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

In the last decade, the Jensen inequality has been intensively used in the context of time-delay or sampled-data systems since it is an appropriate tool to derive tractable stability conditions expressed in terms of linear matrix inequalities (LMIs). However, it is also well-known that this inequality introduces an undesirable conservatism in the stability conditions and looking at the literature, reducing this gap is a relevant issue and always an open problem. In this paper, we propose an alternative inequality based on the Fourier Theory, more precisely on the Wirtinger inequalities. It is shown that this resulting inequality encompasses the Jensen one and also leads to tractable LMI conditions. In order to illustrate the potential gain of employing this new inequality with respect to the Jensen one, two applications on time-delay and sampled-data stability analysis are provided.     

Some generalizations and improvements of discrete Hardy’s inequality

In this paper, by using Jensen’s inequality and Hadamard’s integral inequality for a convex function, some generalizations and improvements of discrete Hardy’s inequality and Carleman’s inequality are obtained.     

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

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.     

常用基本不等式的机器证明

不等式机器证明问题是智能系统领域的难点和热点问题.借助不等式证明软件BOTTEMA,对若干常用的基本不等式成功地实现了机器证明,包括算术、几何与调和平均不等式、排序不等式、Chebyshev不等式、Bernoulli不等式、三角形不等式及Jensen不等式等.所论不等式含有的变元个数是一个不确定的变量,属于Tarski模型外的不等式类型.机器证明得出的结论有时可能是已知结果的推广,其方法本身对同类不等式有示范性,更多的例子表明了该算法和软件的有效性.     

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

Whereas the upper bound lemma for matrix cross-product, introduced by Park (1999) and modified by Moon, Park, Kwon, and Lee (2001), plays a key role in guiding various delay-dependent criteria for delayed systems, the Jensen inequality has become an alternative as a way of reducing the number of decision variables. It directly relaxes the integral term of quadratic quantities into the quadratic term of the integral quantities, resulting in a linear combination of positive functions weighted by the inverses of convex parameters. 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 those based on the Jensen inequality lemma.     

Improved Stability Results for Stochastic Cohen–Grossberg Neural Networks with Discrete and Distributed Delays

This paper is concerned with the exponential stability problem for a class of stochastic Cohen–Grossberg neural networks with discrete and unbounded distributed time delays. By applying the Jensen integral inequality and the generalized Jensen integral inequality, several improved delay-dependent criteria are developed to achieve the exponential stability in mean square in terms of linear matrix inequalities. It is established theoretically that two special cases of the obtained criteria are less conservative than some existing results but including fewer slack variables. As the present conditions involve fewer free weighting matrices, the computational burden is largely reduced. Three numerical examples are provided to demonstrate the effectiveness of the theoretical results.     

Bessel summation inequalities for stability analysis of discrete‐time systems with time‐varying delays

This paper is concerned with the stability analysis problems of discrete‐time systems with time‐varying delays using summation inequalities. In the literature focusing on the Lyapunov‐Krasovskii approach, the Jensen integral/summation inequalities have played important roles to develop less conservative stability criteria and thus have been widely studied. Recently, the Jensen integral inequality was successfully generalized to the Bessel‐Legendre inequalities constructed with arbitrary‐order Legendre polynomials. It was also shown that general inequality contributes to the less conservatism of stability criteria. In the case of discrete‐time systems, however, the Jensen summation inequality are hardly extensible to general ones since there have still not been general discrete orthogonal polynomials applicable to the developments of summation inequalities. Motivated by such observations, this paper proposes novel discrete orthogonal polynomials and then successfully derives general summation inequalities. The resulting summation inequalities are discrete‐time counterparts of the Bessel‐Legendre inequalities but are not based on the discrete Legendre polynomials. By developing hierarchical stability criteria based on the proposed summation inequalities, the effectiveness of the proposed approaches is demonstrated via three numerical examples for the stability analysis of discrete‐time systems with time‐varying delays.     

A Delay-partitioning Approach to the Stability Analysis of 2-D Linear Discrete-time Systems with Interval Time-varying Delays

International Journal of Control, Automation and Systems - Two recent Lyapunov-based methods: delay-partitioning approach and Jensen inequality approach, have reduced the conservatism and the...     

New stability criteria for neutral-type Cohen–Grossberg neural networks with discrete and distributed delays

This paper studies the existence, uniqueness and globally robust exponential stability for a class of uncertain neutral-type Cohen–Grossberg neural networks with time-varying and unbounded distributed delays. Based on Lyapunov–Krasovskii functional, by involving a free-weighting matrix, using the homeomorphism mapping principle, Cauchy–Schwarz inequality, Jensen integral inequality, linear matrix inequality techniques and matrix decomposition method, several delay-dependent and delay-independent sufficient conditions are obtained for the robust exponential stability of considered neural networks. Two numerical examples are given to show the effectiveness of our results.     

时变时滞离散广义Markov 跳变系统的鲁棒稳定性

研究一类具有区间时变时滞的离散不确定广义Markov跳变系统的时滞相关鲁棒稳定性问题.通过将Jensen不等式与一个新的定界不等式相结合,得到了一个新的稳定性判据,该判据中仅含有Lyapunov变量,具有较小的计算负担.进而,基于凸组合方法得到了另一个新的稳定性判据,该判据引入了一些自由矩阵变量,具有较小的保守性.数值算例表明了所提出方法的有效性.     

