Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 537249, pages http://dx.doi.org/10.1155/2013/537249 Research Article Design of 𝐻∞ Filter for a Class of Switched Linear Neutral Systems Caiyun Wu1 and Yue-E Wang2 School of Equipment Engineering, Shenyang Ligong University, Shenyang 110159, China College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, China Correspondence should be addressed to Caiyun Wu; wu cai yun@126.com Received 15 July 2013; Revised 15 September 2013; Accepted 15 September 2013 Academic Editor: Hossein Jafari Copyright © 2013 C Wu and Yue-E Wang This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited 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 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 Mathematical Problems in Engineering state derivatives The resulting filter is of the Luenbergerobserver 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 Section presents our main results Numerical examples are given in Section 4, and we conclude this paper in Section Notation Throughout this paper, 𝑅𝑛 denotes 𝑛-dimensional Euclidean space; 𝑅𝑛×𝑚 is the set of all 𝑛 × 𝑚 real matrices; 𝑃 > means that 𝑃 is positive definite; 𝐿 denotes the space of square integrable vector functions on [0, ∞) with norm ∞ ‖ ⋅ ‖ = (∫0 ‖ ⋅ ‖2 𝑑𝑡)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 constant matrices of appropriate dimensions ℎ(𝑡) and 𝜏(𝑡) are time-varying delays satisfying ≤ ℎ (𝑡) ≤ ℎ, ℎ̇ (𝑡) ≤ ℎ < 1, ≤ 𝜏 (𝑡) ≤ 𝜏, 𝜏̇ (𝑡) ≤ 𝜏 < (3) The objective of this paper is to design a family of filters of Luenberger observer type as follows: ̂̇ (𝑡 − 𝜏 (𝑡)) ̂̇ (𝑡) = 𝐴 0𝜎(𝑡) 𝑥̂ (𝑡) + 𝐴 1𝜎(𝑡) 𝑥̂ (𝑡 − ℎ (𝑡)) + 𝐹𝜎(𝑡) 𝑥 𝑥 + 𝐾𝜎(𝑡) [𝑦 (𝑡) − 𝐶0𝜎(𝑡) 𝑥̂ (𝑡) − 𝐶1𝜎(𝑡) 𝑥̂ (𝑡 − ℎ (𝑡))] , 𝑧̂ (𝑡) = 𝐿 𝜎(𝑡) 𝑥̂ (𝑡) , 𝑥̂ (𝜃) = 𝜓̂ (𝜃) , ∀𝜃 ∈ [−𝐻, 0] , 𝐻 = max {ℎ, 𝜏} , (4) where 𝐾𝑖𝑘 are the filter parameters, which are to be determined Now, we introduce the estimation errors: 𝑥𝑒 (𝑡) = 𝑥(𝑡) − ̂ 𝑧𝑒 (𝑡) = 𝑧(𝑡) − 𝑧̂(𝑡) 𝑥(𝑡), Combining (2) with (4) gives the following filtering error dynamic system: ̃0𝜎(𝑡) 𝑥𝑒 (𝑡) + 𝐴 ̃1𝜎(𝑡) 𝑥𝑒 (𝑡 − ℎ (𝑡)) 𝑥𝑒̇ (𝑡) = 𝐴 + 𝐹𝜎(𝑡) 𝑥𝑒̇ (𝑡 − 𝜏 (𝑡)) + 𝐵̃𝜎(𝑡) 𝜔 (𝑡) , [ 𝑋 𝑌 𝑋 𝑌 ] = [ 𝑇 ] ∗ 𝑍 𝑌 𝑍 (1) 𝑥𝑒 (𝜃) = 𝜓 (𝜃) − 𝜓̂ (𝜃) , Problem Statement 𝑥̇ (𝑡) = 𝐴 0𝜎(𝑡) 𝑥 (𝑡) + 𝐴 1𝜎(𝑡) 𝑥 (𝑡 − ℎ (𝑡)) + 𝐹𝜎(𝑡) 𝑥̇ (𝑡 − 𝜏 (𝑡)) + 𝐵𝜎(𝑡) 𝜔 (𝑡) , (2) 𝑧 (𝑡) = 𝐿 𝜎(𝑡) 𝑥 (𝑡) , 𝑥 (𝜃) = 𝜓 (𝜃) , ∀𝜃 ∈ [−𝐻, 0] , 𝐻 = max {ℎ, 𝜏} , (5) ̃1𝜎 = 𝐴 1𝜎 − 𝐾𝜎 𝐶1𝜎 , 𝐵̃𝜎 = 𝐵𝜎 − ̃0𝜎 = 𝐴 0𝜎 − 𝐾𝜎 𝐶0𝜎 , 𝐴 where 𝐴 𝐾𝜎 𝐷𝜎 The following definitions are introduced, which will play key roles in deriving our main results Consider the following switched linear neutral system: 𝑦 (𝑡) = 𝐶0𝜎(𝑡) 𝑥 (𝑡) + 𝐶1𝜎(𝑡) 𝑥 (𝑡 − ℎ (𝑡)) + 𝐷𝜎(𝑡) 𝜔 (𝑡) , 𝑧𝑒 (𝑡) = 𝐿 𝜎(𝑡) 𝑥𝑒 (𝑡) , ∀𝜃 ∈ [−𝐻, 0] , 𝐻 = max {ℎ, 𝜏} , where 𝑥(𝑡) ∈ 𝑅𝑛 is the state vector; 𝑦(𝑡) ∈ 𝑅𝑚 is the measurements vector; 𝜔(𝑡) ∈ 𝑅𝑝 is the noise signal vector, which belongs to 𝐿 [0, ∞); 𝑧(𝑡) ∈ 𝑅𝑞 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 {𝑥𝑡0 : (𝑖0 , 𝑡0 ), , (𝑖𝑘 , 𝑡𝑘 ), , | 𝑖𝑘 ∈ 𝑀, 𝑘 = 0, 1, }, which means that the 𝑖𝑘 th subsystem is active when 𝑡 ∈ [𝑡𝑘 , 𝑡𝑘+1 ) The system coefficient matrices 𝐴 0𝑖𝑘 , 𝐴 1𝑖𝑘 , 𝐹𝑖𝑘 , 𝐵𝑖𝑘 , 𝐶0𝑖𝑘 , 𝐶1𝑖𝑘 , 𝐷𝑖𝑘 , and 𝐿 𝑖𝑘 are known real Definition (see [26]) The equilibrium 𝑥𝑒∗ = of the filtering error system (5) is said to be exponentially stable under 𝜎(𝑡) if the solution 𝑥𝑒 (𝑡) of system (5) with 𝜔(𝑡) = satisfies ‖𝑥𝑒 (𝑡)‖ ≤ Γ𝑒−𝜆(𝑡−𝑡0 ) ‖𝑥𝑒 (𝑡0 )‖𝐻, for all 𝑡 ≥ 𝑡0 for constants Γ > and 𝜆 > 0, where ‖ ⋅ ‖ denotes the Euclidean norm, and ‖𝑥𝑒 (𝑡)‖𝐻 = sup−𝐻≤𝜃≤0 {𝑥𝑒 (𝑡 + 𝜃), 𝑥𝑒̇ (𝑡 + 𝜃)} Definition (see [26]) For any 𝑇2 > 𝑇1 ≥ 0, let 𝑁𝜎 (𝑇1 , 𝑇2 ) denote the number of switching of 𝜎(𝑡) over (𝑇1 , 𝑇2 ) If 𝑁𝜎 (𝑇1 , 𝑇2 ) ≤ 𝑁0 + (𝑇2 − 𝑇1 )/𝑇𝑎 holds for 𝑇𝑎 > 0, 𝑁0 ≥ 0, then 𝑇𝑎 is called average dwell time As commonly used in the literature, we choose 𝑁0 = The filtering problem addressed in this paper is to seek for suitable filter gain 𝐾𝑖 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) = is exponentially stable; Mathematical Problems in Engineering (2) under the zero initial conditions, that is, 𝑥𝑒 (𝜃) = 0, for all 𝜃 ∈ [−𝐻, 0], the weighted 𝐻∞ performance ∞ ∞ ∫0 𝑒−𝛼𝑠 𝑧𝑒𝑇 (𝑠)𝑧𝑒 (𝑠)𝑑𝑠 ≤ 𝛾2 ∫0 𝜔𝑇 (𝑠)𝜔(𝑠)𝑑𝑠 is guaranteed for all