New Stability Analysis Results for Linear System with Two Additive Time-Varying Delay Components

#is paper focuses on the stability problems of continuous linear systems with two additive time-varying delay components. Firstly, an effective and simple Lyapunov–Krasovskii functional (LKF) is established, which not only takes adequate information of delay components and their upper bounds into consideration but also establishes a simpler form to decrease the computational complexity. Secondly, an improved delay-dependent stability criterion together with its corollary is obtained by employing the generalized free-weighting-matrix-based (GFWM-based) inequality and some other techniques to calculate the derivative of the constructed LKF, which will further reduce the introduced estimation error and make the criteria less conservative. Lastly, a numerical example is presented to illustrate the less conservatism and lower computational complexity of the derived results.


Introduction
It is a phenomenon that time delay often occurs in various dynamic systems such as economics, network transportation, circuit signal system, engineering systems, and so on [1][2][3]. e existence of the time-varying delay has a negative influence on system performance, which will have an adverse effect on the dynamic performance of the system and even make it instable. Moreover, for the remote control systems or networked control systems, signals transmitted between different points may pass different segments of networks, in which there may exist time delays of different characteristics. For instance, the signal transmitted from the sensor to the controller and the other from the controller to the actuator have different properties for the reason of variable transmission conditions [4]. erefore, it is necessary to investigate the stability problem of systems with additive time delays and assess the effects from different delays with various properties, which motivates the research about linear system with two additive time delay components in this paper.
Since the distribution characteristic of delays mentioned above has been modeled firstly as a system with two additive time-varying delay components in [4], many significant results have been devoted to this issue. As a most general form, linear systems with two additive delay components have drawn lots of attention [5][6][7][8][9][10][11][12][13][14][15]. Moreover, neural networks [16][17][18][19][20][21], T-S fuzzy systems [22,23], and Lur'e systems [24], considering two additive delay components in the state, have been also investigated recently. Among these works, two main techniques that contain constructing an appropriate LKF [25,26] and estimating its derivative [27][28][29][30][31][32] are usually used to reduce conservatism of the derived criteria. Additionally, reciprocally convex combination lemma [33] and its extensive form [34] also contribute to analyzing the time-delay system stability [15]. Some criteria with less conservatism are obtained in [8] by taking the delay, its upper bound, and the relationship between them into account. A novel LKF [9] is constructed, for which positive definiteness is ensured by the whole LKF rather than each term. In [10], a delay-partitioningbased LKF was established, which greatly reduces the conservatism. In [11], a novel LKF containing delay-product type terms [12,13] is constructed; meanwhile, both the Wirtinger-based inequality [28] and reciprocally convex combination technique [33] are utilized to estimate the derivative of LKF. Recently, a delay interconnection LKF and its improved form are constructed in [14,15], respectively; meanwhile, the free-matrix-based inequality [30] is utilized to calculate the single integral terms in their derivatives, which is demonstrated to be effective in reducing the conservatism.
For the sake of further study, the relevant effort is to decrease the conservatism and computational complexity of the derived result from the viewpoint of constructing an appropriate LKF and estimating its derivative. Although great development has been devoted to the issue, there still exists room for further study to take both the conservatism and the computational complexity into account. For example, in [14,15], delay interconnection LKF and its improved form are verified for their effectiveness for reducing the conservatism of the derived results; however, the reduction in conservatism is based on the increase in computational complexity. erefore, it is a challenging and meaningful problem to ensure the low conservativeness of the stability criteria and reduce the computational complexity of the result as possible. is paper will study this issue from two aspects, one is to construct an effective and simple LKF which not only considers adequate information of delay components and their upper bounds but also establishes a simpler form in comparison with the existing works to decrease the computational complexity. In addition, the GFWM-based inequality [35] reduces the estimation error of the single integral terms appearing in the derivative of the proposed LKF, which has not been utilized for analyzing stability of the system with additive delay components. Hence, improved results can be derived of less conservatism and lower computational complexity by combining two techniques mentioned above, which motivates the following research.
is paper analyzes delay-dependent stability of the linear systems with two additive time delay components, mainly aiming at considering the conservatism and computational complexity of the results comprehensively. e main contributions are summarized as follows: (i) To simplify the analysis process and derive a stability criterion with less computational complexity, an effective and simple LKF containing adequate information about delay components and their upper bounds is constructed, in which the useless terms that contribute nothing to reducing the introduced conservatism are not included. (ii) GFWM-based inequality together with some other advanced techniques is utilized to estimate the single integral terms appearing in the derivative of the proposed LKF, which further decreases the estimation error and drives less conservative stability criteria. In addition, any vectors in their simplest form are chosen to reduce the computational complexity while using GFWM-based inequality to estimate the single integral terms, which will further reduce the computational complexity. (iii) With the techniques mentioned above employed in this paper, improved results with less conservatism and lower computational complexity can be presented, whose validity and superiority can be demonstrated with a representative numerical example in comparison with the works in [4, 6-9, 14, 15].
is paper is organized as follows. Problem formulation and preliminary are given in Section 2. In Section 3, an improved stability criterion and its corollary are developed. A numerical example is used to demonstrate the validity and superiority of the proposed criteria in Section 4. At last, Section 5 presents the conclusion.
Notations: the notation R n denotes the n-dimensional Euclidean space; P > 0( ≥ 0) means that P is a real symmetric and positive definite (semipositive definite) matrix; I and 0 represent an appropriately dimensioned identify matrix and zero matrix, respectively; * stands for the symmetric term in the symmetric matrix; the transpose and the inverse of a matrix are denoted by the superscripts T and − 1, respectively; and Sym X { } � X + X T .

