Delay dependent stability of linear time-delay systems

European Journal of Control, 2011

2011

In this work, delay-dependent stability conditions for systems described by delayed differential equations are presented. The employment of a special transformation to a state space representation named Benrejeb characteristic arrow matrix permits to determine new asymptotic stability conditions. Illustrative examples are presented to show the effectiveness of the proposed approach.

IEE Proceedings - Control Theory and Applications, 2006

This paper deals with the problem of the delay-dependent stability of linear systems with multiple time delays. A new method is first presented for a system with two time delays, in which free weighting matrices are used to express the relationships among the terms of the Leibniz-Newton formula. Next, this method is used to show the equivalence between a system with two identical time delays and a system with a single time delay. Then, a numerical example verifies that the criterion given in this paper is effective and is a significant improvement over existing ones. Finally, the basic idea is extended to a system with multiple time delays.

IET Control Theory & Applications, 2010

Stability analysis of interconnected systems with finite as well as arbitrary interconnection delays is considered in this study. For the purpose, subsystems interconnected with finite delays are grouped to form larger subsystems with intraconnection delays. However, these larger subsystems are interconnected with arbitrary delays among themselves. By considering a suitable Lyapunov-Krasovskii functional, a stability criterion is derived in the linear matrix inequality (LMI) framework to study simultaneous delay-independent (for the interconnection delays) and delay-dependent (for the intraconnection delays) stability of the reformed interconnected system. This proposed criterion is useful when one attempts to estimate the tolerable bounds of the intraconnection delays. Finally, a numerical example is considered to illustrate the effectiveness of the proposed criterion. In this paper, we consider the stability analysis of a largescale system subject to the finite and arbitrary nature of the interconnection delays. For the purpose, we redefine an interconnected system by grouping the subsystems with finite delays among themselves. In this way, larger subsystems are formed that contain multiple definite intraconnection delays within them but the newly formed subsystems having multiple arbitrary interconnection delays among themselves. For this system, a suitable Lyapunov-Krasovskii functional is defined and correspondingly a 3022

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.

JSME International Journal Series C, 2006

In this paper, the problem of asymptotic stability analysis for a class of linear large-scale systems with time delay in the state of each subsystem as well as in the interconnections is addressed in detail. By utilizing a model transformation and the Lyapunov stability theory, a delay-dependent criterion for stability analysis of the systems is derived in terms of some certain linear matrix inequalities (LMIs). A numerical example is given to illustrate that the proposed result is effective.

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.

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.

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.

In this paper, new stability conditions for time delay system are proposed. They are based on the use of the aggregation techniques and the choice of a state representation as Benrejeb arrow form characteristic matrix. Application cases are treated to illustrate the implementation of the proposed approach.

IEEE Access

This paper gives an overview of the stability analysis of systems with delay-dependent coefficients. Such systems are frequently encountered in various scientific and engineering applications. Most such analyses are generalization of those on systems with delay-independent coefficients. Therefore an introduction on systems with delay-independent coefficients is also given, with an emphasis on the τ-decomposition approach. Methods for two key ingredients of this approach are discussed, namely the identification of imaginary characteristic roots with the corresponding delays, and local behavior analysis of these roots as the delay increases through these critical values. For systems with delay-dependent coefficients, we review the methods of analysis for systems with a single delay and commensurate delays, their application to output feedback control and a geometric perspective that establishes a link between systems with and without delay-dependent coefficients. We provide the main ideas of various stability analysis methods and their advantages and limitations. We also present our perspectives on future directions of research on this interesting topic.

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.

2015

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.

Control and cybernetics

This paper addresses the asymptotic stability analy-sis problem for a class of linear large-scale systems with time delay in the state of each subsystem as well as in the interconnections. Based on the Lyapunov stability theory, a delay-dependent criterion for stability analysis of the systems is derived in terms of a linear matrix inequality (LMI). Finally, a numerical example is given to demonstrate the validity of the proposed result.

Journal of the Franklin Institute, 2013

This paper considers the problem of time delay-dependent exponential stability criteria for the time-delay linear system. Utilizing the linear inequality matrices (LMIs) and slack matrices, a novel criterion based on the Lyapunov-Krasovskii methodology is derived for the exponential stability of the time-delay system. Based on the criteria condition we concluded that the upper bound of the exponential decay rate for the time-delay system can be easily calculated. In addition, an improved sufficient condition for the robust exponential stability of uncertain time-delay system is also proposed. Numerical examples are provided to show the effectiveness of our results. Comparisons between the results derived by our criteria and the one given in Liu (2004) [1], Mondie and Kharitonov (2005) [2], and Xu et al. (2006) [3] show that our methods are less conservative in general. Furthermore, numerical results also show that our criteria can guarantee larger exponential decay rates than the ones derived by Liu (2004) [1] and Mondie and Kharitonov (2005) [2] in all time delay points we have tested and in some of time delay points obtained by Xu et al. (2006) [3].