Lyapunov Direct Method Research Articles

Adaptive Path-Tracking Control Algorithm for Autonomous Mobility Based on Recursive Least Squares with External Condition and Covariance Self-Tuning

This paper introduces an adaptive path-tracking control algorithm for autonomous mobility based on recursive least squares (RLS) with external conditions and covariance self-tuning. The advancement and commercialization of autonomous driving necessitate universal path-tracking control technologies. In this study, we propose a path-tracking control algorithm that does not rely on vehicle parameters and leverages RLS with self-tuning mechanisms for external conditions and covariance. We designed an integrated error for effective path tracking that combines the lateral preview distance and yaw angle errors. The controller design employs a first-order derivative error dynamics model with the coefficients of the error dynamics estimated through the RLS using a forgetting factor. To ensure stability, we applied the Lyapunov direct method with injection terms and finite convergence conditions. Each regression process incorporates external conditions, and the self-tuning of the injection terms utilizes residuals. The performance of the proposed control algorithm was evaluated using MATLAB®/Simulink® and CarMaker under various path-tracking scenarios. The evaluation demonstrated that the algorithm effectively controlled the front steering angle for autonomous path tracking without vehicle-specific parameters. This controller is expected to provide a versatile and robust path-tracking solution in diverse autonomous driving applications.

Analysis of COVID-19’s Dynamic Behavior Using a Modified SIR Model Characterized by a Nonlinear Function

This study develops a modified SIR model (Susceptible–Infected–Recovered) to analyze the dynamics of the COVID-19 pandemic. In this model, infected individuals are categorized into the following two classes: Ia, representing asymptomatic individuals, and Is, representing symptomatic individuals. Moreover, accounting for the psychological impacts of COVID-19, the incidence function is nonlinear and expressed as Sg(Ia,Is)=βS(Ia+Is)1+α(Ia+Is). Additionally, the model is based on a symmetry hypothesis, according to which individuals within the same compartment share common characteristics, and an asymmetry hypothesis, which highlights the diversity of symptoms and the possibility that some individuals may remain asymptomatic after exposure. Subsequently, using the next-generation matrix method, we compute the threshold value (R0), which estimates contagiousness. We establish local stability through the Routh–Hurwitz criterion for both disease-free and endemic equilibria. Furthermore, we demonstrate global stability in these equilibria by employing the direct Lyapunov method and La-Salle’s invariance principle. The sensitivity index is calculated to assess the variation of R0 with respect to the key parameters of the model. Finally, numerical simulations are conducted to illustrate and validate the analytical findings.

Mathematical modeling of containing the spread of heroin addiction via awareness program

As many countries are hit by the social economic impact of heroin addiction, there is an urgent need to have an effective awareness program that focuses on educating the population on the danger of heroin addiction and helping the heroin‐users quit. This paper aims to study the effect of the awareness program on the spread of heroin dependence using a mathematical model with distributed delay. First, we show the existence threshold parameter , which we call the basic reproduction number of the spread of heroin use. We prove, via the Lyapunov direct method, that the drug‐free equilibrium is globally asymptotically stable if . If , the drug dependence persists, and the drug equilibrium is globally asymptotically stable. We give three possible scenarios to find the optimal awareness program strategy that puts the heroin epidemic under control. These scenarios take into consideration the reachability of the population, the immunity against heroin addiction, and the effectiveness of the program to contain the use of heroin in the population and bring below one.

Synchronization on fractional multiplex higher-order networks.

This paper explores the synchronization problem in fractional multiplex higher-order networks. Initially, a fractional multiplex higher-order network model is established, which seamlessly integrates multiplex structures with higher-order interactions. Subsequently, by leveraging a well-crafted Lyapunov function, the Lyapunov direct method, and fractional inequalities, it is demonstrated that the fractional multiplex higher-order network can achieve intra-layer synchronization, inter-layer synchronization, and complete synchronization. Finally, the theoretical findings are validated through two numerical examples featuring a simplicial complex or hypergraph structures within the intra-layer network.

