1. Introduction
Because of the finite speed of data transmission, time delays are unavoidably encountered in a variety of real-world systems, such as multi-agent systems [
1,
2], active suspension systems [
3,
4], chemical engineering systems [
5], and so on. The time delays frequently cause undesirable dynamic behaviors [
6,
7]. Consequently, a considerable amount of attention has been paid to the stability analysis for time-delay systems [
8,
9,
10,
11,
12,
13,
14,
15].
In this area, the Lyapunov–Krasovskii functional (LKF) is the most efficient mathematical tool [
4,
5]. It is the main idea of the LKF based approach to establish a positive definite functional such that its derivative along a solution of the considered system is a definite negative [
7]. However, a certain degree of conservatism is inevitably introduced, since only sufficient conditions can be obtained. As is well known, the reduction of conservatism depends on the construction of LKF to a considerable extent, which aims at making use of more information about the delay [
11]. In this trend, a great deal of effort has been contributed, including augmented LKF [
11], delay decomposition/fractioning [
12,
13], triple-integral terms [
14,
15] and functional including quadratic terms multiplied by a higher degree scalar function [
16].
The reduction of conservatism also benefits from the techniques utilized to estimate the cross terms when differentiating the LKF [
17]. Compared with the choice of LKF, the accurate bounding technique has been considered as a more effective manner to relax the criteria. In this regard, most of the contributions are derived via Jensen inequality thanks to its convenient tractability [
13,
14,
15]. However, as discussed in [
18], the Jensen inequality often leads to undesirable conservatism. Thus, reducing the estimation gap of the Jensen inequality has become an open issue. Recently, an alternative inequality based on the Wirtinger inequality has been proposed in [
19] to achieve a potential gain with respect to Jensen inequality. In [
20], by using information on the double integral of state, a new free-matrix-based integral inequality is developed, which includes the Wirtinger one at the cost of computational burden [
21]. In [
17], a novel modification method is introduced to derive a new integral inequality. If the LKF with triple-integral terms are established, the search for how to provide tighter bounds for double integral functionals is becoming a stringent task. In [
22], the Wirtinger-based inequality is extended to the double integral form. In [
23], the Jensen-based inequalities are refined for both single and double integrals and their potential capacity exhibits obvious advantages over the previous ones. In [
24], by using Wirtinger-based single and double integral inequalities and delay decomposition technique, the stability analysis of neural systems is investigated. Combining advanced integral inequalities and slack variables, a new delay-dependent stability condition for time delay systems is developed in [
25]. Among the literature on this subject, the most noticeable technique is the auxiliary function-based integral inequalities performed in [
26], which are suitable for the LKF in triple integral forms and encompass the Jensen and Wirtinger ones as special cases. Therefore, how to develop new integral inequalities to provide more accurate bounds for both single and double integrals than [
26] motivates the present study.
Based on the above discussions, the main contribution of this paper is that a novel series of integral inequalities independent of slack variables is proposed, which shows significant improvements over the Wirtinger-based, refined Jensen and auxiliary function-based ones. By constructing an augmented LKF with triple-integral terms, the information on the delay range, especially on the lower bound of delay, is fully taken into consideration. By virtue of the improved inequalities, the derived stability conditions are less conservative than some existing ones.
The remainder of the current paper is organized as follows: the problem formulation and preliminary are presented in
Section 2.
Section 3 provides the new integral inequalities. In
Section 4, the stability criteria for interval time-delay systems is obtained. Numerical examples are presented in
Section 5 to demonstrate the effectiveness of the proposed approaches.
Section 6 draws the conclusions.
Notation: In this paper, is the n-dimensional Euclidean space and is the set of all real matrices. * refers to the symmetric term in a symmetric matrix. The superscripts and are the transpose and inverse of a matrix, respectively. The notation means that the matrix is symmetric positive definite (positive semi-definite). I and 0 stand for the identity and zero matrix, respectively. represent the block entry matrices, for example . is a block diagonal matrix, means a column vector, and .
3. New Integral Inequalities
As discussed in the first section, the reduction of conservatism primarily benefits from two factors: the suitable construction of LKF and more accurate estimation for its derivative. Revisiting the literature [
14,
17,
23,
26], the most commonly used LKF consists of the following double and triple integral terms:
where
are the Lyapunov matrices. The derivatives of
and
are given by
In order to obtain a linear matrix inequality (LMI)-based stability condition and less conservatism, it is required to provide closer estimations for the integrals of quadratic functions in Equations (7) and (8). For this purpose, a new set of integral inequalities is developed in the following lemmas.
Lemma 2. For a given matrix the following inequality holds for all continuously differentiable function in where Proof. For any differentiable function
, define a function
for all
given by:
where
are constant vectors to be determined and
For a matrix
, integrating
from
to
leads to
By noting
, it yields
Rewriting the last term of the right-hand side of Equation (12) as sums of squares, one has
where
Equation (13) holds independently of the choice of constant vectors and the last term of the right-hand side of Equation (13) is non-positive. Therefore, choosing leads to the maximum of the right-hand side of Equation (13). Rearranging Equation (13) leads to Equation (9). Thus, the proof is completed. ☐
Remark 1. By the Wirtinger-based inequality [
19], the new inequality [
17] and the auxiliary function-based inequality [
26] (the refined Jensen inequality [
23]), respectively,
can be estimated as follows:
Compared with Equation (14)–(16), one can see that Equation (9) produces a more accurate bound for a single integral term. Thus, the stability criteria by Lemma 2 tend to be less conservative. Unlike the existing ones, not only the relations between , , , , but also between them and are all taken into account, which allows one to make use of extra information on time delay. Moreover, no extra slack matrices are introduced in Equation (9).
