Finite-Time Nonfragile Dissipative Filter Design for Wireless Networked Systems with Sensor Failures

In this study, the problem of finite-time nonfragile dissipative-based filter design for wireless sensor networks that is described by discrete-time systems with time-varying delay is investigated. Specifically, to reduce the energy consumption of wireless sensor networks, it is assumed that the signal is not transmitted at each instant and the transmission process is stochastic. By constructing a suitable Lyapunov-Krasovskii functional and employing discrete-time Jensen’s inequality, a new set of sufficient conditions is established in terms of linear matrix inequalities such that the augmented filtering system is stochastically finite-time bounded with a prescribed dissipative performance level. Meanwhile, the desired dissipative-based filter gain matrices can be determined by solving an optimization problem. Finally, two numerical examples are provided to illustrate the effectiveness and the less conservatism of the proposed filter design technique.


Introduction
In the past few decades, wireless sensor networks (WSNs) have gained considerable attention due to their wide range of applications in various fields, such as mobile communications, target tracking, robotic systems, military, environmental sensing, and monitoring of traffic [1,2].WSNs normally consist of a large number of distributed nodes called sensor nodes, where the communication between the nodes is through radio signals.Since the sensors are battery powered, energy consumption is one of the main issues in WSNs.In recent years, different types of protocols have been proposed to reduce the energy consumption of the sensors in WSNs.For instance, in [3], the nonfragile randomly occurring filter gain variation problem is studied for a class of WSNs with energy constraint by using the Lyapunov technique and linear matrix inequality (LMI) approach.The multirate transmission protocols discussed in [4][5][6][7] are deterministic, since the transmission instant is pre-set which is not allowed to vary and this may lead to poor performance estimation.The authors in [8] considered not only the transmission rate of signals but also the successive nontransmissions, which leads to much conservatism.
In many practical problems, it is important to focus on the stability and filtering issue of a system over a prescribed time interval, in which the state trajectories remain within a predetermined bound over a given finite-time interval under some given initial conditions [9,10].Therefore, much attention has been given to the problem of finite-time filtering for dynamical systems with the use of Lyapunov technique and LMI approach [11][12][13].By constructing a probabilitydependent Lyapunov-Krasovskii functional, a set of sufficient conditions is established in [14] to obtain an energy-topeak filter design for networked Markov switched singular systems over a finite-time interval.Wang et al. [15] studied the finite-time filter design problem of switched impulsive linear systems with parameter uncertainties and sensor induced faults by using the mode-dependent Lyapunov-like function approach.Sathishkumar et al. [16] developed a finite-time  ∞ filter design for a class of uncertain nonlinear discretetime Markovian jump systems represented by Takagi-Sugeno fuzzy model with nonhomogeneous jump process.The 2 Complexity asynchronous resilient controller design problem is investigated in [17], for a class of nonlinear switched systems with time delays and uncertainties in a given finite-time interval.The problems of finite-time stability and finite-time stabilisation for T-S fuzzy system with time-varying delay are investigated in [18].The authors in [19] have designed a finite-time sliding mode controller for a class of conic-type nonlinear systems with time delays and mismatched external disturbances.
On another active research front, dissipativity theory was introduced by Willems in [20], which plays an important role in a wide range of fields, such as systems, circuits, and networks.Compared with passivity and  ∞ performance, dissipative system theory is a more general criterion and it is used for the analysis and the synthesis of dynamical control systems [21,22].Specifically, dissipativity means that the increase in energy storage in the system cannot exceed the energy supplied from outside the systems.Recently, few important results have been reported on dissipative-based filtering for various classes of time-varying delay systems.To mention a few, Feng and Lam [23] discussed the robust reliable dissipative filtering problem of uncertain discretetime singular system with interval time-varying delay and sensor failures, where a set of conditions was derived in terms of LMIs which makes the filtering error singular system regular, causal, asymptotically stable, and strictly (Q, S, R)dissipative.In [24], a set of sufficient conditions is developed by using reciprocally convex approach with Lyapunov technique for reliable dissipativity of Takagi-Sugeno fuzzy systems in the presence of time-varying delays and sensor failures.A new criterion of stability analysis for generalized neural networks subject to time-varying delayed signals is investigated in [25].By employing the LMI approach, a new set of sufficient conditions is obtained in [26] for the existence of reliable dissipative filter which makes the filtering error system stochastically stable and strictly (Q, S, R)-dissipative.