nonzero 𝜔(𝑡) ∈ 𝐿 [0, ∞) and a prescribed level of noise attenuation 𝛾 > Lemma (see [27]) For any constant matrix < 𝑅 = 𝑅𝑇 ∈ 𝑅𝑛×𝑛 , scalar 𝑟 > 0, vector function 𝜔 : [0, 𝑟] → 𝑅𝑛 such that the integrations concerned are well defined; then, 𝑟 𝑟 𝑟 (∫0 𝜔(𝑠)𝑑𝑠)𝑇 𝑅(∫0 𝜔(𝑠)𝑑𝑠) ≤ 𝑟 ∫0 𝜔𝑇 (𝑠)𝑅𝜔(𝑠)𝑑𝑠 Lemma (Schur complement) For given 𝑆 = [ 𝑆∗11 𝑆𝑆1222 ] < 0, 𝑇 𝑇 and 𝑆22 = 𝑆22 , the following is equivalent: where 𝑆11 = 𝑆11 Before concluding this section, we introduce three lemmas which are essential for the development of the results Lemma (see [25]) Let 𝑥(𝑡) ∈ 𝑅𝑛 be a vector-valued function with first-order continuous-derivative entries Then, the following integral inequality holds for any matrices 𝑀1 , 𝑀2 ∈ 𝑅𝑛×𝑛 , and 𝑋 = 𝑋𝑇 > 0, and a scalar ℎ ≥ 0, −∫ 𝑡 (1) 𝑆11 < 0, 𝑇 −1 𝑆22 − 𝑆12 𝑆11 𝑆12 < 0; (2) 𝑆22 < 0, −1 𝑇 𝑆11 − 𝑆12 𝑆22 𝑆12 < (7) Main Results 𝑇 𝑡−ℎ 𝑥̇ (𝑠) 𝑋𝑥̇ (𝑠) 𝑑𝑠 ≤ 𝜉 (𝑡) [ [ 𝑇 𝑀1𝑇 + 𝑀1 −𝑀1𝑇 + 𝑀2 −𝑀2𝑇 − 𝑀2 ∗ ] 𝜉 (𝑡) In this section, we first present a sufficient condition for exponential stability of the filtering error system (5) with 𝜔(𝑡) = Then, it is applied to formulate an approach to design the desired 𝐻∞ filters for switched neutral system (2) (6) ] 𝑀1𝑇 ] 𝑋−1 [𝑀1 𝑀2 ] 𝜉 (𝑡) , [ + ℎ𝜉 (𝑡) 𝑀2𝑇 [ ] 3.1 Stability Analysis 𝑇 Theorem Given 𝛼 > 0, ‖𝐹𝑖𝑘 ‖ < 1, for all 𝑖𝑘 ∈ 𝑀 If there exist matrices 𝑃𝑖𝑘 > 0, 𝑄𝑖𝑘 > 0, 𝑅𝑖𝑘 > 0, 𝑀𝑖𝑘 > 0, 𝑁𝑖𝑘 > 0, and 𝑇1𝑖𝑘 , 𝑇2𝑖𝑘 , 𝑁𝑔𝑖𝑘 (𝑔 = 1, 2, , 7) of appropriate dimensions, and 𝜇 ≥ 1, such that for all 𝑖𝑘 ∈ 𝑀, where 𝜉𝑇 (𝑡) = [𝑥𝑇 (𝑡) 𝑥𝑇(𝑡 − ℎ)] Σ11 [ [∗ [ [ [ [∗ [ [ [ [∗ Σ𝑖𝑘 = [ [ [ [∗ [ [ [ [∗ [ [ [ ∗ [ 𝑃𝑖𝑘 ≤ 𝜇𝑃𝑖𝑗 , Σ12 Σ13 Σ14 Σ15 ̃𝑇 𝑁6𝑖 𝐴 0𝑖𝑘 𝑘 Σ22 Σ23 −𝑁4𝑖𝑘 Σ25 −𝑁6𝑖𝑘 Σ35 ̃𝑇 𝑁6𝑖 𝐴 1𝑖𝑘 𝑘 𝑁4𝑖𝑇𝑘 𝐹𝑖𝑘 𝐹𝑖𝑇𝑘 𝑁6𝑖𝑘 ∗ ̃𝑇 𝑁4𝑖 Σ33 𝐴 1𝑖𝑘 𝑘 ∗ ∗ Σ44 ∗ ∗ ∗ Σ55 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 𝑄𝑖𝑘 ≤ 𝜇𝑄𝑖𝑗 , 𝑅𝑖𝑘 ≤ 𝜇𝑅𝑖𝑗 , − 𝑒−𝛼ℎ 𝑀𝑖𝑘 ℎ ∗ 𝑀𝑖𝑘 ≤ 𝜇𝑀𝑖𝑗 , Σ17 ] −𝑁7𝑖𝑘 ] ] ] ] ̃𝑇 𝑁7𝑖 ] 𝐴 1𝑖𝑘 𝑘 ] ] ] 𝜏𝑇2𝑖𝑇𝑘 ] ] < 0, ] ] 𝑇 𝐹𝑖𝑘 𝑁7𝑖𝑘 ] ] ] ] ] ] ] ] −𝜏𝑒𝛼𝜏 𝑁𝑖𝑘 ] 𝑁𝑖𝑘 ≤ 𝜇𝑁𝑖𝑗 , ∀𝑖𝑘 , 𝑖𝑗 ∈ 𝑀, ̃𝑇 𝑁7𝑖 , Σ17 = 𝜏𝑇1𝑖𝑇𝑘 + 𝐴 0𝑖𝑘 𝑘 where Σ11 = ℎ𝑀𝑖𝑘 + 𝛼𝑃𝑖𝑘 + 𝑄𝑖𝑘 + 𝑒−𝛼𝜏 𝑇1𝑖𝑇𝑘 + 𝑒−𝛼𝜏 𝑇1𝑖𝑘 ̃0𝑖 + 𝐴 ̃𝑇 𝑁1𝑖 , + 𝑁1𝑖𝑇𝑘 𝐴 0𝑖𝑘 𝑘 𝑘 ̃𝑇 𝑁2𝑖 , Σ12 = 𝑃𝑖𝑘 − 𝑁1𝑖𝑇𝑘 + 𝐴 0𝑖𝑘 𝑘 ̃1𝑖 + 𝐴 ̃𝑇 𝑁3𝑖 , Σ13 = 𝑁1𝑖𝑇𝑘 𝐴 0𝑖𝑘 𝑘 𝑘 Σ14 = −𝑒−𝛼𝜏 𝑇1𝑖𝑇𝑘 −𝛼𝜏 +𝑒 𝑇2𝑖𝑘 ̃𝑇 𝑁4𝑖 , +𝐴 0𝑖𝑘 𝑘 ̃𝑇 𝑁5𝑖 , Σ15 = 𝑁1𝑖𝑇𝑘 𝐹𝑖𝑘 + 𝐴 0𝑖𝑘 𝑘 Σ22 = 𝑅𝑖𝑘 + 𝜏𝑁𝑖𝑘 − 𝑁2𝑖𝑇𝑘 − 𝑁2𝑖𝑘 , ̃1𝑖 , Σ23 = −𝑁3𝑖𝑘 + 𝑁2𝑖𝑇𝑘 𝐴 𝑘 Σ25 = −𝑁5𝑖𝑘 + 𝑁2𝑖𝑇𝑘 𝐹𝑖𝑘 , ̃1𝑖 + 𝐴 ̃𝑇 𝑁3𝑖 , Σ33 = − (1 − ℎ) 𝑒−𝛼ℎ 𝑄𝑖𝑘 + 𝑁3𝑖𝑇𝑘 𝐴 1𝑖𝑘 𝑘 𝑘 ̃𝑇 𝑁5𝑖 , Σ35 = 𝑁3𝑖𝑇𝑘 𝐹𝑖𝑘 + 𝐴 1𝑖𝑘 𝑘 (8) (9) Mathematical Problems in Engineering Σ44 = −𝑒−𝛼𝜏 𝑇2𝑖𝑇𝑘 − 𝑒−𝛼𝜏 𝑇2𝑖𝑘 , From Lemma 3, it holds Σ55 = − (1 − 𝜏) 𝑒−𝛼𝜏 𝑅𝑖𝑘 + 𝑁5𝑖𝑇𝑘 𝐹𝑖𝑘 + 𝐹𝑖𝑇𝑘 𝑁5𝑖𝑘 , (10) then the error dynamic system (5) with 𝜔(𝑡) = is exponentially stable for any switching signal with average dwell time satisfying 𝑇𝑎 > 𝑇𝑎∗ = ln 𝜇/𝛼 −∫ 𝑡 𝑥𝑒̇𝑇 (𝑠) 𝑒𝛼(𝑠−𝑡) 𝑁𝑖𝑘 𝑥𝑒̇ (𝑠) 𝑑𝑠 𝑡−𝜏 ≤ −∫ 𝑡−𝜏 −𝛼𝜏 ≤𝑒 Proof Define the piecewise Lyapunov-Krasovskii functional candidate ×[ 𝑉 (𝑡) = 𝑉𝜎(𝑡) (𝑡) = ∑ 𝑉𝑗𝜎(𝑡) (𝑡) , 𝑡 (11) [𝑥𝑒𝑇 (𝑡) 𝑥𝑒𝑇 (𝑡 − 𝜏)] [ [ 𝑇1𝑖𝑇𝑘 + 𝑇1𝑖𝑘 −𝑇1𝑖𝑇𝑘 + 𝑇2𝑖𝑘 −𝑇2𝑖𝑇𝑘 − 𝑇2𝑖𝑘 ∗ [ 𝑇1𝑖𝑇𝑘 𝑇2𝑖𝑇𝑘 𝑥𝑒 (𝑡) ] ] 𝑁−1 [𝑇1𝑖 𝑇2𝑖 ] [ 𝑖𝑘 𝑘 𝑘 𝑥𝑒 (𝑡 − 𝜏) ] [ ] where 𝑉2𝑖𝑘 (𝑡) = ∫ 𝑡−ℎ(𝑡) 𝑉3𝑖𝑘 (𝑡) = ∫ 𝑡 𝑡−𝜏(𝑡) 𝑉4𝑖𝑘 (𝑡) = ∫ ∫ 𝑡 −ℎ 𝑡+𝜃 𝑉5𝑖𝑘 (𝑡) = ∫ ∫ 𝑡 −𝜏 𝑡+𝜃 𝑥𝑒𝑇 (𝑠) 𝑒𝛼(𝑠−𝑡) 𝑄𝑖𝑘 𝑥𝑒 𝜉𝑒𝑇 (𝑡) = [𝜁𝑒𝑇 (𝑡) 𝑥𝑒̇𝑇 (𝑡 − 𝜏 (𝑡)) ∫ (𝑠) 𝑑𝑠, 𝑥𝑒̇𝑇 (𝑠) 𝑒𝛼(𝑠−𝑡) 𝑅𝑖𝑘 𝑥𝑒̇ (𝑠) 𝑑𝑠, (12) 𝑥𝑒̇𝑇 (𝑠) 𝑒𝛼(𝑠−𝑡) 𝑁𝑖𝑘 𝑥𝑒̇ (𝑠) 𝑑𝑠 𝑑𝜃 𝑥𝑒𝑇 (𝑠) 𝑑𝑠] , (15) (16) 𝑃𝑖𝑘 Σ11 [ ∗ 𝑅𝑖 + 𝜏𝑁𝑖 𝑘 𝑘 [ [∗ ∗ − (1 − ℎ) 𝑒−𝛼ℎ 𝑄𝑖𝑘 Σ𝑖𝑘 = [ [∗ ∗ ∗ [ [∗ ∗ ∗ ∗ ∗ [∗ Σ14 0 0 ] ] 0 ] ], Σ44 0 ] ] ∗ Σ55 ] ∗ ∗ Σ66 ] (17) where − (1 − ℎ) 𝑥𝑒𝑇 (𝑡 − ℎ (𝑡)) 𝑒−𝛼ℎ 𝑄𝑖𝑘 𝑥𝑒 (𝑡 − ℎ (𝑡)) + ℎ𝑥𝑒𝑇 (𝑡) 𝑀𝑖𝑘 𝑥𝑒 (𝑡) + 𝜏𝑥𝑒̇𝑇 (𝑡) 𝑁𝑖𝑘 𝑥𝑒̇ (𝑡) + 𝛼𝑥𝑒𝑇 (𝑡) 𝑃𝑖𝑘 𝑥𝑒 (𝑡) − (1 − 𝜏) 𝑥𝑒̇𝑇 (𝑡 − 𝜏 (𝑡)) 𝑒−𝛼𝜏 𝑅𝑖𝑘 𝑥𝑒̇ (𝑡 − 𝜏 (𝑡)) Σ11 = ℎ𝑀𝑖𝑘 + 𝛼𝑃𝑖𝑘 + 𝑄𝑖𝑘 + 𝑒−𝛼𝜏 𝑇1𝑖𝑇𝑘 + 𝑒−𝛼𝜏 𝑇1𝑖𝑘 + 𝜏𝑒−𝛼𝜏 𝑇1𝑖𝑇𝑘 𝑁𝑖−1 𝑇1𝑖𝑘 , 𝑘 Σ14 = −𝑒−𝛼𝜏 𝑇1𝑖𝑇𝑘 + 𝑒−𝛼𝜏 𝑇2𝑖𝑘 + 𝜏𝑒−𝛼𝜏 𝑇1𝑖𝑇𝑘 𝑁𝑖−1 𝑇2𝑖𝑘 , 𝑘 𝑡 𝑒−𝛼ℎ 𝑡 𝑇 ∫ 𝑥𝑒 (𝑠) 𝑑𝑠𝑀𝑖𝑘 ∫ 𝑥𝑒 (𝑠) 𝑑𝑠 ℎ 𝑡−ℎ 𝑡−ℎ 𝑡−𝜏 𝑡−ℎ where ≤ 2𝑥𝑒𝑇 (𝑡) 𝑃𝑖𝑘 𝑥𝑒̇ (𝑡) + 𝑥𝑒𝑇 (𝑡) 𝑄𝑖𝑘 𝑥𝑒 (𝑡) + 𝑥𝑒̇𝑇 (𝑡) 𝑅𝑖𝑘 𝑥𝑒̇ (𝑡) −∫ 𝑡 𝑉𝑖̇ 𝑘 (𝑡) + 𝛼𝑉𝑖𝑘 (𝑡) ≤ 𝜉𝑒𝑇 (𝑡) Σ𝑖𝑘 𝜉𝑒 (𝑡) , 𝑉𝑖̇ 𝑘 (𝑡) + 𝛼𝑉𝑖𝑘 (𝑡) 𝑡 (14) where 𝜁𝑒𝑇 (𝑡) = [𝑥𝑒𝑇(𝑡) 𝑥𝑒̇𝑇 (𝑡) 𝑥𝑒𝑇 (𝑡 − ℎ(𝑡)) 𝑥𝑒𝑇 (𝑡 − 𝜏)] By some algebraic manipulations, it is easy to show that 𝑥𝑒𝑇 (𝑠) 𝑒𝛼(𝑠−𝑡) 𝑀𝑖𝑘 𝑥𝑒 (𝑠) 𝑑𝑠 𝑑𝜃, Now, taking the derivative of 𝑉𝑗𝑖𝑘 (𝑡), 𝑗 = 1, 2, , with respect to 