# MixedH2/H∞ Fault Detection Filter for Markovian Jump Linear Systems

Authors: Leonardo de Paula Carvalho; André Marcorin de Oliveira; Oswaldo Luiz do Valle Costa
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

We tackle the mixedH2/H∞ fault detection filter problem for a Markov jump linear system (MJLS) in the discrete-time domain. We present three distinct formulations: the first one is to minimize an upper bound on the H2 subject to a given upper value on the H∞ norm; the second one is the opposite situation; that is, we minimize an upper bound on H∞, subject to a given restriction on the H2 norm; and the third one is the minimization of a weighted combination of the upper bound of both the H2 and H∞ norms. We present new conditions in the form of linear matrices inequalities (LMI) that provide the design of the fault detection filter. We also present results for the so-called mode-independent case and the design of robust H∞ filters in the sense that the system matrices are uncertain. In order to illustrate the feasibility of the proposed approaches, a numerical example is presented.


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## Body

## 1. Introduction

Dynamic systems that are subject to sudden changes in their dynamics are present in a multitude of situations, ranging from highly complex industrial processes to day by day routine work, and for that reason this class of systems has been extensively studied in the literature. A particular type of system subject to sudden dynamic behavior is the one in which the changes are caused by the occurrence of faults, so that it is highly desirable to provide solutions able to keep the system operational even in the presence of such issues. A possible approach for this problem is the Fault Detection and Isolation (FDI) algorithms, which have the purpose of detecting unusual behaviors in a wide range of fields in engineering including chemical, nuclear, aerospace, and automotive applications; for instance, see the works in [1–4], respectively. Therefore, whenever a fault happens, the main task is to detect the failure and reorganize the control system with the intent of minimizing the operational loss and the possibility of the occurrence of accidents [5].One possible way to implement an FDI is to generate a residual signal using a filter device and predetermine a threshold, and whenever the residue surpasses this established limit, consider that a fault occurred [6, 7]. Bearing in mind this general work scheme of an FDI, it is possible to point out three desirable aspects in the residue generator filter. The first one is the sensibility to abrupt changes; that is, the higher the filter sensibility is, the faster the fault detection will take place. The second one is high robustness against noise in order to prevent the occurrence of false alarms. The last aspect is the consideration of communication failure between components that compose the FDI solution, since the occurrence of packet loss may degrade the performance of the fault detection.The motivation behind the Markov jump linear systems (MJLS) framework usage is the communication between the sensor and the FDI device made via imperfect channels. The communication made through a nonideal network is susceptible to the packet loss, which may be caused, for instance, by collision [8] and channel fading [9]. In the design of a fault detection (FD) filter, it is essential to consider the problems inherent to a communication made through nonideal network. A viable way to model this network characteristic is to use MJLS due to the possibility of modeling the dynamical behavior of systems whose signals are degraded by the possible packet losses in a networked control system; see, for instance, the work in [10].In relation to the MJLS framework applied to FDI theory, we can mention the work in [11] which tackled the problem of designing H∞ residual filters for discrete-time MJLS considering that the Markov chain can be measured. More recently, the works in [12, 13] considered also the synthesis of H∞ residual filters for continuous-time MJLS but with the assumption that the modes of operation of the filter are unmatched in relation to the system being observed. Regarding the aforementioned works, we consider that the fault detection problem is still not completely investigated, since both the works in [11, 12] study a suboptimized H∞ residual filters, and for that reason some alternative design conditions to the ones given in [11] for H∞ residual filters would be desirable in order to cope with potential conservative results.Considering the previous discussions, the main novelty of this paper is the design of mixedH2/H∞ filters for FDI devices within the MJLS framework. This is highly motivated, since the filter structure would combine the desirable aspects of signal robustness that is linked to the H∞ framework, as well as the optimal aspects of the H2 filtering. We provide design conditions based on the linear matrix inequality (LMI) formulation in order to obtain residual filters that impose bounds on both the H2 and H∞ norms of the residual signal. Furthermore, we also investigate the mode-independent residual filter formulation in which the filter matrices do not switch according to the Markov chain and also the design of robust filters with respect to uncertainty on the plant modeling. A numerical example is presented at the end of this paper in order to illustrate our results and compare the proposed approaches.This work is organized as follows. Section2 presents the notation used during the entire work, Section 3 shows the basic theoretical concepts in order to understand the following sections, Section 4 presents the fault detection problem, and Section 5 introduces the main results of this work. Section 6 introduces some secondary results and in Section 7 we present the numerical example that portrays the feasibility of the presented results. Section 8 concludes the paper with some final comments.


## 2. Notation

The notation is standard. The operator(′) denotes the matrix or vector transpose; (•) indicates each symmetric block of a symmetric matrix. We consider the convex set (1)Υ=Q;Q=∑l=1VμlQl,μl≥0,∑l=1Vμl=1where V is the number of vertices in the polytope. The set of Markov chain states is represented by K={1,2,…,N}. The convex combinations of the matrix Xj and the weight ρij are given by εi(X)=∑j=1N‍ρijXj for i∈K. The symbol ε(·) represents mathematical expectations. Considering the stochastic signal z(k), its norm is defined by z22=∑k=0∞‍ε{zk′z(k)}. On a probabilistic space (Ω,F,Fk,P), the set of signals z(k)∈Rp, such that z(k) is Fk measurable, for all k∈N and z2<∞, is indicated by L2.


## 3. Theoretical Background

We define in this section the concepts of mean square stability,H∞ norm and H2 norm.


### 3.1. Markovian Jump Linear System

We initially consider the following general discrete-time Markovian jump linear system (MJLS):(2)G:xk+1=Aθkxk+Jθkwkzk=Cθkxk+Dθkwkwhere x(k)∈Rn is the state vector, y(k)∈Rq is the measured output vector, z(k)∈Rp is the estimated output, w(k)∈Rm is the exogenous input, and {θk} is a Markov chain taking values in K. We define the transition probability matrix of {θk} by P=[ρij], where ρij≥0 is such that ρij=P(θk+1=j∣θk=i) and ∑j=1N‍ρij=1.


### 3.2. Mean Square Stability

We recall the definition of mean square stability presented in [14].Definition. System (2) is mean square stable (MSS) if, for any initial condition x(0)=x0∈Rn and initial distribution θ(0)=θ0∈K, (3)limk→∞εxk′xk∣x0,θ0=0;see, for instance, [15].


### 3.3.H∞ Norm

Assuming that (2) is MSS with x0=0, the H∞ norm of G is given by (see [16]) (4)G∞=sup0≠w∈L2,θ0∈Kz2w2.Notice that the case K={1} corresponds to the deterministic case, that is, the case without jumps.It is possible to calculate theH∞ norm using the so-called Bounded Real Lemma for Markovian jump linear systems, first presented in [17] and stated below.Lemma. System (2) is MSS and satisfies the norm constraint G∞2<δ if and only if there exist matrices Pi=Pi′>0 such that (5)AiJiCiDi′εiP00IAiJiCiDi-Pi00δI<0,∀i∈K.Applying the Schur complement to (5), we get that (6)Pi•••0δI••εiPAiεiPJiεiP•CiDi0I>0and the LMI constraint (6) can also be described by the following inequality: (7)Pi•••0δI••AiJiεθkP-1•CiDi0I>0.See, for instance, [18].