Problem Formulation and Preliminary
Consider the following continuous linear system with two additive time-varying delay components: where x(t) � x 1 (t) x 2 (t) · · · x n (t) T ∈ R n is the state vector; A ∈ R n×n and A d ∈ R n×n are known constant matrices; ϕ(t) is the initial condition; d 1 (t) and d 2 (t) are continuous and differential functions, which stand for two time delays and satisfy the following conditions: where h i , i � 1, 2 and μ i , i � 1, 2 are constants. Let e necessary lemmas are introduced as follows to develop the results in Section 3.
Lemma 1 (GFWM-based inequality [35]). For symmetric positive definite matrix R ∈ R n×n , any matrices L, M, and an arbitrary vector ω: [a, b]↦R n . en, the following inequality holds: where χ 0 is an arbitrary vector and Lemma 2 (Jensen's inequality [27]). For any symmetric positive definite matrix R ∈ R n×n , scalars a and b: a < b, and vector x: [a, b]↦R n , the following inequality holds: Lemma 3 (Wirtinger-based inequality [28]). For any symmetric positive definite matrix R ∈ R n×n , scalars a and b: a < b, and vector x: where χ 1 and χ 2 are defined in Lemma 1.

Results and Discussion
In this section, by constructing a novel LKF and using GFWM-based inequality together with some other advanced techniques to calculate its derivative, a stability criterion and its corollary are derived for system (1). Meanwhile, some discussions are provided to illustrate the superiority of the methods used in this paper.

A Stability Criterion with Less Conservatism and Lower
Computational Complexity. A novel LKF together with the aforementioned lemmas is used to derive the following criterion.