𝑡 along the trajectory of the error system (5) with 𝜔(𝑡) = 0, according to (3) and Lemma 4, we have − ] Define 𝑉1𝑖𝑘 (𝑡) = 𝑥𝑒𝑇 (𝑡) 𝑃𝑖𝑘 𝑥𝑒 (𝑡) , 𝑡 ] 𝑥𝑒 (𝑡) ] + 𝜏𝑒−𝛼𝜏 [𝑥𝑒𝑇 (𝑡) 𝑥𝑒𝑇 (𝑡 − 𝜏)] 𝑥𝑒 (𝑡 − 𝜏) ×[ 𝑗=1 𝑥𝑒̇𝑇 (𝑠) 𝑒−𝛼𝜏 𝑁𝑖𝑘 𝑥𝑒̇ (𝑠) 𝑑𝑠 Σ55 = − (1 − 𝜏) 𝑒−𝛼𝜏 𝑅𝑖𝑘 , Σ44 = −𝑒−𝛼𝜏 𝑇2𝑖𝑇𝑘 − 𝑒−𝛼𝜏 𝑇2𝑖𝑘 + 𝜏𝑒−𝛼𝜏 𝑇2𝑖𝑇𝑘 𝑁𝑖−1 𝑇2𝑖𝑘 , 𝑘 𝑥𝑒̇𝑇 (𝑠) 𝑒𝛼(𝑠−𝑡) 𝑁𝑖𝑘 𝑥𝑒̇ (𝑠) 𝑑𝑠 (13) Σ66 = − 𝑒−𝛼ℎ 𝑀𝑖𝑘 ℎ (18) Mathematical Problems in Engineering Combining (9) with (22), for any 𝑡 ∈ [𝑡𝑘 , 𝑡𝑘+1 ), we have 0.3 𝑉 (𝑡) = 𝑉𝑖𝑘 (𝑡) 0.25 ≤ 𝑒−𝛼(𝑡−𝑡𝑘 ) 𝑉𝑖𝑘 (𝑡𝑘 ) xe (t) 0.2 ≤ 𝜇𝑒−𝛼(𝑡−𝑡𝑘 ) 𝑉𝜎(𝑡𝑘− ) (𝑡𝑘− ) 0.15 ≤ 𝜇𝑒−𝛼(𝑡−𝑡𝑘 ) 𝑒−𝛼(𝑡𝑘 −𝑡𝑘−1 ) 𝑉𝜎(𝑡𝑘−1 ) (𝑡𝑘−1 ) 0.1 ≤ ⋅⋅⋅ 0.05 ≤ 𝜇𝑘 𝑒−𝛼(𝑡−𝑡0 ) 𝑉 (𝑡0 ) −0.05 (23) ≤ 𝑒−(𝛼−(ln 𝜇/𝑇𝑎 ))(𝑡−𝑡0 ) 𝑉 (𝑡0 ) 500 1000 1500 Time 2000 2500 3000 Figure 1: The state responses of the filtering error dynamic system with 𝜔(𝑡) = According to (11), we have 2 2 𝑎𝑥𝑒 (𝑡) ≤ 𝑉 (𝑡) ≤ 𝑏𝑥𝑒 (𝑡0 )𝐻, (24) where 𝑎 = {𝜆 (𝑃𝑖𝑘 )} , ∀𝑖𝑘 ∈𝑀 𝑏 = max {𝜆 max (𝑃𝑖𝑘 )} + ℎ max {𝜆 max (𝑄𝑖𝑘 )} From Lemma 5, Σ𝑖𝑘 < is equivalent to ̃ 11 𝑃𝑖𝑘 Σ [ ∗ 𝑅 + 𝜏𝑁 [ 𝑖𝑘 𝑖𝑘 [ ∗ [∗ ̃𝑖 = [ Σ [∗ ∗ 𝑘 [ [∗ ∗ [ [∗ ∗ ∗ [∗ 0 ̃ 33 Σ ∗ ∗ ∗ ∗ ̃ 14 Σ 0 ̃ 44 Σ ∗ ∗ ∗ ∀𝑖𝑘 ∈𝑀 0 𝜏𝑇1𝑖𝑇𝑘 ] 0 ] ] 0 ] ] 𝑇 0 𝜏𝑇2𝑖𝑘 ] < 0, ] ̃ 55 ] Σ ] ] ̃ ∗ Σ66 ∗ ∗ −𝜏𝑒𝛼𝜏 𝑁𝑖𝑘 ] (19) where ̃ 11 = ℎ𝑀𝑖 + 𝛼𝑃𝑖 + 𝑄𝑖 + 𝑒−𝛼𝜏 𝑇𝑇 + 𝑒−𝛼𝜏 𝑇1𝑖 , Σ 1𝑖𝑘 𝑘 𝑘 𝑘 𝑘 ̃ 14 = −𝑒−𝛼𝜏 𝑇𝑇 + 𝑒−𝛼𝜏 𝑇2𝑖 , Σ 1𝑖𝑘 𝑘 ̃ 44 = −𝑒−𝛼𝜏 𝑇𝑇 − 𝑒−𝛼𝜏 𝑇2𝑖 , Σ 2𝑖𝑘 𝑘 ̃ 33 = − (1 − ℎ) 𝑒−𝛼ℎ 𝑄𝑖 , Σ 𝑘 ̃ 55 = Σ55 , Σ ̃ 66 = Σ66 Σ (20) In addition, the following is true from (5) with 𝜔(𝑡) = 0: 𝑇 2𝜉 𝑇𝑒 (𝑡)[𝑁1𝑖𝑘 𝑁2𝑖𝑘 𝑁3𝑖𝑘 𝑁4𝑖𝑘 𝑁5𝑖𝑘 𝑁6𝑖𝑘 𝑁7𝑖𝑘 ] ̃1𝑖 𝐹𝑖 0] 𝜉𝑒 (𝑡) = 0, ̃0𝑖 −𝐼 𝐴 × [𝐴 𝑘 𝑘 𝑘 (21) where 𝜉 𝑇𝑒 (𝑡) = [𝜉𝑒𝑇 (𝑡) 0] Then, (19) along with (21) gives Σ𝑖𝑘 < 0, which yields Σ𝑖𝑘 < 0; thus, 𝑉𝑖̇ 𝑘 (𝑡) + 𝛼𝑉𝑖𝑘 (𝑡) ≤ (22) ∀𝑖𝑘 ∈𝑀 + 𝜏 max {𝜆 max (𝑅𝑖𝑘 )} + ∀𝑖𝑘 ∈𝑀 + ℎ2 max {𝜆 (𝑀𝑖𝑘 )} ∀𝑖𝑘 ∈𝑀 max (25) 𝜏2 max {𝜆 (𝑁 )} ∀𝑖𝑘 ∈𝑀 max 𝑖𝑘 Considering (23) and (24), it holds ‖𝑥𝑒 (𝑡)‖ ≤ √(𝑏/𝑎)𝑒−(1/2)(𝛼−(ln 𝜇/𝑇𝑎 ))(𝑡−𝑡0 ) ‖𝑥𝑒 (𝑡0 )‖𝐻 Therefore, if 𝛼 − (ln 𝜇/𝑇𝑎 ) > 0, that is 𝑇𝑎 > (ln 𝜇/𝛼), then error dynamic system (5) is exponentially stable Remark When 𝜇 = 1, we have 𝑇𝑎∗ = 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 = in (8) gives asymptotic stability of the filtering error system (5) under arbitrary switching Remark 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 The condition ‖𝐹𝑖𝑘 ‖ < guarantees that Lipschitz ̇ − 𝜏(𝑡)) constant for the right hand of (2) with respect to 𝑥(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 ‖𝐹𝑖𝑘 ‖ < 1, for all 𝑖𝑘 ∈ 𝑀, and if there exists matrices 𝑃𝑖𝑘 > 0, 𝑄𝑖𝑘 > 0, 𝑅𝑖𝑘 > 0, 𝑀𝑖𝑘 > 0, 𝑁𝑖𝑘 > 0, and 𝑇1𝑖𝑘 , 𝑇2𝑖𝑘 , 𝑊𝑖𝑘 , 𝑋𝑖𝑘 of appropriate dimensions, and 𝜇 ≥ 1, such that, for any 𝑖𝑘 ∈ 𝑀, Mathematical Problems in Engineering [ [ [ [ [ [ [ [ [ [ [ Ω𝑖𝑘 = [ [ [ [ [ [ [ [ [ [ [ [ [ 𝑃𝑖𝑘 ≤ 𝜇𝑃𝑖𝑗 , Ω11 Ω12 Ω13 Ω14 𝑊𝑖𝑇𝑘 𝐹𝑖𝑘 𝜏𝑇1𝑖𝑇𝑘 𝑊𝑖𝑇𝑘 𝐹𝑖𝑘 0 ∗ Ω22 Ω23 ∗ ∗ Ω33 𝑊𝑖𝑇𝑘 𝐹𝑖𝑘 0 ∗ ∗ ∗ Ω44 0 𝜏𝑇2𝑖𝑇𝑘 ∗ ∗ ∗ ∗ Ω55 0 ∗ ∗ ∗ ∗ ∗ Ω66 ∗ ∗ ∗ ∗ ∗ ∗ −𝜏𝑒𝛼𝜏 𝑁𝑖𝑘 ∗ ∗ ∗ ∗ ∗ ∗ ∗ 𝑄𝑖𝑘 ≤ 𝜇𝑄𝑖𝑗 , 𝑅𝑖𝑘 ≤ 𝜇𝑅𝑖𝑗 , where Ω11 = ℎ𝑀𝑖𝑘 + 𝛼𝑃𝑖𝑘 + 𝑄𝑖𝑘 + 𝑒−𝛼𝜏 𝑇1𝑖𝑇𝑘 + 𝑒−𝛼𝜏 𝑇1𝑖𝑘 𝑇 + 𝑊𝑖𝑇𝑘 𝐴 0𝑖𝑘 + 𝐴𝑇0𝑖𝑘 𝑊𝑖𝑘 − 𝑋𝑖𝑘 𝐶0𝑖𝑘 − 𝐶0𝑖 𝑋𝑖𝑇𝑘 + 𝐿𝑇𝑖𝑘 𝐿 𝑖𝑘 , 𝑘 Ω12 = 𝑃𝑖𝑘 − 𝑊𝑖𝑇𝑘 𝐴𝑇0𝑖𝑘 𝑊𝑖𝑘 + − 𝑇 𝐶0𝑖 𝑋𝑖𝑇𝑘 , 𝑘 𝑀𝑖𝑘 ≤ 𝜇𝑀𝑖𝑗 , Ω18 ] Ω28 ] ] ] ] Ω38 ] ] ] ] ] ] ] < 0, ] ] ] ] ] ] ] ] ] ] ] −𝛾 𝐼 ] 𝑁𝑖𝑘 ≤ 𝜇𝑁𝑖𝑗 , (26) ∀𝑖𝑘 , 𝑖𝑗 ∈ 𝑀, Proof Consider the piecewise Lyapunov-krasovskii functional candidate as (11) and introduce the vector 𝜂𝑒𝑇 (𝑡) = [𝜉 𝑇𝑒 (𝑡) 𝜔𝑇 (𝑡)], where 𝜉𝑒 (𝑡) is defined in (21) Then, replace (21) with the following 𝑇 2𝜂𝑒𝑇 (𝑡)[𝑊𝑖𝑘 𝑊𝑖𝑘 𝑊𝑖𝑘 0 0 0] ̃1𝑖 𝐹𝑖 0 𝐵̃𝑖 ] 𝜂𝑒 (𝑡) = ̃0𝑖 −𝐼 𝐴 × [𝐴 𝑘 𝑘 𝑘 𝑘 𝑇 Ω13 = 𝑊𝑖𝑇𝑘 𝐴 1𝑖𝑘 + 𝐴𝑇0𝑖𝑘 𝑊𝑖𝑘 − 𝑋𝑖𝑘 𝐶1𝑖𝑘 − 𝐶0𝑖 𝑋𝑖𝑇𝑘 , 𝑘 (27) (29) Ω14 = −𝑒−𝛼𝜏 𝑇1𝑖𝑇𝑘 + 𝑒−𝛼𝜏 𝑇2𝑖𝑘 , Let 𝑋𝑖𝑘 = 𝑊𝑖𝑇𝑘 𝐾𝑖𝑘 and 𝑇(𝑡) = 𝑧𝑒𝑇 (𝑡)𝑧𝑒 (𝑡) − 𝛾2 𝜔𝑇 (𝑡)𝜔(𝑡), similar to the proof of Theorem 6, we have Ω18 = 𝑊𝑖𝑇𝑘 𝐵𝑖𝑘 − 𝑋𝑖𝑘 𝐷𝑖𝑘 , 𝑉̇ (𝑡) + 𝛼𝑉 (𝑡) + 𝑇 (𝑡) ≤ 𝜂𝑒𝑇 (𝑡) Ω𝑖𝑘 𝜂𝑒 (𝑡) Ω22 = 𝑅𝑖𝑘 + 𝜏𝑁𝑖𝑘 − Ω23 = 𝑊𝑖𝑇𝑘 𝐴 1𝑖𝑘 Ω28 = Ω33 = − (1 − ℎ) 𝑒 − 𝑊𝑖𝑘 , 𝑉̇ (𝑡) ≤ −𝛼𝑉 (𝑡) − 𝑇 (𝑡) − 𝑋𝑖𝑘 𝐷𝑖𝑘 , 𝑄𝑖𝑘 + 𝑊𝑖𝑇𝑘 𝐴 1𝑖𝑘 + (31) Integrating both sides of (31) from 𝑡𝑘 to 𝑡, for any 𝑡 ∈ [𝑡𝑘 , 𝑡𝑘+1 ), gives 𝐴𝑇1𝑖𝑘 𝑊𝑖𝑘 𝑡 𝑇 − 𝑋𝑖𝑘 𝐶1𝑖𝑘 − 𝐶1𝑖 𝑋𝑖𝑇𝑘 , 𝑘 𝑉 (𝑡) ≤ 𝑒−𝛼(𝑡−𝑡𝑘 ) 𝑉 (𝑡𝑘 ) − ∫ 𝑒−𝛼(𝑡−𝑠) 𝑇 (𝑠) 𝑑𝑠 𝑡𝑘 Ω38 = 𝑊𝑖𝑇𝑘 𝐵𝑖𝑘 − 𝑋𝑖𝑘 𝐷𝑖𝑘 , Ω66 = − 𝑒−𝛼ℎ 𝑀𝑖𝑘 , ℎ (32) Therefore, similar to the proof method of Theory in [13], we have Ω44 = −𝑒−𝛼𝜏 𝑇2𝑖𝑇𝑘 − 𝑒−𝛼𝜏 𝑇2𝑖𝑘 , Ω55 = − (1 − 𝜏) 𝑒−𝛼𝜏 𝑅𝑖𝑘 , (30) If Ω𝑖𝑘 < 0, it has − 𝑊𝑖𝑘 − 𝑋𝑖𝑘 𝐶1𝑖𝑘 , 𝑊𝑖𝑇𝑘 𝐵𝑖𝑘 −𝛼ℎ 𝑊𝑖𝑇𝑘 𝑉 (𝑡) ≤ 𝑒−𝛼(𝑡−𝑡0 )+𝑁𝜎 (𝑡0 ,𝑡) ln 𝜇 𝑉 (𝑡0 ) (28) 𝑡 − ∫ 𝑒−𝛼(𝑡−𝑠)+𝑁𝜎 (𝑠,𝑡) ln 𝜇 𝑇 (𝑠) 𝑑𝑠 (33) 𝑡0 then the filter problem for the system (2) is solvable for any switching signal with average dwell time satisfying 𝑇𝑎 > 𝑇𝑎∗ = ln 𝜇/𝛼 Moreover, the filter gain 𝐾𝑖𝑘 are given by 𝐾𝑖𝑘 = 𝑊𝑖−𝑇 𝑋𝑖𝑘 𝑘 Under the zero initial condition, (33) gives 𝑡 ≤ − ∫ 𝑒−𝛼(𝑡−𝑠)+𝑁𝜎 (𝑠,𝑡) ln 𝜇 𝑇 (𝑠) 𝑑𝑠 (34) Mathematical Problems in Engineering Multiplying both sides of (34) by 𝑒−𝑁𝜎 (0,𝑡) ln 𝜇 yields 0.15 𝑡 ∫ 𝑒−𝛼(𝑡−𝑠)−𝑁𝜎 (0,𝑠) ln 𝜇 𝑧𝑒𝑇 (𝑠) 𝑧𝑒 (𝑠) 𝑑𝑠 0.1 (35) −𝛼(𝑡−𝑠)−𝑁𝜎 (0,𝑠) ln 𝜇 𝑇 ≤∫ 𝑒 𝛾 𝜔 (𝑠) 𝜔 (𝑠) 𝑑𝑠 Notice that 𝑁𝜎 (0, 𝑠) ≤ (𝑠/𝑇𝑎 ) and 𝑇𝑎 > 𝑇𝑎∗ = (ln 𝜇/𝛼), we have 𝑁𝜎 (0, 𝑠) ln 𝜇 ≤ 𝛼𝑠 Thus, (35) implies that 𝑡 𝑡 0 ze (t) 𝑡 0.05 ∫ 𝑒−𝛼(𝑡−𝑠)−𝛼𝑠 𝑧𝑒𝑇 (𝑠) 𝑧𝑒 (𝑠) 𝑑𝑠 ≤ 𝛾2 ∫ 𝑒−𝛼(𝑡−𝑠) 𝜔𝑇 (𝑠) 𝜔 (𝑠) 𝑑𝑠 (36) Integrating both sides of (36) from 𝑡 = to 𝑡 = ∞, we have ∞ 𝑡 0 ∫ ∫ 𝑒−𝛼𝑡 𝑧𝑒𝑇 (𝑠) 𝑧𝑒 (𝑠) 𝑑𝑠 𝑑𝑡 ∞ 𝑡 0 (37) −0.05 ∞ −𝛼𝑠 𝑇 −𝛼𝑠 𝛼𝑠 𝑇 𝑒 𝑧𝑒 (𝑠) 𝑧𝑒 (𝑠) 𝑑𝑠 ≤ 𝛾2 ∫ 𝑒 𝑒 𝜔 (𝑠) 𝜔 (𝑠) 𝑑𝑠 𝛼 𝛼 (38) Obviously, it follows from (38) that ∫ ∞ 𝑒−𝛼𝑠 𝑧𝑒𝑇 (𝑠) 𝑧𝑒 1500 Time 2000 2500 3000 0.3 ∞ 𝑇 (𝑠) 𝑑𝑠 ≤ 𝛾 ∫ 𝜔 (𝑠) 𝜔 (𝑠) 𝑑𝑠 0.25 xe (t) 1000 0.35 Then, we can obtain ∞ 500 Figure 2: 𝑧𝑒 (t) of the filtering error dynamic system with 𝜔(𝑡) = ≤ 𝛾2 ∫ ∫ 𝑒−𝛼(𝑡−𝑠) 𝜔𝑇 (𝑠) 𝜔 (𝑠) 𝑑𝑠 𝑑𝑡 ∫ (39) 0.2 0.15 0.1 0.05 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 𝑊𝑖 are nonsingular 500 1000 1500 Time 2000 2500 3000 Figure 3: The state responses of the filtering error dynamic system with 𝜔(𝑡) = 0.1𝑒−0.5𝑡 Numerical Examples In this section, we present two numerical examples to illustrate the effectiveness of the results presented previously Example Consider the switched neutral system (2) with two subsystems Subsystem 𝐴 01 −1.5 −0.2 =[ ], 0.2 −1.3 𝐷1 = 0.03, 𝐶01 = [−0.2 0.26] , 0.2 𝐵1 = [ ], −0.3 𝐴 11 𝐹1 = [ 𝐶11 = [−0.2 0.1] , 𝐿 = [0.5 −0.19] , 𝐴 02 = [ −1.4 −0.2 ], 0.2 −1.3 𝐴 12 = [ 𝐷2 = 𝐷1 , 𝐹2 = 𝐹1 , 𝐶02 = [−0.2 0.46] , −0.3 =[ ], 0.1 −0.4 0.1 −0.6 ], 0.09 Subsystem 𝐵2 = 𝐵1 , ℎ (𝑡) = 0.3, (40) −0.2 ], 0.1 −0.4 𝐶12 = 