# RobustH∞ Fault Detection for Networked Markov Jump Systems with Random Time-Delay

Authors: Yanfeng Wang; Zuxin Li; Peiliang Wang; Zhe Zhou
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

This paper investigates the problem of robustH∞ fault detection for networked Markov jump systems with random time-delay which is introduced by the network. The random time-delay is modeled as a Markov process, and the networked Markov jump systems are modeled as control systems containing two Markov chains. The delay-dependent fault detection filter is constructed. Furthermore, the sufficient and necessary conditions which make the closed-loop system stochastically stable and achieve prescribed H∞ performance are derived. The method of calculating controller, fault detection filter gain matrices, and the minimal H∞ attenuation level is also obtained. Finally, one numerical example is used to illustrate the effectiveness of the proposed method.


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## Body

## 1. Introduction

Feedback control systems wherein the control loop is closed through a real-time network are called networked control systems (NCSs) [1, 2]. The information is exchanged among control system components (sensor, controller, actuator, etc.). Due to the advantages such as simple installation, reduced wiring, increased system agility, and high reliability, NCSs have been widely used in broad areas, for example, unmanned aerial vehicles, mobile sensor networks, environment monitoring, and automated highway systems [3–5]. However, the introduction of communication networks also brings communication constraints to the control systems, for example, network-induced time-delays and packet dropouts [6–8]. Fault detection (FD) is very important for practical control systems, especially in safe-critical systems [9–11]. The theory of FD for NCSs is different from that of the traditional control systems due to the limitations induced by the network, such as time-delays and data packet dropouts which should be taken into consideration.In recent years, many results of FD for NCSs have been reported. In [12], the problem of FD for a kind of nonlinear NCS with time-delays and data packet dropouts was investigated, and the sufficient conditions for the existence of FD filter were presented in terms of linear matrix inequalities (LMIs) using Lyapunov function in the continuous domain. In [13], by considering random time-delays, the NCSs were modeled as discrete-time, finite-dimensional Markov jump linear systems (MJLSs). The FD problem was formulated as a robust H∞ FD filter design problem, and the sufficient condition to solve this problem was given in terms of LMIs. In [14], with the presence of stochastic packet dropouts in the network, the problem of FD filter design for NCSs was investigated. A design method for FD filter which made the residual generation system stable in the mean-square sense was proposed by the MJLSs theory. In [15], the problem of robust FD filter design and optimization was investigated for NCSs with random delays. The NCSs were modeled as Markov jump systems by assuming that the random delays obeyed the Markov characteristics. Based on the model, an observer-based residual generator was constructed and the corresponding FD problem was formulated as a filtering problem. A sufficient condition for the existence of the desired FD filter was derived in terms of LMIs. In [16], by employing the multirate sampling method and the augmented state matrix method, the NCSs with long random delays were modeled as MJLSs. Then based on the model, a filter was designed for detecting faults. In [17], two independent Markov chains were introduced to describe the transmission characterization of the data packet dropouts in both channels from sensors to controller and from controller to actuator, and a nonlinear Markov jump system model was established. By employing a mode-dependent FD filter as residual generator, the FD filter design problem of nonlinear NCSs was formulated as a nonlinear H∞ filtering problem. In [18], by use of the augmented matrix approach, the FD error dynamic systems were transformed to the MJLSs. With the established model and using the bounded real lemma (BRL) for MJLSs, a H∞ observer-based FD filter was established in terms of LMIs to guarantee that the error between the residual and the weighted faults was made as small as possible. In [19], the problem of FD was investigated for NCSs with signal quantization and random packet dropouts. A residual generator was constructed, and the corresponding FD problem was converted into a H∞ filtering problem. In [20], the time-delays from sensor to controller and the time-delays from sensor to actuator are both considered which were described by two independent Markov chains. H∞ FD problem for NCSs with time-delays on condition that the transition probabilities were partly unknown was investigated.Markov jump systems are appropriate to model the systems whose structures are subject to the random changes which are widely used in the field of communications systems, power systems, and so on; thus, they have attracted much attention [21–24]. It is significant and necessary to investigate the FD problems for NCSs with the Markov jump controlled plants. However, the controlled plants in most of the existing literature were assumed to be the time-invariant systems (see [12–20]). To the best of the authors’ knowledge, up to now, very limited efforts have been devoted to investigating the FD problem for NCSs with the Markov jump controlled plant, which motivates our investigation.Compared to the previous relevant works, the main contribution of this paper is that, for the Markov jump NCSs, the sufficient and necessary conditions for the stochastically stability of the closed-loop system are derived, and the method of calculating the minimalH∞ attenuation γmin is obtained by constructing proper Lyapunov function candidate.The rest of this paper is organized as follows. The FD filter is constructed and the closed-loop system model is obtained in Section2. The sufficient and necessary conditions which make the closed-loop system stochastically stable and achieve prescribed H∞ performance are derived in Section 3. Section 4 presents the simulation results to show the effectiveness of the proposed method. The conclusions are provided in Section 5.


