Design of 𝐻_∞ Filter for a Class of Switched Linear Neutral Systems

Abstract

This paper is concerned with the 𝐻_∞ filtering problem for a class of switched linear neutral systems with time-varying delays. The time-varying delays appear not only in the state but also in the state derivatives. Based on the average dwell time approach and the piecewise Lyapunov functional technique, sufficient conditions are proposed for the exponential stability of the filtering error dynamic system. Then, the corresponding solvability condition for a desired filter satisfying a weighted 𝐻_∞ performance is established. All the conditions obtained are delay-dependent. Finally, two numerical examples are given to illustrate the effectiveness of the proposed theory.

1. Introduction

Switched time-delay systems have been attracting considerable attention during the recent years [1–9], due to the significance both in theory development and practical applications. However, it is worth noting that only the state time delay is considered, and the time delay in the state derivatives is largely ignored in the existing literature. If each subsystem of a switched system has time delay in the state derivatives, then the switched system is called switched neutral system [10]. Switched neutral systems exist widely in engineering and social systems, many physical plants can be modeled as switched neutral systems, such as distributed networks and heat exchanges. For example, in [11], a switched neutral type delay equation with nonlinear perturbations was exploited to model the drilling system. Compared with the switched systems with state time delay, switched neutral systems are much more complicated [12–15]. As effective tools, the common Lyapunov function method, dwell time approaches, and average dwell time approaches have been extended to study the switched neutral systems, and many valuable results have been obtained for switched neutral systems. On the research of stability analysis for switched neutral systems, the asymptotically stable problem of switched neutral systems was considered in [16]. If there exists a Hurwitz linear convex combination of state matrices, and subsystems are not necessarily stable, switching rules can be designed to guarantee the asymptotical stability of the switched neutral system. The method of Lyapunov-Metzler linear matrix inequalities in [17] was extended to switched neutral systems [18], and some less conservative stability results were obtained.

In contrast with the traditional Kalman filtering, the 𝐻_∞ filtering does not require the exact knowledge of the statistics of the external noise signals, and it is insensitive to the uncertainties both in the exogenous signal statistics and in dynamic models [19, 20]. Because of these advantages, the 𝐻_∞ filtering has attracted much attention in the past decade for nonswitched neutral systems [21–24]. In [22], some sufficient conditions for the existence of an 𝐻_∞ filter of a Luenberger observer type have been provided. However, to the authors' best knowledge, the 𝐻_∞ filtering for switched neutral systems has been rarely investigated and still remains challenging. This motivates our research.

The contribution of this paper lies in three aspects. First, we address the delay-dependent 𝐻_∞ filtering problem for switched linear neutral systems with time-varying delays, which appear not only in the state, but also in the state derivatives. The resulting filter is of the Luenberger-observer type. Second, by using average dwell time approach and the piecewise Lyapunov function technique, we derive a delay-dependent sufficient condition, which guarantees exponential stability of the filtering error system. Then, the corresponding solvability condition for a desired filter satisfying a weighted 𝐻_∞ performance is established. Here, to reduce the conservatism of the delay-dependent condition, we introduce some slack matrix variables and a new integral inequality recently proposed in [25]. Finally, we succeed in transforming the filter design problem into the feasibility problem of some linear matrix inequalities. To show the efficiency of the obtained results, we present two relevant examples.

The remainder of this paper is organized as follows. The 𝐻_∞ filtering problem for switched neutral systems is formulated in Section 2. Section 3 presents our main results. Numerical examples are given in Section 4, and we conclude this paper in Section 5.

Notation. Throughout this paper, 𝑅^𝑛 denotes 𝑛-dimensional Euclidean space; 𝑅^𝑛×𝑚 is the set of all 𝑛×𝑚 real matrices; 𝑃>0 means that 𝑃 is positive definite; 𝐿₂ denotes the space of square integrable vector functions on [0,∞) with norm ‖⋅‖=(∫^∞₀‖⋅‖²𝑑𝑡)^1/2, where ‖⋅‖ denotes the Euclidean vector norm; 𝐼 is the identity matrix with appropriate dimensions; the symmetric terms in a symmetric matrix are denoted by ∗ as for example =𝑌𝑋𝑌∗𝑍𝑋𝑌_𝑇𝑍. (1)

2. Problem Statement

Consider the following switched linear neutral system: 𝑥̇(𝑡)=𝐴_0𝜎(𝑡)𝑥(𝑡)+𝐴_1𝜎(𝑡)𝑥(𝑡−ℎ(𝑡))+𝐹_𝜎(𝑡)𝑥̇(𝑡−𝜏(𝑡))+𝐵_𝜎(𝑡)𝜔(𝑡),𝑦(𝑡)=𝐶_0𝜎(𝑡)𝑥(𝑡)+𝐶_1𝜎(𝑡)𝑥(𝑡−ℎ(𝑡))+𝐷_𝜎(𝑡)𝜔(𝑡),𝑧(𝑡)=𝐿_𝜎(𝑡)𝑥(𝑡),𝑥(𝜃)=𝜓(𝜃),∀𝜃∈[−𝐻,0],𝐻=max{ℎ,𝜏}, (2) where 𝑥(𝑡)∈𝑅^𝑛 is the state vector; 𝑦(𝑡)∈𝑅^𝑚 is the measurements vector; 𝜔(𝑡)∈𝑅^𝑝 is the noise signal vector, which belongs to 𝐿₂[0,∞); 𝑧(𝑡)∈𝑅^𝑞 is the signal to be estimated; 𝜓(𝑡) is the initial vector function that is continuously differentiable on [−𝐻,0];  𝜎(𝑡)∶[0,∞)→𝑀={1,2,…,𝑚} is a piecewise constant function of time 𝑡 called switching signal. Corresponding to the switching signal 𝜎(𝑡), we have the switching sequence {𝑥_𝑡0∶(𝑖₀,𝑡₀),…,(𝑖_𝑘,𝑡_𝑘),…,∣𝑖_𝑘∈𝑀,𝑘=0,1,…}, which means that the 𝑖_𝑘th subsystem is active when 𝑡∈[𝑡_𝑘,𝑡_𝑘+1). The system coefficient matrices 𝐴_0𝑖𝑘, 𝐴_1𝑖𝑘, 𝐹_𝑖𝑘, 𝐵_𝑖𝑘, 𝐶_0𝑖𝑘, 𝐶_1𝑖𝑘, 𝐷_𝑖𝑘, and 𝐿_𝑖𝑘 are known real constant matrices of appropriate dimensions. ℎ(𝑡) and 𝜏(𝑡) are time-varying delays satisfying 0≤ℎ(𝑡)≤ℎ,ℎ̇(𝑡)≤(𝑡)𝜏̇(𝑡)≤ℎ<1,0≤𝜏≤𝜏,𝜏<1. (3)

