# Simultaneous Fault Detection and Control for Discrete-Time Systems via a Switched Scheme

Authors: Jian Li; Xingze Dai
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

This paper is concerned with the problem of simultaneous fault detection and control for linear systems with a switched scheme. The switched detector/controller is designed simultaneously and generates two signals such that it provides fault tolerance, especially including “destabilizing failure” meanwhile, it generates the residual signal to alarm the fault. When the faults are detected, the detector/controller is switched to reduce the effect of the faults. When the faults are removed, the detector/controller is switched to the original detector/controller to guarantee the control objective. In addition, it has time delay
in detection of the faults; then the time-driven switching strategy for asynchronous case is included. Thus a mixed switching strategy is proposed. A two-step procedure is adopted to obtain the solutions through satisfying a set of linear matrix inequalities. Finally, an example is provided to demonstrate the effectiveness of the proposed design method.


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## Body

## 1. Introduction

The problem of designing reliable control systems has attracted strong interest and intensive research activities recently. The objective of this research is focused on designing an appropriate controller such that the closed-loop system can guarantee system stability. During the past decades, there were many results that investigated this important issue. The problem of the reliable control and the reliable filter has been thoroughly investigated in [1–5]. Meanwhile, the simultaneous fault detection and control problem also has attracted a lot of attention in the last two decades. The attention of this study is to unify the control and detection units into a single unit; then it ensures that the reliable control is feasible, and the information of the fault is got. The advantage of this method is far less overall complexity than the design method where the two units are designed separately. Thus, the simultaneous fault detection and control problem has been addressed by several researches, for example, [6–9].On the other hand, there has been a great interest in switching control due to their significance both in theory and in applications [10–14]. The motivation of studying switched systems comes from the fact that many physical plants exhibit the switching feature during multimodels or multicontrollers, and a suitable switching rule is needed to deal with some complex tasks or to overcome the shortcomings of the single controller. Several approaches have been proposed for the control problem or the filtering problem for switched systems [15–18]. In addition, when the faults have been detected, the switched scheme can be applied to the problem of fault detection and control; then the detected information of the fault is the switching signal. This problem has been studied by several researches, for example, [19, 20]. It should be pointed out, however, that there has been “destabilizing failure” in many practical systems; that is, the never-faulty actuator cannot stabilize the considered system. Then, the existing design approaches are not appropriate for these complicated cases; the new technique with the switched scheme should be considered to guarantee the norm of the states of the system to increase within the acceptable limits and then realize the reliable control.In this paper, the detector/controller is designed as a switched scheme. When the actuator faults are detected, the detector/controller is switched to reduce the effect of the faults. When the faults are removed, the detector/controller is converted back to consider the control performance firstly. Meanwhile, the detector/controller is designed as a single unit and generates two signals: a detection signal and a control signal, which are used to detect faults and guarantee fault tolerance, especially including “destabilizing failure,” respectively. The key idea is to view the information of the fault detection to preserve the overall system stability and use the switching strategy to provide fault tolerance, especially including “destabilizing failure.”The contributions of this paper are in two respects. Firstly, the method takes into account the information of the faulty detection to control the system and improves the results in [21, 22], which only employ an average dwell-time switching strategy to stabilize the given system with actuators fault. Subsequently, it has time delay in detection of the faults; thus, the asynchronous case is considered in this switching strategy.


## 2. Problem Statement and Preliminaries

### 2.1. System Model

Consider the following discrete-time linear systems:(1)xk+1=Axk+B1uk+B2dk,yk=Cxk+D2dk,zk=Exk+F1uk+F2dk,where x(k)∈Rn is the state, y(k)∈Rm is the measured output, z(k)∈Rmz is the performance output, u(k)∈Rnu is the control input, and d(k)∈Rnd is the disturbance input which is assumed to belong to l20,∞. The matrices A, B1, B2, C, D2, E, F1, and F2 of each subsystem have appropriate dimensions.


### 2.2. Fault Model

To formulate the fault detection and control problem in this paper, the following fault model type is adopted. The actuator stuck fault defined in [23] is described as follows:(2)ufk=ρiuk+I-ρifk,i=0,…,N,where index i denotes the ith fault mode and N denotes the number of the total fault modes. The diagonal matrices ρi is defined as (3)ρi=diagρi1,…,ρik,…,ρinu,where ρi’s diagonal elements are either 1 or 0. Consider f(k)=f1,…,fk,…,fnu, where fk(k=1,…,nu) is an unknown constant which means the value of the stuck fault for the kth actuator.Remark 1.
Eachρi corresponds to a fault mode. We assume that ρ0=I; then it is fault free case and uf(k)=u(k). Note that when ρik=0 and fik≠0, the kth actuator is stuck. If ρik=0 and fk=0, (2) means that the kth actuator is outages. As [24], the fault f(k) is an intermittent fault, and it can be eliminated in a limited time.Then system (1) with actuator faults (2) can be described as(4)xk+1=Axk+B1uk+B2dk,faultfreecaseAxk+B1ρiuk+B2dk+B1I-ρifk,faultcase,yk=Cxk+D2dk,zk=Exk+F1uk+F2dk,faultfreecaseExk+F1ρiuk+F2dk+F11-ρifk,faultcase.Remark 2.
In this paper, the case that(A,B1ρi) is not stabilizable is also included in this paper; then, all the results which are based on the common assumption are invalid.To detect the fault and control system (4), design the detector/controller as the following form:(5)x^k+1=Akx^k+Bkyk,faultno-detectedcaseA¯kx^k+B¯kyk,faultdetectedcase,uk=Ckx^k,faultno-detectedcaseC¯kx^k,faultdetectedcase,y^k=Cx^k,where x^(k) is the detector/controller state vector and Ak,Bk,Ck, A¯k, B¯k, and C¯k, are real matrices of appropriate dimensions to be determined.Remark 3.
The detector/controller is switched between the fault no-detected case and the fault detected case, which can be seen as event-driven switching control.For fault detection, we can formulate the fault as the weighted faultf^(k)=Wf(z)f(k) with a given stable weighted matrix. Assume Aw, Bw, Cw, and Dw are known constant matrices; then the minimal realization of Wf(z) is(6)xwk+1=Awxwk+Bwfk,f^k=Cwxk+Dwfk.Combining (4), (5), and (6), we have the following augmented model:(7)x~k+1=A~x~k+B~wk,faultfreecaseA~1x~k+B~1wk,faultno-detectedcaseA~2x~k+B~2wk,faultdetectedcase,rek=C~x~k+D~wk,zk=E~x~k+F~wk,faultfreecaseE~1x~k+F~1wk,faultno-detectedcaseE~2x~k+F~2wk,faultdetectedcase,where r(k)=y(k)-y^(k) is the residual signal and re(k)=r(k)-f^(k) is the fault estimate error, x~(k)=x(k)Tx^(k)Txw(k)TT, w(k)=d(k)f(k), (8)A~B~A~1B~1A~2B~2C~D~E~F~E~1F~1E~2F~2=AB1Ck0B20BkCAk0BkD2000Aw0BwAB1ρiCk0B2B1I-ρiBkCAk0BkD2000Aw0BwAB1ρiC¯k0B2B1I-ρiB¯kCA¯k0B¯kD2000Aw0BwC-C-CwD2-DwEF1Ck0F20EF1ρiCk0F2F1I-ρiEF1ρiC¯k0F2F1I-ρi.Remark 4.
Since the faults may not be detected instantaneously, but only after a time period, the detector/controller cannot be available in time to be switched to control the system when the faults occur. Hence, there exists mismatching case between the detector/controller and the system; then it is the asynchronous case.


