Lyapunov Method Research Articles

Equilibrium Problem for the Stochastic Multi-Weighted Urban Public Transportation System with Time Delay: A Graph-Theoretic Method

This paper focuses on the equilibrium problem of an urban public transportation system with time delay. Time delay, multi-weights, and stochastic disturbances are considered in the urban public transportation system. Hence, one can regard the urban public transportation system as a stochastic multi-weighted delayed complex network. By combining graph theory and the Lyapunov method, the global Lyapunov function is constructed indirectly. Moreover, the response system can realize synchronization with the drive system under the adaptive controller. In other words, the urban public transportation system is balanced in the actual running traffic network. Finally, numerical examples about the Chua system and small-world network are presented to confirm the accuracy and validity of the theoretical results.

Study of dynamic stability of MV (middle voltage) Network of Douala within the meaning of Lyapunov

The studies undertaken have used the method of Lyapunov of the system theory to evaluate the stability of the electric network starting from the power function depending on the dephasing between voltage and current in time. To carry out these studies, a look into the dynamic stability of the electrical network, and the stability within the meaning of Lyapunov was effected. Then a modelling of our network with application under MATLAB/Simulink was carried out as well as the tests of simulations without disturbances, and another test with disturbances was made. This reveals that the Douala network suffered falls in voltage, increasing instability depending on the critical disturbances, with the network eliminating the faults slowly.

Analysis of a non-integer order mathematical model for double strains of dengue and COVID-19 co-circulation using an efficient finite-difference method

An efficient finite difference approach is adopted to analyze the solution of a novel fractional-order mathematical model to control the co-circulation of double strains of dengue and COVID-19. The model is primarily built on a non-integer Caputo fractional derivative. The famous fixed-point theorem developed by Banach is employed to ensure that the solution of the formulated model exists and is ultimately unique. The model is examined for stability around the infection-free equilibrium point analysis, and it was observed that it is stable (asymptotically) when the maximum reproduction number is strictly below unity. Furthermore, global stability analysis of the disease-present equilibrium is conducted via the direct Lyapunov method. The non-standard finite difference (NSFD) approach is adopted to solve the formulated model. Furthermore, numerical experiments on the model reveal that the trajectories of the infected compartments converge to the disease-present equilibrium when the basic reproduction number ({mathbb {R}}_0) is greater than one and disease-free equilibrium when the basic reproduction number is less than one respectively. This convergence is independent of the fractional orders and assumed initial conditions. The paper equally emphasized the outcome of altering the fractional orders, infection and recovery rates on the disease patterns. Similarly, we also remarked the importance of some key control measures to curtail the co-spread of double strains of dengue and COVID-19.

Stability of Impaired Humoral Immunity HIV-1 Models with Active and Latent Cellular Infections

This research aims to formulate and analyze two mathematical models describing the within-host dynamics of human immunodeficiency virus type-1 (HIV-1) in case of impaired humoral immunity. These models consist of five compartments, including healthy CD4+ T cells, (HIV-1)-latently infected cells, (HIV-1)-actively infected cells, HIV-1 particles, and B-cells. We make the assumption that healthy cells can become infected when exposed to: (i) HIV-1 particles resulting from viral infection (VI), (ii) (HIV-1)-latently infected cells due to latent cellular infection (CI), and (iii) (HIV-1)-actively infected cells due to active CI. In the second model, we introduce distributed time-delays. For each of these systems, we demonstrate the non-negativity and boundedness of the solutions, calculate the basic reproductive number, identify all possible equilibrium states, and establish the global asymptotic stability of these equilibria. We employ the Lyapunov method in combination with LaSalle’s invariance principle to investigate the global stability of these equilibrium points. Theoretical findings are subsequently validated through numerical simulations. Additionally, we explore the impact of B-cell impairment, time-delays, and CI on HIV-1 dynamics. Our results indicate that weakened immunity significantly contributes to disease progression. Furthermore, the presence of time-delays can markedly decrease the basic reproductive number, thereby suppressing HIV-1 replication. Conversely, the existence of latent CI spread increases the basic reproductive number, intensifying the progression of HIV-1. Consequently, neglecting latent CI spread in the HIV-1 dynamics model can lead to an underestimation of the basic reproductive number, potentially resulting in inaccurate or insufficient drug therapies for eradicating HIV-1 from the body. These findings offer valuable insights that can enhance the understanding of HIV-1 dynamics within a host.

