Discrete Wirtinger-based inequality and its application

Nam, Phan Thanh; Pathirana, Pubudu N.; Trinh, Hieu

doi:10.1016/j.jfranklin.2015.02.004

Cited by 163 publications

(103 citation statements)

References 42 publications

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“…One can see from Table 1 that the results provided by Theorem 1 Ã coincide with those obtained from [16], where Theorem 1 Ã denotes Theorem 1 with P 41 ¼ P 31 , P 42 ¼ P 32 , P 43 ¼ 0 and P 44 ¼ P 33 ; the results are obtained from Theorem 1 in this paper are better than those in [3,6,10,16,17,25] with relatively less demand of computation burden, and some improvements over [10] Table 2 gives the results of the two controller designs under the lower delay bound d 1 ¼ 1: one is based on the method given by [5], and the other is based on Theorem 3 of this paper. It can be seen from Table 2 that the results provided in this paper are less conservative than those by [5].…”

Section: Numerical Examplessupporting

confidence: 63%

Improved Stability and Stabilization Criteria for Uncertain T–S Fuzzy Systems with Interval Time-Varying Delay via Discrete Wirtinger-Based Inequality

Zhang

Peng

et al. 2015

Int. J. Fuzzy Syst.

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This paper introduces a discrete Wirtingerbased inequality to investigate the problem of delay-dependent stability and stabilization of uncertain T-S fuzzy systems with interval time-varying delays. By constructing a novel Lyapunov functional and applying the discrete Wirtinger-based inequality to deal with the sum terms in the derivation of the results, two delay-dependent stability criteria and a stabilization criterion are obtained in terms of matrix inequalities. Numerical examples show that the derived stability and stabilization criteria can provide a larger allowable upper delay bound than some existing results while depending on relatively less scalar decision variables.

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Section: Numerical Examplessupporting

confidence: 63%

Improved Stability and Stabilization Criteria for Uncertain T–S Fuzzy Systems with Interval Time-Varying Delay via Discrete Wirtinger-Based Inequality

Zhang

Peng

et al. 2015

Int. J. Fuzzy Syst.

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“…The LyapunovKrasovskii functional is chosen which utilizes more information about the time delays and a new set of conditions is added to calculate the difference of the designed LyapunovKrasovskii functional. Furthermore, as mentioned in [37], it is clearly seen that the Wirtinger-based inequality can reduce the conservatism and provide a more tighter lower delay bound of the summation terms. Therefore, we revisit the nonfragile state estimation problem for discrete GRNs with time-varying delays and randomly occurring uncertainties by using the discrete-time Wirtinger-based inequality and the reciprocally convex combination inequality.…”

Section: Introductionmentioning

confidence: 80%

Design of Nonfragile State Estimator for Discrete-Time Genetic Regulatory Networks Subject to Randomly Occurring Uncertainties and Time-Varying Delays

2017

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We deal with the design problem of nonfragile state estimator for discrete-time genetic regulatory networks (GRNs) with timevarying delays and randomly occurring uncertainties. In particular, the norm-bounded uncertainties enter into the GRNs in random ways in order to reflect the characteristic of the modelling errors, and the so-called randomly occurring uncertainties are characterized by certain mutually independent random variables obeying the Bernoulli distribution. The focus of the paper is on developing a new nonfragile state estimation method to estimate the concentrations of the mRNA and the protein for considered uncertain delayed GRNs, where the randomly occurring estimator gain perturbations are allowed. By constructing a LyapunovKrasovskii functional, a delay-dependent criterion is obtained in terms of linear matrix inequalities (LMIs) by properly using the discrete-time Wirtinger-based inequality and reciprocally convex combination approach as well as the free-weighting matrix method. It is shown that the proposed method ensures that the estimation error dynamics is globally asymptotically stable and the desired estimator parameter is designed via the solutions to certain LMIs. Finally, we provide two numerical examples to illustrate the feasibility and validity of the proposed estimation results.

