Lyapunov Krasovskii Research Articles

Frobenius Norm-Based Global Stability Analysis of Delayed Bidirectional Associative Memory Neural Networks

The present research investigates the global asymptotic stability of bidirectional associative memory (BAM) neural networks using distinct sufficient conditions. The primary objective of this study is to establish new generalized criteria for the global asymptotic robust stability of time-delayed BAM neural networks at the equilibrium point, utilizing the Frobenius norm and the positive symmetrical approach. The new sufficient conditions are derived with the help of the Lyapunov–Krasovskii functional and the Frobenius norm, which are important in deep learning for a variety of reasons. The derived conditions are not influenced by the system parameter delays of the BAM neural network. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed conclusions regarding network parameters.

Robust Fault Detection for Fuzzy‐Model‐Based Networked Control Systems Under Denial‐of‐Service Attacks and Sensor Faults Within a Stochastic Communication Protocol

ABSTRACTThis paper investigates the robust fault detection problem in non‐linear networked systems, characterized by the Takagi‐Sugeno (T‐S) fuzzy model. The analysis is complicated by Denial‐of‐Service (DoS) attacks and sensor faults. In such systems, DoS attacks disrupt network communications, threatening both system stability and fault detection accuracy. Simultaneously, sensor faults compromise the reliability of the system's measurement outputs. To address these challenges, the stochastic communication protocol (SCP) is employed to optimize the transmission order between multiple sensors and the fault detector, effectively managing the limitations of communication capacity. Consequently, a fault detection scheme based on SCP is developed to generate residual signals, supported by an auxiliary residual system for stability analysis. Using the Lyapunov‐Krasovskii (L‐K) functional approach, sufficient conditions are derived to ensure the asymptotic stability of the residual system with a prescribed performance. Finally, the effectiveness of the proposed fault detection scheme is validated through simulation examples.

Robust fuzzy output feedback control for a magnetorheological semi-active air suspension system with time-delay and actuator saturation

Magnetorheological fluid (MRF) dampers have been utilized in semi-active air suspension (SAAS) due to their attractive advantages. This paper investigates a novel robust fuzzy output feedback controller for semi-active air suspension with MRF damper (SAAS-MRFD), considering time delay and actuator saturation. An aero-thermodynamics theory is first utilized to describe the nonlinear air spring. The damping force of the MRF damper is represented by a hyperbolic tangent model. The interval type-2 (IT-2) Takagi–Sugeno (T-S) fuzzy method is second employed to capture those nonlinearities of the air spring and MRF damper. Considering the difficulty to measure all states, a novel dynamic output feedback (DOF) controller is developed by an appropriate Lyapunov–Krasovskii function and the reciprocally convex technique with consideration of time delay, actuator saturation, and output constraints. The effectiveness of the proposed controller is validated by both simulation and a quarter-car test rig (QCTR). The results indicate that the proposed controller has better performance in vibration reduction than the recent similar controllers in the literature.

Design of a Finite-Time Bounded Tracking Controller for Time-Delay Fractional-Order Systems Based on Output Feedback

This paper focuses on a class of fractional-order systems with state delays and studies the design problem of the finite-time bounded tracking controller. The error system method in preview control theory is first used. By taking fractional-order derivatives of the state equations and error signals, a fractional-order error system is constructed. This transforms the tracking problem of the original system into an input–output finite=time stability problem of the error system. Based on the output equation of the original system and the error signal, an output equation for the error system is constructed, and a memory-based output feedback controller is designed by means of this equation. Using the input–output finite-time stability theory and linear matrix inequality (LMI) techniques, the output feedback gain matrix of the error system is derived by constructing a fractional-order Lyapunov–Krasovskii function. Then, a fractional-order integral of the input to the error system is performed to derive a finite-time bounded tracking controller for the original system. Finally, numerical simulations demonstrate the effectiveness of the proposed method.

Global output feedback stabilization of stochastic uncertain nonlinear systems with large constant delays

This paper studies the problem of output feedback stabilization for a class of stochastic uncertain nonlinear systems with large constant delays in state and input. For stochastic nonlinear systems, the maximal open sector Δ of output function is given. As long as output function belongs to any closed sector included in Δ, by constructing a reduced-order observer and C 2 Lyapunov–Krasovskii functional and using the homogeneous domination method, an output feedback controller can be developed to guarantee the closed-loop system globally asymptotically stable in probability.