### 3.4.H2 Norm

Assuming that (2) is MSS with x0=0, the H2 norm is given by (8)G22=∑s=1m∑i=1Nμizs,i22where zs,i represents the output z(0),z(1),… obtained when (i)
the input is given byw(k)=esδ(k), where es∈Rm is thesth column of the m×m identity matrix and δ(k) is the unitary impulse; see [19];(ii)
θ0=i∈K with probability μi=P(θ0=i).In [20], it is shown that if the Markov chain is ergodic and taking μi=ρi, where ρi=limk→∞P(θ(k)=i), the norm defined in (8) can also be written as (9)G22=limk→∞εzk′zk,where z(k) is the controlled output and w(k) represents a wide-sense white-noise with covariance given by the identity matrix that is independent of the initial condition x0 and the Markov chain {θk}.In [15] or [19], we have that if there is a solution S=(S1,…,SN)>0, P=(P1,…,PN)>0 of the following LMI: (10)∑i=1NμitrSi<γ,(11)Si••εiPJiεiP•Di0I>0,(12)Pi••εiPAiεiP•Ci0I>0,∀i∈K,which can be equivalently written as (13)Si••JiεiP-1•Di0I>0,(14)Pi••AiεiP-1•Ci0I>0,and then G22<γ.


## 3.1. Markovian Jump Linear System

We initially consider the following general discrete-time Markovian jump linear system (MJLS):(2)G:xk+1=Aθkxk+Jθkwkzk=Cθkxk+Dθkwkwhere x(k)∈Rn is the state vector, y(k)∈Rq is the measured output vector, z(k)∈Rp is the estimated output, w(k)∈Rm is the exogenous input, and {θk} is a Markov chain taking values in K. We define the transition probability matrix of {θk} by P=[ρij], where ρij≥0 is such that ρij=P(θk+1=j∣θk=i) and ∑j=1N‍ρij=1.


## 3.2. Mean Square Stability

We recall the definition of mean square stability presented in [14].Definition. System (2) is mean square stable (MSS) if, for any initial condition x(0)=x0∈Rn and initial distribution θ(0)=θ0∈K, (3)limk→∞εxk′xk∣x0,θ0=0;see, for instance, [15].


## 3.3.H∞ Norm

Assuming that (2) is MSS with x0=0, the H∞ norm of G is given by (see [16]) (4)G∞=sup0≠w∈L2,θ0∈Kz2w2.Notice that the case K={1} corresponds to the deterministic case, that is, the case without jumps.It is possible to calculate theH∞ norm using the so-called Bounded Real Lemma for Markovian jump linear systems, first presented in [17] and stated below.Lemma. System (2) is MSS and satisfies the norm constraint G∞2<δ if and only if there exist matrices Pi=Pi′>0 such that (5)AiJiCiDi′εiP00IAiJiCiDi-Pi00δI<0,∀i∈K.Applying the Schur complement to (5), we get that (6)Pi•••0δI••εiPAiεiPJiεiP•CiDi0I>0and the LMI constraint (6) can also be described by the following inequality: (7)Pi•••0δI••AiJiεθkP-1•CiDi0I>0.See, for instance, [18].


## 3.4.H2 Norm

Assuming that (2) is MSS with x0=0, the H2 norm is given by (8)G22=∑s=1m∑i=1Nμizs,i22where zs,i represents the output z(0),z(1),… obtained when (i)
the input is given byw(k)=esδ(k), where es∈Rm is thesth column of the m×m identity matrix and δ(k) is the unitary impulse; see [19];(ii)
θ0=i∈K with probability μi=P(θ0=i).In [20], it is shown that if the Markov chain is ergodic and taking μi=ρi, where ρi=limk→∞P(θ(k)=i), the norm defined in (8) can also be written as (9)G22=limk→∞εzk′zk,where z(k) is the controlled output and w(k) represents a wide-sense white-noise with covariance given by the identity matrix that is independent of the initial condition x0 and the Markov chain {θk}.In [15] or [19], we have that if there is a solution S=(S1,…,SN)>0, P=(P1,…,PN)>0 of the following LMI: (10)∑i=1NμitrSi<γ,(11)Si••εiPJiεiP•Di0I>0,(12)Pi••εiPAiεiP•Ci0I>0,∀i∈K,which can be equivalently written as (13)Si••JiεiP-1•Di0I>0,(14)Pi••AiεiP-1•Ci0I>0,and then G22<γ.


## 4. The Fault Detection Problem Formulation

The MJLS subject to faults we consider in this work is represented by(15)Ga:xk+1=Aθkxk+Bθkuk+Bdθkdk+Bfθkfkyk=Cθkxk+Ddθkdk+Dfθkfkx0=x0,where x(k)∈Rn is the state variable, y(k)∈Rq is the measured output, u(k)∈Rm is the known input, d(k)∈Rp is the exogenous input, and f(k)∈Rt is the fault vector which is considered as an unknown time function. We also consider that f(k),d(k)∈L2.Usually the fault detection system is divided into two distinct stages, a residual generator and a residual evaluation.


