Stability Of Time-varying Delay Systems Research Articles

Novel inequality-based functions for the stability of time-varying delay systems

This paper proposes new inequality-based functions to be Lyapunov functionals for the stability analysis of time-varying delay systems. The novel Lyapunov functionals are developed using a slack-matrices-based integral inequality for the first time. This is unlike most inequality-based functions that have been used as Lyapunov functionals which consist of single-matrices in their integral terms. Based on the new Lyapunov functionals, a new stability criterion is derived in the form of a matrix-valued quadratic function, which is proven to be negative definite using a geometry-based negativeness lemma. Two numerical examples showcase the effectiveness of our presented method.

Stability and stabilisation for time-varying delay systems based on flexible augmented Lyapunov–Krasovskii functional

This paper studies the problems of stability and stabilisation for time-varying delay systems. A flexible augmented Lyapunov–Krasovskii functional (FALKF) including some triple integral terms and delay-product-type quadratic terms is first constructed, in which the upper and lower limits of integral terms are flexible. Compared with some existing ones, the information of time-varying delay can be fully utilised. A parameter-dependent reciprocally convex inequality (PDRCI) is proposed, which covers some existing ones as its special cases. Based on the FALKF and PDRCI, a less conservative stability condition is obtained to ensure the time-varying delay systems to be asymptotically stable. By using a matrix inequality decoupling technique, the corresponding controller for the closed-loop systems is derived. Compared with some existing works, the constraints on introduced slack matrices are avoided. It directly provides extra free dimensions in the solution space. Two examples are employed to illustrate the effectiveness of the proposed methods.

Novel negative-definiteness conditions on the quadratic polynomial function with application to stability analysis of continuous time-varying delay systems

When analyzing the stability of time-varying delay systems in view of the Lyapunov–Krasovskii functional, a quadratic polynomial function with regard to time-varying delay is always generated. And it is particularly crucial to determine the negativeness of the matrix of such a quadratic form function for obtaining an analysis result expressed in linear matrix inequalities. This paper proposes a method of tangent intersection in the delay interval segmentation, producing the generalized quadratic convex conditions by further utilizing the cross point between every two tangent lines. It reduces the conservatism of the existing conditions remarkably without requiring unexplainable adjustable parameters and additional decision variables. Benefiting from the newly proposed quadratic convex conditions, the novel stability conditions are derived, the superiority of which is demonstrated through several widely used numerical instances and single area power system PI control example.

Stability Analysis of Time-Delay Systems via a Delay-Derivative-Partitioning Approach

This paper is devoted to the study of delay-dependent stability of time-varying delay systems. A delay-derivative-partitioning approach is proposed. By constructing an augmented Lyapunov functional that contains two delay-product-dependent terms and using the delay-derivative-partitioning approach, an improved stability condition is established. The derived condition is described as a set of linear matrix inequalities, which can be readily implemented. Finally, four examples are carried out so as to attest the effect and merit of the presented approach.

Exponential stability of time delay systems based on intermediate polynomial-based weighted functions

We investigate exponential stability of time-varying delay systems in this article. By considering intermediate polynomials and introducing slack variables, we develop new intermediate polynomial-based weighted functions (IPWFs). Then an exponential stability criterion, in terms of linear matrix inequalities (LMIs), is obtained by choosing an appropriate Lyapunov–Krasovskii functional (LKF) and some novel techniques on IPWFs. Numerical results show that the criterion obtained in this article is less conservative than some previous results.

Geometry-based Conditions for a Quadratic Function: Application to Stability of Time-Varying Delay Systems

In this paper, novel geometry-based negative conditions are presented for a quadratic function with time-varying delays. The system performance could be optimal by adjusting a parameter value. New stability criteria for the time-varying delayed systems are derived.

Robust stabilization of time-varying delay systems with predictor-observer based controller

Abstract This papers deals with the problem of robust stabilization of a class of discrete systems with time-varying input delay and model uncertainties, where the state is assumed not to be fully accessible. Thus, a predictor-observer based controller is proposed. Furthermore, a control synthesis algorithm based on the Cone Complementarity Linearization (CCL) approach is provided to design the controller and observer gains, with the objective of improving robust performance against uncertainties in the process model and the input delay. Finally, different illustrative simulations are introduced to show the main benefits of this strategy.

