改进的时滞分数阶稳定性分析：线性系统新方法

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"本文主要探讨了区间时滞线性系统的依赖分数阶稳定性的新分析方法。通过对具有时变间隔时滞的线性系统进行研究，文章提出了一个利用延迟变量分解技术的新稳定性准则，该准则降低了保守性，并且不涉及Lyapunov-Krasovskii（LK）泛函的时间导数的直接近似。通过应用詹森积分不等式和凸组合技术，文章能够更精确地处理可变延迟子间隔内的信息，包括Lyapunov矩阵的选择和LK函数中交叉项的上限估计。文章最后通过两个实际案例验证了新方法的有效性和较低的保守性。此外，该文章指出，作者通常被允许在个人网站或机构存储库上发布其文章的版本，但其他用途如复制、分发或销售需遵守Elsevier的版权政策。" 这篇论文的核心内容集中在解决线性系统稳定性分析的问题上，特别是当系统中存在区间时变时滞的情况。传统的稳定性分析方法可能无法有效地处理这种复杂情况，因为它们可能过于保守，即过度估计了系统的不稳定程度。为了克服这个问题，论文提出了一种新的依赖分数阶的稳定性准则。首先，文章引入了延迟变量分解方法，这是一种处理时变间隔时滞的技术，能够区分不同子间隔内的延迟信息，并考虑延迟的上下限。这种方法允许对每个子间隔选择不同的Lyapunov矩阵，以适应延迟变化的特性。其次，通过詹森积分不等式和凸组合技术，研究人员推导出新的稳定性条件。詹森积分不等式是微积分中的一个重要工具，用于处理函数的不等式关系。在本文中，它被用来更精确地估计与延迟相关的LK泛函的性质。同时，通过凸组合，可以将多个不等式组合成一个更强大的不等式，这有助于降低保守性。新提出的稳定性准则的一个显著优点是它不依赖于LK泛函的时间导数的直接近似。这种避免近似的处理方式可以减少误差，提高稳定性分析的准确性。论文最后通过两个具体的数值例子展示了新方法的实际应用和效果，证实了这种方法在降低保守性方面的优势。这些实例不仅证明了新准则的有效性，还为读者提供了如何应用该方法的实际示例。总结来说，这篇论文提供了一个创新的分析工具，对于理解和控制具有区间时变时滞的线性系统具有重要意义，特别是在设计控制系统和评估系统稳定性时。通过改进的分析方法，工程师和科学家可以更准确地评估这类系统的动态性能，从而优化系统设计，减少潜在的不稳定因素。

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excessive free weighting matrices. As a way of reducing the number of decision variables, at the sacriﬁce of conservatism, relaxations

based on the Jensen inequality lemma [1] have been attracted in [7–14,17–19].

On the other hand, the delay partitioning approach has been attracted in the derivation process. Recently, the method of dividing delay

(including the discretized Lyapunov functional approach in [1] and the delay decomposition approach in Gouaisbaut and Peaucelle [20]

and Han [21]) is the best one to handle the stability of system (1) with constant delay, by which the result near analytical delay limit can be

obtained. Since the augmented matrix S has much more room to be adjusted in the criterion, the functional can achieve the beneﬁtof

reducing the conservatism. Furthermore, partitioning of the delay intervals into n4 2 subintervals may lead to further improvements, for

examples, another delay partitioning was introduced in [3,6,20,21,24,29], which corresponded to the partitioning into two subintervals of

½0; h

 and of ½h

; h

. Therefore, it is natural that the less conservative stability results can be obtained when the delay decomposition

approach is extended to the case of time-varying delay. So far, by dividing interval ½ 0; τðtÞ into N non-uniform sub-intervals, the

pioneering studies have been provided, see [21,22]. But, the purpose of reducing conservatism is still limited due to the existence of

multiple coefﬁcients and the impact of subintervals with uniform size [22,25,26]. Moreover, it is still a quite difﬁcult task to divide interval

½0;

τðtÞ in a more reasonable manner [22], so that the functional with the augmented matrix can easily be constructed to obtain less

conservative stability results.

