Seok Young Lee

Auxiliary function-based integral inequalities for quadratic functions and their applications to time-delay systems

Abstract Finding integral inequalities for quadratic functions plays a key role in the field of stability analysis. In such circumstances, the Jensen inequality has become a powerful mathematical tool for stability analysis of time-delay systems. This paper suggests a new class of integral inequalities for quadratic functions via intermediate terms called auxiliary functions, which produce more tighter bounds than what the Jensen inequality produces. To show the strength of the new inequalities, their applications to stability analysis for time-delay systems are given with numerical examples.

Improved criteria on robust stability and H ∞ performance for linear systems with interval time-varying delays via new triple integral functionals

This paper analyzes delay-dependent robust stability and H~ performance of linear systems with an interval time-varying delay, based on a new Lyapunov-Krasovskii functional containing new triple integral terms. The time derivative of the Lyapunov-Krasovskii functional produces not only the strictly proper rational functions but also the non-strictly proper rational functions of the time-varying delays with first-order denominators. The combinations of the rational functions are directly handled via the Jensen inequality lemma and the lower bound lemma for reciprocal convexity, whereas such combinations were approximated in the literature. The proposed criteria become less conservative with the significantly smaller number of decision variables than the existing criteria, which will be demonstrated by some numerical examples.

Improved stability criteria for recurrent neural networks with interval time-varying delays via new Lyapunov functionals

This paper considers the stability problem of recurrent neural networks with interval time-varying delays. Based on a new augmented Lyapunov-Krasovskii functional that contains four triple integral terms and additional terms obtained from the activation function condition, a stability condition is derived in terms of linear matrix inequalities (LMIs). Also, a further improved stability criterion is derived by bounding the derivative of a special case of the proposed Lyapunov-Krasovskii functional based on a new inequality proposed in Seuret and Gouaisbaut (2013) 27]. A numerical example shows the improvement of the proposed approach both in terms of computational complexity and conservatism.

Improved stability criteria for linear systems with interval time-varying delays

This paper suggests first-order and second-order generalized zero equalities and constructs a new flexible Lyapunov-Krasovskii functional with more state terms. Also, by applying various zero equalities, improved stability criteria of linear systems with interval time-varying delays are developed. Using Wirtinger-based integral inequality, Jensen inequality and a lower bound lemma, the time derivative of the Lyapunov-Krasovskii functional is bounded by the combinations of various state terms including not only integral terms but also their interval-normalized versions, which contributes to make the stability criteria less conservative. Numerical examples show the improved performance of the criteria in terms of maximum delay bounds.

Polynomials-based integral inequality for stability analysis of linear systems with time-varying delays

This paper employs polynomial functions to extend a free-matrix-based integral inequality into a general integral inequality, say a polynomials-based integral inequality, which also contains well-known integral inequalities as special cases. By specially designing slack matrices and an arbitrary vector containing state terms, it reduces to an extended version of Wirtinger-based integral inequality or the free-matrix-based integral inequality. Numerical examples for stability analysis of linear systems with interval time-varying delays show the improved performance of the proposed integral inequality in terms of maximum delay bounds and numbers of variables.

A combined reciprocal convexity approach for stability analysis of static neural networks with interval time-varying delays

This paper proposes a novel approach called a combined reciprocal convexity approach for the stability analysis of static neural networks with interval time-varying delays. The proposed approach deals with all convex-parameter-dependent terms in the time derivative of the Lyapunov-Krasovskii functional non-conservatively by extending the idea of the conventional reciprocal convexity approach. Based on the proposed technique and a new Lyapunov-Krasovskii functional, two improved delay-dependent stability criteria are derived in terms of linear matrix inequalities(LMIs). Some numerical examples are given to demonstrate the proposed results.

A combined first- and second-order reciprocal convexity approach for stability analysis of systems with interval time-varying delays

Abstract For the interval time-varying delay systems, Jensen inequality lemma yields some terms with the inverse of convex and squared convex parameters, which often makes it difficult to find their bounds. Recently, reciprocal and second-order reciprocal convexity approaches have been proposed to handle the difficulties with a set of convex parameters and a set of squared convex parameters, respectively, but these approaches do not investigate the relation between the two sets. This paper offers much tighter bounds of those terms utilizing the relations among the two sets with the lower bound lemma. Based on the new approach and Lyapunov theory, less conservative stability criteria for time delay systems are developed. To show the effectiveness of the new approach, two numerical examples are given.

Penetration depth anisotropy in YBa2(Cu1−yMy)3O7−x (M=Zn.Ni) oriented powders by μSR

Abstract Penetrations depth is measured in YBa2(Cu1−xZnx)3O7 and YBa2(Cu1−yNiy)3O7−x oriented powders by Muon Spin Rotation. The observed anisotropy in the effective mass tensor and the relation between Tc and the μSR damping σ point to the existence of additional contributions to the depression of the superconductivity that cannot be accounted for in simple terms of carrier concentration.

Affine Bessel–Legendre inequality: Application to stability analysis for systems with time-varying delays

Abstract Recently, some novel inequalities have been proposed such as the auxiliary function-based integral inequality and the Bessel–Legendre inequality which can be obtained from the former by choosing Legendre polynomials as auxiliary functions. These inequalities have been successfully applied to systems with constant delays but there have been some difficulties in application to systems with time-varying delays since the resulting bounds contain the reciprocal convexity which may not be tractable as it is. This paper proposes an equivalent form of the Bessel–Legendre inequality, which has the advantage of being easily applied to systems with time-varying delays without the reciprocal convexity.

New stability analysis for discrete time-delay systems via auxiliary-function-based summation inequalities

Abstract For the stability analysis of discrete time-delay systems, Jensen inequality has been widely used as the method supporting inequalities for summation quadratic functions. It not only requires a smaller number of decision variables than other approaches but also achieves identical or comparable performance behavior. Based on the analysis for the conservatism of Jensen inequality, however, this paper suggests a new summation inequality say an auxiliary-function-based summation inequality. It is verified that the proposed inequality is a generalized form of the novel summation inequality reported recently. Also, an application to stability analysis for discrete time-delay systems is provided.