Elsevier

Applied Mathematics and Computation

Volume 359, 15 October 2019, Pages 71-83
Applied Mathematics and Computation

Event-based reduced-order fuzzy filtering for networked control systems with time-varying delays

https://doi.org/10.1016/j.amc.2019.04.061Get rights and content

Abstract

This paper investigates the event-based reduced-order fuzzy filtering problem for nonlinear networked control systems with network-induced time-varying delays. A novel fuzzy filter design approach is proposed to analyze the fuzzy systems under a reduced-order scheme. The problem of mismatched membership functions is presented in the fuzzy filtering to simplify implementation complexity and increase design flexibility of the designed reduced-order filter. In addition, the integrands of the Lyapunov–Krasovskii functional are dependent of membership functions and the existence conditions of the fuzzy filtering are given by utilizing a linearization approach. Compared with the existing reduced-order filter design approaches, the time-derivatives of membership functions are considered in the analysis process for reducing the conservativeness of obtaining the maximum delay bounds. Finally, a simulation result is given to show the effectiveness and advantages of the analysis scheme.

Introduction

For nonlinear dynamic systems [1], [2], [3], [4], [5], [6], [7], filter design has become an outstanding approach to estimate unknown system states [8], [9]. So far, a multitude of fuzzy filter methodologies have been published for nonlinear networked control systems (NCSs), for example [10], [11], [12] and references therein. Qiu and Feng in [10] considered robust filter design of multichannel nonlinear NCSs. In [11], Zhang and Wang addressed the fuzzy filtering problem for discrete-time fuzzy NCSs. Distributed filter design result of discrete-time fuzzy NCSs under incomplete measurements was investigated in [12]. Recently, Li and Wu in [13] introduced the fuzzy filtering problem into interval type-2 fuzzy NCSs. However, in the above results [10], [11], [12], [13], a limit is that the filter design is of general time-triggered method in the NCSs, under which all of the transmission signal need to be transmitted within the network.

For saving communication resources, Tabuada in [14] proposed an excellent event-based control technique for NCSs. According to the designed event-based threshold condition [15], [16], [17], [18], [19], [20], [21], [22], the packets are transmitted within the network under specific standards for controller/filter design. Despite its capability and advantages, quiet a few event-based approaches of NCSs have been investigated, for example [23], [24], [25]. The event-based filter and controller coordinated design approach was addressed in [23]. Li and Wang in [24], [25] considered the event-based fuzzy filter problem in the network-based fuzzy systems. However, in the mentioned results [23], [24], [25], the event-based fuzzy filter approach is utilized for nonlinear NCSs without uncertain parameters, which is strict for many nonlinear control systems [26], [27]. For handling the problem of membership functions with uncertain information in the nonlinear dynamic model, an interval type-2 (IT2) modeling technique was proposed in [26], [27]. Under these results in [26], [27], Pan and Yang in [28], [29] extended the fuzzy filtering approach under event-based standard into IT2 fuzzy NCSs. In [28], [29], the asynchronous and mismatched membership functions problems caused by event-based condition and uncertain parameter in the nonlinear dynamic model have been solved.

However, in the complex engineering mathematical models, it is difficulty and complexity to evaluate the performance and analyze the stability of the considered higher-order systems. Hence, approach for simplifying the mathematical models based on possible lower-dimensional filter under certain criteria in practical applications is of important significant. Over the past years, the reduced-order problem has been considered in many strategies, such as reduced-order filter [30], [31] and model reduction problem [32], [33]. It is worth emphasizing that a significant result of reduced-order fuzzy filtering under event-based technique was proposed in [34]. For implementation of practical engineering mathematical applications, reduced-order fuzzy filtering has the advantages of flexibility and simplicity. However, for the existing event-based reduced-order fuzzy filtering result [34], there are some issues need to be considered, such as: how to further increase the advantages of flexibility and simplicity of the designed fuzzy filter under the reduced-order technique and how to apply the information of membership functions and the time-derivatives of membership functions for deriving the stability result in the analysis process?

Motivated by above insight, this paper deals with the reduced-order fuzzy filtering problem for IT2 fuzzy NCSs under an event-based approach. The main contributions and advantages are given as follows:

1) Compared with the event-triggered reduced-order fuzzy filtering approach in [34], a novel filter design result to ensure asymptotic stability and H performance of the filtering error system is proposed by utilizing a membership function dependent Lyapunov–Krasovskii functional which can reduce the conservativeness of obtaining the maximum delay bounds.