The objective of this paper is to design a family of filters of Luenberger observer type as follows: 𝑥̇(𝑡)=𝐴_0𝜎(𝑡)𝑥(𝑡)+𝐴_1𝜎(𝑡)𝑥(𝑡−ℎ(𝑡))+𝐹_𝜎(𝑡)𝑥̇(𝑡−𝜏(𝑡))+𝐾_𝜎(𝑡)𝑦(𝑡)−𝐶_0𝜎(𝑡)𝑥(𝑡)−𝐶_1𝜎(𝑡)𝑥((𝑡,𝑧(𝑡)𝑡−ℎ))=𝐿_𝜎(𝑡)𝑥(𝑡),𝑥(𝜃)=𝜓(𝜃),∀𝜃∈[−𝐻,0],𝐻=max{ℎ,𝜏}, (4) where 𝐾_𝑖𝑘 are the filter parameters, which are to be determined.

Now, we introduce the estimation errors: 𝑥_𝑒(𝑡)=𝑥(𝑡)−𝑥(𝑡), 𝑧_𝑒(𝑡)=𝑧(𝑡)−𝑧(𝑡).

Combining (2) with (4) gives the following filtering error dynamic system: 𝑥̇_𝑒(𝑡)=𝐴_0𝜎(𝑡)𝑥_𝑒(𝑡)+𝐴_1𝜎(𝑡)𝑥_𝑒(𝑡−ℎ(𝑡))+𝐹_𝜎(𝑡)𝑥̇_𝑒(𝑡−𝜏(𝑡))+𝐵_𝜎(𝑡)𝜔(𝑡),𝑧_𝑒(𝑡)=𝐿_𝜎(𝑡)𝑥_𝑒(𝑡),𝑥_𝑒(𝜃)=𝜓(𝜃)−𝜓(𝜃),∀𝜃∈[−𝐻,0],𝐻=max{ℎ,𝜏}, (5) where 𝐴_0𝜎=𝐴_0𝜎−𝐾_𝜎𝐶_0𝜎, 𝐴_1𝜎=𝐴_1𝜎−𝐾_𝜎𝐶_1𝜎, 𝐵_𝜎=𝐵_𝜎−𝐾_𝜎𝐷_𝜎.

The following definitions are introduced, which will play key roles in deriving our main results.

Definition 1 (see [26]). The equilibrium 𝑥^∗_𝑒=0 of the filtering error system (5) is said to be exponentially stable under 𝜎(𝑡) if the solution 𝑥_𝑒(𝑡) of system (5) with 𝜔(𝑡)=0 satisfies ‖𝑥_𝑒(𝑡)‖≤Γ𝑒^{−𝜆(𝑡−𝑡0)}‖𝑥_𝑒(𝑡₀)‖_𝐻, for all 𝑡≥𝑡₀ for constants Γ>0 and 𝜆>0, where ‖⋅‖ denotes the Euclidean norm, and ‖𝑥_𝑒(𝑡)‖_𝐻=sup_{−𝐻≤𝜃≤0}{𝑥_𝑒(𝑡+𝜃),𝑥̇_𝑒(𝑡+𝜃)}.

Definition 2 (see [26]). For any 𝑇₂>𝑇₁≥0, let 𝑁_𝜎(𝑇₁,𝑇₂) denote the number of switching of 𝜎(𝑡) over (𝑇₁,𝑇₂). If 𝑁_𝜎(𝑇₁,𝑇₂)≤𝑁₀+(𝑇₂−𝑇₁)/𝑇_𝑎 holds for 𝑇_𝑎>0, 𝑁₀≥0, then 𝑇_𝑎 is called average dwell time. As commonly used in the literature, we choose 𝑁₀=0.

The filtering problem addressed in this paper is to seek for suitable filter gain 𝐾_𝑖 such that the filtering error system (5) for any switching signal with average dwell time has a prescribed 𝐻_∞ performance 𝛾; that is,(1)the error system (5) with 𝜔(𝑡)=0 is exponentially stable;(2)under the zero initial conditions, that is, 𝑥_𝑒(𝜃)=0, for all 𝜃∈[−𝐻,0], the weighted 𝐻_∞ performance ∫^∞₀𝑒^−𝛼𝑠𝑧^𝑇_𝑒(𝑠)𝑧_𝑒(𝑠)𝑑𝑠≤𝛾²∫^∞₀𝜔^𝑇(𝑠)𝜔(𝑠)𝑑𝑠 is guaranteed for all nonzero 𝜔(𝑡)∈𝐿₂[0,∞) and a prescribed level of noise attenuation 𝛾>0.

Before concluding this section, we introduce three lemmas which are essential for the development of the results.

Lemma 3 (see [25]). Let 𝑥(𝑡)∈𝑅^𝑛 be a vector-valued function with first-order continuous-derivative entries. Then, the following integral inequality holds for any matrices 𝑀₁, 𝑀₂∈𝑅^𝑛×𝑛, and 𝑋=𝑋^𝑇>0, and a scalar ℎ≥0, −^𝑡_𝑡−ℎ𝑥̇^𝑇(𝑠)𝑋𝑥̇(𝑠)𝑑𝑠≤𝜉^𝑇(𝑡)𝑀^𝑇₁+𝑀₁−𝑀^𝑇₁+𝑀₂∗−𝑀^𝑇₂−𝑀₂𝜉(𝑡)+ℎ𝜉^𝑇(𝑡)𝑀^𝑇₁𝑀^𝑇₂𝑋⁻¹𝑀₁𝑀₂𝜉(𝑡), (6) where 𝜉^𝑇(𝑡)=𝑥^𝑇(𝑡)𝑥^𝑇(𝑡−ℎ).

