Abstract
This paper first investigates the event-triggered nonfragile filter design for a class of nonlinear NCSs with interval time-varying delay. An event-triggered scheme is addressed to determine sampled data to be transmitted so that network communication resource can be saved significantly. The nonfragile filter design is assumed to include multiplicative gain variations according to the filter’s implement. Under the event-triggered scheme, the filtering error system is modeled as a system with interval time-varying delay. By constructing a new Lyapunov-Krasovskii functional and employing Wirtinger inequality, a sufficient condition is derived, which guarantees that the filtering error system is asymptotically stable with the prescribed performance. The nonfragile filter parameters are obtained by solving a set of linear matrix inequalities. Two numerical examples are given to show the usefulness and the effectiveness of the proposed method.
1. Introduction
During the past several years, Takagi-Sugeno (T-S) fuzzy model approach has got considerable attention due to its great merits on modeling for nonlinear networked control systems (NCSs) [1]. Many efforts have been proposed based on T-S fuzzy model for nonlinear systems [2–5]. On one hand, some state variables cannot be directly measured in practical system; therefore, the filtering problem attracted the attention of many researchers to estimate the unmeasured states. In comparison with traditional Kalman filtering, the filter does not require statistical assumptions on the exogenous signals. Thus, the filtering theory has got considerable development. The phenomenon of filter design with time-varying delays, sensor faults, and packet dropouts was studied for nonlinear systems [6]. The authors in [7] proposed new results on a delay-derivative-dependent fuzzy filter.
It should be mentioned that all the above works are based on accuracy assumptions that filter can be implemented exactly. However, inaccuracies or uncertainties do occur in filter implementation, which will reduce the performance of systems and lead the filter system to be fragile. Thus, a nonfragile filter must be designed to handle uncertainties and maintain the performance of systems. Up to now, a few results on the nonfragile filter have been proposed. Recently, the nonfragile filtering problem for a class of discrete-time Takagi-Sugeno fuzzy systems with both randomly occurring gain variations and channel fading was investigated in [8]. Based on vertex theory and probabilistic algorithm, deterministic and randomised filtering algorithms are proposed in [9]. Aiming at the fuzzy stochastic systems, the authors in [10] proposed a nonfragile robust filter design and a desired performance level. It should be pointed out that the nonfragile filtering problem has not been fully studied, which is one of motivations of this paper.
On the other hand, one topic that has been increasingly important is how to mitigate the network bandwidth while guaranteeing the stability and other desired performance of systems [11–15]. In traditional time-triggered control systems, the sample data will be transmitted into controller via network communication whatever the data are desired or not, which leads to network resource waste and low efficiency. Therefore, an event-triggered scheme was proposed to reduce data transmission and improve the effective utilities of network resource. Recently, a novel discrete model for networked control systems was introduced in [16], which contained trigger parameters, dynamic quantization, and fault detection. In [17], event-triggering load frequency control was employed in multiarea power systems. It should be noted that although the works on event-triggered NCSs are rich, the limitations still remain and there still exist some problems to be handled [18–20]. To the best of our knowledge, there are no works that investigate how to use the event-triggered scheme in nonfragile filtering systems, which is another motivation of this paper.
According to the above discussion, we first consider the problem of the event-triggered nonfragile filter design for a class of nonlinear networked control systems. Our main contributions of this paper are summarized as follows:
An event-triggered communication scheme is proposed to save network resource significantly. By considering the event-triggered scheme, filter’s multiplicative gain variations, and interval time-varying delays, a novel filtering error system is established.
Different from some existing works, the Wirtinger inequality is used to tackle the integral items of the derivative of Lyapunov-Krasovskii; a more relaxed performance stability criterion is derived.
The rest of this paper is organized as follows. The problem formulation is given in Section 2; under the event-triggered scheme, the filtering error system is modeled as a system with interval time-varying delay. Stability analysis for filtering error system is presented in Section 3. The nonfragile filter design method is first addressed in Section 4. Numerical examples are provided in Section 5 to demonstrate the effectiveness of the proposed method.
