Lyapunov-Krasovskii Functionals Research Articles

Stability analysis of linear delayed systems based on an allowable delay set partitioning approach

This note is concerned with the stability analysis of linear systems with a time-varying delay, where the delay is assumed to be a periodically varying bounded function with restricted derivatives. Combining the allowable delay set (ADS) definition and the region partitioning technique, an ADS partitioning approach is proposed, and then by with aid of this approach, a novel Lyapunov–Krasovskii functional (LKF), namely delay-set-partition-based (DSPB) LKF, is developed. Compared with the previous LKF method, the main advantage of the DSPB LKF lies in allowing the considered system to construct its own LKF in each sub-ADS. By utilizing the proposed LKF, less conservative stability criteria are derived. Finally, two simulation examples are provided to verify the effectiveness and superiority of the proposed method.

Sawtooth-characteristic-based free matrix integral inequality and its application to sampled-data systems

The focus of this article is to present a sawtooth-characteristic-based free-matrix integral inequality and to discuss its application to sampled-data systems (SDSs). Firstly, the free matrix, which is associated with the sawtooth characteristic of the input delay, is presented and incorporated into the integral inequality. In the development of inequality techniques, this is the first time that a free matrix has been associated with the sawtooth characteristic. On this basis, a corresponding sawtooth-characteristic-based free-matrix integral inequality is established, enabling estimation of the integral quadratic terms of the Lyapunov–Krasovskii functional (LKF) derivative. To overcome the challenges posed by second-order terms resulting from the proposed integral inequality, augmented system variables associated with the sawtooth characteristic are also introduced. Thus, the complicated calculation arising from second-order terms and the conservatism caused by the quadratic estimation of the LKF can be avoided. Finally, through the utilization of the sawtooth-characteristic-based free-matrix integral inequality, stability criteria with less conservatism are derived for the SDSs in the form of linear matrix inequalities. The superiority of the proposed approach is illustrated through two numerical examples and a simplified sampled-data based power market.

Delay-dependent Lurie–Postnikov type Lyapunov–Krasovskii functionals for stability analysis of discrete-time delayed neural networks

This paper addresses the influence of time-varying delay and nonlinear activation functions with sector restrictions on the stability of discrete-time neural networks. Compared to previous works that mainly focuses on the influence of delay information, this paper devotes to activation nonlinear functions information to help compensate the analysis technique based on Lyapunov–Krasovskii functional (LKF). A class of delay-dependent Lurie–Postnikov type integral terms involving sector constraints of nonlinear activation function is proposed to complement the LKF construction. The less conservative criteria for the stability analysis of discrete-time delayed networks is given by using improved LKF. Numerical examples show that conservatism can be reduced by the delay-dependent integral terms involving nonlinear activation functions.

General single/multiple integral inequalities and their applications to stability of time-delay systems

This paper is concerned with the stability of time delay systems. A new type of multiple integral inner product and a new class of multiple integral inner product space are firstly introduced. Based on them, general single/multiple integral inequalities are developed, and some previous integral inequalities including single and multiple integral inequalities can be regarded as their special cases. Secondly, suitable Lyapunov–Krasovskii functionals (LKF) are constructed. On the basis of the proposed integral inequalities, the derivative of the LKF can be gauged more accurate than those in the literature. As a result, some improved stability criteria of time-delay systems are formulated in terms of linear matrix inequalities. Through two numerical examples, it is shown that these criteria can provide larger delay upper bounds than some existing ones, showing the effectiveness and superiority of the proposed method.

Output feedback pinning control for complex dynamical networks subjected to multiple attacks

This paper is concentrated on the secure synchronization control issue for complex dynamical networks (CDNs) subjected to multiple attacks. For the sake of conserving resources and being closer to reality, a new output feedback controller based on pinning nodes is firstly designed. Secondly, in contrast with the existent researches on secure network circumstances, we establish a novel multiple attacks model considering the influence of random deception attacks and aperiodic Denial-of-Service (DoS) attacks. Based on the piecewise Lyapunov-Krasovskii function and decoupling method, a new secure synchronization criterion is proposed to realize the mixed H∞ and passive performance of CDNs against multiple attacks, and the control gain is gained. Lastly, the availability of the raised consequence is testified by two concrete examples.

