# Fault Detection for Discrete-Time Nonlinear Impulsive Switched Systems

Authors: Qingyu Su; Xiaolong Jia; Zhengfan Song
Journal: Abstract and Applied Analysis
 (2015)

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## Abstract

This paper investigates the fault detection problem for discrete-time nonlinear impulsive switched systems. Attention is focused on designing the fault detection filters to guarantee the robust performance and the detection performance. Based on these performances, sufficient conditions for the existence of filters are given in the framework of linear matrix inequality; furthermore, the filter gains are characterized by a convex optimization problem. The presented technique is validated by an example. Simulation results indicate that the proposed method can effectively detect the faults.


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## Body

## 1. Introduction

Fault detection (FD) is an important topic in system engineering from the viewpoint of the higher demands for safety and reliability of control systems [1, 2]. The basic idea of the model-based FD is to design observers [3] or filters [4] and generate an residual signal. Since the value counted by the residual evaluation function is larger than the predefined threshold, an alarm is generated. To date, there are many methods to solve the FD problem. As one of the typical methods, the FD problem is converted into a robust filtering problem; then H∞ technique is presented [5, 6]. For another method, FD systems have been directly considered to be sensitive to the faults and simultaneously robust to the unknown disturbance, then the H∞/H- technique investigates this important issue [7, 8].On the other hand, switched systems which belong to hybrid systems consist of a finite number of subsystems and a logical rule that orchestrates switching between these subsystems [9]. The primary motivation for studying switched systems comes partly from the fact that switched systems and switched multicontroller systems have numerous applications in control of flight control [10], missile autopilot design [11], chemical systems [12], networked control systems [13], and many other fields. Until now, a number of recent results are focused on stability and stabilizability under arbitrary switching [14], restricted switching (like dwell time and average dwell time [15, 16]), multiple Lyapunov functions method, and piecewise quadratic Lyapunov functions. As one of the special switched systems, impulsive switched systems produce impulses when the system is switching among subsystems, and There is also a wide range of actual systems such as engineering, economics, and biology. This kind of impulses will cause instability and oscillations and lead to poor performance. Recently, a number of papers have focused on stability problem of the impulsive switched systems [17–21].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 [22–26]. To the best of the authors’ knowledge, the FD problem for impulsive switched systems, especially discrete-time nonlinear impulsive switched systems, has not been investigated yet. It is worth noting that the FD approaches for switched systems without impulses are not appropriate for switched systems with impulses due to the effect of the impulse in switching point. Therefore, a new FD technique is needed to solve the impulsive case. Moreover, even if the mathematics model for the actual system are established with neglecting the impulse in switching point, the inaccurate mathematics model may reduce the robustness of the actual system, and the results may increase the risk of the false alarm. Thus, it is necessary to directly investigate the fault detection problem for impulsive switched systems. As the significance in theory and practice, the FD problem for discrete-time nonlinear