Notes on Stability of Time-Delay Systems: Bounding Inequalities and Augmented Lyapunov-Krasovskii Functionals

doi:10.1109/TAC.2016.2635381

Home
/
Papers
/
Notes on Stability of Time-Delay Systems: Bounding Inequalities and Augmented Lyapunov-Krasovskii Functionals

Journal Article•DOI•

Notes on Stability of Time-Delay Systems: Bounding Inequalities and Augmented Lyapunov-Krasovskii Functionals

Chuan-Ke Zhang¹, Yong He¹, Lin Jiang², Min Wu¹•Institutions (2)

China University of Geosciences (Wuhan)¹, University of Liverpool²

01 Oct 2017-IEEE Transactions on Automatic Control (IEEE)-Vol. 62, Iss: 10, pp 5331-5336

TL;DR: It is proved that, when a simple LKF is applied, the stability criteria obtained by the Wirtinger-based inequality and the Jensen inequality are equivalent, which means that the tighter inequality does not always lead to a less conservative criterion.

read less

Abstract: The bounding inequalities and the Lyapunov-Krasovskii functionals (LKFs) are important for the stability analysis of time-delay systems. Much attention has been paid to develop tighter inequalities for improving stability criteria, while the contribution of the LKFs has not been considered when discussing the relationship between the tightness of inequalities and the conservatism of criteria. This note is concerned with this issue. Firstly, it is proved that, when a simple LKF is applied, the stability criteria obtained by the Wirtinger-based inequality and the Jensen inequality are equivalent although the Wirtinger-based inequality is tighter. It means that the tighter inequality does not always lead to a less conservative criterion. Secondly, it is found that a suitable augmented LKF with necessary integral vectors in its derivative is required to achieve the advantage of the Wirtinger-based inequality. Based on this observation, two delay-product-type terms are introduced into the LKF to establish new stability criteria. Finally, a numerical example is given to verify the equivalence statements and to show the benefit of the proposed criteria.

...read moreread less

Citations

PDF

Open Access

More filters

Journal Article•DOI•

A Survey on Security Communication and Control for Smart Grids Under Malicious Cyber Attacks

[...]

Chen Peng¹, Hongtao Sun¹, Mingjin Yang¹, Yu-Long Wang¹•Institutions (1)

Shanghai University¹

01 Jan 2019-IEEE Transactions on Systems, Man, and Cybernetics

TL;DR: A comprehensive security understanding of the SGs framework, attacks scenarios, detection/protection methods, estimation and control strategies from both communication and control viewpoints are addressed.

...read moreread less

Abstract: Smart grids (SGs), which can be classified into a class of networked distributed control systems, are designed to deliver electricity from various plants through a communication network to serve individual consumers. Due to the complexity of environments, the distribution of the spatial locations and vulnerability of the communication networks, cyber security emerges to be a critical issue because millions of electronic devices are interconnected via communication networks throughout critical power facilities. This paper addresses a comprehensive security understanding of the SGs framework, attacks scenarios, detection/protection methods, estimation and control strategies from both communication and control viewpoints. Also, some potential challenges and solution approaches are discussed to deal with the threat issues of SGs. At last, some conclusions and highlight future research directions are presented.

...read moreread less

235 citations

Cites background from "Notes on Stability of Time-Delay Sy..."

...For examples, besides time delay, packet dropouts, or bandwidth limitation of their own challenges for communication networks [12], a security transmission manner is required so as to avoid malicious jamming or tamper [13]; besides the accuracy or intelligence requirements, fabricating smart meter readings and manipulating energy costs will impose extra limitations for AMIs which are wildly used in SGs [14]....
[...]

Journal Article•DOI•

Overview of recent advances in stability of linear systems with time-varying delays

[...]

Xian-Ming Zhang¹, Qing-Long Han¹, Alexandre Seuret², Frédéric Gouaisbaut², Yong He³ - Show less +1 more•Institutions (3)

Swinburne University of Technology¹, University of Toulouse², China University of Geosciences (Wuhan)³

01 Jan 2019-Iet Control Theory and Applications

TL;DR: In this article, the authors provide an overview and in-depth analysis of recent advances in stability of linear systems with time-varying delays, including stability criteria for three cases of a time varying delay.

