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

)is 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.)emeasurement 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 systemmodeled by a stochastic singular T-S fuzzy system is established and stochastic admissibility for this kind of system is defined firstly. )en, 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 strictly (Q,S,R)-dissipative against randomly occurring deception attacks. Moreover, parameters of the desired filter can be obtained via the solutions of the established conditions. )e validity of our work is illustrated through a mostly used example of the nonlinear system.


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][4][5], sliding mode control for discretetime singular delay systems [6][7][8], and H ∞ control for singular time-delay systems with Markovian jump parameters [9][10][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.
us, 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][14][15][16][17][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][26][27][28][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. erefore, 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 timevarying complex networks with deception attacks, and event-triggered filter design for T-S fuzzy systems with deception attacks had been investigated in [30][31][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. e dissipativity has been analysed for large amounts of nonlinear systems and widely used in control theory and practice [33][34][35]. All the aforementioned facts motivate our research. e 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 delaydependent conditions are established to guarantee the stochastic admissibility and strictly (Q, 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. R n denotes the Euclidean space with n dimension; R n×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 + A T is described by sym(A); L 2 [0, ∞) is the space of integral vector over [0, ∞); for any real function x, y ∈ L 2 [0, ∞) and real matrix M, we define 〈x, My〉 d �

Problem Formulation
For the description of an interval time-varying-delay nonlinear singular system, a delayed T-S fuzzy model is adopted in terms of r plant rules as follows.
where x(t) ∈ R n , y(t) ∈ R m , and ω(t) ∈ R l stand for the state vector, the measurement output, and the disturbance input, respectively; z(t) ∈ R q is the signal to be estimated; ϕ(t) is the initial condition; the premise variable vector is described by , and the fuzzy sets are μ ij (i � 1, . . . , r, j � 1, . . . , p); the time-varying delay d(t) satisfies the conditions of 0 ≤ d 1 ≤ d(t) ≤ d 2 , _ d(t) ≤ ϖ, and d 1 , d 2 , and 0 ≤ ϖ < 1 are constant scalars; E ∈ R n×n may be a singular matrix with rank(E) � g ≤ n; and A i , A di , B i , C i , C di , D i , and L i are known real constant matrices with appropriate dimensions. en, we can generate the model of the T-S fuzzy singular systems when time-varying delay is considered: and μ ij (θ j (t)) stand for the grade of membership for θ j (t) in μ ij . It is easy to verify that In this paper, for calculating the signal z(t), a fuzzy filter is derived as where x f (t) ∈ R n and z f (t) ∈ R q represent the filter state and the filter output, respectively. y a (t) is the sensor 2 Complexity measurement under randomly occurring deception attacks which can be given as σ(t) defines the deception signal imported into the output which is assumed to satisfy ‖σ(t)‖ ≤ ‖Wy(t)‖, and W is considered to be an appropriate dimension matrix; α(t) is a Bernoulli-distributed variable and we have the following assumptions on α(t): Remark 1. e 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 α(t) � 1 or α(t) � 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 By combining systems (2) and (5), we can obtain the following filtering error system: Denoting system (8) can be given as 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 Complexity 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).
(1) System (11) is said to be stochastically regular if where η(s, ϕ ∧ (t)) represents the system solution (4) System (11) with ω(t) � 0 is said to be stochastically admissible if it is stochastically regular, impulse-free, and stable Definition 2 (see [33]). Give real symmetric matrices Q and R and matrix S. For a scalar c > 0, if the equation holds under zero initial state with t * > 0, system (11) is said to be strictly Before discussing the main results, we present a few lemmas that we need to use.

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: Complexity Theorem 1. For given scalars α, c > 0, 0 ≤ d 1 < d 2 , and 0 ≤ ϖ < 1, matrices W and S, symmetric matrices Q and R with Q ≤ 0, and full column rank matrix Γ with E ∧ T Γ � 0, system (11) is said to be stochastically admissible and strictly (Q, S, R)-c-dissipative, if there exist matrices P � P 11 P 12 P 13 P 14 P 15 * P 22 P 23 P 24 P 25 * * P 33 P 34 P 35 * * * P 44 P 45 * * * * P 55 and T 2 such that the following matrix inequalities hold: where Proof. First, we will show the stochastic admissibility for system (11) with ω(t) � 0. It can be obtained from (18) and (19) that where ♯ 1 , ♯ 2 , and ♯ 3 represent matrices which have compatible dimensions and will not be used in the following discussion.

by [I A(t) T ] and [I A(t) T ] T , we can obtain
According to eorem 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 By Definition 1, we can conclude system (11) is stochastically regular and impulse-free. Choose the LKF as follows: Define . Along the trajectory of (11) with ω(t) � 0, we have Using Lemma 1, it can be generated that en, by Lemma 2, we can obtain 8 Complexity us, it is easy to see that for matrices U 1 and U 2 , Given matrices T 1 and T 2 from (11), we have It is easy to obtain from (33) that Noticing E ∧ T Γ � 0, we can obtain that for matrix Υ with appropriate dimension, Considering the definition of deception attacks, it is easy to obtain which means From (28)- (37), it can be verified that where We can obtain from (18) and (19) that E LV(η t ) < 0. Hence, we can find a scalar β > 0 so that E LV(η t ) ≤ − β‖η(t)‖ 2 . Using Dynkin's formula, it can be derived that which implies From Definition 1, it can be concluded that system (11) in the condition of ω(t) � 0 is considered to be stochastically stable.

Furthermore, from LMIs (44) and (45), the parameters of fuzzy filter (5) can be obtained by
Proof. Using Lemma 3 and from (44) and (45), it can be verified that By substituting the filter parameters in (44) into (45) and (46), from eorem 1, we can proof that eorem 2 holds. □ Remark 3. From eorem 1 and eorem 2, we can find that fuzzy filter (5) is designed successfully for a class of timevarying-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)-c-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.

Numerical Example
Consider a time-delay nonlinear system which is borrowed from [39]: ] T , and the nonlinear system in (31) can be exactly modeled as e parameters are given as 12 Complexity 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 [− 0.1 0.1 0] T and the disturbance be ω(t) � 0.1e − 0.5t sin t. Figure 1 shows the system state response x(t).
e deception attack function is given as σ(t) � − tanh(0.09y(t)). e sensor measurement y a (t) under randomly occurring deception attack is given in Figure 2. Figure 3 is the filtering error e f (t). From Figures 1-3, we can find that our results are effective.

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 strictly (Q, 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.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest.