An LMI approach to stability of systems with severe time-delay

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.

European Journal of Control, 2011

1995

This paper considers the problems of robust stability analysis and robust control design for a class of uncertain linear systems with a constant time-delay. The uncertainty is assumed to be norm-bounded and appears in all the matrices of the state space model. We develop methods for robust stability analysis and robust stabilization. The proposed methods are dependent on the size of the delay and are given in terms of linear matrix inequalities

Archives of Control Sciences, 2012

The paper concerns the problem of stabilization of continuous-time linear systems with distributed time delays. Using extended form of the Lyapunov-Krasovskii functional candidate, the controller design conditions are derived and formulated with respect to the incidence of structured matrix variables in the linear matrix inequality formulation. The result give sufficient condition for stabilization of the system with distributed time delays. It is illustrated with a numerical example to note reduced conservatism in the system structure.

This paper addresses exponential stability problem for a class of linear systems with time delay. By constructing a suitable augmented Lyapunov-Krasovskii functional combined with Leibniz-Newton's formula, new sufficient conditions for the exponential stability of the systems are first established in terms of LMIs.

IET Control Theory & Applications, 2010

This study is concerned with the stability analysis of systems with time-varying delay in a given interval. A new type of augmented Lyapunov functional which contains some triple-integral terms is proposed. By introducing free-weighting matrices, a new delay-range-dependent stability criterion is derived in terms of linear matrix inequality. The rate-range of the delay is considered, so the stability criterion is also delay-raterange dependent. Numerical examples are given to illustrate the effectiveness of the proposed method.

Advances in Difference Equations

This paper deals with the state and output feedback stabilization problems for a family of nonlinear time-delay systems satisfying some relaxed triangular-type condition. A new delay-dependent stabilization condition using a controller is formulated in terms of linear matrix inequalities (LMIs). Based on the Lyapunov-Krasovskii functionals, global asymptotical stability of the closed-loop systems is achieved. Finally, simulation results are shown to illustrate the feasibility of the proposed strategy.

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.

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.

Cybernetics and Systems Analysis, 1996

Consider a system of linear differential equations with a delayed deviating argument i(O=ax(O+Bx(t-O. (1) Here A, B are matrices with constant elements; 7-(7" > 0) is a constant delay. The method of Lyapunov-Krasovskii functionals [I] is one of the efficient techniques for stability analysis and calculation of transient~. However, the Lyapunov function has been constructed only for linear stationary systems without deviating argument; in these systems, the Lyapunov function is a quadratic form and its symmetric positive definite matrix is found by solving a matrix equation. The situation is different for delayed systems, where the determination of the Lyapunov function involves fundamental difficulties. As shown in [2], a necessary and sufficient condition of asymptotic stability is the existence of a quadratic Lyapunov-Krasovskii functional with matrix functions dependent on several variables. Its construction requires solving a system of matrix ordinary differential equations and matrix partial differential equations with special boundary conditions. Analytically, this is virtually an impossible undertaking.