Lemma 4 (see [27]). For any constant matrix 0<𝑅=𝑅^𝑇∈𝑅^𝑛×𝑛, scalar 𝑟>0, vector function 𝜔∶[0,𝑟]→𝑅^𝑛 such that the integrations concerned are well defined; then, (∫^𝑟₀𝜔(𝑠)𝑑𝑠)^𝑇𝑅(∫^𝑟₀𝜔(𝑠)𝑑𝑠)≤𝑟∫^𝑟₀𝜔^𝑇(𝑠)𝑅𝜔(𝑠)𝑑𝑠.

Lemma 5 (Schur complement). For given 𝑆=^𝑆11𝑆12_∗𝑆22<0, where 𝑆₁₁=𝑆^𝑇₁₁ and 𝑆₂₂=𝑆^𝑇₂₂, the following is equivalent: (1)𝑆₁₁<0,𝑆₂₂−𝑆^𝑇₁₂𝑆⁻¹₁₁𝑆₁₂(2)𝑆<0;₂₂<0,𝑆₁₁−𝑆₁₂𝑆⁻¹₂₂𝑆^𝑇₁₂<0. (7)

3. Main Results

In this section, we first present a sufficient condition for exponential stability of the filtering error system (5) with 𝜔(𝑡)=0. Then, it is applied to formulate an approach to design the desired 𝐻_∞ filters for switched neutral system (2).

3.1. Stability Analysis

Theorem 6. Given 𝛼>0, ‖𝐹_𝑖𝑘‖<1, for all 𝑖_𝑘∈𝑀. If there exist matrices 𝑃_𝑖𝑘>0, 𝑄_𝑖𝑘>0, 𝑅_𝑖𝑘>0, 𝑀_𝑖𝑘>0, 𝑁_𝑖𝑘>0, and 𝑇_1𝑖𝑘, 𝑇_2𝑖𝑘, 𝑁_𝑔𝑖𝑘 (𝑔=1,2,…,7) of appropriate dimensions, and 𝜇≥1, such that for all 𝑖_𝑘∈𝑀,Σ_𝑖𝑘=Σ₁₁Σ₁₂Σ₁₃Σ₁₄Σ₁₅𝐴^𝑇_0𝑖𝑘𝑁_6𝑖𝑘Σ₁₇∗Σ₂₂Σ₂₃−𝑁_4𝑖𝑘Σ₂₅−𝑁_6𝑖𝑘−𝑁_7𝑖𝑘∗∗Σ₃₃𝐴^𝑇_1𝑖𝑘𝑁_4𝑖𝑘Σ₃₅𝐴^𝑇_1𝑖𝑘𝑁_6𝑖𝑘𝐴^𝑇_1𝑖𝑘𝑁_7𝑖𝑘∗∗∗Σ₄₄𝑁^𝑇_4𝑖𝑘𝐹_𝑖𝑘0𝜏𝑇^𝑇_2𝑖𝑘∗∗∗∗Σ₅₅𝐹^𝑇_𝑖𝑘𝑁_6𝑖𝑘𝐹^𝑇_𝑖𝑘𝑁_7𝑖𝑘1∗∗∗∗∗−ℎ𝑒^−𝛼ℎ𝑀_𝑖𝑘0∗∗∗∗∗∗−𝜏𝑒^𝛼𝜏𝑁_𝑖𝑘<0,𝑃_𝑖𝑘≤𝜇𝑃_𝑖𝑗,𝑄_𝑖𝑘≤𝜇𝑄_𝑖𝑗,𝑅_𝑖𝑘≤𝜇𝑅_𝑖𝑗,𝑀_𝑖𝑘≤𝜇𝑀_𝑖𝑗,𝑁_𝑖𝑘≤𝜇𝑁_𝑖𝑗,∀𝑖_𝑘,𝑖_𝑗∈𝑀, (8)(9)where Σ₁₁=ℎ𝑀_𝑖𝑘+𝛼𝑃_𝑖𝑘+𝑄_𝑖𝑘+𝑒^−𝛼𝜏𝑇^𝑇_1𝑖𝑘+𝑒^−𝛼𝜏𝑇_1𝑖𝑘+𝑁^𝑇_1𝑖𝑘𝐴_0𝑖𝑘+𝐴^𝑇_0𝑖𝑘𝑁_1𝑖𝑘,Σ₁₂=𝑃_𝑖𝑘−𝑁^𝑇_1𝑖𝑘+𝐴^𝑇_0𝑖𝑘𝑁_2𝑖𝑘,Σ₁₃=𝑁^𝑇_1𝑖𝑘𝐴_1𝑖𝑘+𝐴^𝑇_0𝑖𝑘𝑁_3𝑖𝑘,Σ₁₄=−𝑒^−𝛼𝜏𝑇^𝑇_1𝑖𝑘+𝑒^−𝛼𝜏𝑇_2𝑖𝑘+𝐴^𝑇_0𝑖𝑘𝑁_4𝑖𝑘,Σ₁₅=𝑁^𝑇_1𝑖𝑘𝐹_𝑖𝑘+𝐴^𝑇_0𝑖𝑘𝑁_5𝑖𝑘,Σ₁₇=𝜏𝑇^𝑇_1𝑖𝑘+𝐴^𝑇_0𝑖𝑘𝑁_7𝑖𝑘,Σ₂₂=𝑅_𝑖𝑘+𝜏𝑁_𝑖𝑘−𝑁^𝑇_2𝑖𝑘−𝑁_2𝑖𝑘,Σ₂₃=−𝑁_3𝑖𝑘+𝑁^𝑇_2𝑖𝑘𝐴_1𝑖𝑘,Σ₂₅=−𝑁_5𝑖𝑘+𝑁^𝑇_2𝑖𝑘𝐹_𝑖𝑘,Σ₃₃=−1−ℎ𝑒^−𝛼ℎ𝑄_𝑖𝑘+𝑁^𝑇_3𝑖𝑘𝐴_1𝑖𝑘+𝐴^𝑇_1𝑖𝑘𝑁_3𝑖𝑘,Σ₃₅=𝑁^𝑇_3𝑖𝑘𝐹_𝑖𝑘+𝐴^𝑇_1𝑖𝑘𝑁_5𝑖𝑘,Σ₄₄=−𝑒^−𝛼𝜏𝑇^𝑇_2𝑖𝑘−𝑒^−𝛼𝜏𝑇_2𝑖𝑘,Σ₅₅=−(1−𝜏)𝑒^−𝛼𝜏𝑅_𝑖𝑘+𝑁^𝑇_5𝑖𝑘𝐹_𝑖𝑘+𝐹^𝑇_𝑖𝑘𝑁_5𝑖𝑘, (10) then the error dynamic system (5) with 𝜔(𝑡)=0 is exponentially stable for any switching signal with average dwell time satisfying 𝑇_𝑎>𝑇^∗_𝑎=ln𝜇/𝛼.

