Derivative Of Lyapunov-Krasovskii Functional Research Articles

Relaxed mixed convex combination lemmas: Application to stability analysis of time-varying delay systems

Stability analysis of time-varying delay systems is a fundamental issue, and reducing the conservatism of stability criteria is the key. The high-order Bessel-Legendre inequality together with appropriate Lyapunov–Krasovskii functional (LKF) has the potential to reduce conservatism, but the reciprocally convex and quadratic correlation terms of time-varying delay appear in LKF derivative. This paper proposes the relaxed mixed convex combination lemmas, in which the reciprocally convex combination lemma introduces delay square terms in the LKF derivative, and the quadratic convex combination lemma relaxes the negativity condition for quadratic polynomial functions. The less conservative stability criteria of systems with time-varying delay satisfying two cases are derived, making the delay upper bound larger, and the design degree of freedom higher. Finally, two well-known numerical examples are presented to demonstrate the superiority of the lemmas and criteria over the previous works.

Stability and stabilization of systems with a cyclical time-varying delay via delay-product-type looped-functionals

In this paper, the stability analysis and synthesis of systems with a cyclical time-varying delay are investigated. At first, the entire delay interval is divided into intervals of monotonically increasing and monotonically decreasing alternating cycles. Then, a Lyapunov-Krasovskii functional (LKF) with two delay-product-type looped-functionals is constructed based on different delay monotone intervals. For the generated quadratic function with respect to delay in the derivative of LKF, a matrix-injection-based method is utilized to obtain the negative-definite conditions of LKF’s derivative. A delay-dependent stability criterion with less conservatism is derived and the corresponding state-feedback control strategy is given consequently. Finally, a numerical example is utilized to show the superiority and the effectiveness of the proposed method, and a single-area power system under load frequency control is applied to verify the feasibility of the proposed control strategy.

Asymptotic Tracking Control for Uncertain Nonlinear Strict-Feedback Systems With Unknown Time-Varying Delays.

It is nontrivial to achieve asymptotic tracking control for uncertain nonlinear strict-feedback systems with unknown time-varying delays. This problem becomes even more challenging if the control direction is unknown. To address such problem, the Lyapunov-Krasovskii functional (LKF) is used to deal with the time delays, and the neural network (NN) is applied to compensate for the time-delay-free yet unknown terms arising from the derivative of LKF, and then an NN-based adaptive control scheme is constructed on the basis of backstepping technique, which enables the output tracking error to converge to zero asymptotically. Besides, with a milder condition on time delay functions, the notorious singularity issue commonly encountered in coping with time delay problems is subtly settled, which makes the proposed scheme simple in structure and inexpensive in computation. Moreover, all the signals in the closed-loop system are ensured to be semiglobally uniformly ultimately bounded, and the transient performance can be improved with proper choice of design parameters. Both the theoretical analysis and numerical simulation are carried out to validate the relevance of the proposed method.

Improved stabilization condition of delayed T-S fuzzy systems via an extended quadratic function negative-determination lemma

This paper focuses on the stability and stabilization problems of continuous-time T-S fuzzy systems (TSFS) with variable delay. A new augmented Lyapunov–Krasovskii functional (LKF) is established by combining the negative quadratic term with the alterable delay-product-integral term. To further improve the results, an extended quadratic function negative-determination (QFND) lemma is proposed to deal with δ(t)-related quadratic function in the LKF derivative. Subsequently, based on the parallel distributed compensation (PDC) technique, a bounded delay correlation stabilization criterion is derived. Finally, the proposed methods are applied to a numerical example and an inverted pendulum system to verify their effectiveness and superiority.

