Krasovskii Functionals Research Articles

Sliding Mode Fault-Tolerant Control for Nonlinear LPV Systems with Variable Time-Delay

This paper presents a robust sliding mode fault-tolerant control (FTC) strategy for a class of linear parameter variant (LPV) systems with variable time-delays and uncertainties. First fault estimation (FE) is conducted using a robust sliding mode observer, synthesized to simultaneously estimate the states and actuator faults of LPV polytopic delayed systems. Second, a sliding mode FTC is developed, ensuring all states of the closed-loop system converge to the origin. This paper presents an integrated sliding mode FTC strategy to achieve optimal robustness between the observer and controller models. The integrated design approach offers several advantages over traditional separated FTC methods. Our novel approach is based on incorporating adaptive law into the design of the Lyapunov–Krasovskii functional to improve both robustness and performance. This is achieved by combining the concept of sliding mode control (SMC) with the Lyapunov–Krasovskii function under the H∞ criteria, which plays a key role in guaranteeing the stability of this class of system. The effectiveness of the proposed method is demonstrated through a diesel engine example, which highlights the validity and benefits of the integrated and separated FTC strategy for uncertain nonlinear systems with time delays and the sliding mode control.

Gain-scheduled filter design with different time delays for T-S fuzzy systems

In this article, the authors study a problem of parameter-varying systems that gain-scheduled [Formula: see text] filtering T-S fuzzy problems by appearing of time-varying delays. Our paper also aims to establish filters that deal with communication delays. Traditional studies often study delay-free filters, but these filters have a more general form. In addition, a nonlinear fractional transformation structure has been incorporated into this filter, which allows its parameters to vary. The authors implement fuzzy Lyapunov–Krasovskii functionals used to intend the delayed fuzzy filters of this study, and delay-dependent circumstances are obtained for stability and performance analysis. The solution of the presented linear matrix inequalities (LMIs) provides the filter parameters. This method is finally justified with an example that illustrates its effectiveness.

Discrete event-triggered security control for Markovian CVNNs with additive time-varying delays under random deception attacks

This paper is concerned with the security stabilization problem for a class of Complex-valued Neural Networks (CVNNs) with Markov Jump Parameters (MJPs) and Additive Time-varying Delays (ATVDs) under Random Deception Attacks (RDAs). Different from the existing literature, the instant and strength of RDAs considered in this paper is both random, which is more in line with the real situation. Secondly, a general Lyapunov–Krasovskii Functional (LKF) contains more information about MJPs and ATVDs is constructed, and a new Complex-valued Reciprocally Convex Inequality (CVRCI) containing more free matrices and ATVDs parameters is proposed, which play a key role in reducing the conservativeness of security stabilization criteria. Thirdly, a Discrete Event-triggered Mechanism (DETM) is introduced to mitigate the transmission burden of communication networks, in which the triggering condition of DETM mainly relies on the current sampled state and the last triggered state. Then, by combining with the LKF, CVRCI, DETM, and other analysis techniques, some less conservative security stabilization criteria for the underlying systems are provided in terms of Linear Matrix Inequalities (LMIs). Finally, the effectiveness of our results are verified by two numerical examples and a practical example.

Constructing Lyapunov–Krasovskii functionals for linear positive continuous time difference systems

Abstract In this paper, a linear positive functional difference system with discrete and distributed delays is studied. It is proved that if the considered system is exponentially stable, then it admits both linear and diagonal quadratic Lyapunov–Krasovskii functionals. In addition, new exponential stability conditions for switched linear positive functional difference systems are obtained on the basis of a constructive approach to the design of common Lyapunov–Krasovskii functionals for the associated families of subsystems.

Consensus Control for Stochastic Multi-Agent Systems with Markovian Switching via Periodic Dynamic Event-Triggered Strategy

The consensus problem in stochastic multi-agent systems (MASs) with Markovian switching is addressed by proposing a novel distributed dynamic event-triggered (DDET) technique based on periodic sampling to reduce information transmission. Unlike traditional event-triggered control, the proposed periodic sampling-based DDET method is characterized by the following three advantages: (1) The need for continuous monitoring of the event trigger is eliminated. (2) Zeno behavior in stochastic MASs is effectively prevented. (3) Communication costs are significantly reduced. Based on this, sufficient conditions for achieving consensus in the mean-square sense are derived using Lyapunov–Krasovskii functions, providing a solid theoretical foundation for the proposed strategy. The effectiveness of the proposed DDET control is validated through two numerical examples.

