Novel Delay-Decomposing Approaches to Absolute Stability Criteria for Neutral-Type Lur’e Systems

)e problem of absolute stability analysis for neutral-type Lur’e systems with time-varying delays is investigated. Novel delaydecomposing approaches are proposed to divide the variation interval of the delay into three unequal subintervals. Some new augment Lyapunov–Krasovskii functionals (LKFs) are defined on the obtained subintervals. )e integral inequality method and the reciprocally convex technique are utilized to deal with the derivative of the LKFs. Several improved delay-dependent criteria are derived in terms of the linear matrix inequalities (LMIs). Compared with some previous criteria, the proposed ones give the results with less conservatism and lower numerical complexity. Two numerical examples are included to illustrate the effectiveness and the improvement of the proposed method.


Introduction
Over the last 30 years, time delay system has been one of the hottest research areas in control engineering for time delay often appears in many control systems either in the state, the control input, or the measurements [1,2]. Since time delay frequently occurs in practical systems and is often the source of instability, there have been many results for stability of delayed systems [3][4][5][6][7][8][9]. Also stabilization [10][11][12], filtering [13], and adaptive control [14] of time-delay systems have received considerable attention. e neutral systems often appear in the study of automatic control, population dynamics, and vibrating masses attached to an elastic bar [15]. A considerable number of studies related to this topic have been reported (see, for example, [16][17][18][19][20][21], and the references therein).
Lur'e system is originally from a pilot robot [22], which is one of the important classes of nonlinear systems whose nonlinear element satisfies certain sector constraints. Many systems such as Chua's circuits, Goodwin models, Swarm models, n-scroll attractors, and hyperchaotic attractors can be represented as Lur'e-type systems [23,24]. In recent years, stability analysis of the neutral-type Lur'e system with time delays has attracted the attention of many researchers [25][26][27][28][29][30][31][32][33][34][35]. e main purpose of stability analysis is to calculate the maximum allowable delay bounds (MADBs), which is a key index for judging the conservatism of stability criteria, such that the Lur'e system maintains absolute stability for any time delay less than the MADBs by using the LKF method. In [25], absolute stability of Lur'e systems with sector-bounded nonlinearities and constant delay was discussed by using a Lur'e-Postnikov function. To avoid involving a considerable number of free-weighting matrices and leading to a computationally expensive stability criterion in [26], the general free weighting matrix method was proposed and some improved delay-range-dependent stability criteria were obtained [27,28]. By using integral inequality and the Wirtinger integral inequality method instead of free weighting matrix one, some improved delaydependent robust stability criteria were derived in [29,30], respectively. Also, by eliminating nonlinearity and reducing the number of free-weighting matrices, Lur'e systems with interval time-varying delays were discussed in [31]. By discretizing the delay interval into two segmentations with an unequal width, some delay-dependent sufficient conditions were given for the robust stability of neutral-type Lur'e systems in [32,33]. By employing an augmented LKF method, reciprocally convex combination approach, and convex combination technique, several robust absolute stability criteria for uncertain neutral-type Lur'e systems with time-varying delays were presented in [34]. Very recently, by constructing a LKF including both double-integral terms and triple-integral terms, using the piecewise analysis method, Wirtinger-based integral inequality, and the reciprocally convex combination technique, some new stability criteria were obtained in [35]. In fact, one of the term [35] is the special case of the delay-partitioning approach, where T . e delay-partitioning method is an effective one to reduce a criterion's conservativeness, which is widely used in the stability analysis for various systems (see details in [36][37][38][39]). However, when utilizing the delaypartitioning approach, the term and N is the delaypartitioning number. It is clear that the derived conditions become more complicated and the computational burden grows bigger when N increases. In addition, to relate to the Wirtinger-based integral inequality and deal with the derivative of triple-integral terms introduced in LKFs, it is ineluctable that the extra terms t t+θ x T (s)Q dsdθ had to be introduced into the derivation process in [35], which leads to a sharp increase in the dimensions of the LMIs involved.
In particular, to establish the relationship of the vectors such as ds are introduced in LKFs in each of subintervals, which are more general than the ones in [28,35] It is worth mentioning that the merit of the proposed delay-decomposing method lies in that the dimensions of the LMIs involved are independent of the number of subinterval. In addition, to avoid introducing the extra vectors by Wirtinger-based integral inequality, the reciprocally convex combination method [41] and the integral inequality [42] are utilized to deal with the bounds of integral terms. Some novel LKFs related to the above inequalities are constructed on the obtained three subintervals. e presented stability criteria are given in terms of LMIs. Compared with the related literature, the conclusions of this paper have the advantages of less conservatism conservatism and the dimensions of the LMIs. Finally, two well-known numerical examples are given to demonstrate the effectiveness and less conservatism over the existing results. Notation: in this paper, R n denotes n-dimensional Euclidean space and R n×m is the set of all n × m real matrices. For symmetric matrices X and Y, the notation X > Y (respectively, X ≥ Y) means that the matrix X − Y is positive definite (respectively, nonnegative). diag · · · { } denotes the block diagonal matrix. e subscript "T" denotes the transpose of the matrix. I n denotes the identity matrix.