### 2.3. Problem Formulation

The design problem of the detector/controller addressed in this paper can be expressed as follows.The frameworks of the detector/controller: given system (1), we transform system (1) into system (4) which contains the faults. Based on model (4), detector/controller (5) is designed and generates two signals: a residual signal and a control signal. Meanwhile, augmented system (7) is asymptotically stable, and the disturbances and the faults affect the performance output z(k) and the fault estimate errors re(k) are both minimized. When the faults are detected, the controller is switched to reduce the effect of the faults. When the faults are removed, the detector/controller is switched to the original controller, and the residual evaluation function is reset. The proposed fault detection and control scheme is described in Figure 1.Figure 1
Evolution of the procedure.To detect the faults and control system, our design objectives of the detector/controller can now be formulated as the following performance indices:(i)
for the fault detection objective(9)∑k=0∞‍1-ςkrekTrek≤γs12∑k=0∞‍wkTwk,(ii)
for the control objective(10)∑k=0∞‍1-ςkzkTzk≤γs22∑k=0∞‍wkTwk.After designing detector/controller (5), the remaining important task is to evaluate the generated residual. One of the widely adopted residual evaluation functions Jr(k) can be chosen as [25]: (11)Jrk=1k∑l=1k‍rTlrl,where k denotes the evaluation time step. We propose to use the following reasonable threshold for the residual evaluation functions and residual signal:(12)Jthk=supdk∈l20,∞fk=0,Jrk,rthk=supdk∈l20,∞fk=0,⁡rk.Consequently, the occurrence of faults can be detected by the following logical relationship:(13)Jrk>Jthk,rk>rthk,thesystemwithalarm,andswitchingthecontroller,othersthesystemnoalarm.When the fault is removed, the residual evaluation function is reset.


## 2.1. System Model

Consider the following discrete-time linear systems:(1)xk+1=Axk+B1uk+B2dk,yk=Cxk+D2dk,zk=Exk+F1uk+F2dk,where x(k)∈Rn is the state, y(k)∈Rm is the measured output, z(k)∈Rmz is the performance output, u(k)∈Rnu is the control input, and d(k)∈Rnd is the disturbance input which is assumed to belong to l20,∞. The matrices A, B1, B2, C, D2, E, F1, and F2 of each subsystem have appropriate dimensions.


## 2.2. Fault Model

To formulate the fault detection and control problem in this paper, the following fault model type is adopted. The actuator stuck fault defined in [23] is described as follows:(2)ufk=ρiuk+I-ρifk,i=0,…,N,where index i denotes the ith fault mode and N denotes the number of the total fault modes. The diagonal matrices ρi is defined as (3)ρi=diagρi1,…,ρik,…,ρinu,where ρi’s diagonal elements are either 1 or 0. Consider f(k)=f1,…,fk,…,fnu, where fk(k=1,…,nu) is an unknown constant which means the value of the stuck fault for the kth actuator.Remark 1.
Eachρi corresponds to a fault mode. We assume that ρ0=I; then it is fault free case and uf(k)=u(k). Note that when ρik=0 and fik≠0, the kth actuator is stuck. If ρik=0 and fk=0, (2) means that the kth actuator is outages. As [24], the fault f(k) is an intermittent fault, and it can be eliminated in a limited time.Then system (1) with actuator faults (2) can be described as(4)xk+1=Axk+B1uk+B2dk,faultfreecaseAxk+B1ρiuk+B2dk+B1I-ρifk,faultcase,yk=Cxk+D2dk,zk=Exk+F1uk+F2dk,faultfreecaseExk+F1ρiuk+F2dk+F11-ρifk,faultcase.Remark 2.
In this paper, the case that(A,B1ρi) is not stabilizable is also included in this paper; then, all the results which are based on the common assumption are invalid.To detect the fault and control system (4), design the detector/controller as the following form:(5)x^k+1=Akx^k+Bkyk,faultno-detectedcaseA¯kx^k+B¯kyk,faultdetectedcase,uk=Ckx^k,faultno-detectedcaseC¯kx^k,faultdetectedcase,y^k=Cx^k,where x^(k) is the detector/controller state vector and Ak,Bk,Ck, A¯k, B¯k, and C¯k, are real matrices of appropriate dimensions to be determined.Remark 3.
The detector/controller is switched between the fault no-detected case and the fault detected case, which can be seen as event-driven switching control.For fault detection, we can formulate the fault as the weighted faultf^(k)=Wf(z)f(k) with a given stable weighted matrix. Assume Aw, Bw, Cw, and Dw are known constant matrices; then the minimal realization of Wf(z) is(6)xwk+1=Awxwk+Bwfk,f^k=Cwxk+Dwfk.Combining (4), (5), and (6), we have the following augmented model:(7)x~k+1=A~x~k+B~wk,faultfreecaseA~1x~k+B~1wk,faultno-detectedcaseA~2x~k+B~2wk,faultdetectedcase,rek=C~x~k+D~wk,zk=E~x~k+F~wk,faultfreecaseE~1x~k+F~1wk,faultno-detectedcaseE~2x~k+F~2wk,faultdetectedcase,where r(k)=y(k)-y^(k) is the residual signal and re(k)=r(k)-f^(k) is the fault estimate error, x~(k)=x(k)Tx^(k)Txw(k)TT, w(k)=d(k)f(k), (8)A~B~A~1B~1A~2B~2C~D~E~F~E~1F~1E~2F~2=AB1Ck0B20BkCAk0BkD2000Aw0BwAB1ρiCk0B2B1I-ρiBkCAk0BkD2000Aw0BwAB1ρiC¯k0B2B1I-ρiB¯kCA¯k0B¯kD2000Aw0BwC-C-CwD2-DwEF1Ck0F20EF1ρiCk0F2F1I-ρiEF1ρiC¯k0F2F1I-ρi.Remark 4.
Since the faults may not be detected instantaneously, but only after a time period, the detector/controller cannot be available in time to be switched to control the system when the faults occur. Hence, there exists mismatching case between the detector/controller and the system; then it is the asynchronous case.


