Bessel-Legendre Inequality Research Articles

Improved AET Robust Control for Networked T-S Fuzzy Systems With Asynchronous Constraints.

This article proposes a novel improved adaptive event-triggered (AET) control algorithm for networked Takagi-Sugeno (T-S) fuzzy systems with asynchronous constraints. First, taking the limited bandwidth of the network into consideration, an improved AET mechanism is proposed to save the communication resource. Superior to the existing event-triggered mechanism, the improved AET scheme introduces two adjusting parameters, which further contribute to the economization of the communication resource. Second, with consideration of asynchronous premise variables, a reconstructed approach is applied to synchronize the time scales of membership functions of the fuzzy system and the fuzzy controller. Third, to derive a less conservative sufficient condition for the controller design, a new augmented Lyapunov-Krasovskii functional with event-triggered information and triple integral terms is constructed. Meanwhile, by applying a Bessel-Legendre inequality and extended reciprocally convex matrix inequality together, a new control algorithm is derived with less conservatism. Finally, simulations on a cart-damper-spring system are implemented to evaluate and verify the performance and advantages of the proposed algorithm.

Resilient input-to-state stable filter design for nonlinear time-delay systems

This paper is dedicated to the resilient input-to-state stable filter design for nonlinear time-delay systems subject to external disturbance input. Two types of time delay (i.e., time-variant delay and time-invariant delay) are taken into account. First, novel analysis results on input-to-state stability for the filtering error systems are presented on the basis of the Lyapunov-functional method. Both the Bessel-Legendre inequality and the reciprocally convex combination approach are employed to reduce the conservatism of the present conditions. Then, computationally tractable design strategies for the desired filters are developed via using a number of decoupling techniques. Finally, two illustrative examples are given to demonstrate the effectiveness of the input-to-state stable filter approaches for the time-variant delay case and the time-invariant delay case, respectively.

Improved Distributed Event-Triggered Control for Networked Control System under Random Cyberattacks via Bessel–Legendre Inequalities

The stability problem of networked control system (NCS) with cyberattacks and processing delay is considered under an event-triggered scheme. An improved distributed event-triggered mechanism is proposed, which optimizes the performance of system dynamics and decreases the network transmission load simultaneously. By means of Bessel–Legendre inequality method and constructing an active Lyapunov–Krasovskii functional, a series of larger upper bounds of delay are obtained corresponding to the order of N. It is worth mentioning that the upper bound increases with N, which means that the conservatism of the stability criterion lowers. Finally, a distributed event-triggered controller is designed. The validity of the results is verified by numerical examples.

A new result on L2–L∞ performance state estimation of neural networks with time-varying delay

This paper is concerned with the L2–L∞ performance state estimation problem of delayed neural networks. Firstly, the second-order Bessel–Legendre inequality based on reciprocally convex approach is proposed. Secondly, based on the improved integral inequality, a new delay-dependent condition is derived, which ensures the asymptotic stability of estimation error system with L2–L∞ performance. As a result, the estimator gain matrix and the optimal L2–L∞ performance level are obtained. Simulation results are finally shown to illustrate the effectiveness of the proposed approach.

New versions of Bessel–Legendre inequality and their applications to systems with time-varying delay

This paper is concerned with the stability of linear time-delay systems by applying the Lyapunov–Krasovskii (L–K) functional method. Two versions of Bessel–Legendre (B-L) inequality are developed that are more suitable to deal with the stability problem of systems with a time-varying delay. Meanwhile, in order to take full advantage of the interest of the new versions of B-L inequality, a novel L–K functional is properly tailored by integrating the integral information of the state into quadratic and integral terms. As a result, more relaxed stability conditions are obtained, whose effectiveness is illustrated by two commonly-used numerical examples.

Stability analysis and stabilisation of delayed IT2 fuzzy systems based on the Bessel–Legendre inequality

This study is concerned with the problems of asymptotic stability analysis and stabilisation for interval type-2 (IT2) fuzzy systems with time-varying delays. An augmented Lyapunov–Krasovskii functional including triple-integral terms is constructed. By the second-order Bessel–Legendre integral inequality and the reciprocally convex combination, a good estimation of the derivative for the Lyapunov–Krasovskii functional can be derived. Thus, sufficient asymptotic stability conditions for the delayed IT2 fuzzy system are established. The proposed method provides improvements and produces better results than the existing ones in the literature. Furthermore, in light of Finsler's lemma and the obtained asymptotic stability conditions, a state feedback controller for the delayed IT2 fuzzy system is developed. Finally, three numerical examples are given to illustrate the effectiveness of the proposed results.

