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Nonfragile [H.sub.[infinity]] Filter Design for Nonlinear Continuous-Time System with Interval Time-Varying Delay.

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, [L.sub.2] filtering, [H.sub.[infinity]] filtering, and so on. Compared with other filtering methods, [H.sub.[infinity]] filtering does not need the exactly known statistics of the external disturbance [7, 8] and [H.sub.[infinity]] filtering has excellent robustness against unmodeled dynamics. In recent years, [H.sub.[infinity]] filtering system based on the Takagi-Sugeno (T-S) model has attracted much attention from the control community [9, 10], many studies have addressed [H.sub.[infinity]] filtering for TS fuzzy systems with time-varying delay, and the proposed [H.sub.[infinity]] filtering technology has been applied to many actual communications system. Authors in literature [11-13] investigated the problem of [H.sub.[infinity]] filter design for continuous-time via Takagi-Sugeno fuzzy model approach. In literature [14, 15], the problem of [H.sub.[infinity]] 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 [H.sub.[infinity]] 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 [H.sub.[infinity]] filter considering the parameter variation and uncertainty. Some achievements have been reported in journal about nonfragile [H.sub.[infinity]] filtering for T-S fuzzy systems with time-varying delay. In literature [18], design an [H.sub.[infinity]] filter with the gain variations such that the filtering error system was quadratically D stable and guarantees a prescribed [H.sub.[infinity]] performance level. Literature [19] is concerned with the problem of nonfragile [H.sub.[infinity]] filtering for discrete-time nonlinear systems and considered additive interval uncertainty. In literature [20], the designed nonfragile [H.sub.[infinity]] 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 [H.sub.[infinity]] 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 [H.sub.[infinity]] 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 [H.sub.[infinity]] 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 [H.sub.[infinity]] 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 [H.sub.[infinity]] 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 [H.sub.[infinity]] filter. Our objective is to design nonfragile [H.sub.[infinity]] filter which guarantees the filtering error system to be asymptotically stable and satisfies given [H.sub.[infinity]] 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 [H.sub.[infinity]] 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 i. If [[theta].sub.1] (t) is [M.sub.i1] ... and [[theta].sub.r] (t) is [M.sub.ip], then

[mathematical expression not reproducible] (1)

x(t) = [phi](t), where [phi](t) is the continuous initial vector function defined on [-[[tau].sub.2],0], are the fuzzy sets, i = 1, 2, 3, ... r, and j = 1, 2, 3, ..., p is the number of IF-THEN rules. [[theta].sub.1](t), [[theta].sub.2](t)... [[theta].sub.r](t) are the premise variables, x(t) [member of] [R.sup.n] is the state vector, y(t) [member of] [R.sup.m] is the measured output, z(t) [member of] [R.sup.p] is the signal vector to be estimated, [omega](t) [member of] [R.sup.q] is the disturbance signal vector which belongs to [omega](t) [member of] [l.sub.2][0, [infinity]), and [A.sub.i], [A.sub.[tau]i], [B.sub.i], [B.sub.[tau]i], [C.sub.i], [C.sub.[tau]i], [D.sub.i], [D.sub.[tau]i], [E.sub.i] are known constant matrices with appropriate dimensions. [tau](t) is interval time-varying delay that satisfies the following inequality: [[tau].sub.1] [less than or equal to] [tau](t) < [[tau].sub.2], [??](t) [less than or equal to] d where [[tau].sub.1], [[tau].sub.2], and d are constant scalars.

By employing the commonly used center-average defuzzifier, product interference, and singleton fuzzifier, the overall fuzzy model is inferred as follows:

[mathematical expression not reproducible] (2)

where [theta] = [[[[theta].sub.1], ..., [[theta].sub.r]].sup.T], [h.sub.i] ([theta](t)) = [h.sub.i]([theta](t))/[[summation].sup.r.sub.i][h.sub.i]([theta](t)), [h.sub.i]([theta](t)) = [[pi].sup.P.sub.j=1] [M.sub.ij] ([[theta].sub.j] (t)), [M.sub.ij] (*) represents the grade of membership for [[theta].sub.j] (t) in [M.sub.ij], 0 [less than or equal to] [h.sub.i] ([theta](t)) [less than or equal to] 1 (i = 1,2, ..., r), and [[summation].sup.r.sub.i=1] [h.sub.i] ([theta](t)) > 0. It can be seen that [h.sub.i]([theta](t)) [greater than or equal to] 0 (i = 1,2, ..., r), [[summation].sup.r.sub.i=1] [h.sup.i] ([theta](t)) = 1.

Consider the nonfragile fuzzy filter with multiplicative gain uncertainties; we design the following fuzzy [H.sub.[infinity]] filter:

[mathematical expression not reproducible] (3)

Consider the following [H.sub.[infinity]] filter form which is analogous to the fuzzy control form through parallel distributed compensation.