On the other hand, perturbations often appear in the filter gain, which may cause instability in dynamic systems and usually lead to unsatisfactory performances.However, in practical problems, the presence of small uncertainties and inaccuracies during the implementation of filters may provide poor performance of the systems [27].Therefore, it is important and necessary to design a filter that should be reliable and insensitive to some amount of gain fluctuations [28][29][30].Xu et al. [31] studied the problem of passive control for fuzzy Markov jump systems with packet dropouts, where a nonfragile asynchronous controller is designed to guarantee that the closed-loop system is mean-square stable with a satisfactory passivity performance index.A novel method to address a proportional integral observer design for the actuator and sensor faults estimation based on Takagi-Sugeno fuzzy model with unmeasurable premise variables is presented in [32].The nonfragile finite-time filtering problem is studied in [33] for a class of nonlinear Markovian jumping systems with time delays and uncertainties.
It is worth mentioning that, so far in the literature, only few works have been reported on finite-time filter design for wireless sensor networks.However, all the aforementioned works have not unified the external disturbances, time-varying delay, sensor failures, and filter gain variations, despite its practical importance.Motivated by the above, the reliable finite-time dissipative-based nonfragile filtering problem for discrete-time systems with time-varying delays and sensor failures has been investigated in the present study.
The main contributions of this paper are given as follows: (i) Dissipative-based finite-time filter design problem is formulated for a class of WSNs with energy constraint and filter gain variations, which is represented by discrete-time systems with time-varying delay.
(ii) A reliable nonfragile filter is designed such that the augmented filtering system is stochastically finitetime bounded and dissipative.The proposed filter design includes  ∞ filter and passivity filter designs as special cases.
(iii) A set of sufficient conditions is developed in terms of LMIs to obtain the desired nonfragile filter design.
(iv) A unified filter design is proposed to deal with the external disturbances, time-varying delay, sensor failures, and filter gain variations, which makes the system more practical.
Finally, two numerical examples with simulation results are provided to demonstrate the effectiveness of the obtained results.
The brief outline of this paper is as follows.In Section 2, the problem of WSNs with time-varying delay and sensor faults is formulated, and some essential definitions and lemmas are given.The finite-time boundedness of the filtering error system is analyzed and a nonfragile reliable dissipativebased filter is designed in Section 3. Section 4 provides the simulation results to demonstrate the effectiveness of the obtained results.Some conclusions of this work are given in Section 5.
Notations.The following standard notations will be used throughout this paper.The superscript "" stands for matrix transportation; R  denotes the -dimensional Euclidean space; E{⋅} represents the mathematical expectation;  2 [0, ∞) stands for the space of -dimensional square integrable functions over [0, ∞);  > 0 ( ≥ 0) means that  is positive definite (positive-semidefinite);   and   denote the maximum and minimum eigenvalues of the matrix , respectively; diag{⋅ ⋅ ⋅ } stands for a block-diagonal matrix.Moreover, the notion * used in matrix expressions represents a term that is induced by symmetry.

Problem Formulation
In this study, we consider a class of wireless sensor networks (WSNs), which can be described by the discrete-time system with time-varying delay in the following form: where () ∈ R  is the state vector; () ∈ R  is the disturbance signal belonging to  2 [0, ∞); () is the timevarying delay satisfying  1 ≤ () ≤  2 , where  1 > 0 and  2 > 1 +  1 are prescribed integers representing the lower and upper bounds of the delay, respectively; ,   , and  represent system coefficient matrices with appropriate dimensions.Supposing that there are  distributed sensors for the system (1), the measurement of the -th sensor is given by where   () ∈ R  is the observation collected by the -th sensor;   and   are constant matrices with appropriate dimensions.Motivated by the results in [8], in order to save energy in WSNs, in this paper the measurement signal is assumed that it may not be transmitted to the remote filter at each instant.It is to be noted that the measurement signal is transmitted at least once over   (> 0) time steps and the transmission can happen at any time in these   time steps.