Secure memory adaptive event‐triggered filter design of nonlinear network system under hybrid network attacks with application to tunnel diode circuit

AbstractThis article investigates the problem of secure memory adaptive event‐triggered filter design for nonlinear network systems under hybrid network attacks. First, the introduction of memory adaptive event‐triggered mechanism and numerical quantization is employed to enhance system performance, optimize network channel utilization, and prevent network congestion. Second, we consider the impact of deceptive attacks and DoS attacks occurring in the network channel on data transmission, which form the hybrid network attacks studied in this article. Subsequently, based on the established error model, we derive sufficient conditions for the mean‐square exponential stability of the system under a given performance index using Lyapunov–Krasovskii function and linear matrix inequality (LMI) techniques. Furthermore, the filter parameters are determined using the LMI method. Finally, we illustrate the feasibility and generality of the conclusions through numerical simulation examples and a tunnel diode circuit system.

Improved stability and stabilisation conditions of uncertain switched time-delay systems

This article is concerned with the stability and stabilisation of switched time-delay systems (STDSs) with exponential uncertainty. Based on the Hurwitz convex combination and the energy attenuation principle, an improved state-dependent switching strategy is proposed, which switches to the next modes to obey the minimum energy. This approach fully considers the system dynamic of subsystems, which is more general. Considering the complex switching and delay dynamics, a mode-dependent Lyapunov–Krasovskii functional (LKF) that contains a triple integral term is constructed. The generalised free-matrix-based integral inequality (GFMBII) is used to estimate the integral terms in the derivative of the LKF, and an improved delay-dependent stability criterion is established in the form of linear matrix inequalities (LMIs). Further, to guarantee the stability of the STDSs with a large time-varying delay, a controller that considers the time delay and the exponential uncertainty is designed. Under this controller, a less conservative delay-dependent robust stabilisation criterion for STDSs with exponential uncertainty is established. The validity of the proposed methods is validated by two numerical examples and an application in river pollution control.

Fuzzy adaptive containment control for fractional‐order heterogeneous nonlinear multi‐agent systems with mixed time‐varying delays

SummaryThis study investigates the issue of containment control (CC) of multi‐agent systems (MASs) that are heterogeneous, nonlinear, and fractional‐order while taking practical scenarios into account, such as uncertainties, unknown nonlinearities, time‐varying state delays, and distributed time‐varying delays. A fully distributed adaptive observer is created for each follower in accordance with the communication topology information among agents. This observer is designed to estimate the convex combination information of the leaders. The approach utilizes interval type‐2 fuzzy set theory and adaptive methods to design a distributed containment control strategy that only uses the self‐information of the followers and neighbor nodes' information. Sufficient conditions for implementing containment control are given. A new Lyapunov‐Krasovskii functional (LKF) is constructed, and the stability of the system with feedback loop is proven using fractional‐order theory and inequality methods. Lastly, a simulation example is presented to illustrate the efficacy of the proposed approach.

Stability Analysis of Discrete-Time Neural Networks With a Time-Varying Delay: Extended Free-Weighting Matrices Zero Equation Approach.

This research investigates the stability of discrete-time neural networks (DNNs) with a time-varying delay by using the Lyapunov-Krasovskii functional (LKF) method. Recent researches acquired some less conservatism stability criteria for time-varying delayed systems via some augmented LKFs. However, the forward difference of such LKFs resulted in high-degree time-varying delay-dependent polynomials. This research aims to develop some augmented state-related vectors and the corresponding extended free-weighting matrices zero equations to avoid the appearance of such high-degree polynomials and help to provide more freedom for the estimation results. Besides, an augmented delay-product-type LKF is also established for ameliorating the stability conditions of the time-varying delayed DNNs. Then, based on the above methods and Jensen's summation inequality, the auxiliary-function-based summation inequality, and the reciprocally convex matrix inequality, some less conservatism stability criteria for time-varying delayed DNNs are formulated. The validity of the proposed time-varying delay-dependent stability criteria is illustrated by two numerical examples.

Building coercive Lyapunov–Krasovskii functionals based on Razumikhin and Halanay approaches

AbstractIn this article, we provide a systematic and constructive way to build a Lyapunov–Krasovskii functional for time‐delay systems whose stability can be established through the Razumikhin or the Halanay approaches. The constructed Lyapunov–Krasovskii functional turns out to be coercive, meaning sandwiched between functions of the state history norm, and to dissipate in terms of the whole history norm. We present these results in the framework of input‐to‐state stability (ISS) in order to further account for the influence of input disturbances. A special emphasis is also given on exponential stability and exponential ISS. We illustrate our findings though the study of a coupled ODE‐PDE model of a chemical reactor, and show that, unlike most results in that area, our approach happens to ensure ISS in terms of the supremum norm of the state.