Distributed sliding mode control strategy for intelligent connected vehicle platoon in complex media

This study examines the performance of a nonlinear sliding mode controller in cooperative control of the intelligent connected vehicles systems. The study begins by proposing a vehicle-to-vehicle car-following model that takes into account the impacts of disturbances both external and within in the system. Subsequently, an adaptive exponential reaching sliding mode controller based on a non-singular terminal sliding mode variable is proposed. It is analyzed theoretically using Lyapunov method, differential method and difference method and the results show that using the new controller, the system can converge in finite time with enhance dynamic performance. Finally, to verify the effectiveness of the controller proposed in this study, numerical experiments in MATLAB and simulation experiments in the ROS-Gazebo environment are conducted, and the results show that the controller proposed in this study enables the intelligent vehicle system to maintain stable operation in the face of interferences. Experiment results are in accordance with theoretical analysis.

Exponential bipartite synchronisation of stochastic signed networks via intermittent pinning discrete-time state observations control

This paper studies the exponential bipartite synchronisation in mean square (EBS-MS) of stochastic signed networks (SSNs) via periodically intermittent pinning control. It is important to note that intermittent control is an extension of existing literature, which depends on discrete observations instead of continuous observations during the work period. Additionally, only a small percentage of the network nodes are controlled by intermittent control. Several sufficient conditions are established by the Lyapunov method and graph theory. Especially, when the duration τ between two consecutive observations tends to zero, our control will degenerate into common periodically intermittent pinning control. Meanwhile, a corollary about the EBS-MS of SSNs is presented. Besides, theoretical results are applied to a single-link robot arm system and an oscillator system. Finally, numerical simulations are presented to show the feasibility of the theoretical results.

Fully Distributed, Event-Triggered Containment Control of Multi-Agent Systems Based on Wireless Sensor Networks and Time Base Generators

In this study, wireless sensor networks and time base generators are used to solve the fixed-time containment control problem in multi-agent systems with fixed topologies. A new event-triggered control protocol is proposed, which combines a fully distributed method and a time base generator (TBG). The goal is to converge the states of all followers to the convex hull formed by the leader. The controller reduces communication and improves control efficiency by integrating a fully distributed control mechanism using wireless sensor networks. In addition, a time base generator (TBG) is added to ensure that the dwell time continues to be pre-specified and independent of initial conditions. Using matrix theory, the original system is transformed into an error system, and its stability is analyzed by the Lyapunov method. The necessary and sufficient conditions for solving the time consensus containment control problem in multi-agent systems are determined and Zeno behavior is avoided. The effectiveness of the proposed control algorithm is illustrated by numerical examples.

Simultaneously primary and super-harmonic resonance of a van der Pol oscillator with fractional-order derivative

The dynamic characteristics and stability of a fractional-order van der Pol oscillator under simultaneously primary and super-harmonic resonance are analyzed by the multiscale method. At first, the first-order approximate analytical solution of the system is obtained, and the analytical results are verified by numerical simulation. In addition, the concepts of equivalent linear damping and equivalent linear stiffness of simultaneously primary and super-harmonic resonance are proposed to enhance the comprehension of fractional parameters roles. Based on Lyapunov's stability method, qualitative analysis is performed on the stability region and the type of equilibrium points of the steady-state solution. The influences of fractional order and coefficient on system stability, equivalent damping, and equivalent stiffness are discussed. It is found that the change of fractional order and coefficient will cause a deviation of the vibration curve and affect the multiple solutions, resonant frequency, and stability range of the system, which is useful to design or control the similar dynamic system.

