Lyapunov Function Stability Research Articles

On the Novel Auxiliary Lyapunov Function and Uniform Asymptotic Practical Stability of Nonlinear Impulsive Caputo Fractional Differential Equations via New Modelled Generalized Dini Derivative

In this paper, the uniform asymptotic practical stability of nonlinear impulsive Caputo fractional differential equations with fixed moments of impulse is examined using an auxiliary Lyapunov functions which are analogues of vector Lyapunov functions. Together with comparison results, sufficient conditions for the uniform practical stability as well as the uniform asymptotic practical stability of the impulsive Caputo fractional order systems are established. An illustrative example is given to confirm the suitability of the obtained results.

Adaptive neural network based quadrotor UAV formation control under external disturbances

The formation control of a team comprised of multiple quadrotor Unmanned Aerial Vehicles (UAVs) may severely be affected by the unknown external disturbances. The external disturbances are caused by wind forces to create aero-dynamical disturbances. This article addresses the robust formation control problem of multiple UAVs system despite the effect of external disturbances that allow sustaining a stable network connection among the UAVs and maintaining different formations assigned to them. First, a Radial Basis Function Neural Network (RBFNN) based model is developed to reciprocate the external disturbances along the positional and the attitude subsystems. Then incorporating the estimated disturbance values a distributed adaptive formation controller is devised using the Lyapunov theory. It consists of a positional and an attitude controller associated with the translational and the rotational movements of the UAVs. The stability is validated by satisfying the criteria of the Lyapunov stability function. The UAVs are connected through variable adjacency matrix based directed network topology and the network connectivity is established through the properties of the Laplacian Matrix. The robustness of the designed controller is justified via rigorous simulation studies for different sets of desired formations such as triangular, squared, tetrahedron, octahedron and cube shaped. The reference trajectories are considered as spiral, straight line and circular shaped. The time varying external disturbances are considered of sinusoidal waveform of different magnitudes. The simulation results signifies that the proposed RBFNN based formation controller reciprocate different sinusoidal waveforms to achieve the desired formations successfully. Extensive comparative studies demonstrate the efficacy of the proposed adaptive formation controller over the existing controllers presented in the literature for different shapes of trajectories and desired formations.

Mathematical modeling of Ebola using delay differential equations

Nonlinear delay differential equations (NDDEs) are essential in mathematical epidemiology, computational mathematics, sciences, etc. In this research paper, we have presented a delayed mathematical model of the Ebola virus to analyze its transmission dynamics in the human population. The delayed Ebola model is based on the four human compartments susceptible, exposed, infected, and recovered (SEIR). A time-delayed technique is used to slow down the dynamics of the host population. Two significant stages are analyzed in the said model: Ebola-free equilibrium (EFE) and Ebola-existing equilibrium (EEE). Also, the reproduction number of a model with the sensitivity of parameters is studied. Furthermore, the local asymptotical stability (LAS) and global asymptotical stability (GAS) around the two stages are studied rigorously using the Jacobian matrix Routh–Hurwitz criterion strategies for stability and Lyapunov function stability. The delay effect has been observed in the model in inverse relation of susceptible and infected humans (it means the increase of delay tactics that the susceptibility of humans increases and the infectivity of humans decreases eventually approaches zero which means that Ebola has been controlled into the population). For the numerical results, the Euler method is designed for the system of delay differential equations (DDEs) to verify the results with an analytical model analysis.

Distributed State Observer for Systems with Multiple Sensors under Time-Delay Information Exchange.

The issues of state estimations based on distributed observers for linear time-invariant (LTI) systems with multiple sensors are discussed in this paper. We deal with the scenario when the information exchange has known time delays, and aim at designing a distributed observer for each subsystem such that each distributed observer can estimate the system state asymptotically by rejecting the time delay. To begin with, by rewriting the target system in a connecting form, a subsystem which is affected by the time-delay states of other nodes is established. And then, for this subsystem, a distributed observer with time delay is constructed. Moreover, an equivalent state transformation is made for the observer error dynamic system based on the observable canonic decomposition theorem. Further, in order to ensure that the distributed observer error dynamic system is asymptotically stable even if there exists a time delay, a linear matrix inequality (LMI) which is relative to the Laplace matrix is elaborately set up, and a special Lyapunov function candidate based on the LMI is considered. Next, based on the Lyapunov function and Lyapunov stability theory, we prove that the error dynamic system of the distributed observer is asymptotically stable, and the observer gain is determined by a feasible solution of the LMI. Finally, a simulation example is given to illustrate the effectiveness of the proposed method.

Fractional commensurate model on COVID‐19 with microbial co‐infection: An optimal control analysis

AbstractCrossover behaviors have always existed in the history of infectious pandemics due to a few distinct, erratic spread outlines. This research aims to investigate the crossover behavior of the proposed SVICR commensurate fractional model for the COVID‐19 delta variant, considering microbial coinfections. A mathematical model in terms of Atangana–Baleanu Caputo (ABC) category fractional integrals takes into account the co‐infection of mucormycosis in immunocompromised COVID‐19 patients caused by microbial infections. ABC operators preserve the intact history of the happenings under contemplation through its nonsingular kernel. It is observed that the framed five‐compartmental SVICR model is positively bounded on R5, the solution space. Two equilibrium points representing the survival and annihilation of sickness respectively are contributed by the single population N(t), which is counted in five dependent compartments: The bilinear growth rate of new additional infections from the contagious infectives over time ‘t’ is viewed through the threshold metric Lyapunov's stability function examines the parametric influences over the virulent spread globally. The significant focus is to investigate the Mucormycosis cases in COVID‐19 patients with underlying diabetic complications. Diabetes mellitus is the major concern for several coinfections among COVID‐19 recoveries. Aiming to minimalize the critical states, an Lagrangian–Hamiltonian optimum control structure is also performed for the SVICR model by introducing control variables in effect to tri‐control probes of minimized contact rates, persuasive vaccinations, and glycemic control of post recovered diabetic patients. The hike in the Severity of the ailment due to fungal pathogens is studied through numerical convergence of predictor–corrector scheme and simulations. Using estimated parametric values from the statistical data of mucormycosis and infections of COVID‐19 reported cases in India, the prominence of control effects are visualized graphically. To conclude, a complete qualitative analysis of the minimization problem is executed for different levels of control values. We avow that effective control intrusions would almost certainly decline the complexities associated with the viral pathogens.

Notice of Violation of IEEE Publication Principles: Washout Filter-Based Distributed Resilient Current Control for Cyber-Physical Microgrids

Notice of Violation of IEEE Publication Principles: <br><br>“Washout Filter-Based Distributed Resilient Current Control for Cyber-Physical Microgrids," <br>by X. Lu and J. Lai, <br> in IEEE Systems Journal, Early Access <br><br>After careful and considered review of the content and authorship of this paper by a duly constituted expert committee, this paper has been found to be in violation of IEEE’s Publication Principles. <br><br>This paper contains significant portions of original text from the paper cited below. The original text was copied without attribution (including appropriate references to the original author(s) and/or paper title) and without permission. <br><br>"A Combined Distributed Cooperative Method and Washout Filter-Based Method for Power Sharing and Voltage Balancing in Bipolar DC Microgrids" <br> by Ramin Babazadeh-Dizaji, Mohsen Hamzeh, Nima Mahdian Dehkordi, <br>submitted to IEEE Transactions on Energy Conversion, August 2020. <br><br> <br/> This article proposes a washout filter-based distributed resilient current control scheme that can achieve current sharing of distributed generations (DG) and voltage restoration of dc bus almost surely under communication channel noise interferences. Since the cyber channels are exposed to inherent noise interferences, the stability and reliability of cyber-physical dc microgrids may terribly be reduced. To dispel the adverse influences of noise interferences, a novel discrete time washout filter is developed. Accordingly, a discrete-time-distributed noise-resiliency control algorithm is developed for dc microgrids based on washout filters, in which only the current state variable is allowed to exchange among DGs with the local decentralized voltage controller. This makes the proposed control mode different from the conventional parallel mode usually requiring a leader–follower consensus-based voltage control by exchanging two state variables (i.e., terminal voltage and current output). Thus, the proposed control mode decreases the complexity to the formulation and enhances the robustness to the leader communication failure. By introducing stochastic theory and Lyapunov stability function, the sufficient conditions considering noise interferences is derived to ensure the stability operation of the whole closed-loop dc network. The obtained results show the effectiveness of the proposed strategy by a tested dc microgrid in OPAL-RT real-time simulator.

Notice of Violation of IEEE Publication Principles:Consensus-Based Distributed Anti-Disturbance Control of DC Microgrids Over Fading Channels

This paper develops a novel consensus-based distributed anti-disturbance control scheme that can achieve current sharing of distributed generations (DG) and voltage restoration of DC bus in mean square subject to multiplicative measurement noises and communication delays over fading networks. Since the cyber channels are exposed to inherent communication delays, the measurement information exchange with noise interferences among multiple and cooperative DGs may be disturbed in fading communication networks, where fading may either be due to noises, delays, and shadowing from obstacles affecting the wave propagation. As a result, fading makes every DG receives incomplete/imprecise information from its neighbors. To dispel the adverse influences of measurement noises and time delays over fading networks, a washout filter-based distributed anti-disturbance control algorithm is developed for DC microgrids, in which only the current state variable is exchanged among DGs with the local decentralized voltage controller. By introducing stochastic theory and Lyapunov stability function, the criteria for the stability analysis and delays boundedness considering noise interferences is derived to ensure the stability operation of the whole closed-loop DC network. The obtained results show the effectiveness of the proposed strategy by a tested DC microgrid in OPAL-RT real-time simulator.

Global Stability of Equilibria and Lyapunov Functions in a Vaccinating Behavioural SIR Model with Lagged Information

Global Stability of Equilibria and Lyapunov Functions in a Vaccinating Behavioural SIR Model with Lagged Information

A New Discriminant Method for the Stability of Fourth-Order Differential Equation

The stability of nonlinear differential equations has been widely studied based on various methods. In this paper, we transform a class of fourth-order differential equations into a perturbation system and study the stability and asymptotic stability of zero solutions. By utilizing Lyapunov’s second method, we give sufficient conditions for zero-solution stability and asymptotic stability and provide the construction method of the zero-solution asymptotically stable Lyapunov function of such fourth-order differential equations. In addition, this paper removes the requirement of negative definite when the general differential equation is zero solution stable and obtains a looser stability condition. This paper provides an alternative approach to the research of stability of differential systems.

Modeling of PID controlled 3DOF robotic manipulator using Lyapunov function for enhancing trajectory tracking and robustness exploiting Golden Jackal algorithm

In this study, a three degrees of freedom (3 DOF) rigid-link robotic manipulator (RLM) has been simulated by using the Simscape model and the mathematical model derived by Lagrange method. The robot arm has been regulated by an Optimized PID Controller to achieve better tracking performance and reasonable robustness against disturbances and payload uncertainty. To optimize the controller, a novel nature-inspired Golden Jackal Optimization (GJO) algorithm has been used due to its efficient exploration that increases the diversity of the released solutions and its exploitation schemes which enhance the best-explored solutions. The tuning process has utilized a Lyapunov stability function as the objective function (OF) and the efficacy of the proposed algorithm is evaluated through a comprehensive comparison with various state-of-the-art metaheuristic techniques such as Particle Swarm Optimization (PSO), Artificial Bee Colony (ABC), Jellyfish Search Optimizer (JSO), Whale Optimization Algorithm (WOA), Arithmetic Optimization Algorithm (AOA) and Sine Cosine Algorithm (SCA). The assessment has been conducted on benchmark error-based functions, providing rigorous testing and validation of the algorithm's performance. Furthermore, the performance evaluation has focused on the system's robustness against disturbances, noise, and variations in the payload mass, particularly in the context of Pick and Place (PNP) industrial tasks. The results of simulation have demonstrated that the optimized system, employing the Lyapunov function, demonstrated superior performance in minimizing the objective function value compared to other benchmark functions.

Passivity and underactuated modeling-based load energy coupling control for three-dimensional overhead crane systems

The conventional control strategies and methods with the three-dimensional model of overhead crane system are excessively dependent on model parameters and lead to the controllers design complex on position and anti-swing of overhead crane systems. With analysis on the coupling relationship among various state variables of overhead crane, the three-dimensional dynamic model of overhead crane based on mechanical virtual displacement and Lagrangian equation is constructed. A novel control strategy based on load energy coupling is designed to realize system positioning and eliminate the problem of residual load swing. The passivity of the three-dimensional overhead crane system is judged, and the energy composition of the system is deduced. The energy coupling function based on load displacement and swing angle is built. The control strategy of three-dimensional overhead crane system based on load energy coupling is designed by the Lyapunov stability theory and function construction method. The stability of the system is verified by the Lasalle invariance principle. With the experimental and programming, the proposed method is compared with different control methods. The results show that this method can realize the accurate positioning of the trolley more quickly and restrain the load swing more effectively.