## 2.3. Problem Formulation

The design problem of the detector/controller addressed in this paper can be expressed as follows.The frameworks of the detector/controller: given system (1), we transform system (1) into system (4) which contains the faults. Based on model (4), detector/controller (5) is designed and generates two signals: a residual signal and a control signal. Meanwhile, augmented system (7) is asymptotically stable, and the disturbances and the faults affect the performance output z(k) and the fault estimate errors re(k) are both minimized. When the faults are detected, the controller is switched to reduce the effect of the faults. When the faults are removed, the detector/controller is switched to the original controller, and the residual evaluation function is reset. The proposed fault detection and control scheme is described in Figure 1.Figure 1
Evolution of the procedure.To detect the faults and control system, our design objectives of the detector/controller can now be formulated as the following performance indices:(i)
for the fault detection objective(9)∑k=0∞‍1-ςkrekTrek≤γs12∑k=0∞‍wkTwk,(ii)
for the control objective(10)∑k=0∞‍1-ςkzkTzk≤γs22∑k=0∞‍wkTwk.After designing detector/controller (5), the remaining important task is to evaluate the generated residual. One of the widely adopted residual evaluation functions Jr(k) can be chosen as [25]: (11)Jrk=1k∑l=1k‍rTlrl,where k denotes the evaluation time step. We propose to use the following reasonable threshold for the residual evaluation functions and residual signal:(12)Jthk=supdk∈l20,∞fk=0,Jrk,rthk=supdk∈l20,∞fk=0,⁡rk.Consequently, the occurrence of faults can be detected by the following logical relationship:(13)Jrk>Jthk,rk>rthk,thesystemwithalarm,andswitchingthecontroller,othersthesystemnoalarm.When the fault is removed, the residual evaluation function is reset.


