Time-varying Delay Systems Research Articles

New negative-determination conditions for cubic polynomials with applications to time-varying delay systems

This paper studies the stability of time delay systems. The Lyapunov-Krasovskii functional (LKF) method is used for our study, in which a novel negative-determination criterion for cubic polynomials is proposed. An improved stability criterion of time delay system is obtained by the new method. The effectiveness of the proposed method is verified by some numerical examples and reduced conservativeness can be obtained.

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 Analysis of Linear Time-Varying Delay Systems via a Novel Augmented Variable Approach

This paper investigates the stability issues of time-varying delay systems. Firstly, a novel augmented Lyapunov functional is constructed for a class of bounded time-varying delays by introducing new double integral terms. Subsequently, a time-varying matrix-dependent zero equation is introduced to relax the constraints of traditional constant matrix-dependent zero equations. Secondly, for a class of periodic time-varying delays, considering the monotonicity of the delay and combining it with an augmented variable approach, Lyapunov functionals are constructed for monotonically increasing and monotonically decreasing delay intervals, respectively. Based on the constructed augmented Lyapunov functionals and the employed time-varying zero equation, less conservative stability criteria are obtained separately for bounded and periodic time-varying delays. Lastly, three examples are used to verify the superiority of the stability conditions obtained in this paper.

Stability for Markov switching stochastic delay systems binding event-triggered mechanism to activate multi-impulse jumps

This paper focuses on the pth moment exponentially stability for Markov switching and multi-impulse jumps stochastic time-varying delay system, where the switching behavior among subsystems of the target system is determined by Markov chains, and the occurrence of impulsive jumps is decided according to event-triggered impulsive mechanism when certain well-designed conditions are satisfied. By applying the Itô formula, Gronwall inequality and Razumikhin theorem, some novel sufficient criteria are provided to assure the system stability and get rid of Zeno phenomenon. It is worth pointing out that the multi-impulse jumps are our research aim and the range of delays considered is relatively wide, i.e., the daily bounded delay τ(t)∈[0,1) and the unupper bound delay τ(t)∈[1,∞). Subsequently, two diverse event trigger mechanism about impulsive jumps are proposed for such two types of delays, namely the defined event-triggered impulsive mechanism with delay. Finally, the validity and feasibility of the developed theoretical results are verified by two numerical simulations.

Retracted: Global Exponential Stability of Both Continuous-Time and Discrete-Time Switched Positive Time-Varying Delay Systems with Interval Uncertainties and All Unstable Subsystems

Retracted: Global Exponential Stability of Both Continuous-Time and Discrete-Time Switched Positive Time-Varying Delay Systems with Interval Uncertainties and All Unstable Subsystems

Finite-Time Boundedness of Switched Time-Varying Delay Systems With Actuator Saturation: Applications in Water Pollution Control

Finite-Time Boundedness of Switched Time-Varying Delay Systems With Actuator Saturation: Applications in Water Pollution Control

Control for Stochastic Singular Systems With Time-Varying Delays via Sampled-Data Controller.

In this article, control for stochastic singular time-varying delay systems under arbitrarily variable samplings is addressed via designing a sampled-data controller. The first and foremost, a novel time-dependent discontinuous Lyapunov-Krasovskii (L-K) functional is built, which takes good advantage of the factual sampling pattern's available properties. Then, based on the refined input delay method by utilizing the constructed time-dependent L-K functional, the free-weighting matrix method, and the auxiliary vector function approach are adopted to develop conditions ensuring the stochastic admissibility for the studied stochastic singular systems with time-varying delays. On the basis of the derived conditions, the sampled-data control issue is tackled, and an unambiguous expression for the sampled-data controller design method is obtained. Finally, simulation examples manifest that our proposed results are correct and effective.

The finite-time boundedness and control analysis of switched nonlinear time-varying delay systems based on auxiliary matrices

This paper explores the finite-time boundedness and controller problem of switching nonlinear delay systems. The time delay is considered as time-varying delay rather than constant delay. First of all, an appropriate Lyapunov–Krasovskii functional (LKF) which contains three integral terms is selected. The upper and lower bounds of these integral terms are given by segmenting the time delay interval. Meanwhile, the auxiliary matrix method is used to separate the Lyapunov matrix from the system matrix. Moreover, a new state-dependent switching law is designed to switch between subsystems according to the change of the magnitude of the LKF. Based on the above methods, the first step is to research the finite-time bounded analysis of delayed system without control input. Then, a sufficient condition is obtained to ensure the finite-time boundedness of switched systems with control input. Further, by designing an observer-based H∞ controller, the closed-loop system with time-varying delay is bounded and has H∞ performance index ν by using the Finsler’s Lemma and Singular Value Decomposition (SVD). Simultaneously, the controller and observer gains are designed. Finally, two simulations are used to verify the effectiveness and feasibility of the proposed methods.