Positivity for non-Metzler systems and its applications to stability of time-varying delay systems

In 1950, Wazewski obtained the following necessary and sufficient condition for positivity of linear systems of ordinary differential equations: the matrix of coefficients is Metzler (i.e. off-diagonal elements in the matrix of coefficients are nonnegative). No results on the positivity of solutions to delay systems in the case where the matrix is non-Metzler were expected to be obtained. It was demonstrated that for delay systems the Wazewski condition is not a necessary one. We prove results on positivity of solutions for non-Metzler systems of delay differential equations. New explicit tests for exponential stability are obtained as applications of results on positivity for non-Metzler systems. Examples demonstrate possible applications of our theorems to stabilization. For instance, in view of our results, the implicit requirement on dominance of the main diagonal can be skipped. Our approach is based on nonoscillation of solutions and positivity of the Cauchy functions of scalar diagonal delay differential equations.

Novel Lyapunov–Krasovskii functional with delay-dependent matrix for stability of time-varying delay systems

This paper investigates the stability criteria of time-varying delay systems with known bounds of the delay and its derivative. To obtain a tighter bound of integral term, quadratic generalized free-weighting matrix inequality (QGFMI) is proposed. Furthermore, a novel augmented Lyapunov–Krasovskii functional (LKF) are constructed with a delay-dependent matrix, which impose the information for a bound of delay derivative. Relaxed stability condition using QGFMI and LKF provides a larger delay bound with low computational burden. The superiority of the proposed stability condition is verified by comparing to recent results.

A novel Lyapunov functional for stability of time-varying delay systems via matrix-refined-function

This paper is concerned with the development of novel stability criteria for time-varying delayed systems. To this end, a new function which consisted of two quadratic functions with a special structural matrix is established to be a Lyapunov functional candidate. The proposed Lyapunov functional plays key role to decrease the conservatism of derived conditions which is verified by three numerical examples.

Relaxed conditions for stability of time-varying delay systems

In this paper, the problem of delay-dependent stability analysis of time-varying delay systems is investigated. Firstly, a new inequality which is the modified version of free-matrix-based integral inequality is derived, and then by aid of this new inequality, two novel lemmas which are relaxed conditions for some matrices in a Lyapunov function are proposed. Based on the lemmas, improved delay-dependent stability criteria which guarantee the asymptotic stability of the system are presented in the form of linear matrix inequality (LMI). Two numerical examples are given to describe the less conservatism of the proposed methods.

Exponential stabilization of time-varying delay systems with non-linear perturbations

Exponential stabilization of time-varying delay systems with non-linear perturbations

Robust Stability of Time‐Varying Delay Systems: The Quadratic Separation Approach

AbstractIn this article, we are interested in analysing the stability of systems that incorporate time‐varying delays in their dynamic. The Lyapunov‐Krasovskii approach is definitely the most popular method to address this issue and many results have proposed new functionals and enhanced techniques for deriving less conservative stability conditions. In the present work, we propose an original approach: the quadratic separation. To this end, the delay operator properties are exploited to provide delay range stability conditions. In particular, L2‐norm of delay‐dependent operators are computed so as to reduce the conservatism of the approach. Moreover, the main result is able to assess the stability of non‐small delay systems, i.e, it can detect a stability interval for systems that are unstable without any delay. Several examples illustrate the benefit of our methodology.

Robust Absolute Stabilization of Time-Varying Delay Systems with Maximum Admissible Perturbed Bound

This paper is concerned the problem of robust absolute stabilization of time-varying delay systems with admissible perturbation in terms of integral inequality. A linear state-feedback control law is derived for one class of delay systems with sector restriction based on linear matrix inequality (LMI). Especially, this method does not require input terms are absolutely controllable for nonlinear delay systems. Numerical example is used to demonstrate the validity of the proposed method.

Delay-dependent criteria for robust stability of time-varying delay systems

This paper deals with the problem of delay-dependent robust stability for systems with time-varying structured uncertainties and time-varying delays. Some new delay-dependent stability criteria are devised by taking the relationship between the terms in the Leibniz–Newton formula into account. Since free weighting matrices are used to express this relationship and since appropriate ones are selected by means of linear matrix inequalities, the new criteria are less conservative than existing ones. Numerical examples suggest that the proposed criteria are effective and are an improvement over previous ones.

Extension of Robust Stability for Time-Varying Delay Systems with Nonlinear Perturbations

In this paper, earlier results are generalized for robust stability of delay systems with nonlinear time-varying perturbations. Sufficient conditions are stated in terms of the existence of some positive definite matrix functions satisfying certain Riccati-type equations to ensure the robust stability of the system. Dans cet article, des résultats de stabilité robuste sont généralisés au cas des systèmes non linéaires avec retards et avec perturbations non linéaires à temps variant. Des conditions suffisantes, pour guarantir une telle stabilité, sont établies en termes d'existence de matrices définies positives vérifiant une équation du type de Riccati.