Novel stability criteria of generalized neural networks with time-varying delay based on the same augmented LKF and bounding technique

This paper researches the stability issue of generalized neural networks (GNN) with time-varying delay. For the delay, its derivative has an upper bound or is unknown. Firstly, the augmented Lyapunov-Krasovskii functional (LKF) is constructed based on the state vectors of the third order integral inequalities. Then, by introducing two sets of state vectors, the LKF derivative is presented as the quadratic and quintic polynomials of the delay, respectively. Next, the new quadratic and quintic polynomial negative definite conditions (NDCs) are proposed to set up the linear matrix inequalities (LMIs). In addition, based on the same LKF and third order integral inequalities, this paper proves that the introduction of extra state vectors increases the conservatism of the derived stability conditions. Eventually, the advantages of the provided conditions are illustrated by several numerical examples.

Finite time stability of linear time varying delay systems using free matrix based integral inequalities

This paper discusses the problem of finite time stability (FTS) for linear delayed systems where the time-varying delay is bounded. First, based on a Lyapunov–Krasovskii Functional (LKF) which contains terms of triple integrals, delay-dependent FTS conditions are provided by introducing some free-weighting matrices. Then, new approximations of the single and multiple quadratic integrals that appear in the LKF derivative are established using integral inequalities, called free-matrix-based integral inequalities. The dimensions of these free matrices integral in our results are less than those obtained in the literature. This reduction in the number of variables does not mean that our method is a particular case but simply that our approach is completely different from the others and therefore more effective. Finally, less conservative design conditions in terms of Linear Matrix Inequalities (LMIs) are proposed and solved by the LMI Tools of MATLAB to show the advantage and effectiveness of the proposed approach.

Master-slave synchronization for time-varying delay chaotic Lur’e systems based on the integral-term-related free-weighting-matrices technique

In this paper, the master-slave synchronization of the chaotic Lur’e system (MSSCLS) with time-varying delay is investigated. At first, a Lyapunov-Krasovskii functional (LKF) with nonlinear terms and time-varying delay is constructed. Secondly, when the derivative of LKF is estimated, the improved stability criterion is obtained by considering the relationship between the integral terms generated after using the Bessel-Legendre inequality (BLI) and introducing the integral-term-related free-weighting-matrices. Finally, case studies based on Chua’s circuit are carried out to test and verify the effectiveness of the proposed synchronization criterion, and the results are compared with the latest methods to show the advantages of the proposed synchronization criterion.

Dynamic event-triggered control design for a class of [formula omitted]-normal nonlinear time-delay systems with actuator failures

This paper investigates the adaptive tracking control problem for a class of p-normal nonlinear time-delay systems with actuator failures via a dynamic event-triggered strategy. To deal with the time-varying delay, a modified Lyapunov-Krasovskii functional (LKF) with the system powers is constructed. By using radial basis function neural networks and hyperbolic tangent function, the residual terms caused by the derivative of LKF can be handled. Then, to achieve better communication efficiency, a dynamic event-triggered strategy is proposed, in which the threshold parameter is dynamically adjusted with the change of the virtual control signal. With the help of the proposed adaptive controller, all states of the closed-loop system are bounded and a favorable tracking performance is ensured. Finally, two examples are given to demonstrate the effectiveness of the proposed method.

Improved stability criterion for distributed time-delay systems via a generalized delay partitioning approach

<abstract><p>This paper researches the problem of stability analysis for distributed time-delay systems. A newly augmented Lyapunov-Krasovskii functional (LKF) is first introduced via a generalized delay partitioning approach. Then, a less conservative stability criterion is derived by introducing a novel Jensen inequality to estimate the integral terms in the derivative of LKF. The stability condition is given in terms of linear matrix inequality. Finally, the merits of the obtained stability criterion is shown by a well-known example.</p></abstract>

Stochastic stability analysis of Markovian jump systems with additive mode-dependent time-varying delays and partially known transition rates

This paper focuses on the stochastic stability of the Markovian jump systems with two additive mode-dependent time-varying delays and partially known transition rates. First, a novel mode-dependent Lyapunov-Krasovskii functional (LKF) is constructed and the single integral terms in the derivative of LKF are estimated by the generalised free-weighting-matrix-based integral inequality. Then, some stability criteria in terms of linear matrix inequalities are obtained by using the convex combination technique. Finally, two numerical examples are presented to illustrate the feasibility of the proposed method.

