New stability criteria for linear systems with interval time-varying delay

Delay-partitioning approach for systems with interval time-varying delay and nonlinear perturbations

SummaryThe problem of the stability of a linear system with an interval time‐varying delay is investigated. A new Lyapunov–Krasovskii functional that fully uses information about the lower bound of the time‐varying delay is constructed to derive new stability criteria. It is proved that the proposed Lyapunov–Krasovskii functional can lead to less conservative results than some existing ones. Based on the proposed Lyapunov–Krasovskii functional, two stability conditions are developed using two different methods to estimate Lyapunov–Krasovskii functional's derivative. Two numerical examples are given to illustrate that the two stability conditions are complementary and yield a larger maximum upper bound of the time‐varying delay than some existing results. Copyright © 2013 John Wiley & Sons, Ltd.

This paper considers the delay-dependent stability analysis of time-varying delay systems. First of all, the previous Jensen' s like inequality is firstly improved by a new inequality which has less conservatism according to our analysis. Another remarkable approach to reduce the conservatism of stability criteria is the delay-partitioning idea, and the advantage of this method provides larger delay bounds when the delay-partitioning number increases. These technical tools allow to construct efficiently a new class of Lyapunov-Krasovskii functional (LKF) in practice. Combining with the new inequality and reciprocal convex approach and some other inequality techniques, the new less conservative robust stability criteria are shown in form of linear matrix inequalities (LMIs). The effectiveness of the proposed results is illustrated by an example from the literature.

In this paper, new criteria are presented for ascertaining delay-dependent stability of networked multi-area load frequency control (LFC) systems with feedback loop delay in presence of unknown and time-varying exogenous load disturbance. The proposed stability criteria are based on Lyapunov-Krasovskii (LK) functional approach, and they are expressed in linear matrix inequality (LMI) framework. Since the unknown exogenous load disturbance in the power system (cause) ultimately affects the evolution of the state variables of the system (effect), in the reported stability criteria, the load disturbance is mathematically modelled as a norm-bounded nonlinear time-varying function of current and delayed state vectors, and subsequently incorporated into the delay-dependent stability analysis. Two cases of time-delays are considered in the stability analysis: time-varying single feedback-loop delay element and time-invariant multiple feedback-loop delay elements. In addition, to realize less conservative stability criteria with minimal number of decision variables, in the stability analysis, tighter integral inequalities, viz., reciprocal convex combination lemma (for time-varying delay case) and Jenson integral inequality (for time-invariant delay case) are employed. The proposed results are illustrated on standard benchmark LFC systems, and supported suitably by simulation results.

Advanced stability criteria for linear systems with time-varying delays

This paper considers the problem of delay-dependent robust stability for uncertain systems with interval time-varying delays. By using the direct Lyapunov method, a new Lyapunov-Krasovskii (L-K) functional is introduced based on decomposition approach, when dealing with the time derivative of L-K functional, a new tight integral inequality is adopted for bounding the cross terms. Then, a new less conservative delay-dependent stability criterion is formulated in terms of linear matrix inequalities (LMIs), which can be easily solved by optimization algorithms. Numerical examples are given to show the effectiveness and the benefits of the proposed method.

This paper investigates the problem of robust stability for a class of linear uncertain system with interval time-varying delay. Based on delay-central point method and by means of L-K stability theorem, integral inequality method and convex combination technique, a new delay-dependent stability criterion for the system is formulated in terms of linear matrix inequalities (LMIs). Unlike existing methods, when bounding the cross-terms that emerged from the time derivative of the L-K functional, neither superfluous free weighting matrices are introduced nor any useful terms are neglected, only using tighter integral inequalities and a very few free weighting matrices for expressing the relationship of the correlative terms, so that it can reduce the complexity both in theoretical derivation and in computation. As a result, it is effective to reduce the conservatism of the results. Numerical examples are given to illustrate the effectiveness and less conservatism of the proposed method.

This paper presents a global stability criteria of neural networks with neutral-type time-varying delay. By constructing a new Lyapunov-Krasovskii function, where not only upper bound, but also lower bound of time delay is included, some less conservative delay-dependent exponential stability criteria is obtained by employing new technique of dealing with some integral terms. Numerical examples are presented to demonstrate the effectiveness and improvement of our results.

