# Fault Detection Filter Design of Polytopic Uncertain Continuous-Time Singular Markovian Jump Systems with Time-Varying Delays

Authors: Yunling Shi; Xiuyan Peng
Journal: Complexity
 (2020)

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## Abstract

This paper investigates the problem of full-order and reduced-order fault detection filter (FDF) design under unified linear matrix inequality (LMI) conditions for a class of continuous-time singular Markovian jump systems (CTSMJSs) with time-varying delays and polytopic uncertain transition rates. By constructing a new Lyapunov function, sufficient conditions are firstly provided for the singular model error augmented system such that the system is stochastically admissible with anH∞ performance level γ. And then, by applying a novel convex polyhedron technique to decoupled linear matrix inequalities, the full-order and reduced-order fault detection filter parameters can be obtained within a convex optimization frame. The reduced-order fault detection filter (FDF) can not only meet the fault detection accuracy requirements of complex systems but also improve the fault detection efficiency. Finally, a DC motor and an illustrative simulation example are given to verify the feasibility and effectiveness of the proposed algorithms.


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## Body

## 1. Introduction

Markovian jump systems (MJSs) are usually defined as a family of complex random jumping parameter systems [1–4]. The past few decades have witnessed the prosperous research on MJSs because it can simulate the structure and parameter changes of real dynamic systems, such as biological economics systems, network control systems, power electronic systems, and mechanical engineering systems [5–9]. It is known that singular systems are completely different from ordinary systems, and when dealing with the stability of singular systems, we need to consider its regularity, the unique impulse-free solution, and admissibility, which are not considered in ordinary systems [10–17]. However, the singular systems have attracted much attention, and many studied results about singular systems have been reported because of their extensively practical applications, such as the filtering problem investigated in [10, 11], H∞ control problem studied in [12], stabilization problem addressed in [13], dissipative control problem investigated in [14], dynamic output feedback control problem considered in [15], and sliding mode control presented in [16].It is worth mentioning that when the structure or parameters of the singular dynamic hybrid system change, the singular Markovian jump systems (SMJSs) can effectively simulate this phenomenon by using the Markov chain transformation, and many fruitful about SMJSs have been shown in recent years [17–19]. The transition rates (TRs) are the concrete expression of Markov chains, which can express the random transition state of the actual dynamic systems [20–22]. As the soul of SMJSs, TRs are generally considered to be completely known, partly unknown, and completely unknown in most studies [23–25]. It is worth noticing that the TRs with the polytopic uncertain cases are often encountered in practical applications, when the switching mode of the system is uncertain. Due to limitation of measurement conditions under a harsh environment, component failures, and random abrupt changes of system structures, the system parameters changed randomly such that the RTs may be polytopic uncertain systems [26–28], for example, network control systems, VTOL (vertical take-off and landing machine) systems, and power systems. However, the fault detection filter of polytopic uncertain SMJSs has not been studied so far, which is the first motivation of our study.In practice, however, the components or instrument are not always normal, and faults are unavoidable under practical conditions. More generally, the faults are often encountered in various forms. In order to improve the safety performance of the control systems, when the systems fail, we need to quickly detect the faults and avoid accidents. In order to pursue higher security and reliability, many scholars have studied the fault detection problems of Markovian jump systems [29–31]. These scholars only studied the fault detection problems of the ordinary Markovian jump systems. Since the singular Markovian jump systems are more complex than the ordinary Markovian jump systems, the stability, regularity, and nonimpulsiveness need to be considered [32–35]; especially for polytopic uncertain continuous-time singular Markovian jump systems, few results can be found related to the fault detection filter designed problems under this case.Additionally, it has been shown that time-varying delays are universal in real applications because of the signal transmission in communication, and it can lead to systems performance degradation and instability [34–38]. Inevitably, time delays also exist in SMJSs, and it is necessary to investigate the SMJSs with time-varying delays. As a hot topic, many fruits involving time-varying delays of SMJSs fault detection have been addressed [33–35]. It is worth mentioning that the fault detection of polytopic uncertain CTSMJSs with time-varying delays, which contains completely known, partly unknown, and completely unknown TRs, has not been studied. The polytopic uncertain SMJSs with time-varying delays can summarize actual systems more comprehensively, which is the other main motivation of this paper.From what has been discussed above, in this paper, we will tackle the stochastic admissibility problem for polytopic uncertain CTSMJSs with time-varying delays via constructing a novel Lyapunov function. Then, based on a convex polyhedron technique to decoupled linear matrix inequalities, the full-order and reduced-order fault detection filters can be obtained. In particular, full-order and reduced-order FDF parameters can be computed in a convex optimization frame. On the premise of satisfying the requirement of fault detection accuracy, the proposed reduced-order FDF can effectively reduce the data storage and is less sensitive to external disturbance. The effectiveness of the derived results has been demonstrated by two illustrative examples.The main contributions and novelties of this paper are summarized as follows. (I) The full-order and reduced-order FDFs for polytopic uncertain CTSMJSs with time-varying delays are proposed and designed for the first time. (II) The innovation reduced-order FDF design method of polytopic uncertain CTSMJSs with time-varying delays is proposed, which satisfies the stochastically admissible sufficient conditions and theH∞ performance standard γmin for the first time. (III) By applying a novel decoupling technique, it has shown the full-order and reduced-order FDFs designed under a unified framework. Therefore, we can get the parameters of the full-order and reduced-order FDFs by adjusting the dimension of the decoupling matrix parameters, which can effectively improve the full-order and reduced-order FDF design scheme. (IV) In this paper, the proposed method not only can calculate parameters of full-order and reduced-order FDFs as the state TRM is polytopically uncertain but also can obtain the parameters of full-order and reduced-order FDFs when state TRM is completely under known, partly unknown, and completely unknown circumstances, and their performances are compared on conservative by simulation analysis.Notations. The notations presented here are standard. E⋅ stands for the mathematical expectation. symK stands for K+KT. In represents the n-dimensional identity matrix, and 0 represents the zero matrix. R is the set of real number, Rndenotes the n-dimensional Euclidean space, and Rm×n is the set of m×n real matrix. About real symmetric matrix P, P>0P≥0 means that P is the real symmetric positive (semipositive) and ∗ represents the symmetric term that has been ellipsis. ⋅ denotes the Euclidean norm for vectors. l20,∞ is the space of square summable infinite sequence, and for w=wt∈l20,∞, its norm is given by w2=∫0∞w2dt.