Leader-following cluster-delay consensus control for first-order nonlinear multi-agent systems based on event-triggered mechanism

In this paper, the event-triggered-based cluster-delay consensus control problem is investigated for leader-following nonlinear multi-agent systems (MASs). In control design, both state triggering in the agents’ network and constant time-delay in the leaders’ communication network are considered. Under the framework of Lyapunov function stability theory, the Lipschitz condition is used to overcome the influence caused by time-delay. In order to further effectively utilize data transmission resources, and reduce the communication load of the topology network between each agent, an event-triggered mechanism is established. Subsequently, a robust cluster consensus control strategy is proposed based on event-triggered mechanism, which can ensure all signals of the controlled system are bounded, and the tracking errors converge to zero. In addition, it can also effectively avoid the Zeno behavior. Finally, the effectiveness of the presented control method and theory is verified by a simulation example.

Stratonovich framework into Markovian satellite dynamics of Carleman-based filtering using energy function observables approach

In this paper the satellite system in which fluctuating aerodynamic torque and the radial perturbation about the pitch motion of orbit is analyzed. The objectives of the paper are three-fold. The first objective is the ‘Carleman linearization representation into the nonlinear stochastic evolution of the Markov process’. The second is to filter process and measurement noises in the satellite dynamics parameters in the Carleman setting. The third is to find the stability and convergence condition of satellite dynamics. The first is achieved by unifying the generating function, Carleman embedding, Itô stochastic differential rules and the finite closure with Kronecker algebra. Concerning the second objective, we recast the finite-dimensional Stochastic Differential Equations (SDE) of the satellite dynamics into a finite system of bilinear SDE via the Carleman embedding. The third objective is to achieve the Lyapunov function and asymptotic stability condition for the satellite stochastic dynamics involving the ‘Stratonovich differential’. In this paper, we demonstrate the utility of the satellite dynamics filtering in the Carleman-based filtering via ‘its convergence analysis as well as its superiority with available methods’, i.e., the benchmark extended Kalman filter, Gaussian second-order filter and Kushner–Stratonovich higher-order filter. From simulation performed it can be said that the Carleman-based filtering is superior than other benchmark filters in terms of their Absolute Filtering Error (AFE) of conditional means and conditional variances of the satellite dynamics states.

Evaluation of Ride Performance of Active Inerter-Based Vehicle Suspension System with Parameter Uncertainties and Input Constraint via Robust H∞ Control

In this study, we investigate a robust H∞ controller for a quarter-car model of an active inerter-based suspension system under parameter uncertainties and road disturbance. Its main objective is to improve the inherent compromises between ride quality, handling performance, suspension stroke, and energy consumption. Inerters have been extensively used to suppress unwanted vibrations from various kinds of mechanical structures. The advantage of inerter is that the realized ratio of equivalent mass (inertance relative to the mass of the primary structure) is greater than its actual mass ratio, resulting in higher performance for the same effective mass. First, the dynamics and state space of the active inerter-based suspension system were achieved for the quarter-car model with parameter uncertainties. In order to attain the defined objectives, and ensure that the closed-loop system achieves the prescribed disturbance attenuation level, the Lyapunov stability function, and linear matrix inequality (LMI) techniques have been utilized to satisfy the robust H∞ criterion. Furthermore, to limit the gain of the controller, some LMIs have been added. In the case of feasibility, sufficient LMI conditions by solving a convex optimization problem afford the stabilizing gain of the robust state-feedback controller. According to numerical simulations, the active inerter-based suspension system in the presence of parameter uncertainties and external disturbance performs much better than both a passive suspension with inerter and active suspension without inerter.

Adaptive backstepping controller design for DC/DC buck converter optimised by grey wolf algorithm

AbstractA Lypunov‐based Adaptive Backstepping Control (ABSC) approach is designed for a power Buck converter. This strategy is an advanced version of the Backstepping method utilising Lyapunov stability function to reach a higher stability and a better disturbance rejection behaviour in the practical applications. In addition, to reduce the computational burden and increase ease of implantation, Black‐box technique is considered assuming no accurate mathematical model for the system. Nonetheless, in real‐time environments, disturbances with wider ranges including: supply voltage variation, parametric variation, and noise can negatively impact the operation of this method. To compensate for this problem, the gains of the controller should be tuned again for better adaptability with the working condition. Therefore, to satisfy this need and enhance the controller's performance, a metaheuristic algorithm is applied in the control scheme called Grey Wolf Optimisation (GWO) algorithm. GWO is a well‐behaved nature‐inspired algorithm with faster decision‐making dynamics along with more accuracy over different optimisation algorithms. To better elaborate the merits of this approach, conventional BSM and PSO‐based PID schemes are also designed and tested in different situations.