Robust H ∞ model reference tracking control for networked control systems with communication constraints

This paper studies the problem of H _∞ model reference tracking control for continuous time networked control systems (NCSs) with communication constraints. By using the continuous Jensen inequality, linear matrix inequality (LMI)-based H _∞ model reference tracking controller design for nominal NCSs with controller-to-actuator communication constraints is presented, and a new method is proposed to design H _∞ model reference tracking controllers for NCSs with both controller-toactuator and sensor-to-controller communication constraints. The results are also extended to NCSs with norm-bounded parameter uncertainties. The merits of the proposed methods lie in their less computational complexity and less conservatism, which are achieved by adopting continuous Jensen inequality and proposing new controller design methods, respectively. The simulation results illustrate the effectiveness of the proposed H _∞ model reference tracking controller design for NCSs with communication constraints.     

Refined discretized Lyapunov functional method for systems with multiple delays

The previously proposed discretized Lyapunov functional method for systems with multiple delay is refined using variable elimination and Jensen inequality. The resulting new stability criterion is simpler. Numerical examples indicate that the new method is much less conservative for a given discretization mesh. Copyright © 2003 John Wiley & Sons, Ltd.     

An improved approach to delay-dependent robust stabilization for uncertain singular time-delay systems

In this paper, delay-dependent stability analysis and robust stabilization for uncertain singular time-delay systems are addressed. By using Jensen integral inequality, an improved delay-dependent criterion of admissibility for singular time-delay systems is proposed in terms of linear matrix inequality (LMI). Our new proposed criterion is less conservative and the numerical complexity is smaller than the existing ones. Based on this criterion, a state feedback controller is designed to ensure that the uncertain singular time-delay system is admissible. Finally, three numerical examples are employed to illustrate the effectiveness of the proposed method.     

New Result on Delay-dependent Stability for Markovian Jump Time-delay Systems With Partial Information on Transition Probabilities

This paper focuses on the delay-dependent stability for a kind of Markovian jump time-delay systems (MJTDSs), whose transition rates are incompletely known. In order to reduce the computational complexity and achieve better performance, auxiliary function-based double integral inequality is combined with extended Wirtinger's inequality and Jensen inequality to deal with the double integral and the triple integral in augmented Lyapunov-Krasovskii function (ALKF) and their weak infinitesimal generator respectively, the more accurate approximation bounds with a fewer variables are derived. As a result, less conservative stability criteria are proposed in this paper. Finally, numerical examples are given to show the effectiveness and the merits of the proposed method.      

On the triangle inequalities in fuzzy metric spaces

This paper presents level forms of the triangle inequalities in fuzzy metric spaces (X, d, L, R). To aid discussion, a fuzzy pre-metric condition is introduced. It is first pointed out that under the fuzzy pre-metric condition the first triangle inequality is always equivalent to its level form. The second triangle inequality is equivalent to one level form when R is right continuous, and to another level form also when further conditions are imposed on R. In a fuzzy metric space, the level form of the first triangle inequality and one of the level forms of the second triangle inequality are always valid. The other level form of the second triangle inequality holds for all but at most countable α ∈ [0, 1). Finally, a fixed point theorem for fuzzy metric spaces is derived as an application of the preceding results.     

New results on stability analysis of singular time-delay systems

This paper deals with the problem of delay-dependent stability analysis for singular systems with a constant time delay. By employing the Jensen inequality and the Wirtinger-based inequality, new delay-dependent stability criteria are developed. Furthermore, in order to obtain less conservative results, a new method of constructing Lyapunov--Krasovskii functionals (LKFs) is used. It should be pointed out that the positive-definiteness restrictions on some symmetric matrices of the LKFs are removed. All the criteria are proposed in terms of strict linear matrix inequalities. Finally, numerical examples are given to demonstrate the less conservatism of the results.     

Exponential Stability of Semi-Markovian Switching Complex Dynamical Networks with Mixed Time Varying Delays and Impulse Control

This study examines the problem of exponential stability of complex dynamical networks with impulse control and semi-Markovian switching parameters. By utilizing a supplementary variable technique and a plant transformation, the semi-Markovian switching complex dynamical networks can be equivalently expressed as its associated Markovian switching complex dynamical networks. By applying the Lyapunov stability theory, Jensen’s inequality, Dynkins formula, Schur complement and linear matrix inequality technique, some new delay-dependent conditions are derived to guarantee the exponential stability of the equilibrium point. Finally, a numerical example is given to illustrate the feasibility and effectiveness of the results obtained.     

基于直觉模糊粗糙集的属性约简

针对Jensen下近似定义的局限性,提出一种新的等价类形式的近似算子表示,并将其推广到直觉模糊环境.在此基础上,将相对正域、相对约简、相对核等粗糙集的知识约简概念推广到直觉模糊环境,提出一种直觉模糊信息系统的启发式属性约筒算法.实例计算表明.该方法比Jensen的属性约简方法更为合理有效.     

Two general integral inequalities and their applications to stability analysis for systems with time‐varying delay

Integral inequalities have been widely used in stability analysis for systems with time‐varying delay because they directly produce bounds for integral terms with respect to quadratic functions. This paper presents two general integral inequalities from which almost all of the existing integral inequalities can be obtained, such as Jensen inequality, the Wirtinger‐based inequality, the Bessel–Legendre inequality, the Wirtinger‐based double integral inequality, and the auxiliary function‐based integral inequalities. Based on orthogonal polynomials defined in different inner spaces, various concrete single/multiple integral inequalities are obtained. They can produce more accurate bounds with more orthogonal polynomials considered. To show the effectiveness of the new inequalities, their applications to stability analysis for systems with time‐varying delay are demonstrated with two numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.     

一类时滞切换系统的稳定性分析

利用时滞分解和平均驻留时间方法讨论一类时滞切换系统的稳定性问题.定义了更为一般的Lyapunov函数,结合Jensen积分不等式和倒数凸组合技术得到的线性矩阵不等式条件具有更小的保守性和更低的计算复杂性给出的仿真算例进一步验证了所得结果的有效性.