## 2. Problem Formulation

In this section, we consider a class of CTSMJSs with time-varying delays on a complete probability space, which is described as follows:(1)Ex˙t=Artxt+Adrtxt−dt+Brtut+Ertwt+Frtft,yt=Crtxt+Cdrtxt−dt+Drtwt+Hrtft,where ut∈Rnu represents the known input; xt∈Rnx stands for the plant state; wt∈Rnd is the exogenous disturbance signal; and ft∈Rnf is the fault signal to be detected. E∈Rn×n with rankE=r≤nx; ut, wt, and ft, respectively, belong to l20,∞. In addition, dt denotes the time-varying delay and satisfying 0≤d1≤dt≤d2<∞, d˙t≤u<∞, for all t≥0, where d1, d2, and u are scalars, d1,d2∈R+ represents the lower and upper delay bounds, respectively. rt,t≥0 is a right-continuous Markov process, which takes values in a finite state-space set S=1,2,…,N with the mode transition rate matrix (TRM) Π≜λij,i,j∈S, given by(2)Prrt+Δ=jrt=i=λijΔ+0Δ,i≠j,1+λiiΔ+0Δ,i=j,where Δ>0, limΔ⟶00Δ/Δ=0, and λij≥0 is the transition rate from mode i at time t to mode j at time t+Δ for i≠j. For convenience, let rt=i, such that real known constant system matrices Art,Adrt,Brt,Crt,Cdrt,Drt,Ert,Frt,Hrtcan be written as Ai,Adi,Bi,Ci,Cdi,Di,Ei,Fi,Hi. The TRs of the Markovian jump process are considered to be partly accessible and polytopic uncertain; that is, the TRM Λ=λijn×n is hypothesized to belong to a known polytope PΛ with vertices Λs:(3)PΛ≔Λ=∑S=1MαsΛs;αs≥0,∑S=1MαS=1,where Λs=λijn×n, i,j∈I, S=1,2,…,M are known given TRMs containing known, unknown, and polytopic uncertain elements still. For instance, the TRM of system (1) with three operation modes may be expressed as(4)λ11λ˜12λ˜13λ14λ^21λ22λ^23λ24λ31λ32λ^33λ^34λ41λ˜42λ^43λ44,where unknown and polytopic uncertain TRs are tagged with hat “ ˆ ” and “ ˜ ”, respectively, and the others are known as TRs. For notational clarity, we describe S=Ski∪Suki∪Suci as follows:(5)Ski≔j:λijis known,Suki≔j:λ^ijis unknown,Suci≔j:λ˜ijis uncertain.Also, we can denote(6)λukil≔∑j∈Sukiλ^ij=1−∑j∈Skiλij−∑j∈Suciλ˜ijl.Remark 1.
We defineλki≔∑j∈Skiλij and λucil≔∑j∈Suciλ˜ijl. For achievement, the numerical solvability of the performance analysis, when λ^iiis unknown, by providing a lower bound element λbi≤−λki+λucil is provided to restrict the diagonal element of TRM. In this paper, our primary mission is to design FDF for original system (1), and the FDF is considered as follows:(7)x˙Ft=AFixFt+BFiyt,rFt=CFixFt+DFiyt,where rFt∈Rnr is the residual, xFt∈Rnf1≤nf≤nx is the state estimation of filter, and the full-order and reduced-order FDF parameter matrices AFi∈Rnf×nf, BFi∈Rnf×ny, CFi∈Rnr×nf, and DFi∈Rnr×ny can be computed, respectively, when 1≤nf=nx or 1≤nf<nx.
In this paper, we introducef¯s=Wfsfs to improve the performance of the fault detection system by limiting the frequency interval, where fs represents the weighting fault signal and Wfs represents the stable weighting matrix function. fs and f¯s denote Laplace transforms of f¯t and ft, separately. The minimal state-space realization of f¯s=Wfsfs is supposed to be(8)x˙ft=Awfixft+Bwfift,f¯t=Cwfixft+Dwfift,where xft∈Rnw is the state vector and Awfi, Bwfi, Cwfi, and Dwfi are arbitrary constant matrices. Notice that et≔rFt−f¯t denotes the weighted fault estimation error. Considering expressions (1), (7), and (8), the following singular model error augmented system can be obtained:(9)E˜x˜˙t=A˜ix˜t+A˜diΦx˜t−dt+B˜iΨt,et=C˜ix˜t+C˜diΦx˜t−dt+D˜iΨt,where Ψt=uTtwTtfTtT, x˜t≔xTtxFTtxfTtT, and(10)E˜=E000I000I,A˜i=Ai00BFiCiAFi000Awfi,A˜di=AdiBFiCdi0,B˜i=BiEiFi0BFiDiBFiHi00Bwfi,C˜i=DFiCiCFi−Cwfi,C˜di=DFiCdi,D˜i=0DFiDiDFiHi−Dwfi,Φ=I00.Remark 2.
u
t is the known input of the system, which is combined with the disturbance wtand fault ft to form the input of the singular model error augmented system (9) because when ut=0, the H∞ performance γ and conservativeness of the system will decrease, and it can be regarded as the negative input of the system.Remark 3.
In fact, the singular model error augmented system (9) has the characteristics of singular system. In order to better study the stability of system (9), we must ensure the solution is reasonable for the singular model error augmented system (9) when the input Ψt=0. Therefore, we give the following Assumption 1 and Lemma 1.Assumption 1.
For everyi∈I, there exists a pair of nonsingular matrices M,N∈Rnx+nf+nw×nx+nf+nw such that(11)ME˜N=Ir+nf+nw0r+nf+nw×nx−r0nx−r×r+nf+nw0nx−r×nx−r,MA˜iN=A˜i1A˜i2A˜i3A˜i4,MA˜diN=A˜di1A˜di2A˜di3A˜di4.Lemma 1.
Under Assumption1, the unforced singular model error augmented system (9) has a unique (right-continuous) solution.Proof.
Letςt=N−1x˜t=ς1Ttς2Tt, ς1t∈Rr+nf+nw, and ς2t∈Rnx−r, then unforced subsystem (9) can be written as the following form:(12)ς˙1t=A˜i1ς1t+A˜i2ς2t+A˜di1ς1t−dt+A˜di1ς2t−dt,0=A˜i3ς1t+A˜i4ς2t+A˜di3ς1t−dt+A˜di4ς2t−dt,orς˙1t=A˜i1ς1t+A˜di1ς1t−dt,0=A˜i3ς1t++A˜di3ς1t−dt,ς2t=0.
Therefore, according to [35–38], the unforced input system (9) has a unique impulse-free solution for all t≥0 when Assumption 1 holds. Before proceeding to the next step, we introduce the following important definitions to drive our main results.Definition 1 (see [11]).
The model error augmented system (9) is said to be as follows:(1)
Regular, if anyi∈I and Ψt=0, detsE˜−A˜i≠0.(2)
Impulse-free, if anyi∈I and Ψt=0, degdetsE˜−A˜i=rankE˜.(3)
Stochastically stable, if for any initial statex˜0∈Rnx, r0∈I, there exists a scalarMx˜0,r0>0, such that the following inequality holds:(13)E∫0∞x˜t2x˜0,r0≤Mx˜0,r0.(4)
Stochastically admissible, if system (9) is regular, impulse-free, and stochastically stable.Definition 2 (see [11]).
Given a scalarγ>0 and under the condition Ψt∈l20,∞, singular model error augmented system (9) is said to be stochastically admissible and has an H∞ noise attenuation performance index γ, if the following conditions can be satisfied.(1)
WhenΨt=0, t≥0, and condition (3) of Definition 1 is satisfied, system (9) is stochastically stable.(2)
WhenΨt≠0, t≥0, under zero initial conditions, the following inequality holds:(14)E∫0∞eTtetdt<γ2E∫0∞ΨTtΨtdt.We assume that polytopic uncertain CTSMJSs with time-varying delays (1) is stochastic admissible. The main purpose of this paper is to obtain the parameters of the fault detection filtering system (7) under the premise that the model error augmented system (9) is stochastically admissible with the H∞ performance level γ. To improve fault detection performance, we define the following residual estimation function Jr^t and threshold Jth:(15)Jr^t=∫k0k0+Lr^Ttr^t1/2,Jth=sup0≠w∈l2,0≠ut∈l2,f=0EJr^t,where k0 represents the moment of initial evaluation and L represents the evaluation time step. If fault ft has occurred, it can be detected by the following logical relationships Jr^t>Jth⟹fault happened ⟹alarm and Jr^t≤Jth⟹fault free.In order to facilitate subsequent research, we will give the following Jensen inequality.Lemma 2 (see [39]).
For any positive-definite symmetric matrixM∈Rn×n, scalar b≥a, and vector function: v:a,b⟶Rn, such that the following inequality holds:(16)∫abvsdsTM∫abvsds≤b−a∫abvTsMvsds.