## 3. The Fault Detection and Control Design

We letki, i∈N+, denote the starting time when the system resumes normal operations for the ith time and denote by the instants ki1 and ki2 the instant faults that happen and for faults that are detected during the interval ki,ki+1, respectively. Moreover, denote Tn(ki,ki+1), Tnd(ki,ki+1), and Td(ki,ki+1) by the total activation time for fault free case, for fault no-detected case, and for fault detected case in the ith occurred fault, respectively. Then, we have the total time for fault free case Tn(k0,k)=∑l=0i-1Tn(kl,kl+1)+Tn(ki,k), the total activation time for fault no-detected case Tnd(k1,k)=∑l=0i-1Tnd(kl,kl+1)+Tnd(ki,k), the total activation time for fault detected case Td(k1,k)=∑l=0i-1Td(kl,kl+1)+Td(ki,k), and the total activation time for fault case Tf(k1,k)=Tnd(k1,k)+Td(k1,k). The time series of system operation is described in Figure 2.Figure 2
The time series of system operation.In this paper, we focus our study of system (4) for simultaneous fault detection and control on the switching method. The following definition is first introduced.Definition 5.
For anyk0<ks<kv, denote Nσ(k)(ks,kv) by the number for the operations returning to normal during (ks,kv). If Nσ(k)(ks,kv)≤N0+(kv-ks)/τa for τa>0 and N0≥0, then τa and N0 are called the average dwell-time for resumed normal operations and the chatter bound, respectively.Remark 6.
The average dwell-time for resumed normal operations is defined as the average time forTn(ki,ki+1)+Tnd(ki,ki+1)+Td(ki,ki+1).Before the results are obtained, the simultaneous fault detection and control problem design can be solved starting from the weightedL2 performance and the conditions for augmented system (7) with the weighted L2 performance are obtained.Lemma 7.
Consider the discrete-time linear system(14)x˘k+1=A˘x˘k+B˘wk,faultfreecaseA˘1x˘k+B˘1wk,faultno-detectedcaseA˘2x˘k+B˘2wk,faultdetectedcase,y˘k=C˘x˘k+D˘wk,faultfreecaseC˘1x˘k+D˘1wk,faultno-detectedcaseC˘2x˘k+D˘2wk,faultdetectedcaseand let α, β1, β2, β, ς, μ, and γ be some constants satisfying 0<α<1, β1>β>β2>0, 0<ς<1, γ>0, and μ≥1. If there exists Lyapunov-like functions candidate (15)Vx˘k:V1x˘k,faultfreecase,V2x˘k,faultno-detectedcaseV3x˘k,faultdetectedcase,satisfying the following inequalities(16a)ΔV1x˘k<-αV1x˘k-Γ,faultfreecase,(16b)ΔV2x˘k≤β1V2x˘k-Γ,faultno-detectedcase(16c)ΔV3x˘k≤β2V3x˘k-Γ,faultdetectedcase(16d)Vgx˘k-μVhx˘k≤0,where Γk=y˘Tky˘k-γ2wTkwk, g,  h∈{1,2,3}, and g≠h, then system (14) is asymptotically stable with the weighted L2-gain γs=μγ for satisfying the following constraints: (17a)τa≥ceil-3lnμln⁡1-ς,(17b)Tndk0,kTdk0,k≤ln⁡1+β-ln⁡1+β2ln⁡1+β1-ln⁡1+β,(17c)Tnk0,kTfk0,k≥ln⁡1+β-ln⁡1-ςln⁡1-ς-ln⁡1-α,where the function ceil(ν) represents rounding real number ν to the nearest integer greater than or equal to ν.Proof.
Denoteα¯1=(1-α), β¯1=(1+β1), β¯2=(1+β2), β¯=(1+β), and ς¯=1-ς. Firstly, the stability of system (14) with d(k)=0 is considered. When the fault does not occur, that is, k∈Tn(ki,ki+1), i∈N+, it holds from (16a) that(18)V1x˘k+1≤α¯V1x˘k.
When the fault occurs, but the fault is not detected, that is,k∈Tnd(ki,ki+1), i∈N+, it holds from (16b) that(19)V2x˘k+1≤β¯1V2x˘k.
When the fault is detected, that is,k∈Td(ki,ki+1), i∈N+, it holds from (16c) that(20)V3x˘k+1≤β¯2V3x˘k.
Thus, assumingk∈Tn(ki,ki+1), i∈N+, and according to (16d), (18), (19), (20), and Definition 5, we have(21)V1x˘k≤α¯Tnki,kV1x˘ki≤μα¯Tnki,kV3x˘ki-≤⋯≤≤μ3Nσkk0,kβ¯1Tndk01,ki-12β¯2Tdk02,ki×α¯Tnk0,kV1x˘k0.According to (17b), (17c) and Definition 5 can be rewritten as(22)V1x˘k≤μ3Nσkk0,kβ¯Tfk01,kiα¯Tnk0,kV1x˘k0≤μ3Nσkk0,kς¯k-k0V1x˘k0≤μ3N0ς¯μ3/τak-k0V1x˘k0.
If (17a) holds, then ς¯μ3/τa<ς¯μ-3lnς¯/3ln(μ)=1. Therefore, we conclude that V1(x˘(k)) converges to zero as k→∞. Then if (17a), (17b), and (17c) are satisfied, system (14) is stable.
Secondly, establish the weightedL2 performance for system (14). Let any nonzero w(k)∈l20,∞, zero initial condition x˘(k0), and Γ(k)=y˘T(k)y˘(k)-γ2wT(k)w(k) and consider the Lyapunov-like functions for the fault free period, no-detected fault period, and detected fault period, respectively; then one can obtain that(23)V1x˘k≤μ3Nσkk0,kβ¯1Tndk01,ki-12β¯2Tdk02,ki×α¯Tnk0,kV1x˘k0-∑j=0i-1∑l=kj1kj2-1μ3Nσkl,k-1α¯Tnkj+1,k-1β¯1Tndl,ki-12-1×β¯2Tdkj2,ki-1Γl-∑j=0i-1∑l=kj2kj+1-1μ3Nσkl,k-2α¯Tnkj+1,k-1β¯1Tndkj+11,ki-12-1×β¯2Tdl,ki-1Γl-∑j=0i∑l=kjkj1-1μ3Nσkl,kα¯Tnl,k-1β¯1Tndkj1,ki-12-1×β¯2Tdkj2,ki-1Γl.Due to the fact that V1(x˘(k))≥0 and the zero initial condition, we have(24)∑j=0i-1∑l=kj1kj2-1μ3Nσkl,k-1α¯Tnkj+1,k-1β¯1Tndl,ki-12-1β¯2Tdkj2,ki-1Γ(l)+∑j=0i-1∑l=kj2kj+1-1μ3Nσkl,k-2α¯Tnkj+1,k-1β¯1Tndkj+11,ki-12-1×β¯2Tdl,ki-1Γl+∑j=0i∑l=kjkj1-1μ3Nσkl,kα¯Tnl,k-1β¯1Tndkj1,ki-12-1×β¯2Tdkj2,ki-1Γl≤0.According to (17b) and (17c) and from β¯1>0, β¯2>0, β¯>0, and α¯>0, we have(25)β¯1Tndl,ki-12-1β¯2Tdkj2,ki-1≤β¯Tfl,ki-1α¯Tnkj+1,k-1β¯Tfl,ki-1≤ς¯k-l-1,l∈Tndkj1,kj2-1,β¯1Tndkj+11,ki-12-1β¯2Tdl,ki-1≤β¯Tfl,ki-1,α¯Tnkj+1,k-1β¯Tfl,ki-1≤ς¯k-l-1,l∈Tdkj2,kj+1-1,β¯1Tndkj1,ki-12-1β¯2Tdkj2,ki-1≤β¯Tfkj1,ki-1α¯Tnl,k-1β¯Tfkj1,ki-1≤ς¯k-l-1l∈Tnkj,kj1-1.Then, we have(26)∑j=0i-1∑l=kj1kj2-1μ3Nσkl,k-1ς¯k-l-1Γl+∑j=0i-1∑l=kj2kj+1-1μ3Nσkl,k-2ς¯k-l-1Γl+∑j=0i∑l=kjkj1-1μ3Nσkl,kς¯k-l-1Γl≤0.Moreover, it has (27)∑j=0i-1∑l=kj1kj2-2μ3Nσkl,k-1ς¯k-l-1y˘Tky˘k+∑j=0i-1∑l=kj2kj+1-1μ3Nσkl,k-2ς¯k-l-1y˘Tky˘k+∑j=0i∑l=kjkj1-1μ3Nσkl,kς¯k-l-1y˘Tky˘k≤∑j=0i-1∑l=kj1kj2-1μ3Nσkl,k-1ς¯k-l-1γ2wTkwk+∑j=0i-1∑l=kj2kj+1-1μ3Nσkl,k-2ς¯k-l-1γ2wTkwk+∑j=0i∑l=kjkj1-1μ3Nσkl,kς¯k-l-1γ2wTkwk.Thus, from μ>1, we obtain that(28)∑l=k0k-1μ3Nσkl,k-2ς¯k-l-1y˘Tky˘k≤∑l=k0k-1μ3Nσkl,kς¯k-l-1γ2wTkwk.Multiplying both sides of (28) by μ-3Nσ(k)(k0,k), it can be obtained that (29)∑l=k0k-1μ-3Nσkk0,l-2ς¯k-l-1y˘Tky˘k≤∑l=k0k-1μ-3Nσkk0,lς¯k-l-1γ2wTkwk.From the above and μ≥1, we have(30)∑l=k0k-1μ-3Nσkk0,l-2ς¯k-l-1y˘Tky˘k≤∑l=k0k-1μ-3Nσkk0,lς¯k-l-1γ2wTkwk.Moreover, it follows from Definition 5 that 0≤Nσk(k0,j)≤N0+l-k0/τa ≤ +N0-(l-k0)lnς¯/3lnμ. Note that μ3(l-k0)lnς¯/3lnμ=ς¯l-k0 and 0<ς¯<1; then (30) implies that (31)∑l=k0k-1‍ς¯l-k0y˘Tky˘k≤∑l=k0k-1‍μ2γ2wTkwk.
It further implies that(32)∑l=k0∞ς¯l-k0y˘Tky˘k≤∑l=k0∞μγ2dTsds.
Therefore, we conclude that augmented system (14) is asymptotically stable with the weighted L2-gain γs=μγ for satisfying the constraints (17a), (17b), and (17c).