Proof. Define the piecewise Lyapunov-Krasovskii functional candidate 𝑉(𝑡)=𝑉_𝜎(𝑡)(𝑡)=⁵_𝑗=1𝑉_{𝑗𝜎(𝑡)}(𝑡), (11) where 𝑉_1𝑖𝑘(𝑡)=𝑥^𝑇_𝑒(𝑡)𝑃_𝑖𝑘𝑥_𝑒(𝑡),𝑉_2𝑖𝑘(𝑡)=^𝑡_{(𝑡)𝑡−ℎ}𝑥^𝑇_𝑒(𝑠)𝑒^{𝛼(𝑠−𝑡)}𝑄_𝑖𝑘𝑥_𝑒(𝑠)𝑉𝑑𝑠,_3𝑖𝑘(𝑡)=^𝑡_{(𝑡)𝑡−𝜏}𝑥̇^𝑇_𝑒(𝑠)𝑒^{𝛼(𝑠−𝑡)}𝑅_𝑖𝑘𝑥̇_𝑒(𝑠)𝑉𝑑𝑠,_4𝑖𝑘(𝑡)=⁰_−ℎ^𝑡_𝑡+𝜃𝑥^𝑇_𝑒(𝑠)𝑒^{𝛼(𝑠−𝑡)}𝑀_𝑖𝑘𝑥_𝑒(𝑠)𝑉𝑑𝑠𝑑𝜃,_5𝑖𝑘(𝑡)=⁰_−𝜏^𝑡_𝑡+𝜃𝑥̇^𝑇_𝑒(𝑠)𝑒^{𝛼(𝑠−𝑡)}𝑁_𝑖𝑘𝑥̇_𝑒(𝑠)𝑑𝑠𝑑𝜃. (12)
Now, taking the derivative of 𝑉_𝑗𝑖𝑘(𝑡), 𝑗=1,2,…,5 with respect to 𝑡 along the trajectory of the error system (5) with 𝜔(𝑡)=0, according to (3) and Lemma 4, we have 𝑉̇_𝑖𝑘(𝑡)+𝛼𝑉_𝑖𝑘(𝑡)≤2𝑥^𝑇_𝑒(𝑡)𝑃_𝑖𝑘𝑥̇_𝑒(𝑡)+𝑥^𝑇_𝑒(𝑡)𝑄_𝑖𝑘𝑥_𝑒(𝑡)+𝑥̇^𝑇_𝑒(𝑡)𝑅_𝑖𝑘𝑥̇_𝑒(𝑡)−1−ℎ𝑥^𝑇_𝑒(𝑡−ℎ(𝑡))𝑒^−𝛼ℎ𝑄_𝑖𝑘𝑥_𝑒(𝑡−ℎ(𝑡))+ℎ𝑥^𝑇_𝑒(𝑡)𝑀_𝑖𝑘𝑥_𝑒(𝑡)+𝜏𝑥̇^𝑇_𝑒(𝑡)𝑁_𝑖𝑘𝑥̇_𝑒(𝑡)+𝛼𝑥^𝑇_𝑒(𝑡)𝑃_𝑖𝑘𝑥_𝑒(𝑡)−(1−𝜏)𝑥̇^𝑇_𝑒(𝑡−𝜏(𝑡))𝑒^−𝛼𝜏𝑅_𝑖𝑘𝑥̇_𝑒((𝑡−𝑒𝑡−𝜏))_−𝛼ℎℎ^𝑡_𝑡−ℎ𝑥^𝑇_𝑒(𝑠)𝑑𝑠𝑀_𝑖𝑘^𝑡_𝑡−ℎ𝑥_𝑒(𝑠)−𝑑𝑠_{𝑡𝑡−𝜏}𝑥̇^𝑇_𝑒(𝑠)𝑒^{𝛼(𝑠−𝑡)}𝑁_𝑖𝑘𝑥̇_𝑒(𝑠)𝑑𝑠. (13)
From Lemma 3, it holds −^𝑡_𝑡−𝜏𝑥̇^𝑇_𝑒(𝑠)𝑒^{𝛼(𝑠−𝑡)}𝑁_𝑖𝑘𝑥̇_𝑒(𝑠)𝑑𝑠≤−^𝑡_𝑡−𝜏𝑥̇^𝑇_𝑒(𝑠)𝑒^−𝛼𝜏𝑁_𝑖𝑘𝑥̇_𝑒(𝑠)𝑑𝑠≤𝑒^−𝛼𝜏𝑥^𝑇_𝑒(𝑡)𝑥^𝑇_𝑒()𝑇𝑡−𝜏^{𝑇1𝑖𝑘}+𝑇_1𝑖𝑘−𝑇^𝑇_1𝑖𝑘+𝑇_2𝑖𝑘∗−𝑇^𝑇_2𝑖𝑘−𝑇_2𝑖𝑘×𝑥_𝑒(𝑡)𝑥_𝑒(𝑡−𝜏)+𝜏𝑒^−𝛼𝜏𝑥^𝑇_𝑒(𝑡)𝑥^𝑇_𝑒()×𝑇𝑡−𝜏_{𝑇1𝑖𝑘}𝑇^𝑇_2𝑖𝑘𝑁_𝑖⁻¹_𝑘𝑇_1𝑖𝑘𝑇_2𝑖𝑘𝑥_𝑒(𝑡)𝑥_𝑒().𝑡−𝜏 (14)
Define 𝜉^𝑇_𝑒(𝑡)=𝜁^𝑇_𝑒(𝑡)𝑥̇^𝑇_𝑒(𝑡−𝜏(𝑡))^𝑡_𝑡−ℎ𝑥^𝑇_𝑒(𝑠)𝑑𝑠, (15) where 𝜁^𝑇_𝑒(𝑡)=𝑥^𝑇_𝑒(𝑡)𝑥̇^𝑇_𝑒(𝑡)𝑥^𝑇_𝑒(𝑡−ℎ(𝑡))𝑥^𝑇_𝑒(𝑡−𝜏).