## 3. Performance Analysis of FDFS

In this section, theH∞ performance analysis level is firstly derived for system (9). Then, present the FDF design method for CTSMJSs with completely known TRs and time-varying delays.Theorem 1.
Under Assumption1, given scalars d1, d20≤d1≤d2, u≥0, for any time-varying delays dt, the continuous-time singular error augmented system (9) is stochastically admissible with an H∞ performance γ. If there exists symmetric positive-definite matrices Pi∈Rnx+nf+nw×nx+nf+nw, Z1,Z2,R1,R2,Q1i,Q2i,Q3i∈Rnx×nx, appropriate dimensions matrices N˜i,S˜∈Rnx+nf+nw×nx−r, and satisfying E˜TS˜=0, such that the following LMIs hold for any i,j∈I:(17)∑j=1NλijQ1j+Q3j≤Z2,∑j=1NλijQ2j≤Z1,∑j=1NλijQ3j≤Z2,(18)Ωi=Ω11Ω12E˜TΦTR1ΦE˜0Ω15Ω16C˜iT∗Ω22E˜TΦTR2ΦE˜E˜TΦTR2ΦE˜0Ω26ΦTC˜diT∗∗Ω330000∗∗∗Ω44000∗∗∗∗−γ2IΩ56D˜iT∗∗∗∗∗Ω660∗∗∗∗∗∗−I<0,Ω11=∑j=1NλijE˜TPjE˜+symE˜TPiA˜i+symA˜iTS˜N˜iT+ΦTQ1i+Q2i+Q3i+d1Z1+d2Z2Φ−E˜TΦTR1ΦE˜,Ω12=E˜TPiA˜diΦ+N˜iS˜TA˜diΦ,Ω15=N˜iS˜TB˜i+E˜TPiB˜i,Ω16=A˜iTΦTd12R1+d122R2Φ,Ω22=−1−uΦTQ1iΦ−2ETΦTR2ΦE,Ω26=ΦTA˜diTΦTd12R1+d122R2Φ,Ω33=−ΦTQ2iΦ−ETΦTR1+R2ΦE,Ω44=−ΦTQ3iΦ−ETΦTR2ΦE,Ω56=B˜iTΦTd12R1+d122R2Φ,Ω66=−ΦTd12R1+d122R2Φ.Proof.
Firstly, the regularity and impulse-free for system (9) are proved. From (18), we have(19)symE˜TPiA˜i+symA˜iTS˜N˜iT<0,
Suppose that Assumption1 is satisfied, rankE˜=r+nf+nw<nx+nf+nw and nonsingular matrices M,N∈Rnx+nf+nw×nx+nf+nw such that(20)M−TPiM−1=Pi1Pi2Pi3Pi4,NTN˜i=N˜i1N˜i2,M−TS˜=0S,where S∈Rnx+nf+nw−r×nx+nf+nw−r; premultiplying and postmultiplying (19) by NT and N, respectively, we get A˜i4TSN˜i2T+N˜i2STA˜i4<0, and it is easy to see that A˜i4 is the nonsingular matrix. According to Lemma 1 and (1) and (2) in Definition 1, the unforced singular model error augmented system (9) is regular and impulse-free. For the stochastic admissible and H∞ property analysis of system (9), under the zero initial condition and suppose dt is differentiable, that is, Ψt=0, let us define x˜ts=x˜t+s,s∈d1,d2, and choose the following Lyapunov functional:(21)Vx˜t,i,t=∑s=14Vsx˜t,i,t,where(22)V1x˜t,i,t=x˜TtE˜TPiE˜x˜t,V2x˜t,i,t=∫t−dttx˜TαΦTQ1iΦx˜αdα+∫t−d1tx˜TαΦTQ2iΦx˜αdα+∫t−d2tx˜TαΦTQ3iΦx˜αdα,V3x˜t,i,t=∫−d10∫t+stx˜TαΦTZ1Φx˜αdαds+∫−d20∫t+stx˜TαΦTZ2Φx˜αdαds,V4x˜t,i,t=d1∫−d10∫t+stx˜˙TsE˜TΦTR1ΦE˜x˜˙sdαds+d12∫−d2−d1∫t+stx˜˙TsE˜TΦTR2ΦE˜x˜˙sdαds,where parameter-dependent Lyapunov matrices Pi=PiT, Ql=QlT>0, l∈1,2,3, Z1=Z1T>0, Z2=Z2T>0 will be determined. Let δ be the weak infinitesimal generator of random process x˜t,rt, for each rt=i,i∈S, then the infinitesimal operator along the state trajectory of system (9) can be obtained:(23)δV1x˜t,i,t=limΔ⟶0+1Δ∑j=1,j≠iNPrrt+Δrt=ix˜Tt+ΔE˜TPjE˜x˜t+Δ+Prrt+Δ=irt=ix˜Tt+Δ×E˜TPiE˜x˜t+Δ−x˜TtE˜TPiE˜x˜t=limΔ⟶0+1Δ∑j=1,j≠iNλijΔ+0Δx˜t+Δx˜˙t+0ΔTE˜TPjE˜x˜t+Δx˜˙t+0Δ+1+λiiΔ+0Δx˜t+Δx˜˙t+0ΔTE˜TPjE˜x˜t+Δx˜˙t+0Δ−x˜TtE˜TPiE˜x˜t=x˜Tt∑j=1NλijE˜TPjE˜x˜t+x˜TtE˜TPiE˜x˜˙t+x˜˙TtE˜TPiTE˜x˜t=x˜Tt∑j=1NλijE˜TPjE˜x˜t+x˜TtE˜TPiA˜ix˜t+A˜diΦx˜t−dt+A˜ix˜t+A˜diΦx˜t−dtTPiTE˜x˜t,δV2x˜t,i,t=x˜TtΦTQ1i+Q2i+Q3iΦx˜t−1−ux˜Tt−dtΦTQ1iE˜x˜t−dt−x˜Tt−d1ΦTQ2iE˜x˜t−d1−x˜Tt−d2ΦTQ3iE˜x˜t−d2≤x˜TtΦTQ1i+Q2i+Q3iΦx˜t−1−ux˜Tt−dtΦTQ1iE˜x˜t−dt−x˜Tt−d1ΦTQ2iE˜x˜t−d1−x˜Tt−d2ΦTQ3iE˜x˜t−d2+∫t−dttx˜TαΦT∑j=1NλijQ1jΦx˜αdα+∫t−d1tx˜TαΦT∑j=1NλijQ2jΦx˜αdα+∫t−d2t−dtx˜TαΦT∑j=1NλijQ3jΦx˜αdα+∫t−dttx˜TαΦT∑j=1NλijQ3jΦx˜αdα,δV3x˜t,i,t=x˜TtΦTd1Z1+d2Z2Φx˜t−∫t−d1tx˜TαΦTZ1Φx˜αdα−∫t−d2tx˜TαΦTZ2Φx˜αdα,δV4x˜t,i,t=d12x˜˙TtE˜TΦTR1ΦE˜x˜˙t−d1∫t−d1tx˜˙TαE˜TΦTR1ΦE˜x˜˙αdα+d122x˜˙TtE˜TΦTR1ΦE˜x˜˙t−d12∫t−d2t−d1x˜˙TαE˜TΦTR2ΦE˜x˜˙αdα.
According to Lemma1, we can obtain(24)−d1∫t−d1tx˜˙TαE˜TΦTR1ΦE˜x˜˙αdα≤−∫t−d1tΦE˜x˜˙αdαTR1∫t−d1tΦE˜x˜˙αdα=−x˜t−x˜t−d1TE˜TΦTR1ΦE˜x˜t−x˜t−d1−d12∫t−d2t−d1x˜˙TαE˜TΦTR2ΦE˜x˜˙αdα=−d12∫t−dtt−d1x˜˙TαE˜TΦTR2ΦE˜x˜˙αdα−d12∫t−d2t−dtx˜˙TαE˜TΦTR2ΦE˜x˜˙αdα≤−d12dt−d1∫t−dtt−d1ΦE˜x˜˙αdαTR2∫t−dtt−d1ΦE˜x˜˙αdα−d12d2−dt∫t−d2t−dtΦE˜x˜˙αdαTR2∫t−d2t−dtΦE˜x˜˙αdα.
Next, let us estimate the upper bound of the two terms on the right side of the above inequality.
When0<d1<dt<d2, there is(25)dt−d1<d2−d1⟹d2−d1dt−d1>1⟹−d2−d1dt−d1<−1,d2−d1d2−dt>1⟹−d2−d1d2−dt<−1.