Fuzzy adaptive fixed-time control for output-constrained uncertain nonstrict-feedback time-delay systems

This paper deals with the problem of fuzzy adaptive fixed-time control for a class of uncertain nonlinear nonstrict feedback systems with time delays and output constraints. The negative effects of time delays on system performance and stability are overcome by introducing a new Barrier Lyapunov–Krasovskii (BLK) function and an innovative inequality. In addition, the singularity problem that arises in the process of deriving the control and adaptive laws, is solved by the proposed BLK function. The controller is designed in the backstepping framework which ensures that the system output does not violate predefined constraints. The proposed controller needs less information from the system and therefore has less implementation complexity than previous references, which leads to a more straightforward implementation. A fuzzy system approximation is utilized to overcome the uncertainties of the model. It is guaranteed that all closed-loop system signals converge to small neighborhoods around zero in a fixed time. Finally, an example is presented to verify that the proposed method achieves the control objectives.

Bumpless transfer control for switched time-delay systems with application to aero-engine control

For switched time-delay systems, the bumpless transfer control issue is investigated by utilizing the multiple piecewise Lyapunov–Krasovskii function approach. First, a time-delay-dependent bumpless transfer concept is put forward to depict the transient performance of switched time-delay systems, which restricts the bumps in the amplitude of the present state and the time-delay state, simultaneously. Second, a state-dependent switching signal with a prescribed time is designed to avoid switching frequently. A time-varying switching controller is developed to decrease control bumps. Third, under the premise of achieving asymptotic stability, a solvability condition ensuring a satisfactory bumpless transfer performance is derived through relaxing the traditional bumpless transfer condition. Finally, an instance of the aero-engine control system is used for exhibiting the validity of the presented method.

Finite-time adaptive fuzzy event-triggered control for nonlinear time-delay system with input quantization via command filter

This paper investigates the problem of finite-time adaptive fuzzy event-triggered control for a class of uncertain nonlinear systems with input quantization. The nonlinear systems under consideration contain unmodeled dynamics and time-varying delays. Fuzzy logic systems are used to handle unknown nonlinear terms. Additionally, finite-time filters are introduced to solve the problem of “explosion of complexity” as well as to ensure that the derivative of the virtual controller can be approximated in finite time. The Lyapunov–Krasovskii function is used to handle time-varying delays. The hysteresis quantizer is used to ensure adequate precision when there is a low transmission rate requirement. The control strategy combines event-triggered mechanisms and quantization technology to ensure that all signals of the closed-loop system remain bounded in finite time. Finally, two examples are included to demonstrate the effectiveness of the proposed controller.

Stability and convergence rate analysis of continuous-time delay-difference systems with both point and distributed delays

This paper studies the exponential stability and convergence rate analysis of continuous-time delay-difference systems. Firstly, stability and convergence rate analysis of delay-difference systems with both point delays and distributed delays having exponential integral kernels are studied by using the weighted Lyapunov–Krasovskii functionals (LKFs) approach and the state transformation approach, respectively. Different sufficient stability conditions expressed by linear matrix inequalities (LMIs) are presented. Secondly, as a particular case, LMIs based conditions and spectral radius based conditions are given to ensure the exponential stability with a guaranteed convergence rate for delay-difference systems with both point delays and distributed delays having constant integral kernels, respectively. Finally, numerical examples illustrate the effectiveness of the obtained results.

Bounded real lemma and H∞ control for BAM neural networks with unbounded time‐varying delays

AbstractThis article investigates the exponential control issue of bidirectional associative memory neural network (BAMNN) with unbounded time‐varying delays. A bounded real lemma (BRL) is first established via a direct method, which is on the basis of the solutions of BAMNN. Second, based on the obtained BRL, the state feedback controller is designed to guarantee the global exponential stability of the resulting closed‐loop BAMNN with an performance index. Since no Lyapunov–Krasovskii functionals is constructed in the proposed method, the computation burden and complexity are reduced. Lastly, the effectiveness of the theoretical results is illustrated through two numerical examples.

Asynchronous control for nonlinear switched stochastic delayed systems with weighted mode‐dependent average dwell time

AbstractThis article investigates the asynchronous control for a class of nonlinear switched stochastic delayed systems. In response to this issue, the weighted mode‐dependent average dwell time is first applied to switched stochastic systems as a more general switching signal. In addition, the asynchronous phenomenon is considered to address the mismatch in practical applications, where “mismatch” means the switching of the controllers has a delay to the switching of system modes. Then by selecting new multiple Lyapunov–Krasovskii functionals, sufficient conditions for mean‐square exponential stability and an performance are derived. Finally, numerical examples are provided to illustrate the effectiveness of the proposed methods.

Partial Stability in Probability of Nonlinear Stochastic Discrete-Time Systems with Delay

A system of nonlinear stochastic functional-difference equations with finite delay is considered. By assumption, this system has a “partial” trivial equilibrium (with respect to part of the state variables). The problem under study is to analyze partial stability in probability of this equilibrium: stability is considered with respect to part of the variables determining it. The problem is solved using a discrete-stochastic modification of the method of Lyapunov–Krasovskii functionals. Conditions for partial stability in probability are established. An example is provided to illustrate the features of the proposed approach and the rationale for introducing a one-parameter family of functionals.