...read moreread less

Abstract: This study provides an overview and in-depth analysis of recent advances in stability of linear systems with time-varying delays. First, recent developments of a delay convex analysis approach, a reciprocally convex approach and the construction of Lyapunov–Krasovskii functionals are reviewed insightfully. Second, in-depth analysis of the Bessel–Legendre inequality and some affine integral inequalities is made, and recent stability results are also summarised, including stability criteria for three cases of a time-varying delay, where information on the bounds of the time-varying delay and its derivative is totally known, partly known and completely unknown, respectively. Third, a number of stability criteria are developed for the above three cases of the time-varying delay by employing canonical Bessel–Legendre inequalities, together with augmented Lyapunov–Krasovskii functionals. It is shown through numerical examples that these stability criteria outperform some existing results. Finally, several challenging issues are pointed out to direct the near future research.

...read moreread less

210 citations

Journal Article•DOI•

A generalized free-matrix-based integral inequality for stability analysis of time-varying delay systems

[...]

Hong-Bing Zeng¹, Xiao-Gui Liu¹, Wei Wang¹•Institutions (1)

Hunan University of Technology¹

01 Aug 2019-Applied Mathematics and Computation

TL;DR: A generalized free-matrix-based integral inequality (GFMBII) is presented that overcomes the drawback that the Bessel–Legendre inequality is inconvenient to cope with a time-varying delay system as the resultant bound contains a reciprocal convexity.

...read moreread less

184 citations

Journal Article•DOI•

A relaxed quadratic function negative-determination lemma and its application to time-delay systems

[...]

Chuan-Ke Zhang¹, Fei Long¹, Yong He¹, Wei Yao², Lin Jiang³, Min Wu¹ - Show less +2 more•Institutions (3)

China University of Geosciences (Wuhan)¹, Huazhong University of Science and Technology², University of Liverpool³

01 Mar 2020-Automatica

TL;DR: A parameter-adjustable-based lemma is developed, which contains the lemma popularly used in literature as a special case and has potential to reduce the conservatism without requiring extra decision variables.

...read moreread less

170 citations

Journal Article•DOI•

Generalized reciprocally convex combination lemmas and its application to time-delay systems

[...]

Alexandre Seuret, Kun Liu¹, Frédéric Gouaisbaut•Institutions (1)

Beijing Institute of Technology¹

01 Sep 2018-Automatica

TL;DR: It is proved that these existing inequalities can be captured as particular cases of Moon et al.'s inequality and found strong links not only with the most recent and efficient matrix inequalities such as the reciprocally convex combination lemma and also its relaxed version but also with some previous inequalities.

...read moreread less

148 citations

1
2
3
4
…
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48

Collapse

References

PDF

Open Access

More filters

Journal Article•DOI•

Reciprocally convex approach to stability of systems with time-varying delays

[...]

PooGyeon Park¹, Jeong Wan Ko¹, Changki Jeong¹•Institutions (1)

Pohang University of Science and Technology¹

01 Jan 2011-Automatica

TL;DR: This paper suggests the lower bound lemma for such a combination, which achieves performance behavior identical to approaches based on the integral inequality lemma but with much less decision variables, comparable to thosebased on the Jensen inequalityLemma.

...read moreread less

2,248 citations

"Notes on Stability of Time-Delay Sy..." refers background or methods in this paper

...Lemma 2: ([7], [27]) For given vectors β1 and β2 , scalar α in the interval (0, 1), symmetric positive definite matrix R ∈ Rn×n and any matrix X ∈ Rn×n satisfying [ R X ∗ R ] ≥ 0, the following inequality holds...
[...]
...(1) Proof of C1: when Φ1 ≥ 0 holds for any matrix S, using JI (3) and (5) to estimate the integral term in (9) yields [27]...
[...]

Journal Article•DOI•

Wirtinger-based integral inequality: Application to time-delay systems

[...]

Alexandre Seuret¹, Frédéric Gouaisbaut¹•Institutions (1)

Hoffmann-La Roche¹

01 Sep 2013-Automatica

TL;DR: An alternative inequality based on the Fourier Theory, more precisely on the Wirtinger inequalities is proposed and it is shown that this resulting inequality encompasses the Jensen one and also leads to tractable LMI conditions.

