New Free-Matrix-Based Integral Inequality: Application to Stability Analysis of Systems With Additive Time-Varying Delays

Abstract

This paper is concerned with the problem of stability analysis for systems with additive time-varying delays (ATDs). This paper proposes a new free-matrix-based integral inequality that provides estimate of the energy of the vector that contains the state and its derivative at the same time. Consequently, the proposed inequality enables the Lyapunov-Krasovskii functional (LKF) to take into account not only the respective energies of the state and its derivatives but also the correlated effect of them. Then, based on the proposed inequality, this paper derives a new stability criterion of systems with ATDs. This paper constructs LKF by proposing new sets of multiple subintervals of the ATDs, and makes use of the proposed inequality when estimating the derivatives of the LKF. This allows the resulting stability criterion to take full advantage of the information of the multiple subintervals of the ATDs. This paper also successfully applies the proposed free-matrix-based integral inequality to the system with a single time-varying delay. Three numerical examples demonstrate the effectiveness of the proposed methods.

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

“…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.…”

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

“…However, the usage of the augmented vectors is only limited to the quadratic forms of the single integral terms and constant terms in the LKFs, and the double integral terms have been only treated with a single state derivative term. Recently, the LKF which uses state and its derivative term in a double integral is proposed by using a new free-matrixbased integral inequality in [9]. The usage of additional energy from the state term and the cross-term of state and its derivative effectively improve conservatism.…”

“…The proposed GIIBFM 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 GIIBFM includes several famous existing integral inequalities as special cases such as the generalized free-matrix-based integral inequality [20] and the new free-matrix-based integral inequality [9]. For the stability analysis of TVD systems, a new LKF is constructed by including the double integral term with the augmented vector of the state and its derivative to utilize the proposed GIIBFM when estimating the derivative of the LKF.…”