (38) holds if
(40)ℳ-aj+He(ℛℒ)<0
holds for any ℛ with appropriate dimension and ℒ=[0,-𝒞j,-𝒟j2,I]. Let ℛ=[0,0,Rj,0]T and define A^fj=NjTAfj, B^fj=NjTBfj, C^fj=RjTCfj, D^fj=RjTDfj; then (30) is reached by the Schur complement on (40). Hence if the conditions (30) holds, nonlinear impulsive switched system (11) is stable and guarantees the H- performance (8), which completes the proof.Remark 10.
Scalarsa1~a5 are used to reduce conservatism for the same variable Nj in matrix variables 𝒲aj1~𝒲aj3. On the other hand, the coefficient matrices vj are given to adjust the variable dimension. When we set a1~a5, vj as the fixed parameters, the conditions in Theorem 9 become convex.


### 3.2. The Disturbance Attenuation Performance (8)

Consider nonlinear impulsive switched system (4) with f(t)=0, we have
(41)x~˙(t)=𝒜σ(t)x~(t)+ℬσ1d(t)+𝒬σ(t)(t,x~(t)),t≠tk,Δx~=ℋσ(t)x~(t)+𝒪σ(t)(t,x~(t)),t=tk,r(t)=𝒞σx~(t)+𝒟σ1d(t);H
∞ performance for nonlinear impulsive switched system (41) is given.Lemma 11.
Letαj, ε be constants satisfying 0<αj<1, ε>0, j∈𝒩, and Assumptions 1 and 2 hold. Furthermore, suppose that nonlinear impulsive switched system (41) switches from ith subsystem to jth subsystem as switched time point tk. If there exists the Lyapunov functions candidate Vσ(t)(t)=Vj(t)=x~(t)T𝒫jx~(t), j∈𝒩, satisfying
(42)[𝒜jT𝒫j+𝒫j𝒜j+ε-1𝒫j2+αj𝒫j+εgjI+𝒞jT𝒞j𝒫jℬj1+𝒞jT𝒟j1ℬj1T𝒫j+𝒟j1T𝒞j-γ2I+𝒟j1T𝒟j1]<0,[𝒜jT𝒫j+𝒫j𝒜j+ε-1𝒫j2+αj𝒫j+εgjI+𝒞jT𝒞j𝒫jℬj1+ℬj1T𝒫j+𝒟j1T𝒞j-γ2I+]j∈𝒩,
then nonlinear impulsive switched system (41) is stable with H∞-gain γ for any switching signal satisfying (13).Proof.
Following the same lines as those for Lemma7, nonlinear impulsive switched system (41) is stable, for any nonzero d(t)∈l2[0,∞) and zero initial condition x~(t0). Let Γ2(t)=rT(t)r(t)-γ2dT(t)d(t) and consider the Lyapunov function as Lemma 7; we can have
(43)Vj(x~(t))+αjVj(x~(t))+Γ2(t)≤𝒳2TRj2𝒳2,j∈𝒩,
where
(44)𝒳2=[x~(t)Td(t)T]T,Rj2=[𝒜jT𝒫j+𝒫j𝒜j+ε-1𝒫j2+αj𝒫j+εgjI+𝒞jT𝒞j𝒫jℬj1+𝒞jT𝒟j1ℬj1T𝒫j+𝒟j1T𝒞j-γ2I+𝒟j1T𝒟j1].
Equation (42) is equivalent to Rj2<0. Thus, it has
(45)Vj(x~(t))+αjVj(x~(t))+Γ2(t)≤0.
By iteration operation on the above inequality for t∈(tk,tk+1], we have Vj(t)≤e-αj(t-tk)Vj(tk+)-∫tkt‍Γ2(s)ds. As the same lines as those for Lemma 7, it has
(46)Vj(t)≤e-αj(t-tk)Vj(tk+)-∫tkt‍Γ2(s)ds≤⋯≤≤eg(t-t0)Vσ(t0)(t0)-∫t0t‍eg(t-s)Γ2(s)ds≤eg(t-t0)Vσ(t0)(t0)-∫t0t‍Γ1(s)ds.
Under the zero initial condition, (46) implies ∫t0t‍eg(t-s)Γ2(s)ds<0. Because 0<g(t-s)<1, s∈[t0,t), it can further have
(47)∫t0t‍rT(s)r(s)ds≤γ2∫t0t‍dT(s)d(s)ds.
Whent→∞, the nonlinear impulsive switched system (41) has H∞ performance, which completes the proof.Based on Lemma11, the following theorem is given to obtain sufficient conditions by linear matrix inequalities.Theorem 12.
Letαj, γ, and ε be constants satisfying 0<αj<1, γ>0, and ε>0 and Assumptions 1 and 2 hold. If there exist matrix variables Ybj, Xbj,  Mbj,  Sbj, Nj, Rj, A^fj,  B^fj, C^fj, and D^fj and symmetric positive-definite matrices
(48)𝒫bj=[Pbj1Pbj2*Pbj3]>0,j∈𝒩,
satisfying the following inequalities
(49)[Θb1Θb2Θb300*Θb4Θb5𝒫bjΘb6**-γ2I0Dj1TD^fjT***-ε-1I0****I-He(Rj)]<0,j∈𝒩,