In practice, the systems almost contain some uncertainties because it is very difﬁcult to obtain an exact mathematical model due to

environment noise, uncertain or slowly varying parameters, etc. Most recently, the stability analysis for time-delay systems has been

extended to the case of the systems with nonlinear perturbations [25–27] and fuzzy systems [31,32], while the proposed result also has

been expanded to disturbance attenuation analysis [33,38] and H1 controller/ﬁlter design [31,32,36] and robust fault detection [34,37] for

the above systems. Ramakrishnan and Ray [25] studied the problem of stability analysis of the linear systems with nonlinear perturbation

by partitioning the delay interval into two segments of equal length. Zhang et al. [26] employed some novel integral inequalities (Lemmas

1 and 2 in [26]) to derive an improved stability criterion for the above systems. He et al. [33] investigated the disturbance attenuation

problem and H1 controller design for the linear systems. Lin et al. considered the H1 ﬁlter design problem [31] and observer-based H1

control design problem [32] for nonlinear system via T–S fuzzy model approach. Most recently, Liu [35] further studied the stability

analysis of linear systems. However, there also exists further room to investigate the upper bound of the time-derivative of the LK

functional, and the delay interval may be divided into two unequal subintervals or more subintervals [22,24,29]. This motivates the

present research to develop a novel method for stability problem for the concerned systems with less conservatism by making full use of

the information of time-delays and constructing a novel LK functional via variable delay dividing technique.

Motivated by the above discussions, this paper will focus on the delay-fractional-dependent stability problem of linear systems with

interval time-varying delay. Inspired by the reciprocally convex approach in [10], and some integral inequalities in [1,7,13], and then the

work of Kim [12], Zhang [22] and Briat [9],weﬁrstly construct a novel delay-fractional-dependent LK functional by developing a delay

decomposition approach, in which the integral interval ½t h

; 0 is decomposed into ½t h

; t h

ατ, ½t h

ατ; th

 and

½t h

; t ðh

=2Þ, ½t ðh

=2Þ; t with 0o α o 1 and τ 9 h

h

. Secondly, we derive a stability criterion for system (1) by suitably using

the inequalities such as (4)– (8) in Lemma 1 below. Since a tuning parameter

α and different delay partitioning method are introduced, the

information about xðt h

ατÞ and xðt ððh

=2ÞÞ can be taken into full consideration. Meanwhile, based on the convex combination

approach, the cross team in the time derivative of LK functional can be estimated more exactly without any direct approximation of the

delay terms in the derivation process. Therefore, using the variable delay decomposition approach, the Lyapunov matrices in LK functional

of this paper may be different in the delay intervals and the LMIs also may be different in the stability conditions, and thus compared with

the methods using the same Lyapunov matrices and the uniformly dividing delay subintervals, the variable and different Lyapunov

matrices-based method may lead to less conservatism. Finally, two well-known numerical examples are used to compare with some

previous results and demonstrate the effectiveness of the proposed method.

2. Problem formulation

Consider the following linear system with a time-varying state delay:

xðtÞ¼AxðtÞþA

xðt τðtÞÞ

xðtÞ¼

ϕðtÞ; 8t A ½h

; 0;

(

ð1Þ

where xðtÞis the state vector.

ϕðtÞ is the continuous initial vector function deﬁned on ½h

; 0; the matrices A; A

are constant matrices with

appropriate dimensions.

In this paper, the delay

τðtÞ is assumed to be time-varying delay as the following two cases:

Case 1.

τðtÞ is a differentiable function, satisfying for all t Z 0:

0o h

r τðtÞr h

;

τðtÞr h

ð2Þ

Case 2.

τðtÞ is not differentiable or the upper bound of the derivative of τðtÞ is unknown, and τðtÞ satisﬁes

0o h

r τðtÞr h

ð3Þ

where h

; h

and h

are some given values.

The purpose of this paper is to ﬁnd a stability criterion, which is less conservative than the existing results. Such a criterion may be

used to compute the tolerable delay bound h

for given h

or vice versa.

To end this section, we introduce the following lemmas which are useful in stability analysis for the linear system.

J. An et al. / ISA Transactions 53 (2014) 210–219 21 1

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