Notations. Throughout this paper, denote the -dimensional Euclidean space and is the set of real matrices. Superscript stands for the matrix transposition, represents the identity matrix, and denotes the block-diagonal matrix. The notation means that the matrix is a real symmetric positive define matrix. In symmetric block matrices, “” is used as ellipsis for terms induced by symmetry.
2. Problem Formulation
2.1. T-S Fuzzy System
Consider the following nonlinear system, which can be described by T-S fuzzy model with plant rules:
Plant Rule where are the fuzzy sets, , and is the number of IF-THEN rules. are the premise variables. and are the system state vector and the measured output, respectively. is the external disturbance and . is the signal vector to be estimated; , , , and are known parameter matrices with appropriate dimensions.
By using center-average defuzzifier, product inference, and a singleton fuzzifier, the T-S fuzzy system (2) can be rewritten as follows: where ; the fuzzy basis functions are given by represents the grade of membership for in ; .
Therefore, we have
2.2. Event-Triggered Scheme
In this section, we consider the communication networks with limited bandwidth; an event generator under the event-triggered scheme is employed. Inspired by the method in [12], the event-triggered scheme is adopted as follows:for , where is the threshold error among the current sampling data and the last transmitted data. and are the triggering parameters.
Remark 1. It should be mentioned that the current sampling data will be transmitted only when the condition proposed in (6) is satisfied. Therefore, in comparison with periodic transmission communication scheme, the event-triggered mechanism can reduce the transmission rate and utilize the limited bandwidth effectively.
2.3. Nonfragile Fuzzy Filter
In some previous results of filter design, the implicit assumption is made that there are no multiplicative gain uncertainties. However, in fact, there inevitably exist filter parameter uncertainties in filter implementation. Therefore, in this section, we consider a full-order nonfragile fuzzy filter with gain variations as follows:
Plant Rule where is the filter state vector, is the real input of the filter, is the estimated signal vector of , and , , and are the filter parameters to be designed.
Similarly, we represent the nonfragile fuzzy filter aswhere , , and are the multiplicative gain uncertainties, which can be defined aswhere and are constant matrices with appropriate dimension. are uncertain bounded matrices:
Next, consider the effect of the logic ZOH; the last transmission data instant is maintained with the holding interval as follows:
In order to facilitate analysis, the holding interval can be represented by the following subsets:where Then, we define the function of the interval time-varying delays as follows:where , . Therefore, we can easily derivewhere and are lower bound and upper bound of the delays, respectively. is the bound of the delay variation.
To summarize, the inputs of the filter are described as
2.4. Fuzzy Filtering Error System
For simplicity, we let represent and let represent , andBy combining (3), (9), and (17), we can obtain the following filtering error system:whereBefore the end of this section, the following definitions and lemmas are needed for the fuzzy filtering error system.
Definition 2. The fuzzy filtering error system is asymptotically stable with an performance, if the following holds:
The filtering error system (19) is asymptotically stable when .
The filtering error system (19) has a prescribed performance ; under zero initial condition,is satisfied for any nonzero .
Lemma 3 (see [21]). Let , , and be real matrices with appropriate dimensions and . Then, for any constant , one has
Lemma 4 (see [22]). For any vectors and any scalar , matrices , , and are real matrices of appropriate dimensions with ; then the following inequalities hold:
Lemma 5 (see [23]). Let , , , , and be real matrices of appropriate dimensions such that and . Then, for any scalar such that , one has
Lemma 6 (see [24]). Let and let and be any real matrices of appropriate dimensions. Then
Lemma 7 (Seuret and Gouaisbaut [25]). For a given symmetric and positive matrix of appropriate dimensions and different signal in , the following inequality holds:where
3. Stability Analysis
In this section, we first present the performance analysis for the filtering error system (19) under event-triggered scheme. The following results are established to guarantee that the filtering error system (19) is asymptotically stable.
Theorem 8. For given positive parameters , , , , , and , the filtering error system in (19) is asymptotically stable with performance under the event-triggered scheme (16), if there exist matrices , , , , , and and , , and with appropriate dimensions such that the following matrix inequalities hold:for where
Proof. We construct a candidate of Lyapunov-Krasovskii function as follows: whereTaking the time derivation of for , we have whereBy Lemma 3, for , we can obtainBy Lemma 4, if there exist and , we haveFrom (41) to (43), we deriveBy using Lemmas 5 and 6, if there exists , we deriveBy applying Lemma 5, if there exists , we can obtainFurthermore, we use Lemma 7 to deal with the integral items in (40); it follows thatwhereBy combining (44) to (49), we obtainwhereThen, from (28) to (32), under the zero initial condition, we integrate the right and left sides of (51) from 0 to for all ; we can easily deriveThis completes the proof.
