Stability Analysis of Integral Delay Systems With Multiple Delays

Abstract

This note is concerned with stability analysis of integral delay systems with multiple delays. To study this problem, the well-known Jensen inequality is generalized to the case of multiple terms by introducing an individual slack weighting matrix for each term, which can be optimized to reduce the conservatism. With the help of the multiple Jensen inequalities and by developing a novel linearizing technique, two novel Lyapunov functional based approaches are established to obtain sufficient stability conditions expressed by linear matrix inequalities (LMIs). It is shown that these new conditions are always less conservative than the existing ones. Moreover, by the positive operator theory, a single LMI based condition and a spectral radius based condition are obtained based on an existing sufficient stability condition expressed by coupled LMIs. A numerical example illustrates the effectiveness of the proposed approaches.

A systematic approach towards robust stability analysis of integral delay systems with general interval kernels

“…To make the proposed method applicable to a wide range of uncertain systems, system (1) is assumed general enough to include the different classes of integral delay systems in the literature. For example, Li et al (2013Li et al ( , 2016, Mondié and Melchor-Aguilar (2012) and Melchor-Aguilar (2016) consider exponential kernels, Ortiz et al (2020) consider piecewise continuous kernels, Zhou and Li (2016) and Arismendi-Valle and Melchor-Aguilar (2019) consider constant kernels and Melchor-Aguilar et al (2010) consider continuous kernels. All these kernels are bounded and therefore, there exist upper and lower bounds (4) such that (2) and (3) are met.…”

“…For some of the important samples in this regard, see Melchor-Aguilar et al (2010). Lyapunov-Krasovskii (LK) approaches make a standard framework to study the stability of these systems by means of LMIs (Li et al, 2016;Melchor-Aguilar, 2016;Zhou & Li, 2016). When the systems are uncertain, these techniques have also been developed to address the robust stability problem (Melchor-Aguilar & Morales-Sánchez, 2016;Melchor-Aguilar et al, 2008;Morales-Sánchez & Melchor-Aguilar, 2013).…”

“…This has considerably inhibited the usual frequency domain analysis methods for these systems. Lyapunov-Krasovskii (LK) approaches somewhat make up for this shortage by formulating stability conditions in some LMIs (Li et al (2016), Melchor-Aguilar (2016) and Zhou & Li (2016)). However, they are notorious for increased conservatism.…”

Adaptive neural network tracking design for a class of uncertain nonlinear discrete-time systems with unknown time-delay

“…In recent years, we have received great development for uncertain systems which are often to appear in the practical plants [1][2][3][4][5][6][7][8][9][10][11].…”