### 4.1. Residual Generator

For the purpose of generating the residual signalr(k), a Markovian filter is considered with the following definition: (16)F:ηk+1=Aηθkηk+Mηθkuk+Bηθkykrk=Cηθkηk+Dηθkykη0=η0where η(k)∈Rn represents the filter states and r(k)∈Ro is the filter residue. We point out that this filter structure also depends on the Markov mode θk.Similar to the continuous-time case presented in [7] and the discrete-time case in [11], a weighting matrix Wf(z) is used with the intention to increase the fault detection performance (see, e.g., [21]), where f^(z)=Wf(z)f(z). A minimal realization of f^(z)=Wf(z)f(z) is (17)Wf:xfk+1=Awfxfk+Bwffkf^k=Cwfxfk+Dwffkxf0=0where xf(k)∈Rt is the weighting matrix state vector, f^(k)∈Ro is the weighted fault signal, and f(k) is the same fault as in (15).Remark. The matrices that compose system (17) are supposed to be known. The block diagram presented in Figure 1 represents the equivalent system.Figure 1
Block diagram.Consideringre(k)=r(k)-f^(k), the equivalent system can be written in the augmented form as (18)Gaug:x-k+1=A~θkx-k+B~θkw-krek=C~θkx-k+D~θkw-kwhere the augmented state and the input signal are x-(k)=xk′ηk′xfk′′ and w-=uk′dk′f^k′′ with (19)
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.In this paper, we tackle three distinct problems: the H∞ case, the H2 case, and the mixed H2/H∞ case. For the H∞ case, the fault detection (FD) filter problem corresponds to obtaining matrices that compose observer (16) in such a way that system (18) is MSS when u=0, d=0f=0, and δ is as small as possible in the feasibility of (20), meaning that (20)Gaug∞=supw-2≠0,w-∈L2re2w-2<δ1/2.For the H2 case, the FD filter problem has the goal of obtaining matrices that compose the filter as in (16) in such a way that system (18) is MSS, minimizing γ in the equation. (21)Gaug22=∑s=1m∑i=1Nμires,i22<γ.For the mixed H2/H∞ case, a way to describe the mixed problem is by setting the objective function as (22)inf⁡gγ,δ,suchthatGaug22<γandGaug∞2<δ,which considers the restrictions as defined in (20) and (21). By inspection, it is possible to note that there are three possible ways to define the objective function in (22), as described below.First Case. Find a minimum guaranteed cost γ for the H2 norm of system (18), subject to a given upper bound δ>0 on the H∞ norm. In this case, we have (23)gγ,δ=γ.Second Case. Find a minimum guaranteed cost δ for the H∞ norm of system (18), subject to a given upper bound γ>0 on H2. In this case, we have (24)gγ,δ=δ.Third Case. Find a minimum for a weighted combination of the guaranteed cost for both H2 and H∞ norms of system (18). Thus, for given scalars β(∞)≥0 and β(2)≥0, we set (25)gγ,δ=δβ∞+γβ2where β(·) represents the weight for each upper bound. A similar approach is presented in [22].


### 4.2. Residual Evaluation

In the evaluation stage, it is necessary to set an evaluation functionJ(r-(k)) and also a threshold Jth(k), both as defined in [11]. We consider L as the evaluation time, and with that we are able to separate the evaluation process into two distinct cases: the first one is defined by k-L≥0 and the second one by k-L<0. Thus, we define the auxiliary vectors for each case as (26)fork-L≥0,r-k=rk′rk-1′⋯rk-L′fork-L<0,r-k=rk′rk-1′⋯r0′and, given the discrepancy between the intervals, the evaluation functions for each case are set as (27)fork-L≥0,Jr-k=∑σ=kσ=k-Lr-′σr-σ1/2,fork-L<0,Jr-k=∑σ=kσ=0r-′σr-σ1/2.The threshold is defined as (28)Jthk=supd∈L2,f=0εJr-kand the decision rule for the fault detection is taken by analyzing the value of J(r-(k)) as follows: (29)Jr-k<Jthk,meansthatthesystemisinthenominalmode,Jr-k≥Jthk,meansthatafaultoccurredattheinstantk.


## 4.1. Residual Generator

For the purpose of generating the residual signalr(k), a Markovian filter is considered with the following definition: (16)F:ηk+1=Aηθkηk+Mηθkuk+Bηθkykrk=Cηθkηk+Dηθkykη0=η0where η(k)∈Rn represents the filter states and r(k)∈Ro is the filter residue. We point out that this filter structure also depends on the Markov mode θk.Similar to the continuous-time case presented in [7] and the discrete-time case in [11], a weighting matrix Wf(z) is used with the intention to increase the fault detection performance (see, e.g., [21]), where f^(z)=Wf(z)f(z). A minimal realization of f^(z)=Wf(z)f(z) is (17)Wf:xfk+1=Awfxfk+Bwffkf^k=Cwfxfk+Dwffkxf0=0where xf(k)∈Rt is the weighting matrix state vector, f^(k)∈Ro is the weighted fault signal, and f(k) is the same fault as in (15).Remark. The matrices that compose system (17) are supposed to be known. The block diagram presented in Figure 1 represents the equivalent system.Figure 1
Block diagram.Consideringre(k)=r(k)-f^(k), the equivalent system can be written in the augmented form as (18)Gaug:x-k+1=A~θkx-k+B~θkw-krek=C~θkx-k+D~θkw-kwhere the augmented state and the input signal are x-(k)=xk′ηk′xfk′′ and w-=uk′dk′f^k′′ with (19)
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.In this paper, we tackle three distinct problems: the H∞ case, the H2 case, and the mixed H2/H∞ case. For the H∞ case, the fault detection (FD) filter problem corresponds to obtaining matrices that compose observer (16) in such a way that system (18) is MSS when u=0, d=0f=0, and δ is as small as possible in the feasibility of (20), meaning that (20)Gaug∞=supw-2≠0,w-∈L2re2w-2<δ1/2.For the H2 case, the FD filter problem has the goal of obtaining matrices that compose the filter as in (16) in such a way that system (18) is MSS, minimizing γ in the equation. (21)Gaug22=∑s=1m∑i=1Nμires,i22<γ.For the mixed H2/H∞ case, a way to describe the mixed problem is by setting the objective function as (22)inf⁡gγ,δ,suchthatGaug22<γandGaug∞2<δ,which considers the restrictions as defined in (20) and (21). By inspection, it is possible to note that there are three possible ways to define the objective function in (22), as described below.First Case. Find a minimum guaranteed cost γ for the H2 norm of system (18), subject to a given upper bound δ>0 on the H∞ norm. In this case, we have (23)gγ,δ=γ.Second Case. Find a minimum guaranteed cost δ for the H∞ norm of system (18), subject to a given upper bound γ>0 on H2. In this case, we have (24)gγ,δ=δ.Third Case. Find a minimum for a weighted combination of the guaranteed cost for both H2 and H∞ norms of system (18). Thus, for given scalars β(∞)≥0 and β(2)≥0, we set (25)gγ,δ=δβ∞+γβ2where β(·) represents the weight for each upper bound. A similar approach is presented in [22].


## 4.2. Residual Evaluation

In the evaluation stage, it is necessary to set an evaluation functionJ(r-(k)) and also a threshold Jth(k), both as defined in [11]. We consider L as the evaluation time, and with that we are able to separate the evaluation process into two distinct cases: the first one is defined by k-L≥0 and the second one by k-L<0. Thus, we define the auxiliary vectors for each case as (26)fork-L≥0,r-k=rk′rk-1′⋯rk-L′fork-L<0,r-k=rk′rk-1′⋯r0′and, given the discrepancy between the intervals, the evaluation functions for each case are set as (27)fork-L≥0,Jr-k=∑σ=kσ=k-Lr-′σr-σ1/2,fork-L<0,Jr-k=∑σ=kσ=0r-′σr-σ1/2.The threshold is defined as (28)Jthk=supd∈L2,f=0εJr-kand the decision rule for the fault detection is taken by analyzing the value of J(r-(k)) as follows: (29)Jr-k<Jthk,meansthatthesystemisinthenominalmode,Jr-k≥Jthk,meansthatafaultoccurredattheinstantk.