In this paper, the synchronization stability problem for a class of general complex dynamical networks with interval time-varying coupling delay and delay in the dynamical node is investigated. By dividing the delay interval into two variable subintervals, slightly different Lyapunov---Krasovskii functionals are constructed on these two subintervals. Then several less conservative delay-dependent synchronization stability criteria are derived in terms of linear matrix inequality via reciprocally convex approach, which can be easily solved by using the standard numerical software. Numerical examples are given to illustrate the effectiveness and less conservatism of the proposed method.

The delay-dependent stability problem is investigated for discrete-time neural networks with time-varying delays. A new augmented Lyapunov-Krasovskii functional (LKF) with single and double summation terms and several augmented vectors is proposed by decomposing the time-delay interval into two nonequidistant subintervals to derive less conservative stability conditions. Then, by using Wirtinger-based inequality, reciprocally, and extended reciprocally convex combination lemmas, tight estimations for sum terms in the forward difference of the LKF are given. Several zero equalities are introduced to further relax the existing results. Less conservative stability criteria are proposed in terms of linear matrix inequalities (LMIs). Finally, numerical examples are proposed to show the effectiveness and less conservativeness of the proposed method.

This paper focuses on the further improved stability criteria of uncertain T-S fuzzy systems with time-varying delay by the delay-partitioning approach and a new integral inequality, i.e., the Free-Matrix-Based integral inequality. A modified augmented Lyapunov-Krasovskii functional (LKF) is established by partitioning the delay in all integral terms. Then, new results on tighter bounding inequalities such as Peng-Park's integral inequality (reciprocally convex approach) and the Free-Matrix-based integral inequality (which yields less conservative stability criteria than the Wirtinger-based inequality) are introduced to reduce the enlargement in bounding the derivative of LKF as much as possible, therefore, less conservative results can be expected in terms of esand MIs. Finally, a numerical example is included to show that the proposed conditions are less conservative than existing ones.

In this paper, we consider the issue of stability criteria for neural networks systems with variable delays. By introducing two triple integral terms and a new augmented term with more delay information to the Lyapunov-Krasovskii functional(LKF), some less conservative stability criteria are proposed. Finally, the effectiveness of the proposed results are demonstrated by two numerical examples.

This paper addresses the stability and H∞ performance of a single-area discrete-time power system with time-varying transmission delays under Denial-of-Service (DoS) attacks. First, the power system is modeled as a discrete-time delay system that integrates both DoS-induced delays and transmission delays, with PI controllers incorporated for Load Frequency Control (LFC). Using advanced summation inequality techniques, a Lyapunov–Krasovskii Functional (LKF) is constructed to capture comprehensive system state information, enabling the derivation of less conservative stability criteria. The proposed stability criterion based on linear matrix inequalities (LMI) ensures asymptotic stability and meets the H∞ performance index, while considering norm-bounded external load disturbances. Two convex optimization algorithms are designed to obtain optimal controller gains, either for a given H∞ index or by searching within a specified index range. Numerical examples and MATLAB simulations validate the effectiveness of the method. The results demonstrate that the maximum allowable delay upper bounds (MADUBs) estimated by the proposed criterion are larger than those obtained by existing methods, with an increase of at least 1 s. This indicates a reduction in conservatism. Simulation trajectories of frequency deviation (Δf) and area control error (ACE) confirm that the system remains stable under DoS attacks, with responses converging to zero after transient oscillations.

In this paper, we consider the stability of linear systems with an interval time-varying delay. It is known that the adoption of decomposition of delay improves the stability result. For the interval time-delay case, they applied it to the interval of time-delay and got less conservative results. Our basic idea is to apply the general decomposition to the low limit of delay as well as interval of time-delay. Based on this idea, by using the modified Lyapunov-Krasovskii functional and newly derived Lemma, we present a less conservative stability criterion expressed as in the form of linear matrix inequality(LMI). Finally, we show, by well-known two examples, that our result is less conservative than the recent results.

This paper studies the stability of linear systems with interval time-varying delay and nonlinear perturbations. By developing a new delay-dependent decomposition approach, a new delay-fractional-dependent Lyapunov-Krasovskii (LK) function is constructed, and then stability criteria are derived by employing the convex optimization technique. Since that a tuning parameter is employed to variable delay fractional techniques, these criteria can be applied and effected to the case in which the upper bound of the time-derivative of the delay is larger than one. At last, a numerical example is given to show the effectiveness and merits of the present result.