An improved delay-dependent stabilization criterion of linear time-varying delay systems: An iterative method

An improved delay-dependent stabilization criterion of linear time-varying delay systems: An iterative method

Observer based stabilization of linear time delay systems using new augmented LKF

This paper addresses the stabilization problem of linear time-varying delay systems with unmeasurable states. A novel augmented Lyapunov–Krasovskii functional (LKF) is proposed that effectively accounts for the impact of time delays, and an observer based stabilization controller is developed employing linear matrix inequality (LMI) based optimization technique. The utilization of extended reciprocally convex matrix inequality (ERCMI) is employed in this work to establish less conservative stabilization conditions within the framework of linear matrix inequalities (LMIs). By formulating a convex optimization problem, the observer gain and controller gains are determined. Simulation results are used to validate the design, and two numerical examples are considered to prove the usefulness of the proposed method over existing methods.

Finite time SF bifurcation and stability analysis for stochastic time-varying delay system

In this paper, it is found that the systems alternate between oscillatory and steady states in studying the stability of a class of stochastic dynamical systems with fast time-varying periodic delays. We propose a new definition of bifurcation, State Flip (SF) bifurcation, by using the abrupt change in the dwell time ratio of two states. Taking the amplitude of the time-varying delay function δ as the bifurcation parameter, we investigate the dynamics of the evolution of a time-varying delay system driven by Gaussian white noise. Firstly, we convert systems with fast time-varying periodic delays into comparative systems with distributed delays, which in turn are converted into stochastic systems with constant delays for discussion. In order to reduce the stochastic delay equation to the average Itô equation, we adopted the stochastic averaging method and the generalized central manifold theory. It is used to capture the one-dimensional slow variables of a system near the critical state of the transition between two steady states during the evolution of the system. We then find that as the amplitude increases from weak to strong, the stochastic time-varying delay system changes from a two-period oscillation to alternating between oscillation and steady, and finally to a steady state. Lastly, we perform numerical simulations of the original stochastic time-varying delay system and the transformed two-dimensional stochastic constant time-delay system respectively to verify that the ideas and methods in this paper are reasonable and effective.

Enhanced Exponential Stability Analysis for Switched Linear Time-Varying Delay Systems Under Admissible Edge-Dependent Average Dwell-Time Strategy

This article focuses on the exponential stability analysis for the switched linear systems with time-varying delay. By constructing a new augmented multiple Lyapunov–Krasovskii functional, containing time-varying delay information, coupling of current and delayed states, state integral, and state derivative terms the feasibility of the stability condition is considerably enhanced. Besides, the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N$</tex-math> </inline-formula> -order canonical Bessel–Legendre integral inequality and the convex function property are utilized to give more relaxed criteria. On the other hand, for the first time, more flexible switching is employed with an admissible edge-dependent average dwell-time strategy for the switched time-varying delay systems (STDSs). Novel improved delay-dependent stability conditions are expressed by two theorems in the form of linear matrix inequalities which can be used to guarantee the exponential stability of the STDSs in the different delay conditions. Finally, significant improvements over the state-of-the-art in terms of delay upper bound, exponential decay rate, and dwell time are presented by simulating two numerical examples and one practical example including streams water quality preserving.

Stability Analysis of Time-Varying Neutral Stochastic Hybrid Delay System

Stability Analysis of Time-Varying Neutral Stochastic Hybrid Delay System

A Sufficient Condition on Polynomial Inequalities and its Application to Interval Time-Varying Delay Systems

This article examines the stability problem of systems with interval time-varying delays. In the derivation of Lyapunov–Krasovskii functional (LKF), non-convex higher-degree polynomials may arise with respect to interval time-varying delays, making it difficult to determine the negative definiteness of LKF’s derivative. This study was conducted to obtain stability conditions that can be described as linear matrix inequalities (LMIs). By considering the idea of matrix transition and introducing the delay-dependent augmented vector, a novel higher-degree polynomial inequality is proposed under the condition that the lower bound of the polynomial function variable is non-zero, which encompasses the existing lemmas as its special cases. Then, benefiting from this inequality, a stability criterion is derived in terms of LMIs. Finally, several typical examples are presented to verify the availability and strength of the stability condition.

