# Dissipative Filter Design for Nonlinear Time-Varying-Delay Singular Systems against Deception Attacks

Authors: Guobao Liu; Shibin Shen; Xianglei Jia
Journal: Complexity
 (2021)

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## Abstract

This paper applies a T-S fuzzy model to depict a class of nonlinear time-varying-delay singular systems and investigates the dissipative filtering problem for these systems under deception attacks. The measurement output is assumed to encounter random deception attacks during signal transmission, and a Bernoulli distribution is used to describe this random phenomena. In this case, the filtering error system modeled by a stochastic singular T-S fuzzy system is established and stochastic admissibility for this kind of system is defined firstly. Then, by combining some integral inequalities and using the Lyapunov–Krasovskii functional approach, sufficient delay-dependent conditions are presented based on linear matrix inequality techniques, where the system of filtering error can be stochastically admissible and strictlyℚ,S,ℝ-dissipative against randomly occurring deception attacks. Moreover, parameters of the desired filter can be obtained via the solutions of the established conditions. The validity of our work is illustrated through a mostly used example of the nonlinear system.


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## Body

## 1. Introduction

In order to study various actual systems, such as large-scale systems, circuit systems, and biological systems, scholars have widely used the singular system model to describe these systems. In contrast with ordinary state space systems, singular systems can describe the performance characteristics of physical systems better [1]. On the other hand, time delay is a major factor that leads to system instability and performance degradation. Because of the coupling of the delay term and the functional equation, the study of singular time-delay systems is much more difficult than that of the standard time-delay systems [2]. Not only stability but also regularity and absence of impulse (or causality) are involved in the admissible analysis problem for singular time-delay systems. Up to now, various results for studying singular time-delay systems have been published. Reachable set estimation for continuous-time singular delay systems [3–5], sliding mode control for discrete-time singular delay systems [6–8], and H∞ control for singular time-delay systems with Markovian jump parameters [9–11] are just a few examples of so many research works.In the past decade, control researchers discovered that in practical or industrial applications, nonlinearities in system dynamics behavior can be fairly accurately described as a set of locally linear models mixed together with fuzzy membership functions. Thus, any complex nonlinear systems can be fuzzified and effectively approximated as a set of linear models by using the T-S fuzzy model approach [12]. In this case, a similar way to linear systems can be extended to analyze and synthesize nonlinear systems and lots of fruitful studies on the fields of stability theories, adaptive tracking control, H∞ filter design, etc., have been achieved for T-S fuzzy systems [13–18]. Recently, using the T-S fuzzy model to approximate singular nonlinear systems has also been published in many literature studies. To mention a few, the problems of admissibility analysis and controller design for singular T-S fuzzy systems with mismatched membership functions were investigated in [19,20]; the problems of adaptive sliding mode controller design for T-S fuzzy singular systems were considered in [21,22]; and asynchronous filtering problems for T-S fuzzy singular systems with Markovian jump parameters were investigated in [23,24].On another active research frontier, cyber-attacks have become more and more important factors to threaten network security in the network control system since they could lead to the leakage of a large amount of confidential information. So far, denial-of-service attacks and deception attacks are two main kinds of cyber-attacks widely studied by scholars [25–29]. Especially, deception attacks are more secluded and harder to detect which import mendacious data in the process of signal transmission and lead to performance damage of the target system. Therefore, it is very important to design a security filter for the system with deception attacks. For example, the topics include distributed recursive filtering for discrete time-delayed stochastic systems subject to both uniform quantization and deception attacks, recursive filtering for stochastic nonlinear time-varying complex networks with deception attacks, and event-triggered filter design for T-S fuzzy systems with deception attacks had been investigated in [30–32], respectively. As far as we know, control and filtering problems for T-S fuzzy singular systems with time-varying delays against deception attacks have not been fully investigated, not mention to dissipative filter design. Dissipativity, in simple terms, generally indicates that the increase in a system’s internal energy storage does not exceed the external energy supply of the system. The dissipativity has been analysed for large amounts of nonlinear systems and widely used in control theory and practice [33–35]. All the aforementioned facts motivate our research.The main contribution of this paper is centered on dealing with the problem of dissipative filtering for T-S fuzzy singular systems with time-varying delays subject to deception attacks, where the measurement output is assumed to encounter random deception attacks based on a Bernoulli distribution during signal transmission. In this case, the filtering error system modeled by a stochastic singular T-S fuzzy system is established and the stochastic admissibility for this kind of system is defined. By using the Lyapunov–Krasovskii functional (LKF) approach and based on linear matrix inequality (LMI) techniques, sufficient delay-dependent conditions are established to guarantee the stochastic admissibility and strictlyℚ,S,ℝ-dissipativity of the filtering error with randomly occurring deception attacks. Furthermore, based on these feasible conditions, parameters of the desired filter can be obtained. At last, we give an example of the nonlinear system to show the effectiveness of our result.Notations. In this work, the dimensions of all the matrices are generally considered to be compatible. Rn denotes the Euclidean space with n dimension; Rn×m represents the real matrices with n×m dimension; I represents an identity matrix, and 0 denotes a zero matrix with appropriate dimension; ⋅ denotes the Euclidean norm of a vector and its induced norm of a matrix; A+AT is described by symA; ℒ20,∞ is the space of integral vector over 0,∞; for any real function x,y∈ℒ20,∞ and real matrix M, we define x,Myd=∫0dxTtMytdt; the mathematical expectation operator is denoted as E⋅.


## 2. Problem Formulation

For the description of an interval time-varying-delay nonlinear singular system, a delayed T-S fuzzy model is adopted in terms ofr plant rules as follows.Rule 1.
Ifθ1t is μi1, and …, and θpt is μip, then(1)Ex˙t=Aixt+Adixt−dt+Biωt,yt=Cixt+Cdixt−dt+Diωt,zt=Lixt,xt=ϕt,∀t∈−d2,0,where xt∈Rn, yt∈Rm, and ωt∈Rl stand for the state vector, the measurement output, and the disturbance input, respectively; zt∈Rq is the signal to be estimated; ϕt is the initial condition; the premise variable vector is described by θt=θ1t,θ2t,…,θpt, and the fuzzy sets are μiji=1,…,r, j=1,…,p; the time-varying delay dt satisfies the conditions of 0≤d1≤dt≤d2, d˙t≤ϖ, and d1, d2, and 0≤ϖ<1 are constant scalars; E∈Rn×n may be a singular matrix with rankE=g≤n; and Ai, Adi, Bi, Ci, Cdi, Di, and Li are known real constant matrices with appropriate dimensions.
Then, we can generate the model of the T-S fuzzy singular systems when time-varying delay is considered:(2)Ex˙t=∑i=1rhiθtAixt+Adixt−dt+Biωt,yt=∑i=1rhiθtCixt+Cdixt−dt+Diωt,zt=∑i=1rhiθtLixt,where(3)hiθt=∏j=1pμijθjt∑i=1r∏j=1pμijθjtand μijθjt stand for the grade of membership for θjt in μij. It is easy to verify that(4)hiθt≥0,∑i=1rhiθt=1.
In this paper, for calculating the signalzt, a fuzzy filter is derived as(5)Ex˙ft=∑i=1rhiθtAfixft+Bfiyat,zft=∑i=1rhiθtCfixft,where xft∈Rn and zft∈Rq represent the filter state and the filter output, respectively. yat is the sensor measurement under randomly occurring deception attacks which can be given as(6)yat=yt+αt−yt+σt.
σt defines the deception signal imported into the output which is assumed to satisfy σt≤Wyt, and W is considered to be an appropriate dimension matrix; αt is a Bernoulli-distributed variable and we have the following assumptions on αt:(7)Probαt=1=Eαt=α,Probαt=0=1−α.Remark 1.
The model of deception attacks considered in this paper is established in (6). A Bernoulli distribution is applied to describe the random property of deception attacks. It is easy to say that αt=1 or αt=0 means sensor measurement is under deception attacks or not, respectively. Furthermore, the deception attacks are supposed to be norm bounded in (6), since there usually exists an upper bound for the attack signals to avoid detection.We define the initial condition of system (5) as xf0=xf0, and for any t∈−d2,0, assume that xft=xf0. Afi, Bfi, and Cfi are filter gains to be determined.Then, defineeft=zt−zft and ηt=xTtxfTtT. By combining systems (2) and (5), we can obtain the following filtering error