Therefore, the following can be obtained:(26)δV4x˜t,i,t<x˜˙TtE˜TΦTd12R1+d122R2ΦE˜x˜˙t−∫t−d1tΦE˜x˜˙αdαTR1∫t−d1tΦE˜x˜˙αdα−∫t−dtt−d1ΦE˜x˜˙αdαTR2∫t−dtt−d1ΦE˜x˜˙αdα−∫t−d2t−dtΦE˜x˜˙αdαTR2∫t−d2t−dtΦE˜x˜˙αdα=x˜˙TtE˜TΦTd12R1+d122R2ΦE˜x˜˙t−x˜t−x˜t−d1TE˜TΦTR1ΦE˜x˜t−x˜t−d1−x˜t−d1−x˜t−dtTE˜TΦTR2ΦE˜x˜t−d1−x˜t−dt−x˜t−dt−x˜t−d2TE˜TΦTR2ΦE˜x˜t−dt−x˜t−d2=A˜ix˜t+A˜diΦx˜t−dtTΦTd12R1+d122R2ΦA˜ix˜t+A˜diΦx˜t−dt−x˜t−x˜t−d1TE˜TΦTR1ΦE˜x˜t−x˜t−d1−x˜t−d1−x˜t−dtTE˜TΦTR2ΦE˜x˜t−d1−x˜t−dt−x˜t−dt−x˜t−d2TE˜TΦTR2ΦE˜x˜t−dt−x˜t−d2.
Noting thatE˜TS˜=0 and E˜N˜=0, we have(27)∇Vx˜t,i,t=x˜˙TtE˜TS˜N˜Tx˜t+x˜TtN˜S˜TE˜x˜˙t=A˜ix˜t+A˜diΦx˜t−dtTS˜N˜Tx˜t+x˜TtN˜S˜TA˜ix˜t+A˜diΦx˜t−dt=0.
Add the weak infinitesimal operators to get the following equation:(28)δVx˜t,i,t=δV1x˜t,i,t+δV2x˜t,i,t+δV3x˜t,i,t+δV4x˜t,i,t+∇Vx˜t,i,t≤x˜Tt∑j=1NλijE˜TPjE˜x˜t+x˜TtE˜TPiA˜ix˜t+A˜diΦx˜t−dt+A˜ix˜t+A˜diΦx˜t−dtTPiTE˜x˜t+x˜TtΦTQ1i+Q2i+Q3iΦx˜t−1−ux˜Tt−dtΦTQ1iE˜x˜t−dt−x˜Tt−d1ΦTQ2iE˜x˜t−d1−x˜Tt−d2ΦTQ3iE˜x˜t−d2+∫t−dttx˜TαΦT∑j=1NλijQ1jΦx˜αdα+∫t−d1tx˜TαΦT∑j=1NλijQ2jΦx˜αdα+∫t−d2t−dtx˜TαΦT∑j=1NλijQ3jΦx˜αdα+∫t−dttx˜TαΦT∑j=1NλijQ3jΦx˜αdα+x˜TtΦTd1Z1+d2Z2Φx˜t−∫t−d1tx˜TαΦTZ1Φx˜αdα−∫t−d2tx˜TαΦTZ2Φx˜αdα+A˜ix˜t+A˜diΦx˜t−dtTΦTd12R1+d122R2ΦA˜ix˜t+A˜diΦx˜t−dt−x˜t−x˜t−d1TE˜TΦTR1ΦE˜x˜t−x˜t−d1−x˜t−d1−x˜t−dtTE˜TΦTR2ΦE˜x˜t−d1−x˜t−dt−x˜t−dt−x˜t−d2TE˜TΦTR2ΦE˜x˜t−dt−x˜t−d2+x˜˙TtE˜TS˜N˜Tx˜t+x˜TtN˜S˜TE˜x˜˙t=ξ1TtΩ1iξ1t,where(29)ξ1t=x˜Ttx˜Tk−dtx˜Tt−d1x˜Tt−d2T,Ω1i=Ω11′Ω12′E˜TΦTR1ΦE˜0∗Ω22′E˜TΦTR2ΦE˜E˜TΦTR2ΦE˜∗∗Ω33′0∗∗∗Ω44′,Ω11′=∑j=1NλijE˜TPjE˜+symE˜TPiA˜i+symA˜iTS˜N˜iT+ΦTQ1i+Q2i+Q3i+d1Z1+d2Z2Φ−E˜TΦTR1ΦE˜+A˜iTΦTd12R1+d122R2ΦA˜i,Ω12′=E˜TPiA˜diΦ+N˜iS˜TA˜diΦ+A˜iTΦTd12R1+d122R2ΦA˜diΦ,Ω22′=−1−uΦTQ1iΦ−2E˜TΦTR2ΦE˜+ΦTA˜diTΦTd12R1+d122R2ΦA˜diΦ,Ω33′=−ΦTQ2iΦ−E˜TΦTR1+R2ΦE˜,Ω44′=−ΦTQ3iΦ−E˜TΦTR2ΦE˜.
IfδVx˜t,i,t=ξ1TtΩ1iξ1t<0 is true, system (9) is stochastically stable. Next, we derive the conditions for δVx˜t,i,t<0 to be true. Let the minimum eigenvalue of matrix −Ψi be λmin−Ψi. According to equation (28), the following equation can be obtained:(30)δVx˜t,i,t<−λmin−ΨiTx˜Ttx˜t.According to Dynkin formula, we can get(31)EVx˜t,rt−Vx˜0,r0=E∫0tδVx˜s,rs,sds≤−λmin−ΨiE∫0tx˜Tsx˜sdsx˜0,r0,from Vxt,rt,t≥0; when δVx˜t,rt,t<0, the following formula is true:(32)E∫0tx˜Tsx˜sdsx˜0,r0≤Vx˜0,r0λmin−Ψi=Mx˜0,r0.When the upper limit of the integration tends to infinity, it can be deduced:(33)E∫0∞x˜Tsx˜sdsx˜0,r0≤Mx˜0,r0.
As can be seen from the definition of 1.1 (3), system (9) is stochastically stable. System (9) is regular, impulse-free, and stochastically stable; thus, it is stochastically admissible.
WhenΨt≠0, set(34)ξ2t=x˜Ttx˜Tk−dtx˜Tt−d1x˜Tt−d2ΨTtT.
According to (2) in Definition2, the following formula can be obtained:(35)JT≤E∫0TeTtet−γ2ΨTtΨt+δVx˜t,rt,tdt−E∫0TδVx˜t,rt,tdt≤E∫0TeTtet−γ2ΨTtΨt+δVx˜t,rt,tdt≤E∫0Tξ2TtΩ2iξ2tdt,Ω2i=Ω⌣11Ω⌣12E˜TΦTR1ΦE˜0Ω15∗Ω⌣22E˜TΦTR2ΦE˜E˜TΦTR2ΦE˜Ω25∗∗Ω33′00∗∗∗Ω44′0∗∗∗∗Ω55,Ω⌣11=Ω11′+C˜TiC˜i,Ω⌣12=Ω12′+C˜TiC˜diΦ,Ω⌣22=Ω22′+ΦTC˜TdiC˜diΦ,Ω15=E˜TPiTB˜i+A˜iTΦTd12R1+d122R2ΦB˜i+C˜iTD˜i+N˜iS˜TB˜i,Ω25=ΦTA˜diTΦTd12R1+d122R2ΦB˜i+ΦTC˜diTD˜i,Ω55=B˜iTΦTd12R1+d122R2ΦB˜i+D˜TiD˜i−γ2I.
Applying the Schur theorem to formula (35), formula (18) can be obtained. Theorem 1 is proved.Theorem 2.
Considering singular model error augmented system (9) with polytopic uncertain transition rates and time-varying delays is stochastic admissible with an H∞ performance index γ. If there exist positive-definite symmetric matrices Pi∈Rnx+nf+nw×nx+nf+nw, Z1,Z2,R1,R2,Q1i,Q2i,Q3i∈Rnx×nx, any appropriate dimensions matrices S˜,N˜i∈Rnx+nf+nw×nx−r, and satisfying E˜TS˜=0, such that the following LMIs hold for any ∀i,j∈I:(36)δ1ki=∑j∈SkiλijQ1j+∑j∈Suciλ˜ijlQ1j,δ2ki=∑j∈SkiλijQ2j+∑j∈Suciλ˜ijlQ2j,δ3ki=∑j∈SkiλijQ3j+∑j∈Suciλ˜ijlQ3j.
Ifi∈Ski∪Suci, j∈Suki,(37)δ1ki+δ3ki−λki+λucilQ1j+Q3j<Z2,δ2ki−λki+λucilQ2j<Z1,δ3ki−λki+λucilQ3j<Z2.
Ifi∈Suki, j∈Suki,(38)δ1ki+δ3ki+λbiQ1j+Q3j−λbi+λki+λucilQ1j+Q3j<Z2,δ2ki+λbiQ2j−λbi+λki+λucilQ2j<Z1,δ3ki+λbiQ3j−λbi+λki+λucilQ3j<Z2,(39)Ω⌢i=Ω⌢11Ω12E˜TΦTR1ΦE˜0Ω15Ω16C˜iT∗Ω22E˜TΦTR2ΦE˜E˜TΦTR2ΦE˜0Ω26ΦTC˜diT∗∗Ω330000∗∗∗Ω44000∗∗∗∗−γ2IΩ56D˜iT∗∗∗∗∗Ω660∗∗∗∗∗∗−I<0,Ω^11=symE˜TPiA˜i+symA˜iTS˜N˜iT+ΦTQ1i+Q2i+Q3i+d1Z1+d2Z2Φ−E˜TΦTR1ΦE˜,Ω⌢11=Ω^11+E˜TΞki+Ξucil−λki+λucilPjE˜,ifi∈Ski∪Suci,j∈Suki,Ω⌢11=Ω^11+E˜TΞki+Ξucil+λbiPi−λbi+λki+λucilPjE˜,ifi∈Suki,j∈Suki,Ξki=∑j∈SkiλijPj,Ξucil=∑j∈Suciλ˜ijlPj,λki=∑j∈Skiλij,λucil=∑j∈Suciλ˜ijl.Proof.