where(50)Θb1=[-He(Xbj)-Ybj-b1NjT*-He(b2Nj)],Θb3=[XbjTBj1+b1B^fjDj1YbjTBj1+b2B^fjDj1],Θb2=[Pbj1-Mbj+XbjTAj+b1B^fjCjPbj2-Sbj+b1A^fjPbj2T-b3Nj+YbjTAj+b2B^fjCjPbj3-b4Nj+b2A^fj],Θb4=[αjPbj1+He(MbjTAj+b3B^fjCj)+εgjIαjPbj2+b3A^fj+AjTSbj+b4CjTB^fjT*αjPbj3+εgjI+He(b4A^fj)],Θb5=[MbjTBj1+b3B^fjDj1SbjTBj1+b4B^fjDj1],Θb6=[CjTD^fjTC^fjT],then, for any switching signal satisfying (13), nonlinear impulsive switched system (41) is stable and guarantees the H∞ performance (9). Moreover, if (49) is feasible, then the FD filter gains in form of (3) can be given by
(51)[AfjBfjCfjDfj]=[Nj00Rj]-T[A^fjB^fjC^fjD^fj].Proof.
To establish the convex condition, (42) can be rewritten as follows:
(52)[𝒜jℬj1I0]T[0𝒫bj𝒫bjα𝒫bj][𝒜jℬj1I0]+[ε-1𝒫bj2+εgjI+𝒞jT𝒞j𝒞jT𝒟j1𝒟j1T𝒞j-γ2I+𝒟j1T𝒟j1]<0.
Denote
(53)ℳbj=[0𝒫bj0𝒫bjα𝒫bj+ε-1𝒫bj2+εgjI+𝒞jT𝒞j𝒞jT𝒟j10𝒟j1T𝒞j-γ2I+𝒟j1T𝒟j1].
From (52), we have
(54)[𝒜jℬj1I00I]Tℳbj[𝒜jℬj1I00I]<0.
On the other hand,
(55)[00I]Tℳbj[00I]=-γ2I+𝒟j1T𝒟j1<0.
Based on Projection Lemma, it follows from (54) and (55) that
(56)ℳbj+He([-IA~jTB~j1T]𝒲bj[I000I0])<0,
where 𝒲bj introduced by Projection Lemma is the matrix variable of appropriate dimensions. Partition 𝒲bj as 𝒲bj=[𝒲bj1𝒲bj2]. By the Schur complement, (56) is equivalent to(57)[-He(𝒲bj1)𝒫bj-𝒲bj2+𝒲bj1T𝒜j𝒲bj1Tℬj100*αj𝒫bj+εgjI+He(𝒜j𝒲bj2)𝒲bj2Tℬj1𝒫bj𝒞jT**-γ2I0𝒟j1T***-ε-1I0****-1]<0.Then pre- and postmultiply diag{I,I,I,I,RjT} to (57); one obtains(58)[-He(𝒲bj1)𝒫bj-𝒲bj2+𝒲bj1T𝒜j𝒲bj1Tℬj100*αj𝒫bj+εgjI+He(𝒜jT𝒲bj2)𝒲bj2Tℬj1𝒫bj𝒞jTRj**-γ2I0𝒟j1TRj***-ε-1I0****-RjTRj]<0.we partition 𝒲bj1, 𝒲bj2, respectively, as
(59)𝒲bj1=[XbjYbja1Nja2Nj],𝒲bj2=[MbjSbja3Nja4Nj]
and b1~b4 are set to be fixed parameters. Define A^fj=NjTAfj, B^fj=NjTBfj, C^fj=RjTCfj, and D^fj=RjTDfj; -RjTRj≤I-He(Rj) then (58) becomes (49). Hence if the condition (30) holds, nonlinear impulsive switched system (41) is stable and guarantees the H∞ performance (8), which completes the proof.Remark 13.
Scalarsb1~b4 are used to reduce conservatism for the same variable Nj in matrix variables 𝒲bj1 and 𝒲bj2. When we set b1~b4 as the fixed parameters, the conditions in Theorem 12 become convex.


### 3.3. Algorithm

In the previous sections, Theorems9 and 12 have formulated the inequality conditions for the performances (8)-(9), respectively. Summarily, we have the following algorithm.It is noted that conditions (30) and (49) are all convex. Hence, the problem of FD filter design can directly translate into the following optimization problem:
(60)maxβs.t.(30)and(49),j∈𝒩.Remark 14.
TheH- performance and H∞ performance are considered to describe the fault sensitiveness performance and the disturbance attenuation performance, respectively. Due to the multiobjective optimization problem, each performance has some LMI conditions. However, by analyzing the theorems and their proofs, it can be discovered that 𝒲aj1,  𝒲aj2 in Theorem 9 and 𝒲bj1, 𝒲bj2 in Theorem 12 can be defined as the same variable to decrease the computational complexity.Finally, the gain matricesAfj, Bfj, Cfj, and Dfj of the filters can be derived as
(61)Afj=Nj-TA^fj,Bfj=Nj-TB^fj,Cfj=Rj-TC^fj,Dfj=Rj-TD^fj.


