The problem of robust fault detection filter (FDF) design and optimization is investigated for a class of networked control systems (NCSs) with random delays. The NCSs are modeled as Markovian jump systems (MJSs) by assuming that the random delays obey a Markov chain. Based on the model, an observer-based residual generator is constructed and the corresponding fault detection problem is formulated as an H∞ filtering problem by which the error between the residual signal and the fault is made as small as possible. A sufficient condition for the existence of the desired FDF is derived in terms of linear matrix inequalities (LMIs). Furthermore, to improve the performance of the robust fault detection systems, a time domain optimization approach is proposed. The solution of the optimization problem is given in the form of Moore-Penrose inverse of matrix. A numerical example is provided to illustrate the effectiveness and potential of the proposed approach.
1. Introduction
Networked control systems (NCSs) are feedback control systems in which sensors, controllers, actuators, and other system components are connected with real-time networks [1, 2]. The new structure has many advantages over conventional control systems, such as low cost, simple installation and maintenance, reliability, and enhanced resource utilization, which make NCSs a promising structure for control systems [3]. However, this structure also brings challenges on NCSs analysis and design, for instance, network induced delay and packet dropout [4–8], which inevitably increase the complexity of system design and degrade the system performance [9]. As an important essential to improve the performance, safety, and reliability of dynamic systems, fault detection technique for NCSs has recently attracted considerable attention [10, 11].
Network induced delay is an active field of NCSs research. So far, there are fruitful results in fault detection for NCSs with various network induced delays [12–16]. In [12], the influence caused by unknown network induced delays is transformed into time-varying polytopic uncertainty. Assisted by parameter-dependent Lyapunov function matrix, an optimal fault detection filter (FDF) is designed to detect faults. In many cases, network induced delays are random and can be modeled as Markov chains [17–20]. In the literature [17], by employing the multirate sampling method together with the augmented state matrix method, NCSs with long random delays are modeled as Markovian jump systems (MJSs). Then based on the model, an H∞ filter is designed for detecting faults. Since all or part of the elements in the desired transition probabilities matrix are hard or costly to obtain, a robust FDF for discrete-time MJSs with partially known transition probabilities is designed in the literature [21]. Moreover, in order to improve the performance of fault detection systems, time domain optimization approaches are proposed for observed-based fault detection systems [22–24].
To the best of authors’ knowledge, the problem of robust FDF design and optimization for a class of NCSs, which can be modeled as MJSs, has not been fully investigated yet. This motivates us to study this interesting and challenging problem, which has great potential in practical applications.
This paper addresses the problem of robust FDF design and optimization for a class of NCSs with random delays and the main achievement is composed of the following four steps. Firstly, the NCSs are modeled as MJSs and the partially known transition probabilities of the Markov process are taken into account. Secondly, an observer-based residual generator is constructed and the robust fault detection problem is formulated as an H∞ filtering problem. A sufficient condition for the existence of the desired FDF is derived in terms of linear matrix inequalities (LMIs). Thirdly, a time domain optimization approach for detecting smaller faults is proposed for the robust fault detection systems. The optimal solution of the problem is given in terms of Moore-Penrose inverse of matrix. Lastly, a numerical example is provided to illustrate the effectiveness and potential of the proposed approach.
2. Problem Formulation
Consider the following continuous-time, state-space model of the linear time-invariant plant dynamics:(1)x˙t=Acxt+Bcut+Bcddt+Bcfftyt=Cxt+Dddt+Dfft,where x∈Rn, u∈Rm, y∈Rl, and f∈Rq denote the state, the control input, the output, and the latent fault, respectively, d∈Rp is the external disturbance belonging to l20,∞, and the real matrices Ac, Bc, Bcd, Bcf, C, Dd, and Df are of appropriate dimensions.
Consider the NCSs as [17]; we introduce the following assumptions.
Assumption 1 (see [<xref ref-type="bibr" rid="B1">1</xref>, <xref ref-type="bibr" rid="B14">14</xref>]).
The sampling period of the NCSs is T. The sensors are clock-driven, and the controllers and actuators are time-division-driven with the same time-division. There are no packet dropout and packet disordering. The control law is fixed. The sensor-to-controller delay is τksc and the controller-to-actuator delay is τkca. τk=τksc+τkca is introduced to denote the network induced delay at time instant k and supposed to be smaller than the sampling period T in this paper.
Assumption 2 (see [<xref ref-type="bibr" rid="B17">17</xref>]).
The sampling interval kT,(k+1)T is divided into N pieces.