Adaptive control of networked control systems under DoS attacks

This research introduces a new adaptive control approach tailored for networked control systems, aimed at addressing the challenges of Denial of Service (DoS) attacks and round-trip time delays. Initially, a compensation mechanism based on estimated values is introduced to counter DoS attacks; subsequently, for round-trip time delay issues, a predictive delay compensation control strategy is designed. The proposed control strategy does not rely on the specific structure or a particular model of the system, but instead solely updates the controller using the system's input and output values. Moreover, we successfully demonstrate the bounded stability of the system using the direct Lyapunov method. Ultimately, the effectiveness and practicality of this method are confirmed through simulation results.

Robust higher order sliding mode control of grid‐forming converters with LCL filter in weak grid scenarios for fast frequency support

AbstractThis paper presents a nonlinear higher‐order sliding mode control (HO‐SMC) designed for a droop control‐based grid‐forming converter. In weak grid scenarios, where the rate of change of frequency is notably high, achieving a rapid frequency response becomes imperative. The stable operation of a grid‐forming converter using droop control, coupled with classical vector control that employs cascaded voltage and current loops (multiloop) with PI controllers, faces limitations when higher droop coefficients are applied. This constraint on the application of classical vector control in weak grid conditions necessitates alternative solutions. Operating as a grid‐forming converter, the grid‐connected converter with an LCL filter represents a second‐order system. HO‐SMC mitigates the switching challenges associated with conventional SMC by integrating robust feedback linearized control. A graphical method is proposed for designing the switching gain using Lyapunov's direct method to counteract the impact of a matched disturbance. The study demonstrates that the implementation of HO‐SMC in the grid‐forming converter enhances fast frequency response by increasing the gain margin of the power frequency loop. Finally, it is illustrated that the proposed control method also improves the transient response of the converter.

A spatio-temporal infection epidemic model with fractional order, general incidence, and vaccination analysis

The paper investigates a spatio-temporal SEIR epidemic model on a global level with fractional order. It employs four partial differential equations incorporating fractional derivatives and account for diffusion to characterize infection dynamics. By applying fixed-point theorem results, the paper establishes the existence, uniqueness, and boundedness of the solution. Equilibrium points are determined based on vaccination values and the basic reproduction number R0. Global stability of each equilibrium is confirmed using the Lyapunov direct method. Through simulations with a predictor–corrector algorithm, key insights into epidemiological dynamics are provided, elucidating the impact of vaccination on reducing disease transmission and altering R0. Trajectories of various compartments closely align with theoretical equilibrium points, affirming the model’s predictive precision. Furthermore, simulations indicate the potential for attaining disease-free equilibria with heightened vaccination rates, underscoring the pivotal role of vaccination strategies in epidemic control and disease eradication. It was shown that the fractional derivative order has no effect on the equilibrium stability but rather only on the convergence speed towards the equilibrium points.

Academy Transaction note “matrix gains selection for three-axis magnetic attitude control of dynamically elongated satellite”

The paper proposes algorithm for angular motion control of a dynamically elongated spacecraft. The algorithm is based on the direct Lyapunov method using matrix control coefficients. The calculated mechanical torque is implemented using magnetorquers. Control parameters are selected using Floquet theory to ensure convergence to the required motion.

On discrete FitzHugh–Nagumo reaction–diffusion model: Stability and simulations

This research paper focuses on the analysis of a discrete FitzHugh–Nagumo reaction–diffusion system. We begin by discretizing the FitzHugh–Nagumo reaction–diffusion model using the second-order and L1-difference approximations. Our study examines the local stability of the equilibrium points of the system. To identify conditions that ensure the global asymptotic stability of the steady-state solution, we employ various techniques, with a primary focus on the direct Lyapunov method. Theoretical results are supported by numerical simulations that demonstrate the practical validity of the asymptotic stability conclusions. Our findings provide new insights into the stability characteristics of discrete FitzHugh–Nagumo reaction–diffusion systems and contribute to the broader understanding of such systems in mathematical biology.

An LMI backstepping generalization via [formula omitted] dynamic control and Takagi-Sugeno models

Convex optimization and dynamic control are employed in this work to overcome limitations of the backstepping control technique, regarding handling of uncertain MIMO non-affine-in-control systems subject to additive disturbances. To this end, a novel set of conditions for H∞ attenuation in the form of linear matrix inequalities is developed based on the direct Lyapunov method and exact Takagi-Sugeno models. The proposal is put at test both in academic and physical systems that cannot otherwise be treated by standard backstepping techniques.

Inverse compensation mechanism-based adaptive fuzzy-neural fault-tolerant control for an uncertain quadrotor UAV

Actuator faults, system uncertainties, and external disturbances can affect tracking control performance of the quadrotor UAV, and even cause the instability of the system. To address this issue, an adaptive fuzzy-neural fault-tolerant control (AFNFTC) scheme is proposed for the quadrotor UAV integrated with inverse compensation strategy and fuzzy neural networks (FNNs) technology. Firstly, we design an inverse compensation mechanism to address actuator faults and utilize the approximation capabilities of the FNNs to compensate for system uncertainties. Then, based on the Lyapunov direct method, the AFNFTC scheme is designed to ensure the stability of the quadrotor UAV. Under the proposed scheme, the closed-loop system for the quadrotor UAV is proven to be uniformly bounded, and the tracking error of the quadrotor UAV is demonstrated to converge to a small compact set ultimately by appropriately selecting design parameters. Finally, the feasibility and effectiveness of the proposed scheme is validated through simulation results.

Stability and synchronization of fractional-order reaction–diffusion inertial time-delayed neural networks with parameters perturbation

This study is centered around the dynamic behaviors observed in a class of fractional-order generalized reaction–diffusion inertial neural networks (FGRDINNs) with time delays. These networks are characterized by differential equations involving two distinct fractional derivatives of the state. The global uniform stability of FGRDINNs with time delays is explored utilizing Lyapunov comparison principles. Furthermore, global synchronization conditions for FGRDINNs with time delays are derived through the Lyapunov direct method, with consideration given to various feedback control strategies and parameter perturbations. The effectiveness of the theoretical findings is demonstrated through three numerical examples, and the impact of controller parameters on the error system is further investigated.

Global behavior and optimal control of a dengue transmission model with standard incidence rates and self‐protection

It takes into account that conscious, susceptible individuals with self‐protection cannot be infected by mosquito bites, a seven‐dimensional dengue transmission model with self‐protection, and two different standard incidence rates are proposed. Our focus is on discussing the global behavior and the optimal control of the model. We first calculate the control reproduction number and determine that if , the model has a unique dengue equilibrium . Furthermore, we obtain the local stability of dengue‐free equilibrium and the dengue equilibrium by using the method of proof by contradiction. By utilizing the limit system of the model and the Lyapunov direct method, we acquire that is globally stable if ; is globally attractive if , and if , the model is weakly persistent with a thorough analysis, and then is globally stable. Finally, in order to minimize investment in dengue transmission, an optimal control strategy with four control variables is presented using Pontryagin's minimum principle.

Adaptive Dynamic Programming for Robust Event-Driven Tracking Control of Nonlinear Systems With Asymmetric Input Constraints.

This article considers the robust dynamic event-driven tracking control problem of nonlinear systems having mismatched disturbances and asymmetric input constraints. Initially, to tackle the asymmetric constraints, a novel nonquadratic value function is constructed for the original system. This makes the asymmetrically constrained tracking control problem transformed into an unconstrained optimal regulation problem. Then, a dynamic event-driven mechanism is proposed. Meanwhile, the event-driven Hamilton-Jacobi-Bellman equation (ED-HJBE) is developed for the optimal regulation problem in order to acquire the optimal control with distinctly decreased computational burden. To solve the ED-HJBE, a single critic neural network (CNN) is designed in the adaptive dynamic programming framework. Meanwhile, the gradient descent method is employed to update the CNN's weights. After that, both the weight estimation error and the tracking error are proved to be uniformly ultimately bounded via Lyapunov's direct method. Finally, simulations of the spring-mass-damper system and the pendulum plant are separately utilized to validate the established theoretical claims.

Lag projective synchronization of discrete-time fractional-order quaternion-valued neural networks with time delays

This paper deals with the lag projective synchronization (LPS) problem for a class of discrete-time fractional-order quaternion-valued neural networks(DTFO QVNNs) systems with time delays. Firstly, a DTFOQVNNs system with time delay is constructed. Secondly, linear and adaptive feedback controllers with sign function are designed respectively. Furthermore, through Lyapunov direct method, DTFO inequality technique and Razumikhin theorem, some sufficiency criteria are obtained to ensure that the system in this article can achieve LPS. At last, the significance of the theoretical part of this paper is verified through numerical simulation.

Asymptotic stability results of generalized discrete time reaction diffusion system applied to Lengyel-Epstein and Dagn Harrison models

In this research paper, we delve into the analysis of a generalized discrete reaction-diffusion system. Our study begins with the discretization of a generalized reaction-diffusion model, achieved through second-order and L1-difference approximations. We explore the local stability of its unique solution, both in the absence and presence of the diffusion term. To determine the conditions for global asymptotic stability of the steady-state solution, we employ suitable techniques including the direct Lyapunov method. To illustrate the practical application of this theoretical framework, we provide several numerical simulations that examine both the Lengyel-Epstein reaction-diffusion model and the discrete Degn-Harrison reaction-diffusion model. These simulations serve to validate the predictions of asymptotic stability.

Finite-time synchronization of fractional-order uncertain quaternion-valued neural networks via slide mode control

This paper investigates finite-time synchronization of fractional-order quaternion-valued neural networks (F-QV-NNs), involving uncertainties in the system parameters. We adopt a combined approach of sliding mode control (SMC) and a non-separation strategy to achieve finite-time synchronization. Based on SMC theory, we construct a specific sliding surface and design a controller to ensure the occurrence of sliding motion. To achieve the desired sliding motion, we apply the fractional Lyapunov direct method to guide the system's states to the designed sliding surface. Moreover, the derived sufficient conditions ensure finite-time synchronization within the models. Lastly, we give a numerical example to validate the effectiveness of the acquired results.

Conditions for ultimate boundedness of solutions and permanence for a hybrid Lotka–Volterra system

In the paper, a generalized Lotka–Volterra – type system with switching is considered. The conditions for the ultimate boundedness of solutions and the permanence of the system are studied. With the aid of the direct Lyapunov method, the requirements for the switching law are established to guarantee the necessary dynamics of the system. An attractive compact invariant set is constructed in the phase space of the system, and a given region of attraction for this set is provided. A distinctive feature of the work is the use of a combination of two different Lyapunov functions, each of which plays its own special role in solving the problem.

Neural network-based adaptive optimal tracking control for hypersonic morphing aircraft with appointed-time prescribed performance

In this paper, an adaptive optimal attitude tracking controller for hypersonic morphing aircraft with appointed-time prescribed performance is presented. A proper error transformation methodology is developed to ensure the performance constraints are met without knowing initial tracking conditions. Subsequently, an adaptive composite control scheme is devised for the transformed system, comprising a feedforward neuro-backstepping controller and a feedback optimal controller. The former is employed to convert the optimal tracking problem into an equivalent optimal regulation problem, while the latter is constructed using adaptive dynamic programming (ADP) with a single critic network to achieve optimality. A novel weight-tuning law for the critic network is introduced, incorporating a stability indicator and the concurrent learning technique. This approach relaxes the underlying restrictive conditions in ADP, improving the robustness and availability of the system. The stability of the closed-loop system is analyzed using the Lyapunov direct method. Finally, the effectiveness and improved performance of the proposed control scheme are validated through numerical simulations.

A novel chaotic oscillator with a half-line of unstable equilibria: Basins of attraction, chaos control, chaos synchronization, and encryption applications

A novel 3D chaotic oscillator that incorporates quadratic and absolute-function nonlinearities is introduced in this paper. The system dynamics are explored using the Lyapunov direct method, phase plane trajectories, time response, Lyapunov exponents, bifurcation diagrams, and basins of attraction. The uniqueness and existence of the system solution have been proven. The analytical investigations show that the system has two stable equilibrium points along with one unstable equilibrium point and a line of equilibria. The positive half of this line represents unstable equilibria, while the negative half is associated with stable equilibria. Additionally, it is found that the oscillator exhibits a chaotic basin of attraction centered along the line of equilibria and surrounded by a fixed-point attractor. An electronic circuit using Multisim software is designed to demonstrate the possibility of physical implementation for the considered mathematical model. Chaos control is addressed using adaptive and sliding mode control strategies. The performance of both control methods is compared, with the sliding mode control demonstrating superior results in both fast response and small transient overshoot. Furthermore, a novel sliding mode controller for master–slave synchronization is introduced, showing high performance compared to other sliding-mode and adaptive control methods applied in the literature. Finally, the proposed oscillator is employed as a Pseudo Random Number Generator (PRNG) for image encryption applications using a new encryption model. The experimental results confirm the high security and robustness of the proposed encryption algorithm against various attack methods.