Abstract
This paper investigates nonfragile filter design for a class of continuous-time delayed Takagi-Sugeno (T-S) fuzzy systems with interval time-varying delays. Filter parameters occur multiplicative gain variations according to the filter’s implementation, to handle this variations, a nonfragile filter is presented and a novel filtering error system is established. The nonfragile filter guarantees the filtering error system to be asymptotically stable and satisfies given performance index. By constructing a novel Lyapunov-Krasovskii function and using the linear matrix inequality (LMI), delay-dependent conditions are exploited to derive sufficient conditions for nonfragile designing filter. Using new matrix decoupling method to reduce the computational complexity, the filter parameters can be obtained by solving a set of linear matrix inequalities (LMIs). Finally, numerical examples are given to show the effectiveness of the proposed method.
1. Introduction
As we all known, in practical control systems, nonlinearity and time delay phenomena are often encountered in various industry and control system, such as networked control system and mechanical drive control system. The control of nonlinear systems has been explored and studied by many scholars in related fields. T-S fuzzy model is a powerful tool to deal with nonlinearity; much effort has been devoted on the networked control system for T-S fuzzy system or time-delayed (see [1–4]). The actuator and sensor faults estimation based on T-S fuzzy model with unmeasurable premise variables were investigated in [5]. The problem of exponential stabilization for sampled-data T-S fuzzy control systems with packet dropouts was investigated in [6]; a switched system approach is proposed to model the data-missing phenomenon. There always exist many kinds of noise interference in the process of transmission among real industrial control system’s signal, causing the error between the obtained signals and the desired signals; in order to obtain the accurate data information about the control signal and eliminate the influence of disturbances on the system, it is essential to be filtering. At present, there are the Kalman filtering, fault detection filter, filtering, filtering, and so on. Compared with other filtering methods, filtering does not need the exactly known statistics of the external disturbance [7, 8] and filtering has excellent robustness against unmodeled dynamics. In recent years, filtering system based on the Takagi-Sugeno (T-S) model has attracted much attention from the control community [9, 10], many studies have addressed filtering for T-S fuzzy systems with time-varying delay, and the proposed filtering technology has been applied to many actual communications system. Authors in literature [11–13] investigated the problem of filter design for continuous-time via Takagi-Sugeno fuzzy model approach. In literature [14, 15], the problem of filtering for a class of discrete fuzzy system has been reported. Based on discrete inequality technique and the Lyapunov-Krasovskii functional approach, sufficient conditions for the existence of admissible filters are established in terms of linear matrix inequalities. In literature [16], the event-triggered filtering for networked control systems with quantization and network-induced delays was investigated; it improved the usage of network resource.
However, in practical system, it is difficult for an exactly implemented filter to meet the actual requirements because inaccuracies or uncertainties, which include collection error and component aging, may occur during filter implementation. It often degrades the performance of the control system and even instability; the filter has a higher sensitivity to the parameter uncertainty [17]. Thus, we need to design nonfragile filter considering the parameter variation and uncertainty. Some achievements have been reported in journal about nonfragile filtering for T-S fuzzy systems with time-varying delay. In literature [18], design an filter with the gain variations such that the filtering error system was quadratically D stable and guarantees a prescribed performance level. Literature [19] is concerned with the problem of nonfragile filtering for discrete-time nonlinear systems and considered additive interval uncertainty. In literature [20], the designed nonfragile filter was in standard form and the filter was designed, which have two types of multiplicative gain variations; these models were in standard form. In literature [21], the problem of nonfragile filter design for linear continuous-time systems was studied; it proposed a notion of structured vertex separator. In literature [22], this paper studied the nonfragile filtering problem for a class of discrete-time T-S fuzzy systems with both randomly occurring gain variations and channel fading. In literature [23], the problem of nonfragile filter design for linear continuous-time systems has been studied. The filter has been designed; it included additive gain variations. In literature [24], it studied the nonfragile filtering design for a kind of fuzzy stochastic system with time-varying delay and parameter uncertainties. Sufficient conditions for stochastic input-to-state stability (SISS) of the fuzzy stochastic systems were obtained. Papers proposed the filter design methods with occurring additive gain variations according to the filter’s implementation.