Plant Rule i. If [[theta].sub.1] (t) is [M.sub.i1] ... and [[theta].sub.p] (t) is [M.sub.ip], then

[mathematical expression not reproducible] (4)

[x.sub.f] (0) = [x.sub.f0], where [x.sub.f0] is the continuous initial vector function, [x.sub.f] (t) [member of] [R.sup.n] is the filter state vector, the estimated signal vector z(t) is [z.sub.f] (t), and [A.sub.fi], [B.sub.fi], [C.sub.fi], i = 1,2,3, ..., r are the filter parameters. [DELTA][A.sub.fi] (t), [DELTA][B.sub.fi] (t), [DELTA][C.sub.fi] (t) represent the gain variations.

The multiplicative gain uncertainties are defined as

[mathematical expression not reproducible] (5)

where [M.sub.1i], [N.sub.1i], [M.sub.2i], [N.sub.2i], [M.sub.3i], [N.sub.3i] are constant matrices with appropriate dimensions and [K.sub.A] (t), [K.sub.B] (t), [K.sub.C] (t) are uncertain matrices bounded, such that

[mathematical expression not reproducible] (6)

By combining (2) with (4), we can obtain the following filtering error system:

[mathematical expression not reproducible] (7)

where

[mathematical expression not reproducible] (8)

In this paper, our purpose is to design the fuzzy [H.sub.[infinity]] filter in the form of (3), meanwhile, satisfying the following requirements.

(1) The filtering error system (7) with [omega](t) = 0 is said to be asymptotic stability for any initial condition

(2) For a given positive scalar [gamma] >0, the filtering error system (7) is said to be asymptotically stable with guaranteed [H.sub.[infinity]] performance [gamma], if it is asymptotic stability and the filtering error e(t) satisfies

[[integral].sup.L.sub.0] [[parallel]e(t)[parallel].sup.2] dt[less than or equal to][[gamma].sup.2] [[integral].sup.L.sub.0] [[parallel][omega](t)[parallel].sup.2] dt (9)

for all L >0 and nonzero [omega](t) [member of] [L.sub.2] [0, [infinity]) subject to the zero initial condition.

3. Stability and [H.sub.[infinity]] Filtering Performance Analysis

The purpose of this paper is to design nonfragile [H.sub.[infinity]] filter such that the filtering error system (7) is asymptotically stable with [H.sub.[infinity]] 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 X, Y, and K be real matrices with appropriate dimensions and [K.sup.T.sub.i] (t)[K.sub.i] (t) < I. Then, for any scalar [delta] > 0,

XK(t)Y + [Y.sup.T] K(t)[X.sup.T] [less than or equal to] [epsilon]X[X.sup.T] + [[epsilon].sup.-1] [Y.sup.T] Y (10)

Lemma 2. For any vectors x,y [member of] [R.sup.n] and any scalar [epsilon] >0, matrices D, F, E are real matrices of appropriate dimensions with [F.sup.T] (t)F(t) < I; then the following inequalities hold:

2xDF (t) Ey [less than or equal to] e[x.sup.T] D[D.sup.T] x + [[epsilon].sup.-1] [y.sup.T] E[E.sup.T] y (11)

Lemma 3. Let V, H, E, Q, and F be real matrices of appropriate dimensions such that Q >0 and [F.sup.T] F [less than or equal to] I. Then, for any scalar [epsilon] >0 such that [Q.sup.-1] - [epsilon]H[H.sup.T] > 0, we have

[(V + HFE).sup.T] Q(V + HFE)

(12)

[less than or equal to] [V.sup.T] [([Q.sup.-1] - [epsilon]H[H.sup.T]).sup.-1] V + [[epsilon].sup.-1] [E.sup.T] E

Lemma 4. Let P > 0 and [A.sub.i], [A.sub.l] be any real matrices of appropriate dimensions. Then

[A.sup.T.sub.i] P[A.sub.l] + [A.sup.T.sub.l] P[A.sub.i] [less than or equal to] [A.sup.T.sub.i] P[A.sub.i] + [A.sup.T.sub.l] P[A.sub.l] (13)

Lemma 5 (Jenson's inequality). Suppose [[tau].sub.1] [less than or equal to] [tau](t) [less than or equal to] [[tau].sub.2] and x(t) [member of] [R.sup.n]; for any positive matrix R [member of] [R.sup.nxn], the following inequality holds:

[mathematical expression not reproducible] (14)

Theorem 6. For nonlinear systems (1) and the filtering error system (7), the given positive scalar [H.sub.[infinity]] performance y and the filtering error system (7) are asymptotically stable with [H.sub.[infinity]] performance [gamma] if there exist symmetric positive scalars [[epsilon].sub.2] > 0, [[epsilon].sub.6] > 0, [[epsilon].sub.10] > 0, [[epsilon].sub.29] > 0, [[epsilon].sub.15] > 0, [[epsilon].sub.25] > 0, [[epsilon].sub.20] > 0, [[epsilon].sub.21] > 0 [[epsilon].sub.22] > 0, [[epsilon].sub.11] > 0, [[epsilon].sub.12] > 0 and symmetric positive definite matrices [mathematical expression not reproducible] such that we have the following inequality.