From Assumption 1, system (1) can be transformed into the following discrete time model:(2)xk+1=Axk+∑i=01Biτkuk-i+Bdd(k)+Bffk,yk=Cxk+Dddk+Dffk,where A=eAcT, B0(τk)=∫0T-τkeActBcdt, B1(τk)=∫T-τkTeActBcdt, Bd=∫0TeAc(T-t)Bcddt, and Bf=∫0TeAc(T-t)Bcfdt.
Remark 3.
In this paper, we only consider the network induced delay τk<T. In more general case, the network induced delay τk<d1T and τksc<d2T with d1 and d2 being nonnegative integers. System (1) can be written as(3)xk+1=Axk+∑i=0d1Biτkuk-i+Bdd(k)+Bffkyk=Cxk-floorτkscT+Dddk+Dffk,where floor(·) stands for the function rounding towards minus infinity.
From Assumption 2 and similar to [10, 17], we can obtain that the random network induced delay τk=(h(k)-1)T/N, where the sequence {h(k)} can be considered as a discrete-time homogeneous Markov chain taking values in the following finite state space Γ=1,2,…,N and π=[πij] is the stationary transition probability matrix with its elements defined as πij=Probhk+1=j∣h(k)=i>0 and ∑j∈Γπij=1. In addition, for all or part of the elements in π are hard or costly to obtain, we assume that the stationary transition probabilities of the Markov chain in this paper are partially known. For notation clarity [21], ∀i∈Γ, we denote Γki=j:πijisknown, Γuki=j:πijisunknown. Also, we denote that πki=∑j∈Γkiπij throughout the paper.
Considering the control input signal u(k)=-Kx(k), then (2) can be equivalently written as the following MJSs:(4)zk+1=Ai,kzk+B-ddk+B-ffk,yk=C-zk+Dddk+Dffk,where one has z(k)=xT(k)xT(k-1)T, Ai,k=A-B0(τk)K-B1(τk)KI0, B-d=BdT0T, B-f=BfT0T, C-=C0, and the subscript i,k of Ai,k denotes h(k)=i∈Γ at time instant k.
An observer-based FDF is adopted to generate residual signal:(5)z^k+1=Ai,kz^k+Li,kyk-C-z^k,rk=yk-C-z^k,where z^∈R2n is the state estimation vector of z(k), r∈Rl is the generated residual signal, and Li,k is the filter’s gain matrix to be designed.
Set the filter error e(k)=z(k)-z^(k), and then the overall fault detection system is given by(6)ek+1=A-i,kek+B-i,kwkrk=C-ek+D-wk,where A-i,k=Ai,k-Li,kC-, B-i,k=BLdi,kBLfi,k, BLdi,k=B-d-Li,kDd, BLfi,k=B-f-Li,kDf, D-=DdDf, and w(k)=dT(k)fT(k)T.
Remark 4.
In (5) and (6), the subscripts or superscripts i, k have the same meanings as the subscript of Ai,k in system (4).
After the above manipulations, the original robust FDF problem for system (1) can be further converted to find a series of filter gain matrices Li,k such that the MJSs (6) are stochastically stable and under zero initial condition, and the H∞ performance index γ is made as small as possible in the feasibility of [21](7)supw(k)≠0Er(k)2w(k)2<γ2,γ>0.
For improving the performance of the fault detection system (6), we use a time domain optimization approach to optimize the fault detection system (6). Let ξ(k)=Vk(z)r(k)=(Vs,k+Vs-1,kz-1+⋯+V0,kz-s)r(k) denote the modified residual signal [25], where matrix Vk(z) is called the postfilter [22, 25]. Then the residual evaluation function can be selected as(8)Jk=ξke=1β+1∑i=k-βkξTiξi1/2,where β denotes the detection window.
Then the fault can be observed by comparing J(k) with a threshold Jth according to the following logic:(9)Jk>Jth⟹alarmforfaultJk≤Jth⟹nofault.
Remark 5.
Note that the threshold Jth in (9) is the minimum threshold that prevents false alarms and it is also an adaptive threshold which will be shown in the following section.
3. Main Results
In this section, we will discuss the robust FDF design problem of system (4) with partially known transition probabilities and the time domain optimization of fault detection systems (6).
3.1. Filter Gain Design
To finish the filter design based on the MJSs model (6) with partially known transition probabilities, we first introduce the following lemma which will help us to derive the gain of the FDF (5).
Lemma 6 (see [<xref ref-type="bibr" rid="B21">21</xref>]).