## 3.1. The Fault Sensitiveness Performance (8)

Considering nonlinear impulsive switched system (4) with d(t)=0, we have
(11)x~˙(t)=𝒜σ(t)x~(t)+ℬσ(t)2f(t)+𝒬σ(t)(t,x~(t)),t≠tk,Δx~=ℋσ(t)x~(t)+𝒪σ(t)(t,x~(t)),t=tk,r(t)=𝒞σ(t)x~(t)+𝒟σ(t)2f(t);H- performance for nonlinear impulsive switched system (11) is given.Lemma 7.
Letαj, ε be constants satisfying 0<αj<1, ε>0, j∈𝒩, and Assumptions 1 and 2 hold. Furthermore, suppose that nonlinear impulsive switched system (11) switches from ith subsystem to jth subsystem as switched time point tk. If there exists the Lyapunov functions candidate Vσ(t)(t)=Vj(t)=x~(t)T𝒫jx~(t),j∈𝒩, satisfying the following inequalities
(12)[𝒜jT𝒫j+𝒫j𝒜j+ε-1𝒫j2+αj𝒫j+εgjI-𝒞jT𝒞j𝒫jℬj2-𝒞jT𝒟j2ℬj2T𝒫j-𝒟j2T𝒞jβ2I-𝒟j2T𝒟j2]<0,j∈𝒩,
then nonlinear impulsive switched system (11) is stable with H--gain β for any switching signal satisfying
(13)ln(μij|k)-αi(tk-tk-1)<0,i∈𝒩,j∈𝒩,i≠j,k=1,2,…,
where μij|k=((δ+1)λmax{(ℋj+I)T𝒫j(ℋj+I)}+(δ-1+1)ρj2λmax{𝒫j})/λmin{𝒫i},δ>0.Proof.
Whent∈(tk,tk+1] and jth subsystem is activated, evaluating the time derivative of Vj(t) along the trajectory of nonlinear impulsive switched system (11) gives
(14)V˙j(t)=x~T(t)(𝒜jT𝒫j+𝒫j𝒜j)x~(t)+x~T(t)𝒫jℬj2f(t)+f(t)Tℬj2T𝒫jx~(t)+𝒬j(t,x~(t))T𝒫jx~(t)+x~T(t)𝒫𝒬j(t,x~(t)).
By Lemma 5, it is clear that
(15)2𝒬j(t,x~(t))T𝒫jx~(t)≤ε𝒬j(t,x~(t))T𝒬j(t,x~(t))+ε-1x~T(t)𝒫j2x~(t).
Therefore, when assuming the zero input (i.e., f(t)=0) and using Assumptions 1, we have the following condition from (14) and (15):
(16)V˙j∥f(t)=0(t)≤x~T(t)Qjx~(t),
where Qj=𝒜jT𝒫j+𝒫j𝒜j+εgjI+ε-1𝒫j2. If (12) holds, then Qj+αj𝒫j≤0, which implies that Qj≤-αj𝒫j≤0. Thus, from (16), we have V˙j(t)|f(t)=0(t)≤-αjVj(t)≤0. Integrating this inequality gives
(17)Vj(t)≤e-αj(t-tk)Vj(tk+),t∈(tk,tk+1].
At the impulsive switching time point tk, it has
(18)Vj(tk+)=(Δx~(tk)+x~(tk))T𝒫j(Δx~(tk)+x~(tk))=[(ℋj+I)x~(tk)]T𝒫j[(ℋj+I)x~(tk)]+2𝒪jT(t,x~(t))𝒫j[(ℋj+I)x~(tk)]+𝒪jT(t,x~(t))𝒪j(t,x~(t)).
By Lemma 6, it has
(19)Vj(tk+)≤(δ+1)[(ℋj+I)x~(tk)]T𝒫j(ℋj+I)x~(tk)+(δ-1+1)𝒪jT(t,x~(t))𝒫j𝒪j(t,x~(t))≤{(δ+1)λmax{(ℋj+I)T𝒫j(ℋj+I)}+(δ-1+1)ρj2λmax{𝒫j}}x~T(tk)x~(tk)≤μij|kVi(tk),
where μij|k=((δ+1)λmax{(ℋj+I)T𝒫j(ℋj+I)}+(δ-1+1)ρj2λmax{𝒫j})/λmin{𝒫i}, δ>0. Therefore, from (17) and (19), we have
(20)Vj(t)≤e-αj(t-tk)Vj(tk+)≤μij|ke-αj(t-tk)Vi(tk)≤μij|ke-αj(t-tk)e-αi(tk-tk-1)Vi(tk-1+)≤⋯≤≤(∏l=1,q≠pq∈𝒩,p∈𝒩kμqp|le-αq(tl-tl-1))e-αj(t-tk)×Vσ(t0)(t0),
where μqp|l means the lth switching impulsive value from qth subsystem to pth subsystems, and αq denotes the decay rate of the Lyapunov function for qth subsystem. Since (13) holds; that is, there exists μqp|l>0 such that ln(μqp|l)-αq(tl-tl-1)<0, then
(21)g(t-t0)=-αj(t-tk)+∑l=1,q≠pq∈𝒩,p∈𝒩kln(μqp|l)-αq(tl-tl-1)<0.
It follows that
(22)Vσ(t)(t)≤eg(t-t0)Vσ(t0)(t0).
Therefore, we conclude that Vσ(t)(t) converges to zero as t→∞ then nonlinear impulsive switched system (11) with f(t)=0 is stable.
Secondly, establish theH- performance defined in (7) for nonlinear impulsive switched system (11). Consider the following performance index:
(23)J=∫t0∞‍(β2∥f(s)∥2-∥r(s)∥2)ds.
For any nonzerof(t)∈l2[t0,∞) and zero initial condition x~(t0), let Γ1(t)=β2fT(t)f(t)-rT(t)r(t) and consider the Lyapunov functions as Vj(t)=x~(t)T𝒫jx~(t). For jth subsystem, it has
(24)Vj(x~(t))+αjVj(x~(t))+Γ1(t)≤𝒳1TRj1𝒳1,j∈𝒩,
where
(25)𝒳1=[x~(t)Tf(t)T]T,Rj1=[𝒜jT𝒫j+𝒫j𝒜j+ε-1𝒫j2+αj𝒫j+εgjI-𝒞jT𝒞j𝒫jℬj2-𝒞jT𝒟j2ℬj2T𝒫j-𝒟j2T𝒞jβ2I-𝒟j2T𝒟j2].
Equation (12) is equivalent to Rj1<0. Then, it has Vj(x~(t))+αjVj(x~(t))+Γ1(t)≤0. By iteration operation on the above inequality for t∈(tk,tk+1], we have
(26)Vj(t)≤e-αj(t-tk)Vj(tk+)-∫tkt‍Γ1(s)ds.
Along the same lines as the impulsive switching time point tk, it has from (19) and (26) that