Problem Statements and Preliminaries
Consider a class of Lur'e systems of neutral type with timevarying delays and sector-bound nonlinearities described as follows: , where A, B, C, and D are constant matrices with appropriate dimensions.
where d M , τ M , μ τ , and μ d are the known constant scalars.
. . , f m (·)] T ∈ R m is the nonlinear function and f i (σ i (t)) · (i � 1, 2, . . . , m) is assumed to satisfy the finite sector restriction: with known positive scalar k i or the infinite sector restriction: e objective of this paper is to formulate the delaydependent stability conditions of system (1). e following lemmas will play important roles in deriving the criteria.
Lemma 1 (see [41]). Mathematical Problems in Engineering Lemma 2 (see [42]). For any matrices Q > 0, M, N, X with compatible dimensions, any continuous time vector function x(t) and η(t) with compatible dimensions, and any scalar τ M satisfying 0 ≤ τ(t) ≤ τ M , the following integral inequality holds: if , the integral inequality reduces the one in [43].

Main Result
In this section, some new delay-dependent stability criteria are proposed for system (1).
In the following, we will propose some criteria for the three subintervals. Now, we give the stability criteria for system (1) with conditions (2) and (4) when τ(t) ∈ Δ 1 as follows. (2) and (4) is absolutely stable if there exist n × n symmetric positive definite matrices P,

Theorem 1. For given scalars
. . , l m , any n × n matrix X 12 , and 2n × n matrices Y 12 , Z 12 , such that LMIs (5) and (6) hold when τ(t) ∈ Δ 1 : where Proof. For positive diagonal matrices L � diag l 1 , l 2 , . . . , l m and positive definite matrices P, R j , S i , and Q i · (i � 1, 2, j � 1, 2, 3), let us consider the following LKF candidates for the case 0 ≤ τ(t) ≤ τ α : where Mathematical Problems in Engineering 3 e time derivative of V(x(t)) along the trajectory of system (4) is given by where If 0 ≤ τ(t) ≤ τ α holds, one can compute out the following according to Lemma 1: where Φ 1 is given in (6). Based on Lemma 2 (in which M � N � Y 12 and and using Lemma 2 again (in which M � N � Z 12 and
By using Lemma where

Remark 3.
e terms t t− τ α 9 T 3 (t)Q 1 9 3 (t)ds and t t− τ β 9 T 4 (t) Q 2 9 4 (t)ds have been introduced in LKFs in the work instead of the terms [28,35] and And the relations between x(t), x(t − τ α ), x(t − τ β ), and x(t − τ M ) have been effectively expressed by the derivative of V 2 (x(t)). One can easily see that t t− τ α 9 T 3 (t)Q 1 9 3 (t)ds or t t− τ β 9 T 4 (t)Q 2 9 4 (t)ds deduces into t t− τ M /2 9 T 1 (t)Q9 1 (t)ds when α � β � 0.5. is is to say the established LKFs are more general than ones in [28,35]. e idea is expected to reduce the conservatism of obtained criteria. On the other hand, it can effectively overcome the weakness of the computational burden growing bigger when delay-partitioning number increases in [36][37][38][39].
Remark 4. In [35], the Wirtinger-based integral inequality and double-integral inequality were used to deal with bounds of the derivative of double-integral and triple-integral terms in LKFs.
us, the extra terms t t+θ x T (s)Q ds dθ were unavoidably introduced in the derivation process, which leads the dimensions of the LMIs involved to increase sharply. A detailed comparison is given in Example 1.