Consider system (6) with partially known transition probabilities and let γ>0 be a given scalar. If there exist matrices Pi>0,Gi>0, ∀i∈Γ, such that (10)-Pi0A-i,kΤGiTC-T*-γ2IB-i,kΤGiTD-T**-Gi-GiT+P-j0***-I<0,where asterisk (*) is used to represent a term that is induced by symmetry in symmetric block matrices, and(11)P-j=∑j∈ΓkiπijPjπkij∈ΓkiP-j=Pjj∈Γukithen the system (6) is stochastically stable with an H∞ performance index γ.
As an application of Lemma 6, the following theorem provides a sufficient condition for the existence of an admissible H∞ FDF in the form of (5).
Theorem 7.
Consider system (6) with partially known transition probabilities and let γ>0 be a given scalar. If there exist matrices Pi>0, Gi>0, and ∀i∈Γ and matrices Ki, ∀i∈Γ, such that(12)-Pi00Ai,kΤGiT-C-TKiTC-T*-γ2I0B-dΤGiT-DdΤKiTDdT**-γ2IB-fΤGiT-DfTKiTDfT***-Gi-GiT+P-j0****-I<0,where Ai,k, C-, and P-j are defined in (4) and (11), then the system (6) is stochastically stable with an H∞ performance index γ. Moreover, the filter gains of an admissible H∞ FDF in the form of (5) are given by Li,k=Gi-1Ki, ∀i∈Γ.
Proof.
From (6), we can replace A-i,k,B-i,k,D- in (10) by A-i,k=Ai,k-Li,kC-, B-i,k=B-d-Li,kDdB-f-Li,kDf, and D-=DdDf, respectively. Then, (10) in Lemma 6 can be written as(13)-Pi00Ai,kTGiT-C-TLi,kTGiTC-T*-γ2I0B-dTGiT-DdTLi,kTGiTDdT**-γ2IB-fTGiT-DfTLi,kTGiTDfT***-Gi-GiT+P-j0****-I<0.Define Ki=GiLi,k; (13) can be transformed into (12). This completes the proof.
Remark 8.
The optimal H∞ performance index γ* and the corresponding filter gains can be obtained by setting δ=γ2 and solving the following optimization problem:(14)Minimize:δ,s.t.(12).
3.2. Determination of Threshold
From Section 3.1, we can obtain the residual signal r(k), and then the modified residual signal ξ(k) can be generated by using r(k) as the input of the postfilter. According to the system (6) and the definition of ξ(k), we can rewrite ξ(k) in the following compact form [22]:(15)ξk=VkHd,kdsk+Hf,kfsk,where(16)Hf,k=Df0⋯0gf(k-s+1,k-s)Df⋯0⋮gf(k,k-s)⋯gf(k,k-1)Df,fsk=fk-sfk-s+1⋮fk,Hd,k=ge(k-s,k-s)Dd0⋯0ge(k-s+1,k-s)gd(k-s+1,k-s)Dd⋯0⋮ge(k,k-s)gd(k,k-s)⋯gd(k,k-1)Dd,ds(k)=e(k-s)d(k-s)⋮d(k),gfk,j=C-Φk,j+1BLfi,j,gek,j=C-Φk,j,gdk,j=C-Φk,j+1BLdi,j,Φk,j=∏a=jk-1A-i,a,Φk,k=Ι,k-s≤j≤k-1,Vk=V0,kV1,k⋯Vs,k.
Remark 9.
From (16), we know that the matrices Hf,k and Hd,k are time-varying matrices since they are constructed by BLfi,j, BLdi,j, and A-i,j(i∈Γ,k-s≤j≤k-1) which are influenced by the mode of system (6). We can obtain the mode and these matrices online at each time instant. It also should be noted that the selection of index s which is the order of the postfilter Vk(z) is arbitrary in principle, but, in this paper, considering the computational complexity of online implementation, we set it equal to 2n.
Once the modified residual signal has been generated and the residual evaluation function has been selected, we can determine the threshold Jth. From (8) and (15), we have(17)Jth=supd,f=0Jk=supd,f=0ξke=supd,f=01β+1∑i=k-βkξTiξi1/2.
It should be pointed out that the threshold defined in (17) is the minimum threshold that prevents false alarms. It follows from (15) and (17) that(18)Jth=supd1β+1∑j=k-βkVkHd,kdsjTVkHd,kdsj1/2≤σ-VkHd,ksupd1β+1∑j=k-βkdsTjdsj1/2,where σ-(·) denotes the maximum singular value.
From (16), we can obtain(19)∑j=k-βkdsTjdsj=∑j=k-βkeTj-sej-s+∑n=0sdTj-ndj-n.
According to (6), we have [22](20)∑j=k-βkeTj-sej-s1/2≤β+11/2ejωI-A-i,k-1BLdi,k∞dke≤β+11/2supi∈Γ,ωσ-ejωI-A-i,k-1BLdi,kΔd,where Δd≥d(k)e, and note that(21)∑j=k-βk∑n=0sdTj-ndj-n1/2≤β+1s+11/2Δd.
So according to (18)–(21), the threshold can be defined as(22)Jth=σ-VkHd,ks+1+λd1/2Δd,where λd=supi∈Γ,ωσ-((ejωI-A-i,k)-1BLdi,k)2.
Note that Vk,Hd,k vary with the mode of system (6), so Jth is an adaptive threshold which can be obtained online.
3.3. Optimization of Fault Detection Systems
The objective of optimizing the fault detection system (6) is to seek a performance index in order to detect faults as small as possible. For describing the performance index, we first give the following definitions [22].
Definition 10.
The set of detectable faults which are denoted by Sf can be expressed by(23)Sf=f∣infdξe≥Jth=f∣VkHf,kfske≥2Jth.
Definition 11.
Minimum detectable faults, denoted by fmin, are faults which belong to Sf and minimize infdξe. So an fmin can be obtained by solving the following extreme problem:(24)inff∈Sf,dξe=Jth.
Definition 12.
Maximal minimum detectable faults, denoted by fmmin, are defined by(25)fmmin2,[0,β]=∑a=-β0fmminT(k+a)fmmin(k+a)1/2=maxfmin∑a=-β0fminT(k+a)fmin(k+a)1/2.Note that the smaller fmmin becomes, the more faults can be detected. So our objective of optimizing can be formulated as(26)minVkJ=minVkfmmin2,[0,β].Followed from (23) and (24), the minimum detectable faults fmin ensure that(27)1β+1∑a=-β0VkHf,kfs,mink+aTVkHf,kfs,mink+a1/2=2Jth,where one has fs,min(k+a)=fminT(k+a-s)fminΤ(k+a-s+1)⋯fminΤ(k+a)T.
Since (28)1β+1∑a=-β0VkHf,kfs,mink+aTVkHf,kfs,mink+a1/2≥σ_VkHf,k×1β+1∑a=-β0fs,mink+aTfs,mink+a1/2=σ_VkHf,k×1β+1∑a=-β0∑b=0sfminTk+a-bfmink+a-b1/2=σ_VkHf,k×1β+1∑b=0s∑a=-β0fminTk+a-bfmink+a-b1/2,where σ_(·) denotes the minimum singular value, then we have(29)1β+1∑b=0s∑a=-β0fminT(k+a-b)fmin(k+a-b)1/2≤2Jthσ_VkHf,k.
It is evident that the equality in (29) holds true only if vectors fmin(k+a-b), b=0,…,s, satisfy(30)fmink+a=fmink+a-1=⋯=fmink+a-sand are equal to the eigenvector of matrix (VkHf,k)TVkHf,k corresponding to σ_2(VkHf,k). According to the definition of fmmin, we finally have(31)maxfmin∑a=-β0fminTk+afmink+a1/2=2Jthσ_VkHf,kβ+1s+11/2=2Δdβ+11/21+λds+11/2σ-VkHd,kσ_VkHf,k.
Thus, we can know that the objective of optimizing system (6) is reduced to finding matrices Vk at each time instant that solve the following optimization problem:(32)minVkJ=minVkfmmin2,[0,β]=2Δdβ+11/21+λds+11/2minVkσ-VkHd,kσ_VkHf,k.
Next, we give the following lemma that plays a key role in deriving the solution of optimization problem (32).
Lemma 13 (see [<xref ref-type="bibr" rid="B22">22</xref>]).
Given matrices H,P of appropriate dimensions, then the optimal solution X for optimization problem minXσ-(P+XH) is given by(33)X=-PH+,and furthermore(34)minXσ-P+XH=σ-P-XH+H,where H+ denotes the pseudoinverse or Moore-Penrose inverse of matrix H.
Based on Lemma 13, we have the following theorem to determine the optimal solution for problem (32).
Theorem 14.
Given time-varying matrices Hd,k, Hf,k with rank(Hf,k)=q(s+1)=α which are defined as (16) at each time instant, then the optimal solution Vk* for (32) is given by(35)Vk*=Hf,k--Hf,k-Hd,kHfn,kHd,k+Hfn,kwithHf,k-Hf,k=Iα×α,Hfn,kHf,k=0.
Furthermore,(36)minVkJ=minVkfmmin2,0,β=2Δdβ+11/21+λds+11/2×σ-Hf,k-Hd,kI-Hfn,kHd,k+Hfn,kHd,k,where (·)+ denotes the Moore-Penrose inverse.
Proof.
From Theorem 7 and (22), we know that λd is a constant which can be calculated offline. So the original optimization problem (32) is equivalent to the following time-varying optimization problem:(37)minVkσ-(VkHd,k)σ_(VkHf,k).
For deriving the solution of problem (37), at each time instant, we set(38)Vk*=X1,kHf,k-+XkHfn,k,and substitute it into (37)(39)σ-Vk*Hd,kσ_Vk*Hf,k=σ-X1,kHf,k-+XkHfn,kHd,kσ_X1,kHf,k-+XkHfn,kHf,k=σ-X1,kHf,k-+XkHfn,kHd,kσ_X1,k,where X1,k∈Rα×α and rank(X1,k)=α, Xk is arbitrarily selectable. Note that(40)σ-X1,kHf,k-+XkHfn,kHd,kσ_X1,k≥σ_X1,kσ-Hf,k-Hd,k+X1,k-1XkHfn,kHd,kσ_X1,k=σ-Hf,k-Hd,k+X1,k-1XkHfn,kHd,k,and the equality holds true if and only if(41)σ-X1,k=σ_X1,k⟺X1,k=Iα×α.Thus, we finally have(42)minVkσ-VkHd,kσ_VkHf,k=minX1,k,Xkσ-X1,kHf,k-+XkHfn,kHd,kσ_X1,k=minXkσ-Hf,k-Hd,k+X1,k-1XkHfn,kHd,k.Using Lemma 13, we can obtain(43)Xk=-Hf,k-Hd,kHfn,kHd,k+.Hence, the optimal solution Vk* for (32) is given by(44)Vk*=Hf,k--Hf,k-Hd,kHfn,kHd,k+Hfn,k.Furthermore, substituting Vk* into (32) leads to (36). This completes the proof.
Remark 15.
From Remark 9 and (35), we can know that the optimal solution Vk* for (32) is time-varying and can be obtained online at each time instant. As a result, the postfilter Vk(z) is time-varying as well. That is different from the conventional approach in [22], in which the postfilter Vk(z) is time-invariant.
Remark 16.
Note that if Hf,k is a full rank square matrix which is a special case that is often met, we have Hf,k-=Hf,k-1, Hfn,k=0. Thus, the optimal solution Vk*=Hf,k-1.
3.4. Summary
The following Algorithm 17 summarizes the essential parts of this section and the approach proposed above for the FDF system design.
Algorithm 17.
Consider the following steps.
Step 1. Solve the optimal H∞ problem in Theorem 7 and Remark 8 for Li,k,∀i∈Γ.
Step 2. Generate residual signal r(k) from FDF (5).
Step 3. From (16), form Hf,k,Hd,k, and Hy,k.
Step 4. Find Hf,k-,Hfn,k with Hf,k-Hf,k=Iα×α, Hfn,kHf,k=0, and calculate Hfn,kHd,k+.
Step 5. Set the optimal postfilter Vk*=Hf,k--Hf,k-Hd,k(Hfn,kHd,k)+Hfn,k.
Step 6. Establish the adaptive threshold Jth=σ-(VkHd,k)(s+1+λd)1/2Δd.
Step 7. Let ξ(k)=Vk*r(k) denote the modified residual signal; then the residual evaluation function is(45)J(k)=ξke=1β+1∑i=k-βkξTiξi1/2.From Algorithm 17, it can be easily known that these steps are implemented online except Step 1.
4. Numerical Example
In this section, a numerical example is given to show the effectiveness of the proposed method. Consider the following continuous dynamics model:(46)x˙t=01-3-4xt+21ut+0.10.05dt+21ft,yt=10xt+0.1dt+0.8ft.