(27)Vj(t)≤e-αj(t-tk)Vj(tk+)-∫tkt‍Γ1(s)ds≤μij|ke-αj(t-tk)Vi(tk)-∫tkt‍Γ1(s)ds≤μij|ke-αj(t-tk-1)Vi(tk-1+)-μij|ke-αj(t-tk)×∫tk-1tk‍Γ1(s)ds-∫tkt‍Γ1(s)ds≤⋯≤≤eg(t-t0)Vσ(t0)(t0)-∫t0t‍eg(t-s)Γ1(s)ds,
where g(t-t0) and g(t-s) are defined as (21).
Under the zero initial condition, (27) implies ∫t0t‍eg(t-s)Γ1(s)ds<0. Because 0<g(t-s)<1,  s∈[t0,t), it has ∫t0t‍Γ1(s)ds<∫t0t‍eg(t-s)Γ1(s)ds<0; that is,
(28)β2∫t0t‍fT(s)f(s)ds≤∫t0t‍rT(s)r(s)ds.
When t→∞, nonlinear impulsive switched system (11) has H- performance, which completes the proof.Remark 8.
For switched systems, the Lyapunov function is discontinuous at the time instant of switching; that is to say, this Lyapunov function does not have its derivative in the whole time domain. Then the piecewise Lyapunov function was utilized in Lemma7. Each subsystem has one corresponding Lyapunov function, and each Lyapunov function has its derivative in the corresponding subsystem time domain.Based on Lemma7, the following theorem is given to obtain the fault sensitiveness performance conditions in terms of linear matrix inequalities.Theorem 9.
Letαj, β, and ε be constants satisfying 0<αj<1, β>0, ε>0 and Assumptions 1 and 2 hold. If there exist matrix variables Yaj, Xaj, Maj, Saj, Zaj, Nj, Rj, A^fj, B^fj, C^fj, D^fj, and symmetric positive-definite matrices
(29)𝒫aj=[Paj1Paj2*Paj3]>0,j∈𝒩,
satisfying the following inequalities
(30)[Θa1Θa2Θa300*Θa4Θa5𝒫aj0**Θa60RjT***-ε-1I0****-I]<0,j∈𝒩,
where(31)Θa1=[-He(Xaj)-Yaj-a1NjT*-He(a2Nj)],Θa3=[-Zaj+XajTBj2+a1B^fjDj2-a5Njvj+YajTBj2+a2B^fjDj2],Θa2=[Paj1-Maj+XajTAj+a1B^fjCjPaj2-Saj+a1A^fjPaj2T-a3Nj+YajTAj+a2B^fjCjPaj3-a4Nj+a2A^fj],Θa4=[αjPaj1+εgjI+He(MajTAj+a3B^fjCj)αjPaj2+a4A^fj+AjTSaj+a4CjTB^fjT*αjPaj3+εgjI+He(a4A^fj)],Θa5=[MajTBj2+a3B^fjDj2+AjTZaj+a5CjTB^fjTvj-CjTD^fjTSajTBj2+a4B^fjDj2+a5A^fjTvj-C^fjT],Θa6=β2I+He(Bj2TZaj+a5Dj2TB^fjTvj-D^fjDj2),then, for any switching signal satisfying (13), nonlinear impulsive switched system (11) is stable and guarantees the H- performance (8). Moreover, if (30) is feasible, then the FD filter gains in form of (3) can be given by
(32)[AfjBfjCfjDfj]=[Nj00Rj]-T[A^fjB^fjC^fjD^fj].Proof.
Now, to establish the convex condition, (12) can be rewritten as
(33)𝒰Tℳaj𝒰<0,
where
(34)𝒰=[𝒜jℬj2I00I],ℳaj=[0𝒫aj0𝒫ajαj𝒫aj+ε-1𝒫aj2+εgjI-𝒞jT𝒞j-𝒞jT𝒟j20-𝒟j2T𝒞jβ2I-𝒟j2T𝒟j2].
On the other hand, we also can have
(35)[-I𝒜jℬj2]𝒰=0.
Based on Finsler’s Lemma, it has(36)ℳaj+He([-IA~jTB~j2T]𝒲aj)<0,
where 𝒲aj introduced by Finsler’s Lemma is the matrix variable of appropriate dimensions. Partition 𝒲aj as 𝒲aj=[𝒲aj1𝒲aj2𝒲aj3], and
(37)𝒲aj1=[XajYaja1Nja2Nj],𝒲aj2=[MajSaja3Nja4Nj],𝒲aj3=[Zaja5Njvj].
Due to -𝒞jT𝒞j, -𝒞jT𝒟j2 and -𝒟j2T𝒟j2 in ℳaj, then the condition (36) is non-convex. To establish the convex condition, (36) can be rewritten as
(38)𝒰-Tℳ-aj𝒰-<0,
where(39)𝒰-=[I000I000I0𝒞j𝒟j2],ℳ-aj=[-He(𝒲aj1)𝒫aj-𝒲aj2+𝒲aj1T𝒜j-𝒲aj3+𝒲aj1Tℬj20*αj𝒫aj+εgjI+ε-1𝒫aj2+He(𝒜jT𝒲aj2)𝒜jT𝒲aj3+𝒲aj2Tℬj20**β2I+He(ℬj2T𝒲aj3)0***0***-I].Using Finsler’s Lemma again, (38) holds if
(40)ℳ-aj+He(ℛℒ)<0
holds for any ℛ with appropriate dimension and ℒ=[0,-𝒞j,-𝒟j2,I]. Let ℛ=[0,0,Rj,0]T and define A^fj=NjTAfj, B^fj=NjTBfj, C^fj=RjTCfj, D^fj=RjTDfj; then (30) is reached by the Schur complement on (40). Hence if the conditions (30) holds, nonlinear impulsive switched system (11) is stable and guarantees the H- performance (8), which completes the proof.Remark 10.
Scalarsa1~a5 are used to reduce conservatism for the same variable Nj in matrix variables 𝒲aj1~𝒲aj3. On the other hand, the coefficient matrices vj are given to adjust the variable dimension. When we set a1~a5, vj as the fixed parameters, the conditions in Theorem 9 become convex.


