A Novel Generalized Integral Inequality Based on Free Matrices for Stability Analysis of Time-Varying Delay Systems

Abstract

This paper proposes a novel generalized integral inequality based on free matrices and applies it to stability analysis of time-varying delay systems. The proposed integral inequality estimates the upper bound of the augmented quadratic term of the state and its derivative term by utilizing not only the single integral term but also the higher-order multiple integral terms. The proposed integral inequality includes several well-known integral inequalities as special cases. For the stability analysis of time-varying delay systems, a new Lyapunov-Krasovskii functional is constructed by including the double integral term with the augmented vector of the state and its derivative to utilize the proposed integral inequality when estimating the derivative of the Lyapunov-Krasovskii functional. Furthermore, to fully exploit the information on the time-varying delay, this paper divides the interval of the double integral term into two parts. Two numerical examples show that the results of the proposed method outperform those of the existing methods. INDEX TERMS Stability analysis, Time-varying delays, Lyapunov-Krasovskii functional, Free matrices, Generalized integral inequality.

An Affine Integral Inequality of an Arbitrary Degree for Stability Analysis of Linear Systems With Time-Varying Delays

“…Remark 1: This remark discusses the relation among the proposed inequality and the existing inequalities in the literature [11], [12], [30], [33], [35], [36]. Compared to the inequalities [35], [36], the proposed integral inequality is constructed by fully utilizing Legendre polynomials and thus provides the upper bound of an arbitrary degree N . When N = 1, 2, the proposed inequality reduces to the freematrix based integral inequalities of [35], [36], respectively.…”

“…Compared to the inequalities [35], [36], the proposed integral inequality is constructed by fully utilizing Legendre polynomials and thus provides the upper bound of an arbitrary degree N . When N = 1, 2, the proposed inequality reduces to the freematrix based integral inequalities of [35], [36], respectively. In [11], it was shown that an orthogonal-polynomials-based integral inequality [11] is more general than the B-L inequality [30] and affine B-L inequality [12].…”

“…A common feature of theses integral inequalities is that the integral quadratic function only consists of a system state or its derivative, separately. Recently, new free-matrix-based integral inequalities [35], [36] for integral quadratic functions containing both a system state and its time derivative have been proposed. In these papers, with a construction of LKF containing correlated integral terms of a system state and its derivative, the new free-matrix-based integral inequalities have effectively reduced the conservatism of stability criteria.…”

“…In these papers, with a construction of LKF containing correlated integral terms of a system state and its derivative, the new free-matrix-based integral inequalities have effectively reduced the conservatism of stability criteria. However, since the integral inequalities proposed in [35], [36] are constructed only with use of zero-, first-, and second-degree orthogonal polynomials, there still exists a room for improvement. Such observations motivate our work.…”

“…Further, a note on the relation among exiting inequalities and the proposed one is provided for effective comparisons. In this note, it is shown that the proposed integral inequality is a generalized version of the free-matrix-based integral inequalities [35], [36] and the orthogonal polynomialsbased integral inequalities [11], [33]. Further, it is proven that increasing a degree of the proposed integral inequality only reduces a bounding gap between an integral quadratic function and its affine upper bounds.…”