Wirtinger-based Integral Inequality 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.

An investigation into the stability of helicopter control system under the influence of time delay using an improved LKF

This paper presents an improved Lyapunov–Krasovskii functional (LKF)-based stability analysis of time-delay systems (TDS) and its application to helicopter control systems. A novel LKF is proposed for the stability analysis of linear TDS after thoroughly examining the effects of each integral associated with a general class of LKF. The suggested stability results in the form of linear matrix inequalities (LMIs) are obtained using the Wirtinger-based integral inequality (WII), reciprocally convex combination lemma (RCCL), and extended reciprocally convex matrix inequality (ERCMI). Numerical examples are utilized to evaluate the impact of the presented theorems on stability. The proposed theorems are then used to examine the stability of helicopter control systems, and the results of these stability analyses are verified using a laboratory model of a helicopter control system.

Observer-based fuzzy integral sliding mode control for bilateral teleoperation systems with time-varying delays

This paper aims to examine the stability and tracking performance of a bilateral teleoperation system. The Takagi–Sugeno fuzzy method is utilized, through which the nonlinear master–slave dynamics is converted as a fuzzy-based system. The state observers are designed for the linearized fuzzy teleoperation systems, and the corresponding estimation errors are formulated. Importantly, a novel observer-based fuzzy integral sliding mode control is developed by deliberately introducing the delay term into the sliding surfaces. As such, advanced delay-product type of Lyapunov–Krasovskii functionals are constructed for the augmented state vectors, in order to acquire the additional delay information. In addition, the Wirtinger-based integral inequality along with an extended reciprocally convex matrix inequality is applied to the Lyapunov derivatives to establish the delay-dependent stability conditions. Numerical results are provided to demonstrate efficacy of the developed control mechanism.

Sampled-data output tracking control based on T–S fuzzy model for cancer-tumor-immune systems

This study investigates the tracking control issue for a cancer-tumor immune nonlinear model, which explains the interaction between cancer cells and the immune system in the body. To achieve this, the nonlinear model is first converted into linear sub-models using the Takagi–Sugeno (T–S) fuzzy methodology. The tracking and control of cancer cell proliferation while preserving immune cells is proposed using a sample-data output tracking control technique. Time-dependent discontinuous Lyapunov Krasovskii functional is recommended based on the improved Wirtinger inequality, this may provide additional information about the sawtooth structure of the sampling interval. Less conservative stability criteria are extracted having the form of linear matrix inequalities (LMIs) in consequence of extended reciprocally convex matrix inequality and Wirtinger-based integral inequality. The proposed approach achieves the stabilization of an augmented time delay system that connects with a given fuzzy nonlinear model and tracking error system. Furthermore, the proposed approach reduces the impact of external disturbances under the H∞ norm bound.

Nonfragile Memory Sampled-Data Control for Exponential Synchronization of Complex Dynamical Networks With Time-Varying Coupling Delay

This article addresses the issue of exponential synchronization in complex dynamical networks. First, the nonfragile memory sampled-data control is used to mitigate the impact of controller uncertainties. Second, an augmented Lyapunov–Krasovskii functional (LKF) is built using a modified two-sided looped-functional that requires the positive definite only at sampling instants instead of throughout the sampling period. Moreover, the constructed LKF takes into account the whole sampling period and encompasses information on both the current states and the delayed states. Some exponential synchronization criteria are obtained by combining the Wirtinger-based integral inequality with the optimal reciprocally convex technology, and this scheme results in a larger sampling upper bound. At last, two extensively used numerical simulation examples confirm the proposed method's validity and advantage.

Mixed dissipative filtering design for uncertain singular Markovian jump time-varying delay systems with nonlinear perturbations and generally uncertain transition rates

This paper studies the dissipative filtering of uncertain singular time-varying delay Markovian jump systems (MJSs) with nonlinear perturbations and generally uncertain transition rates (GUTRs). We focus on the design of a mixed filter, which can include mode-dependent and mode-independent filters in a unified and simple structure. First, a new Lyapunov–Krasovskii functional is proposed through the delay-decomposition approach, and improved Wirtinger-based integral inequalities are used to obtain a sufficient condition to ensure the stochastic admissibility of the considered uncertain nonlinear singular time-varying delay MJSs with dissipative performance. Based on these ingredients, the explicit expression of the desired mixed filter is obtained. Finally, some numerical examples and the filter design of a two-loop circuit network reflect the less conservatism and effectiveness of the method in this article.