By some algebraic manipulations, it is easy to show that 𝑉̇_𝑖𝑘(𝑡)+𝛼𝑉_𝑖𝑘(𝑡)≤𝜉^𝑇_𝑒(𝑡)Σ_𝑖𝑘𝜉_𝑒(𝑡), (16) where Σ_𝑖𝑘=Σ₁₁𝑃_𝑖𝑘0Σ₁₄00∗𝑅_𝑖𝑘+𝜏𝑁_𝑖𝑘0000∗∗−1−ℎ𝑒^−𝛼ℎ𝑄_𝑖𝑘000∗∗∗Σ₄₄00∗∗∗∗Σ₅₅0∗∗∗∗∗Σ₆₆, (17) where Σ₁₁=ℎ𝑀_𝑖𝑘+𝛼𝑃_𝑖𝑘+𝑄_𝑖𝑘+𝑒^−𝛼𝜏𝑇^𝑇_1𝑖𝑘+𝑒^−𝛼𝜏𝑇_1𝑖𝑘+𝜏𝑒^−𝛼𝜏𝑇^𝑇_1𝑖𝑘𝑁_𝑖⁻¹_𝑘𝑇_1𝑖𝑘,Σ₁₄=−𝑒^−𝛼𝜏𝑇^𝑇_1𝑖𝑘+𝑒^−𝛼𝜏𝑇_2𝑖𝑘+𝜏𝑒^−𝛼𝜏𝑇^𝑇_1𝑖𝑘𝑁_𝑖⁻¹_𝑘𝑇_2𝑖𝑘,Σ₅₅=−(1−𝜏)𝑒^−𝛼𝜏𝑅_𝑖𝑘,Σ₄₄=−𝑒^−𝛼𝜏𝑇^𝑇_2𝑖𝑘−𝑒^−𝛼𝜏𝑇_2𝑖𝑘+𝜏𝑒^−𝛼𝜏𝑇^𝑇_2𝑖𝑘𝑁_𝑖⁻¹_𝑘𝑇_2𝑖𝑘,Σ₆₆1=−ℎ𝑒^−𝛼ℎ𝑀_𝑖𝑘. (18)
From Lemma 5, Σ_𝑖𝑘<0 is equivalent to Σ_𝑖𝑘=Σ₁₁𝑃_𝑖𝑘0Σ₁₄00𝜏𝑇^𝑇_1𝑖𝑘∗𝑅_𝑖𝑘+𝜏𝑁_𝑖𝑘Σ00000∗∗₃₃Σ0000∗∗∗₄₄00𝜏𝑇^𝑇_2𝑖𝑘∗∗∗∗Σ₅₅Σ00∗∗∗∗∗₆₆0∗∗∗∗∗∗−𝜏𝑒^𝛼𝜏𝑁_𝑖𝑘<0, (19) where Σ₁₁=ℎ𝑀_𝑖𝑘+𝛼𝑃_𝑖𝑘+𝑄_𝑖𝑘+𝑒^−𝛼𝜏𝑇^𝑇_1𝑖𝑘+𝑒^−𝛼𝜏𝑇_1𝑖𝑘,Σ₁₄=−𝑒^−𝛼𝜏𝑇^𝑇_1𝑖𝑘+𝑒^−𝛼𝜏𝑇_2𝑖𝑘,Σ₃₃=−1−ℎ𝑒^−𝛼ℎ𝑄_𝑖𝑘,Σ₄₄=−𝑒^−𝛼𝜏𝑇^𝑇_2𝑖𝑘−𝑒^−𝛼𝜏𝑇_2𝑖𝑘,Σ₅₅=Σ₅₅,Σ₆₆=Σ₆₆. (20)
In addition, the following is true from (5) with 𝜔(𝑡)=0: 2𝜉^𝑇_𝑒(𝑡)𝑁_1𝑖𝑘𝑁_2𝑖𝑘𝑁_3𝑖𝑘𝑁_4𝑖𝑘𝑁_5𝑖𝑘𝑁_6𝑖𝑘𝑁_7𝑖𝑘^𝑇×𝐴_0𝑖𝑘−𝐼𝐴_1𝑖𝑘0𝐹_𝑖𝑘00𝜉_𝑒(𝑡)=0, (21) where 𝜉^𝑇_𝑒(𝑡)=𝜉^𝑇_𝑒(𝑡)0.
Then, (19) along with (21) gives Σ_𝑖𝑘<0, which yields Σ_𝑖𝑘<0; thus, 𝑉̇_𝑖𝑘(𝑡)+𝛼𝑉_𝑖𝑘(𝑡)≤0. (22) Combining (9) with (22), for any 𝑡∈[𝑡_𝑘,𝑡_𝑘+1), we have 𝑉(𝑡)=𝑉_𝑖𝑘(𝑡)≤𝑒^{−𝛼(𝑡−𝑡𝑘)}𝑉_𝑖𝑘𝑡_𝑘≤𝜇𝑒^{−𝛼(𝑡−𝑡𝑘)}𝑉_𝜎(𝑡⁻_𝑘)𝑡⁻_𝑘≤𝜇𝑒^{−𝛼(𝑡−𝑡𝑘)}𝑒^{−𝛼(𝑡𝑘−𝑡𝑘−1)}𝑉_{𝜎(𝑡𝑘−1)}𝑡_𝑘−1≤⋯≤𝜇^𝑘𝑒^{−𝛼(𝑡−𝑡0)}𝑉𝑡₀≤𝑒^{−(𝛼−(ln𝜇/𝑇𝑎))(𝑡−𝑡0)}𝑉𝑡₀. (23)
According to (11), we have 𝑎𝑥_𝑒(𝑡)²≤𝑉(𝑡)≤𝑏𝑥_𝑒(𝑡₀)²_𝐻, (24) where 𝑎=min_{∀𝑖𝑘∈𝑀}𝜆_min𝑃_𝑖𝑘,𝑏=max_{∀𝑖𝑘∈𝑀}𝜆_max𝑃_𝑖𝑘+ℎmax_{∀𝑖𝑘∈𝑀}𝜆_max𝑄_𝑖𝑘+𝜏max_{∀𝑖𝑘∈𝑀}𝜆_max𝑅_𝑖𝑘+ℎ²2max_{∀𝑖𝑘∈𝑀}𝜆_max𝑀_𝑖𝑘+𝜏²2max_{∀𝑖𝑘∈𝑀}𝜆_max𝑁_𝑖𝑘. (25)
Considering (23) and (24), it holds ‖𝑥_𝑒(𝑡)‖≤(𝑏/𝑎)𝑒^{−(1/2)(𝛼−(ln𝜇/𝑇𝑎))(𝑡−𝑡0)}‖𝑥_𝑒(𝑡₀)‖_𝐻.
Therefore, if 𝛼−(ln𝜇/𝑇_𝑎)>0, that is 𝑇_𝑎>(ln𝜇/𝛼), then error dynamic system (5) is exponentially stable.

