Asymptotic Stability Criteria Research Articles

Enhancing Stability Criteria for Linear Systems with Interval Time-Varying Delays via an Augmented Lyapunov–Krasovskii Functional

This work investigates the stability conditions for linear systems with time-varying delays via an augmented Lyapunov–Krasovskii functional (LKF). Two types of augmented LKFs with cross terms in integrals are suggested to improve the stability conditions for interval time-varying linear systems. In this work, the compositions of the LKFs are considered to enhance the feasible region of the stability criterion for linear systems. Mathematical tools such as Wirtinger-based integral inequality (WBII), zero equalities, reciprocally convex approach, and Finsler’s lemma are utilized to solve the problem of stability criteria. Two sufficient conditions are derived to guarantee the asymptotic stability of the systems using linear matrix inequality (LMI). First, asymptotic stability criteria are induced by constructing the new augmented LKFs in Theorem 1. Then, simplified LKFs in Corollary 1 are proposed to show the effectiveness of Theorem 1. Second, asymmetric LKFs are shown to reduce the conservatism and the number of decision variables in Theorem 2. Finally, the advantages of the proposed criteria are verified by comparing maximum delay bounds in four examples. Four numerical examples show that the proposed Theorems 1 and 2 obtain less conservative results than existing outcomes. Particularly, Example 2 shows that the asymmetric LKF methods of Theorem 2 can provide larger delay bounds and fewer decision variables than Theorem 1 in some specific systems.

Complex Dynamics of a Discrete-Time Food Chain Model

In a food chain, the role of intake patterns of predators is very influential on the survival and extinction of the interacting species as well as the entire dynamics of the ecological system. In this study, we investigate the affluent and intricate dynamics of a simple three-species food chain model in a discrete-time framework by analyzing the parameter plane of the system with simultaneous changes of two crucial parameters, the predation rates of middle and top predators. From the theoretical viewpoint, we study the model by determining the fixed points’ biological feasibility and local asymptotic stability criteria, and performing some analyses of local bifurcations, namely, transcritical, flip, and Neimark–Sacker bifurcations. Here, we initiate the numerical simulation by plotting the changes of the prey population density in terms of a vital parameter of the system, and we observe the switching among different dynamical behaviors of the system. We also draw some phase portraits and plot the time series solutions to show the diverse characteristics of the system dynamics. Further, we move one step ahead to explore the intricate dynamical scenarios appearing in the parameter plane by forming Lyapunov exponent and isoperiodic diagrams. In the parameter plane of the system, we see the emergence of innumerable Arnold tongues. All these Arnold tongues are organized along a particular direction, and the beautiful arrangement of these tongues forms several kinds of period-adding sequences. The study sheds more light on various types of multistability occurring in the model system. We see the coexistence of three periodic attractors in the parameter plane. In this study, the most striking observation is the coexistence of four periodic attractors, which occurs infrequently in ecological systems. We draw the basins of attraction for the tristable and tetrastable attractors, which are complex Wada basins. A system with Wada basin is very sensitive to initial conditions and more erratic in nature than a system with fractal basin. Also, we plot the density of all interacting species in terms of the predation rates of middle and top predators and observe the variation in the population densities of all species with the variability of these two key parameters. In the parameter plane created by the simultaneous changes of two parameters, the system exhibits a variety of intricate and subtle dynamics, which cannot be found by changing only a single parameter.

NOVEL UNIFIED STABILITY CRITERION FOR FRACTIONAL-ORDER TIME DELAY SYSTEMS WITH STRONG RESISTANCE TO FRACTIONAL ORDERS

In this study, a novel unified stability criterion is first proposed for general fractional-order systems with time delay when the fractional order is from 0 to 1. Such a new unified criterion has the advantage of having an initiative link with the fractional orders. A further advantage is that the corresponding asymptotic stability theorem, derived from the proposed criterion used to analyze the asymptotic stability, is only slightly affected by the change of the fractional order. In addition, the unified stability criterion is applied to general multi-dimensional nonlinear fractional-order systems with time delays, the corresponding asymptotic stability criterion is applied by combining the vector Lyapunov function with the M-matrix method. Compared with the traditional stability criterion, the unified stability criterion is slightly influenced by the changing fractional order and large time delays. The reliability and effectiveness of the novel uniform stability criterion were verified through three representative examples.