Event-Triggered Exponential Synchronization of the Switched Neural Networks With Frequent Asynchronism.

The synchronization for a class of switched uncertain neural networks (NNs) with frequent asynchronism based on event-triggered control is researched in this article. Compared with existing works that require one switching during an inter-event interval, frequent switching is allowed in this article. By employing controller-mode-dependent Lyapunov-Krasovskii functionals (LKFs), we devise the control strategy to guarantee that the switched NNs can be synchronized. The proposed LKFs can make full use of system information. Using an improved integral inequality, some sufficient stability conditions formed by linear matrix inequalities (LMIs) are derived for the synchronization of switched uncertain NNs. Average dwell time (ADT) is obtained in the form of inequality that includes the maximum inter-event interval. In addition, the existence of lower bound of inter-event interval is discussed to avoid Zeno behavior. At last, the feasibility of the proposed method is proven by a numerical example.

Dynamic Event-Triggered H∞ Control for Networked Control Systems With Denial-of-Service Attacks

A dynamic event-triggered H∞ control for networked control systems (NCSs) subject to denial-of-service (DoS) attacks is investigated in this paper. Firstly, the DoS attacks model in communication channel are established based on stochastic variables that obey the Bernoulli distribution, and the attacks rate over network along with switching rules are described. Secondly, the dynamic event-triggered mechanism (DETM) is applied to weaken the influence of DoS attacks and avoid excessive transmission. The threshold parameters under the event-trigger condition can be adjusted by the dynamic characteristics of the systems. Thirdly, by using Lyapunov-Krasovskii functional (LKF) and Linear matrix inequality (LMI) methods, a sufficient condition for NCSs to satisfy the asymptotic stability of given H∞ is obtained. In addition, a cooperative design scheme of DETM parameters and controller gain is proposed. Finally, a RLC series circuit systems is used as an example to show the effectiveness of the method.

Fault-Tolerant Fuzzy-Resilient Control for Fractional-Order Stochastic Underactuated System With Unmodeled Dynamics and Actuator Saturation.

This article is considered on underactuated fractional-order stochastic systems (FOSSs) with actuator saturation and incrementally conic nonlinear terms, whose fractional-order α ∈ (0,1) . First, to bring FO dynamic signals, solving the unmodeled dynamics, in the meantime, the saturated nonlinear term of the control input is taken into account. At the time, to cope with the stability issue of FOSS under such situation, the fault tolerant resilient controller based on underactuated condition is designed. Then, according to the method of the Lyapunov and It∧ o differential formulation to design proper multiple Lyapunov-Krasovskii (L-K) functions, such that, a novel sufficient condition of the robustly asymptotically stability of fuzzy FOSS under underactuated conditions is rigorously proved in terms of linear matrix inequality (LMI). Furthermore, in order to research the mean square stability of the above-mentioned system, so the solution of FOSS is obtained to achieve this purpose. By applying the above method, which is proposed in this work that the controlled system can be obtained with faster response and higher control accuracy. At last, to display the superiority of the above-mentioned scheme is effective, tethered satellite system and numerical results are presented.

Membership-difference-dependent switching control for time-delay T–S fuzzy systems

The switching control issue in time-delay Takagi–Sugeno fuzzy systems (TSFS) based on membership difference (MD) is investigated in this paper. For the asymmetric Lyapunov-Krasovskii functional (LKF) selected in this paper, the zero equation based on MD is introduced in the analysis process, and combined with inequality lemmas, the stability conditions of linear matrix inequality (LMI) form are derived. According to the characteristics of MD, the corresponding switching controller is designed and a more concise controller form is given. Compared with the existing literature, the stability condition presented in this paper is less conservative. Finally, two examples are provided to demonstrate the efficacy and superiority of the suggested method.