Enhanced Results on Stability Criteria for Linear Time Delay Systems with Distributed Delay via Relaxed Double Integral Inequality

This paper investigates the matter of stability criteria for linear time delay systems with distributed delay. Firstly, a relaxed double integral inequality is established to estimate the double integral terms appearing within the derivative of Lyapunov-Krasovskii functionals (LKFs) with a triple integral term. Unlike the recently introduced Jensen's inequalities, Wirtinger based integral inequalities, refined Jensen's inequalities and therefore the auxiliary function based integral inequalities the proposed relaxed integral inequality provides large feasible solution region and fewer conservative results. Secondly, by constructing an augmented Lyapunov-Krasovskii functional with a triple integral term, the robust stability criteria for linear time delay systems with distributed delay are given in terms of linear matrix inequalities (LMIs), which may be easily computed by the LMI toolbox of MATLAB. Finally, two numerical examples are performed to indicate the effectiveness of the proposed criterion.

Improved Synchronization Analysis for Delayed Lur’e Systems Using Improved Technique

The master-slave synchronization (MSS) issue of chaotic Lur’e systems (LSs) with time-varying delay (TVD) is explored in this work by means of feedback control. After taking the nonlinear function into full consideration, a creative Lyapunov-Krasovskii functional (LKF) is attained. Then novel criteria which ensure the synchronization of the studied systems are obtained with the aid of the improved inequality technique and the reciprocally convex combination method to cope with the derivative of LKF. The effectiveness and virtues of the synchronization conditions are testified via the notable Chua’s circuit in three different cases.

Novel stability criterion for linear system with two additive time-varying delays using general integral inequalities

<abstract> The stability analysis strategy for continuous linear system with two additive time-varying delays is proposed in this paper. First, for the purpose of analysis, the novel Lyapunov-Krasovskii functional (LKF) consisting of integral terms based on the first-order derivative of the system state is constructed. Second, the derivative of LKF is estimated by utilizing the Wirtinger-based integral inequality and extended reciprocally convex matrix inequality. The delay-dependent stability criterions are established in terms of linear matrix inequalities (LMIs) framework. The results show that the system performances are improved based on both enlarging the maximum allowable upper bound of the time-delays and reducing the number of decision variables. Furthermore, the conservatism of obtained delay-dependent stability criterion is reduced. Finally, a numerical simulation is given to demonstrate the effectiveness of obtained theoretical results. </abstract>

Stability analysis of linear systems with time‐varying delay via a quadratic function negative‐definiteness determination method

This study aims to carry out the stability analysis of linear systems with a time-varying delay. It is known that the negative-definite condition of the derivative of a Lyapunov–Krasovskii functional (LKF) can be determined using the convex combination method if the convexity requirement is satisfied by the derivative of the LKF. However, this method is not feasible in cases where the LKF's derivative is a quadratic function. To address this problem, this study proposes a novel negative-definiteness determination lemma that encompasses the previous lemmas as its special cases and shows less conservatism. Then, this lemma is employed to derive a stability criterion, and its superiority is demonstrated using three examples.

Event-Triggered H∞ Filtering for Networked Systems Under Hybrid Probability Deception Attacks

In this article, the deception attacks are the research object, which are widely concerned in cyber attacks, and the feasibility of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_\infty $ </tex-math></inline-formula> filtering for networked system based on event-triggered mechanism (ETM) is discussed. Firstly, in order to avoid waste of limited bandwidth, the ETM are introduced. At the same time, considering the impact of two types of different deception attacks, we introduce respectively two sets of random variables satisfying the Bernoulli distribution to depict the probability of the data transmitted by the network being subjected to deception attacks and the switching law of two types of deception attacks. Furthermore, a function model in accordance with the above situation is established. Then, by using Lyapunov-Krasovskii functional (LKF) and linear matrix inequalities (LMIs) techniques, the stability criterion of the established system model and the display expression of filter parameters are obtained. For the single integral terms in the derivative of LKF, we utilize Jensen’s Inequality and reciprocally convex combination lemma (RCCL) to process them. Finally, the feasibility of the proposed method is verified by a simulation example.

