时变间隔与非线性摄动系统鲁棒稳定性条件
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更新于2024-08-28
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"这篇研究论文探讨了具有时变间隔时滞和非线性摄动的线性系统的鲁棒稳定性问题。作者W. Zhang, X.-S. Cai, 和 Z.-Z. Han提出了一种基于Lyapunov-Krasovskii函数的新型稳定性分析方法，并利用线性矩阵不等式（LMI）来建立稳定性标准。此方法考虑了时变时滞的范围信息，并能处理系统中的非线性干扰因素。通过数值实例验证了该方法的有效性。文章已被《计算与应用数学》期刊接受，并给出了相关的DOI和接收及修订日期。"
在控制系统理论中，鲁棒稳定性是关键概念，它关注的是系统在面对参数不确定性、外部扰动或内部非线性时仍能保持稳定的能力。本文特别关注的是一类特殊的系统，这类系统具有时变间隔时滞，即系统的延迟不是恒定的，而是在一个范围内变化。时滞的存在会增加系统分析的复杂性，因为它可能导致系统的不稳定。
Lyapunov-Krasovskii函数是分析稳定性的一种常用工具，它是一种能量函数，能够度量系统的状态距离平衡点的远近。在本文中，作者提出的新Lyapunov-Krasovskii函数不仅考虑了系统的动态特性，还考虑了时变时滞的动态范围，使得分析更加精确。他们引入的方法估计了Lyapunov-Krasovskii泛函的时间导数的上限，这是确定系统稳定性的重要步骤。
为了处理非线性摄动，作者利用了线性矩阵不等式（LMI）技术。LMI是一种强大的工具，它将稳定性条件转化为一组易于求解的代数不等式。通过调整自由加权矩阵，作者能够建立一个依赖于时变时滞范围的稳定性标准，这使得在存在不确定性的情况下也能评估系统的稳定性。
数值算例的应用证明了所提出方法的实际效用，这可能包括模拟不同条件下的系统行为，以及比较新方法与传统方法的性能。这项工作提供了一种新的、实用的工具，用于分析和设计具有时变时滞和非线性干扰的复杂控制系统的鲁棒稳定性。这一贡献对于工程应用和理论研究都具有重要意义，特别是在航空航天、自动化、电力系统和其他涉及延迟效应的领域。
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researchers have fo c used o n the delay-dependent stability. Numbers of significant results have been reported
in recent litera tur e [1]- [25]. For example, a novel Lyapunov-Krasovskii functional was introduced in [11].
Augmented Lyapunov-K rasovskii functional appro ach was developed in [13] and [17]. Juensen integral
inequality approach was employed in [12], [15], [16], [18] and [24]. A novel piecewise analysis method was
proposed in [22]. Delay-range-dependent stability was inves tig ated in [8] by using the free-weighting matrix
approach [9], [20]. The stability problem of discrete-time systems with interval time-varying delay was
studied in [4] a nd [23].
In prac tice, real systems usually present some uncertainties due to environmental noise, uncerta in
or slowly varying parameters, e tc. Therefore, the stability pro blem of time-delay sy stems with nonlinear
perturbations has received increasing attention (see e.g., [1], [7], [25], [26]). A model transfo rmation method
was used in [1]. Bounding technique for some cross terms was proposed in [14]. A descriptor model
transformation together with a decomposition technique of the delay term matrix wa s employed in [7].
Recently, a less cons e rvative delay-dependent stability criterion was provided in [25] by employing the
free-weighting matrix approach. Robust stabilization for nonlinear discrete-time systems was studied in
[26]. In the above references, reducing the conservatism of the existing stability c riteria is a central issue.
As we know, b ounding technique [14] or model transformation [2] may increase the conservatism. The
free-weighting matrix method, by contrast, is helpful to reduce the conser vatism of stability criteria [20].
On the other hand, choosing appropriate Lyapunov-Krasovskii functional and estimating the upper bound
of its time derivative are very important in deriving the stability criteria.
In this paper, we deal with the delay-dependent stability problem for a class of linear systems with
nonlinear perturbations and interval time-varying delay. We first introduce a new Lyapunov-Krasovskii
functional by taking the range information of the delay into account. The delay-dependent stability of
systems is then analyzed by using the functional. An approach is proposed in estimating the upp e r bound
of the time derivative of the functional. New delay-range-dependent stability criteria are obtained by
intr oducing fre e-weighting matrices a nd free-weighting parameters. The propo sed stability criteria are
formulated in terms of a set of linear matrix inequalities (LMIs). Finally, two numerical examples are given
to show the effectiveness of the proposed approach.
Notations: R
n
denotes the n-dimensional Euclidean space. The superscript “T ” stands for matrix
transposition. X > Y (respectively, X ≥ Y ), where X and Y are real symmetric matrices, means that
the matrix X − Y is positive definite (respectively, positive semi-definite). I is an identity matrix with
appropriate dimension. In symmetric block matrices, we use an asterisk (∗ ) to represent a term that is
induced by symmetry. Matrices, if not explicitly stated, are assumed to have compatible dimensions.
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