Load Frequency Stability Analysis of Time-Delayed Multi-Area Power Systems With EV Aggregators Based on Bessel-Legendre Inequality and Model Reconstruction Technique

This paper investigates the stability of time-delayed load frequency control (LFC) systems with electric vehicles (EV) aggregators based on Lyapunov theory and linear matrix inequality (LMI) technology. A novel Lyapunov-Krasovskii functional (LKF) is constructed. Then, based on Bessel-Legendre (B-L) inequality and model reconstruction technique, two stability criteria are derived, respectively. Some simulations are carried out to verify the effectiveness of the proposed methods. And the interaction of time-delays between areas and the effect of gains of EVs on delay margins are discussed.

IEEE Access Special Section: Analysis and Synthesis of Time-Delay System

Time-delay is unavoidable, encountered in many practical systems such as networked control systems, communication networks, manufacturing, and biology. On the one hand, it often represents a source of instability and oscillation. On the other hand, the appearance of delay can have a stabilizing effect, and delayed feedback control can be used to stabilize some unstable systems. Considering its theoretical and practical importance, increasing attention has been focused on the analysis and synthesis of time-delay systems, and various new approaches have been proposed to reduce their conservatism. The aim of reducing the conservatism of delay-dependent stability criteria is to establish new stability criteria to provide a maximal allowable delay as large as possible. Among different techniques, the reduced conservatism is mainly obtained by constructing an improved Lyapunov functional and employing a tighter bound on some weighted cross products. A great number of effective methods have been provided such as delay-partitioning method, convex combination method, Wirtinger-based inequality, and Bessel–Legendre inequality.

Synchronization Control Design Based On Observers For Time-delay Lur’e Systems

This paper researches synchronization control design for chaotic Lur'e systems (CLSs) with time-varying delay. Considering of the unmeasurable state variables, Luenberger-type observers are designed for the master-slave systems. The synchronization has been achieved by using the proposed controller with the estimated system states. Based on Bessel-Legendre inequality, improved reciprocally convex combination approach and a novel double integral inequality, new Lyapunov Krasovskii functionals(LKFs) are tailored and a synchronization control criterion with less conservatism is obtained. Some simulations are given to verify that this control strategy we proposed is effective.

Co-Design of Dual Security Control and Communication for Nonlinear CPS Under DoS Attack

The less conservative co-design problem of dual security control and communication for CPS was studied for a class of nonlinear CPS with an actuator fault and DoS attack based on the DETCS. First, combining with the edge computing thought, data filtering and partial computing tasks were assigned to the network edge devices. The DoS attack with limited energy is converted into a special time delay; thus, the discrete event trigger condition, the actuator fault estimation and regulation, and the robustness for DoS attack can be considered uniformly. A dual security control framework of non-uniformly sampled data system was proposed under the DETCS for CPS, and the T-S fuzzy model of closed-loop nonlinear CPS with the co-existence of fault and attack was established. Second, with the help of time-delay system theory, a robust observer for state and fault estimation and the co-design method of dual security control and communication with the actuator fault and DoS attack were obtained by constructing an appropriate Lyapunov-Krasovskii function and using affine Bessel-Legendre inequality. Finally, the validity and feasibility of the theoretical results were verified by a classical quadruple-tank system.

Parallel distributed compensation controller design for Markovian jump system with time-varying delays using Bessel-Legendre inequality method and improved positive definite rule

In this paper, based on time-varying delay Markovian jump system (MJS), Bessel-Legendre inequality method and improved positive definite condition of Lyapunov function are used, and parallel distributed compensation (PDC) state feedback controller is also introduced. The conservatism of the system is reduced by using this inequality method and positive definite conditions, and further decreases with the increase of the Legendre parameter N. The PDC controller considers two control parameters simultaneously, which can represent the actual system more truthfully. Finally, Example 1 proves the effectiveness of the improved method in this paper. Example 2 considers time-varying probability transition perturbations. Example 3 obtains the parameters of PDC controller with different parameters N. Example 4 illustrates the practical significance of methods in this paper by introducing a time-delay inverted pendulum system with Markovian parameters.