2) Compared with the technique in [34], the reduced-order fuzzy filtering does not need to share the same membership functions with the original fuzzy model to further increase the advantages of flexibility and simplicity of the designed fuzzy filter. And, by introducing slack matrices, the stability result is derived subject to mismatched membership functions in the filtering error systems.

3) Unlike the time-based fuzzy filter result for IT2 fuzzy NCSs [13], a novel event-based fuzzy filter design approach under a reduced-order technique is presented, which has the advantage of simplicity in the implementation of nonlinear dynamic systems and can save network resources in the NCSs.

Notation: He(S)= S+ST for simplicity. w2=t=0|w(t)|2dt.

Section snippets

IT2 T-S fuzzy model

Consider the following nonlinear dynamic systems with uncertain parameters, which can be represented by the IT2 modeling technique [26], [27].

Plant Rule i: IF P1(x(t)) is φ1i AND ⋅⋅⋅ AND PT(x(t)) is φTi,THEN:x˙(t)=Aix(t)+Biw(t),z(t)=Dix(t),y(t)=Eix(t),where x(t)Rn is the state-space vector, w(t) ∈ L2[0, ∞) is the external disturbance, and z(t)Rnz stands for the output to be estimated, y(t)Rny denotes the measurement output. Ai, Bi, Di, and Ei are known system matrices. In addition, the

Event-based communication technique under switching threshold

In this section, an event-based communication approach under switching threshold is considered to save communication resources. By utilizing the zero-order holder within communication network, the sampled output y(qkh) is transmitted if the following event-based condition under switching threshold satisfieswk+1h=wkh+minl0{(l+1)h|[y(qkh)y(wkh)]Tϕ[y(qkh)y(wkh)]>g(qkh)yT(wkh)ϕy(wkh)},where ϕ > 0 is a weighting matrix and qkh=wkh+(l+1)h.(lN). g(qkh) is given under the following switching

Reduced-order fuzzy filtering

In this section, a reduced-order fuzzy filtering is structured, which has the advantages of simplicity and flexibility to analyze the complexity nonlinear dynamic model. To further increase the simplicity and flexibility, the designed reduced-order fuzzy filtering and the fuzzy dynamic model do not require to share same membership functions. The details of reduced-order fuzzy filtering are described by

Filter Rule j: IF ϑ1(x^(t)) is V1j AND ⋅⋅⋅ AND ϑa(x^(t)) is Vaj,THEN:x^˙(t)=C^jx^(t)+D^jy^(t),z

Main results

By employing membership function dependent Lyapunov–Krasovskii functional strategy, sufficient conditions of asymptotically stable and H performance are presented in this section for system (17) under reduced-order fuzzy filtering technique.

Theorem 1

For given constants g2 > 0 and m¯>0, the event-based system (17) under reduced-order fuzzy filtering satisfies asymptotically stable and H performance level γ, if the membership functions hold for ujβjhj0 (0 < βj ≤ 1) and there exist matrices S > 0, Hj

Simulation example

In this example, a tunnel diode circuit shown in Fig. 1 with following dynamics is introduced to show the feasibility of the designed event-based reduced-order fuzzy filtering.iD(t)=0.002υD(t)+δυD3(t),with the uncertain parameter δ ∈ [0.01, 0.03].

Let x1(t)=υC(t), x2(t)=iL(t) and Γ=0.002+δυC2(t). Then, the circuit system is represented by:Cˇx˙1(t)=gˇx1(t)+x2(t),Lˇx˙2(t)=x1(t)Rˇx2(t)+w(t),The parameters of the system are same as the Example 2 in [34]. Using the technique in [13] with x1(t)[3,

Conclusion

The reduced-order fuzzy filtering problem with event-based approach has been solved. In the analysis scheme, a novel filter design result to ensure asymptotic stability and H performance of the filtering error system has been proposed by utilizing a membership function dependent Lyapunov–Krasovskii functional which can reduce the conservativeness of obtaining the maximum delay bounds. In addition, the reduced-order fuzzy filtering does not need to share the same membership functions with the

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    This work was supported in part by the Funds of the National Natural Science Foundation of China (Grant Nos. 61621004 and 61420106016), the Fundamental Research Funds for the Central Universities (No. N160406003), and the Research Fund of State Key Laboratory of Synthetical Automation for Process Industries (Grant no. 2018ZCX03).

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