## 3.2. The Disturbance Attenuation Performance (8)

Consider nonlinear impulsive switched system (4) with f(t)=0, we have
(41)x~˙(t)=𝒜σ(t)x~(t)+ℬσ1d(t)+𝒬σ(t)(t,x~(t)),t≠tk,Δx~=ℋσ(t)x~(t)+𝒪σ(t)(t,x~(t)),t=tk,r(t)=𝒞σx~(t)+𝒟σ1d(t);H
∞ performance for nonlinear impulsive switched system (41) is given.Lemma 11.
Letαj, ε be constants satisfying 0<αj<1, ε>0, j∈𝒩, and Assumptions 1 and 2 hold. Furthermore, suppose that nonlinear impulsive switched system (41) switches from ith subsystem to jth subsystem as switched time point tk. If there exists the Lyapunov functions candidate Vσ(t)(t)=Vj(t)=x~(t)T𝒫jx~(t), j∈𝒩, satisfying
(42)[𝒜jT𝒫j+𝒫j𝒜j+ε-1𝒫j2+αj𝒫j+εgjI+𝒞jT𝒞j𝒫jℬj1+𝒞jT𝒟j1ℬj1T𝒫j+𝒟j1T𝒞j-γ2I+𝒟j1T𝒟j1]<0,[𝒜jT𝒫j+𝒫j𝒜j+ε-1𝒫j2+αj𝒫j+εgjI+𝒞jT𝒞j𝒫jℬj1+ℬj1T𝒫j+𝒟j1T𝒞j-γ2I+]j∈𝒩,
then nonlinear impulsive switched system (41) is stable with H∞-gain γ for any switching signal satisfying (13).Proof.
Following the same lines as those for Lemma7, nonlinear impulsive switched system (41) is stable, for any nonzero d(t)∈l2[0,∞) and zero initial condition x~(t0). Let Γ2(t)=rT(t)r(t)-γ2dT(t)d(t) and consider the Lyapunov function as Lemma 7; we can have
(43)Vj(x~(t))+αjVj(x~(t))+Γ2(t)≤𝒳2TRj2𝒳2,j∈𝒩,
where
(44)𝒳2=[x~(t)Td(t)T]T,Rj2=[𝒜jT𝒫j+𝒫j𝒜j+ε-1𝒫j2+αj𝒫j+εgjI+𝒞jT𝒞j𝒫jℬj1+𝒞jT𝒟j1ℬj1T𝒫j+𝒟j1T𝒞j-γ2I+𝒟j1T𝒟j1].
Equation (42) is equivalent to Rj2<0. Thus, it has
(45)Vj(x~(t))+αjVj(x~(t))+Γ2(t)≤0.
By iteration operation on the above inequality for t∈(tk,tk+1], we have Vj(t)≤e-αj(t-tk)Vj(tk+)-∫tkt‍Γ2(s)ds. As the same lines as those for Lemma 7, it has
(46)Vj(t)≤e-αj(t-tk)Vj(tk+)-∫tkt‍Γ2(s)ds≤⋯≤≤eg(t-t0)Vσ(t0)(t0)-∫t0t‍eg(t-s)Γ2(s)ds≤eg(t-t0)Vσ(t0)(t0)-∫t0t‍Γ1(s)ds.
Under the zero initial condition, (46) implies ∫t0t‍eg(t-s)Γ2(s)ds<0. Because 0<g(t-s)<1, s∈[t0,t), it can further have
(47)∫t0t‍rT(s)r(s)ds≤γ2∫t0t‍dT(s)d(s)ds.
Whent→∞, the nonlinear impulsive switched system (41) has H∞ performance, which completes the proof.Based on Lemma11, the following theorem is given to obtain sufficient conditions by linear matrix inequalities.Theorem 12.
Letαj, γ, and ε be constants satisfying 0<αj<1, γ>0, and ε>0 and Assumptions 1 and 2 hold. If there exist matrix variables Ybj, Xbj,  Mbj,  Sbj, Nj, Rj, A^fj,  B^fj, C^fj, and D^fj and symmetric positive-definite matrices
(48)𝒫bj=[Pbj1Pbj2*Pbj3]>0,j∈𝒩,
satisfying the following inequalities
(49)[Θb1Θb2Θb300*Θb4Θb5𝒫bjΘb6**-γ2I0Dj1TD^fjT***-ε-1I0****I-He(Rj)]<0,j∈𝒩,
where(50)Θb1=[-He(Xbj)-Ybj-b1NjT*-He(b2Nj)],Θb3=[XbjTBj1+b1B^fjDj1YbjTBj1+b2B^fjDj1],Θb2=[Pbj1-Mbj+XbjTAj+b1B^fjCjPbj2-Sbj+b1A^fjPbj2T-b3Nj+YbjTAj+b2B^fjCjPbj3-b4Nj+b2A^fj],Θb4=[αjPbj1+He(MbjTAj+b3B^fjCj)+εgjIαjPbj2+b3A^fj+AjTSbj+b4CjTB^fjT*αjPbj3+εgjI+He(b4A^fj)],Θb5=[MbjTBj1+b3B^fjDj1SbjTBj1+b4B^fjDj1],Θb6=[CjTD^fjTC^fjT],then, for any switching signal satisfying (13), nonlinear impulsive switched system (41) is stable and guarantees the H∞ performance (9). Moreover, if (49) is feasible, then the FD filter gains in form of (3) can be given by
(51)[AfjBfjCfjDfj]=[Nj00Rj]-T[A^fjB^fjC^fjD^fj].Proof.
To establish the convex condition, (42) can be rewritten as follows:
(52)[𝒜jℬj1I0]T[0𝒫bj𝒫bjα𝒫bj][𝒜jℬj1I0]+[ε-1𝒫bj2+εgjI+𝒞jT𝒞j𝒞jT𝒟j1𝒟j1T𝒞j-γ2I+𝒟j1T𝒟j1]<0.
Denote
(53)ℳbj=[0𝒫bj0𝒫bjα𝒫bj+ε-1𝒫bj2+εgjI+𝒞jT𝒞j𝒞jT𝒟j10𝒟j1T𝒞j-γ2I+𝒟j1T𝒟j1].
From (52), we have
(54)[𝒜jℬj1I00I]Tℳbj[𝒜jℬj1I00I]<0.
On the other hand,
(55)[00I]Tℳbj[00I]=-γ2I+𝒟j1T𝒟j1<0.
Based on Projection Lemma, it follows from (54) and (55) that
(56)ℳbj+He([-IA~jTB~j1T]𝒲bj[I000I0])<0,
where 𝒲bj introduced by Projection Lemma is the matrix variable of appropriate dimensions. Partition 𝒲bj as 𝒲bj=[𝒲bj1𝒲bj2]. By the Schur complement, (56) is equivalent to(57)[-He(𝒲bj1)𝒫bj-𝒲bj2+𝒲bj1T𝒜j𝒲bj1Tℬj100*αj𝒫bj+εgjI+He(𝒜j𝒲bj2)𝒲bj2Tℬj1𝒫bj𝒞jT**-γ2I0𝒟j1T***-ε-1I0****-1]<0.Then pre- and postmultiply diag{I,I,I,I,RjT} to (57); one obtains(58)[-He(𝒲bj1)𝒫bj-𝒲bj2+𝒲bj1T𝒜j𝒲bj1Tℬj100*αj𝒫bj+εgjI+He(𝒜jT𝒲bj2)𝒲bj2Tℬj1𝒫bj𝒞jTRj**-γ2I0𝒟j1TRj***-ε-1I0****-RjTRj]<0.we partition 𝒲bj1, 𝒲bj2, respectively, as
(59)𝒲bj1=[XbjYbja1Nja2Nj],𝒲bj2=[MbjSbja3Nja4Nj]
and b1~b4 are set to be fixed parameters. Define A^fj=NjTAfj, B^fj=NjTBfj, C^fj=RjTCfj, and D^fj=RjTDfj; -RjTRj≤I-He(Rj) then (58) becomes (49). Hence if the condition (30) holds, nonlinear impulsive switched system (41) is stable and guarantees the H∞ performance (8), which completes the proof.Remark 13.
Scalarsb1~b4 are used to reduce conservatism for the same variable Nj in matrix variables 𝒲bj1 and 𝒲bj2. When we set b1~b4 as the fixed parameters, the conditions in Theorem 12 become convex.


## 3.3. Algorithm

In the previous sections, Theorems9 and 12 have formulated the inequality conditions for the performances (8)-(9), respectively. Summarily, we have the following algorithm.It is noted that conditions (30) and (49) are all convex. Hence, the problem of FD filter design can directly translate into the following optimization problem:
(60)maxβs.t.(30)and(49),j∈𝒩.Remark 14.
TheH- performance and H∞ performance are considered to describe the fault sensitiveness performance and the disturbance attenuation performance, respectively. Due to the multiobjective optimization problem, each performance has some LMI conditions. However, by analyzing the theorems and their proofs, it can be discovered that 𝒲aj1,  𝒲aj2 in Theorem 9 and 𝒲bj1, 𝒲bj2 in Theorem 12 can be defined as the same variable to decrease the computational complexity.Finally, the gain matricesAfj, Bfj, Cfj, and Dfj of the filters can be derived as
(61)Afj=Nj-TA^fj,Bfj=Nj-TB^fj,Cfj=Rj-TC^fj,Dfj=Rj-TD^fj.


## 4. Thresholds Computation

After the gain matrices of the filtersAfj, Bfj, Cfj, and Dfj are designed, similarly to that proposed in [32], the residual evaluation function Jr(τ) can be chosen as
(62)Jr(τ)=1τ∫slsk‍rT(s)r(s)ds,
where Jr(τ) is square value which means the average energy of residual signal over a time interval (sl,sk), sl denotes the initial evaluation time instant, and sk stands for the evaluation time.We propose to use the following threshold:(63)Jth=supd(t)∈l2[0,∞)f(t)=0,j∈𝒩Jr.
Consequently, the occurrence of faults can be detected by the following logic rule:
(64)∥Jr(τ)∥≤Jth⟹noalarm∥Jr(τ)∥>Jth⟹alarm.