## 5. Main Results

In this section, we introduce the three main results of this paper: theH∞ FD filter, the H2 FD filter, and the mixed H2/H∞ FD filter. A theorem is presented for each case.


### 5.1. FD Filter for theH∞ Case

Theorem 1.
There exists a mode-dependent FD filter as in (16) satisfying Gaug∞2<δ if there exist symmetric matrices Zi, Xi, and Wi and matrices Hi, Δi, Oi, Fi, and Gi with compatible dimensions that satisfy the following LMI constraint: (30)Zi•••••••••ZiXi••••••••00Wi•••••••000δI••••••0000δI•••••00000δI••••εiZAiεiZAi0εiZBiεiZBdiεiZBfiεiZ•••εiXAi+ΔiCi+OiεiXAi+ΔiCi0εiXBi+HiεiXBdi+ΔiDdiεiXBfi+ΔiDfiεiZεiX••00εiWAwf00εiWBwf00εiW•GiCi+FiGiCi-Cwf0GiDdiGiDfi-Dwf000I>0for all i∈K. If a feasible solution for (30) is obtained, then a suitable FD filter is given by Aηi=(εi(Z)-εi(X))-1Oi, Bηi=(εi(Z)-εi(X))-1Δi, Cηi=Fi, Dηi=Gi, and Mηi=(εi(Z)-εi(X))-1Hi, for all i∈K.Proof.
The first step to derive the result is to impose the following structure, similar to the structure in [23], for the matrices Pi and Pi-1: (31)Pi=XiUi0Ui′X^i000Wi,Pi-1=YiVi0Vi′Y^i000Hi,and also consider the following structure for the matrices εi(P) and εi(P)-1: (32)εiP=εiXεiU0εiU′εiX000εiW,εiP-1=R1iR2i0R2i′R3i000εiW-1.We define the matrices π and ζ as follows: (33)π=II0Vi′Yi-10000I,ζ=R1i-1εiX00εiU′000εiG.
ConsideringUi=Zi-Xi in (31), we get from (31) and (33) that Yi=Vi′ and Yi=Zi-1. Also considering Ui=-X^i, we get R1i-1=εi(X+U)=ε(Z). Moreover, we have that R1i-1=εi(Z), and so we have that (34)π′Piπ=Yi-1Yi-10Yi-1Xi000Wi,ζ′A~iπ=R1i-1AiR1i-1Ai0εiXAi+εiUBηiCi+εiUAηiVi′Yi-1εiXAi+εiUBηiCi000εiWAwf,ζ′B~i=R1i-1BiR1i-1BdiR1i-1BfiεiXBi+εiUMηiεiXBdi+εiUBηiDdiεiXBfi+εiUBηiDfi00εiWBwf,ζ′εiP-1ζ=R1i-1εiZ0εiZεiX000εiW,C~iπ=DηiCi+CηiVi′ZiDηiCi-Cwf,D~i=0DηiDdiDηiDi-Dwf.Applying the change of variables εi(U)Bηi=Δi, εi(U)AηiVi′Zi=Oi, εi(U)Mηi=Hi, CηiVi′Zi=Fi, and Dηi=Gi and also substituting εi(Z)=R1i-1 in (30), we get the following inequality: (35)π′P~iπ•••0δI••ζ′A~iπζ′B~iζ′εiP-1ζ•C~iπD~i0I>0,and it is easy to see that inequality (35) is equivalent to inequality (30). Multiplying to the right by diag[π-1,I,ζ-1,I] and to the left by its transpose, we get inequality (7) and with that we can guarantee that G∞2<δ.


### 5.2. FD Filter for theH2 Case

Theorem 2.
There exists a mode-dependent FD filter in the form of (16) satisfying Gaug22<γ if there exist symmetric matrices Zi, Xi, Si, and Ti and matrices Hi, Δi, Oi, Fi, and Gi with compatible dimensions that satisfy the LMI constraints (36), (37), and (38):(36)∑i=1NμitrSi<γ(37)••••Si••••••••εiZBiεiZBdiεiZBfiεiZ•••εiXBi+HiεiXBdi+ΔiDdiεiXBfi+ΔiDfiεiZεiX••00εiTBwf00εiT•0GiDdiGiDfi-Dwf000I>0(38)Zi••••••ZiXi•••••00Ti••••εiZAiεiZAi0εiZ•••εiXAi+ΔiCi+OiεiXAi+ΔiCi0εiZεiX••00εiTAwf00εiT•GiCi+FiGiCi-Cwf000I>0for all i∈K. If a feasible solution for (36), (37), and (38) is obtained, then a suitable FD filter is given by Aηi=(εi(Z)-εi(X))-1Oi, Bηi=(εi(Z)-εi(X))-1Δi, Cηi=Fi, Dηi=Gi, and Mηi=(εi(Z)-εi(X))-1Hi, for all i∈K.Proof.
In the same way as presented for theH∞ case, the structures for the matrices Ti and Ti-1 are as shown in (31) for, respectively, Pi and Pi-1. For the matrices εi(T) and εi(T)-1, the structures are equal to the one in (32) for, respectively, εi(P) and εi(P)-1. Furthermore, the matrices π and ζ are as shown in (33). Applying the change of variables εi(U)Bηi=Δi, εiUAηiVi′Zi=Oi, εi(U)Mηi=Hi, CηiVi′Zi=Fi, and Dηi=Gi and also substituting εi(Z)=R1l-1 in (37) and (38), we get the following inequalities: (39)∑i=1NμitrSi<γ,(40)Si••ζ′B~iζ′εiP-1ζ•D~i0I>0,(41)π′Piπ••ζ′A~iπζ′εiP-1ζ•C~iπ0I>0.Multiplying (40) to the right by diag[I,ζ-1,I] (resp. (41) by diag[π-1,ζ-1,I]) and to the left by its transpose, we get inequalities (13) and (14) that, combined with (39), yield that Gaug22<γ.


### 5.3. FD Filter for the MixedH2/H∞ Case

In this subsection, we consider the mixedH2/H∞ case. The set of variables is defined as (42)ψ=Zi>0,Xi>0,Wi>0,Ti>0,Si,Hi,Δi,Oi,Fi,Gi∪ζwhere ζ represents a set that contains γ, δ, or both, depending on whether these parameters γ and δ are assumed to be given or a variable of the problem. See Section 4.1 and (23), (24), and (25) for the possible cases to design the FD filter. We also define (43)Ψ=ψasin42suchthattheLMIs30,36,37,38aresimultaneouslyfeasible.The next theorem provides a sufficient condition for the FD filter design for the mixed H2/H∞ case.Theorem 3.
There exists a mode-dependent FD filter as in (16) such that Gaug22<γ and Gaug∞2<δ if there exists ψ∈Ψ, where ψ is defined as in (43). If a feasible solution is obtained, then a suitable FD filter is given by Aηi=(εi(Z)-εi(X))-1Oi, Bηi=(εi(Z)-εi(X))-1Δi, Cηi=Fi, Dηi=Gi, and Mηi=(εi(Z)-εi(X))-1Hi, for all i∈K.Proof.
The proof follows directly from the proofs for Theorems1 and 2.