Let   () denote the measurement signal sequence and ỹ () is the transmitted measurement.The above transmission protocol shows that there is no transmission at some time instants.In such situations, there is no input to the filter and the input to the filter has to be predefined by some rules.Thus, it is reasonable to assume that the filter may use the last transmitted measurement signal as its input [8].Therefore, the input to the filter must be one member of the transmitted subset {  (),   ( − 1), . . .,   ( −   − 1)}.Moreover, to reflect the random selection of filter input, a set of stochastic variables  , () ∈ {0, 1},  = 0, 1, . . .,   − 1 is introduced such that  , () = 1, if   ( − ) is selected at time  as the filter input, and  , () = 0, otherwise.Furthermore, it is assumed that the expectations of the stochastic variables are known, that is, E{ , () = 1} =  , , where  , is the transmission probability and satisfies ∑   −1 =0  , = 1.Based on the above transmission protocol, the filter input can further be expressed by   () =  ,0 ()   () +  ,1 ()   ( − 1) + ⋅ ⋅ ⋅ +  ,  −1 ()   ( −   + 1) .  .Now, by substituting (2) into (3), we can get where  , and  , , respectively, are  ×  0 and  ×  0 matrices containing an identity matrix at the ( + ⋅ ⋅ ⋅   ,  () ]  .Here, it is noted that the total number of possible realizations of () is  =  1 ×  2 × ⋅ ⋅ ⋅ ×   and () could be viewed as the signal that specifies one particular case of ( () ,  () ).Again, introduce a new set of stochastic variables   () ∈ {0, 1},  ∈ {1, 2, . . ., }, which could be designed in such a way that if   () = [1, 0, . . ., 0] for  = 1, 2, . . ., , then  1 () = 1, if   () = [1, 0, . . ., 0] for  = 1, 2, . . .,  − 1 and   () = [0, 1, . . ., 0], then  2 () = 1, and so on.Therefore, at any time instant, there is only one realization of () such that ∑  =1   () = 1.By using the probabilities of sensor transmissions  , , the probability E{  () = 1} =   can be determined.For example, let us consider two sensors and assume that the measurement is transmitted within two time steps stochastically and their probabilities are  1,0 ,  1,1 and  2,0 ,  2,2 , respectively.Now, using probability rules, it can be seen that The main objective of this study is to design an appropriate reliable filter such that the considered WSNs (1) with sensor failures are stochastically finite-time bounded and (Q, S, R) −  dissipative.For this purpose, the sensor failure model in the following form is adopted in this paper:   () = (), where  is a diagonal matrix representing sensor fault range defined in the interval 0 ≤  ≤  and () is the filter input vector received from sensor and is expressed as . .,   }, and   and   are some appropriate matrices obtained from  () and  () .On the other hand, define () = (), where () ∈ R  is the output signal to be estimated and  is a constant matrix with appropriate dimension.Now, it is the right time to consider the filter equation consisting of gain fluctuations and sensor faults to be designed for the system (1) and that is given by where   () ∈ R  is the filter's state;   () ∈ R  is the estimate of ();   ,   , and   are filter gain parameters to be determined later.Further, the matrices Δ  () and Δ  () represent the fluctuations in the filter gains and are assumed to satisfy the following structures: where   ,   ,   , and   are known constant matrices with appropriate dimensions; Δ() is an unknown time-varying matrix function satisfying Δ  ()Δ() ≤ .
In order to derive the augmented filtering system, we rewrite the discrete-time system (1) and the output signal as follows: ( + 1) =  () +    ( −  ()) +  () , where  = [  0 By defining a new augmented state vector as and the estimation error as () = () −   (), the augmented filtering system and the corresponding error system can be formulated as where In order to derive the main results in the forthcoming section, we need the following assumption, definitions, and lemmas.

Definition
(see [16]).The augmented filtering system (8) is stochastically finite-time bounded with respect to ( 1 ,  2 , M, N, ), where 0 <  1 <  2 and M is a positive definite matrix, if Definition (see [26]).The augmented filtering system ( 8) is (Q, S, R) −  dissipative with respect to ( 1 ,  2 , M, N, , ), where 0 <  1 <  2 ,  > 0 and M is a positive definite matrix, and if the system is stochastically finite-time bounded with respect to ( 1 ,  2 , M, N, ) and under the zero initial condition, the output () satisfies for any non-zero () satisfying Assumption 1, where Q, S and R are real constant matrices in which Q and R are symmetric.Also, for convenience, we assume that Q ≤ 0, then we can have Lemma 4 (see [16]).For given matrices ,  =   and  > 0, the inequality    −  < 0 holds if and only if there exists a matrix  such that Lemma 5 (see [9]).For any two matrices  and  with appropriate dimensions,    +    ≤    +  −1    holds for any scalar  > 0.