impulsive switched systems should be investigated, which motivates us to study this interesting issue.In this paper, we consider a general class nonlinear impulsive switched system with nonlinear impulsive increments, and the fault detection problem for this class of systems is investigated. Firstly, a weightedl2 performance for discrete-time nonlinear impulsive switched systems is presented; meanwhile, the H- performance of discrete-time nonlinear impulsive switched systems is derived to reflect the effect on the residual signal from the faults. Subsequently, sufficient conditions for the weighted l2 performance and the H- performance are formulated by linear matrix inequalities (LMIs) with less conservatism. 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. An example is given in Section 4 to demonstrate the proposed method. Conclusions of this paper are given in the last section.Notation. The superscripts T,-1 stand for the transposition and the inverse of a matrix, respectively. The matrices A>0  (A≥0) and A<0  (A≤0) denote positive-definiteness (positive semidefinite matrix) and negative-definiteness (negative semidefinite matrix). I and 0 represent the identity matrix and the zero matrix with appropriate dimensions, 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 discrete-time nonlinear impulsive switched systems:(1)xk+1=∑j=1Nξjk·Ajxk+Bj1dk+Bj2fk+Υjk,xk,k≠ki,Δxk=∑j=1NξjkHjxk+Ωjk,xk,k=ki,yk=∑j=1NξjkCjxk+Dj1dk+Dj2fk,xk0+=x0,where x(k)∈Rn is the state and y(k)∈Rm is the measured output. d(k)∈Rnd and f(k)∈Rnf are the disturbance input and the fault, respectively, which are energy bounded; then they are demanded to belong to l20∞. ki,i=0,1,2,…, is impulsive switching time points. Denote N=1⋯N. The switching signal ξj(k):Z+→{0,1} specifies that jth subsystem is activated when ξj(k)=1, and ∑j=1Nξj(k)=1. Υj(k,x(k)):k0,∞×Rn→Rn, which is globally Lipschitz continuous, and Υj(k,0)≡0 for all k∈k0,∞Δx(ki)=x(ki+)-x(ki-)=x(ki+)-x(ki), with x(ki+)=limk→ki+x(k) and x(ki)=x(ki-)=limk→ki-x(k); that is, the solution x(t) is left continuous. Ωj(k,x(k)):k0,∞×Rn→Rn is nonlinear functions, and Ωk(k,0)≡0 for all t∈k0,∞. The jth subsystem is denoted by the matrices Aj,Bj1,Bj2,Hj,Cj,Dj1, and Dj2 with appropriate dimensions.The following assumptions for nonlinear impulsive switched system (1) are introduced.Assumption 1.
There exist nonnegative scalarsgj>0, such that (2)ΥjTk,xkΥjk,xk≤gjxTkxk,j∈N.Assumption 2.
Denote byρ(·) the spectral radius for each subsystem and ρh=maxj∈Nρ(Hj+I); then Ωj(ki,x(ki))≤ρhx(ki).Remark 3.
Note that owing to the presence of system nonlinearity, the above assumptions essentially draw from the analysis of the stability for the system. Moreover, they are two basic conditions for this kind of systems (see [27] for further discussion). Therefore we consider the nonlinear impulsive switched systems, which satisfies Assumptions 1 and 2.For the purpose of the fault detection, the following FD filters are designed:(3)xfk+1=∑j=1NξjkAfjxfk+Bfjyk,rk=∑j=1NξjkCfjxfk+Dfjyk,where xf(k) is the state of the filter, r(k) is the residual signal, and j∈N. The matrices Afj,Bfj,Cfj, and Dfj with appropriate dimensions are to be determined.Denoting the augmented state vectorx~(k)=x(k)T,xf(k)TT and augmenting the model of system (1) to include the states of (3), we can obtain the augmented system as follows:(4)x~k+1=∑j=1NξjkAjx~k+Bj1dk+Bj2fk+Υ~jk,x~k,k≠ki,Δx~k=∑j=1NξjkHjxk+Ω~jk,x~k,k=ki,rk=∑j=1NξjkCjx~k+Dj1dk+Dj2fk,x~k0+=x~0,where j∈N,(5)Aj=Aj0BfjCjAfj,Bj1=Bj1BfjDj1,Bj2=Bj2BfjDj2,Hj=Hj000,Υ~jk,x~k=Υjk,xk0,Ω~jk,x~k=Ωjk,xk0,Cj=DfjCjCfj,Dj1=DfjDj1,Dj2=DfjDj2.To present the purpose of this paper more precisely, the following definition is introduced.Definition 4.
Let Assumptions1 and 2 be satisfied and d(k)=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(6)∑k=0∞rkTrk≥β2∑k=0∞fkTfk.


### 2.2. Problem Formulation

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 be formulated as the following performances:(7)∑k=0∞1-αkrkTrk≤γ2∑k=0∞dkTdk,(8)∑k=0∞rkTrk≥β2∑k=0∞fkTfk.Remark 5.
Condition (7) is used for the disturbance attenuation performance, which minimizes the disturbance effects on the residual output and ensures that the disturbance is not disastrous. Condition (8) is expressed to maximize the effects of the fault f(k) on the residual output r(k). That is, the residual output r(k) is sensitive to the fault f(k).After designing the residual generator, how to evaluate the generated residual is considered. One of the widely adopted approaches is to select an appropriate threshold and an appropriate residual evaluation function. Similar to that proposed in [28], the residual evaluation function Jr(k)(k) can be chosen as (9)Jrkk=1k∑s=1krTsrs,where k denotes the evaluation time step.LetJth=supd(k)∈l2,f(k)=0Jr(k)(k) be the threshold. Based on this, the occurrence of faults can be detected by comparing Jr(k)(k) and Jth according to the following logical relationship:(10)Jrk≤Jth⟹no fault⟹no alarm,Jrk>Jth⟹fault⟹alarm.


## 2.1. System Model

Consider the following discrete-time nonlinear impulsive switched systems:(1)xk+1=∑j=1Nξjk·Ajxk+Bj1dk+Bj2fk+Υjk,xk,k≠ki,Δxk=∑j=1NξjkHjxk+Ωjk,xk,k=ki,yk=∑j=1NξjkCjxk+Dj1dk+Dj2fk,xk0+=x0,where x(k)∈Rn is the state and y(k)∈Rm is the measured output. d(k)∈Rnd and f(k)∈Rnf are the disturbance input and the fault, respectively, which are energy bounded; then they are demanded to belong to l20∞. ki,i=0,1,2,…, is impulsive switching time points. Denote N=1⋯N. The switching signal ξj(k):Z+→{0,1} specifies that jth subsystem is activated when ξj(k)=1, and ∑j=1Nξj(k)=1. Υj(k,x(k)):k0,∞×Rn→Rn, which is globally Lipschitz continuous, and Υj(k,0)≡0 for all k∈k0,∞Δx(ki)=x(ki+)-x(ki-)=x(ki+)-x(ki), with x(ki+)=limk→ki+x(k) and x(ki)=x(ki-)=limk→ki-x(k); that is, the solution x(t) is left continuous. Ωj(k,x(k)):k0,∞×Rn→Rn is nonlinear functions, and Ωk(k,0)≡0 for all t∈k0,∞. The jth subsystem is denoted by the matrices Aj,Bj1,Bj2,Hj,Cj,Dj1, and Dj2 with appropriate dimensions.The following assumptions for nonlinear impulsive switched system (1) are introduced.Assumption 1.
There exist nonnegative scalarsgj>0, such that (2)ΥjTk,xkΥjk,xk≤gjxTkxk,j∈N.Assumption 2.
Denote byρ(·) the spectral radius for each subsystem and ρh=maxj∈Nρ(Hj+I); then Ωj(ki,x(ki))≤ρhx(ki).Remark 3.
Note that owing to the presence of system nonlinearity, the above assumptions essentially draw from the analysis of the stability for the system. Moreover, they are two basic conditions for this kind of systems (see [27] for further discussion). Therefore we consider the nonlinear impulsive switched systems, which satisfies Assumptions 1 and 2.For the purpose of the fault detection, the following FD filters are designed:(3)xfk+1=∑j=1NξjkAfjxfk+Bfjyk,rk=∑j=1NξjkCfjxfk+Dfjyk,where xf(k) is the state of the filter, r(k) is the residual signal, and j∈N. The matrices Afj,Bfj,Cfj, and Dfj with appropriate dimensions are to be determined.Denoting the augmented state vectorx~(k)=x(k)T,xf(k)TT and augmenting the model of system (1) to include the states of (3), we can obtain the augmented system as follows:(4)x~k+1=∑j=1NξjkAjx~k+Bj1dk+Bj2fk+Υ~jk,x~k,k≠ki,Δx~k=∑j=1NξjkHjxk+Ω~jk,x~k,k=ki,rk=∑j=1NξjkCjx~k+Dj1dk+Dj2fk,x~k0+=x~0,where j∈N,(5)Aj=Aj0BfjCjAfj,Bj1=Bj1BfjDj1,Bj2=Bj2BfjDj2,Hj=Hj000,Υ~jk,x~k=Υjk,xk0,Ω~jk,x~k=Ωjk,xk0,Cj=DfjCjCfj,Dj1=DfjDj1,Dj2=DfjDj2.To present the purpose of this paper more precisely, the following definition is introduced.Definition 4.
Let Assumptions1 and 2 be satisfied and d(k)=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(6)∑k=0∞rkTrk≥β2∑k=0∞fkTfk.