Asymptotic stability of impulsive stochastic switched system with double state-dependent delays and application to neural networks and neural network-based lecture skills assessment of normal students

<abstract><p>This article investigates the stability problem of impulsive stochastic switched systems with double state-dependent delays. In the designed system, unstable and stable impulses are taken into consideration, respectively, and they do not need to function simultaneously with switching behavior. Additionally, two new ideas, i.e., mode-dependent switching density and mode-dependent impulsive density, are developed. Based on the Lyapunov function method and comparison principle, the asymptotic stability criteria for an impulsive stochastic switched system with state-dependent delays are given. Moreover, the application of theoretical results to neural networks and the neural network-based lecture skills assessment of normal students is analyzed. Finally, two numerical examples are provided to illustrate the effectiveness and reliability of the theoretical criteria.</p></abstract>

Switching-like dynamic event-triggered load frequency control for multi-area power systems with electric vehicles under DoS attacks

The load frequency control (LFC) for multi-area power systems with electric vehicles (EVs) is investigated under Denial of Service (DoS) attacks. First, a new dynamic event-triggered mechanism is proposed, where the event parameters can be switched according to whether DoS attacks occur or not. Then, a time-delay model with parameter uncertainty is developed by considering the state of charge (SOC) of EVs batteries and DoS attacks. Particularly, asymptotic stability criterion with prescribed H∞ performance is derived, and the relationship between the packet losses induced by DoS attacks and the event parameters is revealed. Further, a co-design scheme of the controller gain and the event-triggered weighting matrix is provided. Finally, simulation results show that the proposed method is advantageous to resist DoS attacks and save communication resources.

Improved Adaptive Dynamic Event-Triggered Consensus of Multi-Agent Systems

This brief exploits the adaptive dynamic event-triggered (ADET) consensus problem for multi-agent systems (MASs). Firstly, we propose a novel ADET protocol which improves the existing dynamic event triggered (DET) protocol and adaptive event-triggered (AET) protocol based on sampling control. The possibility that linear matrix inequalities (LMIs) has no feasible solution is reduced by presetting an adaptive maximum threshold. The asymptotic stability criterion is obtained. Subsequently, the controller gain and ADET parameters are obtained. Finally, the validity of the theoretical results in this brif is verified by a numerical simulation.

Impulsively fractional‐order time‐delayed gene regulatory networks: Existence and asymptotic stability

AbstractThis paper presents asymptotic stability criteria for fractional‐order gene regulatory networks (FOGRNs) with impulses, time delays, and two numerical cases to illustrate the applicability of the results. The established system's boundedness, existence, and uniqueness are discussed using the Mittag–Leffler function, homeomorphism theory, and Cauchy–Schwartz inequality. The delay‐independent asymptotic stability criteria for FOGRNs are derived using algebraic and LMI methods, famous inequality techniques, and Lyapunov stability theory.

A sampled-data stabilisation analysis on memristive BAM neural networks for memductance function

In this paper, we explore a stabilisation problem for two different class of memductance functions for memristive bidirectional associative memory neural networks (MBAMNNs) with time-varying delay. A new Lyapunov-Krasovskii function (LKF) with double and triple integral terms is derived with some sufficient conditions to guarantee the global asymptotic stability criteria using the Lyapunov stability theory. The controller gain matrix for sampled-data controller is obtained for the state-dependent switched system and state-dependent continuous system in terms of linear matrix inequalities (LMIs). Finally, we stabilise the considered model by visualising the simulation part using standard MATLAB software which demonstrates the validness of the theoretical results.

Global Asymptotic Stability Analysis of Fixed Points for a Density-Dependent Single-Species Population Growth Model

In a density-dependent single-species population growth model, a simple method is proposed to explicitly and directly derive the analytic expressions of reliable regions for local and global asymptotic stability. Specifically, first, a reliable region ΛLAS is explicitly represented by solving the fixed point and utilizing the asymptotic stability criterion, over which the fixed point is locally asymptotically stable. Then, two types of auxiliary Liapunov functions are constructed, where the variation of the Liapunov function is decomposed into the product of two functions and is always negative at the non-equilibrium state. Finally, based on the Liapunov stability theorem, a closed-form expression of reliable region ΛGAS is obtained, where the fixed point is globally asymptotically stable in the sense that all the solutions tend to fixed point. Numerical results show that our analytic expressions of reliable regions are accurate for both local and global asymptotic stability.

Asymptotical Stability Criteria for Exact Solutions and Numerical Solutions of Nonlinear Impulsive Neutral Delay Differential Equations

In this paper, the idea of two transformations is first proposed and applied. Some new different sufficient conditions for the asymptotical stability of the exact solutions of nonlinear impulsive neutral delay differential equations (INDDEs) are obtained. A new numerical scheme for INDDEs is also constructed based on the idea. The numerical methods that can preserve the stability and asymptotical stability of the exact solutions are provided. Two numerical examples are provided to demonstrate the theoretical results.

Qualitative analysis of TB transmission dynamics considering both the age since latency and relapse

Since the beginning of time, tuberculosis (TB) has been a fatal illness that predominantly affects the human lungs before spreading to other organs including the brain, spine, etc. The main elements of TB mitigation are age-dependent heterogeneity, identifying those who are latently infected, and treating them using the right diagnostic strategy. In this present work, the complex transmission mechanism of this disease in a population is described by a coupled system of integro-partial differential equations (IDE-PDE). The system’s well-posedness requirement is confirmed. The proposed system’s basic reproduction number (R0) is obtained. This work provides a complete analysis of the qualitative properties of the model, including steady state existence, asymptotic smoothness of the solution semi-flow, uniform persistence of the endemic equilibrium, and the global asymptotic stability criterion. It is observed that in assessing the severity of the pandemic, the value of R0 is crucial. Additionally, the stability results are visually illustrated by solving the model equations numerically while assuming two hypothetical cases. The current work also suggests several methods for reducing the value of the basic reproductive number (R0) by manipulating a few parameter values, which may help to lessen the prevalence of TB in a community.

Distributed fault-tolerant PI load frequency control for power system under stochastic event-triggered scheme

This article investigates the load frequency control problem of interconnected power systems by proposing a distributed fault-tolerant proportional integral (PI) control strategy. Firstly, a unified actuator fault model is established, and a distributed fault-tolerant PI control strategy is proposed on the basis of the actuator fault model. Then, a stochastic event-triggered scheme (SETS) based on stochastic sampling period is developed to relieve the redundant occupation of network communication resources. Further, an asymptotical stability criterion with H∞ performance is established by means of the Lyapunov method. Finally, illustrative examples are presented from two cases, isolated power system and interconnected power systems, to demonstrate the effectiveness of the designed control approach and the superiority of SETS compared with periodic event-triggered scheme.

Practical Input-to-State Stability of Nonlinear Systems with Time Delay

An investigation of the input-to-state practical stability (ISpS) and the integral input-to-state practical stability (iISpS) of nonlinear systems with time delays (NSWTDs) is presented in this paper. The ISpS and iISpS of the systems are obtained by using a continuously differentiable Lyapunov–Krasovskii functional (LKF) within-definite derivative, which generalizes the classical LKF with a positive definite derivative. As a result, the uniform practical asymptotic stability (UPAS) criteria of NSWTD were also given by using the proposed LKF.

Consensus-based Distributed Bus Voltage Control of in DC Microgrid Clusters

This paper proposes a consensus-based voltage control strategy for the bus of a DC microgrid cluster. Firstly, the closed-loop dynamics of a DC microgrid cluster bus voltage control system based on the consensus distributed control architecture is described, taking into full account the information exchange delay among the microgrid subsystems. Secondly, by constructing the Lyapunov-Krasovskii functional, the voltage control system asymptotic stability criteria under parameter uncertainties are rigorously derived, and the corresponding controller design criteria are obtained. Simulation results show that compared with the centralized control schemes, the proposed consensus distributed control method can reduce the average settling time by 24.39% and the average integral of absolute error of aggregate deviations by 23.92%.

Overflow Oscillation Elimination in Fixed-Point 2D Digital Filters Based on the Roesser Model

This paper investigates the overflow oscillation elimination problem in the fixed-point two-dimensional (2D) digital filter (DF) based on Roesser model subjected to two’s complement overflow (TCO) nonlinearities. By utilizing the system information and behavior of TCO nonlinearities more effectively, a new global asymptotic stability (GAS) criterion for 2D DFs is presented. A comparison of the obtained criterion with existing criteria is made with the help of examples.

Necessary and sufficient criteria for existence, regularity, and asymptotic stability of enhanced pullback attractors with applications to 3D primitive equations

We introduce several new concepts called enhanced pullback attractors for nonautonomous dynamical systems by improving the compactness and attraction of the usual pullback attractors in strong topology spaces uniformly over some infinite time intervals. Then we establish several necessary and sufficient criteria for the existence, regularity and asymptotic stability of these enhanced pullback attractors under general theoretical frameworks which are expected to be applied to many kinds of nonautonomous PDEs. These abstract results are applied to the famous 3D primitive equations modeling the large-scale ocean and atmosphere dynamics. We establish the existence, regularity and asymptotic stability of the enhanced pullback attractors for the nonautonomous primitive equations in [Formula: see text] and [Formula: see text] when the forces satisfy some mild conditions. The uniform pullback asymptotic compactness of the solution operators in [Formula: see text] and [Formula: see text] are established by deriving the uniform “flattening effects” of the solutions in [Formula: see text] and the uniform [Formula: see text]-Hölder continuity of the solutions from [Formula: see text] to [Formula: see text] with respect to the initial data, respectively. Our approaches are applicable to prove the existence of global attractors for the autonomous primitive equations in [Formula: see text] provided [Formula: see text].

State feedback controller design of fractional ordinary differential equations coupled with a fractional reaction–advection–diffusion equation with delay

The boundary control problem of fractional ordinary differential equations coupled with a time fractional reaction–advection–diffusion equation with delay is studied in this paper. To ensure the asymptotic stability of the system we studied, a state feedback boundary controller is proposed. By backstepping method, we transform the fractional coupled system into a chosen target system under a controller. Furthermore, we obtain the existence and uniqueness of the state solution of the considered system. A Lyapunov functional is constructed to show the asymptotic stability of the fractional coupled system by the special fractional Halanay inequality. The asymptotic stability criterion of the fractional coupled system is described by Linear Matrix Inequalities (LIMs). Which can be easily solved and verified. Finally, the applicability of our theoretical results is showed by a numerical simulation.

Quantization-Uncertainty-Dependent Analysis and Control of Linear Systems With Multi-Input–Multi-Output Quantization

This paper proposes a novel quantized control strategy for network-based linear systems subject to multi-input-multi-output (MIMO) quantization. A logarithmic quantization scheme is adopted for characterizing the quantization effect on system dynamics. A sufficient and necessary condition on the asymptotic stability is established for quantized MIMO systems. To improve the numerical testability of the obtained results, a polytopic approach approximating the MIMO quantization uncertainties is developed. By constructing a novel Lyapunov function that has dependence on the MIMO quantization uncertainties, asymptotic stability criteria are established for closed-loop quantized MIMO systems. The conditions on the existence of state-feedback controllers that guarantee the closed-loop stability are derived based on the proposed technique that decouples the controller gains and the parameters of MIMO quantization uncertainties. The proposed method and the associated theoretical results are extended to the disturbance attenuation case. Finally, the theoretical results are applied to a benchmark example and a converter circuit to illustrate their efficacy and superiority.

Dynamic memory event-triggered proportional-integral-based [formula omitted] load frequency control for multi-area wind power systems

Aiming at ensuring the stability of multi-area doubly fed induction generator (DFIG)-integrated wind power systems, a dynamic memory event-triggered H∞ load frequency control (LFC) scheme is presented in this article. Firstly, a unified LFC model of multi-area DFIG-integrated wind power systems is constructed considering the grid connection of the DFIG-based wind turbine. Secondly, a dynamic memory event-triggered scheme (DMETS) is proposed to reduce the utilization of communication bandwidth by introducing a dynamic threshold parameter and historical data. Furthermore, the designed multi-area LFC model is reconstructed as a time-delay system under DMETS and proportional-integral (PI) controller with transmission delays introduced by open communication network. Then, asymptotic stability criteria meeting the H∞ attenuation performance index and an H∞ controller design criterion are obtained according to a selected Lyapunov–Krasovskii functional. Finally, numerical examples are presented to demonstrate the availability of the designed control scheme and the advantages of the proposed DMETS in reducing communication burden.

Stability and Finite-Time Synchronization Analysis for Recurrent Neural Networks with Improved Integral-Type Time-Varying Delays

This paper studies the stability criterion of integral time-varying recurrent neural networks (RNNs) with zero lower bound and finite-time synchronization based on improved sliding mode control (SMC). Firstly, a sufficient criterion for universal asymptotic stability of RNNs with integral time-varying delays is obtained by estimating a tight upper bound of augmented Lyapunov-Krasovskii functional (LKF) derivative with inequality scaling technique and mutually convex combined inequality. Secondly, in order to eliminate the time that error system state trajectory slides along sliding mode flow pattern until convergence at the origin, based on drive response and SMC theory, a suitable sliding mode controller is designed by considering that sliding mode flow pattern is equal to synchronization error. Finally, maximum allowable upper bound of delay under different delay derivatives are obtained by considering trajectory change of input function under different initial value. Synchronization trajectory of drive and response systems with mismatched parameters and activation functions under the influence of controller are studied, and synchronization time which is required for error system to reach stability is obtained. Simulation results show that the introduction of integral delay can be more comprehensive from both difference and area, so that drive system state is eventually steady at equilibrium point and synchronized with response system. Stability criterion of this paper not only has less conservative and computation complexity but also has shorter synchronization control time.