## 2. Problem Formulation

Without loss of generality, we assume that the time-delayτk only exists between sensor and controller, and τk is modeled as a homogeneous Markov chin which takes value in the set M≜{0,…,τ}, and the transition probability matrix is Λ=[λij]. That is, τk jumps from mode i to j with probability λij, which is defined by λij=Pr(τk+1=j∣τk=i), where λij≥0 and ∑j=0τλij=1, for all i,j∈M.In this paper, the following Markov jump controlled plant is considered:(1)xk+1=Aθkxk+Buθkuk+Bdθkdk+Bfθkfk,yk=Cθkxk,where xk∈Rn is the state vector, uk∈Rm is the input vector, yk∈Rr is the measured output vector, dk∈Rd is the external disturbance noise belonging to l2∈[0,∞), and fk∈Rf is the fault to be detected. Aθk,  Buθk,  Bdθk,  Bfθk, and Cθkare all known real constant matrices with appropriate dimensions. {θk,k≥0} is a discrete-time homogeneous Markov chain, which takes values in a finite set G≜{1,…,N} with a transition probability matrix Π=[πpq]; namely, for θk=p,  θk+1=q, one has πij=Pr(θk+1=q∣θk=p), where πpq≥0 and ∑q=1Nπpq=1, for all p,q∈G.It is noticed that the information ofθk is not available for the controller at the time instant k duo to the time-delay τk; however, the information of τk is known to the controller. Consequently, the controller gain can be designed depending on τk; that is, (2)uk=Kτkxk-τk.Construct a full-order FD filter at the side of controller as follows:(3)x^k+1=Aθkx^k+BuθkKτkxk-τk+Lτkyk-τk-y^k-τk,y^k=Cθkx^k,rk=Vyk-τk-y^k-τk,where x^k∈Rn is the filter state vector, rk∈Rg is the residual vector which is sensitive to the fault, Lτk is the filter gain matrix to be determined, and V is the gain matrix of the residual rk.Define the state estimation error and residual error as follows:(4)ek=xk-x^k,rek=rk-fk.The closed-loop systems can be obtained as(5)ξk+1=A~θkξk+B~uθkKτkI1-I2LτkC~θkξk-τk+Bωθkωk,rek=VC~θkξk-τk-I3ωk,ξk=ηk,k∈-τ,…,0,where (6)A~θk=Aθk00Aθk,B~uθk=Buθk0,C~θk=0Cθk,B~ωθk=BdθkBfθkBdθkBfθk,I1=I0,I2=0I,I3=0I,ξkT=xkTekT,ωkT=dkTfkT.Definition 1 (see [25]).
System (5) is stochastically stable if for ωk=0 and every initial mode τ0∈M,  θ0∈G, there exists a finite matrix W>0 such that(7)E∑k=0∞ξk2∣ξ0,τ0,θ0<ξ0TWξ0.In this paper, our objective is to design controller (2) and the FD filter (3), such that one has the following:
(a) The closed-loop system (5) is stochastically stable for ωk=0.
(b) Under the zero-initial conditions, the residual errorrek satisfies the following H∞ noise attenuation performance:(8)E∑k=0∞rekTrek<γ2E∑k=0∞ωkTωk,where γ>0 is the attenuation level.
For the purpose of FD, an evaluation function and a threshold should be provided, and in this paper the evaluation functionJk and a threshold Jth are selected as (9)Jk=E∑ρ=l0l0+krρTrρ,Jth=supωk∈L2,fk=0⁡E∑ρ=l0l0+L0rρTrρ,where l0 is the initial evaluation time instant and L0 is the evaluation step length. The occurrence of fault can be detected by comparing Jk and Jth with the following rule: (10)Jk≤Jth⟹normal,Jk>Jth⟹fault.Remark 2.
It should be pointed out that if time-delay also exists between controller and actuator which is written asνk, the control input of the controlled plant (1) should be Kτkxk-τk-νkwhich is different from the control input of the FD filter which is Kτkxk-τk.Remark 3.
If there is no time-delay in system (5), the FD filter (3) can still detect the fault effectively.Remark 4.
In almost all the existing literatures related to the FD for NCSs, the standard infinite impulse response (IIR) filter (3) is commonly used. However, the researches about FD for NCSs using finite impulse response (FIR) filter including deadbeat dissipative FIR filtering, hybrid particle FIR filtering, and composite particle FIR filtering have not been reported, which is a completely new research area.