𝐶11 , (41) 𝐿 = [0.5 −0.09] , 𝜏 (𝑡) = 0.3, 𝛼 = 0.0376 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 𝑇𝑎 ≥ 𝑇𝑎∗ = ln 𝜇min /𝛼 = 2.6596𝑒−004 (here, the allowable minimum of 𝜇 is 𝜇min = 1.00001) 8 Mathematical Problems in Engineering 0.15 Table 𝛾 0.80 0.54 0.48 0.46 0.44 ze (t) 0.1 0.05 −0.05 500 1000 1500 Time 2000 2500 3000 Figure 4: 𝑧𝑒 (𝑡) of the filtering error dynamic system with 𝜔(𝑡) = 0.1𝑒−0.5𝑡 0.9 0.8 0.7 0.6 w(t) 𝜏max 3.2 1.9 0.2 1.3 1.1 𝐾1 [−4.7398 1.1340]𝑇 [−4.1874 1.1606]𝑇 [−3.9220 1.1308]𝑇 [−3.8076 1.0948]𝑇 [−3.6949 1.0686]𝑇 𝐾2 [−2.8595 0.9824]𝑇 [−2.5316 0.9494]𝑇 [−2.3354 0.8774]𝑇 [−2.2459 0.8288]𝑇 [−2.1560 0.7834]𝑇 Example Consider the switched neutral system in Example with constant delays; that is, ℎ = 0, 𝜏 = 0, 𝛼 = 0.0376, and 𝜇 = 1.0001 We calculate the admissible maximum value ℎmax of ℎ, 𝜏max 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 ℎmax , 𝜏max , and the filter gain 𝐾𝑖 are listed in Table Conclusions 0.5 0.4 0.3 0.2 0.1 ℎmax 2.0 1.1 0.7 0.5 0.3 500 1000 1500 Time 2000 2500 3000 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 Figure 5: The noise signal 𝑤(𝑡) References 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, 𝑇𝑎 = 0.2646, and the corresponding filter parameters are computed as 𝐾1 = [ −5.0768 ], 1.2264 𝐾2 = [ −2.8724 ] 0.9217 (42) In the following, we illustrate the effectiveness of the designed 𝐻∞ filter through simulation Let the initial condition be 𝑥𝑒 (𝑡) = [ 0.3 0.2 ], 𝑡 ∈ [−0.3, 0] Figures and are, respectively, the simulation results on 𝑥𝑒 (𝑡) and 𝑧𝑒 (𝑡); we can see that the filtering error dynamic system with 𝜔(𝑡) = is stable 𝑥𝑒 (𝑡) and 𝑧𝑒 (𝑡) of the filtering error dynamic system with 𝑤(𝑡) = 0.1𝑒−0.5𝑡 are given in Figures and Figure shows 𝑤(𝑡) [1] X M Sun and W Wang, “Integral input-to-state stability for hybrid delayed systems with unstable continuous dynamics,” Automatica, vol 48, no 9, pp 2359–2364, 2012 [2] H Jia, H R Karimi, and Z Xiang, “Dynamic output feedback passive control of uncertain switched stochastic systems with 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