Theorem1 has given the sufficient condition that system (9) is stochastically admissible and has an H∞performance indexes γ when the TRs are completely known. Next, we discuss the case that the state TRM contains unknown elements.Case 1.
Ifi∈Ski∪Suci, j∈Suki.
Wheni∈Ski∪Suci, it is indicated that diagonal elements λiiare known or uncertain, then λki+λucil≤0. Since λki+λucil=0, all the elements in the ith row of the vertex Λl,l=1,2,⋯M are known, we only need to consider the case λki+λucil<0 and rewrite the term ∑j=1NλijPjin formula (18) as(40)∑j=1NλijPj=∑j∈SkiλijPj+∑j∈Sukiλ^ijPj+∑j∈Suci∑l=1Mαlλ˜ijlPj=∑j∈SkiλijPj+−λki−λucil∑j∈Sukiλ^ij−λki−λucilPj+∑l=1Mαl∑j∈Suciλ˜ijlPj=Ξki+−λki−λucil∑j∈Sukiλ^ij−λki−λucilPj+∑l=1MαlΞucil=Ξki+−λki−λucil∑j∈Sukiλ^ij−λki−λucilPj+∑l=1MαlΞucil,where λ^ijj∈Suki and λ˜ijj∈Sucirepresent unknown and polytopic uncertain elements, respectively, Ξki=∑j∈SkiλijPj, and Ξucil=∑j∈Suciλ˜ijlPj. Since 0≤αl≤1, ∑l=1Mαl=1, and 0≤λ^ij/−λki−λucil≤1, ∑j∈Sukiλ^ij/−λki−λucil=1, the right side (RS) of formula (40) can be written as(41)RS50=∑l=1Mαl∑j∈Sukiλ^ij−λki−λucilΞki+Ξucil−−λki−λucilPj.
Similarly, when0≤αl≤1 and 0≤λ^ij≤−λki+λucil, the left side (LS) of formula (18) can be transformed into the following form:(42)LHS18=∑l=1Mαl∑j∈Sukiλ^ij−λki−λucilΩ⌢i,j∈Suki,l=1,2,⋯M,where Ω⌢i has been defined in formula (39), which means that when not all elements are known in the TRM, the inequality set (17) is directly converted into the inequality set (37).Case 2.
Ifi∈Suki, j∈Suki.
When0≤αl≤1, λ^ii=−λki+λucil, the elements in the ith row are all known, so 0≤αl≤1, λ^ii<−λki+λucil is true if at least one unknown element in the ith row exists besides the diagonal. Under this condition, the 0≤αl≤1, ∑j=1NλijPj term in formula (18) can be rewritten as(43)∑j=1NλijPj=∑j∈SkiλijPj+λ^iiPi+∑j∈Suki,j≠iλ^ijPj+∑j∈Suci∑l=1Mαlλ˜ijlPj=Ξki+λ^iiP+−λ^ii−λki−λucil∑j∈Suki,j≠iλ^ij−λ^ii−λki−λucilPj+∑l=1MαlΞucil.
Since0≤αl≤1, ∑l=1Mαl=1, and 0≤λ^ij/−λ^ii−λki−λucil≤1, ∑j∈Suki,j≠iλ^ij/−λ^ii−λki−λucil=1; the following formula can be deduced:(44)∑j=1NλijPj=∑l=1Mαl∑j∈Suki,j≠iλ^ij−λ^ii−λki−λucilΞki+λ^iiPi+Ξucil−λ^ii+λki+λucilPj.
By rewriting the left side (LS) of inequality (18) according to equation (44), we can get(45)LS18=∑l=1Mαl∑j∈Suki,j≠iλ^ij−λ^ii−λki−λucilΩ⌢i,i∈Suki,j∈Suki,(46)Ω⌢11=Ω^11+E˜TΞki+Ξucil+λ^iiPi−λ^ii+λki+λucilPjE˜,where Ω⌢i has been defined in formula (39) and contains the term Ω⌢11. In order to achieve the solvability of the stability analysis, the lower bound of the unknown element λ^ii is defined as λbi, and λbi≤λ^ii<−λki−λucil can be obtained. It indicates that, for any small ε<0, λ^ii can take value in the interval λbi,−λki−λucil+ε, then λ^ii can be written as the form of the following convex combination λ^ii=−βλki−βλucil+βε+1−βλbi, where 0≤β≤1, and it can be seen that λ^iidepends linearly on β. Therefore, if only β takes the boundary value of 0 or 1 that the linear solvable condition of inequality (45) can be satisfied. At this time, equation (46) can be rewritten as(47)Ω⌢11=Ω^11+E˜TΞki+Ξucil+λbiPi−λbi+λki+λucilPjE˜,β=0,j≠i,(48)Ω⌢11=Ω^11+E˜TΞki+Ξucil−λki+λucilPj+εPi−PjE˜,β=1,j≠i.
Whenε is infinitely small and j≠i, j∈Suki, formula (48) can be converted to(49)Ω⌢11=Ω^11+E˜TΞki+Ξucil−λki+λucilPjE˜,j≠i.
Whenj=i, j∈Suki,equation (47) can be written as (49), so wheni∈Suki and j∈Suki, the term Ω⌢11 can be expressed by (47). The inequality group (17) can be directly converted into the inequality group (38). Therefore, when the elements in the state TRM are not completely known, as long as the inequality set (37)–(39) exists, system (9) is still stochastically admissible and has an H∞performance γ. The prove is completed.Remark 4.
In this section, stochastically admissible analysis criteria for a family of polytopic uncertain singular continuous-time Markovian jump error augmented systems with time-varying delays are given. Because the coupling terms in the system matrices will affect the performance of the FDFs, we will use a special method to design the full-order and reduced-order FDFs of polytopic uncertain CTSMJSs with time-varying delays. It can improve the effectiveness and practicability of the FDFs.