Synchronization of complex dynamical networks subjected to actuator faults and periodic scaling attacks using probabilistic time-varying delays

In this paper, the synchronization problem of complex dynamical networks (CDNs) subjected to actuator faults and periodic scaling attacks using probabilistic time-varying delays is investigated. In order to handle the fault effects in actuators of proposed complex networks, an actuator fault model is considered. The major objective of this paper is to construct a memory state feedback controller with time-varying actuator faults that are asymptotically synchronized in the mean square sense to defend against periodic scaling attacks. By establishing suitable Lyapunov-Krasovskii functionals and applying Wirtinger-based integral inequality, sufficient conditions are derived in terms of linear matrix inequalities (LMIs) for the proposed system. Moreover, a time-varying delay is supposed to depend on probabilistic distribution conditions. Thus, the resulting criteria that emerge are based on the amount of the delay as well as the probability that the delay will have values at certain intervals. At last, the efficiency of our proposed work is demonstrated by two numerical simulations.

Stability Criteria of Delayed Memristor-Based Neural Networks via Continuous-Time Model and Interval Matrix Approach

Modeling and stability analysis of memristor-based neural networks (MNNs) are the premise of designs and applications. Different from most previous research, delayed MNNs are described by continuous differential equations with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2^{2n^{2}+n}$</tex-math> </inline-formula> variables where the memductances of memristors are continuously dependent on the fluxes. Both system delays and input delays are considered, and the delays and their derivatives may vary in intervals whose lower bounds are not restricted to be zero. The systems are further reduced to continuous-time neural networks (NNs) with interval matrix uncertainties, and a unified method is developed to solve the stability of delayed NNs and MNNs. Stability criteria are obtained for delayed MNNs by augmented Lyapunov functionals, Wirtinger-based integral inequality, reciprocally convex approach, and linear matrix inequalities. In the end, two numerical simulations are used to demonstrate the validity of our theorems.

Dissipative Sliding-Mode Synchronization Control of Uncertain Complex-Valued Inertial Neural Networks: Non-Reduced-Order Strategy

This study investigates the dissipative synchronization of uncertain complex-valued inertial neural networks with external disturbances using sliding mode control (SMC). In the absence of both variable substitution as well as the equivalent transformations of real- and complex-valued systems, this study focuses directly on the original complex-valued inertial system. First, a suitable integral switching surface (SS) function is proposed. Second, by constructing innovative Lyapunov-Krasovskii functionals and applying the Wirtinger-based integral inequality and reciprocally convex approach, a synchronization criterion is derived on the basis of the linear matrix inequality technique to ensure that the sliding mode dynamics are stable and dissipative. Then, an SMC law and an adaptive SMC law are designed, and the accessibility analysis of the predefined SS is provided. Numerical verifications as well as the superiority and practicality analysis of the proposed approach are provided through four examples.

New delay-range-dependent exponential stability criterion and $ H_\infty $ performance for neutral-type nonlinear system with mixed time-varying delays

&lt;abstract&gt;&lt;p&gt;For a neutral system with mixed discrete, neutral and distributed interval time-varying delays and nonlinear uncertainties, the problem of exponential stability is investigated in this paper based on the $ H_\infty $ performance condition. The uncertainties are nonlinear time-varying parameter perturbations. By introducing a decomposition matrix technique, using Jensen's integral inequality, Peng-Park's integral inequality, Leibniz-Newton formula and Wirtinger-based integral inequality, utilization of a zero equation and the appropriate Lyapunov-Krasovskii functional, new delay-range-dependent sufficient conditions for the $ H_\infty $ performance with exponential stability of the system are presented in terms of linear matrix inequalities. Moreover, we present numerical examples that demonstrate exponential stability of the neutral system with mixed time-varying delays, and nonlinear uncertainties to show the advantages of our method.&lt;/p&gt;&lt;/abstract&gt;

Direct quaternion method-based stability criteria for quaternion-valued Takagi-Sugeno fuzzy BAM delayed neural networks using quaternion-valued Wirtinger-based integral inequality

&lt;abstract&gt;&lt;p&gt;This paper investigates the global asymptotic stability problem for a class of quaternion-valued Takagi-Sugeno fuzzy BAM neural networks with time-varying delays. By applying Takagi-Sugeno fuzzy models, we first consider a general form of quaternion-valued Takagi-Sugeno fuzzy BAM neural networks with time-varying delays. Then, we apply the Cauchy-Schwarz algorithm and homeomorphism principle to obtain sufficient conditions for the existence and uniqueness of the equilibrium point. By utilizing suitable Lyapunov-Krasovskii functionals and newly developed quaternion-valued Wirtinger-based integral inequality, some sufficient criteria are obtained to guarantee the global asymptotic stability of the considered networks. Further, the results of this paper are presented in the form of quaternion-valued linear matrix inequalities, which can be solved using the MATLAB YALMIP toolbox. Two numerical examples are presented with their simulations to demonstrate the validity of the theoretical analysis.&lt;/p&gt;&lt;/abstract&gt;

Event-triggered control for uncertain stochastic systems under triple network attacks

This paper studies the control problem of uncertain stochastic systems, which takes into account the impact of network attacks. The types of network attacks considered are denial-of-service (DoS) attacks, deception attacks and replay attacks. In order to save network resources and improve communication utilization, the static event-triggered mechanism and adaptive event-triggered mechanism are cited respectively. Firstly, a new Lyapunov-Krasovskii functional is constructed, employing improved Wirtinger-based integral inequality and Jensens inequality, the criteria on stochastic stability in the mean square for uncertain stochastic systems are proposed. Secondly, the design methods of static event-triggered controller and adaptive event-triggered controller are given respectively. Finally, a practical example is given to manifest the effectiveness of the theoretical results.

An ESETM based robust synchronizing control on master-slave neural network with multiple time-varying delays

This paper focuses on the synchronization issue of master-slave neural network with multiple time-varying delays by way of an enhanced stretchy event-triggered mechanism (ESETM). Firstly, multiple time-varying delays are considered to express the generalization and complexity of system behavior. Secondly, the ESETM is designed by employing an internal dynamic variable and a modified adaptive function, which can provide the expected robustness performance during synchronization on the premise of saving limited network resources. Thirdly, less conservative stabilization criteria are derived in terms of an asymmetric Lyapunov-Krasovskii functional (LKF) candidate. In addition, Wirtinger-based integral inequality and reciprocally convex combination are utilized to estimate derivative of the constructed LKF. Meanwhile, a general activation function method is developed to introduce more information of system states on cross terms of neuron activation function to further reduce the conservatism. Finally, two numerical examples are illustrated to demonstrate the merits and effectiveness of the development.

Finite-time H∞ synchronization control for coronary artery chaos system with input and state time-varying delays.

This is the first time for studying the issue of finite-time H∞ synchronization control for the coronary artery chaos system (CACS) with input and state time-varying delays. Feedback control is planned for finite-time of synchronization CACS. By constructing the Lyapunov-Krasovskii functional (LKF) is derived for finite-time stability criteria of CACS with interval and continuous differential time-varying delays. We use Wirtinger-based integral inequality to evaluate the upper bound of the time derivative of the LKF. We apply the single integral form and the double integral form of the integral inequality, according to Wirtinger-based integral inequality, to ensure that the feedback controller for synchronization has good performance with disturbance and time-varying delay. The new sufficient finite-time stability conditions have appeared in the form of linear matrix inequalities (LMIs). Numerical checks can be performed using the LMI toolbox in MATLAB. A numerical example is presented to demonstrate the success of the proposed methods. This resultant is less conservative than the resultants available in the previous works.

Reachable set bounding for linear systems with mixed delays and state constraints

This paper studies the reachable set estimation for systems with mixed delays and state constraints. Firstly, by constructing an appropriate maximal Lyapunov–Krasovskii functional, combined with the wirtinger-based integral inequality and the extended reciprocally convex combination method, we derive a new criterion about reachable set estimation which reduces the related decision variables without increasing the computational burden. Then, the results are extended to polytopic uncertainties systems and we consider the condition of uncertain differentiable parameters. Three examples are presented to demonstrate the validity of our theorems.