system:(8)E∧η˙t=∑i=1r∑j=1rhiθthjθtA∧ij−α∧tB∧f1ijηt+A∧dij−α∧tB∧f2ijηt−dt+B∧ij−α∧tB∧f3ijωt+αB∧f4jσt+α∧tB∧f4jσt,eft=∑i=1r∑j=1rhiθthjθtL∧ijηt,ηt=ϕ∧t=ϕTtxf0TT,∀t∈−d2,0,where(9)E∧=E00E,A∧ij=Ai01−αBfjCiAfj,A∧dij=Adi01−αBfjCdi0,B∧ij=Bi1−αBfjDi,B∧f1ij=00BfjCi0,B∧f2ij=00BfjCdi0,B∧f3ij=0BfjDi,B∧f4j=0Bfj,L∧ij=Li−Cfj,α∧t=αt−α.Denoting(10)A∧t=∑i=1r∑j=1rhiθthjθtA∧ij−α∧tB∧f1ij,A∧dt=∑i=1r∑j=1rhiθthjθtA∧dij−α∧tB∧f2ij,B∧t=∑i=1r∑j=1rhiθthjθtB∧ij−α∧tB∧f3ij,B∧1t=∑j=1rhjθtαB∧f4j+α∧tB∧f4j,L∧t=∑i=1r∑j=1rhiθthjθtL∧ij,system (8) can be given as(11)E∧η˙t=A∧tηt+A∧dtηt−dt+B∧tωt+B∧1tσt,eft=L∧tηt.Remark 2.
Due to a stochastic variable of Bernoulli distribution, filtering error system (11) is established as a stochastic T-S fuzzy singular system. In this condition, the definition of admissibility in [19] does not work for the stochastic T-S fuzzy singular system in (11) anymore. Motivated by the stochastic admissibility defined in [36] of the discrete time case, we can generalize the definition of admissibility in [19] naturally and have the following definition of stochastic admissibility for system (11).Definition 1 (see [36]).
(1)
System (11) is said to be stochastically regular if detEsE∧−A∧tis not identically zero(2)
System (11) is said to be stochastically impulse-free if degdetEsE∧−A∧t=rankE∧(3)
System (11) with ωt=0 is said to be stochastically stable if there exists a scalar ℳ>0 such that(12)limt⟶∞E∫0tηTs,ϕ∧tηs,ϕ∧tds≤ℳ,whereηs,ϕ∧t represents the system solution(4)
System (11) with ωt=0 is said to be stochastically admissible if it is stochastically regular, impulse-free, and stableDefinition 2 (see [33]).
Give real symmetric matricesℚ and ℝ and matrix S. For a scalar γ>0, if the equation(13)Eef,ℚeft∗+2Eef,Sωt∗+Eω,ℝωt∗≥γEω,ωt∗holds under zero initial state with t∗>0, system (11) is said to be strictly ℚ,S,ℝ-γ-dissipative. Furthermore, we suppose ℚ≤0 and −ℚ=ℚ−Tℚ− for some ℚ−.Before discussing the main results, we present a few lemmas that we need to use.Lemma 1 (see [37]).
For a matrixZ>0, the following inequality holds:(14)−k2−k1∫k1k2x˙TsZx˙sds≤−ΠTdiagZ,3Z,5ZΠ,where Π=xk2−xk1xk2+xk1−2/k2−k1∫k1k2xsdsxk2−xk1−6/k2−k1∫k1k2ςk1,k2sxsds and ςk1,k2s=2s−k1/k2−k1−1.Lemma 2 (see [38]).
Given a real scalarβ∈0,1, matrices Z1>0 and Z2>0, and any matrices W1 and W2, the inequality(15)1βZ10011−βZ2≥Z1+1−βT11−βW1+βW2åZ2+βT2holds, where T1=Z1−W2Z2−1W2T and T2=Z2−W1TZ1−1W1.Lemma 3 (see [13]).
If(16)Ξii<0,i=1,2,…,r,2r−1Ξii+Ξij+Ξji<0,i≠j,i,j=1,2,…,r,then ∑i=1r∑j=1rhiθthjθtΞij<0, where hiθt,i=1,2,…,r, satisfy (3) and (4).