Illustrative Example
In this section, we will use two well-known numerical examples to show the effectiveness and benefits of our results. Example 1. Consider the following nominal neutral-type Lur'e system (1) subject to (2) and (4) with the parameters: (45) e purpose of this example is to compute MADBs of τ M such that the neutral-type Lur'e system (1) remains stable for different μ τ and μ d . For given μ d � 0.9, α � 0.45, μ d � 0.5, α � 0.45, and μ d � 0.1, α � 0.4, the acceptable upper bounds of τ M is 0.271, 2.277, and 2.652 when μ τ ≥ 1 by using Corollary 1 (τ(t) ∈ Δ 1 ) in our work, respectively. In order to make a comparison with some existing stability criteria, we calculate the MADBs and list them in Tables 1-3. Together with all derived MADBs listed in Tables 1-3, one can check that Corollary 1 can be superior over some present ones. e number of decision variables, maximal order of LMIs between our work, and the criteria in [29,30,33,35] are listed in Table 4. It shows that our proposed method involves smaller decision variables or lower maximal order of LMIs than the relative ones. To confirm the obtained result, a simulation result is shown in Figure 1 when µ is unknown, x(0) � [0, 0.4] T , τ M � 5.886, and h(t) � 0.5 + 0.1 sin t. Example 2. Consider Chua's circuit example discussed in [35]: e nonlinear function g(θ 1 ) � m 1 θ 1 + 0.5(m 0 − m 1 ) (|θ 1 + c| − |θ 1 − c|). Let m 0 � − 1/7, m 1 � 2/7, a � 9, b � 14.28, and c � 1; then, Chua's circuit can be expressed as a Lur'e-type system with

Mathematical Problems in Engineering
e feedback nonlinear function belongs to K [0,1] . Now, we calculate MADBs of τ M . For different μ τ , the obtained results are given in Table 5. From this table, it is clear to see that eorems 1 and 2 offer larger MADBs of τ M than those methods in existing references.

Conclusions
is paper has investigated the absolute stability analysis for neutral-type Lur'e systems with time-varying delays. Based on the new delay-decomposition approaches in combination   Method μ τ � 0.2 μ τ � 0.4 μ τ � 0.6 μ τ � 0.8 [31] 0.1122 0.1088 0.1086 0.1086 [27] 0.1122 0.1089 0.1086 0.1086 [32] 0.1126 0.1102 0.1102 0.1102 [28] 0.1130 0.1105 0.1105 0.1105 [44] 0.1221 0.1087 0.1083 0.1083 [29] 0.1227 0.1197 0.1197 0.1197 [35] 0.1799 0.1697 0.1660 0.1658 [30] 0.2495 0.2366 0.2300 0.2285 [33] 0.2906 0.2745 0.2653 0.2621 Cor 1 (τ(t) ∈ Δ 1 , α � 0. 35   with the integral inequality and reciprocally convex technique, several improved stability criteria have been derived by constructing some appropriate LKFs on the subintervals. e merit of the obtained stability criteria lies in the significant less conservativeness and lower computational complexity than some existing ones. Finally, two examples have been given to demonstrate the effectiveness and less conservatism of the proposed method. In the future works, we will be dedicated to study the stability analysis for systems with infinite delays and devote to the study of output feedback, tracking control and filtering of the neutral-type Lur'e systems with time-varying delays based on the method proposed in this paper.

Data Availability
All data generated or analyzed during this study are included in this article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.