## 2.2. Problem Formulation

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 be formulated as the following performances:(7)∑k=0∞1-αkrkTrk≤γ2∑k=0∞dkTdk,(8)∑k=0∞rkTrk≥β2∑k=0∞fkTfk.Remark 5.
Condition (7) is used for the disturbance attenuation performance, which minimizes the disturbance effects on the residual output and ensures that the disturbance is not disastrous. Condition (8) is expressed to maximize the effects of the fault f(k) on the residual output r(k). That is, the residual output r(k) is sensitive to the fault f(k).After designing the residual generator, how to evaluate the generated residual is considered. One of the widely adopted approaches is to select an appropriate threshold and an appropriate residual evaluation function. Similar to that proposed in [28], the residual evaluation function Jr(k)(k) can be chosen as (9)Jrkk=1k∑s=1krTsrs,where k denotes the evaluation time step.LetJth=supd(k)∈l2,f(k)=0Jr(k)(k) be the threshold. Based on this, the occurrence of faults can be detected by comparing Jr(k)(k) and Jth according to the following logical relationship:(10)Jrk≤Jth⟹no fault⟹no alarm,Jrk>Jth⟹fault⟹alarm.


## 3. The Fault Detection Filter Design

Before beginning this section, the following lemmas are needed to present our main results.Lemma 6 (see [27]).
Letϵ>0 be a given scalar and Ξ∈Rp×q 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 7 (see [29]).
LetP∈Rn×n be a given symmetric positive definite matrix and let Q∈Rn×n be a given symmetric matrix. Then λminP-1Qx(t)TPx(t)≤x(t)TQx(t)≤λmaxP-1Qx(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 filter gains can be obtained.


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

Considering discrete-time nonlinear impulsive switched system (4) with f(k)=0, we have(11)x~k+1=∑j=1NξjkAjx~k+Bj1dk+Υ~jk,x~k,k≠ki,Δx~k=∑j=1NξjkHjxk+Ω~jk,x~k,k=ki,rk=∑j=1NξjkCjx~k+Dj1dk,x~k0+=x~0.Firstly, the weightedl2 performance for nonlinear impulsive switched system (11) is given.Lemma 8.
Letα,εj1,εj2, and εj3 be constants satisfying 0<α<1,εj1>0,εj2>0, and εj3>0, and Assumptions 1 and 2 hold. Furthermore, suppose that discrete-time nonlinear impulsive switched system (11) switches from pth subsystem to jth subsystem as switched time point ki. If there exist λj>0 and Lyapunov functions candidate Vj(k)=x~(k)TPjx~(k) satisfying(12)0≤Pj≤λjI,(13)ΘjAjTPjBj1+CjTDj1∗-γ2I+Bj1TPj+εj2-1Pj2Bj1+Dj1TDj1<0,j∈N,where Θj=AjT(Pj+εj1-1Pj2)Aj-(1-α)Pj+(εj1+εj2+λj)gjI+CjTCj, then nonlinear impulsive switched system (11) is stable with the weight l2-gain γ for any switching signal satisfying(14)τp≥ceil⁡-ln⁡μpjln⁡1-α,where μpj=(εj3+1)λmax(Hj+I)TPj(Hj+I)+(εj3-1+1)ρh2λmaxPj/λminPp,p,j∈N, and the function ceil(ν) represents rounding real number ν to the nearest integer greater than or equal to ν.Proof.
See the appendices.Subsequently, inequality conditions for the disturbance attenuation performance (8) are constructed.Theorem 9.
Letγ, α, εj1, and εj2 be constants satisfying γ>0,0<α<1,εj1>0, and εj2>0. If there exist matrix variables A^fj,B^fj,C^fj,D^fj,Rj, and λj and symmetric positive-definite matrices(15)Pj=Pj1Pj2∗Pj2>0,j∈N,satisfying the following inequalities:(16)0≤Pj≤λjI,(17)-α¯Pj+φajI0Ξa13Ξa140Ξa16∗-γ2IΞa230Ξa25Dj1TD^fjT∗∗-Pj000∗∗∗-εj1I00∗∗∗∗-εj2I0∗∗∗∗0I-HeRj<0,where α¯=1-α,φaj=(εj1+εj2+λj)gj, (18)Ξa13=Ξa14=AjTPj1+CjTB^fjTAjTPj2+CjTB^fjTA^fjTA^fjT,Ξa23=Ξa25=Bj1TPj1+Dj1TB^fjTBj1TPj2+Dj1TB^fjT,Ξa16=CjTD^fjTC^fjT,then switched system (11) is asymptotically stable for any switching signal satisfying (14) and guarantees the weighted l2 performance ∑k=0∞(1-α)kr(k)Tr(k)≤γ2∑k=0∞d(k)Td(k).Proof.
See the appendices.