Remark 7. When 𝜇=1, we have 𝑇^∗_𝑎=0, which means that the switching signal 𝜎(𝑡) can be arbitrary. In this case, condition (9) implies that there exists a common Lyapunov functional for all subsystems. Moreover, setting 𝛼=0 in (8) gives asymptotic stability of the filtering error system (5) under arbitrary switching.

Remark 8. The filters of Luenberger observer type has been adopted in the literatures, see [17]. The Luenberger-type observer can produce an approximation to the system state that is independent of the system trajectory, and it only depends on the initial value of the system state.

Remark 9. The condition ‖𝐹_𝑖𝑘‖<1 guarantees that Lipschitz constant for the right hand of (2) with respect to 𝑥̇(𝑡−𝜏(𝑡)) is less than one.

3.2. Filter Design

Now, we design the desired 𝐻_∞ filter for the switched neutral system (2).

Theorem 10. Given 𝛼>0, if ‖𝐹_𝑖𝑘‖<1, for all 𝑖_𝑘∈𝑀, and if there exists matrices 𝑃_𝑖𝑘>0, 𝑄_𝑖𝑘>0, 𝑅_𝑖𝑘>0, 𝑀_𝑖𝑘>0, 𝑁_𝑖𝑘>0, and 𝑇_1𝑖𝑘, 𝑇_2𝑖𝑘, 𝑊_𝑖𝑘, 𝑋_𝑖𝑘 of appropriate dimensions, and 𝜇≥1, such that, for any 𝑖_𝑘∈𝑀,Ω_𝑖𝑘=Ω₁₁Ω₁₂Ω₁₃Ω₁₄𝑊^𝑇_𝑖𝑘𝐹_𝑖𝑘0𝜏𝑇^𝑇_1𝑖𝑘Ω₁₈∗Ω₂₂Ω₂₃0𝑊^𝑇_𝑖𝑘𝐹_𝑖𝑘00Ω₂₈∗∗Ω₃₃0𝑊^𝑇_𝑖𝑘𝐹_𝑖𝑘00Ω₃₈∗∗∗Ω₄₄00𝜏𝑇^𝑇_2𝑖𝑘0∗∗∗∗Ω₅₅000∗∗∗∗∗Ω₆₆00∗∗∗∗∗∗−𝜏𝑒^𝛼𝜏𝑁_𝑖𝑘0∗∗∗∗∗∗∗−𝛾²𝐼<0,𝑃_𝑖𝑘≤𝜇𝑃_𝑖𝑗,𝑄_𝑖𝑘≤𝜇𝑄_𝑖𝑗,𝑅_𝑖𝑘≤𝜇𝑅_𝑖𝑗,𝑀_𝑖𝑘≤𝜇𝑀_𝑖𝑗,𝑁_𝑖𝑘≤𝜇𝑁_𝑖𝑗,∀𝑖_𝑘,𝑖_𝑗∈𝑀, (26)(27)where Ω₁₁=ℎ𝑀_𝑖𝑘+𝛼𝑃_𝑖𝑘+𝑄_𝑖𝑘+𝑒^−𝛼𝜏𝑇^𝑇_1𝑖𝑘+𝑒^−𝛼𝜏𝑇_1𝑖𝑘+𝑊^𝑇_𝑖𝑘𝐴_0𝑖𝑘+𝐴^𝑇_0𝑖𝑘𝑊_𝑖𝑘−𝑋_𝑖𝑘𝐶_0𝑖𝑘−𝐶^𝑇_0𝑖𝑘𝑋^𝑇_𝑖𝑘+𝐿^𝑇_𝑖𝑘𝐿_𝑖𝑘,Ω₁₂=𝑃_𝑖𝑘−𝑊^𝑇_𝑖𝑘+𝐴^𝑇_0𝑖𝑘𝑊_𝑖𝑘−𝐶^𝑇_0𝑖𝑘𝑋^𝑇_𝑖𝑘,Ω₁₃=𝑊^𝑇_𝑖𝑘𝐴_1𝑖𝑘+𝐴^𝑇_0𝑖𝑘𝑊_𝑖𝑘−𝑋_𝑖𝑘𝐶_1𝑖𝑘−𝐶^𝑇_0𝑖𝑘𝑋^𝑇_𝑖𝑘,Ω₁₄=−𝑒^−𝛼𝜏𝑇^𝑇_1𝑖𝑘+𝑒^−𝛼𝜏𝑇_2𝑖𝑘,Ω₁₈=𝑊^𝑇_𝑖𝑘𝐵_𝑖𝑘−𝑋_𝑖𝑘𝐷_𝑖𝑘,Ω₂₂=𝑅_𝑖𝑘+𝜏𝑁_𝑖𝑘−𝑊^𝑇_𝑖𝑘−𝑊_𝑖𝑘,Ω₂₃=𝑊^𝑇_𝑖𝑘𝐴_1𝑖𝑘−𝑊_𝑖𝑘−𝑋_𝑖𝑘𝐶_1𝑖𝑘,Ω₂₈=𝑊^𝑇_𝑖𝑘𝐵_𝑖𝑘−𝑋_𝑖𝑘𝐷_𝑖𝑘,Ω₃₃=−1−ℎ𝑒^−𝛼ℎ𝑄_𝑖𝑘+𝑊^𝑇_𝑖𝑘𝐴_1𝑖𝑘+𝐴^𝑇_1𝑖𝑘𝑊_𝑖𝑘−𝑋_𝑖𝑘𝐶_1𝑖𝑘−𝐶^𝑇_1𝑖𝑘𝑋^𝑇_𝑖𝑘,Ω₃₈=𝑊^𝑇_𝑖𝑘𝐵_𝑖𝑘−𝑋_𝑖𝑘𝐷_𝑖𝑘,Ω₄₄=−𝑒^−𝛼𝜏𝑇^𝑇_2𝑖𝑘−𝑒^−𝛼𝜏𝑇_2𝑖𝑘,Ω₅₅=−(1−𝜏)𝑒^−𝛼𝜏𝑅_𝑖𝑘,Ω₆₆1=−ℎ𝑒^−𝛼ℎ𝑀_𝑖𝑘, (28) then the filter problem for the system (2) is solvable for any switching signal with average dwell time satisfying 𝑇_𝑎>𝑇^∗_𝑎=ln𝜇/𝛼.
Moreover, the filter gain 𝐾_𝑖𝑘 are given by 𝐾_𝑖𝑘=𝑊_𝑖^−𝑇_𝑘𝑋_𝑖𝑘.