## 5. Examples

In this section, we present examples derived from a liquid level control system to illustrate the effectiveness of FD design approach.Consider a continuous stirred tank reactor (CSTR) control system [33] shown in Figure 1. It consists of a constant-volume CSTR fed by a single inlet stream through a selector valve which is connected to two different source streams.Figure 1
Schematic diagram of the process.Assuming constant liquid volume, negligible heat losses, perfectly mixing, and a first-order reaction in reactantA, the dynamical equations of the CSTR at each operating mode are described by
(65)C˙A=qiV(CAfi-CA)-a0exp(-ERT)CA,T˙=qiV(Tfi-T)-a1exp(-ERT)CA+a2(Ta-T),
where CA is the reactant A concentration (moL/L), qi is the ith mode’s feed flow rate (L/min), CAfi is the ith mode’s concentration of the feed stream (moL/L), V is the volume of the reactor (L), E is the activation energy, R is the gas constant (J/(moL·K)), T is the reactor temperature (K), Tfi is the ith mode’s reactor temperature of the feed stream (K), and Ta is the coolant temperature (K). These nominal values of the parameters are described in [34], and we choose the nominal operating conditions corresponding to an unstable equilibrium point as TA*=300K, CA*=0.5moL/L, and T*=350K for both modes.Since the objective in the example is to testify the FD filter design techniques for the switched system and sustain the main theoretical results, we make design as [33] and   the closed-loop system can be obtained with matrices(66)Aa1=[0.51-3-3],Aa2=[21-8-3].
Suppose other system matrices in (1) as
(67)B11=[0.24-0.10],B12=[0.95-1.29],B21=[0.1-0.31],B22=[-0.521.34],Υ1(t,x(t))=[sin(x1(t))e-0.05t0],Υ2(t,x(t))=[sin(x1(t))e-0.05tsin(x2(t))e-0.05t],H1=H2=-[0.58000.58],Ω1(t,x(t))=Ω2(t,x(t))=[0.3sin(x1(ti))0.3sin(x2(ti))],C1=C2=[1,1],D11=D21=0.1,D12=D12=0.94.
Let q1=q3=-1, q2=q4=1,  q∈{a,b},  vj=[1,1]T, j∈𝒩. By giving γ=1, αj=0.5, and ε=0.02and obtaining gj=1, ρj=0.32, we solve the convex optimization problem (60) and get the optimal sensitivity performance gain β=1.0884. The gain matrices of the fault detection filters and the matrix in the Lyapunov function are
(68)[Af1Bf1Cf1Df1]=[-0.9011-0.26861.2677-0.3316-0.7128-1.52270.2561-0.49823.2893],[Af2Bf2Cf2Df2]=[2.90381.6812-0.6576-10.1086-4.66031.6406-1.9562-1.38631.4977],𝒫1=[35.267624.1501-6.6667-1.726024.150122.2633-3.1844-4.7123-6.6667-3.184419.791211.8843-1.7260-4.712311.884312.3490],𝒫2=[28.415110.2415-22.8368-8.196910.24154.8161-9.0122-4.0940-22.8368-9.012224.47909.1073-8.1969-4.09409.10734.1803].
Then by (13), the dwell time for each subsystem is obtained with δ=1:
(69)dwelltimeforsubsystem1⟹τ1>ln(μ12|k1)α1=9.6813,dwelltimeforsubsystem2⟹τ2>ln(μ21|k2)α2=13.8568k1,k2∈{1,2,…},k1≠k2.To illustrate the simulation results of the FD objective, the two cases which include the fault for subsystem 1 and subsystem 2, respectively, are considered. The disturbance is assumed to bed(t)=0.8cos(0.53t)exp(-0.05t). The switching signal in this paper is shown in Figure 2.Figure 2
Switch signal.Case 1.
The fault for subsystem 1 with the unit amplitude occurs from 100 sec to 200 sec. The generated residualr(t) and the evolution of the residual evaluation function Jr(t) are shown in Figure 3. The simulation results show that when the fault for subsystem 1 occurs, the residual signal varies sharply, and Jr(t)>Jth at 109 sec, which means that the fault for subsystem 1 can be detected 9 sec after the fault of subsystem 1 occurs. Hence, the fault for subsystem 1 can be detected.(a) The residual signal. (b) The residual evaluation function and threshold when the fault for subsystem 1 occurs from 100 sec to 200 sec.
(a)
(b)Case 2.
The fault for subsystem 2 with the unit amplitude also occurs from 200 sec to 300 sec. The generated residualr(t) and the evolution of the residual evaluation function Jr(t) are shown in Figure 4. It can be seen that when subsystem 2 are activated at 210 sec, the residual signal is changed sharply and Jr(t)>Jth at 234 sec. Thus, the fault for subsystem 2 can be detected.(a) The residual signal. (b) The residual evaluation function and threshold when the fault for subsystem 2 occurs from 200 sec to 300 sec.
(a)
(b)From Cases1 and 2, we see that both the faults for subsystem 1 and subsystem 2, respectively, can be detected, and they can demonstrate the effectiveness of the proposed design method.


## 6. Conclusion

In this paper, the problem of FD filter design for nonlinear impulsive switched systems has been investigated. Firstly, theH- performance and the H∞ performance are presented for nonlinear impulsive switched systems, and sufficient conditions to characterize given performances have been obtained. Subsequently, the design of FD filters is formulated as a multiobjective optimization problem, and the filter gains are characterized in term of the solution of LMI conditions. Finally, an example has been given to illustrate the effectiveness of the proposed method.Future works on fault detection for nonlinear impulsive switched systems are to focus on how to cope with stochastic switched systems as in [35, 36] and switched systems with time delay as in [37–39].



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*Source: 815329-2013-07-29.xml*