Asymptotic Stability Research Articles

Asymptotic behavior of solutions of nonlinear discrete systems in critical cases

AbstractThe paper is dedicated to the analysis of the asymptotic behavior of essentially nonlinear discrete systems whose linearization possesses eigenvalues on the unit circle. For these systems, the paper establishes sufficient conditions for the asymptotic stability regardless of terms higher than the third order. Moreover, it derives an asymptotic stability criterium for a reduced critical subsystem. The proof incorporates a constructive approach for deriving Lyapunov functions using the center manifold reduction and normal form method. The key result of the paper asserts that, given the obtained stability conditions, the solutions of the system converge to the origin with a polynomial decay rate. The paper also introduces a method for evaluating the coefficients of the decay rate estimate. To illustrate these findings, we apply the obtained result to analyzing nonlinear business cycle and predator-pray models.

MPPT control for PMSG based wind turbine: Adaptive dynamic programming approach

This paper proposes an online adaptive dynamic programming-based controller for the variable-speed wind turbine system, ensuring it tracks the maximum power point under uncertain conditions. This control scheme is composed of two components: the adaptive optimal control component and the high-order disturbance observer-based adaptive sliding mode control component. The adaptive optimal control part is designed using the online adaptive dynamic programming technique, which is one of the algorithms of the reinforcement learning method. This control component has the role of achieving the optimal characteristics for the nonlinear system. Likewise, the high-order disturbance observer is established to estimate system uncertainties and external disturbances. These approximated disturbances are then fed to the adaptive sliding mode control component for system disturbance compensation. As a result, the system, which is affected by the unknown disturbances, achieves robust optimal performance. In addition, the convergence of the adaptive laws and the stability of the overall system are ensured through the Lyapunov theory. Finally, comparative simulations are conducted to validate the advantages of the proposed control scheme in comparison to the existing methods.

Image encryption algorithm based on matrix projective combination-combination synchronization of an 11-dimensional time delayed hyperchaotic system

Abstract Secure image transmission is critical to protect sensitive data from unauthorized access, especially in an era of increasing digital threats. Chaotic systems with their inherent complexity and unpredictability, provide a promising solution for enhancing encryption security. To contribute to this field, we investigate a new 11-dimensional hyperchaotic system by taking advantage of its complex dynamical properties to strengthen security. The high dimensional of the system intensifies chaotic behaviors such as stability, attractors and sensitive to initial conditions, making it particularly suitable for encrypted transmission.
 Time delay is an important factor to be considered affecting the control and synchronization in nonlinear system. Additionally, time delays include the effects of past states, further increasing the unpredictability of the system. To explore these dynamics, we analyze the Lyapunov exponents, stability of equilibrium points, symmetry and dissipation. A matrix projective combination-combination synchronization scheme is proposed to synchronize four identical 11-dimensional hyperchaotic systems with time delay. Nonlinear active controllers designed based on Lyapunov stability theory are used to achieve this synchronization. This work advances an important idea for encryption and decryption algorithms, which is the secure transmission of images using affine encryption. In the affine encryption algorithm, the key is based on the solution of synchronized chaotic delayed systems and the private message of the sender and receiver. This proposed encryption and decryption algorithms have been applied on plain images. Numerical simulations and security analysis including key space, histogram, information entropy and correlation analysis are conducted to validate the theoretical results and encryption algorithm. Experimental analysis and comparisons with existing literature confirm the effectiveness and security of the proposed approach for cryptographic purposes.

Fractional order slide mode droop control for simultaneous voltage and frequency regulation of AC microgrid

AbstractThis research proposes the application of fractional‐order sliding mode control (FOSMC) at the primary controller level to improve the stability of an islanded microgrid by adjusting its voltage and frequency. The control strategies used in the microgrid are performed in two levels (primary and secondary) in the islanded mode. Practically, most previous studies have worked to improve the primary controller. Droop control is one of the most commonly used methods at the primary level and is adopted in this study as well. The sliding mode control (SMC) strategy is normally used to control linear equations. Thus, the non‐linear microgrid equations were transformed into some linear ones using the input‐output feedback linearization technique. Further, a fractional sliding surface was acquainted. The sliding surface and FOSMC were designed to reject system uncertainties and organize the voltage and frequency. Design parameters were chosen using the Lyapunov stability theorem. The validation of the proposed method using Simulink‐MATLAB confirms its effectiveness in enhancing level power sharing, regulating frequency, and maintaining voltage stability across the system.

Distributed Bipartite Consensus of Multi-Agent Systems via Disturbance Rejection Control Strategy

This work aims to focus on analyzing the consensus control problem in cooperative–competitive networks in the occurrence of external disturbances. The primary motive of this work is to employ the equivalent input-disturbance estimation technique to compensate for the impact of external disturbances in the considered multi-agent system. In particular, a suitable low-pass filter is implemented to enhance the accuracy of disturbance estimation performance. In addition, a specific signed, connected, and structurally balanced undirected communication graph with positive and negative edge weights is considered to express the cooperation–competition communication among neighboring agents. The cooperative–competitive multi-agent system reaches its final state with same magnitude and in opposite direction under the considered structurally balanced graph. By utilizing the properties of Lyapunov stability theory and graph theory, the adequate conditions assuring the bipartite consensus of the examined multi-agent system are established as linear matrix inequalities. An illustrative example is delivered at the end to check the efficacy of the designed control scheme.

Fixed‐Time State Observer‐Based Robust Adaptive Neural Fault‐Tolerant Control for a Quadrotor Unmanned Aerial Vehicle

ABSTRACTThis paper presents a fixed‐time state observer‐based robust adaptive neural fault‐tolerant control (RANFTC) for attitude and altitude tracking and control of quadrotor unmanned aerial vehicles (UAVs), considering multiple actuator faults, parametric uncertainty, and unknown external disturbances simultaneously. A novel fixed‐time state error estimation based on sliding mode observer is designed, which is independent of initial conditions. A proportional–integral–derivative (PID) based sliding mode control (SMC) is proposed to handle actuator faults and unknown disturbances in combination with the fixed‐time observer within the fault‐tolerant control (FTC) design scheme. The radial basis function neural network (RBFNN) is employed with the controller to approximate the uncertain parameters of the system. Furthermore, two new adaptive laws are designed to estimate the sudden actuator fault and the unknown upper bound of disturbances independently. Implementing these estimation schemes avoids overestimation, enhances the robustness of the presented controller, and substantially eliminates the control chattering problem. By applying the Lyapunov stability concept, the suggested control strategy guarantees that the states of the quadrotor UAV converge to the origin in a finite time. Finally, simulation studies are conducted to demonstrate the tracking performance and highlight the effectiveness of the proposed FTC design compared to the existing FTC methods.

Event-triggered finite-time adaptive neural network control for quadrotor UAV with input saturation and tracking error constraints

In this article, an event-triggered finite-time adaptive neural network control tracking strategy is proposed for quadrotor unmanned aerial vehicle (UAV) with input saturation and error constraints. Firstly, the radial basis function neural networks (RBFNNs) are adopted to identify the unknown uncertainty of quadrotor UAV model from the installation errors, gyroscope errors and so on. An auxiliary equation is constructed to deal with input physical saturation from the actuator motors. Additionally, by combining the performance function and error transformation, the issue of error constraint is solved. Based on the Lyapunov stability theory and event-triggered mechanisms, a finite-time adaptive neural network scheme is developed to ensure that the closed-loop quadrotor UAV system is semi-globally practically finite-time stable, and save the computation, resources, and transmission load. Finally, the simulation results illustrate the good tracking performance of quadrotor UAV by using the proposed control strategy.

Control synchronization of the anti-resonance vibration system with two eccentric rotors driven by the motors

The present research works focuses on the control synchronization of anti-resonance vibration system (ARVS) with two eccentric rotors driven by the motors. The small parameters average method and Hamilton’s criterion are employed to analyze the self-synchronous theory of the proposed system from the views of mechanics and mathematics. The research results show that the zero-phase synchronization between two motors is difficult to carry out owing to its own weak coupling property; the ideal dynamic characteristics of the ARVS are easily destroyed by the material quality fluctuation. Therefore, the controllers of the angular speed and phase angle are designed by using the sliding mode control (SMC) algorithm with master-slave control structure. And then Lyapunov theory is used to prove the stability of the controllers. Moreover, some simulation results are given to explore the feasibility and validity of the proposed controllers. The results indicate that the zero-phase synchronization between two motors can be performed by using the proposed control strategy; meanwhile, the ideal dynamic characteristics of the ARVS can be maintained by the proposed controllers with the material quality fluctuation; additionally, a strong robustness of the control system is carried out to reject the structural parametric interferences to the dynamic characteristics. The results can provide a control method for the engineering design of the anti-resonance vibration machine and help anti-resonance vibration machine adapt to more complex working environment.

Distributed robust tracking control for multiple unmanned aerial vehicles: theory and experimental verification

The distributed trajectory tracking problem for multiple UAVs under unknown external disturbance is investigated. A new nonlinear robust trajectory tracking strategy for multiple UAVs is proposed. The dynamic model for the UAVs' formation is illustrated in the Tangent frame. For the trajectory tracking problem, a Robust formation tracking control strategy based on robust integral of the signum of error (RISE) is designed to compensate for the effects of unknown external disturbances, which improves the robustness of the UAVs' formation system. Based on the Lyapunov stability analysis, it is proved that the global asymptotic convergence of the coordination errors and the semi-global asymptotic convergence of the UAVs' position tracking errors are achieved. The proposed formation control strategy is validated via real-time experiments that are performed on the self-build UAVs' formation flight control testbed. Flights without wind disturbance and flights with disturbance are all performed. The performance comparison experiment with the conventional sliding mode control algorithm is performed. The experimental results show that the designed control strategy can achieve good coordinated trajectory tracking of multiple UAVs, and has better control performance compared with normal sliding mode control law.

Adaptive output feedback control of a bidirectional AC/DC totem‐pole PFC converter for wireless charging of electric vehicles

AbstractThis paper deals with the control problem of a single‐phase bidirectional alternating current/direct current (AC/DC) totem‐pole power factor correction (PFC) converter intended to be used in wireless power transfer (WPT) chargers for electric vehicles (EVs). The converter is operating as a bridgeless boost rectifier with the PFC functionality during grid to vehicle (G2V) mode and as a PFC inverter during vehicle to grid (V2G) mode. Firstly, the operating principle is presented. Secondly, an overall mathematical model governing all the operating modes of the converter is elaborated. Thirdly, due to the nonlinearity of the studied system, a nonlinear controller–based Lyapunov technique is designed to achieve the following control objectives: (i) unity power factor (UPF) during both power transfer modes G2V and V2G, (ii) regulation of the DC‐link voltage, and (iii) asymptotic stability of the closed‐loop system. Further, as the developed control law requires the measurement of all voltages and currents, an adaptive observer–based Kalman‐like technique is designed to estimate all the required signals and parameters. The performances of the proposed output feedback control technique are validated through several simulation scenarios showing that all control objectives are achieved. It should be noted that the use of the Kalman observer reduces the complexity of the systems as it requires only two measurements (grid voltage and current) making it an advantageous solution over existing solutions where three measurements are generally used. Furthermore, due to the low‐pass filtering behavior of the observer and its susceptibility to estimate the converter's parameters, the robustness of the controller and its insensitivity to the parameter uncertainties are greatly improved.

Neuroadaptive Sliding Mode Tracking Control for an Uncertain TQUAV With Unknown Controllers

ABSTRACTIn this article, a neuroadaptive sliding mode control (NSMC) strategy based on recurrent neural network (RNN) for robustly and adaptively tracking the desired position and attitude of an uncertain tilting quadrotor unmanned aerial vehicle (TQUAV) with unknown controllers is presented. The main contribution of this article is the real‐time adjustment of unknown flight controllers using the approximation characteristics of RNN, in which the derived approximation errors of RNN are sufficiently estimated by adaptive control method that can reduce or eliminate the impact of error terms on the evolution of closed‐loop systems. Especially, Lyapunov stability analysis is greatly simplified compared to existing methods and does not require amplification or reduction. Finally, the superior performance of the NSMC strategy was verified by comparing simulation results.

Robust fixed‐time cooperative control for nonlinear multi‐agent systems via event‐triggered communication

AbstractThis article presents a robust fixed‐time cooperative control scheme for high‐order nonlinear multi‐agent systems under event‐triggered communication. A fixed‐time leader‐states estimator with event‐triggered communication is first designed to reconstruct the essential information, since only a fraction of agents can access the leader. Subsequently, we develop a disturbance observer‐based fixed‐time dynamic surface controller that utilizes the estimated leader state. The fixed‐time convergence of the whole closed‐loop system is proved by Lyapunov stability theory. Finally, numerical simulations elucidate the validity of the theoretical result.

ADP-based adaptive control of ship course tracking system with prescribed performance

In this paper, an adaptive dynamic programing (ADP)-based adaptive control problem is studied for the ship course tracking system with prescribed performance. Firstly, the complex sea conditions will bring difficulty to ship course tracking when unmanned ship sailing on sea. In order to overcome this difficulty, a prescribed performance feedforward term is designed in the controller to improve the course tracking ability of ship. Secondly, by defining a cost function, an ADP technology is exploited to achieve an optimal control of ship course tracking system. Finally, the stability of the closed loop system is turned out by Lyapunov stability theory and all signals are uniformly ultimately bounded (UUB). Simulation results validate the effectiveness of the proposed method.

Adaptive vibration control for stabilisation of the Euler–Bernoulli beam with an unknown payload

In this paper, an adaptive boundary controller is designed to solve the boundary control problem of Euler–Bernoulli beams with an unknown payload. Therefore, a boundary controller with an adaptive law is designed to compensate for the parameter uncertainty of the system. Partial Differential Equations (PDEs) are used to describe the Euler–Bernoulli beam system. The well-posedness of the system under the action of an adaptive boundary controller is proved by using the linear operator semigroup method. Meanwhile, the asymptotic stability of the closed-loop system is derived from the extended Krasovskii–LaSalle invariance principle. The effectiveness and superiority of the proposed method are illustrated by simulation in comparison with the existing results.

Compensation Function Observer-Based Backstepping Sliding-Mode Control of Uncertain Electro-Hydraulic Servo System

Observer-based control is the most commonly used method in the control of electro-hydraulic servo system (EHSS) with uncertainties, but it suffers from the drawback of low accuracy under the influence of large external load forces and disturbances. To address this problem, this paper proposes a novel compensation function observer-based backstepping sliding-mode control (BSMC) approach to achieve high-accuracy tracking control. In particular, the model uncertainties, including nonlinearities, parameter perturbations and external disturbances are analyzed and treated together as a lumped disturbance. Then, a fourth-order compensation function observer (CFO) is constructed, which fully utilizes the system state information to accurately estimate the lumped disturbance. On this basis, the estimate of the lumped disturbance is incorporated into the design of a backstepping sliding-mode controller, allowing the control system to compensate for the disturbance effect. The stability of the closed-loop control system under the CFO and BSMC is rigorously proven through the use of the Lyapunov theory, which guarantees that all the tracking error signals converge exponentially to the origin. Comparative simulations are carried out to show the effectiveness and efficiency of the proposed approach, i.e., compared with PID and ESO-based BSMC methods, the tracking accuracy is respectively improved by 94.86% and 88.19% under the influence of large external load forces and disturbances.

Finite-time optimal control for a class of nonlinear systems with performance constraints via critic-only ADP: Theory and experiments

This paper addresses the optimal control problem within the framework of adaptive dynamic programming (ADP) for a class of nonlinear systems subjected to performance constraints. A new finite-time optimal control scheme is developed to stabilize the system by using the critic-only neural network ADP method. Compared with the existing ADP-based optimal control methods with uniformly ultimately bounded stability, the provided control scheme ensures that the controlled system's state and neural network weight estimation error are finite-time stable. It can ensure optimality, prescribed performance, and finite-time stability of the closed-loop control system simultaneously through an integration of ADP, the prescribed performance control technique, and Lyapunov theory. The designed adaptive neural network weight update law can relax the persisting excitation condition. The proposed control scheme is implemented on a robotic experiment platform to achieve trajectory tracking and verify its effectiveness.

Stability and Bifurcation Analysis of a Symmetric Fractional-Order Epidemic Mathematical Model with Time Delay and Non-Monotonic Incidence Rates for Two Viral Strains

This study investigates a symmetric fractional-order epidemic model with time delays and non-monotonic incidence rates, considering two viral strains. By confirming the existence, uniqueness, and boundedness of the system’s solutions, the research ensures the model’s well-posedness, guaranteeing its mathematical soundness and practical relevance. The study calculates and evaluates the equilibrium points and the basic reproduction numbers R01 and R02 to understand the dynamic behavior of the model under different parameter settings. Through the application of the Lyapunov method, the research examines the asymptotic global stability of the system, determining whether it will converge to a particular equilibrium state over time. Furthermore, Hopf bifurcation theory is employed to investigate potential periodic solutions and bifurcation scenarios, highlighting how the system might shift from stability to periodic oscillations under certain conditions. By utilizing the Adams-Bashforth-Moulton numerical simulation method, the theoretical results are validated, reinforcing the conclusions and demonstrating the model’s applicability in real-world contexts. It emphasizes the importance of fractional-order models in addressing epidemiological issues related to time delays (τ), individual heterogeneity (m, k), and memory effects (θ), offering greater accuracy compared with traditional integer-order models. In summary, this research provides a theoretical foundation and practical insights, enhancing the understanding and management of epidemic dynamics through fractional-order epidemic models.

Dynamic output feedback robust MPC hybrid fault-tolerant control for discrete uncertain systems: a switching control strategy

To guarantee the control performance of the discrete system under unmeasurable state and partial actuator failure, a robust dynamic output feedback predictive fault-tolerant control method derived from switching strategy is developed. A new iterative error switching model is established, which both ensures tracking performance and lays the foundation for the development of the next switching controller. Based on this model, output feedback switching fault-tolerant controller is developed. Compared with conventional fault-tolerant control methods, the designed controller can be used under the case of unmeasurable states, and it is more fault tolerant and less conservative. According to the Lyapunov stability theory, the robust stability analysis is carried out under normal and fault conditions, respectively. Then, depending on the findings of the stability analysis, sufficient conditions to guarantee the stability of the closed-loop normal and faulty system are derived. Finally, the suggested method's efficacy is confirmed using the injection moulding process as an example.

Enhanced control of grid-connected multi-machine wind power generation systems using fuzzy backstepping approaches

This research paper presents an approach for enhancing the performance of a multi-machine wind power generation system (WPGS) through the combination of nonlinear and intelligent control techniques. The suggested approach focuses on optimizing the WPGS, which utilizes permanent magnet synchronous generators (PMSGs) interconnected with the grid. To enable grid connection, a back-to-back configuration is employed for both the rectifier and inverter in a fifteen-switch configuration, with control achieved through pulse width modulation. The designed control algorithm is designed to effectively regulate the system and reduce the active and reactive power ripples. Initially, the WPGS model is provided. Subsequently, the fuzzy backstepping control (FBC) law is detailed, incorporating the Lyapunov stability technique. Fuzzy logic is used to adapt the gains in the backstepping approach, ensuring the control system can accommodate disturbances and system variations. As a result, this adaptive control approach enables optimal effusiveness of the entire system in both static and dynamic operational regimes. Experimental tests were conducted using MATLAB to compare the efficacy of the designed strategy against the traditional backstepping control approach. The obtained results demonstrated the robustness and effective reference tracking capabilities of the designed FBC approach, along with the complete elimination of speed overshoot across diverse wind conditions. These findings validate the designed control approach's effectiveness and superiority over the BC approach under varying conditions.

The role of asymptomatic cattle for leptospirosis dynamics in a herd with imperfect vaccination

Leptospirosis is an emerging zoonotic disease with high health and economic damage. In this study, we developed a deterministic mathematical model that describes the dynamics of leptospirosis transmission within a cattle herd, incorporating asymptomatic infected and vaccinated compartments. The study examined the transmission role of asymptomatic cattle that contaminate herds without farmers’ knowledge. We proved the well-posedness of the proposed model and found the basic reproduction number using the next-generation matrix. Analytically, we demonstrated that the disease-free equilibrium point is locally and globally asymptotically stable when R0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr {R}}_0$$\\end{document} is less than unity and is otherwise unstable. Graphically, we further established the local asymptotic stability of disease-free and endemic equilibria. Sensitivity analysis showed that the contact rate with asymptomatic infected cattle, βA\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta _A$$\\end{document}, is the most sensitive parameter in the stated model, followed by the recovery rate of asymptomatic infected cattle, σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma $$\\end{document}, and the vaccination rate of susceptible cattle, τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document}. Numerical simulations revealed that a reduction in contact rate with asymptomatic infected cattle significantly reduced pathogen Leptospira transmission in the herd. In addition, fostering the recovery rate of asymptomatic infected cattle can significantly reduce new infections in the herd. Furthermore, augmenting the vaccination rate among susceptible cattle resulted in a notable decrease in disease prevalence within the herd. Findings of this study underscore the remarkable importance of targeted interventions, such as reducing contact rates with asymptomatic infected cattle, increasing recovery rates using proper treatments, and enhancing vaccination efforts to manage leptospirosis transmission in cattle herds.