Complexity
Proof. A vector ξ(t) ∈ R 13n is defined as follows: A novel LKF containing adequate information about delay components and their derivatives with a simper form is constructed as follows: where with Firstly, the matrices declared in eorem 1 including P, Q i (i � 1, 2, 3, 4), R i (i � 1, 2, 3), Z, Z a are positive definite, which insures the positive definitiveness of the proposed LKF.
Secondly, the derivative of the proposed LKF with respect to time along the trajectory of system (1) is calculated as follows: Calculating the derivatives of Complexity 5 Applying Lemma 1 to estimate the R 1 -dependent single integral term in equation (16) yields where η 1 (t) is selected as and the other vectors are derived based on Lemma 1: which implies Similarly, the R 2 -dependent single integral term in equation (16) can be estimated via Lemma 1: where η 2 (t) is selected as us, inequality (21) can be rewritten as e R 3 -dependent single integral term in equation (16) is also calculated with Lemma 1, which yields where η 3 (t) is selected as Complexity 7 erefore, inequality (25) can be rewritten as Combining inequalities (16), (20), (24), and (28), the estimation of _ V 3 (x t ) can be derived as follows: where Ξ 31 and Ξ 32 (d 1 (t), d 2 (t)) are defined in equation (8).
From what has been mentioned above, the positive definiteness of the matrices declared in eorem 1 and the inequalities in (7) ensure V(x t ) ≥ ε 1 ‖x(t)‖ and _ V(x t ) ≤ − ε 2 ‖x(t)‖, respectively, for any existing sufficient small scalars ε 1 , ε 2 > 0, which implies system (1) with two time-varying delay components satisfying (2) is asymptotically stable. is completes the proof. □ 3.2. Discussion. By constructing a novel LKF and using GFWM-based inequality to estimate its derivative, eorem 1 is derived with less conservatism and lower computational complexity. Moreover, its superiority can be reflected in the following discussion.
Firstly, among the conditions of eorem 1, four matrix inequalities included in equation (7) are not linear matrix inequalities (LMIs); hence, LMI Toolbox in MATLAB cannot solve these inequalities directly on account of the existence of the inverse matrices in Ω(d 1 (t), d 2 (t)). Nevertheless, based on Schur complement, the matrix inequalities in equation (7) can be transformed into LMIs. erefore, the stability problem of system (1) is equivalent to the feasibility checking problem of LMIs.
Secondly, some advanced techniques are applied to derive eorem 1, which will be conducive to reducing its conservativeness and computational complexity.
(i) e LKF constructed in this paper plays an important role to decrease the computational complexity. It not only contains adequate information about delay components and their upper bounds but also has a simpler form in comparison with the existing works. It is obvious that the nonintegral terms, quadratic single integral terms, and quadratic double integral terms in the LKF constructed in this paper have been simplified compared with the latest works [14,15], and the useless terms that contribute nothing to reducing the introduced conservatism are not included. e simplification of LKF will undoubtedly greatly reduce the computational complexity of eorem 1; moreover, it ensures the low conservatism of eorem 1.
(ii) GFWM-based inequality is used to estimate the single integral term in _ V 3 (x t ), which further decreases the conservatism and the computational complexity of eorem 1. e superiority of GFWM-based inequality in comparison with FMBII utilized in [14,15] is verified theoretically in [35]. Moreover, in the proof process of eorem 1 in this paper, three arbitrary vectors η 1 (t), η 2 (t), and η 3 (t) defined in equations (18), (22), and (26) are chosen to estimate the R 1 -, R 2 -, and R 3 -dependent single integral terms in _ V 3 (x t ), respectively, which are verified to be the simplest form to minimize the conservatism introduced in equations (20), (24), and (28). In other words, adding additional terms to η 1 (t), η 2 (t), and η 3 (t) will not contribute to the reduction of conservatism. Hence, both the conservatism and the computational complexity will be further reduced with the techniques used in estimating the derivative of the LKF.
Last but not the least, both the conservatism and the computational complexity are considered in this paper, while the only objective is to decrease the conservatism in most existing works. e effect of LKF components is fully investigated; thus, an effective and simple LKF constructed in this paper will greatly reduce the decision variables without adversely affecting the conservatism introduced. Moreover, GFWM-based inequality is utilized to deal with three important single integral terms appearing in _ V 3 (x t ), which reduces both the introduced conservatism and decision variables. erefore, the method proposed in this Complexity 9 paper contributes to analyzing stability of system (1) with delay components satisfying condition (2). Triple integral term in LKF is verified beneficial to analyzing stability of the systems with additive delays. erefore, to further illustrate the effect of the GFWM-based inequality used in this paper, the following corollary is derived from eorem 1 directly without considering V 4 (x t ) in LKF.

A Numerical Example
In this section, a typical numerical example is provided to illustrate both the less conservativeness and the lower computational complexity of the derived criteria in comparison with the existing works. Example 1. Consider system (1) with the following parameters: Suppose that _ d 1 (t) ≤ 0.1 and _ d 2 (t) ≤ 0.8, which means μ 1 � 0.1 and μ 2 � 0.8.
is is a representative number example to analyze the stability of system (1) satisfying condition (2), which is also used in [4, 6-9, 14, 15] to illustrate the conservatism of the derived criteria. As known, the stability criterion obtained with Lyapunov stability theory is usually a sufficient condition. Hence, the calculated upper bounds of delays based on different criteria are less than their analytical value, which represent the conservatism of criteria. In other words, the criterion with bigger calculated values of delays is less conservative.
To verify the validity of the method presented in this paper, Tables 1 and 2    obtained by eorem 1 and Corollary 1 in this paper and other existing works is listed in Table 2. Moreover, the number of decision variables (NoVs) of the stability criteria is also presented in tables to demonstrate the superiority of the proposed method. Based on the comparison of the calculated upper bounds and NoVs for different cases, it is obvious that the method used in this paper contributes to the stability analysis of system (1) on account of not only the further decrease of the conservatism but also the significant reduction of the decision variables. Compared with the existing works, for instance, eorem 1 in [15], the derived criteria in this paper have greatly reduced the computational complexity while being less conservative, which is also the main contribution of this paper.
Moreover, based on the comparison of the calculated upper bounds and NOVs from eorem 1 and Corollary 1, the triple integral term constructed in V 4 (x t ) is verified beneficial to decrease the conservativeness. Compared with the existing works, Corollary 1 is less conservative without the triple integral terms introduced in LKF, which further illustrate the effect of the method used in this paper.

Conclusions
is paper investigates the stability of linear system with two additive time-varying delays. e main contribution of this paper is that the derived results have greatly reduced the computational complexity while being less conservative with comparison with the existing works. To achieve the goal, an effective and simple LKF is established in this paper, which is verified to be a great contribution to reduce the computational complexity. Additionally, GFWM-based inequality together with some other advanced techniques is utilized in this paper to calculate the derivative of the proposed LKF, which is verified to be significant to further decrease the conservatism of the derived criteria. A representative numerical example is presented to illustrate the validity and superiority of the method used in this paper.

Data Availability
e data used to support the findings of the study are included within the article.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.