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“…On the other hand, time delays and unknown inputs are inherent issues in many dynamical systems [2,8,10,16,20,21,22,25,26,33]. The presence of delays and disturbances often causes unsatisfactory performance and instability of dynamical systems.…”

Section: Introductionmentioning

confidence: 99%

“…In [18], Ibrir, Xie, and Su designed observers for a class of Lipschitz discrete-time systems. To deal with the time-delay issue, we utilize the recently developed discrete Wirtinger-based inequality [21,25], which encompasses the wellknown Jensen inequality [10], and the reciprocally convex approach [22] to derive less conservative conditions. In [34], Zemouche and Boutayeb presented both full-and reduced-order observer designs for a class of Lipschitz nonlinear discrete-time systems.…”

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confidence: 99%

Observer Design for One-Sided Lipschitz Discrete-Time Systems Subject to Delays and Unknown Inputs

Nguyen¹,

Trinh²

2016

SIAM J. Control Optim.

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In this paper, we address the problem of observer design for a class of nonlinear discrete-time systems in the presence of delays and unknown inputs. The nonlinearities studied in this work satisfy the one-sided Lipschitz and quadratically inner-bounded conditions which are more general than the traditional Lipschitz conditions. Both H∞ observer design and asymptotic observer design with reduced-order are considered. The designs are novel compared to other relevant nonlinear observer designs subject to time delays and disturbances in the literature. In order to deal with the time-delay issue as well as the bilinear terms which usually appear in the problem of designing observers for discrete-time systems, several mathematical techniques are utilized to deduce observer synthesis conditions in the linear matrix inequalities form. A numerical example is given to demonstrate the effectiveness and high performance of our results. MINH CUONG NGUYEN AND HIEU TRINHis of both theoretical and practical importance. UIO designs can be employed in many applications, e.g., robust fault detection and isolation schemes [29]. In order to design UIOs, the H ∞ filtering approach [3,13,14,24,37] provides a robust solution when the UIO matching condition [8,16] is not satisfied.Since the pioneering works by Thau [27] and Rajamani [23], observer designs for Lipschitz nonlinear systems have been widely studied. In recent years, designing observers for Lipschitz systems subject to delays or disturbances has received considerable attention from researchers. In [28], Trinh, Aldeen, and Nahavandi considered a class of uncertain nonlinear time-delay systems where the delayed portion was decomposed into matched and mismatched parts. In [11], Ha and Trinh estimated simultaneously the states and inputs of Lipschitz nonlinear systems. In [18], Ibrir, Xie, and Su designed observers for a class of Lipschitz discrete-time systems. Later on, the authors extended their work by adding delays into the nonlinear discrete-time systems in [19]. In [34], Zemouche and Boutayeb presented both full-and reduced-order observer designs for a class of Lipschitz nonlinear discrete-time systems. The same authors introduced an observer design for a class of Lipschitz discrete-time delayed systems with extension to H ∞ performance analysis in [35]. In [24], Sayyaddelshad and Gustafsson presented a robust H ∞ observer for a class of uncertain nonlinear discrete time-delay systems with multiobjective optimization, i.e., maximizing the Lipschitz constant and minimizing the disturbance attenuation level.Nevertheless, nonlinear observer designs based on the traditional Lipschitz conditions expose a major limitation, which is the inability to deal with large-Lipschitzconstant systems. One solution to overcome this drawback is to reformulate the conventional Lipschitz conditions so that all the properties of the nonlinearities can be taken into account [14,36]. Another solution is to use more general conditions, which include the well-known Lipschitz condition as a special...

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Discrete Wirtinger-based inequality and its application

Cited by 163 publications

References 42 publications

Improved Stability and Stabilization Criteria for Uncertain T–S Fuzzy Systems with Interval Time-Varying Delay via Discrete Wirtinger-Based Inequality

Improved Stability and Stabilization Criteria for Uncertain T–S Fuzzy Systems with Interval Time-Varying Delay via Discrete Wirtinger-Based Inequality

Design of Nonfragile State Estimator for Discrete-Time Genetic Regulatory Networks Subject to Randomly Occurring Uncertainties and Time-Varying Delays

Observer Design for One-Sided Lipschitz Discrete-Time Systems Subject to Delays and Unknown Inputs

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