Resilient Adaptive Event-Triggered Control for Singular Networked Cascade Control Systems Under DoS Attacks

This paper addresses the problem of resilient adaptive event-triggered control for singular networked cascade control systems (SNCCSs) under time-varying actuator faults, actuator saturation, DoS attacks, external disturbances, and time-varying delay. The purpose of DoS attacks is to obstruct network communication from time to time, which happens aperiodically. According to the adaptive threshold technique, a resilient adaptive event-triggered mechanism (AETM) is developed to reduce the transmission frequency and also combating the aperiodic DoS attacks. Moreover, the discussed SNCCS is modeled as a switched system due to the presence of attacks that is closely restricted by DoS frequency and duration. Then, the exponential admissible analysis and controller synthesis conditions of the resulting switched SNCCS are obtained and the extended dissipative performance is satisfied by using the Wirtinger-based integral inequality and Lyapunov-Krasovskii functional (LKF) approach. Additionally, a co-design method of the primary and secondary controllers and triggering parameters for the system under consideration is given. Simulation results of a boiler-turbine power plant are presented to validate the proposed method.

New insights on fuzzy sampled-data stabilization of delayed nonlinear systems

This paper explores an improved T-S fuzzy stabilization criteria for nonlinear systems with time-delays via memory sampled-data strategy in the looped-functional context. The sampling period is assumed to be varying within an interval. A new integral inequality is developed to minimize conservatism in the integral term. In addition, free-matrix-based integral inequality (FMBII), and Wirtinger-based integral inequality (WII) are employed. Improved stabilization conditions are derived in linear matrix inequalities (LMIs) using the FMBII, which is based on the looped-functional approach (LFA). A fuzzy memory sampled-data controller design is devised to ensure the asymptotic stability of the closed-loop system. The numerical investigation is carried out with the different approaches, and the resulting outcomes justify that the derived results are less conservative and advantageous over the existing systems.

Leader-following consensus of multi-agent systems with sampled-data control and looped functionals

Leader-following consensus of multi-agent systems is highly influenced by the sampling effects of the agents’ actuators. In this paper, the multi-agent systems involving leader-following strategy is considered for the consensus analysis via sampled-data control using time-dependent Lyapunov–Krasovskii functionals. A directed graph topology is assumed to address the communication network between the agents and each agent obeys a class of stochastic nonlinear dynamics via Bernoulli distribution. A looped Lyapunov Krasovskii functional (LKF) is constructed for the sake of the consensus analysis with suitable controller design to achieve closed-loop dynamics. Using Wirtinger-based integral inequalities to the integral terms of LKF derivatives, less conservative consensus conditions are established in the form of linear matrix inequalities (LMIs). Finally, two illustrative examples are given to show the effectiveness of developed theoretical results.

Improved Results on Delay dependent Stability Criteria of Neural Networks with Interval Time Varying Delay

This paper examines the stability issue of continuous Neural Networks with a time varying delay. A Lyapunov Krasovskii functional consisting of some simple augmented terms and delay dependent terms is constructed. While calculating the derivative of Lyapunov functional, various integral inequalities such as Auxiliary Function Based Integral Inequality, Wirtinger-based integral inequality and an extended Jensen double integral inequality are jointly adopted and hence in terms of linear matrix inequality a new delay dependent stability criterion is obtained. Two numerical examples are taken to show that the derived result is less conservative than some existing ones.

Observer-based synchronization of fractional-order Markovian jump multi-weighted complex dynamical networks subject to actuator faults

This paper is concerned with an observer-based robust H∞ synchronization problem for a class of fractional-order multi-weighted complex dynamical networks subject to Markovian jump parameters, actuator faults and time-varying coupling delay by using fault-tolerant control scheme. The considered actuator fault model is a general one, which consisting of both linear and nonlinear fault terms. Further, by employing a continuous frequency distributed equivalent model and using indirect Lyapunov approach, a new set of sufficient conditions is derived to ensure the asymptotic synchronization of the addressed system in the mean square sense with a prescribed H∞ performance index. In particular, by constructing monochromatic Lyapunov functional and utilizing extended Wirtinger-based integral inequality the required sufficient conditions are developed in terms of linear matrix inequalities. Finally, two numerical examples including financial model are provided to illustrate the effectiveness of the obtained results.