Stability estimates for a class of neutral type systems with distributed time-varying delay components

This paper investigates new stability estimates for a class of neutral type systems with distributed time-varying delay components. Initially, an appropriate Lyapunov–Krasovskii functional (LKF) is constructed to generate some solution estimates for the considered system. Then, using the estimates obtained in the employed method framework, it is determined whether the solutions are stable or not. The stabilization rates of the solutions at infinity are discussed in the context of asymptotic and exponential stability (ES). Finally, two numerical examples with simulations are presented to demonstrate the applicability of the proposed method on the current results.

Fault-Tolerant Time-Varying Formation Trajectory Tracking Control for Multi-Agent Systems with Time Delays and Semi-Markov Switching Topologies

The fault-tolerant time-varying formation (TVF) trajectory tracking control problem is investigated in this paper for uncertain multi-agent systems (MASs) with external disturbances subject to time delays under semi-Markov switching topologies. Firstly, based on the characteristics of actuator faults, a failure distribution model is established, which can better describe the occurrence of the failures in practice. Secondly, switching the network topologies is assumed to follow a semi-Markov stochastic process that depends on the sojourn time. Subsequently, a novel distributed state-feedback control protocol with time-varying delays is proposed to ensure that the MASs can maintain a desired formation configuration. To reduce the impact of disturbances imposed on the system, the H∞ performance index is introduced to enhance the robustness of the controller. Furthermore, by constructing an advanced Lyapunov–Krasovskii (LK) functional and utilizing the reciprocally convex combination theory, the TVF control problem can be transformed into an asymptotic stability issue, achieving the purpose of decoupling and reducing conservatism. Furthermore, sufficient conditions for system stability are obtained through linear matrix inequalities (LMIs). Eventually, the availability and superiority of the theoretical results are validated by three simulation examples.

CONSTRUCTING DIAGONAL LYAPUNOV–KRASOVSKII FUNCTIONALS FOR A CLASS OF POSITIVE DIFFERENTIAL-ALGEBRAIC SYSTEMS

A coupled system describing the interaction of a differential subsystem with nonlinearities of a sector type and a linear difference subsystem is considered. It is assumed that the system is positive. A diagonal Lyapunov–Krasovskii functional is constructed and conditions are determined under which the absolute stability of the investigated system can be proved with the aid of such a functional. In the case of nolinearities of the power form, estimates for the convergence rate of solution to the origin are derived. The stability analysis of the corresponding system with parameter switching is fulfiled. Sufficient conditions guaranteeing the asymptotic stability of the zero solution for any admissible switching law are obtained.

ESTIMATES OF SOLUTIONS FOR A CLASS OF NONAUTONOMOUS SYSTEMS OF NEUTRAL TYPE WITH CONCENTRATED AND DISTRIBUTED DELAYS

A class of systems of nonautonomous differential equations of neutral type with concentrated and distributed delays is considered. By using a Lyapunov–Krasovskii functional, estimates are established imply whether the solutions are stable. In the case of exponential stability, estimates for the stabilization rate of the solutions at infinity are given.

ESTIMATES FOR SOLUTIONS OF A BIOLOGICAL MODEL WITH INFINITE DISTRIBUTED DELAY

A model of competition between several microorganism species described by a system of nonlinear differential equations with infinite distributed delay is considered. The case of asymptotic stability of the equilibrium point corresponding to the survival of only one species and the extinction of all others is studied. The conditions for the initial numbers of species and the initial concentration of the nutrient at which the system reaches an equilibrium state are specified, and estimates of the stabilization rate are established. The results are obtained using the Lyapunov–Krasovskii functional.

Synchronisation of complex nonlinear dynamical networks with coupling delays via dual hybrid-triggered control design

The synchronisation problem for a class of complex nonlinear dynamical networks (CNDNs) that are subject to time-varying coupling delays and nonlinearities is explored in this work. This is accomplished by designing a dual hybrid-triggered communication technique based on the dynamic output feedback controller which is the main intent of this study. In contrast to the existing hybrid-triggered mechanism, a dual hybrid-triggered technique is being employed with the utilisation of outputs of considered CNDNs and the dynamic output feedback controller. To be precise, with an aim to effectively lower the network transmission rate in both the sensor-to-controller and controller-to-actuator channels, the triggering mechanism is implemented to both the outputs. Later, the Lyapunov–Krasovskii functional is utilised to establish the stability criterion and the necessary conditions for ensuring the synchronisation of chosen CNDNs are derived in terms of linear matrix inequalities. Eventually, two numerical examples, including the aircraft model have been rendered to reveal the potency of acquired theoretical results.

Decentralized Finite‐Time Adaptive Neural Output‐Feedback Quantized Control for Switched Nonlinear Large‐Scale Delayed Systems

ABSTRACTThis paper considers the problem of decentralized finite‐time adaptive neural output‐feedback quantized control for a class of switched nonlinear large‐scale delayed systems. A switched high‐gain quantized state observer is therefore constructed for each subsystem to estimate unavailable system states. Different from the traditional Lyapunov–Krasovskii functional method, multiple Lyapunov–Krasovskii functions are introduced to develop the decentralized adaptive output‐feedback control strategy with neural network approximation for the switched nonlinear large‐scale delayed systems. Under a category of switching signals with persistent dwell time, all signals in the closed‐loop switched system are semi‐globally uniformly ultimate bounded. Meanwhile, the tracking errors can remain in a small domain of origin in finite time. Case studies are finally used to illustrate the flexibility and effectiveness of the proposed control approach, including the switched two continuous stirred tank reactor delayed systems.

Robust finite-time H2/H∞ control for stochastic Markovian jump systems via state feedback

The robust finite-time [Formula: see text] control problems for linear discrete-time stochastic Markovian jump systems (MJSs) with external disturbance are investigated. Firstly, the definitions of robust finite-time boundedness, and robust finite-time [Formula: see text] boundedness are presented. Subsequently, by using the stochastic Lyapunov–Krasovskii functional method and linear matrix inequality technique, sufficient conditions for the existence of a robust finite-time [Formula: see text] controller are provided for stochastic MJSs. A state feedback controller is designed. Furthermore, the finite-time guaranteed cost bounds are given. More specifically, the designed controller not only ensures the finite-time [Formula: see text] bounded but also makes the [Formula: see text] cost function less than the bound given by the performance index of the systems. Finally, two examples are given to illustrate the validity of the proposed approach.

Relaxed Positivity, η-Exponential Stabilization and l1-Gain Performance of Polynomial Discrete-Time Fuzzy System with Time-Delay and External Disturbance

The positivity constraint makes the stability for positive systems under the precise convergence decay and anti-disturbance performance quite challenging considering the current advanced control theory. Furthermore, the existence of time delay will disturb the dynamics of non-linear positive systems and even lead to a series of complex behaviours, such as oscillations and instability. With respect to the above problems, an approach of membership function-dependent positivity, η-exponential stability and l1-gain performance investigations which consider the dedicated membership function shape for the polynomial fuzzy discrete-time model (PFDM) with time-delay is proposed in this article. Firstly, we use a novel co-positive polynomial η-exponential Lyapunov–Krasovskii functional candidate to investigate the systems exponential stability under the decay η-rate and l1-gain anti-disturbance performance. Secondly, to reduce the conservativeness of conditions, a novel piecewise-polynomial membership function is used to introduce the Lagrange approximation knowledge into stability and performance analysis. Finally, the synthesized state-feedback fuzzy controller with polynomial terms guarantees these control objectives under positivity constraints. Two simulation examples validate the developed new method.

Stability and [formula omitted]-gain analysis of switched positive systems with unstable subsystems and sector nonlinearities

In this paper, we are concerned with exponential stability and L1-gain performance of a type of switched nonlinear positive systems (SNPSs) with time-varying delay made up of partially unstable subsystems. What sets it apart from other articles is that nonlinear parts of this article are confined in a certain region. Besides, the adequacy criteria are obtained in a linear programming format via designing the novel nonlinear Lyapunov-Krasovskii (L-K) functionals and a set of special switching sequences under average dwell time (ADT) switching rule. Eventually, with the help of comparison principle, the theoretical results can be further expanded to time-varying switched systems. As practical applications, we apply the theoretical achievements to stability of switched neural networks to confirm the feasibility of the conclusions.

Asymptotic Stability of Delayed Complex Balanced Reaction Networks with Non-Mass Action Kinetics

We consider delayed chemical reaction networks with non-mass action monotone kinetics and show that complex balancing implies that within each positive stoichiometric compatibility class there is a unique positive equilibrium that is locally asymptotically stable relative to its class. The main tools of the proofs are respectively a version of the well-known classical logarithmic Lyapunov function applied to kinetic systems and its generalization to the delayed case as a Lyapunov–Krasovskii functional. Finally, we demonstrate our results through illustrative examples.

Event-triggered [formula omitted] control for partial unknown Markov jump discrete-time complex networks with distributed delay

Periodic event triggering, an important control mechanism, effectively saves communication resources and energy, making it highly valuable in networked control systems. This study focuses on the comprehensive utilization of periodic sampling pattern and event triggering mechanism to achieve synchronization control of a class of discrete-time Markov jump complex dynamic networks (CDNs) with H∞ performance index. Firstly, a comprehensive CDNs model is established incorporating Markovian jump parameters with incomplete transition probabilities, time-varying delays, and external random disturbances, making it more readily implementable in practical engineering scenarios. Secondly, a novel periodic event-triggered control strategy is proposed. In this strategy, the threshold parameters and weighting matrix in the event-triggering condition are mode-dependent, which can achieve better resource efficiency and effectively avoid the Zeno phenomenon. Thirdly, a discontinuous Lyapunov–Krasovskii functional is designed to capture information about discrete periodic sampling patterns. By applying free-weighting matrix, Jensen’s inequality, and Wirtinger’s inequality, sufficient conditions and controller for CDNs synchronization are derived with lower conservativeness. Finally, a simulation example is provided to demonstrate the feasibility and effectiveness of the theoretical results obtained.

Accuracy-preassigned fixed-time synchronization of switched inertial neural networks with time-varying distributed, leakage and transmission delays

In this paper, the accuracy-preassigned fixed-time synchronization problem of a class of switched inertial neural networks with time-varying distributed, leakage and transmission delays is studied. To this end, a parameterized system solution-based direct analysis method is proposed for the first time. Unlike existing works, this method sets out from the definition of accuracy-preassigned fixed-time synchronization, and does not require variable substitution for inertial item or the construction of any Lyapunov–Krasovskii functional. This not only simplifies the proof process, but also reduces the computational complexity for solving synchronization conditions. Significantly, this paper introduced the time-varying leakage delay into switched inertial neural networks for the first time. Furthermore, the approach utilized in this manuscript stands apart from all previous techniques for achieving fixed-time synchronization. Finally, the reliability of the theoretical results is verified by numerical simulation.

Adaptive Neural Control Design for Strict-Feedback Time-Delay Nonlinear Systems Based on Fast Finite-Time Stabilization: A Case Study of Synchronous Generator Systems

This study aims to investigate the finite-time control problem for a class of strict-feedback time-delay nonlinear systems with unknown functions. The control design is based on a fast finite-time practical stability criterion. Unknown nonlinear functions can be estimated using the universal approximation performance of neural networks. Finite-time control design is performed using adaptive backstepping technology. By performing closed-loop stability analyses and choosing appropriate Lyapunov–Krasovskii functionals, all signals in a closed-loop system can be bounded within a finite time. Subsequently, the proposed control method can be applied for the excitation control of synchronous generators. The effectiveness of the proposed method is verified using a numerical model of a single-machine power system.

Anti-disturbance reliable memory feedback tracking control for unmanned aerial vehicle systems

The problem of memory feedback tracking control for unmanned aerial vehicle systems subject to nonlinear actuator faults and disturbances is investigated in this paper. First, a generalized extended state observer is constructed to get the estimates of the system states and the disturbances. Then, by incorporating the obtained estimates in the control design, an anti-disturbance memory feedback tracking control law is designed such that the resulting closed-loop augmented system is asymptotically stable, and thereby the states of the unmanned aerial vehicle system track the given reference trajectories with satisfactory disturbance rejection performance. Notably, by constructing appropriate asymmetric Lyapunov–Krasovskii functional, the stability and stabilization conditions for guaranteeing the desired tracking and disturbance attenuation performance are derived in terms of linear matrix inequalities. Further, the observer gains are evaluated using the pole placement approach. Finally, numerical simulations are given to verify that the proposed control strategy is viable.