Relaxed Stability Criteria for Neural Networks With Time-Varying Delay Using Extended Secondary Delay Partitioning and Equivalent Reciprocal Convex Combination Techniques.

This article investigates global asymptotic stability for neural networks (NNs) with time-varying delay, which is differentiable and uniformly bounded, and the delay derivative exists and is upper-bounded. First, we propose the extended secondary delay partitioning technique to construct the novel Lyapunov-Krasovskii functional, where both single-integral and double-integral state variables are considered, while the single-integral ones are only solved by the traditional secondary delay partitioning. Second, a novel free-weight matrix equality (FWME) is presented to resolve the reciprocal convex combination problem equivalently and directly without Schur complement, which eliminates the need of positive definite matrices, and is less conservative and restrictive compared with various improved reciprocal convex inequalities. Furthermore, by the present extended secondary delay partitioning, equivalent reciprocal convex combination technique, and Bessel-Legendre inequality, two different relaxed sufficient conditions ensuring global asymptotic stability for NNs are obtained, for time-varying delays, respectively, with unknown and known lower bounds of the delay derivative. Finally, two examples are given to illustrate the superiority and effectiveness of the presented method.

Resilient Control Design Based on a Sampled-Data Model for a Class of Networked Control Systems Under Denial-of-Service Attacks.

This article is concerned with designing resilient state feedback controllers for a class of networked control systems under denial-of-service (DoS) attacks. The sensor samples system states periodically. The DoS attacks usually prevent those sampled signals from being transmitted through a communication network. A logic processor embedded in the controller is introduced to not only receive sampled signals but also capture information on the duration time of each DoS attack. Note that the duration time of DoS attacks is usually both lower and upper bounded. Then the closed-loop system is modeled as an aperiodic sampled-data system closely related to both lower and upper bounds of duration time of DoS attacks. By introducing a novel looped functional, which caters for the N -order canonical Bessel-Legendre inequalities, some N -dependent stability criteria are presented for the resultant closed-loop system. It is worth pointing out that a number of identity formulas are uncovered, which enable us to apply the notable free-weighting matrix approach to derive less conservative stability criteria. A linear-matrix-inequality-based criterion is provided to design stabilizing state-feedback controllers against DoS attacks. A satellite control system is given to demonstrate the effectiveness of the proposed method.

New Results on Stability Analysis for Sampled-Data Control Systems With Nonuniform Sampling and Communication Delays

This paper investigates the stability analysis of sampled-data control systems with nonuniform sampling and communication delays. First, by taking full advantage of effective information of the sampling intervals [t <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sub> , t <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k+1</sub> ) and [t <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sub> -τ, t <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k+1</sub> -τ), we propose a new looped Lyapunov functional called two-sided sampling-interval-dependent looped Lyapunov functional, which is tailored for the use of generalized free-matrix-based inequality and canonical Bessel-Legendre inequality. Second, by using the proposed looped Lyapunov functional and the integral inequality technique, some novel N-dependent stability criteria with less conservatism are derived for the considered systems. The obtained stability criteria are shown to have a hierarchical character: the higher level of hierarchy, the less conservatism of the derived result. Finally, two examples are provided to illustrate the effectiveness of the proposed approaches.

Synchronization control of Markov jump neural networks with mixed time-varying delay and parameter uncertain based on sample point controller

This paper put forward an improved synchronization problem for neural networks with Markov jump parameters. The traditional Markov jump neural network (MJNN) only considers the basic external time-varying delays, ignoring both the distributed and leakage delays in the internal transmission of the neural network and the small time-varying errors in the mode switching of Markov probability transition rates. In this paper, we focus on the synchronization of MJNN with mixed time-varying delay. And an improved Lyapunov–Krasovskii functional is constructed. The convergence of inequalities is solved by using affine Bessel–Legendre inequalities and Wirtinger double integral inequalities. At the same time, a new method is used to optimize the mathematical geometric area of the time-varying delay and reduce the conservativeness of the system. Finally, a sample point controller is constructed to synchronize the driving system and the corresponding system.

