Research Article | Open Access
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.
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 , 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 [4–7] 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  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 [11–13]. By constructing a probability-dependent Lyapunov-Krasovskii functional, a set of sufficient conditions is established in  to obtain an energy-to-peak filter design for networked Markov switched singular systems over a finite-time interval. Wang et al.  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.  developed a finite-time filter design for a class of uncertain nonlinear discrete-time Markovian jump systems represented by Takagi-Sugeno fuzzy model with nonhomogeneous jump process. The asynchronous resilient controller design problem is investigated in , 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 . The authors in  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 , 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  discussed the robust reliable dissipative filtering problem of uncertain discrete-time 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 -dissipative. In , 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 . By employing the LMI approach, a new set of sufficient conditions is obtained in  for the existence of reliable dissipative filter which makes the filtering error system stochastically stable and strictly -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 . Therefore, it is important and necessary to design a filter that should be reliable and insensitive to some amount of gain fluctuations [28–30]. Xu et al.  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 . The nonfragile finite-time filtering problem is studied in  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 finite-time 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 dissipative-based 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; denotes the -dimensional Euclidean space; represents the mathematical expectation; stands for the space of -dimensional square integrable functions over ; means that is positive definite (positive-semidefinite); and denote the maximum and minimum eigenvalues of the matrix , respectively; stands for a block-diagonal matrix. Moreover, the notion used in matrix expressions represents a term that is induced by symmetry.
2. 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 is the state vector; is the disturbance signal belonging to ; is the time-varying delay satisfying , where and 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 bywhere is the observation collected by the -th sensor; and are constant matrices with appropriate dimensions. Motivated by the results in , 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 time steps and the transmission can happen at any time in these time steps. Let denote the measurement signal sequence and 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 . Therefore, the input to the filter must be one member of the transmitted subset . Moreover, to reflect the random selection of filter input, a set of stochastic variables , is introduced such that , if is selected at time as the filter input, and , otherwise. Furthermore, it is assumed that the expectations of the stochastic variables are known, that is, , where is the transmission probability and satisfies .
Based on the above transmission protocol, the filter input can further be expressed byLet and define , and . Now, by substituting (2) into (3), we can getwhere and , respectively, are and matrices containing an identity matrix at the -th block and the rest of elements are zero.
Let , and , then the possible realizations of is . Moreover, define , and . Here, it is noted that the total number of possible realizations of is and could be viewed as the signal that specifies one particular case of . Again, introduce a new set of stochastic variables , , which could be designed in such a way that if for , then , if for and , then , and so on. Therefore, at any time instant, there is only one realization of such that .
By using the probabilities of sensor transmissions , the probability 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 and , respectively. Now, using probability rules, it can be seen that , , and . 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 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 and is the filter input vector received from sensor and is expressed as with , , and and are some appropriate matrices obtained from and . On the other hand, define , where 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 bywhere is the filter’s state; is the estimate of ; , and are filter gain parameters to be determined later. Further, the matrices and represent the fluctuations in the filter gains and are assumed to satisfy the following structures:where , , , and are known constant matrices with appropriate dimensions; is an unknown time-varying matrix function satisfying .
In order to derive the augmented filtering system, we rewrite the discrete-time system (1) and the output signal as follows:where , , , and
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 aswhere
In order to derive the main results in the forthcoming section, we need the following assumption, definitions, and lemmas.
Assumption 1. The disturbance input vector is time-varying and satisfies , where .
Definition 2 (see ). The augmented filtering system (8) is stochastically finite-time bounded with respect to , where and is a positive definite matrix, if , holds for any non-zero satisfying Assumption 1.
Definition 3 (see ). The augmented filtering system (8) is dissipative with respect to , where , and is a positive definite matrix, and if the system is stochastically finite-time bounded with respect to and under the zero initial condition, the output satisfies for any non-zero satisfying Assumption 1, where and are real constant matrices in which and are symmetric. Also, for convenience, we assume that , then we can have .
Lemma 4 (see ). For given matrices and , the inequality holds if and only if there exists a matrix such that
Lemma 5 (see ). For any two matrices and with appropriate dimensions, holds for any scalar .
Lemma 6 (see ). For any symmetric constant matrix and two positive integers and satisfying , the condition holds.
3. 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 discrete-time system (1) for known filter gains without any fluctuations is discussed. Second, the finite-time 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).
3.1. Stochastic Finite-Time Boundedness Analysis
By constructing a suitable Lyapunov-Krasovskii functional, a new set of sufficient conditions is obtained to ensure the stochastic finite-time boundedness of the augmented filtering system (8).
Theorem 7. Let Assumption 1 hold, , and let be given scalars and be a positive definite matrix. Then, the augmented filtering system (8) is stochastically finite-time bounded subject to , if there exist a scalar and symmetric matrices , , , , , such that the following matrix inequalities hold:where , , , , , , , , and + .
Proof. Consider the Lyapunov-Krasovskii functional for the augmented filtering system (8) in the following form:where Computing the forward differences of along the solution of augmented filtering system (8) and taking the mathematical expectation, we can getNow, applying Lemma 6 to the summation terms in (20), we can haveThen, it follows from (18) to (22) thatwhere = , and the elements of and are defined in the theorem statement.
By applying Lemma 2.3 in  to the matrix terms in the right-hand side of (23), we can obtain the matrix term in (13). If the matrix inequality in (13) holds, it is obvious that Thus, we can get . Further, if , it follows from Assumption 1 that Moreover, from (16), we can also get Let , , , , , and . Then, we can have On the other hand, it follows from (16) that . Then, it is easy to obtain that . From inequality (14), it is clear that for all . Hence, according to Definition 3, the augmented filtering system (8) is finite-time bounded with respect to , which completes the proof.
3.2. Dissipativity-Based Stochastic Finite-Time Boundedness Analysis
To increase the robustness of the obtained results in Theorem 7, the dissipative performance index is taken into account in the following theorem.
Theorem 8. Let Assumption 1 hold. For given scalars , , , , and positive definite matrix , the augmented filtering system (8) is stochastically finite-time bounded and dissipative with respect to if there exist a constant and symmetric matrices , , , , , such that the following matrix inequalities together with (15) are satisfied:where , and the rest of elements of are the same as of defined in Theorem 7.
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 . Thus, . Further, if , it follows that At the same time, under zero initial condition and , , we can have This implies thatThen, 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 dissipative. This completes the proof of the theorem.
3.3. Dissipativity-Based Finite-Time Nonfragile Reliable Filter Design
Theorem 9. Consider the discrete-time system (1). Let Assumption 1 hold, , , , be given constants, and , , , be known matrices. If there exist positive scalars , , , , symmetric matrices , , , , , , and any matrices , , , with appropriate dimensions such that the following LMI together with (29) holds:where , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , = , = , = , , = , = , then there exists a filter (5) such that the discrete-time system (1) is stochastically finite-time bounded and dissipative with respect to . Furthermore, if the above said matrix inequalities have feasible solutions, the filter gains in (5) can be obtained by , , and .
Proof. For convenience, define the matrices as follows: , , , , , , and . Letting , , and , using Lemma 4 and the partition matrices and in (28) together with the parameter uncertainties definition in (6), we can get