[mathematical expression not reproducible] (15)

[mathematical expression not reproducible] (16)

[mathematical expression not reproducible] (17)

[mathematical expression not reproducible] (18)

[mathematical expression not reproducible] (19)

[mathematical expression not reproducible] (20)

[mathematical expression not reproducible] (21)

Proof. We construct a novel Lyapunov-Krasovskii function as follows:

[mathematical expression not reproducible] (22)

[mathematical expression not reproducible] (23)

[mathematical expression not reproducible] (24)

By Lemma 2, for [for all][[epsilon].sub.2] > 0, [for all][[epsilon].sub.21] > 0, we can obtain

[mathematical expression not reproducible] (25)

Similar to (25), if there exists [for all][[epsilon].sub.20] > 0, we can obtain

[mathematical expression not reproducible] (26)

By Lemma 2, if there exists [for all][[epsilon].sub.6] > 0, we can obtain

[mathematical expression not reproducible] (27)

Similar to (25), if there exists [for all][[epsilon].sub.10] > 0, we can obtain

[mathematical expression not reproducible] (28)

Combining with inequalities (25), (27), and (28) gives

[mathematical expression not reproducible] (29)

Lemma 4 gives

[mathematical expression not reproducible] (30)

where

[mathematical expression not reproducible] (31)

[mathematical expression not reproducible] (32)

[mathematical expression not reproducible] (33)

By Lemma 3, if there exist [mathematical expression not reproducible], from (31), it follows that

[mathematical expression not reproducible] (34)

By Lemma 3, if there exist [mathematical expression not reproducible], from (32), it follows that

[mathematical expression not reproducible] (35)

By Lemma 3, if there exist [mathematical expression not reproducible], from (33), we can get

[mathematical expression not reproducible] (36)

Combining with formulas (31)-(36), from (30), we have

[mathematical expression not reproducible] (37)

Similarly, for formula (30) we have

[mathematical expression not reproducible] (38)

By Lemma 5, according to Jensen's inequality, we have that

[mathematical expression not reproducible] (39)

[mathematical expression not reproducible] (40)

Combining with formulas (25)~(30), from (23), we have

[??]([xi](t))[less than or equal to][[eta].sup.T] (t)[bar.[phi]][eta](t) (41)

[mathematical expression not reproducible] (42)

By Schur complement and formula (26), we have

[??] ([xi](t)) + [e.sup.T] (t) e (t) - [[gamma].sup.2] [[omega].sup.T] (t) [omega] (t) [less than or equal to] [[eta].sup.T] (t) [phi][eta](t) (43)

where

[[eta].sup.T] (t)

(44)

= [[[epsilon].sup.T] (t) [[epsilon].sup.T] (t- [[tau].sub.1]) [[epsilon].sup.T] (t - [[tau].sup.2]) [[omega].sup.T] (t)]

Consequently, it follows from inequality (20), V[([xi](t))|.sub.t=0] = 0 and V[([xi](t))|.sub.t=L] [greater than or equal to] 0, and we have [[integral].sup.L.sub.0] ([parallel]e(t)[parallel].sup.2] - [[gamma].sub.2] [[parallel][omega](t)[parallel].sup.2])df + V[([xi](t))|.sub.t=L] - V[([xi](t))|.sub.t=0] [less than or equal to] 0, which implies that (9) holds.

Thus, [H.sub.[infinity]] performance is verified. In addition, when the zero disturbance input [omega](t) = 0, by Schur complement, we can obtain that the time derivative of Lyapunov-Krasovskii [??]([xi](t)) [less than or equal to] 0; that means that the filtering error system (7) with [omega](t) = 0 is asymptotically stable.

4. Fuzzy [H.sub.[infinity]] Filter Design

Theorem 6 provides a sufficient condition for [H.sub.[infinity]] filter design with time delay and satisfied the [H.sub.[infinity]] performance. However, there exist some coupled matrix variables in the matrix inequality (15); [H.sub.[infinity]] 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 [H.sub.[infinity]] filter parameter.