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

Considering discrete-time nonlinear impulsive switched system (4) with d(k)=0, we have(19)x~k+1=∑j=1NξjkAjx~k+Bj2fk+Υ~jk,x~k,k≠ki,Δx~k=∑j=1NξjkHjxk+Ω~jk,x~k,k=ki,rk=∑j=1NξjkCjx~k+Dj2fk,x~k0+=x~0.H
- performance for discrete-time nonlinear impulsive switched system (19) is given.Lemma 10.
Letα,εj1, and εj4 be constants satisfying 0<α<1,εj1>0, and εj4>0, and Assumptions 1 and 2 hold. Furthermore, suppose that nonlinear impulsive switched system (19) switches from pth subsystem to jth subsystem as switched time point ki. If there exist λj>0 and Lyapunov functions candidate Vj(k)=x~(k)TPjx~(k) satisfying(20)0≤Pj≤λjI,(21)ΦjBj2TPjAj-CjTDj2∗β2I+Bj2TPj+εj4-1Pj2Bj2-Dj2TDj2<0,j∈N,where Φj=AjT(Pj+εj1-1Pj2)Aj-(1-α)Pj+(εj1+εj4+λj)gjI-CjTCj, then discrete-time nonlinear impulsive switched system (19) is stable with the H--gain β for any switching signal satisfying (14).Proof.
See the appendices.Based on Lemma10, the following theorem is given to obtain sufficient conditions by linear matrix inequalities.Theorem 11.
Letβ, α, εj1, and εj4 be constants satisfying β>0, 0<α<1, εj1>0, and εj4>0. If there exist matrix variables A^fj,B^fj,C^ffj=[C^fj0], D^ffj=[D^fj0], Rj=[Rj,0], and λj and symmetric positive-definite matrices(22)Pj=Pj1Pj2∗Pj2>0,j∈N,satisfying the following inequalities:(23)0≤Pj≤λjI,(24)-IΞb12-Rj000∗Ξb22+φbjIΞb23Ξb24Ξb250∗∗β2I+HeDj2TD^fjΞb340Ξb36∗∗∗-Pj00∗∗∗∗-εj1I0∗∗∗∗∗-εj4I<0,where α¯=1-α,φbj=(εj1+εj4+λj)gj, (25)Ξb12=-Rj-Rj,Ξb22=-α¯Pj1+HeCjTD^ffjT-α¯Pj2+CjTD^ffjT+C^ffj∗-α¯Pj2+HeC^ffjT,Ξb23=CjTD^fjT+D^ffjDj2C^fjT+D^ffjDj2,Ξb24=Ξb25=AjTPj1+CjTB^fjTAjTPj2+CjTB^fjTA^fjTA^fjT,Ξb34=Ξb36=Bj2TPj1+Dj2TB^fjTBj2TPj2+Dj2TB^fjT,then discrete-time nonlinear impulsive switched system (19) is asymptotically stable for any switching signal satisfying (14) and guarantees H- performance: (26)∑k=0∞β2fkTfk≤∑k=0∞rkTrk.Proof.
See the appendices.