## 3. Main Results

In this section, we will design a dissipative filter for system (1) with randomly occurring deception attacks. First, we give some notations in order to simplify the presentation:(17)ς1s=2s+d1d1−1,ς2s=2s+dtdt−d1−1,ς3s=2s+d2d2−dt−1,ς4s=2s+d2d2−d1−1,ζt=ηTtE∧T∫t−d1tηTsE∧Tds∫t−d1tς1sηTsE∧Tds∫t−d2t−d1ηTsE∧Tdsd2−d1∫t−d2t−d1ς4sηTsE∧TdsT,ξ1t=ηTtη˙TtE∧TηTt−d1ηTt−dtηTt−d2T,ξ2t=1d1∫t−d1tηTsE∧Tds∫t−d1tς1sηTsE∧TdsT,ξ3t=1dt−d1∫t−dtt−d1ηTsE∧Tds∫t−dtt−d1ς2sηTsE∧TdsT,ξ4t=1d2−dt∫t−d2t−dtηTsE∧Tds∫t−d2t−dtς3sηTsE∧TdsT,ξ5t=dt−d1ξ3t,ξ6t=d2−dtξ4t,ξt=ξ1Ttξ2Ttξ3Ttξ4Ttξ5Ttξ6TtσTtωTtT,Π1=Ψ3Ψ4T2R˜2W1åR˜2Ψ3Ψ4,Π2=Ψ3TW2R˜2−1W2TΨ3,Π3=Ψ3Ψ4TR˜2W2å2R˜2Ψ3Ψ4,Π4=Ψ4TW1TR˜2−1W1Ψ4,Ψ1dt=ℐ1TE∧Td1ℐ6Td1ℐ7Tℐ12T+ℐ14Td2−dtℐ12T+ℐ15T+dt−d1ℐ13T−ℐ14TT,Ψ2=ℐ2Tℐ1T−ℐ3TE∧Tℐ1T+ℐ3TE∧T−2ℐ6Tℐ3T−ℐ5TE∧Td2−d1ℐ3T+ℐ5TE∧T−2ℐ12T+ℐ14TT,Ψ3=ℐ3T−ℐ4TE∧Tℐ3T+ℐ4TE∧T−2ℐ8Tℐ3T−ℐ4TE∧T−6ℐ9TT,Ψ4=ℐ4T−ℐ5TE∧Tℐ4T+ℐ5TE∧T−2ℐ10Tℐ4T−ℐ5TE∧T−6ℐ11TT,f1dt=dt−d1ℐ8ℐ9−ℐ12ℐ13,f2dt=d2−dtℐ10ℐ11−ℐ14ℐ15,ℐi=02n×i−12nI2n02n×15−i2n02n×m+l,i=1,…,15,ℐ16=0m×30nIm0m×l,ℐ17=0l×30n+mIl,ℐ18=In0n×29n+m+l,ℐ19=0n×6nIn0n×23n+m+l,R˜2=diagR2,3R2,5R2.Theorem 1.
For given scalarsα, γ>0, 0≤d1<d2, and 0≤ϖ<1, matrices W and S, symmetric matrices ℚ and ℝ with ℚ≤0, and full column rank matrix Γ with E∧TΓ=0, system (11) is said to be stochastically admissible and strictly ℚ,S,ℝ-γ-dissipative, if there exist matrices P=P11P12P13P14P15∗P22P23P24P25∗∗P33P34P35∗∗∗P44P45∗∗∗∗P55>0, Q1>0, Q2>0, Q3>0, R1>0, R2>0, Υ, U1, U2, W1, W2, T1, and T2 such that the following matrix inequalities hold:(18)Ωd1−Π1+Π2<0,(19)Ωd2−Π3+Π4<0,where(20)Ωdt=symΨ1TdtPΨ2+ℐ1TΥΓTℐ2+U1f1dt+U2f2dt+ℐ1TT1+ℐ2TT2Φ−L∧tℐ1TSℐ17+αCtℐ18+Cdtℐ19+Dtℐ17TWTWCtℐ18+Cdtℐ19+Dtℐ17−αℐ16Tℐ16−L∧tℐ1TℚL∧tℐ1−ℐ17Tℝ−γIℐ17+ℐ1TQ1+Q2+Q3ℐ1−ℐ3TQ1ℐ3−1−ϖℐ4TQ2ℐ4−ℐ5TQ3ℐ5+ℐ2Td12R1+d2−d12R2ℐ2−ℐ1−ℐ3TE∧TR1E∧ℐ1−ℐ3−3E∧ℐ1+E∧ℐ3−2ℐ6TR1E∧ℐ1+E∧ℐ3−2ℐ6−5E∧ℐ1−E∧ℐ3−6ℐ7TR1E∧ℐ1−E∧ℐ3−6ℐ7,Φ=A˜t−I2n02nA˜dt02n×22nB˜1tB˜t,A˜t=∑i=1r∑j=1rhiθthjθtA∧ij,A˜dt=∑i=1r∑j=1rhiθthjθtA∧dij,B˜1t=∑j=1rhjθtαB∧f4j,B˜t=∑i=1r∑j=1rhiθthjθtB∧ij,Ct=∑i=1rhiθtCi,Cdt=∑i=1rhiθtCdi,Dt=∑i=1rhiθtDi.Proof.