Lemma 3. For a given matrix the following inequalities hold for all continuously differentiable function in where Proof. In order to prove Equation (17), consider a function
for all
defined as
where
and
are constant vectors to be determined and
satisfying
,
and
For a matrix
, one has
From Equation (20), one can obtain
where
By setting
to rearrange Equation (21), one can obtain Equation (17). On the other hand, choose
where
and
are constant vectors to be determined and
,
By proceeding the proof process similar to that of Equations (17) and (18) is derived. This completes the proof. ☐
Remark 2. By the double-integral form of Wirtinger-based inequality [
22] and the auxiliary function-based inequality [
26], respectively,
can be bounded in the following two forms:
It is obvious that the proposed inequality Equation (17) achieves the desirable effect in reducing the estimation gaps of Equations (23) and (24). Furthermore, the additional signal of triple integral is utilized in Lemma 3, which could offer more information in the criteria and thus yield better performance. Then, the combination of Lemmas 2 and 3 is suitable for the LKF consisting of functionals in single, double and triple integral forms.
4. Stability Analysis Criteria for Interval Time-Delay Systems
In this section, in order to demonstrate the merits of the new integral inequalities, the proposed ones are applied to the stability analysis of the interval time-delay systems. For simplicity of presentation, for a matrix , define and .
Theorem 1. Given scalars and , the system (1) with an interval time-delay subject to (2) is asymptotically stable if there exist matrices ,
,
,
,
,
(
i = 1, 2;
j = 1, 2)
and any matrices of appropriate dimensions such that the following LMIs hold:where
Proof. Consider the following LKF candidate,
where the individual functionals are defined as follows:
Differentiating
along the solution of system (1) leads to
where
Then, the applications of Lemmas 2 and 3 to the cross terms in Equations (31)–(33) yields
By Lemma 1, if there exist matrices
satisfying Equation (25), the following inequality is obtained, which is similar to the treatment in [
26],
By combing Equations (27)–(39), one has
where
Since is convex in , and ensure , which means that the system (1) is asymptotically stable. This completes the proof. ☐
Remark 3. By establishing a novel LKF, a stability criterion is derived in terms of LMIs. Various functionals in triple integral forms are introduced in the proposed LKF, which is effective in improving the feasible stability region [
14]. Unlike the existing literatures [
14,
20,
21,
22,
23,
26],
,
and
are utilized as elements of different augmented vectors such that the information on the lower bound of delay is fully exploited. Thus, the derived criterion could result in less conservatism, especially for the systems with larger lower bounds of delay.
Remark 4. More accurate bounds for the cross terms are obtained by the improved integral inequalities. As a result, it can be expected that the resulting condition has the potential to achieve more desirable performance. Unlike [
14,
15], handling the double integrals as a whole, more of a relationship between time-varying delay and delay range is taken into account by elaborately dividing the derivatives of triple integrals into several parts in Equation (33). By the Lemma 3, triple integrals of the state are obtained in the final result to provide more flexibility in finding the solutions of matrix variables than the approaches in [
19,
20,
22,
23,
26]. Consequently, a great many of cross terms among double and triple integrals are obtained in the criterion, which are conducive to reducing the conservatism.
Remark 5. Applying Lemma 3 to double integrals divided as shown in Equation (33) results in the functions weighted by
and
. In [
28], they are enlarged as
and
, respectively. In Theorem 1, using
,
, the above terms and the terms with inverses of convex parameters are handled as a combination, avoiding direct approximation (see Equation (39)), which is similar to the treatment in [
26]. The advantage of Theorem 1 over [
26] is the substituting the Auxiliary-function based inequality by the Lemma 2. On the other hand, when the information on the delay derivative is unavailable, by setting
, one can easily arrive at the following corollary.
Corollary 1. For given scalars , the system (1) with an interval time-varying delay is asymptotically stable if there exist matrices P = [Pij]2×2>0, R = [Rij]2×2>0, Q2>0, Ui>0, Si>0, Wi>0 (i = 1, 2; j = 1, 2), and any matrices Zl (l = 1, 2, 3, 4) of appropriate dimensions such that the LMIs (25)–(26) hold with .
Proof. Eliminating the term with in LKF Equation (27) and proceeding the similar way as Theorem 1 yield Corollary 1. As the deriving process is directly following Theorem 1, it is omitted for the sake of simplicity. ☐
Remark 6. When , the interval is missing. It yields different selection of the LKF from the one in Theorem 1, as the in integral limit in Equation (27) becomes zero. Thus, it is more reasonable to reevaluate the Theorem 1 for and the following corollary is derived.
Corollary 2. Given scalars and , the system (1) with a time delay subject to (2) is asymptotically stable if there exist matrices , ,
(
i = 1, 2;
j = 1, 2),
and any matrices of appropriate dimensions such that the following LMIs are feasible:where Proof. Choose the following LKF candidate
where
The proof follows a similar fashion as that of Theorem 1, which ultimately leads to
where
Thus, if Equations (41)–(42) hold, the system (1) subject to (2) with
is asymptotically stable. This ends the proof. ☐
Remark 7. Besides the number of decision variables, the dimension of the LMI-based condition is also considered as a key factor for computational complexity [
29]. Compared to
in Theorem 1, the dimensions of
in Corollary 2 are reduced from
to
. Thus, Corollary 2 is more reasonable than Theorem 1 for systems with zero lower bounds.