## 3. Main Results

In this section, the sufficient and necessary conditions which make system (5) stochastically stable will be derived. Further, we will present the calculation method of the controller gain matrix Kτk, the FD filter gain matrix Lτk, and the minimal H∞ attenuation γmin in terms of matrix inequalities. To proceed, we will need the following lemma.Lemma 5 (see [26]).
For any positive-definite matrixR, scalars δ,δ0 satisfying δ≥δ0≥1, and vector function υl, one always has ∑l=δ0δυlTR∑l=δ0δυl≤(δ-δ0+1)∑l=δ0δυlTRυl.Theorem 6.
Whenωk=0, the closed-loop system (5) is stochastically stable if and only if there exist positive-definite matrices Pi,p>0,Pj,q>0,Q1>0,Q2>0,Z1>0and matrices Ki,  Li such that the inequality (11)Ω≜Ω11∗∗Ω21Ω22∗0Z1-Q1-Z1<0,where (12)Ω11=A~pTP-j,qA~p+τ2A~pTZ1A~p+Q1+τ+1Q2-Z1-Pi,p,Ω21=B~upKiI1-I2LiC~pTP-j,qA~p+τ2B~upKiI1-I2LiC~pTZ1A~p+Z1,Ω22=B~upKiI1-I2LiC~pTP-j,qB~upKiI1-I2LiC~p+τ2B~upKiI1-I2LiC~pTZ1B~upKiI1-I2LiC~p-Q2-2Z1,P-j,q=∑j=0τ∑q=1NλijπpqPj,q,holds for all i,j∈M and p,q∈G.Proof.
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Sufficiency. Choose the Lyapunov function candidate as(13)Vξk,τk,θk≜ξkTΨτk,θkξk=∑μ=14Vμξk,τk,θk,where(14)V1ξk,τk,θk=ξkTPτk,θkξk,V2ξk,τk,θk=∑m=k-τk-1ξmTQ1ξm,V3ξk,τk,θk=∑m=k-τkk-1ξmTQ2ξm+∑n=-τ+10∑m=k+nk-1ξmTQ2ξm,V4ξk,τk,θk=∑n=-τ+10∑m=k+nk-1τφmTZ1φm,φm=ξm+1-ξm.Apparently, we have Ψτk,θk>0.
Along the solution of system (5), we have(15)EΔV1=Eξk+1TPτk+1,θk+1ξk+1∣τk=i,θk=p-ξkTPτk,θkξk=A~pξk+B~upKiI1-I2LiC~pξk-τkT∑j=0τ∑q=1NλijπpqPj,qA~pξk+B~upKiI1-I2LiC~pξk-τk-ξkTPi,pξk,EΔV2=ξkTQ1ξk-ξk-τTQ1ξk-τ,EΔV3=ξkTQ2ξk-ξk-iTQ2ξk-i+∑l=k+1-τk+1k-1ξlTQ2ξl-∑l=k+1-τkk-1ξlTQ2ξl+τξkTQ2ξk-∑l=k+1-τkξlTQ2ξl.Note that(16)∑l=k+1-τk+1k-1ξlTQ2ξl=∑l=k+1-τkk-1ξlTQ2ξl+∑l=k+1-τk+1k-τkξlTQ2ξl≤∑l=k+1-τkk-1ξlTQ2ξl+∑l=k+1-τkξlTQ2ξl.Hence, we can obtain (17)EΔV3≤ξkTQ2ξk-ξk-iTQ2ξk-i+τξkTQ2ξk,EΔV4=τ2φkTZ1φk-∑l=k-τk-1τφlTZ1φl=τ2φkTZ1φk-∑l=k-ik-1τφlTZ1φl-∑l=k-τk-i-1τφlTZ1φl≤τ2φkTZ1φk-∑l=k-ik-1iφlTZ1φl-∑l=k-τk-i-1τ-iφlTZ1φl.By Lemma 5, one can obtain (18)-∑l=k-ik-1iφlTZ1φl-∑l=k-τk-i-1τ-iφlTZ1φl≤-ξk-ξk-iTZ1ξk-ξk-i-ξk-i-ξk-τTZ1ξk-i-ξk-τ.From (15)–(18), we have(19)EΔVξk,τk,θk∣τk=i,θk=p≤ζkTΩζk≤-λmin-ΩζkTζk=-λmin-ΩξkTξk+ξk-iTξk-i+ξk-τTξk-τ≤-αξk2,where(20)ζkT=ξkTξk-iTξk-τT,α=inf⁡-λmin-Ω>0.From (19), we can see that for any T≥1(21)E∑k=0∞ξk2≤1αEVξ0,τ0,θ0-1αEVξT+1,τT+1,θT+1≤1αEVξ0,τ0,θ0=ξ0TΨτ0,θ0ξ0.That is, the closed-loop system (5) is stochastically stable according to Definition 1.
Necessity. Assume that the closed-loop system (5) is stochastically stable. Thus, we have(22)E∑k=0∞ξk2∣ξ0,τ0<ξ0TWξ0.Let(23)ξkTΨ^τk,θkξk=E∑t=kTξtTHτt,θtξt,where Hτt,θt>0.
Assumeξk≠0, from (23), it can be easily inferred that Ψ^τk,θk is bounded, and the following limit exists:(24)ξkTΨτk,θkξk≜limT→∞⁡ξkTΨ^τk,θkξk=limT→∞⁡E∑t=kTξtTHτt,θtξt.Since (24) holds for any ξk, we have Ψτk,θk=limT→∞⁡Ψ^τk,θk. Since Hτt,θt>0, it can be seen that Ψτk,θk>0 from (24).
Let us consider(25)EξkTΨ^τk,θkξk-ξk+1TΨ^τk+1,θk+1ξk+1∣τk=i,θk=p=ξkTHτk,θkξk>0.Letting T→∞, we have(26)EξkTΨτk,θkξk-ξk+1TΨτk+1,θk+1ξk+1∣τk=i,θk=p=-EΔVξk,τk,θk∣τk=i,θk=p≥-ζkTΩζk>0,which completes the proof.Corollary 7.
Whenωk≠0, consider the closed-loop system (5) and let γ>0 be a given real scalar. If there exist Pi,p>0,Pj,q>0,Fj,q>0,Q1>0,Q2>0,Y1>0,Z1>0 and matrices Ki,Li such that(27)Ξ11∗∗Ξ21Ξ22∗Ξ310Ξ33<0,(28)Pj,q-1Fj,q=I,Z1-1Y1=I,where (29)Ξ11=Q1+τ+1Q2-Z1-Pi,p∗∗Z1-Q2-2Z1∗00-γ2I,Ξ21=0Z10A~p-IB~upKiI1-I2LiC~pB~ωp,Ξ22=Q1-Z1∗0-Y1,Ξ31=λi0πp1A~pλi0πp1B~upKiI1-I2LiC~pλi0πp1B~ωp⋮⋮⋮λiτπpNA~pλiτπpNB~upKiI1-I2LiC~pλiτπpNB~ωp,Ξ33=diag⁡-F0,1,…,-Fτ,N,holds for all i,j∈M,p,q∈G, system (5) is stochastically stable with H∞ performance index γ.Proof.
From (19), we can obtain(30)EΔVξk,τk,θk∣τk=i,θk=p+ErekTrek-γ2EωkTωk≤ςkTΩ-ςk,where(31)Ω-=Ω11∗∗∗Ω21Ω-22∗∗Ω31Ω32Ω33∗0Z10-Q1-Z1,Ω-22=B~upKiI1-I2LiC~pTP-j,qB~upKiI1-I2LiC~p+τ2B~upKiI1-I2LiC~pTZ1B~upKiI1-I2LiC~p-Q2-2Z1+VC~pTVC~p,Ω31=B~pTP-j,qA~p,Ω32=B~pTP-j,qB~upKiI1-I2LiC~p-I3TVC~p,Ω33=B~pTP-j,qB~p+I3TI3-γ2I,ςkT=ξkTξk-iTωkTξk-τT.If Ω-<0, from (30) and under zero-initial condition, we have E{∑k=0∞rekTrek}<γ2E{∑k=0∞ωkTωk}.
By Schur compliment,Ω-<0 is equivalent to (32)Ξ11∗∗Ξ21Ξ-22∗Ξ310Ξ-33<0,where(33)Ξ-22=Q1-Z1∗0-Z1-1,Ξ-33=diag⁡-P0,1-1,…,-Pτ,N-1.Letting Pj,q-1=Fj,q,j∈M,q∈G,Z1-1=Y1, (27) and (28) can be obtained, which completes the proof.In Corollary7, the conditions are a set of LMIs with some inversion constraints. Though they are nonconvex which cannot be solved by using the existing convex optimization tool, we can use the cone complementarity linearization (CCL) algorithm [27] to transform this problem into the nonlinear minimization problem as follows:(34)mintr⁡∑j=0τ∑q=1NPj,qFj,q+Z1Y1s.t.27,35,(35)Pj,qIIFj,q>0,j∈M,q∈G,Z1IIY1>0.Furthermore, the iterative algorithm which can be used to calculate the controller gain Ki, FD gain matrix Li, and the minimal H∞ attenuation γmin is given bellow.Algorithm 8.
Step  1. Let γ=γ0 and set the maximum iterations number as nmax.
Step  2. Find a feasible solution satisfying (27), (35) and set it as (Pj,s0,Fj,s0,Z10,Y10,Ki0,Li0). Let k=0.
Step  3. Solve the following LMI optimization problem for variables (Pj,q,Fj,q,Z1,Y1,Ki,Li):(36)mintr⁡∑j=0τ∑q=1NPj,qkFj,q+Pj,qFj,qk+Z1kY1+Z1Y1k,subject  to27,35;set (Pj,sk=Pj,s,Fj,sk=Fj,s,Z1k=Z1,Y1k=Y1,Kik=Ki,Lik=Li).
Step  4. If (27) and (28) are satisfied, let γ=γ-δ,δ>0and return to Step 3. If the number of iterations exceeds nmax, the iteration is terminated.
Step  5. Check γ: if γ=γ0, the optimization problem has no solutions within the maximum iterations number nmax. Otherwise, γmin=γ+δ.Remark 9.
In this paper, we assume that the transition probabilities ofτk and θk are completely known. When transition probabilities of τk and θk are partly unknown, we can separate the unknown ones from the known ones; see [28].


## 4. Numerical Example

In this section, we present an example to demonstrate the effectiveness of the proposed method. Consider the controlled plant with the following parameter:(37)A1=100.250.15,Bu1=0.30.5,Bd1=0.20.1,Bf1=0.230.81,C1=0.10.3,A2=0.650.050.10.3,Bu2=0.20.6,Bd2=0.10.3,Bf2=0.370.32,C2=0.20.3,the system mode θk∈{1,2}, and the transition probability matrix of θk is Ω=0.20.80.90.1. The random time-delay τk∈{0,1} and the transition probability matrix is Λ=0.70.30.60.4. Given V=0.1, by Corollary 7, we can obtain the delay-dependent controller gain matrix Ki, filter gain matrix Li, and γmin as follows:(38)K0=-0.00040.003,L0=0.0481-0.1333,K1=0.0036-0.0020,L1=-0.0342-0.0906,γmin=1.6125.The initial value x-1=00T,x0=-1.11T,x^-1=00T,x^0=1-1.1T,τ-1=τ0=0,θ-1=θ0=0. Assume that the external disturbance dk is uniformly distributed random signal on -0.150.15; when there is no fault, the trajectories of the closed-loop system’s states and the corresponding estimated value are shown in Figures 1 and 2.Figure 1
The statex1 and its estimated value x^1.Figure 2
The statex2 and its estimated value x^2.We can see that the filter can track the states of the system closely. Assume the fault signal is(39)fk=0.5,k=10,…,200,others.The residual evaluation function is adopted as Jk=E{∑ρ=0krρTrρ}, and the FD threshold can be obtained as Jk=supωk∈L2,fk=0⁡E{∑ρ=0krρTrρ}=0.0122. Figures 3 and 4 show one simulation run of the time-delay and the system mode under the transition probability matrices Π and Λ, respectively.Figure 3
The time-delayτk.Figure 4
The system modeθk.Figures5 and 6 show the residual signal rk and the residual evaluation function Jk, respectively, from which we can see that when fault occurs, rk and Jk change obviously. Moreover, it is noticed that J11=0.0114<Jth<J12=0.0170. This means that the fault has been detected at the third time period after it occurs.Figure 5
The residual signalrk.Figure 6
The residual evaluation functionJk and the threshold Jth.


## 5. Conclusion

With the presence of random time-delay introduced by the network, the problem of robustH∞ FD for networked Markov jump systems is investigated in this paper. By constructing delay-dependent FD filter, the closed-loop systems are established. The sufficient and necessary conditions which make the closed-loop system stochastically stable and achieve prescribed H∞ performance are derived. The method of calculating controller, FD filter gain matrices, and the minimal H∞ attenuation level is also obtained. The numerical example shows that the proposed FD filter is both sensitive to the fault and also robust to the exogenous disturbance.



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