First, we will show the stochastic admissibility for system (11) with ωt=0. It can be obtained from (18) and (19) that(21)M11−L∧tTℚL∧t+Q1+Q2+Q3M12♯1åM22+d12R1+d2−d12R2♯2åå♯3<0,where(22)M11=symT1A˜t+E∧TP12E∧+E∧TP13E∧−E∧TR1E∧−3E∧TR1E∧−5E∧TR1E∧,M12=E∧TP11+ΥΓT−T1+A˜tTT2T,M22=−T2+T2T.
♯1, ♯2, and ♯3 represent matrices which have compatible dimensions and will not be used in the following discussion. Since Q1>0, Q2>0, Q3>0, R1>0, R2>0, and −ℚ≥0, we have(23)M11M12åM22<0.
Premultiplying and postmultiplying (23) by IA˜tT and IA˜tTT, we can obtain(24)symE∧TP11+ΥΓTA˜t+E∧TP12E∧+E∧TP13E∧−E∧TR1E∧−3E∧TR1E∧−5E∧TR1E∧<0.
According to Theorem 10.1 of [1] and from (24), we have that the pair E∧,A˜t is regular and impulse-free. It should be noted that(25)detEsE∧−A∧t=detsE∧−A˜t,degdetEsE∧−A∧t=degdetsE∧−A˜t.
By Definition1, we can conclude system (11) is stochastically regular and impulse-free. Choose the LKF as follows:(26)Vηt=ζTtPζt+∫t−d1tηTsQ1ηsds+∫t−dttηTsQ2ηsds+∫t−d2tηTsQ3ηsds+d1∫−d10∫t+θtη˙TsE∧TR1E∧η˙sdsdθ+d2−d1∫−d2−d1∫t+θtη˙TsE∧TR2E∧η˙sdsdθ.
Set(27)ξ∧t=ξ1Ttξ2Ttξ3Ttξ4Ttξ5Ttξ6TtT,ℐ∧i=02n×i−12nI2n02n×15−i2n,i=1,…,15,ℐ∧16=0m×30nIm,ℐ∧18=In0n×29n+m,ℐ∧19=0n×6nIn0n×23n+m,Ψ∧1dt=ℐ∧1TE∧Td1ℐ∧6Td1ℐ∧7Tℐ∧12T+ℐ∧14Td2−dtℐ∧12T+ℐ∧15T+dt−d1ℐ∧13T−ℐ∧14TT,Ψ∧2=ℐ∧2Tℐ∧1T−ℐ∧3TE∧Tℐ∧1T+ℐ∧3TE∧T−2ℐ∧6Tℐ∧3T−ℐ∧5TE∧Td2−d1ℐ∧3T+ℐ∧5TE∧T−2ℐ∧12T+ℐ∧14TT,Ψ∧3=ℐ∧3T−ℐ∧4TE∧Tℐ∧3T+ℐ∧4TE∧T−2ℐ∧8Tℐ∧3T−ℐ∧4TE∧T−6ℐ∧9TT,Ψ∧4=ℐ∧4T−ℐ∧5TE∧Tℐ∧4T+ℐ∧5TE∧T−2ℐ∧10Tℐ∧4T−ℐ∧5TE∧T−6ℐ∧11TT,f∧1dt=dt−d1ℐ∧8ℐ∧9−ℐ∧12ℐ∧13,f∧2dt=d2−dtℐ∧10ℐ∧11−ℐ∧14ℐ∧15,Φ∧=A˜t−I2n02nA˜dt02n×22nB˜1t,Π∧1=Ψ∧3Ψ∧4T2R˜2W1åR˜2Ψ∧3Ψ∧4,Π∧2=Ψ∧3TW2R˜2−1W2TΨ∧3,Π∧3=Ψ∧3Ψ∧4TR˜2W2å2R˜2Ψ∧3Ψ∧4,Π∧4=Ψ∧4TW1TR˜2−1W1Ψ∧4.
Define the infinitesimalℒ as ℒVηt=limΔ⟶0+1/ΔEVηt+Δ|ηt−Vηt. Along the trajectory of (11) with ωt=0, we have(28)EℒVηt≤ξ∧TtsymΨ∧1TdtPΨ∧2+ℐ∧1TQ1+Q2+Q3ℐ∧1−ℐ∧3TQ1ℐ∧3−1−ϖℐ∧4TQ2ℐ∧4−ℐ∧5TQ3ℐ∧5+ℐ∧2Td12R1+d2−d12R2ℐ∧2ξ∧t−d1∫t−d1tη˙TsE∧TR1E∧η˙sds−d2−d1∫t−d2t−d1η˙TsE∧TR2E∧η˙sds.
Using Lemma1, it can be generated that(29)−d1∫t−d1tη˙TsE∧TR1E∧η˙sds≤−ξ∧Ttℐ∧1−ℐ∧3TE∧TR1E∧ℐ∧1−ℐ∧3+3E∧ℐ∧1+E∧ℐ∧3−2ℐ∧6TR1E∧ℐ∧1+E∧ℐ∧3−2ℐ∧6+5E∧ℐ∧1−E∧ℐ∧3−6ℐ∧7TR1E∧ℐ∧1−E∧ℐ∧3−6ℐ∧7ξ∧t,(30)−d2−d1∫t−d2t−d1η˙TsE∧TR2E∧η˙sds=−d2−d1∫t−dtt−d1η˙TsE∧TR2E∧η˙sds+∫t−d2t−dtη˙TsE∧TR2E∧η˙sds≤−ξ∧Ttd2−d1dt−d1Ψ∧3TR˜2Ψ∧3+d2−d1d2−dtΨ∧4TR˜2Ψ∧4ξ∧t.