4. Fuzzy Filter Design
It is worth mentioning that the condition in (32) cannot be directly used for filter design. Therefore, in this section, we provide a sufficient condition for the fuzzy filter design, and a suitable filter parameter matrix is obtained.
Theorem 9. For given positive parameters , , , , , , and , if there exist symmetric positive definite matrices , , , , , , and and , , and with appropriate dimensions, the following hold:forwhereMoreover, if inequality is feasible, the filter parameters can be obtained by
Proof. By applying Schur complement lemma, the matrix inequality conditions (32) in Theorem 8 can be rewritten as the following matrix inequality:forwhereThen, we define the following matrices:We haveMoreover, we can derive by the inequality for any scalar . Then, by applying Schur complement lemma and combining (32) and (62)–(66), we can easily derive inequality (58). This completes the proof.
5. Simulation Results
In this section, we provide two examples to demonstrate the merit of the nonfragile filter based event-triggered scheme.
Example 1. Consider the following fuzzy systems:whereAnd the parameter uncertainties are given asThe membership function , , and the external disturbance signal are given asConsider the interval time-varying delay with and . And choose , , , and sample period . By Theorem 9, we derive the desired nonfragile parameter as follows:Meanwhile, we obtain the event-triggered parameter and the minimum performance level is .
It is assumed that the initial condition is . The responses of the original state and the filter state are shown in Figures 1 and 2. The system output signal is depicted in Figure 3. As depicted in Figure 4, we can clearly see that the response of the filtering error is stable after about three seconds, which demonstrates the effective of the proposed approach. Note that, in Figure 5, the transmission rate is mitigated from 100% to 7%, which demonstrates that the event-triggered scheme is effective in reducing data transmission and saving the network resource.
Example 2. To further demonstrate the effectiveness of the proposed method, in this example, we consider a truck-trailer system; the practical example is borrowed from [26]. The system parameters are given as follows:Then, we suppose that the parameter uncertainties are given by We assume that the lower bound and upper bound of the interval time-varying delay are and , respectively. The derivative of the time delay is . Furthermore, we suppose that and . By solving the LMIs in Theorem 9, we obtain minimum performance level , which is less than previously studied in [26]. Meanwhile, the event-triggered argument is obtained, and the corresponding filter parameters can be derived as follows:Figures 6, 7, and 8 show the original state and the filter state; it is clear that the filter state is asymptotically stable. The curve of the system output signal is depicted in Figure 9.
The filtering error response is depicted in Figure 10, which shows that the designed nonfragile filter is feasible.
Finally, we choose the sample period ; under the event-triggered scheme, it can be seen that Figure 11 shows that only 24 times in all 900 sampling instants are transmitted, which illustrates that the network bandwidth is greatly mitigated.
6. Conclusions
In this paper, the problem of the event-triggered fuzzy nonfragile filter design has been investigated for nonlinear NCSs with interval time-varying delay. The resultant filtering error system has been modeled as a system with an interval time-varying delay. By constructing a new Lyapunov-Krasovskii functional and employing Wirtinger inequality, we have obtained a sufficient condition for designing the fuzzy nonfragile filter such that the fuzzy filtering error system is asymptotically stable with a prescribed performance level. Two numerical examples have been given to demonstrate the effectiveness of the proposed method. Future research includes nonfragile fuzzy fault filter design for nonlinear control systems considering time-varying delays. In addition, type-2 nonfragile fuzzy filter design for NCSs with packet dropout also can be further considered for the future investigation.
Data Availability
The data used to support this study are currently under embargo, while the research findings are commercialized. Requests for data will be considered by the corresponding author.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported in part by Natural Science Foundation of Heilongjiang Province (Grant no. LC2015024) and the Fundamental Research Funds in Heilongjiang Provincial Universities (Grant no. 135109238).