## 4. Design of Full-Order and Reduced-Order FDFS for Polytopic Uncertain CTSMJSS with Time-Varying Delays

Theorem 3.
On the premise that the error augmented system (9) is stochastically admissible and has an H∞performance index γ, for a given scalar γ>0, if exist positive definite symmetric matrices Pi≔Pi1KPi2Pi4∗Pi20∗∗Pi5∈Rnx+nf+nw×nx+nf+nw, Z1,Z2,R1,R2,Q1i,Q2i,Q3i∈Rnx×nx, any appropriate dimensions matrices S˜,N˜i∈Rnx+nf+nw×nx−r, and satisfying E˜TS˜=0, ∀i,j∈I, K≔Inf0nf×nx−nfT, the following linear matrix inequality sets are satisfied.(50)Ω⌢i′=Ω⌢11Ω⌢12E˜TΦTR1ΦE˜0Ω⌢15Ω16C˜iT∗Ω22E˜TΦTR2ΦE˜E˜TΦTR2ΦE˜0Ω26ΦTC˜diT∗∗Ω330000∗∗∗Ω44000∗∗∗∗−γ2IΩ56D˜iT∗∗∗∗∗Ω660∗∗∗∗∗∗−I<0,Ω⌢11=Ω¯1iΩ¯2iΩ¯4i∗Ω¯3i0∗∗Ω¯5iΩ⌢12=Ω12′TΩ12″TΩ12‴TTΩ⌢15=Ω15′TΩ15″TΩ15‴TTΩ12′=ETPi1Adi+ETKPi2Cdi+NiSTAdi,Ω12″=Pi2TKTAdi+Pi2BFiCdi,Ω12‴=Pi4TAdi,Ω15′=NiSTBi+ETPi1BiTNiSTEi+ETPi1TEi+ETKPi2BFiDiTETPi1Fi+ETKPi2BFiHi+ETPi4BwfiTT,Ω15″=Pi2TKTBiPi2TKTEi+Pi2BFiDiPi2TKTFi+Pi2BFiHi,Ω15‴=Pi4TBiPi4TEiPi4TFi+Pi5Bwfi,Ω¯1i=Ω¯1i′+ETΞk1i+Ξuc1il−λki+λucilPj1E,ifi∈Ski∪Suci,j∈Suki,Ω¯1i′+ETΞk1i+Ξuc1il+λbiPi1−λbi+λki+λucilPj1E,ifi∈Suki,j∈Suki,Ω¯1i′=symETPi1Ai+symAiTSNiT+symETKB¯FiCi+Q1i+Q2i+Q3i+d1Z1+d2Z2−ETR1E,Ω¯2i=Ω¯2i′+ETΞk2i+Ξuc2il−λki+λucilPj2E,ifi∈Ski∪Suci,j∈Suki,Ω¯2i′+ETΞk2i+Ξuc2il+λbiPi2−λbi+λki+λucilPj2E,ifi∈Suki,j∈Suki,Ω¯2i′=ETKA¯Fi+AiTKPi2+CiTB¯FiT,Ω¯3i=Ω¯3i′+Ξk2i+Ξuc2il−λki+λucilPj2,ifi∈Ski∪Suci,j∈Suki,Ω¯3i′+Ξk2i+Ξuc2il+λbiPi2−λbi+λki+λucilPj2,ifi∈Suki,j∈Suki,Ω¯3i′=A¯Fi+A¯FiT,Ω¯4i=Ω¯4i′+ETΞk4i+Ξuc4il−λki+λucilPj4E,ifi∈Ski∪Suci,j∈Suki,Ω¯4i′+ETΞk4i+Ξuc4il+λbiPi4−λbi+λki+λucilPj4E,ifi∈Suki,j∈Suki,Ω¯4i′=ETPi4Awfi+AiTPi4,Ω¯5i=Ω¯5i′+Ξk5i+Ξuc5il−λki+λucilPj5,ifi∈Ski∪Suci,j∈Suki,Ω¯5i′+Ξk5i+Ξuc5il+λbiPi5−λbi+λki+λucilPj5,ifi∈Suki,j∈Suki,Ω¯5i′=Pi5TAwfi+AwfiPi5,Ξk1i=∑j∈SkiλijPj1,Ξk2i=∑j∈SkiλijPj2,Ξk4i=∑j∈SkiλijPj4,Ξk5i=∑j∈SkiλijPj5,Ξuc1il=∑j∈Suciλ˜ijlPj1,Ξuc2il=∑j∈Suciλ˜ijlPj2,Ξuc4il=∑j∈Suciλ˜ijlPj4,Ξuc5il=∑j∈Suciλ˜ijlPj5.The gain matrices AFi,BFi,CFi,DFi of fault detection filter (7) of system (1) can be obtained, and its parameter matrices are expressed as follows:(51)AFi≔Pi2−1A¯Fi,BFi≔Pi2−1B¯Fi,CFi≔C¯Fi,DFi≔D¯Fi.Proof.
In order to design high-performance full-order and reduced-order FDFs that meets the requirements, the Lyapunov matrixPi in Theorem 1 has the following form:(52)Pi≔Pi1KPi2Pi4∗Pi30∗∗Pi5,where K≔Inf0nf×nx−nfT, Pi1∈Rnx×nx, Pi2∈Rnf×nf, Pi3∈Rnf×nf, Pi4∈Rnx×nw, and Pi5∈Rnw×nw.
By using diagonal matricesdiagInx,Pi2Pi3−1,Inf and diagInx,Pi3−TPi2T,Inf to perform the congruent transformation of (52), we can get(53)Inx00∗Pi2Pi3−10∗∗InfPi1KPi2Pi4∗Pi30∗∗Pi5Inx00∗Pi3−TPi2T0∗∗Inf=Pi1KPi2Pi3−TPi2TPi4∗Pi3Pi3−TPi2T0∗∗Pi5≕Pi1KP¯i2Pi4∗P¯i20∗∗Pi5.
Therefore, without loss of generality, Lyapunov matrixPi can be specified as the following general form:(54)Pi≔Pi1KPi2Pi4∗Pi20∗∗Pi5.
The matrix variablePi2 in (54) can be combined with the gain matrices AFiand BFiof the filter to form new matrices, as follows:(55)A¯Fi≔Pi2AFi,BFi≔Pi2BFi.
Equation (50) can be obtained by substituting (54) into equation (39). The proof is completed.Remark 5.
Theorem3 presents a high efficiency method for designing fault detection filter for polytopic uncertain CTSMJSs with time-varying delays that full-order and reduced-order FDFs can be expressed in a unified formula. In order to illustrate the effectiveness of the proposed approach, the following example will be used to verification.