Predefined-time hierarchical controller-estimator time-varying formation-containment tracking for multiple Euler-Lagrange systems with two-layer leaders

This paper mainly discusses the problem of the predefined-time time-varying formation-containment tracking (TVFCT) for the multiple Euler-Lagrange systems (MELSs), where the external disturbances are taken into consideration. To cope with the above problem, a two-layer formation-containment tracking (FCT) framework is established, namely, the real leaders’ formation layer and the followers’ containment layer, where the analysis of the system is executed step by step. Meanwhile, the corresponding hierarchical control algorithms based on nonsingular terminal sliding mode are designed for different layers to facilitate the implementation of the TVFCT control problem. With the help of the Lyapunov function and predefined-time stability theory, some sufficient criteria for the convergence of the proposed algorithms are provided. In the end, the simulation results demonstrate the effectiveness of the theoretical results and the designed controller.

Event‐triggered adaptive stabilization control of stochastic nonlinear systems with unmodeled dynamics

AbstractIn this paper, we present a novel controller design method for stochastic nonlinear systems with unmodeled dynamics, uncertain parameters, and unknown covariance noise. In order to deal with these uncertainties, a new event‐based small gain controller is designed. The event‐triggered scheme can reduce the computational burden caused by disturbance and noise. By combining the technique of changing supply rate with small gain condition, a stochastic input‐to‐state practically stability Lyapunov function is obtained for the subsystem. Simultaneously, the changing supply function is used to treat with unmodeled dynamics. Based on backstepping process and adaptive control approach, the influence of unknown covariance noise is overcome. It is proved that the designed state‐feedback controller can ensure the overall system is stochastic input‐to‐state practically stability.

On input‐to‐state stability Lyapunov functions for mechanical systems

AbstractFor two canonical models of mechanical systems with disturbances presented by Rayleigh‐ and Liénard‐type equations, several designs of Lyapunov functions are investigated. Under given restrictions, the existence of these functions implies global or local input‐to‐state stability property for the systems. The proposed stability conditions are constructive, which is demonstrated in applications.

H∞ Robust LMI-Based Nonlinear State Feedback Controller of Uncertain Nonlinear Systems with External Disturbances

In this paper, we propose a nonlinear state feedback controller based on linear matrix inequality (LMI) for a class of nonlinear systems with parametric uncertainties and external disturbances. The primary goals of the proposed controller are to guarantee system stability and performance in the presence of system uncertainties and time-dependent disturbances. To meet the specified objectives, the LMI form is calculated as a hierarchical control structure. Using the Lyapunov stability function, the asymptotic stability of the nominal system obtained from the nonlinear state feedback is proven, and the LMI condition is attained. After applying the nonlinear state feedback controller, asymptotic stability conditions for the nominal system are constructed using the Lyapunov function, and the nonlinear state-feedback control mechanism is determined accordingly. Considering the external disturbance as input, the terms of the state matrices are substituted in the obtained LMI, and the LMI condition for a nominal system is achieved in the presence of disturbances. The asymptotic stability condition of the uncertain system in the presence of external disturbances is determined by adding uncertainties to the system. The proposed approach yields a simple control mechanism representing an independent of system order. The performance of the proposed approach was assessed using a simulation study of a ball and beam system.

Safe and sub-optimal CAV platoon longitudinal control protocol accounting for state constraints and uncertain vehicle dynamics

State constraints and uncertain vehicle dynamics severely affect control stability and performance of connected and autonomous vehicle (CAV). To this end, this study puts forward a safe and sub-optimal longitudinal control protocol for CAV platoon with uncertain vehicle dynamics and state constraints. For platoon leader, a second order disturbance observer with L2 stability is presented to estimate lumped uncertainty coupled in vehicle dynamics. By iteratively utilizing control barrier functions and control Lyapunov function, state constraints and speed trajectory tracking stability condition are encoded into control constraints. Based on disturbance observer and encoded constraints, an extended quadratic programming is established as trajectory control law for platoon leader. For platoon followers, backstepping method and disturbance observer accounting for forward communication network are synthesized as formation control law. Besides, conditions of individual vehicle stability and string stability for formation control law are analyzed. Simulation results show that the leader of platoon can automatically switch its drive mode between speed cruising and safe headway keeping, respectively. Furthermore, each follower in platoon can follow its predecessors coordinatively and precisely.