Main Results
This section pays attention to solve the problem of robust finite-time nonfragile filter design for the discrete-time system (1) by employing the LMI approach.For this purpose, first, the stochastic finite-time boundedness of the discretetime system (1) for known filter gains without any fluctuations is discussed.Second, the finite-time (Q, S, R) −  dissipative performance of the system (1) is analyzed.Third, by taking the filter gain fluctuations into account, the result is extended to obtain the desired finite-time nonfragile reliable filter for the considered system.Precisely, all the aforementioned results are investigated for the system (1) by means of the augmented filtering system (8).
Proof.To discuss the dissipativity of the system (8), we consider the following performance index: Following the similar steps carried out in the proof of Theorem 7, it is easy to get that E{Δ()−(−1)()−} ≤ 0. Thus, E{( + 1)} < E{() At the same time, under zero initial condition and () ≥ 0, ∀ = 1, 2, . . ., N, we can have This implies that Then, from (33), the inequality in Definition 3 can be easily obtained.Hence, it can be concluded that the augmented filtering system (8) is stochastically finite-time bounded and (Q, S, R) −  dissipative.This completes the proof of the theorem.
. .Dissipativity-Based Finite-Time Nonfragile Reliable Filter Design.In this subsection, we design a finite-time nonfragile reliable filter in the form of (5) for the discrete-time system (1) according to the conditions established in the previous section.
Proof.The proof of this corollary is similar to that of Theorem 9 and thus it is omitted here.
Remark .It is noted that the construction of Lyapunov-Krasovskii functional plays a constructive role in reducing the conservatism of the developed results.Moreover, the complex Lyapunov-Krasovskii functional with multiple summation terms can bring more number of decision variables to the LMI.Also, when the number of decision variable increases, the computational complexity also increases.So, there should be a trade-off between the summation terms in the construction of Lyapunov-Krasovskii functional and the LMI constraints.However, in this article we have chosen an appropriate LKF of the form (16) without using any free weighting matrix technique, which results in less conservative conditions.
Remark .It should be mentioned that the fragility is an important factor in the design of controllers due to the uncertainty and disturbances in control systems.Due to the occurrence of uncertainties, most of the controllers proposed in the existing literature are sensitive to small inaccuracies in their implementation.To overcome these facts, the controller should be designed in such a way that it is insensitive or nonfragile to its own parameter uncertainties.Moreover, dissipativity is an effective concept in designing the feedback controllers for linear and nonlinear systems.Specifically, in Definition 3, if we take Q = −, S = 0, and R =  2 , then it corresponds to the finite-time  ∞ performance index and it minimizes the closed-loop impact of an external perturbation; when Q = 0, S = , and R = , we can obtain the finite-time filter with passivity performance index and use it to measure the excess or shortage of passivity of the system; if Q = − −1 , S = (1 − ), and R = , we can have the finite-time filter with mixed  ∞ and passivity performance index, where  ∈ [0, 1] is the weight parameter which deals with the trade-off between the performances of  ∞ and passivity concepts.Also, Theorem 8 provides a filter such that the closed-loop system is not only finite-time bounded, but also (Q, S, R) −  dissipative.Moreover, the proposed filter is more general, which includes  ∞ performance, passivity and mixed  ∞ and passivity as its special cases.

Numerical Simulations
In this section, two numerical examples are presented to show the effectiveness of the proposed filter design technique.Specifically, the first example validates the efficiency of finitetime nonfragile reliable filter design proposed in Theorem 9 and the second example presents a comparison result to illustrate the conservativeness of the proposed method with the existing ones.
Example .Let us consider the discrete-time system (1) and filter system (5) with parameters as follows: Furthermore, to demonstrate the effectiveness of the filtering performance, the initial conditions for the considered system and the filter are set to be [−0.30.1]  and [0 0]  , respectively.Moreover, the disturbance signal is taken as () = 0.05 exp(−0.1)sin().Based on the filter parameters mentioned above, the simulation results are presented in Figure 1.To be more specific, Figures 1(a Example .Consider the modified continuous stirred tank reactor (CSTR) system as in [8,34], where the production of cyclopentanol (B) from cyclopentadiene (A) is considered.The complete reaction is given as follows: Cyclopentadiene (A) → Cyclopentanol (B) → Cyclopentanediol (C), and 2 Cyclopentadiene (A) → Dicyclopentadiene (D).By assuming constant density and an ideal residence time distribution within the reactor, the balance equations can be described in the following form: where   denotes the concentration of educt ;   represents the concentration of the desired product  within the reactor;  is the reactor temperature.The rate components  1 ,  2 , and  3 depend exponentially on the reactor temperature  via Arrhenius law given by   () =  0 exp(−  /) ( = 1, 2, 3).