### 3.3. Algorithm

In the previous sections, Theorems9 and 11 have formulated the inequality conditions for the performances (7) and (8), respectively. Summarily, we have the following algorithm.It is noted that conditions (16), (17), (23), and (24) are all convex. Hence, the problem of FD filter design can directly translate into the following optimization problem:(27)max⁡β,s.t.16,17,23and24,j∈N.Moreover, if (27) is feasible, then the FD filter gains can be given by(28)AfjBfjCfjDfj=Pj200RjT-1A^fjB^fjC^fjD^fj.


## 3.1. The Disturbance Attenuation Performance (8)

Considering discrete-time nonlinear impulsive switched system (4) with f(k)=0, we have(11)x~k+1=∑j=1NξjkAjx~k+Bj1dk+Υ~jk,x~k,k≠ki,Δx~k=∑j=1NξjkHjxk+Ω~jk,x~k,k=ki,rk=∑j=1NξjkCjx~k+Dj1dk,x~k0+=x~0.Firstly, the weightedl2 performance for nonlinear impulsive switched system (11) is given.Lemma 8.
Letα,εj1,εj2, and εj3 be constants satisfying 0<α<1,εj1>0,εj2>0, and εj3>0, and Assumptions 1 and 2 hold. Furthermore, suppose that discrete-time nonlinear impulsive switched system (11) switches from pth subsystem to jth subsystem as switched time point ki. If there exist λj>0 and Lyapunov functions candidate Vj(k)=x~(k)TPjx~(k) satisfying(12)0≤Pj≤λjI,(13)ΘjAjTPjBj1+CjTDj1∗-γ2I+Bj1TPj+εj2-1Pj2Bj1+Dj1TDj1<0,j∈N,where Θj=AjT(Pj+εj1-1Pj2)Aj-(1-α)Pj+(εj1+εj2+λj)gjI+CjTCj, then nonlinear impulsive switched system (11) is stable with the weight l2-gain γ for any switching signal satisfying(14)τp≥ceil⁡-ln⁡μpjln⁡1-α,where μpj=(εj3+1)λmax(Hj+I)TPj(Hj+I)+(εj3-1+1)ρh2λmaxPj/λminPp,p,j∈N, and the function ceil(ν) represents rounding real number ν to the nearest integer greater than or equal to ν.Proof.
See the appendices.Subsequently, inequality conditions for the disturbance attenuation performance (8) are constructed.Theorem 9.
Letγ, α, εj1, and εj2 be constants satisfying γ>0,0<α<1,εj1>0, and εj2>0. If there exist matrix variables A^fj,B^fj,C^fj,D^fj,Rj, and λj and symmetric positive-definite matrices(15)Pj=Pj1Pj2∗Pj2>0,j∈N,satisfying the following inequalities:(16)0≤Pj≤λjI,(17)-α¯Pj+φajI0Ξa13Ξa140Ξa16∗-γ2IΞa230Ξa25Dj1TD^fjT∗∗-Pj000∗∗∗-εj1I00∗∗∗∗-εj2I0∗∗∗∗0I-HeRj<0,where α¯=1-α,φaj=(εj1+εj2+λj)gj, (18)Ξa13=Ξa14=AjTPj1+CjTB^fjTAjTPj2+CjTB^fjTA^fjTA^fjT,Ξa23=Ξa25=Bj1TPj1+Dj1TB^fjTBj1TPj2+Dj1TB^fjT,Ξa16=CjTD^fjTC^fjT,then switched system (11) is asymptotically stable for any switching signal satisfying (14) and guarantees the weighted l2 performance ∑k=0∞(1-α)kr(k)Tr(k)≤γ2∑k=0∞d(k)Td(k).Proof.
See the appendices.


## 3.2. The Fault Sensitiveness Performance (8)

Considering discrete-time nonlinear impulsive switched system (4) with d(k)=0, we have(19)x~k+1=∑j=1NξjkAjx~k+Bj2fk+Υ~jk,x~k,k≠ki,Δx~k=∑j=1NξjkHjxk+Ω~jk,x~k,k=ki,rk=∑j=1NξjkCjx~k+Dj2fk,x~k0+=x~0.H
- performance for discrete-time nonlinear impulsive switched system (19) is given.Lemma 10.
Letα,εj1, and εj4 be constants satisfying 0<α<1,εj1>0, and εj4>0, and Assumptions 1 and 2 hold. Furthermore, suppose that nonlinear impulsive switched system (19) switches from pth subsystem to jth subsystem as switched time point ki. If there exist λj>0 and Lyapunov functions candidate Vj(k)=x~(k)TPjx~(k) satisfying(20)0≤Pj≤λjI,(21)ΦjBj2TPjAj-CjTDj2∗β2I+Bj2TPj+εj4-1Pj2Bj2-Dj2TDj2<0,j∈N,where Φj=AjT(Pj+εj1-1Pj2)Aj-(1-α)Pj+(εj1+εj4+λj)gjI-CjTCj, then discrete-time nonlinear impulsive switched system (19) is stable with the H--gain β for any switching signal satisfying (14).Proof.
See the appendices.Based on Lemma10, the following theorem is given to obtain sufficient conditions by linear matrix inequalities.Theorem 11.
Letβ, α, εj1, and εj4 be constants satisfying β>0, 0<α<1, εj1>0, and εj4>0. If there exist matrix variables A^fj,B^fj,C^ffj=[C^fj0], D^ffj=[D^fj0], Rj=[Rj,0], and λj and symmetric positive-definite matrices(22)Pj=Pj1Pj2∗Pj2>0,j∈N,satisfying the following inequalities:(23)0≤Pj≤λjI,(24)-IΞb12-Rj000∗Ξb22+φbjIΞb23Ξb24Ξb250∗∗β2I+HeDj2TD^fjΞb340Ξb36∗∗∗-Pj00∗∗∗∗-εj1I0∗∗∗∗∗-εj4I<0,where α¯=1-α,φbj=(εj1+εj4+λj)gj, (25)Ξb12=-Rj-Rj,Ξb22=-α¯Pj1+HeCjTD^ffjT-α¯Pj2+CjTD^ffjT+C^ffj∗-α¯Pj2+HeC^ffjT,Ξb23=CjTD^fjT+D^ffjDj2C^fjT+D^ffjDj2,Ξb24=Ξb25=AjTPj1+CjTB^fjTAjTPj2+CjTB^fjTA^fjTA^fjT,Ξb34=Ξb36=Bj2TPj1+Dj2TB^fjTBj2TPj2+Dj2TB^fjT,then discrete-time nonlinear impulsive switched system (19) is asymptotically stable for any switching signal satisfying (14) and guarantees H- performance: (26)∑k=0∞β2fkTfk≤∑k=0∞rkTrk.Proof.
See the appendices.


## 3.3. Algorithm

In the previous sections, Theorems9 and 11 have formulated the inequality conditions for the performances (7) and (8), respectively. Summarily, we have the following algorithm.It is noted that conditions (16), (17), (23), and (24) are all convex. Hence, the problem of FD filter design can directly translate into the following optimization problem:(27)max⁡β,s.t.16,17,23and24,j∈N.Moreover, if (27) is feasible, then the FD filter gains can be given by(28)AfjBfjCfjDfj=Pj200RjT-1A^fjB^fjC^fjD^fj.


## 4. Examples

In this section, we present a numerical example to illustrate the effectiveness of FD design approach. Consider the discrete-time nonlinear impulsive switched systems (1) with two subsystems and two parameters:(29)A1=-0.120.53-0.230.59,A2=-0.600.23-0.49-0.05,B11=B21=-0.17-0.16,B12=B22=-0.140.05,Ω1ki,xki=Ω2t,xt=0.2sin⁡x1ki0.2sin⁡x2ki,Υ1k,xk=0.1sin⁡x1ke-0.5k0,Υ2k,xk=00.1sin⁡x2ke-0.5k,H1=H2=0.3000.3,C1=C2=0.3-0.1,D11=D21=0.1.Given γ=1,αj=0.4, and εj1=εj2=εj3=εj4=1 and choosing gj=1,ρj=0.32, we solve the convex optimization problem (27) and get the optimal sensitivity performance gain β=0.3561. The gain matrices of fault detection filters and matrix in Lyapunov functions are obtained as(30)Af1=0.06640.2438-0.10530.3415,Bf1=0.6063-0.2671,Cf1=-0.40740.0660,Df1=-2.0153,Af2=-0.28780.1567-0.3807-0.0119,Bf2=1.06650.0108,Cf2=-0.42650.0314,Df2=-2.1530,P1=1.5526-1.00700.6676-0.2132-1.00701.8900-0.21320.44650.6676-0.21320.6676-0.2132-0.21320.4465-0.21320.4465,P2=2.5529-1.03511.1314-0.3521-1.03511.3644-0.35210.55261.1314-0.35211.1314-0.3521-0.35210.5526-0.35210.5526.Then, according to (14), the dwell time for each subsystem is obtained:(31)dwell time for subsystem 1⟹τ1≥11,dwell time for subsystem 2⟹τ2≥10.To illustrate the simulation results of the FD objective, two cases which include the fault for subsystem 1 and subsystem 2, respectively, are considered. Furthermore, the switching signal is generated by satisfying (31) as shown in Figure 1. The threshold can be determined as Jth=0.1842.Figure 1
Switching signal.Case 1.
The fault for subsystem 1 is simulated as a pulse signal with amplitude 1 that occurred from 90 to 120 steps. The generated residual signalr(k) and evaluation of residual evaluation function Jr(k) are shown in Figure 2. The simulation results show that when the fault for subsystem 1 occurs, the subsystem 1 is not activated. Since subsystem 1 is activated at 93 steps the residual signal varies sharply, and Jr(t)>Jth at 99 steps, which means that the fault for subsystem 1 can be detected 6 steps after subsystem 1 is activated. Hence, the fault for subsystem 1 can be detected.Figure 2
Residual signal and residual evaluation function for (Case1).Case 2.
The fault for subsystem 2 is simulated as a pulse signal with amplitude 1 that occurred from 170 to 220 steps. The generated residual signalr(k) and evaluation of residual evaluation function Jr(k) are shown in Figure 3. It can be seen that when the fault for subsystem 2 occurs at 170 steps, the residual signal is changed sharply and Jr(k)>Jth at 182 steps. Thus, the fault for subsystem 2 can be detected.Figure 3
Residual signal and residual evaluation function for (Case2).From Cases1 and 2, we see that both faults for subsystem 1 and subsystem 2, respectively, can be detected, and they demonstrate the effectiveness of the proposed design method.


## 5. Conclusion

In this paper, the problem of FD filter design for discrete-time nonlinear impulsive switched systems has been investigated. Firstly, the weightl2 performance and the H- performance are presented, and sufficient conditions to characterize given performances have been formulated as the form of LMI. Subsequently, the gains of FD filters are obtained by a multiobjective optimization problem. Finally, the effectiveness of the proposed method for discrete-time nonlinear impulsive switched systems is illustrated by the example.



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*Source: 694319-2015-08-02.xml*