Then, by Lemma2, we can obtain(31)−d2−d1∫t−d2t−d1η˙TsE∧TR2E∧η˙sds≤−d2−dtd2−d1ξ∧TtΠ∧1−Π∧2ξ∧t−dt−d1d2−d1ξ∧TtΠ∧3−Π∧4ξ∧t.
Note thatξ∧5t=dt−d1ξ∧3t and ξ∧6t=d2−dtξ∧4t. Thus, it is easy to see that for matrices U1 and U2,(32)2ξ∧TtU1f∧1dt+U2f∧2dtξ∧t=0.
Given matricesT1 and T2 from (11), we have(33)2EηTtT1+E∧η˙tTT2−E∧η˙t+A∧tηt+A∧dtηt−dt+B∧1tσt=0.
It is easy to obtain from (33) that(34)2ξ∧Ttℐ∧1TT1+ℐ∧2TT2Φ∧ξ∧t=0.
NoticingE∧TΓ=0, we can obtain that for matrix Υ with appropriate dimension,(35)2ξ∧Ttℐ∧1TΥΓTℐ∧2ξ∧t=0.
Considering the definition of deception attacks, it is easy to obtain(36)yTtWTWyt−σTtσt≥0,which means(37)ξ∧TtαCtℐ∧18+Cdtℐ∧19TWTWCtℐ∧18+Cdtℐ∧19−αℐ∧16Tℐ∧16ξ∧t≥0.
From (28)–(37), it can be verified that(38)EℒVηt≤ξ∧TtΩ∧dt−d2−dtd2−d1Π∧1−Π∧2−dt−d1d2−d1Π∧3−Π∧4ξ∧t,where(39)Ω∧dt=symΨ∧1TdtPΨ∧2+ℐ∧1TΥΓTℐ∧2+U1f∧1dt+U2f∧2dt+ℐ∧1TT1+ℐ∧2TT2Φ∧−ℐ∧3TQ1ℐ∧3+αCtℐ∧18+Cdtℐ∧19TWTWCtℐ∧18+Cdtℐ∧19−αℐ∧16Tℐ∧16+ℐ∧1TQ1+Q2+Q3ℐ∧1−1−ϖℐ∧4TQ2ℐ∧4−ℐ∧5TQ3ℐ∧5+ℐ∧2Td12R1+d2−d12R2ℐ∧2−ℐ∧1−ℐ∧3TE∧TR1E∧ℐ∧1−ℐ∧3−3E∧ℐ∧1+E∧ℐ∧3−2ℐ∧6TR1E∧ℐ∧1+E∧ℐ∧3−2ℐ∧6−5E∧ℐ∧1−E∧ℐ∧3−6ℐ∧7TR1E∧ℐ∧1−E∧ℐ∧3−6ℐ∧7.
We can obtain from (18) and (19) that EℒVηt<0. Hence, we can find a scalar β>0 so that EℒVηt≤−βηt2. Using Dynkin’s formula, it can be derived that(40)EVηt−EVη0≤−βE∫0tηs2ds,which implies(41)E∫0tηs2ds≤β−1EVη0.
From Definition1, it can be concluded that system (11) in the condition of ωt=0 is considered to be stochastically stable.
To verify the dissipativity for system (11), by (18), (19), and (25), it can be derived that(42)ℒVηt−efTtℚeft−2efTtSωt−ωTtℝ−γIωt=ξTtΩdt−d2−dtd2−d1Π1−Π2−dt−d1d2−d1Π3−Π4ξt<0.
Thus, for anyt∗≥0, it is easy to see that(43)EVηt∗−Vη0−∫0t∗efTtℚeftdt−∫0t∗2efTtSωtdt−∫0t∗ωTtℝ−γIωtdt<0.
ConsideringEVηt∗≥0 and Vη0=0 under zero initial condition, by (43), we can obtain that (13) holds, which shows that system (11) is strictly ℚ,S,ℝ−γ−dissipative.Theorem 2.
For given scalarsα, γ>0, 0≤d1<d2, and 0≤ϖ<1, matrices W and S, symmetric matrices ℚ and ℝ with ℚ≤0 and −ℚ=ℚ−Tℚ−, and full column rank matrix Γ with E∧TΓ=0, if there are matrices P=P11P12P13P14P15∗P22P23P24P25∗∗P33P34P35∗∗∗P44P45∗∗∗∗P55>0, Q1>0, Q2>0, Q3>0, R1>0, and R2>0, an invertible matrix Y, matrices Υ, U1, U2, W1, W2, Y1, Y2, Y3, Y4, A¯fj, B¯fj, and C¯fj, j=1,2,…,r, such that the following linear matrix inequalities hold:(44)Ξiidk<0,i=1,2,…,r,k=1,2,(45)Ξijdk+Ξjidk<0,i<j,i=1,2,…,r,k=1,2,where(46)Ξijd1=Ωijd1−Π1Ψ3TW2L¯ije1Tℚ−å−R˜20åå−I,Ξijd2=Ωijd2−Π3Ψ4TW1TL¯ije1Tℚ−å−R˜20åå−I,Ωijdt=symΨ1TdtPΨ2+ℐ1TΥΓTℐ2+U1f1dt+U2f2dt+ℐ1TΘ¯ij+ℐ2TΘ⌣ij−L¯ijℐ1TSℐ17−ℐ17Tℝ−γIℐ17+αCiℐ18+Cdiℐ19+Diℐ17TWTWCiℐ18+Cdiℐ19+Diℐ17−αℐ16Tℐ16+ℐ1TQ1+Q2+Q3ℐ1−ℐ3TQ1ℐ3−1−ϖℐ4TQ2ℐ4−ℐ5TQ3ℐ5+ℐ2Td12R1+d2−d12R2ℐ2−ℐ1−ℐ3TE∧TR1E∧ℐ1−ℐ3−3E∧ℐ1+E∧ℐ3−2ℐ6TR1E∧ℐ1+E∧ℐ3−2ℐ6−5E∧ℐ1−E∧ℐ3−6ℐ7TR1E∧ℐ1−E∧ℐ3−6ℐ7,Θ¯ij=A¯ij−T102nA¯dij02n×22nB¯1jB¯ij,Θ⌣ij=A⌣ij−T202nA⌣dij02n×22nB⌣1jB⌣ij,A¯ij=Y1Ai+1−αB¯fjCiA¯fjY2Ai+1−αB¯fjCiA¯fj,A⌣ij=Y3Ai+1−αB¯fjCiA¯fjY4Ai+1−αB¯fjCiA¯fj,A¯dij=Y1Adi+1−αB¯fjCdi0Y2Adi+1−αB¯fjCdi0,A⌣dij=Y3Adi+1−αB¯fjCdi0Y4Adi+1−αB¯fjCdi0,B¯1j=B⌣1j=αB¯fjαB¯fj,B¯ij=Y1Bi+1−αB¯fjDiY2Bi+1−αB¯fjDi,B⌣ij=Y3Bi+1−αB¯fjDiY4Bi+1−αB¯fjDi,L¯ij=Li−C¯fj,T1=Y1YY2Y,T2=Y3YY4Y.Then, fuzzy filter (5) can guarantee system (11) to be stochastically admissible and strictly ℚ,S,ℝ-γ-dissipative. Furthermore, from LMIs (44) and (45), the parameters of fuzzy filter (5) can be obtained by(47)Afj=Y−1A¯fj,Bfj=Y−1B¯fj,Cfj=C¯fj.Proof.
Using Lemma3 and from (44) and (45), it can be verified that(48)∑i=1r∑j=1rhiθthjθtΩijd1−L¯ijℐ1TℚL¯ijℐ1−Π1+Π2<0,(49)∑i=1r∑j=1rhiθthjθtΩijd2−L¯ijℐ1TℚL¯ijℐ1−Π3+Π4<0.
By substituting the filter parameters in (44) into (45) and (46), from Theorem 1, we can proof that Theorem 2 holds.Remark 3.
From Theorem1 and Theorem 2, we can find that fuzzy filter (5) is designed successfully for a class of time-varying-delay T-S fuzzy singular systems subject to randomly occurring deception attacks based on the LKF approach and LMI techniques. According to Definitions 1 and 2, the filtering error system is guaranteed to be stochastically admissible and satisfy strictly ℚ,S,ℝ-γ-dissipativity by conditions (44) and (45). Furthermore, Lemmas 1 and 2 are applied in (29)–(31) to lessen the estimation gaps of (47), which have been shown to reduce the conservatism in dealing with time delays in [37,38], respectively.


## 4. Numerical Example

Consider a time-delay nonlinear system which is borrowed from [39]:(50)1+acosφtφ¨t=−bφ˙3t+cφt+cdφt−dt+fωt,where φ˙t is supposed to satisfy φ˙t<ψ. Choose ψ=2, a=−0.01, b=0.5, c=−0.5, cd=−0.05, f=−0.2, and dt=0.3+0.2sin0.5t. Set x1t=φt, x2t=φ˙t, and x3t=φ¨t. Introduce a new variable xt=x1Ttx2Ttx3TtT, and the nonlinear system in (31) can be exactly modeled as(51)Ex˙t=∑i=13hiAixt+Adixt−dt+Biωt,yt=∑i=3rhiCixt+Cdixt−dt+Diωt,zt=∑i=3rhiLixt.The parameters are given as(52)h1=x22tψ2+2,h2=1+cosx1tψ2+2,h3=ψ2−x22t+1−cosx1tψ2+2,E=100010000,A1=010001c−bψ2+2a−1,A2=010001c0−a−1−aψ2,A3=010001c0a−1,Ad1=Ad2=Ad3=000000cd00,B1=B2=B3=00f,C1=010,C2=0.60−0.4,C3=−0.20.10.55,Cd1=−0.250.30,Cd2=0.12−0.60,Cd3=00.80,L1=0.1−0.51,L2=10.5−0.5,L3=−102,D1=−0.36,D2=2.5,D3=1,W=0.09,α=0.2,ℚ=−0.01,S=−1.1,ℝ=3.Forγ=0.5, we can obtain from conditions (44)–(47) in Theorem 2 that the desired filter parameters are(53)Af1=−0.25020.5137−0.00010.5429−1.85291.0002−0.28680.4149−0.9900,Af2=−0.2839−0.10280.02900.3353−0.28480.9769−0.3028−0.0022−1.0297,Af3=−0.3269−0.19420.37350.3252−0.37900.9439−0.27170.1739−1.0781,Bf1=0.1762−1.64140.3855,Bf2=−0.07240.05770.0994,Bf3=0.6792−0.1021−0.1602,Cf1=−0.23130.8816−1.0000,Cf2=−2.38802.73100.5015,Cf3=0.29351.2410−2.0008.Next, we will provide the simulation results to demonstrate the effectiveness of our filter design method against randomly occurring deception attacks. Let the initial condition be−0.10.10T and the disturbance be ωt=0.1e−0.5tsint. Figure 1 shows the system state response xt. The deception attack function is given as σt=−tanh0.09yt. The sensor measurement yat under randomly occurring deception attack is given in Figure 2. Figure 3 is the filtering error eft. From Figures 1–3, we can find that our results are effective.Figure 1
System statext.Figure 2
Sensor measurementyat under randomly occurring deception attack.Figure 3
Filtering erroreft.


## 5. Conclusions

In this work, a method for dealing with the problem of dissipative filtering associated with T-S fuzzy singular systems with time-varying delays subject to deception attacks has been developed. Since the measurement output is supposed to encounter random deception attacks based on a Bernoulli distribution during signal transmission, the filtering error system is modeled by a stochastic singular T-S fuzzy system and the definition of stochastic admissibility for this kind of system has been presented. By using the LKF approach and LMI techniques, sufficient delay-dependent results have been generated, where the filtering error system can be stochastically admissible and strictlyℚ,S,ℝ-dissipative against randomly occurring deception attacks. Besides, the desired filter parameters can be obtained by these solvable conditions. Finally, a frequently used example of the nonlinear system has been given to show the effectiveness of our work.



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