Proof. Consider the piecewise Lyapunov-krasovskii functional candidate as (11) and introduce the vector 𝜂^𝑇_𝑒(𝑡)=𝜉^𝑇_𝑒(𝑡)𝜔^𝑇(𝑡), where 𝜉_𝑒(𝑡) is defined in (21). Then, replace (21) with the following 2𝜂^𝑇_𝑒(𝑡)𝑊_𝑖𝑘𝑊_𝑖𝑘𝑊_𝑖𝑘00000^𝑇×𝐴_0𝑖𝑘−𝐼𝐴_1𝑖𝑘0𝐹_𝑖𝑘00𝐵_𝑖𝑘𝜂_𝑒(𝑡)=0. (29)
Let 𝑋_𝑖𝑘=𝑊^𝑇_𝑖𝑘𝐾_𝑖𝑘 and 𝑇(𝑡)=𝑧^𝑇_𝑒(𝑡)𝑧_𝑒(𝑡)−𝛾²𝜔^𝑇(𝑡)𝜔(𝑡), similar to the proof of Theorem 6, we have 𝑉̇(𝑡)+𝛼𝑉(𝑡)+𝑇(𝑡)≤𝜂^𝑇_𝑒(𝑡)Ω_𝑖𝑘𝜂_𝑒(𝑡). (30)
If Ω_𝑖𝑘<0, it has 𝑉̇(𝑡)≤−𝛼𝑉(𝑡)−𝑇(𝑡). (31)
Integrating both sides of (31) from 𝑡_𝑘 to 𝑡, for any 𝑡∈[𝑡_𝑘,𝑡_𝑘+1), gives 𝑉(𝑡)≤𝑒^{−𝛼(𝑡−𝑡𝑘)}𝑉𝑡_𝑘−^𝑡_𝑡𝑘𝑒^{−𝛼(𝑡−𝑠)}𝑇(𝑠)𝑑𝑠. (32)
Therefore, similar to the proof method of Theory 2 in [13], we have 𝑉(𝑡)≤𝑒^{−𝛼(𝑡−𝑡0)+𝑁𝜎(𝑡0,𝑡)ln𝜇}𝑉𝑡₀−^𝑡_𝑡0𝑒^{−𝛼(𝑡−𝑠)+𝑁𝜎(𝑠,𝑡)ln𝜇}𝑇(𝑠)𝑑𝑠. (33)
Under the zero initial condition, (33) gives 0≤−^𝑡₀𝑒^{−𝛼(𝑡−𝑠)+𝑁𝜎(𝑠,𝑡)ln𝜇}𝑇(𝑠)𝑑𝑠. (34)
Multiplying both sides of (34) by 𝑒^{−𝑁𝜎(0,𝑡)ln𝜇} yields ^𝑡₀𝑒^{−𝛼(𝑡−𝑠)−𝑁𝜎(0,𝑠)ln𝜇}𝑧^𝑇_𝑒(𝑠)𝑧_𝑒(𝑠)≤𝑑𝑠_𝑡0𝑒^{−𝛼(𝑡−𝑠)−𝑁𝜎(0,𝑠)ln𝜇}𝛾²𝜔^𝑇(𝑠)𝜔(𝑠)𝑑𝑠. (35)
Notice that 𝑁_𝜎(0,𝑠)≤(𝑠/𝑇_𝑎) and 𝑇_𝑎>𝑇^∗_𝑎=(ln𝜇/𝛼), we have 𝑁_𝜎(0,𝑠)ln𝜇≤𝛼𝑠. Thus, (35) implies that ^𝑡₀𝑒^{−𝛼(𝑡−𝑠)−𝛼𝑠}𝑧^𝑇_𝑒(𝑠)𝑧_𝑒(𝑠)𝑑𝑠≤𝛾^2𝑡₀𝑒^{−𝛼(𝑡−𝑠)}𝜔^𝑇(𝑠)𝜔(𝑠)𝑑𝑠. (36)
Integrating both sides of (36) from 𝑡=0 to 𝑡=∞, we have ^∞₀^𝑡₀𝑒^−𝛼𝑡𝑧^𝑇_𝑒(𝑠)𝑧_𝑒(𝑠)𝑑𝑠𝑑𝑡≤𝛾^2∞₀^𝑡₀𝑒^{−𝛼(𝑡−𝑠)}𝜔^𝑇(𝑠)𝜔(𝑠)𝑑𝑠𝑑𝑡. (37)
Then, we can obtain ^∞₀1𝛼𝑒^−𝛼𝑠𝑧^𝑇_𝑒(𝑠)𝑧_𝑒(𝑠)𝑑𝑠≤𝛾^2∞₀1𝛼𝑒^−𝛼𝑠𝑒^𝛼𝑠𝜔^𝑇(𝑠)𝜔(𝑠)𝑑𝑠. (38)
Obviously, it follows from (38) that ^∞₀𝑒^−𝛼𝑠𝑧^𝑇_𝑒(𝑠)𝑧_𝑒(𝑠)𝑑𝑠≤𝛾^2∞₀𝜔^𝑇(𝑠)𝜔(𝑠)𝑑𝑠. (39)