## 5.1. FD Filter for theH∞ Case

Theorem 1.
There exists a mode-dependent FD filter as in (16) satisfying Gaug∞2<δ if there exist symmetric matrices Zi, Xi, and Wi and matrices Hi, Δi, Oi, Fi, and Gi with compatible dimensions that satisfy the following LMI constraint: (30)Zi•••••••••ZiXi••••••••00Wi•••••••000δI••••••0000δI•••••00000δI••••εiZAiεiZAi0εiZBiεiZBdiεiZBfiεiZ•••εiXAi+ΔiCi+OiεiXAi+ΔiCi0εiXBi+HiεiXBdi+ΔiDdiεiXBfi+ΔiDfiεiZεiX••00εiWAwf00εiWBwf00εiW•GiCi+FiGiCi-Cwf0GiDdiGiDfi-Dwf000I>0for all i∈K. If a feasible solution for (30) is obtained, then a suitable FD filter is given by Aηi=(εi(Z)-εi(X))-1Oi, Bηi=(εi(Z)-εi(X))-1Δi, Cηi=Fi, Dηi=Gi, and Mηi=(εi(Z)-εi(X))-1Hi, for all i∈K.Proof.
The first step to derive the result is to impose the following structure, similar to the structure in [23], for the matrices Pi and Pi-1: (31)Pi=XiUi0Ui′X^i000Wi,Pi-1=YiVi0Vi′Y^i000Hi,and also consider the following structure for the matrices εi(P) and εi(P)-1: (32)εiP=εiXεiU0εiU′εiX000εiW,εiP-1=R1iR2i0R2i′R3i000εiW-1.We define the matrices π and ζ as follows: (33)π=II0Vi′Yi-10000I,ζ=R1i-1εiX00εiU′000εiG.
ConsideringUi=Zi-Xi in (31), we get from (31) and (33) that Yi=Vi′ and Yi=Zi-1. Also considering Ui=-X^i, we get R1i-1=εi(X+U)=ε(Z). Moreover, we have that R1i-1=εi(Z), and so we have that (34)π′Piπ=Yi-1Yi-10Yi-1Xi000Wi,ζ′A~iπ=R1i-1AiR1i-1Ai0εiXAi+εiUBηiCi+εiUAηiVi′Yi-1εiXAi+εiUBηiCi000εiWAwf,ζ′B~i=R1i-1BiR1i-1BdiR1i-1BfiεiXBi+εiUMηiεiXBdi+εiUBηiDdiεiXBfi+εiUBηiDfi00εiWBwf,ζ′εiP-1ζ=R1i-1εiZ0εiZεiX000εiW,C~iπ=DηiCi+CηiVi′ZiDηiCi-Cwf,D~i=0DηiDdiDηiDi-Dwf.Applying the change of variables εi(U)Bηi=Δi, εi(U)AηiVi′Zi=Oi, εi(U)Mηi=Hi, CηiVi′Zi=Fi, and Dηi=Gi and also substituting εi(Z)=R1i-1 in (30), we get the following inequality: (35)π′P~iπ•••0δI••ζ′A~iπζ′B~iζ′εiP-1ζ•C~iπD~i0I>0,and it is easy to see that inequality (35) is equivalent to inequality (30). Multiplying to the right by diag[π-1,I,ζ-1,I] and to the left by its transpose, we get inequality (7) and with that we can guarantee that G∞2<δ.


## 5.2. FD Filter for theH2 Case

Theorem 2.
There exists a mode-dependent FD filter in the form of (16) satisfying Gaug22<γ if there exist symmetric matrices Zi, Xi, Si, and Ti and matrices Hi, Δi, Oi, Fi, and Gi with compatible dimensions that satisfy the LMI constraints (36), (37), and (38):(36)∑i=1NμitrSi<γ(37)••••Si••••••••εiZBiεiZBdiεiZBfiεiZ•••εiXBi+HiεiXBdi+ΔiDdiεiXBfi+ΔiDfiεiZεiX••00εiTBwf00εiT•0GiDdiGiDfi-Dwf000I>0(38)Zi••••••ZiXi•••••00Ti••••εiZAiεiZAi0εiZ•••εiXAi+ΔiCi+OiεiXAi+ΔiCi0εiZεiX••00εiTAwf00εiT•GiCi+FiGiCi-Cwf000I>0for all i∈K. If a feasible solution for (36), (37), and (38) is obtained, then a suitable FD filter is given by Aηi=(εi(Z)-εi(X))-1Oi, Bηi=(εi(Z)-εi(X))-1Δi, Cηi=Fi, Dηi=Gi, and Mηi=(εi(Z)-εi(X))-1Hi, for all i∈K.Proof.
In the same way as presented for theH∞ case, the structures for the matrices Ti and Ti-1 are as shown in (31) for, respectively, Pi and Pi-1. For the matrices εi(T) and εi(T)-1, the structures are equal to the one in (32) for, respectively, εi(P) and εi(P)-1. Furthermore, the matrices π and ζ are as shown in (33). Applying the change of variables εi(U)Bηi=Δi, εiUAηiVi′Zi=Oi, εi(U)Mηi=Hi, CηiVi′Zi=Fi, and Dηi=Gi and also substituting εi(Z)=R1l-1 in (37) and (38), we get the following inequalities: (39)∑i=1NμitrSi<γ,(40)Si••ζ′B~iζ′εiP-1ζ•D~i0I>0,(41)π′Piπ••ζ′A~iπζ′εiP-1ζ•C~iπ0I>0.Multiplying (40) to the right by diag[I,ζ-1,I] (resp. (41) by diag[π-1,ζ-1,I]) and to the left by its transpose, we get inequalities (13) and (14) that, combined with (39), yield that Gaug22<γ.


## 5.3. FD Filter for the MixedH2/H∞ Case

In this subsection, we consider the mixedH2/H∞ case. The set of variables is defined as (42)ψ=Zi>0,Xi>0,Wi>0,Ti>0,Si,Hi,Δi,Oi,Fi,Gi∪ζwhere ζ represents a set that contains γ, δ, or both, depending on whether these parameters γ and δ are assumed to be given or a variable of the problem. See Section 4.1 and (23), (24), and (25) for the possible cases to design the FD filter. We also define (43)Ψ=ψasin42suchthattheLMIs30,36,37,38aresimultaneouslyfeasible.The next theorem provides a sufficient condition for the FD filter design for the mixed H2/H∞ case.Theorem 3.
There exists a mode-dependent FD filter as in (16) such that Gaug22<γ and Gaug∞2<δ if there exists ψ∈Ψ, where ψ is defined as in (43). If a feasible solution is obtained, then a suitable FD filter is given by Aηi=(εi(Z)-εi(X))-1Oi, Bηi=(εi(Z)-εi(X))-1Δi, Cηi=Fi, Dηi=Gi, and Mηi=(εi(Z)-εi(X))-1Hi, for all i∈K.Proof.
The proof follows directly from the proofs for Theorems1 and 2.