On adaptive identification of systems having multiple nonlinearities

Objectives. The solution to the relevant problem of identifying systems with multiple nonlinearities depends on such factors as feedback, ways of connecting nonlinear links, and signal properties. The specifics of nonlinear systems affect control systems design methods. As a rule, the basis for the development of a mathematical model involves the linearization of a system. Under conditions of uncertainty, the identification problem becomes even more relevant. Therefore, the present work sets out to develop an approach to the identification of nonlinear dynamical systems under conditions of uncertainty. In order to obtain a solution to the problem, an adaptive identification method is developed by decomposing the system into subsystems.Methods. Methods applied include the adaptive identification method, implicit identified representation, S-synchronization of a nonlinear system, and the Lyapunov vector function method.Results. A generalization of the excitation constancy condition based on fulfilling the S-synchronizability for a nonlinear system is proposed along with a method for decomposing the system in the output space. Adaptive algorithms are obtained on the basis of the second Lyapunov method. The boundedness of the adaptive system trajectories in parametric and coordinate spaces is demonstrated. Approaches for self-oscillation generation and nonlinear correction of a nonlinear system are considered along with obtained exponential stability conditions for the adaptive system.Conclusions. Simulation results confirm the possibility of applying the proposed approach to solving the problems of adaptive identification while taking the estimation of the structural identifiability (S-synchronization) of the system nonlinear part into account. The influence of the structure and relations of the system on the quality of the obtained parametric estimates is investigated. The proposed methods can be used in developing identification and control systems for complex dynamic systems.

Adaptive event-triggered control for stability of fractional-order T-S fuzzy multi-links complex networks with random coupling delay

The main concern of this article is to address the exponential stability problem of fractional-order T-S fuzzy multi-links complex networks through designing the novel intermittent adaptive event-triggered control (IAE-TC) scheme. To reduce unnecessary transmission burden as much as possible in an allowable range, dynamic variable is introduced into fuzzy intermittent adaptive control and then fuzzy intermittent adaptive event-triggered control is established. In this control scheme, the controller is only activated when the event triggering conditions are not met, and has adaptability to the evolution results of the controlled network. Furthermore, by applying the Lyapunov method, the graph theory and the IAE-TC strategy, exponential stability criteria of the proposed networks are deduced. Meanwhile, we prove that Zeno behavior does not exist. Ultimately, an application about fractional-order T-S fuzzy multi-links competitive neural networks model is given to demonstrate the validity of the results, simultaneously, we show some numerical simulations.

Observer-based finite-time robust control of nonlinear time-delay uncertain systems with different power Hamiltonian function

Using the Hamiltonian function method with different powers, this paper studies the observer-based finite-time robust control problem of general nonlinear uncertain systems with time delay and presents some new results on this issue. First, by applying Hamiltonian realization method, the paper develops an equivalent Hamiltonian form of original system and designs its observer system. Then, based on the observer system, we study finite-time control problem and present several finite-time stabilization and finite-time robust stabilization results by using the augmented technology and the Lyapunov method. Finally, a real unmanned vehicle is used to verify the performance of the observer-based finite-time robust stabilization controller. Different from the existing literature on Hamiltonian method, the Hamiltonian function in the paper has different powers, which implies that the results developed in the paper have a wider range of application.

Novel General Regression Neural Networks for Improving Control Accuracy of Nonlinear MIMO Discrete-Time Systems.

In this article, a novel version of the general regression neural network (Imp_GRNN) is developed to control a class of multiinput and multioutput (MIMO) nonlinear discrete-time (DT) systems. The improvements retain the features of the original GRNN along with a significant improvement of the control accuracy. The enhancements include developing a method to set the input-hidden weights of GRNN using the inputs recursive statistical means, introducing a new output layer and adaptable forward weighted connections from the inputs to the new layer, and suggesting an interval-type smoothing parameter to eradicate the need for selecting the parameter beforehand or adapting it online. Also, controller stability is studied using Lyapunov's method for DT systems. The controller performance is tested with different simulation examples and compared with the original GRNN to verify its superiority over it. Also, Imp_GRNN performance is compared with an adaptive radial basis function network controller, an adaptive feedforward neural-network (NN) controller, and a proportional-integral-derivative (PID) controller, where it demonstrated higher accuracy in comparison with them. In comparison with the formerly proposed control methods for MIMO DT systems, our controller is capable of producing high control accuracy while it is model free, does not require complex mathematics, has low computational complexity, and can be utilized for a wide range of DT dynamic systems. Also, it is one of the few methods that aims to improve the control system accuracy by improving the NN structure.