We choose the sampling period of NCSs as 0.3 s and the division of the sample interval as N=3; then it is easily obtained that the Markov chain h(k)∈123. The initial mode is set to be τ0=0, and the detection window β=10. For k=0,1,…,200, the external disturbance d(k) is supposed to be a random noise uniformly distributed over -0.5,0.5, and the fault signal f(k) is given as(47)f(k)=0.13,fork=100,101,…,2000others.The discrete control law and the transition probability matrices are given as(48)K=1.6918-0.0623T,π1=0.50.40.10.20.60.20.20.30.5,π2=0.5????0.20.20.30.5,π3=?????????,where “?” denotes the inaccessible elements of the matrices. So π1, π2, and π3 denote the transition probability matrix with completely known transition probabilities (Case 1), partially known transition probabilities (Case 2), and completely unknown transition probabilities (Case 3), respectively.
Considering Case 1 as the practical one for the other two cases, we can generate a possible evolution of system modes as shown in Figure 1.
Mode evolution.
Then according to Theorem 7, the filter’s gain matrices for the three cases of the observer (5) are, respectively, given by(49)Case1:L1,k=0.7505-0.02310.00740.0053,L2,k=0.7533-0.02650.00980.0023,L3,k=0.7550-0.02680.0098-0.0001,Case2:L1,k=0.7487-0.02310.01130.0129,L2,k=0.7505-0.02770.01200.0007,L3,k=0.7552-0.02960.01430.0002,Case3:L1,k=0.7312-0.03340.02880.0028,L2,k=0.7431-0.03860.03170.0040,L3,k=0.7548-0.04450.03930.0010.
Accordingly, Figure 2 shows the generated residual signals r(k) for three different cases, and Figures 3, 4, and 5 present the evolution of J(k) and the corresponding threshold Jth, respectively, for three different transition probability matrices. In order to show the time steps Nd for the fault detection in different case, the corresponding enlarged figures are given in Figures 3–5 as well.
Residual signals r(k) for three different cases.
Evaluation of J(k) and threshold Jth for case 1.
Evaluation of J(k) and threshold Jth for case 2.
Evaluation of J(k) and threshold Jth for case 3.
In order to compare the performance of detection systems in three different cases before and after optimization, the minimum detectable faults fmin for the six different conditions are obtained by 500 times simulation. For example, when the fault signal f(k) is given as(50)fk=0.12,fork=100,101,…,200,0others,the evolution of J(k) and the corresponding threshold Jth in Case 3 are shown in Figure 6. It is obvious from Figure 6 that the FDF system with optimization can detect the given fault but the FDF system without optimization cannot.
Evaluation of J(k) and threshold Jth for case 3 when f(k) is given as (50).
Based on the path in Figure 1 and the selected threshold Jth, the optimal H∞ performance index γ* by Theorem 7, the time steps Nd for the fault detection by the evaluation function J(k) and logic (9), and the minimum detectable faults fmin can be obtained and given in Table 1.
The optimal H∞ performance index γ*, time steps Nd, and the minimum detectable faults fmin for three different cases.
Case
γ*
With optimization
Without optimization
Nd
fmin
Nd
fmin
1
0.8108
4
0.1053
5
0.1204
2
0.8121
5
0.1056
6
0.1211
3
0.8225
5
0.1057
6
0.1215
Obviously, it can be seen from Figures 3–6 and Table 1 that the fault detection systems with optimization can detect the smaller faults and need less time steps than the system without optimization; that is, the fault detection systems with optimization have a better performance than the system without optimization for each case. It also can be depicted from Figures 3–5 and Table 1 that the more transition probability knowledge we have, the better H∞ performance index can be achieved, the less time is needed, and the smaller faults can be detected. Therefore, our design and optimization approaches for robust fault detection systems actually build a tradeoff in practice between the complexity to obtain transition probabilities and the performance benefits and efficiency of detection.
5. Conclusion
The problem of observer-based robust FDF design and optimization for NCSs with random delays is investigated in this paper. A MJSs model has been developed by assuming the random delays to obey a Markov chain, and the partially known transition probabilities of the Markov process are taken into account. Based on the developed model, an H∞ FDF is derived in terms of LMIs. Furthermore, to improve the performance of the FDF, a time domain optimization approach is proposed for the robust fault detection systems. The optimal solution of the problem is given in the form of Moore-Penrose inverse of matrix. Finally, a numerical example has been given to demonstrate the effectiveness and potential of the proposed approach. Some extensions of the present method are under investigation. For example, the problem of FDF design and optimization for NCSs with unknown delays or packet dropout needs to be further studied. Besides, in order to reduce the computation of the fault detection algorithm and enhance the engineering practicability of the time domain optimization approach, a recursive algorithm is worth forthcoming investigation.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work was supported by the National Natural Science Foundation of China (60974014 and 61074027).
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