Remark 11. Theorem 10 provides a sufficient condition for the solvability of the 𝐻_∞ filtering problem for switched neutral system with time-varying delay. If the condition is satisfied, then matrices 𝑊_𝑖 are nonsingular.

4. Numerical Examples

In this section, we present two numerical examples to illustrate the effectiveness of the results presented previously.

Example 1. Consider the switched neutral system (2) with two subsystems.
Subsystem 1. 𝐴₀₁=−1.5−0.20.2−1.3,𝐴₁₁=,𝐷−0.300.1−0.4₁=0.03,𝐹₁=,𝐶0.1−0.600.09₀₁=−0.20.26,𝐶₁₁=,𝐵−0.20.1₁=0.2−0.3,𝐿₁=0.5−0.19, (40)
Subsystem 2. 𝐴₀₂=−1.4−0.20.2−1.3,𝐴₁₂=,𝐷−0.200.1−0.4₂=𝐷₁,𝐹₂=𝐹₁,𝐶₀₂=−0.20.46,𝐶₁₂=𝐶₁₁,𝐵₂=𝐵₁,𝐿₂=,ℎ(𝑡)(𝑡)0.5−0.09=0.3,𝜏=0.3,𝛼=0.0376. (41)
By using the LMI toolbox, it can be checked that the conditions given in Theorem 10 are satisfied. Therefore, the previously switched neutral system has the given 𝐻_∞ performance 𝛾, when 𝑇_𝑎≥𝑇^∗_𝑎=ln𝜇_min/𝛼=2.6596𝑒⁻⁰⁰⁴ (here, the allowable minimum of 𝜇 is 𝜇_min=1.00001).
Setting 𝜇=1.01 and solving LMIs (26) using the LMI Toolbox in MATLAB, it follows that the minimized feasible 𝛾 is 𝛾^∗=2.0, 𝑇_𝑎=0.2646, and the corresponding filter parameters are computed as 𝐾₁=−5.07681.2264,𝐾₂=.−2.87240.9217 (42)
In the following, we illustrate the effectiveness of the designed 𝐻_∞ filter through simulation. Let the initial condition be 𝑥_𝑒(𝑡)=^0.3_0.2, 𝑡∈[−0.3,0]. Figures 1 and 2 are, respectively, the simulation results on 𝑥_𝑒(𝑡) and 𝑧_𝑒(𝑡); we can see that the filtering error dynamic system with 𝜔(𝑡)=0 is stable. 𝑥_𝑒(𝑡) and 𝑧_𝑒(𝑡) of the filtering error dynamic system with 𝑤(𝑡)=0.1𝑒^−0.5𝑡 are given in Figures 3 and 4. Figure 5 shows 𝑤(𝑡).

Figure 1 

The state responses of the filtering error dynamic system with 𝜔(𝑡)=0.

Figure 2 

𝑧𝑒(t) of the filtering error dynamic system with 𝜔(𝑡)=0.

Figure 3 

The state responses of the filtering error dynamic system with 𝜔(𝑡)=0.1𝑒−0.5𝑡.

Figure 4 

𝑧𝑒(𝑡) of the filtering error dynamic system with 𝜔(𝑡)=0.1𝑒−0.5𝑡.

Figure 5 

The noise signal 𝑤(𝑡).

Example 2. Consider the switched neutral system in Example 1 with constant delays; that is, ℎ=0, 𝜏=0, 𝛼=0.0376, and 𝜇=1.0001. We calculate the admissible maximum value ℎ_max of ℎ, 𝜏_max of 𝜏, which ensures that the resulting filtering error system is exponentially stable with a prescribed level 𝛾 of noise attenuation. For the different values, 𝛾, the obtained ℎ_max, 𝜏_max, and the filter gain 𝐾_𝑖 are listed in Table 1.

Table 1 

5. Conclusions

We have addressed the 𝐻_∞ filtering problem for a class of switched neutral systems with time-varying delays which appear in both the state and the state derivatives. For switched neutral systems with average dwell time scheme, we have provided a condition, in terms of upper bounds on the delays and in terms of a lower bound on the average dwell time, for the solvability of the 𝐻_∞ filtering problem. The piecewise Lyapunov functional technique has been used, which makes the proposed conditions are both delay-dependent and neutral delay-dependent. The design of filters for switched neutral systems is a difficult issue that is far from being well explored. Since multiple Lyapunov functional approach is commonly considered less conservative, the design of  filters for switching neutral system with an appropriate switching law using multiple Lyapunov functionals is of great significance which deserves further study.