Stability analysis of systems with time-varying delays via the second-order Bessel–Legendre inequality

Liu, Kun; Seuret, Alexandre; Xia, Yuanqing

doi:10.1016/j.automatica.2016.11.001

Cited by 233 publications

(150 citation statements)

References 12 publications

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“…Remark As did in the work of Seuret et al, since we are now considering an extensional version of the Wirtinger inequality (ie, second‐order Bessel‐Legendre inequality), the term η T ( t ) Pη ( t ) also needs to contain the two additional signals

\int_{t - d_{1}}^{0} λ_{- d_{1}, 0} false(s false) x false(t + s false) d s

and

\int_{- d_{2}}^{- d_{1}} λ_{- d_{2}, - d_{1}} false(s false) x false(t + s false) d s

. A similar augmented state vector was considered in the work of Liu et al and has demonstrated its potential strength in conservatism reduction. Four triple integrals that are not taken into account in other works are also introduced into the L‐K functional to acquire further less conservative results.…”

Section: Resultsmentioning

confidence: 99%

“…A similar augmented state vector was considered in the work of Liu et al and has demonstrated its potential strength in conservatism reduction. Four triple integrals that are not taken into account in other works are also introduced into the L‐K functional to acquire further less conservative results. In addition, the B‐L inequality that encompasses the Wirtinger inequality tighter than Jensen inequality is fully exploited during estimating the single integral terms from time derivative of L‐K functional in this paper.…”

Section: Resultsmentioning

confidence: 99%

“…There also exist the “Input‐to‐State” or “Input‐to‐Output” approaches, which are related to the robust analysis . In recent years, many scholars have been paying attention to systems with interval time‐varying delay d ( t ) ∈ [ d 1 , d 2 ] . Concurrently, some articles have exploited full information on both the lower and upper bounds of

\overset{d}{true} false(t false)

of unequal variation for positive and negative directions as

μ_{1} \leq \overset{d}{true} false(t false) \leq μ_{2}

to deduce a delay‐derivative‐dependent stability criterion …”

Section: Introductionmentioning

confidence: 99%

“…Liu et al have proposed more accurate conditions using the second‐order B‐L inequality. Unfortunately, there is no restriction on the derivative of the delay function in the works of Park et al and Liu et al, and the only linear convex combination property on d ( t ) enables them to handle the resultant LMIs. In addition, the inequalities in the works of Liu et al Kim only deal with single integral terms of quadratic functions while upper bounds of double integral terms are not estimated because there are no triple integral terms introduced into the LKF to reduce the conservatism.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Delay range‐and‐rate dependent stability criteria for systems with interval time‐varying delay via a quasi‐quadratic convex framework

Yang

Kang

2019

Intl J Robust & Nonlinear

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Summary This paper concerns the stability analysis of systems with interval time‐varying delay. A Lyapunov‐Krasovskii functional containing an augmented quadratic term and certain triple integral terms is constructed to integrate features of the truncated Bessel‐Legendre inequality less conservative than Wirtinger inequality that encompasses Jensen inequality, respectively, and to exploit merits of the newly developed double integral inequalities tighter than auxiliary function‐based, Wirtinger, and Jensen double integral inequalities. A new quadratic convex lemma is proposed to derive delay and its derivative dependent sufficient stability conditions in terms of linear matrix inequalities synthetically with reciprocal convex approach and affine convex combination. The efficiency of the presented method is illustrated on some classical numerical examples.

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\int_{t - d_{1}}^{0} λ_{- d_{1}, 0} false(s false) x false(t + s false) d s

and

\int_{- d_{2}}^{- d_{1}} λ_{- d_{2}, - d_{1}} false(s false) x false(t + s false) d s

Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

\overset{d}{true} false(t false)

of unequal variation for positive and negative directions as

μ_{1} \leq \overset{d}{true} false(t false) \leq μ_{2}

to deduce a delay‐derivative‐dependent stability criterion …”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Delay range‐and‐rate dependent stability criteria for systems with interval time‐varying delay via a quasi‐quadratic convex framework

Yang

Kang

2019

Intl J Robust & Nonlinear

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show abstract

“…On the other hand, time delay usually exists in many practical dynamical systems, and the influence on system performance of time delay cannot be ignored. Many researchers have been devoting to obtaining less conservative delay‐dependent results, and there have been a number of methods concerned with time‐delay systems; see, eg, other works . Recently, the study of time‐delay SMJSs has been extensively considered.…”

Section: Introductionmentioning

confidence: 99%

Reliable exponential filtering for singular Markovian jump systems with time‐varying delays and sensor failures

Liu

Park

et al. 2018

Intl J Robust & Nonlinear

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Summary This paper deals with the problem of exponential H∞ filtering for singular Markovian jump systems with time‐varying delays subject to sensor failures. The main objective is to design a reliable filtering such that the considered filtering error system in the presence of a time‐varying delay and sensor failures is mean‐square exponentially admissible with a specified decay rate and simultaneously satisfies an H∞ performance. First, the delay interval is partitioned into m subintervals and a novel mode‐dependent stochastic Lyapunov‐Krasovskii functional is constructed. By using the reciprocally convex inequality in each subinterval, sufficient conditions of exponential H∞ performance analysis are developed for the considered filtering error system. Then, based on these conditions, the existence conditions of the desired reliable filter are derived and the filter parameters are obtained. It should be mentioned that all the results presented here are not only dependent on the time delay but also dependent on the decay rate and the partitioning size. Furthermore, all the conditions are established in terms of strict linear matrix inequalities. Finally, two numerical examples are given to illustrate the less conservatism and effectiveness of the proposed methods.

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Delay‐dependent conditions for finite time stability of linear time‐varying systems with delay

Chen

Sun

2019

Asian Journal of Control

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This paper investigates the problem of finite time stability of linear time‐varying system with delay. By constructing an augmented time‐varying Lyapunov functional and using the Wirtinger‐type inequality deductively, delay‐dependent finite time stability conditions are derived and presented in terms of differential linear matrix inequalities (DLMIs). Then, the DLMIs are transformed into a series of recursive linear matrix inequalities (RLMIs) by discretizing the time interval into equally spaced time distances, and an algorithm is given to solve the RLMIs. Examples illustrate the feasibility and effectiveness of the proposed method.

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Stability analysis of systems with time-varying delays via the second-order Bessel–Legendre inequality

Cited by 233 publications

References 12 publications

Delay range‐and‐rate dependent stability criteria for systems with interval time‐varying delay via a quasi‐quadratic convex framework

Delay range‐and‐rate dependent stability criteria for systems with interval time‐varying delay via a quasi‐quadratic convex framework

Reliable exponential filtering for singular Markovian jump systems with time‐varying delays and sensor failures

Delay‐dependent conditions for finite time stability of linear time‐varying systems with delay

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