### 3.1. Condition for the Fault Detection Objective

Theorem 8.
Consider system (7), and let α, β1, β2, μ, and γ be some constants satisfying 0<α<1, β1>β2>0, γ1>0, and μ≥1. If there exist matrix variables Ya, Ga, Za, N, Ya1, Ga1, Za1, N1, Ya2, Ga2, Za2A^B^, C^, A-^B-^, and C-^, and symmetric positive-definite matrices(33)Pa1=Pa11Pa12Pa13∗Pa14Pa15∗∗Pa16>0,Pa2=Pa21Pa22Pa23∗Pa24Pa25∗∗Pa26>0,Pa3=Pa31Pa32Pa33∗Pa34Pa35∗∗Pa36>0such that(34)Θa1Θa2Θa30∗Θa40Θa5∗∗-γ12IΘa6∗∗∗-I<0,(35)Ωa1Ωa2Ωa30∗Ωa40Ωa5∗∗-γ12IΩa6∗∗∗-I<0,(36)Ξa1Ξa2Ξa30∗Ξa40Ξa5∗∗-γ12IΞa6∗∗∗-I<0,(37)Pag-μPah≤0,where g,h∈{1,2,3}, g≠h, α¯=1-α, β¯1=1+β1, and β¯2=1+β2,(38)Θa1=Pa11-HeYaPa12+NT-GaPa13∗Pa14-HeNPa15∗∗Pa16-HeZa,Θa5=CT-CT-CwT,Θa2=YaTA-B^CYaTB1C^-A^0GaTA+B^CGaTB1C^+A^000ZaTAw,Θa6=D2T-DwT,Θa3=YaTB2-B^D20GaTB2+B^D200ZaTBw,Θa4=-α¯Pa11-α¯Pa12-α¯Pa13∗-α¯Pa14-α¯Pa15∗∗-α¯Pa16,Ωa1=Pa21-HeYa1Pa22+NT-Ga1Pa23∗Pa24-HeNPa25∗∗Pa26-HeZa1,Ωa5=CT-CT-CwT,Ωa2=Ya1TA-B^CYa1TB1ρiC^-A^0Ga1TA+B^CGa1TB1ρiC^-A^000Za1TAw,Ωa6=D2T-DwT,Ωa3=Ya1TB2-B^D2Ya1TB1I-ρiGa1TB2+B^D2Ga1TB1I-ρi0Za1Bw,Ωa4=-β¯1Pa21-β¯1Pa22-β¯1Pa23∗-β¯1Pa24-β¯1Pa25∗∗-β¯1Pa26,Ξa1=Pa31-HeYa2Pa32+N1T-Ga2Pa33∗Pa34-HeN1Pa35∗∗Pa36-HeZa2,Ξa5=CT-CT-CwT,Ξa2=Ya2TA-B-^CYa2TB1ρiC-^-A-^0Ga2TA+B-^CGa2TB1ρiC-^+A-^000Za2TAw,Ξa6=D2T-DwT,Ξa3=Ya2TB2-B-^D2Ya2TB1I-ρiGa2TB2+B-^kD2Ga2TB1I-ρi0Za2Bw,Ξa4=-β¯2Pa31-β¯2Pa32-β¯2Pa33∗-β¯2Pa34-β¯2Pa35∗∗-β¯2Pa26,then system (7) is asymptotically stable and guarantees the weighted L2 performance (9) with the weighted L2 gain γs1=μγ1 when system (7) satisfies the constraints (17a), (17b), and (17c).Proof.
Whenk∈Tn(ki,ki+1), according to Lemma 7 and along the trajectory of system (7), one has(39)ΔV1x~k+αV1x~k-γ2wkTwk+y˘kTy˘k=x~kwkT×A~TPa1A~-1-αPa1+C~TC~A~TPa1B~+C~TD~B~TPa1A~+D~TC~B~TPa1B~-γ12I+D~TD~×x~kwk.
Whenk∈Tnd(ki,ki1), we obtain(40)ΔV2x~k-β2V2x~k-γ2wkTwk+y˘kTy˘k=x~kwkT×A~1TPa2A~1-1+β1Pa2+C~TC~A~1TPa2B~1+C~TD~B~1TPa2A~1+D~TC~B~1TPa2B~1-γ12I+D~TD~×x~kwk.
Whenk∈Td(ki1,ki2), it can be obtained that(41)ΔV3x~k-β2V3x~k-γ2wkTwk+y˘kTy˘k=x~kwkT×A~2TPa3A~2-1+β2Pa3+C~TC~A~2TPa3B~2+C~TD~B~2TPa3A~2+D~TC~B~2TPa3B~2-γ12I+D~TD~×x~kwk.
In addition, due to the relation of the Lyapunov-like functions at the moment the fault happened, we have thatVg(x~(k))-μVh(x~(k))=x~(k)T(Pg-μPh)x~(k), where g,h∈{1,2,3}, g≠h. Thus, if the inequalities(42)A~TPa1A~-1-αPa1+C~TC~A~TPa1B~+C~TD~B~TPa1A~+D~TC~B~TPa1B~-γ12I+D~TD~<0,(43)A~1TPa2A~1-1+β1Pa2+C~TC~A~1TPa2B~1+C~TD~B~1TPa2A~1+D~TC~B~1TPa2B~1-γ12I+D~TD~<0,(44)A~2TPa3A~2-1+β2Pa3+C~TC~A~2TPa3B~2+C~TD~B~2TPa3A~2+D~TC~B~2TPa3B~2-γ12I+D~TD~<0,(45)Pg-μPh≤0,
hold, system (14) is asymptotically stable with the weighted L2 performance for satisfying constraints (17a), (17b), and (17c) according to Lemma 7.
To reduce the conservatism, (42) can be rewritten as follows:(46)A~B~I0TPa100-1-αPa1A~B~I0+C~TC~C~TD~D~TC~-γ12I+D~TD~<0.Then, we can obtain that (46) is equivalent to(47)A~B~I00ITPa1000-1-αPa1+C~TC~C~TD~0D~TC~-γ12I+D~TD~×A~B~I00I<0.On the other hand,(48)00I00ITPa1000-1-αPa1+C~TC~C~TD~0D~TC~-γ12I+D~TD~×00I00I=-1-αPa1+C~TC~C~TD~D~TC~-γ12I+D~TD~<0.Applying Projection Lemma, it follows from (47) and (48) that(49)Pa1000-1-αPa1+C~TC~C~TD~0D~TC~-γ12I+D~TD~+He-IA~TB~TWaI00<0.
Consider the structure of the matrixA~; we construct the matrix variable Wa as (50)Wa=YaGa0-NaNa000Za.Define A^=NTAk, B^=NTBk, and C^=Ck. By Schur complement, (49) becomes (34). In a similar way, we can obtain the conditions (35) and (36). Hence, if the conditions (34), (35), (36), and (37) hold, system (7) is asymptotically stable and has the weighted L2 performance gain γs1, which completes the proof.