## 6. Secondary Results

In this section, we derive the mode-independent case and the case with parametric uncertainties.


### 6.1. Mode-Independent Case

We tackle in this subsection the mode-independent case, in which the FD filter does not depend on the Markov chain parameterθ(k). In this particular case, we design a single FD filter suitable for all system modes.For the sake of obtaining a single FD filter for theN modes, it is necessary to fix the variables, Δi=Δ, Oi=O, Fi=F, Gi=G, and Hi=H, in the LMI constraints (30) or (36), (37), and (38) or all the set of LMI constraints, depending on whether the constraints are related to the problems analyzed in Theorem 1, Theorem 2, or Theorem 3. Furthermore, it is also necessary to assume that (44)ρij=ρj,∀i,j∈Kwhich corresponds to the Bernoulli case. Notice that the addition of this assumption is necessary due to the terms in Theorems 1 and 2 (εi(Z)-εi(X)), which in the Bernoulli case become (ε(Z)-ε(X)), where ε(X)=∑j=1N‍ρjXj, similarly for ε(Z). Theorem 4 presents this result.Theorem 4.
There exists a mode-independent FD filter as in (16) satisfying the constraint presented in Theorem 1 or Theorem 2 or Theorem 3, if there are Zi, Xi, and Wi (and Si and Ti for the H2 and mixed cases) and matrices H, Δ, O, F, and G (independent of i) satisfying (30) for the H∞ case, (36)-(38) for the H2 case, and simultaneously (30), (36), (37), and (38) for the mixed case. If a feasible solution is obtained, a suitable FD filter is given by Aη=(ε(Z)-ε(X))-1O, Bη=(ε(Z)-ε(X))-1Δ, Cη=F, Dη=G, and M=(ε(Z)-ε(X))-1H.Proof.
The proof can be derived directly from the proofs for Theorems1 and 2.


### 6.2. Parametric Uncertain Case

The last special case we work on is the procedure to add parametric uncertainties in Theorems1, 2, and 3. In order to describe system (15) with polytopic uncertainties, we consider that for vertex matrices (45)
A
i
l
B
i
l
C
i
l
0
,
i
∈
Kwe have that (46)
A
θ
k
B
θ
k
C
θ
k
0
∈
Υwhere Υ is the polytope as described in (1) and l∈{1,…,V} represents the uncertain polytopic vertex. We replace in (30) and (37) and (38) the matrices Ai, Bi, and Ci by, respectively, Ail, Bil, and Cil, so that adding this new index implies adding V new constraints in (30) for Theorem 1 or (37) and (38) for Theorem 2 or (30), (37), and (38) for Theorem 3. We have the following result.Theorem 5.
There exists a mode-dependent FD filter as in (16) satisfying the constraints in Theorem 1 or Theorem 2 or Theorem 3 if there exist symmetric matrices Zi, Xi, and Wi, (Si and Ti for the H2 and mixed cases) and matrices Hi, Δi, Oi, Fi, and Dηi satisfying (30) for the H∞ case, (36)-(38) for the H2 case, and simultaneously (30), (36), (37), and (38) for the mixed case. If a feasible solution is obtained, a suitable FD filter is given by Aηi=(εi(Z)-εi(X))-1Oi, Bηi=(εi(Z)-εi(X))-1Δi, Cηi=Fi, Dηi=Gi=, and Mηi=(εi(Z)-εi(X))-1Hi, for all i∈K.Proof.
The proof derives directly from the proofs for Theorems1 and 2.


## 6.1. Mode-Independent Case

We tackle in this subsection the mode-independent case, in which the FD filter does not depend on the Markov chain parameterθ(k). In this particular case, we design a single FD filter suitable for all system modes.For the sake of obtaining a single FD filter for theN modes, it is necessary to fix the variables, Δi=Δ, Oi=O, Fi=F, Gi=G, and Hi=H, in the LMI constraints (30) or (36), (37), and (38) or all the set of LMI constraints, depending on whether the constraints are related to the problems analyzed in Theorem 1, Theorem 2, or Theorem 3. Furthermore, it is also necessary to assume that (44)ρij=ρj,∀i,j∈Kwhich corresponds to the Bernoulli case. Notice that the addition of this assumption is necessary due to the terms in Theorems 1 and 2 (εi(Z)-εi(X)), which in the Bernoulli case become (ε(Z)-ε(X)), where ε(X)=∑j=1N‍ρjXj, similarly for ε(Z). Theorem 4 presents this result.Theorem 4.
There exists a mode-independent FD filter as in (16) satisfying the constraint presented in Theorem 1 or Theorem 2 or Theorem 3, if there are Zi, Xi, and Wi (and Si and Ti for the H2 and mixed cases) and matrices H, Δ, O, F, and G (independent of i) satisfying (30) for the H∞ case, (36)-(38) for the H2 case, and simultaneously (30), (36), (37), and (38) for the mixed case. If a feasible solution is obtained, a suitable FD filter is given by Aη=(ε(Z)-ε(X))-1O, Bη=(ε(Z)-ε(X))-1Δ, Cη=F, Dη=G, and M=(ε(Z)-ε(X))-1H.Proof.
The proof can be derived directly from the proofs for Theorems1 and 2.


## 6.2. Parametric Uncertain Case

The last special case we work on is the procedure to add parametric uncertainties in Theorems1, 2, and 3. In order to describe system (15) with polytopic uncertainties, we consider that for vertex matrices (45)
A
i
l
B
i
l
C
i
l
0
,
i
∈
Kwe have that (46)
A
θ
k
B
θ
k
C
θ
k
0
∈
Υwhere Υ is the polytope as described in (1) and l∈{1,…,V} represents the uncertain polytopic vertex. We replace in (30) and (37) and (38) the matrices Ai, Bi, and Ci by, respectively, Ail, Bil, and Cil, so that adding this new index implies adding V new constraints in (30) for Theorem 1 or (37) and (38) for Theorem 2 or (30), (37), and (38) for Theorem 3. We have the following result.Theorem 5.
There exists a mode-dependent FD filter as in (16) satisfying the constraints in Theorem 1 or Theorem 2 or Theorem 3 if there exist symmetric matrices Zi, Xi, and Wi, (Si and Ti for the H2 and mixed cases) and matrices Hi, Δi, Oi, Fi, and Dηi satisfying (30) for the H∞ case, (36)-(38) for the H2 case, and simultaneously (30), (36), (37), and (38) for the mixed case. If a feasible solution is obtained, a suitable FD filter is given by Aηi=(εi(Z)-εi(X))-1Oi, Bηi=(εi(Z)-εi(X))-1Δi, Cηi=Fi, Dηi=Gi=, and Mηi=(εi(Z)-εi(X))-1Hi, for all i∈K.Proof.
The proof derives directly from the proofs for Theorems1 and 2.