Delay-variation-dependent summation inequality and its application to stability analysis of discrete-time systems with time-varying delay

This work tends to research the stability of the discrete-time systems with a time-varying delay based on the Lyapunov-Krasovskii functional (LKF) method. In order to acquire a less conservative stability criterion, some techniques in this work are refined and taken into use. Firstly, to reduce the conservatism generated when estimating the forward difference of the LKF, a newly delay-variation-dependent summation inequality is constructed, which includes the existing free-matrix-based and Bessel function-based summation inequalities, using delay-variation-product relaxed matrices to provide more freedom for the estimation results. Secondly, to further show the influence of the introduced variation information in the time-varying delay, we give another selection of the allowable delay set for the delayed discrete-time systems. Thirdly, by taking advantages of the above method and by constructing a delay-product-type LKF, using extended free-weighting-matrices zero equations, and considering different allowable delay sets, two improved linear matrix inequality (LMI)-based delay-variation-dependent criteria for delayed discrete-time systems are formulated. Some classical numerical instances are presented to explain the effectiveness of these proposed stability criteria.

New results on stability analysis for a class of generalized delayed neural networks

This paper addresses the stability analysis of neural networks with a time-varying delay, where the delay is periodically varying bounded function with constrained derivatives. In order to obtain less conservative stability criteria, two novel Lyapunov-Krasovskii functionals (LKFs) with delay-derivative-variation-dependent matrices are constructed. In one LKF, two delay-product terms are double utilized and more free matrices are incorporated. Additionally, a new type of the LKF with delay-derivative-variation-dependent matrices, namely, the neuronal-activation-function-based looped functional, is developed by accounting for the periodic nature of the delay function and the coupling information of the system state and the neural activation function. Then, several stability conditions of delayed neural networks are developed by combining some integral inequalities. Finally, the superiority and validity of developed stability conditions are validated on two benchmark examples.

Learning-based asynchronous sliding mode control for semi-Markov jump systems with time-varying delay using relaxed negative-determination lemma

This article figures out the asynchronous sliding mode control (SMC) issue for a class of semi-Markov jump systems (SMJSs) with time-varying delay and actuator faults. Firstly, to tackle the chattering produced by asynchronous phenomenon and guarantee smooth reachability, the hidden Markov model and the reduced-order theory are used to design the learning-based SMC law. Besides, the mode-dependent augmented Lyapunov–Krasovskii functional (LKF) is constructed. By using the relaxed negative definite lemma and the generalized free matrix integral inequality (GFMBII), sufficient conditions for ensuring the stochastic stability of the SMJSs are provided. Furthermore, the conservatism of the maximum time delay is effectively reduced by adjusting the value of β. Finally, simulation examples prove the validity and superiority of the proposed method.

L2$$ {L}_2 $$‐gain analysis and fault‐tolerant control for uncertain nonlinear switched system with actuator saturation and time‐varying delay

AbstractFor the nonlinear continuous‐time switched system with input saturation and time‐varying delay, the problem of actuator failure is investigated in the presence of external disturbances. First, the sufficient conditions for the tolerance disturbances of system are obtained by using the multiple augmented Lyapunov Krasovskii functionals method combined with the Wirtinger integral inequality method and the reciprocally convex combination inequality. Second, the restricted ‐gain is analyzed. Third, the switching law and the disturbance suppression fault‐tolerant controllers are designed to obtain the maximum value of tolerance disturbance and the minimum upper bound on the restricted ‐gain. Finally, the numerical examples are given to verified the effectiveness of the proposed method by comparing the state trajectories under the action of the fault‐tolerant controllers and the standard controllers.

An improved sampled-data control for a nonlinear dynamic positioning ship with Takagi-Sugeno fuzzy model.

This article considered the sampled-data control issue for a dynamic positioning ship (DPS) with the Takagi-Sugeno (T-S) fuzzy model. By introducing new useful terms such as second-order term of time, an improved Lyapunov-Krasovskii function (LKF) was constructed. Additionally, the reciprocally convex method is introduced to bound the derivative of LKF. According to the constructed LKF, the sampling information during the whole sampling period was fully utilized, and less conservatism was obtained. Then, the stability condition, robust performance, mode uncertainty and sampled-data controller design were analyzed by means of the linear matrix inequality (LMI). Finally, an example was given to demonstrate the effectiveness of the proposed method.

On Master-Slave Synchronization of Chaotic Lur’e Systems Using Sampled-Data Control

This brief aims to achieve a less conservative master-slave synchronization criterion for chaotic Lur’e systems using sampled-data control. To this end, a novel piecewise differentiable Lyapunov-Krasovskii functional (LKF) is constructed to fully capture the characteristics about real sampling pattern and nonlinearities of the system. By using the new LKF, a less conservative synchronization criterion is presented to guarantee global exponential stability about synchronization error system. Chua’s circuit system is adopted to verify effectiveness and advantage of obtained approach.