A Fuzzy-Logic-System-Based Cooperative Control for the Multielectromagnets Suspension System of Maglev Trains With Experimental Verification

A maglev train is a sustainable public transport method with the characteristics of being green, pollution-free, and low-noise, as well as providing environmental protection. However, the performance of existing maglev control strategies for maglev trains may be deteriorated by various challenges, including disregard of the coordination and synchronization between multiple electromagnets, control input unidirectionality, dead zones, saturation, and finite time stability ability, etc. In this paper, an adaptive fuzzy-based suspension control method based on a multi-electromagnets dynamic coupling model is proposed that can cope with dead zone and saturation problems and guarantee the finite-time of the airgap tracking errors of multiple electromagnets simultaneously. Specifically, a fuzzy-logic system (FLS) is utilized to compensate for the nonlinear input unidirectionality, dead zone, saturation and unmodeled dynamics. Moreover, considering the coupling dynamic characteristics of adjacent electromagnet control modules, a fuzzy-based cooperative suspension controller with adaptive update law is designed. The finite time stability of the presented control strategy is proven with the Lyapunov method. Finally, the suspension frame experimental results are illustrated to validate the effectiveness and robustness of the developed method, whose superior performance is shown by being experimentally compared with some baseline methods.

High-Performance Position Control for Repetitive Tasks of Motor-Driven Servo Systems Based on Periodic Disturbance Observer

Repetitive position control of the motor-driven servo system (MDSS) plays an important role in numerous industrial processing and production. In this article, based on the Euler–Lagrange equation, a dynamics model was established by analyzing the relationship of the energy conversion to describe the motion of an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> -degree-of-freedom MDSS. For repetitive position control tasks of the considered system, a robust controller was designed to improve the performance of the closed-loop system, where the disturbance observer was employed to approximate the periodic dynamics of the system to improve both the transient and steady-state performances. The stability of the closed-loop system was analyzed based on the Lyapunov method. Moreover, to validate the control effect of the proposed method, the coordinated contouring task was considered for a dual-linear-motor-driven gantry system, where a B-spline wavelet series was employed as the periodic disturbance observer. The results of the comparative experiment show not only the effectiveness of the designed control scheme but also the superiority of the B-spline wavelet series on the approximation of periodic functions.

Ways to increase the stability of the movement of cars at small and large angles of rotation of the controlled wheels

The controllability and stability of the car are the most important operational properties and components of the active safety of its movement. Currently, the issues of creating active security systems are actively engaged all over the world. The purpose of the work is to analyze ways to increase the stability of the movement of a two—axle wheeled vehicle at small and large angles of rotation of the controlled wheels. A new method is proposed for analyzing ways to increase the stability of a two-axle wheeled vehicle, characterized in that it allows, based on the application of the second Lyapunov method, to identify methods to increase the stability of movement at small and large angles of rotation of controlled wheels. The analysis of the stability of the movement of two-axle wheeled vehicles at large rotation angles of controlled wheels with various traction and steering drive schemes is carried out. It is proved that for a two-axle wheeled vehicle with all driving and steerable wheels at small angles, the corrective steering angle on the rear axle should be directed towards reducing the angle of rotation of the front wheels of the rear axle modulo and, in addition, the rotation of the rear steerable wheels should begin later than the rotation of the front, i.e. with a delay. Keywords: сontrollability and stability of wheeled vehicles; large rotation angles of the driven wheels; redistribution of torques

Finite-Time Synchronization of Fractional-Order Fuzzy Time-Varying Coupled Neural Networks Subject to Reaction–Diffusion

In this article, finite-time synchronization is investigated for fractional-order fuzzy time-varying coupled neural networks subject to reaction-diffusion by establishing a new framework under fuzzy-based feedback control and fuzzy-based adaptive control. For the considered networks, we put forward an innovative graph-theory-based time-varying Lyapunov function. To overcome the difficulty of estimating the fractional derivative of this function, this paper proposes a novel fractional derivative rule. Through graph theory and the Lyapunov method, several finite-time synchronous criteria are obtained for the considered networks, and the estimation of the settling time is derived. Finally, the numerical results are shown to demonstrate the practicability of the given results.