Hierarchical conditions for exponential stabilisation of time-varying delay systems using Bessel–Legendre inequalities and Finsler's lemma

This paper addresses the exponential stabilisation of time-varying delay systems by means of Lyapunov–Krasovskii approaches, Bessel–Legendre inequalities and Linear Matrix Inequalities (LMIs). The key aspect of the proposed control synthesis method is that the LMI conditions are proved for the first time to be hierarchical under time-varying delays, that is to say, a progressive conservatism reduction is achieved as long as the degree of Legendre polynomials is increased. To this end, Finsler's lemma has been properly applied to decouple the Lyapunov matrices from the controller gain. Finally, simulation results are provided to illustrate the effectiveness of the proposed method.

Conditions of stability and guaranteed convergence rate for linear time-varying discrete-time delay systems

This work introduces novel results on the stability of linear time-varying discrete-time delay systems, addressing both internal stability and input-to-state stability. We start from the special case of positive delay systems and provide conditions of global exponential stability, both delay-dependent and delay-independent. Moreover, we formulate an explicit bound on the exponential decay rate of the state evolution, as a function of the maximum delay. We then show that the aforementioned stability conditions also prove input-to-state stability. The stability conditions consist of simple linear inequalities, albeit infinite-many due to the time-varying nature of the system. To overcome the computational burden, we also introduce a simple relaxation that allows to check the stability with a finite test, at the expense of conservatism. Finally, leveraging comparison systems and results from the theory of non-negative matrices, we generalize all the aforementioned results to systems with no sign constraint, for which we still provide stability conditions by means of linear inequalities and guaranteed convergence rate. Some consequences and a comparative example are provided for the well investigated special case of systems with constant matrices and time-varying delays.

Mathematical Model for Fault Handling of Singular Nonlinear Time-Varying Delay Systems Based on T-S Fuzzy Model

In this paper, a mathematical model based on the T-S fuzzy model is proposed to solve the fault estimation (FE) and fault-tolerant control (FTC) problem for singular nonlinear time-varying delay (TVD) systems with sensor fault. TVD is is extremely difficult to solve and the Laplace transform is devised to build an equal system free of TVD. Additionally, the sensor fault is changed to actuator fault by the developed coordinate transformation. A fuzzy learning fault estimator is first built to estimate the detailed sensor fault information. Then, a PI FTC scheme is suggested aiming at minimizing the damage caused by the fault. Simulation results from multiple faults reveal that the FE and FTC algorithms are able to estimate the fault and guarantee the system performance properly.

Sampled Memory-Event-Triggered Fuzzy Load Frequency Control for Wind Power Systems Subject to Outliers and Transmission Delays.

This study is devoted to event-triggered fuzzy load frequency control (LFC) for wind power systems (WPSs) with measurement outliers and transmission delays. Due to the integration of wind turbine (WT) with nonlinearity, the T-S fuzzy model of WPS is established for stability analysis and controller design. To mitigate the network burden, a new sampled memory-event-triggered mechanism (SMETM) related to historical system information is presented. It has the following two merits: 1) the utilization of continuous memory outputs over a given interval is useful to reduce the information loss in the period of samples and the redundant triggering events induced by disturbances and noises and 2) an extra upper constraint is added in the triggering condition to generate a new event only when the error signal belongs to a bounded range, thus, the false events caused by measurement outliers can be differentiated out and then be dropped. By representing the memory signal with transmission delay as a time-varying distributed delay term, a T-S fuzzy time-varying distributed delay system is built up to model the H∞ LFC WPS. With the help of the Lyapunov method and the integral inequality relying on distributed delay, some criteria are derived to solve the triggering matrix and fuzzy controllers. Finally, the merits of the proposed SMETM are tested by simulation results.

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.

Unknown input observers for time-varying delay Takagi–Sugeno fuzzy systems with unmeasured nonlinear consequents

This paper investigates the problem of unknown input observer (UIO) design for Takagi–Sugeno (TS) fuzzy systems with time-varying delay. For observer design, all the unmeasured nonlinearities are isolated in the nonlinear consequent parts of TS fuzzy systems. This allows dealing with the well-known issue related to unmeasured premise variables for TS fuzzy observer design in a more effective fashion via the mean value theorem. Using the unknown input (UI) decoupling technique, no a priori information on the UI is needed for the proposed TS fuzzy observer design, which can guarantee the asymptotic estimations of both the system state and the unknown input. Employing a fuzzy Lyapunov–Krasovskii functional with a suitable augmented vector, the effect of time-varying delay on the estimation error dynamics is taken into account in the observer design. Numerical examples are presented to demonstrate the effectiveness of the proposed fuzzy UIO design method for nonlinear time-varying delay systems.