Let us assume that the first and second rate components are equal for the reaction system; that is, The product concentration   may be needed in practice and the signal processing approaches are used to estimate the concentration.By deploying two sensors for measuring the concentration of educt  and the reactor temperature, the product concentration   is estimated.Hence,  1 = [1 0 0] ,  2 = [0 0 1] ,  1 = 0.3,  2 = 0.2, and  = [0 1 0] .Let the unknown input signal () lie in the interval [−1, 1] and assume that the signal is transmitted if one of the following conditions is satisfied: ‖  () −  last, ‖ ≥  , , − last, >  , , where  last, is the last transmitted signal of the th sensor at time instant  last, and  , and  , are the magnitude and time threshold values, respectively.Moreover, it is also assumed that there is no packet dropouts and set  ,1 = 0.1,  ,2 = 0.2,  ,1 =  ,2 = 1.From this setting, it is seen that  1 =  2 = 2. Further, choose the values of the transmission probabilities as  1,0 = 0.69,  1,1 = 0.31,  2,0 = 0.44, and  2,1 = 0.56.It should be mentioned that based on the method proposed in [8], the minimum value of  is 1.9069, and while using the proposed filter design in this paper, the minimum value of  is 1.6686, which reveals that the proposed filter design technique in this paper is better than that in [8].Furthermore, for the simulation purposes, we choose  = 1.01,N = 30, M = ,  1 = 0.4, and  = 0.5.Then, by Corollary 10, the optimal value of  2 can be calculated as 838.1240.By solving the LMI-based conditions in Corollary 10, the filter gain parameters are obtained as Based on these values, the state responses of the discretetime system (44) with the proposed nonfragile reliable filter are plotted in Figure 2(a) and the associated filter state responses are presented in Figure 2(b).The system output signal together with its estimation is given in Figure 2(c), respectively.Furthermore, it can be viewed from Figure 2(d) that under the chosen initial condition and the obtained filter parameters, the state responses of the corresponding augmented filtering system satisfy the condition   ()M() <  2 = 838.1240.Then, it directly follows that the discrete-time system (44) is stochastically finite-time bounded with respect to (0.4, 838.1240, , 30, 0.5).

Conclusion
In this paper, the problem of dissipative-based finite-time robust filter design has been discussed for a class of WSNs which is described by discrete-time systems with timevarying delay.More precisely, a reliable nonfragile filter has been designed such that the augmented filtering system is stochastically finite-time bounded and (Q, S, R) −  dissipative.In this connection, a set of sufficient conditions in terms of LMIs has been developed for obtaining the desired nonfragile reliable filter for the system under consideration, wherein the filter gain parameters have been obtained by solving the developed LMIs.Finally, two numerical examples including CSTR model have been presented to demonstrate the effectiveness of the proposed filter design.The problem of finite-time dissipative-based filtering for nonlinear stochastic system with actuator saturation is an untreated topic which will be the future work.
) and 1(b) show the actual state responses and the designed filter state responses, respectively.The filtering error signal () is shown in Figure1(c).The trajectories of the system output signal and its estimation are shown in Figure1(d), wherein the effectiveness of the proposed filter design is clearly exhibited.Moreover, the time history of   ()M() is depicted in Figure1(e).From these simulations, it can be concluded that the considered discrete-time system with time-varying delay (1) is stochastically finite-time bounded with respect to (0.1, 45.1135, , 30, 0.5) under the proposed dissipative-based filter (5) even in the presence of sensor failures and gain fluctuations.
Time history of   ()M()
and the matrices   and   are given by The steady-state values of the main operating point of the reactor are given by   = 1.235/,   = 0.9/,   = 407.29,V/  = 18.83ℎ −1 ,  0 = 5.1/.It should be noted that the control input can be treated as the unknown input signal in the state estimation problem.According to this point, the discrete-time state representation of (42) with the sampling period  0 = 1min can be represented by the following: