Lyapunov-Krasovskii Functionals Research Articles

Low conservative stability criteria for discrete-time Lur’e systems with sector and slope constrained nonlinearities

Lur’e system is an important nonlinear system in engineering, which is composed of a feedback connection of a linear dynamic system and a nonlinear system that satisfies the sector constraint. In this paper, for nominal and uncertain discrete Lur’e systems with time-varying delays, the absolute and robustly absolute stability are studied in detail with the nonlinearity satisfying the sector and slope constraints. Firstly, an augmented Lyapunov–Krasovskii functional (LKF) is designed, in which some augmented vectors are selected to increase the coupling information between the delay intervals and other system state variables. At the same time, some effective integral terms of nonlinear state variables are extended in the LKF according to the characteristics of sector constraint and slope constraint. Secondly, a general lemma of summation inequality for free-weighted matrices is given, which adds some coupling information between vectors and extends the search range for solving linear matrix inequalities (LMIs). Thirdly, by synthesizing the general summation inequality and the newly constructed LKF, different absolute stability criteria are derived respectively, and the absolute stability criteria are extended to robustly absolute stability cases with uncertain parameters. The stability criteria reduce the conservatism of some previously proposed results. Finally, some common numerical examples and simulations are used to verify the results of this paper.

Stability analysis of delayed load frequency control system based on a novel augmented functional

This article is concerned with the stability issue of PI-based load frequency control (LFC) systems with time-varying delays and load interference. Firstly, a novel augmented Lyapunov–Krasovskii functional (LKF) is developed, in which the single integral term includes the augmented s-dependent integral term of delay-partition. For the purpose of coordinating with the constructed LKF effectively, a generalised free-matrix-based integral inequality and quadratic inequality are utilised to estimate the functional derivatives accurately, so that the stability criterion of less conservative is obtained. Finally, the relationship between the PI gains and the delay margins is presented, and the effects of time delays and PI gains on the system dynamic performance are discussed simultaneously by simulation. The simulation results verify the validity of the proposed method.

Improved stability criteria of neural networks with time-varying delay based on the variable-augmented-based free-weighting-matrices

This paper discusses the stability problem of neural networks with time-varying delays. Firstly, a delay-product-type term is added to the Lyapunov–Krasovskii functionals(LKF) to take into account extra time-delay-related information. Then, by employing the variable-augmented-based free-weighting-matrices, the higher-order terms of the time-varying delay are successfully avoided in estimating the derivative of the LKF. Nextly, two stability criteria with less conservatism are established. Finally, two examples are used to illustrate the effectiveness of the presented criteria.

Designing asynchronous resilient sampled data controller for nonlinear Markov jump systems via hidden Markov model and quantum genetic algorithm

The article delves into the problem of stochastic stability analysis for nonlinear Markov jump systems (MJSs) with fuzzy asynchronous resilient sampled-data controllers (SDCs) using hidden Markov model (HMM) and quantum genetic algorithms (QGAs). A novel framework using QGA has been developed to obtain the control gain matrices by solving the linear matrix inequality (LMI). A new criterion is proposed to guarantee the stochastic stability of fuzzy nonlinear MJSs, relying on the mode-dependent Lyapunov-Krasovskii functional (LKF) approach. Standard Chua’s circuit has been used to show the reliability and value of the developed results. Also, the Lorentz system has been explored to demonstrate the non-stochastic case.

On stability and H∞ control synthesis of sampled‐data systems: A multiple convex function approximation approach

AbstractThis article deals with the stability and control synthesis for a class of linear sampled‐data systems. First, a novel integral inequality with the cubic term of integral interval length instead of reciprocal one of integral interval length is constructed by non‐orthogonal polynomials. This can make full use of slack matrix variables and additional information about sawtooth structural sampling pattern. Then, on the basis of the constructed integral inequality, a cubic‐term‐dependent discontinuous exponent Lyapunov–Krasovskii Functional (LKF) is developed containing additional information about system states to design an sampled‐data state feedback controller. To better estimate the upper bound of the derivative of LKF, a kind of multiple convex function approximation approach is developed, which can be used to deal with the two‐variable polynomial negative definite issue by dividing the variable interval into multiple subintervals. By the constructed integral inequality and multiple convex function approximation approach, sufficient conditions with less conservatism are deduced for the feasibility of sampled‐data state feedback controller. Moreover, by the inner convex approximation solution technique and proposed iterative algorithm, the bilinear matrix inequalities can be transformed into linear matrix ones, which can be easily tackled, to obtain the desired sampled‐data controller gain. Finally, by comparing with existing results, the advantages and superiority of our proposed approaches can be confirmed via two numerical examples.

Event-based nonfragile state estimation for memristive recurrent neural networks with stochastic cyber-attacks and sensor saturations

Abstract This paper addresses the issue of nonfragile state estimation (SE) for memristive recurrent neural networks (MRNNs) with proportional delay and sensor saturations. In practical engineering, numerous unnecessary signals are transmitted to the estimator through the networks, which increases the burden of communication bandwidth. In this paper, a dynamic event-triggered mechanism (DETM) is employed to select useful data instead of a static event-triggered mechanism. By constructing a meaningful LyapunovKrasovskii functional (LKF), a delay-dependent criterion is derived in terms of linear matrix inequalities (LMIs) for ensuring the global asymptotic stability of the augmented system. In the end, two numerical simulations are employed to illustrate the feasibility and validity of the proposed theoretical results.

Finite-time interval stabilization for time-varying stochastic delayed systems via interval matrix method by piecewise controllers

This paper presents a comprehensive study addressing the problem of finite-time interval stabilization (FTIS) for two distinct types of systems: linear time-invariant stochastic delayed systems (LTISDSs) and linear time-varying stochastic delayed systems (LTVSDSs). To begin, the definition of FTIS is established, which is based on employing piecewise state feedback controllers. Subsequently, a time-varying Lyapunov-Krasovskii functional (LKF) is proposed and a novel optimization algorithm is designed to address the challenge of FTIS for LTISDSs. Additionally, this paper investigates the equivalence of LTVSDSs to uncertain stochastic delayed systems with interval parameters, represented as a series of convex combinations. Notably, both the optimization algorithm and the linear matrix inequalities (LMIs) technique are employed to establish sufficient conditions for achieving FTIS in LTVSDSs. Furthermore, when compared to traditional state feedback controllers, the piecewise controllers not only enhance the precision of system control but also lead to reduced control gains and fewer control effort requirements. Finally, the findings are substantiated through the presentation of two numerical examples, providing practical validation for the proposed methodologies and their effectiveness.

Integrated fault estimation and control for multi-agent systems with external disturbance using delayed relative information

This paper is dedicated to the problem of actuator fault estimation and fault-tolerant control for general linear leader–follower multi-agent systems subject to external disturbance and communication delay. First, the intermediate variable observer is constructed to estimate faults using relative information with symmetric time delays. A contradiction between the convergence speed of estimation error and the time-delay margin is revealed, where quick convergence implies small time-delay margin. As such, the augmented system observer is employed to improve the convergence speed. In the next, integrated scheme is adopted to handle the bi-directional interaction between fault estimation and fault-tolerant control arising from external disturbance and time-varying faults. In addition, we also consider the asymmetric delay case with only transmission delay. The delay-independent stability condition is derived with the use of augmented system observer by virtue of Lyapunov–Krasovskii function and [Formula: see text] robust theories. Finally, the simulation results are presented to verify the effectiveness of proposed method.

Containment control for second-order multi-agent systems with time-varying delays via variable-augmented-based free-weighting matrices

This paper investigates the containment control problem for multi-agent systems (MASs), where the time-varying communication delays are considered. The aim of this paper is to introduce new methods for the containment control research of MASs with time-varying delays. To obtain better results on maximum admissible upper bounds (MAUBs) of the delays to ensure the containment control, the variable-augmented-based free-weighting matrices (VABFWMs) method is introduced to avoid the higher-order terms of time-varying delays in estimating the derivative of the Lyapunov-Krasovskii functionals (LKFs). So our method no longer requires the quadratic function negative-determination lemma. In addition, by utilizing the proposed method, we design a containment controller to ensure that all followers are included into the convex hull formed by leaders. Finally, a numerical example is presented to illustrate the feasibility and superiority of our method.

Sampled-Data Control for Exponential Synchronization of Delayed Inertial Neural Networks With Aperiodic Sampling and State Quantization.

This article is devoted to dealing with exponential synchronization for inertial neural networks (INNs) with heterogeneous time-varying delays (HTVDs) under the framework of aperiodic sampling and state quantization. First, by taking the effect of aperiodic sampling and state quantization into consideration, a novel quantized sampled-data (QSD) controller with time-varying control gain is designed to tackle the exponential synchronization of INNs. Second, considering the available information of the lower and upper bounds of each HTVD, a refined Lyapunov-Krasovskii functional (LKF) is proposed. Meanwhile, an improved looped-functional method is utilized to fully capture the characteristic of practical sampling patterns and further relax the positive definiteness requirement for LKF. Consequently, less conservative exponential synchronization conditions with extra flexibility are derived. Finally, a numerical example is employed to demonstrate the effectiveness and advantages of the proposed synchronization method.

Observer-based finite-time H∞ control for discrete-time singular Markovian jump systems with asynchronous modes

The asynchronous observer-based finite-time [Formula: see text] control problem for discrete-time singular Markovian jump systems with time-varying delays is studied, where the asynchronism of the controller and observer with respect to the original system is fully taken into consideration. By the use of finite-time analysis theory and the construction of suitable Lyapunov-Krasovskii functionals, sufficient conditions to determine the closed-loop error system is singular stochastic finite-time boundedness are given with [Formula: see text] performance level. Then, by applying linear matrix inequalities, the desired observer parameters and controller parameters can be computed. Finally, the effectiveness of our proposed method is confirmed by simulation.

Finite-Time H∞ Controllers Design for Stochastic Time-Delay Markovian Jump Systems with Partly Unknown Transition Probabilities.

This paper concentrates on the finite-time H∞ control problem for a type of stochastic discrete-time Markovian jump systems, characterized by time-delay and partly unknown transition probabilities. Initially, a stochastic finite-time (SFT) H∞ state feedback controller and an SFT H∞ observer-based state feedback controller are constructed to realize the closed-loop control of systems. Then, based on the Lyapunov-Krasovskii functional (LKF) method, some sufficient conditions are established to guarantee that closed-loop systems (CLSs) satisfy SFT boundedness and SFT H∞ boundedness. Furthermore, the controller gains are obtained with the use of the linear matrix inequality (LMI) approach. In the end, numerical examples reveal the reasonableness and effectiveness of the proposed designing schemes.

Extended dissipative criteria for delayed semi-discretized competitive neural networks

This brief investigates the extended dissipativity performance of semi-discretized competitive neural networks (CNNs) with time-varying delays. Inspired by the computational efficiency and feasibility of implementing the networks, we formulate a discrete counterpart to the continuous-time CNNs. By employing an appropriate Lyapunov–Krasovskii functional (LKF) and a relaxed summation inequality, sufficient conditions ensure the extended dissipative criteria of discretized CNNs are obtained in the linear matrix inequality framework. Finally, to refine our prediction, two numerical examples are provided to demonstrate the sustainability and merits of the theoretical results.

Stability analysis of delayed Takagi–Sugeno fuzzy systems via a membership-dependent reciprocally convex inequality

This article investigates the stability analysis of delayed Takagi–Sugeno (T–S) fuzzy systems. To begin with, a membership-dependent reciprocally convex inequality is proposed. Different from the frequently employed reciprocally convex inequalities, the proposed inequality can introduce the membership functions and provide a smaller bounding gap. Then, a suitable Lyapunov–Krasovskii functional (LKF) combined with the delay-product-type technique is established. As a consequence, two enhanced stability criteria for the delayed T–S fuzzy systems are developed. Lastly, the benefits of membership-dependent inequality are demonstrated by utilizing two well-known examples.

Synchronization of delayed stochastic reaction–diffusion Hopfield neural networks via sliding mode control

Synchronization of stochastic reaction–diffusion Hopfield neural networks with s-delays via sliding mode control is investigated in this article. To begin with, we choose suitable functional space for state variables, then the system is transformed into a functional differential equation in an infinite-dimensional Hilbert space by using appropriate functional analysis technique. Based on above preliminary preparation, sliding mode control (SMC) is constructed to drive the error trajectory into the designed switching surface. Specifically, the switching surface is constructed as linear combination of state variables, which is related to control gains. Then novel SMC law is designed which involving delay, reaction diffusion term, and reaching law. Furthermore, the criterion of mean-square exponential synchronization for stochastic delayed reaction–diffusion Hopfield neural networks with s-delays is given in the form of matrix form. This criterion is less restrictive and easy to check in computer. Meanwhile, a different novel Lyapunov–Krasovskii functional (LKF) mixed with Itô’s formula, Young inequality, Hanalay inequality is employed in this proof procedure. At last, a numerical example is presented to validate the availability of theoretical result. The simulation is based on the finite difference method, and numerical result coincides with the theoretical result proposed.

Consensus of stochastic multi-agent systems with time-delay and Markov jump

In this paper, the H ∞ consensus problem of stochastic nonlinear multi-agent systems with time-delay, Markov jump and ( x , u , v ) -dependent noises is studied. Firstly, the H ∞ consensus problem is transformed into a standard H ∞ control problem by model transformation. Then, a dynamic output feedback control protocol is constructed by solving a set of linear matrix inequalities to ensure that the closed-loop system achieves the mean square consensus and meets the specified H ∞ performance level. After that, both delay-independent and delay-dependent stochastic bounded real lemmas are established by taking advantage of the Lyapunov–Krasovskii function method and the generalised It o ^ formula. Finally, we illustrate the validity of the developed method with numerical simulations.

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.

Global exponential synchronization of switching neural networks with leakage time-varying delays

In this paper, the synchronization problem of a class of switching neural networks with leakage time-varying delays is studied. A system solution-based direct analysis method is proposed to derive the sufficient conditions of global exponential synchronization for master–slave systems. Firstly, the state variable expression of the error system is derived by constructing a suitable regulation function, in which the leakage delays are explicitly transformed outside the state variable. Then, on this basis, the corresponding synchronization conditions is obtained by using the transition condition ingeniously. In addition, the obtained sufficient conditions contain only some simple linear scalar inequalities, which is different from most publications and greatly reduces the computational complexity. Finally, the reliability of the theoretical results is verified by numerical simulation. It is worth noting that the synchronization problem of switching neural networks with time-varying leakage delays is studied for the first time in this paper, and the method adopted does not need to construct any Lyapunov–Krasovskii functionals, which simplifies the proof process.

Stability analysis of discrete-time delayed systems via matrix-injection-based transformation method for bivariate quadratic functions

In this paper, the stability analysis of discrete-time systems with a time-varying delay is studied. First, a selectable matrix-injection-based transformation lemma is proposed to guarantee the negative (or positive) definiteness of bivariate quadratic function. This lemma allows a selection of conditions that combine different delay sets, which is more relaxed than that in most previous studies. Then, an improved Lyapunov–Krasovskii functional (LKF) with delay-squared-product summation is given, which introduces extra cross-terms about delay-variation square for stability analysis. Considering the role of injection matrices in the proposed lemma, together with two types of allowable delay sets, four stability criteria are established. Finally, the less conservatism of the proposed criteria is verified by two numerical examples, which demonstrate the superiority of the presented methods.

Non-separation Method-Based Global Stability Criteria for Takagi–Sugeno Fuzzy Quaternion-Valued BAM Delayed Neural Networks Using Quaternion-valued Auxiliary Function-Based Integral Inequality

This paper focuses on the global asymptotic stability (GAS) problem for Takagi–Sugeno (T-S) fuzzy quaternion-valued bidirectional associative memory neural networks (QVBAMNNs) with discrete, distributed and leakage delays by using non-separation method. By applying T-S fuzzy model, we first consider a general form of T-S fuzzy QVBAMNNs with time delays. Then, by constructing appropriate Lyapunov–Krasovskii functionals and employing quaternion-valued integral inequalities and homeomorphism theory, several delay-dependent sufficient conditions are obtained to guarantee the existence and GAS of the considered neural networks (NNs). In addition, these theoretical results are presented in the form of quaternion-valued linear matrix inequalities (LMIs), which can be verified numerically using the effective YALMIP toolbox in MATLAB. Finally, two numerical illustrations are presented along with their simulations to demonstrate the validity of the theoretical analysis.