## 7. Numerical Example

In this section, some numerical examples are presented. The first analysis is on theH∞ and H2 norm behavior when we consider variation on some of the parameters in the transition matrix P.The second analysis is related to the mixedH2/H∞ case, and we consider the three cases presented in Section 4.1. For the first case, defined by (23), the upper bound for the H∞ norm δ is fixed and the upper bound for the H2 norm γ is minimized. For the second case, defined by (24), the opposite situation is considered; that is, the upper bound γ is fixed and the upper bound δ is minimized. For the third case, defined by (25), we vary the scalars β∞ and β2 and consider the behavior of both upper bounds γ and δ.The third analysis is a temporal simulation comparing theH∞, H2, and all three mixed H2/H∞ cases when two kinds of failures occur: an abrupt failure and a smooth failure. This analysis is divided into two stages: the first one with a single sample and the second one with a Monte Carlo simulation with 2000 random samples. The system used for this example was extracted from [11] and its matrices are given by (47)A1=0.101000.100.5000.200000.2,A2=0.30-10-0.10.20-0.500-0.20000-0.5,Bd=0.8-2.41.60.8,Bf=112-2,C=01011010,Dd=0.20.4Df=2-1,Awf=0.5,Bwf=0.25,Cwf=1,Dwf=0.5.The matrixP is given by (48)P=ρ11-ρ11-ρ2ρ2where 0≤ρi≤1,i∈K.


### 7.1. Norm Behavior

The norm behavior is important to be analyzed in order to identify the worst case scenario and the system sensibility to the parameters variation in the transition matrixP.


#### 7.1.1. FD Filter for theH∞ Case and the H2 Case

Considering thatρ1=ρ2∈[0.1,0.9] and P as in (48), the obtained curves are presented in Figure 2. It is possible to observe that in both cases the variation range is small, showing that the system sensibility is small in terms of the variation of the probability of jumps. Another important information that could be extracted from the graphics is that the system has a good level of robustness, given that, from Figure 2(a), the H∞ norm’s maximum value is less than 0.6. The variation of the values for the H2 norm, which represents the total energy dissipated by the impulsive inputs, is shown in Figure 2(b), which indicates that the system is sensible to this class of signals.Figure 2
Norm behavior forH∞ and H2 cases.
(a)
H ∞ norm (b)
H 2 norm


### 7.2. FD Filter for the MixedH2/H∞ Case

In this subsection, we present the norm behavior for each case below:(i)
First case: the behavior of γ when δ varies between (3.6,10)(ii)
Second case: the behavior of δ when γ varies between (3.6,10)(iii)
Third case: the behavior of γ and δ when β2 varies between (0.2,0.8), considering that β∞=1-β2It is possible to observe that, in the first case, shown in Figure3(a), as the value of δ increases, the upper bound γ decreases, which is expected, since the conservatism in the optimization problem is reduced. A similar situation occurs in Figure 3(b), since that, as the value of γ increases, the value of δ decreases. For the third case (Figure 3(c)), it is important to observe that when β2=0 the optimization problem falls in the first case, and for β∞=0 the optimization problem falls in the second case. Bearing this information in mind, the third graphic (Figure 3(c)) represents exactly what it was expected; that is, when β is closer to zero the δ value is closer to the value obtained in the second case; otherwise, the value of γ gets closer to the value obtained in the first case.Figure 3
Norm behavior for the mixed cases.
(a)
First case (b)
Second case (c)
Third case


### 7.3. Temporal Simulation

Hereafter, the matrixP is given by (49)P=0.80.20.40.6.Two distinct faults were selected: the first one an abrupt fault and the second one a smooth signal, and both are presented in Figure 4.Figure 4
Failure behavior.
(a)
Abrupt failure (b)
Smooth failureIn order to analyze the applicability of our approaches, we present in Figure5 the simulation results considering a single sample for the cases H2, H∞, and mixed H∞/H2. Observing the values in Table 1, we notice that the H∞ approach has a small advantage when compared to the other approaches. Other information obtained is that the smooth failure, as expected, demands more time to be detected.Table 1
Amount of time until detection in a single simulation.
Abrupt failure Smooth failure H ∞ case 104 125 H 2 case 105 122 Mixed first case 107 123 Mixed second case 110 132 Mixed third case 112 123Figure 5
Temporal simulation.
(a)
For the abrupt failure (b)
For the smooth failureUsing the same parameter as in the previous simulation and performing a Monte Carlo simulation with 2000 samples, we obtain the curves as in Figure6.Figure 6
Monte Carlo simulation.
(a)
For abrupt failure (b)
For smooth failureObserving the graphics presented in Figure6, it is possible to see that the H2 and second mixed H2/H∞ cases, explained in Section 4.1 and represented by (24), are the most sensible approaches, as can also be observed from Table 2.Table 2
Average amount of time until detection and the standard deviation.
Abrupt failure Smooth failure H ∞ case 105.2750 ± 2.1336 127.4950 ± 7.0018 H 2 case 104.7400 ± 2.2489 127.0150 ± 7.2773 Mixed first case 104.6000 ± 2.0100 127.9650 ± 6.5688 Mixed second case 106.4350 ± 3.0795 128.0850 ± 7.2893 Mixed third case 105.1900 ± 3.0922 127.9400 ± 9.3075As explained in [7], the main goal is to provide a FD filter with a maximum ratio between fault sensitivity (r/f) and disturbance sensitivity (r/d), and, as observed, the solution that fits better in this condition is the first mixed H2/H∞ case, explained in the Section 4.1 and represented by (23), where the robustness is provided by the H∞ norm and the fault sensibility is provided by the H2 norm.According to the graphic presented in Figure6, the approach with higher level of fault sensibility is the H2 case, but this approach has the lowest level of exogenous input resilience. Thus, the H2 approach does not provide the best option when considering the desirable aspect of a FD filter of high sensibility to the fault and low sensibility to the disturbance. For this reason, analyzing Table 3, the best performance is provided by the first mixed H2/H∞ case, explained in Section 4.1 and represented by (23), due to the high level of sensibility against the fault and low sensibility to the disturbance. The H∞ case provides the lowest sensibility to disturbance, but the level of sensibility against the fault does not get close to the levels presented by the H2 norm and the first mixed H2/H∞ case.Table 3
The sensibility against the failure and resilience against exogenous input.
r / f r / d H ∞ case 0.1979 0.0145 H 2 case 0.7130 0.2163 Mixed first case 0.5357 0.1648 Mixed second case 0.0450 0.0567 Mixed third case 0.1288 0.0389


## 7.1. Norm Behavior

The norm behavior is important to be analyzed in order to identify the worst case scenario and the system sensibility to the parameters variation in the transition matrixP.