H ∞ consensus control via intermittent communication for nonlinear multi-agent systems with unknown parameters and external disturbances

The consensus control problem is investigated for time-delayed nonlinear leader-following multi-agent systems, in which the unknown parameters and external disturbances are considered. Instead of using only one control method, a control scheme combining periodically intermittent control with H ∞ performance is employed. By manipulating the algebraic graph theory and constructing appropriate Lyapunov–Krasovskii functionals, sufficient solutions assuring the exponential H ∞ consensus of the considered systems are established in the form of linear matrix inequalities. The communication graph among the followers is directed, which removes the restrictions on undirected graph or balanced graph in the literature. Simulation analyses are finally presented to verify the theoretical results.

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.

Improved Command Filtered Backstepping Control for Uncertain Nonlinear Systems with Time-Delay

This paper presents an alternative method to solve the control problem of an uncertain nonlinear system in strict-feedback form with a time delay. Instead of using Lyapunov–Krasovskii functionals, ordinary Lyapunov functionals are used to design the controllers. In order to address the completely unknown uncertainties of the system, including the unmodeled dynamics, time-delay nonlinearities, and external disturbances, command filters are applied to reconstruct the estimations of such uncertainties, and the negative feedback of these estimations can be used to reduce the influence of such uncertainties on the system. With the help of the backstepping technique and the Lyapunov stability criterion, it is proved that the system output tracks the target signal with a small error, and contrastive simulation results verify our method’s effectiveness.

Event‐triggered fault detector design in networked fuzzy control systems under denial‐of‐service attacks

AbstractThis article addresses the problem of designing an event‐triggered fault detector within a networked nonlinear system described by a Takagi–Sugeno (T–S) fuzzy model, accounting for time‐varying delays and denial‐of‐service (DoS) attacks characterized by a binary Markov chain. Our focus is on constructing an event‐triggered residual system, which aims to detect malfunctions in the original fuzzy system while optimizing communication resource allocation. Specifically, it is tailored to combat DoS attacks in the event generator‐to‐fault detector channel and mitigate network‐induced time‐varying delays. Utilizing the Lyapunov–Krasovskii function approach, we formulate linear matrix inequalities to establish the necessary conditions ensuring the asymptotic mean‐square stability of the residual system, meeting a predefined performance criterion. Finally, the simulation results are presented to validate the effectiveness of the proposed algorithms.

Constructing Lyapunov–Krasovskii functionals for some classes of switched positive systems with infinite delay

Constructing Lyapunov–Krasovskii functionals for some classes of switched positive systems with infinite delay

Stochastic Intermittent Control with Uncertainty

In this article, we delve into the exponential stability of uncertainty systems characterized by stochastic differential equations driven by G-Brownian motion, where coefficient uncertainty exists. To stabilize the system when it is unstable, we consider incorporating a delayed stochastic term. By employing linear matrix inequalities (LMI) and Lyapunov–Krasovskii functions, we derive a sufficient condition for stabilization. Our findings demonstrate that an unstable system can be stabilized with a control interval within (θ*,1). Some numerical examples are provided at the end to validate the correctness of our theoretical results.

Multi-synchronization of coupled multi-stable memristive Cohen–Grossberg neural networks with mixed time-delays

This paper investigates the multi-synchronization issue for a class of coupled multi-stable memristive Cohen–Grossberg neural networks with both additive time delays and distributed delays. First, the multi-stability of the isolated subnetwork with a general class of activation function is analyzed. By employing state-space decomposition, M-matrix and fixed point theory, some sufficient conditions are deduced to ensure that each subnetwork of the coupled networks has (r+1)n locally exponential stable equilibrium states. In the scenario where the time delays are continuously differentiable, a unique Lyapunov–Krasovskii function is established and some sufficient conditions are derived for the coupled system to achieve multi-synchronization under feedback control. Additionally, in circumstances where the time delays are immeasurable or discontinuous, a new feedback controller is constructed based on the system’s model parameters, and a delay-independent finite-time multi-synchronization criterion is proposed. Finally, three numerical examples are provided to illustrate the effectiveness of the proposed methods.

Adaptive neural dynamic surface control for nonstrict feedback systems with state delays

AbstractThis article focuses on a class of nonstrict feedback systems with input delay, state delays and time‐varying full‐state constraints by proposing an adaptive neural control scheme. To overcome the problems of all state variables effected by time‐varying constraints, the asymmetric time‐varying barrier Lyapunov functions are constructed. The influence of state delays and input delay is eliminated by employing suitable Lyapunov–Krasovskii functionals. Additionally, the process of controller design is based on backstepping method and the unknown functions can be approximated by radial basis function neural networks. Moreover, the problem of repeated differentiations for nonlinear components during controller design is hugely simplified by taking advantage of the dynamic surface control method. The boundness of all the closed‐loop signals can be ensured by the designed controller. Finally, two numerical simulations illustrate that the proposed adaptive neural control scheme is effective.