An Overview of Stability Analysis of Neural Networks with Time-Delays

Stability of neural networks (NNs) with time-delays (TDs) has become a popular research object. The Lyapunov-Krasovskii functional (LKF) method is an extremely effective technique for the stability research of NNs with TDs. Three primary issues, including ideas for constructing a reasonable LKF method, investigation of methods to estimate the LKF's derivative and reviewing approaches for establishing the stability criteria are discussed in details. Finally, some challenges are proposed for the future research directions.

Mixed‐delay‐dependent stability for time‐delay neutral system: an improved dynamic Lyapunov method

This work studies the mixed-delay-dependent asymptotical stability for a class of time-delay neutral systems and three less conservative criteria are presented in terms of linear matrix inequalities. Firstly, by proposing an improved dynamic Lyapunov method, an augmented Lyapunov-Krasovskii functional (LKF) is constructed, which does not only benefit from the information on each time-delay but also heavily depends on the interconnection between time-delays. Then, two effective Wirtinger-based integral inequalities and an extended reciprocal convex technique are employed to give much tighter bound on the LKF's derivative. In all, these treatments above can result in much less conservatism. Finally, two numerical examples are presented to demonstrate the effectiveness and advantages of our proposed criteria.

Auxiliary function-based integral inequality approach to robust passivity analysis of neural networks with interval time-varying delay

Abstract In this paper, we study the problem of passivity for uncertain neural networks with interval time-varying delay. Firstly, a suitable augmented Lyapunov–Krasovskii functional (LKF) containing two triple integral terms is constructed and an auxiliary function-based integral inequality (AFBI) is used to manipulate the augmented single integral terms in the derivative of LKF. Secondly, a special form of the AFBI is applied to deal with the delay-product-type term, which was used to be ignored in the time derivative of a triple integral term. As a result, less conservative delay-dependent passivity criteria are derived for normal delayed neural networks (DNNs) in the form of linear matrix inequalities (LMIs). In addition, with the same LKF, delay-dependent passivity criteria are obtained for normal DNNs without the delay-product-type term. Subsequently, these criteria are extended to DNNs with parameter uncertainties. Finally, four numerical examples and simulations are provided to illustrate the effectiveness of the proposed criteria.

Nonfragile Sampled-Data Control for Uncertain Networked Control Systems With Additive Time-Varying Delays.

This paper investigates the stabilization problem of uncertain networked control systems with additive time-varying delays by using nonfragile sampled-data control. Suitable Lyapunov-Krasovskii functional (LKF) is constructed which includes more information about the additive time-varying delays. The main aim of this paper is to design a nonfragile sampled-data control scheme which guarantees asymptotic stability of the considered system. Besides that, the Jensen's and improved integral inequalities are used for estimating the single and double integral terms in the derivative of LKFs and the sufficient conditions are expressed in terms of linear matrix inequalities. Finally, the effectiveness of proposed theoretical results are validated and compared with existing literature by numerical examples and their simulation results.

New stability criteria of singular systems with time-varying delay via free-matrix-based integral inequality

ABSTRACTThis paper concerns the stability problem of singular systems with time-varying delay. First, the singular system with time-varying delay is transformed into the neutral system with time-varying delay. Second, a more proper Lyapunov–Krasovskii functional (LKF) is constructed by adding some integral terms to quadratic forms. Then, to obtain less conservative conditions, the free-matrix-based integral inequality is adopted to estimate the derivative of LKF. As a result, some delay-dependent stability criteria are given in terms of linear matrix inequalities. Finally, two numerical examples are provided to demonstrate the effectiveness and superiority of the proposed method.