A novel observer-predictor control for uncertain systems with unknown time-varying input and output delays

In this paper, a novel observer-predictor control is proposed to stabilise uncertain systems subject to unknown but bounded arbitrarily-fast time-varying input and output delays. Due to the presence of time-varying model uncertainties and delay mismatches, the separation principle cannot be used to design the observer and control gains separately, which becomes a challenging problem. By applying small gain theory and Lyapunov-Krasovskii functionals (LKF), in combination with Bessel-Legendre inequalities and the Projection Lemma, we develop sufficient conditions based on Linear Matrix Inequalities (LMIs) for control design. As a result, the closed-loop robust performance can be improved whilst satisfying a prescribed decay rate in the closed-loop dynamics at the expense of introducing more LMI decision variables without increasing the complexity of the control scheme. Also, the Extended State Observer (ESO) is integrated in the proposed control scheme to improve the rejection performance of mismatched disturbances. The effectiveness of the proposed method is illustrated by simulation results.

Improved Delay-Dependent Stability and Stabilization Conditions of T-S Fuzzy Time-Delay Systems via an Augmented Lyapunov-Krasovskii Functional Approach

This paper develops some improved stability and stabilization conditions of T-S fuzzy system with constant time-delay and interval time-varying delay with its derivative bounds available, respectively. These conditions are presented by linear matrix inequalities (LMIs) and derived by applying an augmented Lyapunov-Krasovskii functional (LKF) approach combined with a canonical Bessel-Legendre (B-L) inequality. Different from the existing LKFs, the proposed LKF involves more state variables in an augmented way resorting to the form of the B-L inequality. The B-L inequality is also applied in ensuring the positiveness of the constructed LKF and the negativeness of derivative of the LKF. By numerical examples, it is verified that the obtained stability conditions can ensure a larger upper bound of time-delay, the larger number of Legendre polynomials in the stability conditions can lead to less conservative results, and the stabilization condition is effective, respectively.

New results on stability analysis of systems with time-varying delays using a generalized free-matrix-based inequality

This paper addresses the delay-dependent stability problem of linear systems with interval time-varying delays. A generalized free-matrix-based inequality is proposed and employed to derive stability conditions, which are less conservative than the Bessel–Legendre inequality. An augmented Lyapunov–Krasovskii functional is tailored for the generalized free-matrix-based inequality. Then, some items in the Lyapunov–Krasovskii functionals are integrated so as to relax its positive definite condition, which provides a more accurate lower bound for the Lyapunov–Krasovskii functionals. Therefore, some less conservative stability criteria are presented. Two numerical examples illustrate the effectiveness of the method.

Generalised state estimation of Markov jump neural networks based on the Bessel–Legendre inequality

In the study, the authors are interested in investigating the stability analysis and state estimation of Markov jump static neural networks subject to time delays by the feat of Bessel–Legendre inequality. A canonical Bessel–Legendre inequality, which converts the limited interval [ − h , 0 ] required in traditional Bessel–Legendre inequality to a general interval [ a , b ] is employed. Accordingly, compared with the existing results, the restriction is naturally relaxed and the less conservative criterion is presented. The stability analysis is complicated after constructing an enhanced Lyapunov–Krasovskii functional suitable for the canonical Bessel–Legendre inequality. Furthermore, taking account of the information of system mode, the mode-dependent scheme is applied to the design of a state estimator. Corresponding results to the stability of the estimation error system and the acquisition of the desired observer are presented. In the end, an example, which proves the validity of the method is given.

Delay-dependent robust stability analysis of power systems with PID controller

This study examines the robust stability of a power system, which is based on proportional-integral-derivative load frequency control and involves uncertain parameters and time delays. The model of the system is firstly established, following which the system is transformed into a closed-loop system with feedback control. On this basis, a new augmented Lyapunov-Krasovskii (LK) functional is established for using the new Bessel-Legendre inequality to estimate the derivative of the functional, which can provide a maximum lower bound. A stability criterion is then derived by employing the LK functional and Bessel-Legendre inequality. Finally, numerical examples are used to demonstrate the validity and superiority of the proposed method.