Quasi-synchronization of stochastic heterogeneous networks via intermittent pinning sampled-data control

This paper is mainly concerned with the quasi-synchronization of stochastic heterogeneous networks (SHNs). Furthermore, the control strategy adopted here combines the advantages of aperiodically intermittent pinning control (AIPC) and sampled-data control, namely intermittent pinning sampled-data control. Different from intermittent control referred to in existing results, the restriction of the lower bound of control rate for AIPC, which is called the quasi-periodicity condition, is removed. As the main results, through an auxiliary timer, Lyapunov method, and inequality techniques, several sufficient conditions for quasi-synchronization of SHNs are derived. Especially, two different estimations of sampling error are given and the obtained results indicate that the maximum allowable bound of the sampling interval can be calculated easily. Then, some easily-verified quasi-synchronization criteria are provided. Finally, to show the feasibility of our theoretical results, two numerical examples are provided.

Decentralized formation of multi‐agent conformable fractional nonlinear robot systems

AbstractA novel approach to address the asymptotic stability problem of the decentralized formation control of multi‐agent robot systems considering a cyclic pursuit formation graph is derived by using the classical Lyapunov's stability method and LaSalle's invariance principle. Theoretical development of a local potential function‐based control law and the formulation of the position and formation error dynamics, from the viewpoint of the multi‐variable conformable fractional‐order calculus, is given. In addition, a comparison between the integer‐order case and conformable fractional‐order case is provided. Finally, a numerical example illustrates the performance of the developed results.

Adaptive Super-Twisting Sliding Mode Control of Underwater Mechanical Leg with Extended State Observer

Underwater manipulation is one of the most significant functions of the deep-sea crawling and swimming robot (DCSR), which relies on the high-accuracy control of the body posture. As the actuator of body posture control, the position control performance of the underwater mechanical leg (UWML) thus determines the performance of the underwater manipulation. An adaptive super-twisting sliding mode control method based on the extended state observer (ASTSMC-ESO) is proposed to enhance the position control performance of the UWML by taking into account the system’s inherent nonlinear dynamics, uncertainties, and the external disturbances from hydrodynamics, dynamic seal resistance, and compensation oil viscous resistance. This newly designed controller incorporates sliding mode (SMC) feedback control with feedforward compensation of the system uncertainties estimated by the ESO, and the external disturbances of the hydrodynamics by fitting the parameters, the dynamic seal resistance, and the compensation oil viscous resistance to the tested results. Additionally, an adaptive super-twisting algorithm (AST) with integral action is introduced to eliminate the SMC’s chattering phenomenon and reduce the system’s steady-state error. The stability of the proposed controller is proved via the Lyapunov method, and the effectiveness is verified via simulation and comparative experimental studies with SMC and the adaptive fuzzy sliding mode control method (AFSMC).

Primary frequency regulation for multi‐infeed HVDC system using model predictive control and local estimation

AbstractThis paper proposes a novel primary frequency regulation (PFR) scheme for multi‐infeed high voltage direct current (HVDC) (MIDC) system. First, a dynamic model for frequency response in the MIDC system is carried out to involve line‐commutated‐coveter (LCC)‐HVDCs in frequency regulation. Then, a disturbance observer (DO) is proposed to estimate both the power imbalance and the full order states of the system. By combining a Kalman filter, this DO gets an accurate estimation quickly with local frequency measurement. On the basis of the estimated disturbance and state variables, feedforward control schemes can be concluded to fully utilize the fast adjustability of LCC‐HVDCs. In this paper, a model predictive control based scheme is designed to coordinate LCC‐HVDCs under constraints of HVDC capacity and frequency boundaries. As a result, the optimal power‐sharing property of LCC‐HVDCs is achieved to enhance the PFR dynamic performance of the AC main system. Further, the stability of the proposed method is analysed using the Lyapunov method. Finally, numerical simulations are concluded to verify the effectiveness and the merits of the proposed method, by comparing it to typical droop controls.