### 3.2. Condition for the Control Objective

Theorem 9.
Consider system (7), and let α, β1, β2, μ, and γ be some constants satisfying 0<α<1, β1>β2>0, γ2>0, and μ≥1. If there exist matrix variables Yb, Gb, Zb, N, Yb1, Gb1, Zb1, Yb2, Gb2, Zb2, N1, A^B^, C^, A-^B-^, and C-^ and symmetric positive-definite matrices(51)Pb1=Pb11Pb12Pb13∗Pb14Pb15∗∗Pb16>0,Pb2=Pb21Pb22Pb23∗Pb24Pb25∗∗Pb26>0,Pb3=Pb31Pb32Pb33∗Pb34Pb35∗∗Pb36>0,such that(52)Θb1Θb2Θb30∗Θb40Θb5∗∗-γ22Θb6∗∗∗-I<0,(53)Ωb1Ωb2Ωb30∗Ωb40Ωb5∗∗-γ22IΩb6∗∗∗-I<0,(54)Ξb1Ξb2Ξb30∗Ξb40Ξb5∗∗-γ22IΞb6∗∗∗-I<0,(55)Pbg-μPbh≤0,where g,h∈{1,2,3}, g≠h, α¯=1-α, β¯1=1+β1, β¯2=1+β2,(56)Θb1=Pb11-HeYbPb12+NT-GbPb13∗Pb14-HeNPb15∗∗Pb16-HeZb,Θb5=ET-C^TF1T0,Θb2=YbTA-B^CYbTB1C^-A^0GbTA+B^CGbTB1C^+A^000ZbTAw,Θb6=F2T0,Θb3=YbTB2-pb1B^D20GbTB2+B^D200ZbTBw,Θb4=-α¯Pb11-α¯Pb12-α¯Pb13∗-α¯Pb14-α¯Pb15∗∗-α¯Pb16,Ωb1=Pb21-HeYb1Pb22+NT-Gb1Pb23∗Pb24-HeNPb25∗∗Pb26-HeZb,Ωb5=ET-C^TρiTF1T0,Ωb2=Yb1TA-B^CYb1TB1ρ1C^-A^0Gb1TA+B^CGb1TB1ρ1C^+A^000Zb1TAw,Ωb6=F2TI-ρiTF1T,Ωb3=Yb1TB2-B^D2Yb1TB1I-ρiGb1TB2+B^D2Gb1TB1I-ρi0Zb1Bw,Ωb4=-β¯1Pb21-β¯1Pb22-β¯1Pb23∗-β¯1Pb24-β¯1Pb25∗∗-β¯1Pb26,Ξb1=Pb31-HeYb1Pb32+N1T-Gb1Pb33∗Pb34-HeN1Pb35∗∗Pb36-HeZb1,Ξb5=ET-C-^TρiF1T0,Ξb2=Yb2TA-B-^CYb2TB1ρiC-^-A-^0Gb2TA+B-^CGb2TB1ρiC-^+A-^000Zb2TAw,Ξb6=F2TI-ρiTF1T,Ξb3=Yb2TB2-B-^D2Yb2TB1ρiGb2TB2+B-^D2Gb2TB1ρi0Zb2Bw,Ξb4=-β¯2Pb31-β¯2Pb32-β¯2Pb33∗-β¯2Pb34-β¯2Pb35∗∗-β¯2Pb36,then system (7) is asymptotically stable and guarantees the weighted L2 performance with the weighted L2 gain γs2=μγ2 when system (7) satisfies the constraints (17a), (17b), and (17c).Proof.
Similar to the process in Theorem8, the conditions (52), (53), (54), and (55) can be obtained for system (7) with the weighted L2 performance (10).


### 3.3. Algorithm

The performances (9)-(10) have been formulated as the inequality conditions in Theorems 8 and 9, respectively. But the conditions (34), (35), (52), and (53) are all nonconvex owing to the product terms YqTB1C^, GqTB1C^, Yq1TB1ρiC^, Gq1TB1ρiC^, Yq2TB1ρiC-^, and Gq2TB1ρiC-^, q∈{a,b}. Then a two-step procedure is presented, which follows from the idea proposed as [26]; it can be summarized as follows.Step 1.
Design robust state feedback gain matricesCk, C¯k for the closed-loop system with the state feedback control u(k)=Ckx(k) for fault free case and u(k)=C¯kx(k) for fault detected case; then the performance index (10) is satisfied. This problem is convex and see the appendix for the design of Ck, C¯k.Step 2.
Put the state feedback gain matricesCk, C¯k into switched system (7) and search for a solution that satisfies the conditions in Theorems 8 and 9. Given γ2, solving the following problem(57)minγ1s.t.34,35,36,37,iiiiiiiii52,53,54,55,denote A^B^, A-^B-^, and N, N1 by the optimal solution of (57); the gain matrices Ak, Bk, A¯k, B¯k can be obtained as (58)Ak=N-1A^,Bk=N-1B^,A¯k=N1-1A-^,B¯k=N1-1B-^.


## 3.1. Condition for the Fault Detection Objective

Theorem 8.
Consider system (7), and let α, β1, β2, μ, and γ be some constants satisfying 0<α<1, β1>β2>0, γ1>0, and μ≥1. If there exist matrix variables Ya, Ga, Za, N, Ya1, Ga1, Za1, N1, Ya2, Ga2, Za2A^B^, C^, A-^B-^, and C-^, and symmetric positive-definite matrices(33)Pa1=Pa11Pa12Pa13∗Pa14Pa15∗∗Pa16>0,Pa2=Pa21Pa22Pa23∗Pa24Pa25∗∗Pa26>0,Pa3=Pa31Pa32Pa33∗Pa34Pa35∗∗Pa36>0such that(34)Θa1Θa2Θa30∗Θa40Θa5∗∗-γ12IΘa6∗∗∗-I<0,(35)Ωa1Ωa2Ωa30∗Ωa40Ωa5∗∗-γ12IΩa6∗∗∗-I<0,(36)Ξa1Ξa2Ξa30∗Ξa40Ξa5∗∗-γ12IΞa6∗∗∗-I<0,(37)Pag-μPah≤0,where g,h∈{1,2,3}, g≠h, α¯=1-α, β¯1=1+β1, and β¯2=1+β2,(38)Θa1=Pa11-HeYaPa12+NT-GaPa13∗Pa14-HeNPa15∗∗Pa16-HeZa,Θa5=CT-CT-CwT,Θa2=YaTA-B^CYaTB1C^-A^0GaTA+B^CGaTB1C^+A^000ZaTAw,Θa6=D2T-DwT,Θa3=YaTB2-B^D20GaTB2+B^D200ZaTBw,Θa4=-α¯Pa11-α¯Pa12-α¯Pa13∗-α¯Pa14-α¯Pa15∗∗-α¯Pa16,Ωa1=Pa21-HeYa1Pa22+NT-Ga1Pa23∗Pa24-HeNPa25∗∗Pa26-HeZa1,Ωa5=CT-CT-CwT,Ωa2=Ya1TA-B^CYa1TB1ρiC^-A^0Ga1TA+B^CGa1TB1ρiC^-A^000Za1TAw,Ωa6=D2T-DwT,Ωa3=Ya1TB2-B^D2Ya1TB1I-ρiGa1TB2+B^D2Ga1TB1I-ρi0Za1Bw,Ωa4=-β¯1Pa21-β¯1Pa22-β¯1Pa23∗-β¯1Pa24-β¯1Pa25∗∗-β¯1Pa26,Ξa1=Pa31-HeYa2Pa32+N1T-Ga2Pa33∗Pa34-HeN1Pa35∗∗Pa36-HeZa2,Ξa5=CT-CT-CwT,Ξa2=Ya2TA-B-^CYa2TB1ρiC-^-A-^0Ga2TA+B-^CGa2TB1ρiC-^+A-^000Za2TAw,Ξa6=D2T-DwT,Ξa3=Ya2TB2-B-^D2Ya2TB1I-ρiGa2TB2+B-^kD2Ga2TB1I-ρi0Za2Bw,Ξa4=-β¯2Pa31-β¯2Pa32-β¯2Pa33∗-β¯2Pa34-β¯2Pa35∗∗-β¯2Pa26,then system (7) is asymptotically stable and guarantees the weighted L2 performance (9) with the weighted L2 gain γs1=μγ1 when system (7) satisfies the constraints (17a), (17b), and (17c).Proof.
Whenk∈Tn(ki,ki+1), according to Lemma 7 and along the trajectory of system (7), one has(39)ΔV1x~k+αV1x~k-γ2wkTwk+y˘kTy˘k=x~kwkT×A~TPa1A~-1-αPa1+C~TC~A~TPa1B~+C~TD~B~TPa1A~+D~TC~B~TPa1B~-γ12I+D~TD~×x~kwk.
Whenk∈Tnd(ki,ki1), we obtain(40)ΔV2x~k-β2V2x~k-γ2wkTwk+y˘kTy˘k=x~kwkT×A~1TPa2A~1-1+β1Pa2+C~TC~A~1TPa2B~1+C~TD~B~1TPa2A~1+D~TC~B~1TPa2B~1-γ12I+D~TD~×x~kwk.
Whenk∈Td(ki1,ki2), it can be obtained that(41)ΔV3x~k-β2V3x~k-γ2wkTwk+y˘kTy˘k=x~kwkT×A~2TPa3A~2-1+β2Pa3+C~TC~A~2TPa3B~2+C~TD~B~2TPa3A~2+D~TC~B~2TPa3B~2-γ12I+D~TD~×x~kwk.
In addition, due to the relation of the Lyapunov-like functions at the moment the fault happened, we have thatVg(x~(k))-μVh(x~(k))=x~(k)T(Pg-μPh)x~(k), where g,h∈{1,2,3}, g≠h. Thus, if the inequalities(42)A~TPa1A~-1-αPa1+C~TC~A~TPa1B~+C~TD~B~TPa1A~+D~TC~B~TPa1B~-γ12I+D~TD~<0,(43)A~1TPa2A~1-1+β1Pa2+C~TC~A~1TPa2B~1+C~TD~B~1TPa2A~1+D~TC~B~1TPa2B~1-γ12I+D~TD~<0,(44)A~2TPa3A~2-1+β2Pa3+C~TC~A~2TPa3B~2+C~TD~B~2TPa3A~2+D~TC~B~2TPa3B~2-γ12I+D~TD~<0,(45)Pg-μPh≤0,
hold, system (14) is asymptotically stable with the weighted L2 performance for satisfying constraints (17a), (17b), and (17c) according to Lemma 7.
To reduce the conservatism, (42) can be rewritten as follows:(46)A~B~I0TPa100-1-αPa1A~B~I0+C~TC~C~TD~D~TC~-γ12I+D~TD~<0.Then, we can obtain that (46) is equivalent to(47)A~B~I00ITPa1000-1-αPa1+C~TC~C~TD~0D~TC~-γ12I+D~TD~×A~B~I00I<0.On the other hand,(48)00I00ITPa1000-1-αPa1+C~TC~C~TD~0D~TC~-γ12I+D~TD~×00I00I=-1-αPa1+C~TC~C~TD~D~TC~-γ12I+D~TD~<0.Applying Projection Lemma, it follows from (47) and (48) that(49)Pa1000-1-αPa1+C~TC~C~TD~0D~TC~-γ12I+D~TD~+He-IA~TB~TWaI00<0.
Consider the structure of the matrixA~; we construct the matrix variable Wa as (50)Wa=YaGa0-NaNa000Za.Define A^=NTAk, B^=NTBk, and C^=Ck. By Schur complement, (49) becomes (34). In a similar way, we can obtain the conditions (35) and (36). Hence, if the conditions (34), (35), (36), and (37) hold, system (7) is asymptotically stable and has the weighted L2 performance gain γs1, which completes the proof.


## 3.2. Condition for the Control Objective

Theorem 9.
Consider system (7), and let α, β1, β2, μ, and γ be some constants satisfying 0<α<1, β1>β2>0, γ2>0, and μ≥1. If there exist matrix variables Yb, Gb, Zb, N, Yb1, Gb1, Zb1, Yb2, Gb2, Zb2, N1, A^B^, C^, A-^B-^, and C-^ and symmetric positive-definite matrices(51)Pb1=Pb11Pb12Pb13∗Pb14Pb15∗∗Pb16>0,Pb2=Pb21Pb22Pb23∗Pb24Pb25∗∗Pb26>0,Pb3=Pb31Pb32Pb33∗Pb34Pb35∗∗Pb36>0,such that(52)Θb1Θb2Θb30∗Θb40Θb5∗∗-γ22Θb6∗∗∗-I<0,(53)Ωb1Ωb2Ωb30∗Ωb40Ωb5∗∗-γ22IΩb6∗∗∗-I<0,(54)Ξb1Ξb2Ξb30∗Ξb40Ξb5∗∗-γ22IΞb6∗∗∗-I<0,(55)Pbg-μPbh≤0,where g,h∈{1,2,3}, g≠h, α¯=1-α, β¯1=1+β1, β¯2=1+β2,(56)Θb1=Pb11-HeYbPb12+NT-GbPb13∗Pb14-HeNPb15∗∗Pb16-HeZb,Θb5=ET-C^TF1T0,Θb2=YbTA-B^CYbTB1C^-A^0GbTA+B^CGbTB1C^+A^000ZbTAw,Θb6=F2T0,Θb3=YbTB2-pb1B^D20GbTB2+B^D200ZbTBw,Θb4=-α¯Pb11-α¯Pb12-α¯Pb13∗-α¯Pb14-α¯Pb15∗∗-α¯Pb16,Ωb1=Pb21-HeYb1Pb22+NT-Gb1Pb23∗Pb24-HeNPb25∗∗Pb26-HeZb,Ωb5=ET-C^TρiTF1T0,Ωb2=Yb1TA-B^CYb1TB1ρ1C^-A^0Gb1TA+B^CGb1TB1ρ1C^+A^000Zb1TAw,Ωb6=F2TI-ρiTF1T,Ωb3=Yb1TB2-B^D2Yb1TB1I-ρiGb1TB2+B^D2Gb1TB1I-ρi0Zb1Bw,Ωb4=-β¯1Pb21-β¯1Pb22-β¯1Pb23∗-β¯1Pb24-β¯1Pb25∗∗-β¯1Pb26,Ξb1=Pb31-HeYb1Pb32+N1T-Gb1Pb33∗Pb34-HeN1Pb35∗∗Pb36-HeZb1,Ξb5=ET-C-^TρiF1T0,Ξb2=Yb2TA-B-^CYb2TB1ρiC-^-A-^0Gb2TA+B-^CGb2TB1ρiC-^+A-^000Zb2TAw,Ξb6=F2TI-ρiTF1T,Ξb3=Yb2TB2-B-^D2Yb2TB1ρiGb2TB2+B-^D2Gb2TB1ρi0Zb2Bw,Ξb4=-β¯2Pb31-β¯2Pb32-β¯2Pb33∗-β¯2Pb34-β¯2Pb35∗∗-β¯2Pb36,then system (7) is asymptotically stable and guarantees the weighted L2 performance with the weighted L2 gain γs2=μγ2 when system (7) satisfies the constraints (17a), (17b), and (17c).Proof.
Similar to the process in Theorem8, the conditions (52), (53), (54), and (55) can be obtained for system (7) with the weighted L2 performance (10).


## 3.3. Algorithm

The performances (9)-(10) have been formulated as the inequality conditions in Theorems 8 and 9, respectively. But the conditions (34), (35), (52), and (53) are all nonconvex owing to the product terms YqTB1C^, GqTB1C^, Yq1TB1ρiC^, Gq1TB1ρiC^, Yq2TB1ρiC-^, and Gq2TB1ρiC-^, q∈{a,b}. Then a two-step procedure is presented, which follows from the idea proposed as [26]; it can be summarized as follows.Step 1.
Design robust state feedback gain matricesCk, C¯k for the closed-loop system with the state feedback control u(k)=Ckx(k) for fault free case and u(k)=C¯kx(k) for fault detected case; then the performance index (10) is satisfied. This problem is convex and see the appendix for the design of Ck, C¯k.Step 2.
Put the state feedback gain matricesCk, C¯k into switched system (7) and search for a solution that satisfies the conditions in Theorems 8 and 9. Given γ2, solving the following problem(57)minγ1s.t.34,35,36,37,iiiiiiiii52,53,54,55,denote A^B^, A-^B-^, and N, N1 by the optimal solution of (57); the gain matrices Ak, Bk, A¯k, B¯k can be obtained as (58)Ak=N-1A^,Bk=N-1B^,A¯k=N1-1A-^,B¯k=N1-1B-^.


## 4. Example

Consider a discrete-time system (1) consisting of two subsystems(59)xk+1=Axk+B1uk+B2dk,yk=Cxk+D2dk,zk=Exk+F1uk+F2dk,where (60)A1B1B2CD2EF1F2=0.50-0.26-0.090.20.50.10-0.590.64-0.24-1.30.5-0.10.54-0.250.110.5-0.50.40.36-0.010.320.1-0.010.10.30.10.20.2.Here, we consider the following possible fault modes, and, choosingα=0.2, β1=60, β2=5, and μ=1.2, we can obtain the gain matrices of the detector/controller as algorithm in Section 3:(61)Ak1=-0.5060-0.3372-1.00511.51120.14012.7485-0.24250.1207-0.5719,Bk1=2.3951-5.92802.1928,A¯k1=-0.0523-0.1325-0.30560.40730.20410.3789-0.3925-0.0723-0.6093,B¯k1=1.5261-3.54482.7507,Ck1=-0.2413-0.0327-0.77150.1707-0.4837-1.1142,C¯k1=0.0142-0.1561-0.58870.0429-0.4220-1.2056.Choosing β=15 and ς=0.05, it is obtained that(62)τa≥τa∗=ceil-3lnμln1-ς=11,(63)Tndk0,kTdk0,k≤ln⁡1+β-ln⁡1+β2ln⁡1+β1-ln⁡1+β=0.7329,(64)Tnk0,kTfk0,k≥ln1+β-ln1-ςln1-ς-ln1-α=16.4322.Thus, if the normal operations are larger than the regular intervals11 steps, meanwhile, it satisfies Tnd/Td≤0.7329 and Tn/Tf>16.4322, then the designed detector/controller makes the closed-loop system asymptotically stable and guarantees the weighted L2 performance.Here, fork=0,…,500, the disturbances for each subsystem are d(k)=0.5sin(0.1k)e-0.001k. We consider the two kinds of fault mode to demonstrate the effectiveness of the method proposed by this paper.Case 1.
The first actuator is stuck and the second actuator is normal; that is,ρ1=0001, and the 1st actuator has the fault from step 75 to step 100 with the unit amplitude. Suppose the total runtime is 500 steps; the condition (64) has been satisfied; moreover, the closed-loop matrix A~1 is not Hurwitz. We consider the method proposed in this paper which considers the switching strategy. The actuator stuck fault can be detected at step 76; then the controller is switched at step 76. The condition (63) holds; then the closed-loop system with the switching strategy is asymptotically stable and guarantees the weighted L2 performance. The generated residual r(k) and the evolution of the residual evaluation function Jr(k) are shown in Figure 3.Figure 3
The detection objectives for Case1.
(a)
The residual signal
(b)
The evolution of the residual evaluation functionJrFor the detection objectives, it can be seen from Figure3 that the residual signal r(k) is larger than the threshold rth(k) of the residual in step 76 and the residual evaluation function Jr(k) is larger than the threshold Jth(k) in step 76. It can be deduced that the detector/controller with the parameters A¯k1, B¯k1, and C¯k1 is switched at step 76 to guarantee that the norm of the states increases within acceptable limits. Until the fault is removed in step 100, the residual evaluation function is reset to complete fault detection objective.For the control objective, we will first demonstrate that the performance outputz(k) is shown in Figure 4(a) if the detector/controller is not switched when the fault is detected, while the performance output z(k) is shown in Figure 4(b) if the detector/controller is switched when the fault is detected. Comparing these two figures, it can been seen that the effects of disturbances d(k) and the fault f(k) on the performance output z(k) in our method are not disastrous and the performance output z(k) slowly varies.Figure 4
Performance outputz(k) when the fault is detected for Case 1.
(a)
z
(
k
) when the detector/controller is not switched
(b)
z
(
k
) when the detector/controller is switchedCase 2.
Both of the two actuators are stuck, that is,ρ2=0000, and the fault from step 250 to step 300. As Case 1, we can also obtain that the conditions (63) and (64) hold, and the closed-loop matrix A~1 is not Hurwitz.The performance outputz(k) for the detector/controller both no-switched and switched is shown in Figure 5. From Figure 5, we can see that the performance output z(k) for the no-switched case varies more sharply from -12 to 4, while the performance output z(k) for the switched case only changes form from -1.1 to 0.4. The advantage for our method is obvious for the control objective.Figure 5
Performance outputz(k) when the fault is detected for Case 2.
(a)
z
(
k
) when the detector/controller is not switched
(b)
z
(
k
) when the detector/controller is switchedFrom Cases1 and 2, we see that when the conditions (62), (63), and (64) are satisfied, no matter whether the faults are in the first or in both actuators, all of them can be detected, and the closed-loop system is asymptotically stable and guarantees the given weighted L2 performance. The fault detection and control for the system with the switching strategy are effective.


## 5. Conclusion

In this paper, the simultaneous fault detection and control problem has been presented for discrete-time systems. The detector/controller is designed as a single unit such that when the fault is detected, the controller is switched to stabilize “destabilizing failure.” The switching strategy which combines event-driven switching with time-driven switching not only makes the closed-loop system asymptotically stable, but also guarantees the weightedL2 performance. An example is provided to demonstrate the effectiveness of the proposed design method.



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*Source: 841031-2015-07-22.xml*