D1.1 -Overview of Lyapunov methods for time-delay systems

This paper overviews the research investigations pertaining to stability and stabilization of control systems with time-delays. The prime focus is the fundamental results and recent progress in theory and applications. The overview sheds light on the contemporary development on the linear matrix inequality (LMI) techniques in deriving both delay-independent and delay-dependent stability results for time-delay systems. Particular emphases will be placed on issues concerned with the conservatism and the computational complexity of the results. Key technical bounding lemmas and slack variable introduction approaches will be presented. The results will be compared and connections of certain delay-dependent stability results are also discussed.

In this paper, a procedure for construction of quadratic Lyapunov–Krasovskii functionals for linear time-delay systems is proposed. It is shown that these functionals admit a quadratic low bound. The functionals are used to derive robust stability conditions.

European Journal of Control, 2011

Proceedings of the 45th IEEE Conference on Decision and Control, 2006

This paper presents a Lyapunov-Krasovskii methodology for studying the input-to-state stability of nonlinear time-delay systems. The methodology is feasible by the use, for instance, of the M 2 norm (that is the norm induced by the inner product in the Hilbert space known in literature as M 2 , or Z) in the space of continuous functions, and by the use of functionals which have a suitable (simple) integral term with strictly increasing kernel. The proposed results can be seen as a preliminary step towards extending some existing stability criteria to nonlinear time-delay systems with disturbance inputs.

IEEE/CAA Journal of Automatica Sinica, 2019

This paper investigates the stability problem for time-varying delay systems. To obtain a larger delay bound, this paper uses the second-order canonical Bessel-Legendre (B-L) inequality. Secondly, using four couples of integral terms in the augmented Lyapunov-Krasovskii function (LKF) to enhance the relationship between integral functionals and other vectors. Furthermore, unlike the construction of the traditional LKF, a novel augmented LKF is constructed with two new delay-product-type terms, which adds more state information and leads to less conservative results. Finally, two numerical examples are provided to demonstrate the effectiveness and the significant improvement of the proposed stability criteria.

2007 46th IEEE Conference on Decision and Control, 2007

Stability analysis of linear systems with time-varying delay is investigated. In order to highlight the relations between the variation of the delay and the states, redundant equations are introduced to construct a new modeling of the delay system. New types of Lyapunov Krasovskii functionals are then proposed allowing to reduce the conservatism of the stability criterion. Delay dependent stability conditions are then formulated in terms of linear matrix inequalities (LMI). Finally, an example shows the effectiveness of the proposed methodology.

Journal of Systems Science and Complexity, 2020

This paper focuses on the problem of delay-dependent stability of linear systems with time-varying delay. A new delay-product-type augmented Lyapunov-Krasovskii functional (LKF) is constructed. Based on the LKF and by employing a generalized free-matrix-based integral inequality, less conservative delay-dependent stability criteria are obtained. Finally, two well-known numerical examples are used to confirm the effectiveness and the superiority of the presented stability criteria.

2012 IEEE 10th Jubilee International Symposium on Intelligent Systems and Informatics, 2012

This paper provides sufficient conditions for the asymptotic practical and finite time stability of linear continuous time delay systems mathematically described as

Theoretical and Applied Mechanics, 2013

This paper deals with the problem of delay dependent stability for both ordinary and large-scale time-delay systems. Some necessary and sufficient conditions for delay-dependent asymptotic stability of continuous and discrete linear time-delay systems are derived. These results have been extended to the large-scale time-delay systems covering the cases of two and multiple existing subsystems. The delay-dependent criteria are derived by Lyapunov's direct method and are exclusively based on the solvents of particular matrix equation and Lyapunov equation for non-delay systems. Obtained stability conditions do not possess conservatism. Numerical examples have been worked out to show the applicability of results derived.

IEEE Transactions on Automatic Control, 1994

This paper addresses the problem of stability analysis of a class of linear systems with time-varying delays. We develop conditions for robust stability that can be tested using Semidefinite Programming using the Sum of Squares decomposition of multivariate polynomials and the Lyapunov-Krasovskii theorem. We show how appropriate Lyapunov-Krasovskii functionals can be constructed algorithmically to prove stability of linear systems with a variation in delay, by using bounds on the size and rate of change of the delay. We also explore the quenching phenomenon, a term used to describe the difference in behaviour between a system with fixed delay and one whose delay varies with time. Numerical examples illustrate changes in the stability window as a function of the bound on the rate of change of delay.