Motivated by the aforementioned discussion, in this paper, a nonfragile filter design method is proposed to enhance the nonfragility of the filter. By considering the multiplicative gain variations and interval time-varying delays according to the filter implementation, a novel filtering error system is established. Different from some existing works, Jensen’s inequality is used to tackle the integral items of the derivative of Lyapunov-Krasovskii; a more relaxed performance stability criterion is derived. By constructing a novel Lyapunov-Krasovskii function and using the linear matrix inequality (LMI), delay-dependent conditions are exploited to derive sufficient conditions for nonfragile designing filter. Our objective is to design nonfragile filter which guarantees the filtering error system to be asymptotically stable and satisfies given performance index. The filter parameters can be obtained by solving a set of linear matrix inequalities (LMIs).
The rest of this paper is organized as follows. The problem formulation is stated in Section 2; nonfragile filter scheme and filtering error system are employed to enhance system’s stabilization. Stability analysis and fuzzy filter design are obtained in Section 3; by constructing a Lyapunov-Krasovskii functional, a new stability criterion is proposed to prove being less conservative than the existing ones. An applicable filter is designed in Section 4, which guarantees stability and a desire performance of the filtering error system. In order to show the effectiveness of the proposed method, simulation results are presented in Section 5.
2. Problem Formulation
Consider a nonlinear system with time-varying delay which could be approximated by a class of T-S fuzzy systems with time-varying delays. The T-S fuzzy model with plant rules can be described by the following.
Plant Rule . If is ... and is , then
, where is the continuous initial vector function defined on , are the fuzzy sets, , and is the number of IF-THEN rules. are the premise variables, is the state vector, is the measured output, is the signal vector to be estimated, is the disturbance signal vector which belongs to , and , , , , , , , , are known constant matrices with appropriate dimensions. is interval time-varying delay that satisfies the following inequality: , where , , and are constant scalars.
By employing the commonly used center-average defuzzifier, product interference, and singleton fuzzifier, the overall fuzzy model is inferred as follows:
where , , , represents the grade of membership for in , , and . It can be seen that , .
Consider the nonfragile fuzzy filter with multiplicative gain uncertainties; we design the following fuzzy filter:
Consider the following filter form which is analogous to the fuzzy control form through parallel distributed compensation.
Plant Rule . If is ... and is , then
, where is the continuous initial vector function, is the filter state vector, the estimated signal vector is , and , , , are the filter parameters. , , represent the gain variations.
The multiplicative gain uncertainties are defined aswhere , , , , , are constant matrices with appropriate dimensions and , , are uncertain matrices bounded, such that
By combining (2) with (4), we can obtain the following filtering error system:where
In this paper, our purpose is to design the fuzzy filter in the form of (3), meanwhile, satisfying the following requirements.
The filtering error system (7) with is said to be asymptotic stability for any initial condition
For a given positive scalar , the filtering error system (7) is said to be asymptotically stable with guaranteed performance , if it is asymptotic stability and the filtering error satisfiesfor all and nonzero subject to the zero initial condition.
3. Stability and Filtering Performance Analysis
The purpose of this paper is to design nonfragile filter such that the filtering error system (7) is asymptotically stable with performance index. A sufficient condition is presented in the following theorem to guarantee the existence of the filter in form of (3).
Lemma 1. Let , , and be real matrices with appropriate dimensions and . Then, for any scalar ,
Lemma 2. For any vectors and any scalar , matrices are real matrices of appropriate dimensions with ; then the following inequalities hold:
Lemma 3. Let , , , , and be real matrices of appropriate dimensions such that and . Then, for any scalar such that , we have
Lemma 4. Let and be any real matrices of appropriate dimensions. Then
Lemma 5 (Jenson’s inequality). Suppose and ; for any positive matrix , the following inequality holds:
Theorem 6. For nonlinear systems (1) and the filtering error system (7), the given positive scalar performance and the filtering error system (7) are asymptotically stable with performance if there exist symmetric positive scalars , , , , , , , , , , and symmetric positive definite matrices , , , , , such that we have the following inequality.