### 7.1.1. FD Filter for theH∞ Case and the H2 Case

Considering thatρ1=ρ2∈[0.1,0.9] and P as in (48), the obtained curves are presented in Figure 2. It is possible to observe that in both cases the variation range is small, showing that the system sensibility is small in terms of the variation of the probability of jumps. Another important information that could be extracted from the graphics is that the system has a good level of robustness, given that, from Figure 2(a), the H∞ norm’s maximum value is less than 0.6. The variation of the values for the H2 norm, which represents the total energy dissipated by the impulsive inputs, is shown in Figure 2(b), which indicates that the system is sensible to this class of signals.Figure 2
Norm behavior forH∞ and H2 cases.
(a)
H ∞ norm (b)
H 2 norm


## 7.1.1. FD Filter for theH∞ Case and the H2 Case

Considering thatρ1=ρ2∈[0.1,0.9] and P as in (48), the obtained curves are presented in Figure 2. It is possible to observe that in both cases the variation range is small, showing that the system sensibility is small in terms of the variation of the probability of jumps. Another important information that could be extracted from the graphics is that the system has a good level of robustness, given that, from Figure 2(a), the H∞ norm’s maximum value is less than 0.6. The variation of the values for the H2 norm, which represents the total energy dissipated by the impulsive inputs, is shown in Figure 2(b), which indicates that the system is sensible to this class of signals.Figure 2
Norm behavior forH∞ and H2 cases.
(a)
H ∞ norm (b)
H 2 norm


## 7.2. FD Filter for the MixedH2/H∞ Case

In this subsection, we present the norm behavior for each case below:(i)
First case: the behavior of γ when δ varies between (3.6,10)(ii)
Second case: the behavior of δ when γ varies between (3.6,10)(iii)
Third case: the behavior of γ and δ when β2 varies between (0.2,0.8), considering that β∞=1-β2It is possible to observe that, in the first case, shown in Figure3(a), as the value of δ increases, the upper bound γ decreases, which is expected, since the conservatism in the optimization problem is reduced. A similar situation occurs in Figure 3(b), since that, as the value of γ increases, the value of δ decreases. For the third case (Figure 3(c)), it is important to observe that when β2=0 the optimization problem falls in the first case, and for β∞=0 the optimization problem falls in the second case. Bearing this information in mind, the third graphic (Figure 3(c)) represents exactly what it was expected; that is, when β is closer to zero the δ value is closer to the value obtained in the second case; otherwise, the value of γ gets closer to the value obtained in the first case.Figure 3
Norm behavior for the mixed cases.
(a)
First case (b)
Second case (c)
Third case


## 7.3. Temporal Simulation

Hereafter, the matrixP is given by (49)P=0.80.20.40.6.Two distinct faults were selected: the first one an abrupt fault and the second one a smooth signal, and both are presented in Figure 4.Figure 4
Failure behavior.
(a)
Abrupt failure (b)
Smooth failureIn order to analyze the applicability of our approaches, we present in Figure5 the simulation results considering a single sample for the cases H2, H∞, and mixed H∞/H2. Observing the values in Table 1, we notice that the H∞ approach has a small advantage when compared to the other approaches. Other information obtained is that the smooth failure, as expected, demands more time to be detected.Table 1
Amount of time until detection in a single simulation.
Abrupt failure Smooth failure H ∞ case 104 125 H 2 case 105 122 Mixed first case 107 123 Mixed second case 110 132 Mixed third case 112 123Figure 5
Temporal simulation.
(a)
For the abrupt failure (b)
For the smooth failureUsing the same parameter as in the previous simulation and performing a Monte Carlo simulation with 2000 samples, we obtain the curves as in Figure6.Figure 6
Monte Carlo simulation.
(a)
For abrupt failure (b)
For smooth failureObserving the graphics presented in Figure6, it is possible to see that the H2 and second mixed H2/H∞ cases, explained in Section 4.1 and represented by (24), are the most sensible approaches, as can also be observed from Table 2.Table 2
Average amount of time until detection and the standard deviation.
Abrupt failure Smooth failure H ∞ case 105.2750 ± 2.1336 127.4950 ± 7.0018 H 2 case 104.7400 ± 2.2489 127.0150 ± 7.2773 Mixed first case 104.6000 ± 2.0100 127.9650 ± 6.5688 Mixed second case 106.4350 ± 3.0795 128.0850 ± 7.2893 Mixed third case 105.1900 ± 3.0922 127.9400 ± 9.3075As explained in [7], the main goal is to provide a FD filter with a maximum ratio between fault sensitivity (r/f) and disturbance sensitivity (r/d), and, as observed, the solution that fits better in this condition is the first mixed H2/H∞ case, explained in the Section 4.1 and represented by (23), where the robustness is provided by the H∞ norm and the fault sensibility is provided by the H2 norm.According to the graphic presented in Figure6, the approach with higher level of fault sensibility is the H2 case, but this approach has the lowest level of exogenous input resilience. Thus, the H2 approach does not provide the best option when considering the desirable aspect of a FD filter of high sensibility to the fault and low sensibility to the disturbance. For this reason, analyzing Table 3, the best performance is provided by the first mixed H2/H∞ case, explained in Section 4.1 and represented by (23), due to the high level of sensibility against the fault and low sensibility to the disturbance. The H∞ case provides the lowest sensibility to disturbance, but the level of sensibility against the fault does not get close to the levels presented by the H2 norm and the first mixed H2/H∞ case.Table 3
The sensibility against the failure and resilience against exogenous input.
r / f r / d H ∞ case 0.1979 0.0145 H 2 case 0.7130 0.2163 Mixed first case 0.5357 0.1648 Mixed second case 0.0450 0.0567 Mixed third case 0.1288 0.0389


## 8. Conclusion

In this paper, the fault detection problem under the discrete-time Markovian jump linear systems framework is studied. We propose a convex formulation in terms of LMI for the design of mixedH2/H∞ FD filters such that the H2 and H∞ norms with respect to the residual signal are bounded. Three possible design formulations are presented: (1) an upper bound of the H2 norm is minimized while guaranteeing that the H∞ norm is bounded; (2) the inverse scheme (minimizing an upper bound of the H∞ norm by restricting its H2 norm); (3) a weighed combination of both bounds on H2 and H∞ norms is minimized. Design conditions for mode-independent filters in the scope of the Bernoulli case, as well as robust filters (in the sense of uncertain system matrices), are also given. Regarding the simulation results, it is important to point out that all approaches provided a plausible solution to the fault detection problem. However, following the comparison criterion presented in [7], the approach with the best ratio between fault sensitivity and disturbance sensitivity was the first mixed H2/H∞ case as in (23), since this solution presented the best results as can be seen in Table 3.Along the lines of the present paper, a possible next step would be to consider that the Markov chain mode is not accessible, as considered in [22], which is an assumption that brings new challenges to the fault detection problem.



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*Source: 4239696-2018-07-16.xml*