Theorem 7. For given scalars [delta] > 0, [[epsilon].sub.2i] > 0, [[epsilon].sub.6i] > 0, [[epsilon].sub.10i] > 0, [[epsilon].sub.19i] > 0, [[epsilon].sub.29i] > 0, [[epsilon].sub.15i] > 0, [[epsilon].sub.25i] > 0, [[epsilon].sub.20i] > 0, [[epsilon].sub.2i] > 0, [[epsilon].sub.22i] > 0, [[epsilon].sub.11i] > 0, and [[epsilon].sub.12i] > 0, the filtering error system (6) is asymptotically stable as well as with the [H.sub.[infinity]] performance level [gamma], if there exist matrices [mathematical expression not reproducible] such that the following linear matrix inequalities are satisfied:

[mathematical expression not reproducible] (45)

[mathematical expression not reproducible] (46)

[mathematical expression not reproducible] (47)

[mathematical expression not reproducible] (48)

[mathematical expression not reproducible] (49)

where

[mathematical expression not reproducible] (50)

We can obtain the filter parameters as follows:

[mathematical expression not reproducible] (51)

Proof. By Schur complement formula, the matrix inequality conditions (20) in Theorem 6 can be described as the following matrix inequalities:

[mathematical expression not reproducible] (52)

where

[mathematical expression not reproducible] (53)

According to [mathematical expression not reproducible], we have that [mathematical expression not reproducible] holds for any scalar [delta] >0. 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:

[mathematical expression not reproducible] (54)

where

[mathematical expression not reproducible] (55)

The known matrices in (5) are given by

[mathematical expression not reproducible] (56)

The disturbance signal u>(t) is given as follows:

[mathematical expression not reproducible] (57)

We can get the desired nonfragile filter by solving LMIS (45)-(49) in Theorem 7; the nonfragile parameter matrices are given as follows:

[mathematical expression not reproducible] (58)

In addition, the new method provides less conservative design result; we can obtain a smaller [gamma] = 0.4996. Figures 1 and 2 show the response of system's states v(t) and filter's states [x.sub.f] (t), respectively. Figure 3 shows the trajectories of z(t) and its estimates [z.sub.f] (t). The estimation error e(t) is depicted in Figure 4.

The random disturbance signal is given as follows:

[mathematical expression not reproducible] (59)

where [alpha](t) denotes Bernoulli's random event. The simulation results are obtained as shown in Figures 5-9.

The stochastic variables a(t) are Bernoulli-distributed white sequences; Figure 5 shows the event occurrence probability. Figures 6 and 7 show the response of system's states [x.sub.1] (t), [x.sub.2] (t) and filter's states [x.sub.f1] (t), [x.sub.f2] (t), respectively Figure 8 shows the trajectories of z(t) and its estimates [z.sub.f](t). The system error e(t) is depicted in Figure 9.

The results show that the [H.sub.[infinity]] 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 [H.sub.[infinity]] filter design problem for a class of nonlinear systems with an interval time-varying delays; meanwhile, the designing [H.sub.[infinity]] filter with multiplicative gain variations was considered. By constructing a new Lyapunov-Krasovskii functional, we obtained a sufficient condition for designing the nonfragile [H.sub.[infinity]] filter such that the filtering error system is asymptotically stable and satisfied the given [H.sub.[infinity]] 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 [H.sub.[infinity]] 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.

https://doi.org/10.1155/2018/8198603

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-JS175).

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Zhongda Lu, (1,2) Guoliang Zhang, (1) Yi Sun, (3) Jie Sun, (3) Fangming Jin, (3) and Fengxia Xu (1)

(1) College of Mechatronic Engineering, Qiqihar University, Qiqihar 161006, China

(2) College of Computer Science and Technology, Harbin University of Science and Technology, Harbin 150080, China

(3) State Grid Heilongjiang Electric Power Company Qiqihar Power Supply Company, Qiqihar 161006, China

Correspondence should be addressed to Zhongda Lu; luzhongda@163.com

Received 9 May 2018; Revised 18 July 2018; Accepted 2 August 2018; Published 5 September 2018

Academic Editor: Jongrae Kim

Caption: FIGURE 1: The response curve of system states [x.sub.1], [x.sub.2].

Caption: FIGURE 2: The response curve of system states [x.sub.f1], [x.sub.f2].

Caption: FIGURE 3: The response curve of vector z(t), [z.sub.f] (t).

Caption: FIGURE 4: The response curve of vector e(t).

Caption: FIGURE 5: The event occurrence probability [alpha](t).

Caption: FIGURE 6: The response curve of system states [x.sub.1], [x.sub.2].

Caption: FIGURE 7: The response curve of system states [x.sub.f1], [x.sub.f2].

Caption: FIGURE 8: The response curve of vector z(t), [z.sub.f] (t).

Caption: FIGURE 9: The response curve of vector e(t).
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Article Details
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Title Annotation:Research Article
Author:Lu, Zhongda; Zhang, Guoliang; Sun, Yi; Sun, Jie; Jin, Fangming; Xu, Fengxia
Publication:Journal of Control Science and Engineering
Article Type:Report
Date:Jan 1, 2018
Words:4531
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