Proof. We construct a novel Lyapunov-Krasovskii function as follows: By Lemma 2, for , , we can obtainSimilar to (25), if there exists , we can obtainBy Lemma 2, if there exists , we can obtainSimilar to (25), if there exists , we can obtainCombining with inequalities (25), (27), and (28) givesLemma 4 giveswhereBy Lemma 3, if there exist , such that , from (31), it follows thatBy Lemma 3, if there exist , such that , from (32), it follows thatBy Lemma 3, if there exist such that , from (33), we can getCombining with formulas (31)~(36), from (30), we haveSimilarly, for formula (30) we haveBy Lemma 5, according to Jensen’s inequality, we have thatCombining with formulas (25)~(30), from (23), we havewhereBy Schur complement and formula (26), we havewhereConsequently, it follows from inequality (20), and , and we have , which implies that (9) holds.
Thus, performance is verified. In addition, when the zero disturbance input , by Schur complement, we can obtain that the time derivative of Lyapunov-Krasovskii ; that means that the filtering error system (7) with is asymptotically stable.
4. Fuzzy Filter Design
Theorem 6 provides a sufficient condition for filter design with time delay and satisfied the performance. However, there exist some coupled matrix variables in the matrix inequality (15); filter parameter can not be calculated directly. In order to decouple the variables in (15), we will use decoupling technique. Using this method, inequality (15) can be equivalently expressed in another form; hence, we can obtain filter parameter.
Theorem 7. For given scalars , , , , , , , , , , , , and , the filtering error system (6) is asymptotically stable as well as with the performance level , if there exist matrices , , , , , , , , and such that the following linear matrix inequalities are satisfied:whereWe can obtain the filter parameters as follows:
Proof. By Schur complement formula, the matrix inequality conditions (20) in Theorem 6 can be described as the following matrix inequalities:whereLet , , , , , , , , , , , , , , , , , , , , , , , where , .
According to , we have that holds for any scalar . The filter parameters in (3) can be designed as (51). This completes the proof.
5. Simulation Example
Consider the following nonlinear systems with time-varying delays:
where
The known matrices in (5) are given by
The disturbance signal is given as follows:
We can get the desired nonfragile filter by solving LMIS (45)-(49) in Theorem 7; the nonfragile parameter matrices are given as follows:
In addition, the new method provides less conservative design result; we can obtain a smaller . Figures 1 and 2 show the response of system’s states and filter’s states , respectively. Figure 3 shows the trajectories of and its estimates . The estimation error is depicted in Figure 4.
The random disturbance signal is given as follows: where denotes Bernoulli’s random event. The simulation results are obtained as shown in Figures 5–9.
The stochastic variables are Bernoulli-distributed white sequences; Figure 5 shows the event occurrence probability. Figures 6 and 7 show the response of system’s states and filter’s states , respectively. Figure 8 shows the trajectories of and its estimates . The system error is depicted in Figure 9.
The results show that the nonfragile filter can make the system have good stability. The system generates small overshoot; this filter can efficiently reduce the influence of external disturbance and uncertainty; it can enhance control precision and dynamic qualities of the system.
6. Conclusions
This paper studied the fuzzy nonfragile filter design problem for a class of nonlinear systems with an interval time-varying delays; meanwhile, the designing filter with multiplicative gain variations was considered. By constructing a new Lyapunov-Krasovskii functional, we obtained a sufficient condition for designing the nonfragile filter such that the filtering error system is asymptotically stable and satisfied the given performance index. This nonfragile filter design method enhances the nonfragility of the filter and reduces some conservatism. A numerical example has shown the effectiveness of the proposed method. Future research includes event-triggered nonfragile filter design for nonlinear system considering packet dropout and interval time-varying delays. Moreover, type 2 fuzzy filter design for nonlinear system with time-varying delays 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, 12 months after initial publication, 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 Science and Technology project of State Grid corporation of China (SGTYHT/13-JS-175).