## 5. Numerical Example

In this section, two examples are used to illustrate the practicability and effectiveness of our results.Example 1.
Consider the DC motor driving a load that switching is driven by a continuous-time Markov processrt,t>0 [40, 41]. Similarly, by neglecting the DC motor inductance Lm, let it, vt, and w¯˙t represent the electric current, the voltage, and the speed of the shaft at time t, respectively. According to the basic electrical and mechanic laws, we have(56)w¯˙t=−biJiw¯t+KtJiit,vt=Kw¯w¯t+Rit,where Kt, Kw¯, and R denote the torque constant, the electromotive force, and the electric resistor, respectively. Ji and bi are defined as Ji=Jm+Jci/n2 and bi=bm+bci/n2, where Jci and Jm represent the moments of the load and the motor, respectively, and bmand bcistand for the damping ratios with gear ratio n. Set x1t=w¯t and x2t=it, and system (56) is expressed as(57)1000x˙t=−biJiKtJiKw¯Rxt+01vt.
To guarantee the system stability, in this paper, we use the state feedback controllervt=Kixt and value K1=−1.90832−2.7715 and K2=−2.3635−2.8652 in [39, 40]. Without loss of generality, as used in [41], it is assumed that time delay dt=0.1+0.05sin2t and system (57) can be written as(58)1000x˙t=−biJiKtJiKw¯R+01Kixt+000.50xt−dt.
To ensure the validity of the actual example, we will use the parameters in [40, 41], where Jm=0.5kg⋅m, Jc1=50kg⋅m, Jc2=150kg⋅m, bc1=100, bc2=240, R=1, bm=1, Kt=3Nm/A, Kw¯=1Vs/rad, n=10, and Π=−0.01930.01930.0307−0.0307, and the following closed-loop system can be obtained:(59)Ex˙t=Artxt+Adrtxt−dt,where E=1000, A1=−2.00003.0000−0.9083−1.7715, A2=−1.70001.5000−1.3635−1.8652, and Ad1=Ad2=000.50.
In order to design the full-order and reduced-order FDFs about system (59), the other parameters in [41] are given as(60)B1=−2.5;−0.3,B2=1.5;−0.2,E1=−0.2;−0.1,E2=0.15;−0.12,F1=−0.5;0.4,F2=0.6;0.3,C1=−0.51.2,C2=−0.350.7,Cd1=−0.5−1.6,Cd2=−0.360.8,D1=1.2,D2=0.25,H1=2,H2=−0.5,Awfi=−5,Bwfi=3,Cwfi=1,Dwfi=0.
By Theorem3, lettingS=0;1 and γ=2, we can get the parameters of the desired full-order and reduced-order FDFs, which are given as follows.(1)
Parameter matrices of full-order FDF:(61)AF1=−0.85900.2076−0.0525−1.3138,BF1=0.2362;0.0058,CF1=−0.06020.0866,DF1=0.1134,AF2=−0.84470.7543−0.5808−1.3574,BF2=0.1068;0.4773,CF2=−0.01520.0299,DF2=−0.3351.(2)
Parameter matrices of reduced-order FDF:(62)AF1=−4.9187,BF1=1.3578,CF1=−0.0714,DF1=0.1160,AF2=−2.6482,BF2=0.3304,CF2=−0.0165,DF2=−0.3374.Remark 6.
Compared with [41], the proposed approach in this paper can not only solve the parameters of full-order FDF for DC motor but also figure out the parameters of reduced-order FDF. However, when the DC motor driving a load that switching is driven to start with the TRs have unknown elements and many uncertain elements, the proposed method is also practical. The following examples will further illustrate the effectiveness of the proposed method for FDFs design for different TRMs. We compare and analyse the H∞ performance index γmin and the number of decision variables of full-order and reduced-order FDF for four different TRMs.Example 2.
In this section, we select a system with four modals to verify the effectiveness of the designed full-order and reduced-order FDFs of CTSMJSs with time-varying delays and four different TRMs. The parameters matrices can be described as follows:(63)A1=−2.80.4−3.1−1.6,A2=2.50.5−0.6−1.0,A3=−2.5−0.50.2−1.8,A4=−1.500−0.9,Ad1=0.1−0.2−0.1−0.3,Ad2=0.10.200.1,Ad3=0.1−0.2−0.1−0.3,Ad4=0.1−0.2−0.1−0.1,B1=0.6;0,B2=0.4;−0.2,B3=0.3;0.8,B4=0.3;−0.2,C1=0.50.3,C2=0.4−0.5,C3=−0.3−0.4,C4=0.5−0.2,Cd1=0.1−0.3,Cd2=0.2−0.3,Cd3=0.3−0.4,Cd4=0.5−0.1,D1=0.5,D2=0.3,D3=−0.3,D4=0.5,E1=0.4;−0.3,E2=0;−0.3,E3=−0.3;0.1,E4=−0.5;0,F1=0.2;0,F2=0.1;−0.3,F3=0.3;−0.2,F4=0.2;−0.1,H1=0.2,H2=−0.2,H3=0.3,H4=−0.4,E=10;00,S=0;1,Awfi=−5,Bwfi=3,Cwfi=1,Dwfi=0.
To make the convenience of simulation, the weighted fault frequency function and known control input signal to be set asWfs=3/s+5 and ut=0.1+0.05cost, the Gaussian white noise signal with amplitude less than 0.2 is selected as the exogenous disturbance input wt, respectively. The range of time-varying delay dt satisfies 0.1≤dt≤0.6 and d˙t≤0.3 for 0≤t≤60. In this paper, the simulation example selects the fault expression ft that was used in many classic papers as the potential fault signal [31–34]. The fault detection graph simulated by this fault expression is not only simple but also clear; we can clearly see whether the fault detection filter is effective or not. The fault signal ft is(64)ft=1.5,20≤t≤40,0,others.
Table1 lists the four different cases TRMs, and Figure 1 shows the simulation results of four different cases Markov chain rtevolution processes.
For Case2, all the rows remain the same except for the third row, it contains two uncertain TRs, which means the TRM includes two vertices Λ3s and s=1,2 in third rows, and the TRM in the third row can be expressed as(65)Λ31=0.4λ^32−1.3λ^34,Λ32=0.6λ^32−1.1λ^34.
At the same time, we set the lower boundsλ^2b44=−1 of unknown diagonal elements λ^44 in Case 2; set λ^3b44=−1 in Case 3; and assign λ^4b11=−0.8, λ^4b22=−1.2, λ^4b33=−1.3, and λ^4b44=−0.7 in Case 4, respectively.
When the system parameters are presented, the linear matrix inequalities in Theorem3 can be solved by MATLAB, and the full-order and reduced-order FDFs parameter matrices for four different cases of TRMs can be obtained as follows.Table 1
TRMs of four different cases.
Case 1: completely known TRM
Case 2: polytopic uncertain TRM
Case 3: partly unknown TRM
Case 4: completely unknown TRM
−
0.8
0.3
0.4
0.1
0.5
−
1.2
0.3
0.4
0.6
0.5
−
1.3
0.2
0.2
0.1
0.4
−
0.7
−
0.8
λ
^
12
0.4
λ
^
14
0.5
−
1.2
0.3
0.4
λ
˜
31
λ
^
32
λ
˜
33
λ
^
34
λ
^
41
λ
^
42
λ
^
43
λ
^
44
−
0.8
λ
^
12
0.4
λ
^
14
0.5
−
1.2
0.3
0.4
0.6
λ
^
32
−
1.3
λ
^
34
λ
^
41
λ
^
42
λ
^
43
λ
^
44
λ
^
11
λ
^
12
λ
^
13
λ
^
14
λ
^
21
λ
^
22
λ
^
23
λ
^
24
λ
^
31
λ
^
32
λ
^
33
λ
^
34
λ
^
41
λ
^
42
λ
^
43
λ
^
44Figure 1
The modes random jumping process of four different TRMs cases. (a) Case 1. TRM is completely known. (b) Case 1. TRM is polytopic uncertain. (c) TRM is partly unknown. (d) TRM is completely unknown.
(a)
(b)
(c)
(d)Case 1.
TRM is completely known.(1)
Parameter matrices of full-order FDF with completely known TRM:(66)AF1=−2.39220.18860.3139−3.5091,BF1=−0.8997;1.2641,CF1=−0.14490.1438,DF1=0.1801,AF2=−2.31142.0323−1.7332−2.7174,BF2=0.5977;−0.7239,CF2=−0.0860−0.1613,DF2=−0.0504,AF3=−2.3830−0.34800.1603−3.8508,BF3=−1.3546;−0.6935,CF3=−0.1584−0.1482,DF3=0.2610,AF4=−1.46100.0310−0.0857−2.3989,BF4=1.5553;−0.4436,CF4=−0.2164−0.0215,DF4=−0.2857.(2)
Parameter matrices of reduced-order FDF with completely known TRM:(67)AF1=−2.8377,BF1=−1.0126,CF1=−0.1645,DF1=0.1884,AF2=−2.6302,BF2=0.4328,CF2=−0.0973,DF2=−0.0482,AF3=−2.6505,BF3=−1.2617,CF3=−0.2041,DF3=0.2795,AF4=−0.9532,BF4=1.0858,CF4=−0.2087,DF4=−0.3146.Case 2.
TRM is polytopic uncertain.(1)
Parameter matrices of full-order FDF with polytopic uncertain TRM:(68)AF1=−1.01260.0893−0.0672−1.8273,BF1=−0.0499;0.8281,CF1=−0.14240.0691,DF1=0.0739,AF2=−0.95540.4547−0.4769−1.2196,BF2=−0.0562;0.4387,CF2=−0.0964−0.0333,DF2=0.0367,AF3=−1.1803−0.09170.0210−1.8310,BF3=−0.6052;−0.3351,CF3=−0.0943−0.0407,DF3=0.1409,AF4=−0.90070.1558−0.1743−2.3419,BF4=−0.2081;−0.1826,CF4=−0.14040.0017,DF4=−0.0472.(2)
Parameter matrices of reduced-order FDF with polytopic uncertain TRM:(69)AF1=−4.1974,BF1=0.1968,CF1=−0.1456,DF1=0.579,AF2=−3.8601,BF2=0.2387,CF2=−0.1003,DF2=0.0377,AF3=−4.5124,BF3=0.1336,CF3=−0.0944,DF3=0.1374,AF4=−0.9109,BF4=−0.2188,CF4=−0.1296,DF4=−0.0409.Case 3.
TRM is partly unknown.(1)
