Assessing stability of time-delay systems using rational systems (original) (raw)

Stability independent of delay using rational functions

2009

This paper is concerned with the problem of assessing the stability of linear systems with a single timedelay. Stability analysis of linear systems with time-delays is complicated by the need to locate the roots of a transcendental characteristic equation. In this paper we show that a linear system with a single time-delay is stable independent of delay if and only if a certain rational function parameterized by an integer k and a positive real number T has only stable roots for any finite T ≥ 0 and any k ≥ 2. We then show how this stability result can be further simplified by analyzing the roots of an associated polynomial parameterized by a real number δ in the open interval (0, 1). The paper is closed by showing counterexamples where stability of the roots of the rational function when k = 1 is not sufficient for stability of the associated linear system with time-delay. We also introduce a variation of an existing frequency-sweeping necessary and sufficient condition for stability independent of delay which resembles the form of a generalized Nyquist criterion. The results are illustrated by numerical examples.

New delay-dependent stability conditions for linear system with delay

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.

New delay-dependent stability conditions for linear systems with delay

In this work, delay-dependent stability conditions for systems described by delayed differential equations are presented. The employment of a special transformation to another state space representation named Benrejeb characteristic arrow matrix permit to determine a practical asymptotic stability condition. An Illustrative example is presented permitting to understand the application of the proposed methods.

A test for stability of linear differential delay equations

Quarterly of Applied Mathematics, 1982

The changes in the stability of a system of linear differential delay equations resulting from the delay are studied by analyzing the associated eigenvalues of the characteristic equation. A specific contour is mapped by the characteristic equation into the complex plane to give an easy test for stability from an application of the argument principle. When the real part of an eigenvalue is positive, the contour gives bounds on the imaginary part which are important in certain applications to nonlinear problems.

Delay dependent stability of linear time-delay systems

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.

Stability analysis of linear systems with time delay

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.

Stability Analysis of Linear Systems with Time Delays

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.

Stabilization of time-delay systems using finite-dimensional compensators

IEEE Transactions on Automatic Control, 2000

For linear time-invariant systems with one or more noncommensnrate time delays, necessary and sufficient conditions are given for the existence of, a fiite-dimensional stabilizing feedback compensator. In particular, it is shown that a stabilizable time-delay system can always be stabilized using a finite-dimensional compensator. The problem of explicitly constructing finite-dimensional stabilizing compensators is also considered.

Stability Analysis of Systems With Delay-Dependent Coefficients: An Overview

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.

Computation of Stability for Linear Time-Delay Systems

This paper presents a computational approach to stability determination of a class of time-delay systems. Using a parametrization of quadratic functionals, we construct a nested sequence of sufficient conditions, expressible as semidefinite programs, which are of non-decreasing accuracy. In addition, using a construction based on Putinar's representation, we also address the case of parametric uncertainty and uncertain delay.

Exponential Stability of Linear Systems with Multiple Time Delays

Mathematical Researches, 2016

In this paper, a class of linear systems with multiple time delays is studied. The problem of exponential stability of time-delay systems has been investigated by using Lyapunov functional method. We will convert the system of multiple time delays into a single time delay system and show that if the old system is stable then the new one is so. Then we investigate the stability of converted new system, by using matrix decomposition and linear matrix inequality (LMI) technique. Some numerical examples are given to illustrate the efficiency of our method.

An improved criterion for exponential stability of linear systems with multiple time delays

Applied Mathematics and Computation, 2008

Exponential stability of linear systems with multiple time delays is studied in this paper. By using an improved Lyapunov functional, we have obtained an improved criterion, which is strictly less conservative than the very recent criterion in Ren and Cao [Fengli Ren, Jinde Cao, Novel a-stability of linear systems with multiple time delays, Appl. Math. Comput. 181 (2006), 282-290], and an estimation of Lyapunov factor. A numerical example is also given to shown the superiority of our result to those in the literature.

Asymptotic practical stability of time delay systems

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

Exponential stabilizability of linear systems with multiple delays on states and controls

Exponential stabilizability of linear control systems with multiple delays on states and controls is studied in this paper. Based on the Lyapunov method, we establish new criteria that ensure the exponential stability of the closed-loop system with memoryless state feedback control or instantaneous feedback control. The criteria are derived in terms of linear matrix inequalities, which allows to compute simultaneously two bounds that characterize the exponential stability rate of the solution. Our results are illustrated with numerical examples.

Stability and Robust Stability of Linear Time-Invariant Delay Differential-Algebraic Equations

SIAM Journal on Matrix Analysis and Applications, 2013

Necessary and sufficient conditions for exponential stability of linear time invariant delay differential-algebraic equations (DDAEs) are presented. The robustness of this property is studied when the equation is subjected to structured perturbations and a computable formula for the structured stability radius is derived. The results are illustrated by several examples.

Stability in a Linear Delay System without Instantaneous Negative Feedback

SIAM Journal on Mathematical Analysis, 2002

It is shown that every solution of a linear differential system with constant coefficients and time delays tends to zero if a certain matrix derived from the coefficient matrix is a nonsingular M-matrix and the diagonal delays satisfy the so-called 3/2 condition.

Stability of Linear Systems with Delayed State: A Guided Tour

IFAC Proceedings Volumes, 1998

In this paper, some recent stability results on linear time-delay systems are outlined. The goal is to give an overview of the state of the art of the techniques used in delay system stability analysis. In particular, two specific problems (delay-independent / delay-dependent) are considered and some references where the reader can find more details and proofs are pointed out. This paper is based on Niculescu et al. (1997).