...read moreread less

1,791 citations

"Notes on Stability of Time-Delay Sy..." refers background or methods in this paper

...(2) Proof of C2: when Ψ1 ≥ 0 holds for any matrix U , using WBI (4) and (5) to estimate the integral term in (10) yields [7]...
[...]
...By comparing the proof of Theorem 1 and Theorem 7 of [7], the key difference is that the LKF used in [7] includes an augmented non-integral term, which activates the advantage of the WBI....
[...]
...An alternative type of method is to directly estimate the integral term via bounding inequalities, among which Jensen inequality (JI) [15] and Wirtinger-based inequality (WBI) [7] are frequently applied....
[...]
...Lemma 1: (Jensen Inequality [15] and Wirtinger-Based Inequality [7]): For given symmetric positive definite matrix R ∈ Rn×n , scalars a and b satisfying a b, and vector ω : [a, b] → Rn such that the integrations concerned are well defined, the following inequalities hold (b − a) ∫ b a ωT (s)Rω(s)ds ≥ χT1 Rχ1 (3) (b − a) ∫ b a ωT (s)Rω(s)ds ≥ χT1 Rχ1 + 3χT2 Rχ2 (4) where χ1 = ∫ b a ω(s)ds and χ2 =χ1 − 2b−a ∫ b a ∫ s a ω(u)duds....
[...]
...Lemma 2: ([7], [27]) For given vectors β1 and β2 , scalar α in the interval (0, 1), symmetric positive definite matrix R ∈ Rn×n and any matrix X ∈ Rn×n satisfying [ R X ∗ R ] ≥ 0, the following inequality holds...
[...]

Proceedings Article•DOI•

An integral inequality in the stability problem of time-delay systems

[...]

Keqin Gu¹•Institutions (1)

Southern Illinois University Edwardsville¹

12 Dec 2000

TL;DR: In this paper, an integral inequality is derived, and applied to the stability problem of time-delay systems using discretized Lyapunov functional formulation, and a simpler stability criterion is derived.

...read moreread less

Abstract: An integral inequality is derived, and applied to the stability problem of time-delay systems using discretized Lyapunov functional formulation. As the result, a simpler stability criterion is derived.

...read moreread less

1,521 citations

"Notes on Stability of Time-Delay Sy..." refers background or methods in this paper

...Lemma 1: (Jensen Inequality [15] and Wirtinger-Based Inequality [7]): For given symmetric positive definite matrix R ∈ Rn×n , scalars a and b satisfying a < b, and vector ω : [a, b] → Rn such that the integrations concerned are well defined, the following inequalities hold...
[...]
...Lemma 1: (Jensen Inequality [15] and Wirtinger-Based Inequality [7]): For given symmetric positive definite matrix R ∈ Rn×n , scalars a and b satisfying a b, and vector ω : [a, b] → Rn such that the integrations concerned are well defined, the following inequalities hold (b − a) ∫ b a ωT (s)Rω(s)ds ≥ χT1 Rχ1 (3) (b − a) ∫ b a ωT (s)Rω(s)ds ≥ χT1 Rχ1 + 3χT2 Rχ2 (4) where χ1 = ∫ b a ω(s)ds and χ2 =χ1 − 2b−a ∫ b a ∫ s a ω(u)duds....
[...]
...An alternative type of method is to directly estimate the integral term via bounding inequalities, among which Jensen inequality (JI) [15] and Wirtinger-based inequality (WBI) [7] are frequently applied....
[...]

Journal Article•DOI•

Technical Communique: Delay-dependent criteria for robust stability of time-varying delay systems

[...]

Min Wu¹, Yong He¹, Jinhua She², Guo-Ping Liu³•Institutions (3)

Central South University¹, Tokyo University of Technology², Chinese Academy of Sciences³

01 Aug 2004-Automatica

TL;DR: Some new delay-dependent stability criteria are devised by taking the relationship between the terms in the Leibniz-Newton formula into account, which are less conservative than existing ones.

...read moreread less

1,069 citations

"Notes on Stability of Time-Delay Sy..." refers methods in this paper

...Model transformations [8]–[10] and various free-weighting-matrix approach [11]–[14] are applied to estimate the integral term in the early research....
[...]

Journal Article•DOI•

Technical communique: Delay-range-dependent stability for systems with time-varying delay

[...]

Yong He¹, Qing-Guo Wang¹, Chong Lin¹, Min Wu²•Institutions (2)

National University of Singapore¹, Central South University²

01 Feb 2007-Automatica

TL;DR: The present results may improve the existing ones due to a method to estimate the upper bound of the derivative of Lyapunov functional without ignoring some useful terms and the introduction of additional terms into the proposed Lyap unov functional, which take into account the range of delay.

...read moreread less

899 citations