Parameter matrices of full-order FDF with partly unknown TRM:(70)AF1=−1.00780.0714−0.0417−1.8167,BF1=−0.0856;0.8378,CF1=−0.12930.0672,DF1=0.0757,AF2=−0.95930.4838−0.4977−1.2134,BF2=0.0874;−0.4544,CF2=−0.0874−0.0316,DF2=0.0309,AF3=−1.0948−0.05320.0106−1.7874,BF3=−0.6223;−0.3788,CF3=−0.1368−0.0554,DF3=0.1112,AF4=−0.94410.1718−0.1679−2.3823,BF4=−0.1383;−0.1917,CF4=−0.1312−0.0015,DF4=−0.0585.(2)
Parameter matrices of reduced-order FDF with partly unknown TRM:(71)AF1=−3.2699,BF1=−0.0283,CF1=−0.1212,DF1=0.0680,AF2=−3.9731,BF2=0.3550,CF2=−0.0852,DF2=−0.0278,AF3=−3.8903,BF3=−0.1837,CF3=−0.1315,DF3=0.1106,AF4=−0.9442,BF4=−0.0697,CF4=−0.1344,DF4=−0.0503.Case 4.
TRM is completely unknown.(1)
Parameter matrices of full-order FDF with completely unknown TRM:(72)AF1=−1.2535−0.22730.3095−3.1673,BF1=−0.3551;0.1637,CF1=−0.18400.0571,DF1=0.0036,AF2=−1.37330.1410−0.1867−3.1945,BF2=−0.0225;−0.2367,CF2=−0.0935−0.0711,DF2=−0.0171,AF3=−1.71440.1364−0.3325−3.6592,BF3=−1.0059;−0.3267,CF3=−0.0368−0.0623,DF3=0.1701,AF4=−0.93800.1069−0.1388−2.4802,BF4=−0.6428;−0.0735,CF4=0.1001−0.0100,DF4=−0.0231.(2)
Parameter matrices of reduced-order FDF with completely unknown TRM:(73)AF1=−3.6041,BF1=−0.9925,CF1=−0.1896,DF1=0.0061,AF2=−2.7577,BF2=−0.1842,CF2=−0.1007,DF2=−0.0224,AF3=−3.6929,BF3=−0.2183,CF3=−0.0513,DF3=0.0967,AF4=−1.2794,BF4=−0.9063,CF4=−0.1050,DF4=−0.0387.Remark 7.
According to Theorem3 and the simulation results, we can obtain that the selection dimension of Pi2∈Rnf×nf is the key to obtain the parameters of the full-order and reduced-order FDFs. When nf=nx, the parameters of the full-order FDF can be determined. When nf<nx, the parameters of the reduced-order FDF can be determined. For simplicity, the parameter matrices selected in this paper is two-dimensional, so only one-dimensional reduced-order FDF parameters are simulated. The method for determining the parameters of the reduced-order FDF is suitable for multidimensional matrices. By adjusting the dimensions of Pi2, the parameters of the reduced-order FDF of different dimensions can be obtained. For example, when the parameter matrices of the full-order fault detection filter are three-dimensional, the one-dimensional and two-dimensional parameter matrices of reduced-order FDFs can be obtained, respectively.
For convenience of simulation, the shortest time stepLsmin is set as Jr^t>Jth, NDV represents the number of decision variables, where Jth=sup0≠w∈l2,0≠uk∈l2,f=0E∫k0k0+Lr^Ttr^tdt, and the minimum H∞ performance is γmin. Next, we will illustrate the effectiveness of the full-order and reduced-order FDFs for four diffident cases of TRM by simulation analysis and parameter comparison lists.
Figures2 and 3 demonstrate residual generation of full-order and reduced-order FDFs for four different TRMs cases, respectively. Obviously, it can be seen by residual curve of comparison in Figures 3 and 4 that the estimated residual value of the full-order filter is nearer to the weighting fault signal than the estimated residual value of the reduced-order filter. The more the known transition rates are in the TRM, the more accurate the estimated residual values are. Case 2 with polytopic uncertainty TRM is obviously closer to the actual residual value than Case 3 and Case 4. The simulation results further indicate that no matter which case TRM is to be chosen, the designed FDFs are effective and feasible.
Figures4 and 5 demonstrate the residual evaluation function curves of full-order and reduced-order FDFs for four different TRMs cases, respectively. It can be seen from Figures 4 and 5 that the shortest time step Lsminis close to 20 (s), when Jr^t>Jth, that the weighting fault signal is effective and can improve the performance of the fault detection system. By comparing and analysing residual evaluation function curves of different FDFs and different cases of TRMs, we can draw a conclusion that Case 2 with polytopic uncertain TRM is closer to Case 1 with completely known TRM than Case 3 and Case 4, and the residual evaluation function value of the reduced-order FDF is obviously smaller than that of the full-order FDF and the evaluation time is obviously shortened.Figure 2
Residual generation of full-order FDF for four different TRMs cases. (a) Case 1. TRM is completely known. (b) Case 1. TRM is polytopic uncertain. (c) TRM is partly unknown. (d) TRM is completely unknown.
(a)
(b)
(c)
(d)Figure 3
Residual generation of reduced-order FDF for four different TRMs cases. (a) Case 1. TRM is completely known. (b) Case 1. TRM is polytopic uncertain. (c) TRM is partly unknown. (d) TRM is completely unknown.
(a)
(b)
(c)
(d)Figure 4
Residual evaluation function curves of full-order FDF for four different TRMs cases. (a) Case 1. TRM is completely known. (b) Case 1. TRM is polytopic uncertain. (c) TRM is partly unknown. (d) TRM is completely unknown.
(a)
(b)
(c)
(d)Figure 5
Residual evaluation function curves of reduced-order FDF for four different TRMs cases. (a) Case 1. TRM is completely known. (b) Case 1. TRM is polytopic uncertain. (c) TRM is partly unknown. (d) TRM is completely unknown.
(a)
(b)
(c)
(d)Remark 8.
Thistimes represents the shortest time for obtaining H∞ performance level γmin, but the value of this time is not obtained at one time, by calculating the average value that 50 times of data are obtained on the same computer.
Tables2 and 3 show the performance comparing full-order and reduced-order FDFs for the cases of different delay differentials. By comparing Table 2 with Table 3 filters characteristics, we can conclude the following:(1)
Under the condition of the same-order FDFs for different TRMs, the more known transition rates are in the TRM that theH∞ performance level γmin and the conservativeness are less(2)
Under the condition of the same-order FDFs and same TRMs, as the time-varying delays differential value increases, the time to solveγmin significantly increased(3)
Compared with different-order FDFs of the same TRMs, theH∞, performance index γmin, and the conservativeness are almost the same, but the number of decision variables (NDV ) reduced-order FDF is obviously less than full-order FDF; moreover, the processing speed of reduced-order FDF is faster than full-order FDFTable 2
Performance comparison results for four different cases of full-order FDFs with time-varying delays.
TRM
d
˙
t
=
0.4
t
=
60
d
˙
t
=
0.6
t
=
60
γ
min
time
s
N
D
V
γ
min
time
s
N
D
V
Case 1
0.2946
80.0568
164
0.2980
96.0701
164
Case 2
0.3373
137.7648
164
0.3524
144.4304
164
Case 3
0.4095
101.4276
164
0.4368
109.8631
164
Case 4
0.5749
82.8212
164
0.6078
98.6324
164Table 3
Performance comparison results for four different cases of reduced-order FDFs with time-varying delays.
TRM
d
˙
t
=
0.4
t
=
60
d
˙
t
=
0.6
t
=
60
γ
min
time
s
N
D
V
γ
min
time
s
N
D
V
Case 1
0.2946
50.0306
136
0.2980
67.4760
136
Case 2
0.3373
83.2772
136
0.3524
93.1037
136
Case 3
0.4095
70.7835
136
0.4368
82.7555
136
Case 4
0.5749
52.8212
136
0.6079
71.9304
136


## 6. Conclusions

In this paper, we propose full-order and reduced-order FDF design methods for CTSMJSs with time-varying delays and polytopic uncertain TRM. Based on the Lyapunov–Krasovskii functional method and convex optimization technique, some delay-dependent sufficient conditions are obtained such that the singular model error augmented system is stochastically admissible with anH∞ performance γmin. Finally, two examples are presented to illustrate the effectiveness of the proposed design approach. Moreover, the issue of semi-Markovian jump systems [42, 43] is a promising research topic which has a strong practical background. In future, we will study the fault detection of semi-Markovian jump systems.



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