Convergence Of Error Research Articles

Two highly accurate and efficient numerical methods for solving the fractional Liénard’s equation arising in oscillating circuits

In this paper, we investigate the fractional model of the Liénard and Duffing equations with the Liouville-Caputo fractional derivative. These equations grow with the evolution of radio and vacuum tube technology, which describe oscillating circuits and generalize the spring–mass device equation. We compare two numerical approaches, namely Jacobi and Haar wavelet collocation methods. The given approaches are used to discretize and transform the equation into a system of algebraic equations, and the Broyden-Quasi Newton algorithm is applied to solve the resulting nonlinear system of equations. A complete error analysis and convergence rates for different grid sizes are derived for both methods, which are used to compare the accuracy and efficiency of the two approaches. While both approaches produce correct solutions, according to the numerical findings, the Jacobi collocation method is more efficient and accurate than the Haar wavelet collocation method.

Compensation control of commercial vehicle platoon considering communication delay and response lag

The real-time accurate control of vehicle is of great significance for the stability of platoon driving. To address the impact of random time-varying communication delays, communication packet loss, data disorder, and significant response lag of commercial vehicle on platoon control performance in non-ideal communication environment, a novel compensation control method for commercial vehicle platoon is proposed to overcome the abovementioned issues. The proposed method significantly improves the impact of communication issues and vehicle response lag on platoon control, ensuring platoon safety while reducing inter-vehicle spacing and spacing error, and maintaining platoon stability. In addition, this method combines the comprehensive compensation mechanism with the Distributed Model Predictive Controller (DMPC), avoiding the establishment of additional compensation control modules and effectively reducing the computational burden. The results under emergency braking condition show that, compared with the method without compensation, the proposed method significantly reduces inter-vehicle spacing and spacing error, reducing the average maximum spacing error from 29.25 m to 1.44 m, shortening the error convergence time of the platoon by 28 %, and ensuring platoon stability. Additionally, compared to the compensation method based on Kalman Filtering (KF) and Smith predictor, the proposed method can reduce the average maximum spacing error by 60 % while maintaining similar convergence time and platoon stability.

Prescribed Performance Formation Tracking Control for Underactuated AUVs under Time-Varying Communication Delays

Achieving formation tracking control of underactuated autonomous underwater vehicles (AUVs) under communication delays presents a significant challenge. To address this challenge, a distributed prescribed performance control protocol based on a real-time state information online predictor (RSIOP) is proposed in this paper. First, we innovatively designed an RSIOP to achieve active compensation for the delayed state information of neighboring AUVs. Next, considering formation performance and safety, a low-complexity and practical nonlinear mapping function was used to implement prescribed performance tracking control for the AUV formation. Additionally, the adverse effects of external disturbance uncertainties and input saturation are also considered. Finally, the simulation tests demonstrated that the proposed formation control protocol can successfully achieve the predetermined formation tracking tasks in the presence of time-varying communication delays and external disturbances, while also enabling real-time changes in formation configuration during the process. Throughout, the protocol maintains input saturation limits, and the actual control inputs remain smooth, with no significant oscillations. Furthermore, comparative simulation tests verified the necessity of the RSIOP developed in this study and quantitatively demonstrated that the proposed control method exhibits superior performance in terms of formation control accuracy, error convergence speed, and transient-state constraints.

Mode-wise principal subspace pursuit and matrix spiked covariance model

Abstract This paper introduces a novel framework called Mode-wise Principal Subspace Pursuit (MOP-UP) to extract hidden variations in both the row and column dimensions for matrix data. To enhance the understanding of the framework, we introduce a class of matrix-variate spiked covariance models that serve as inspiration for the development of the MOP-UP algorithm. The MOP-UP algorithm consists of two steps: Average Subspace Capture (ASC) and Alternating Projection. These steps are specifically designed to capture the row-wise and column-wise dimension-reduced subspaces which contain the most informative features of the data. ASC utilizes a novel average projection operator as initialization and achieves exact recovery in the noiseless setting. We analyse the convergence and non-asymptotic error bounds of MOP-UP, introducing a blockwise matrix eigenvalue perturbation bound that proves the desired bound, where classic perturbation bounds fail. The effectiveness and practical merits of the proposed framework are demonstrated through experiments on both simulated and real datasets. Lastly, we discuss generalizations of our approach to higher-order data.

Finite-Time Fault-Tolerant Control via Fully Actuated System Approaches.

In this article, a finite-time fault-tolerant controller based on the fully actuated system (FAS) theory is presented to realize system stabilization and trajectory tracking. Paralleling to first-order nonlinear state space theory, the high-order FAS (HOFAS) theory contains rich controller design approaches. The existing FAS approaches can only give general global asymptotic stability results. In order to enhance the applicability of FAS approaches in fast control systems, a parameterized FAS stabilization controller based on the homogeneity principle is established for global finite-time stability. Moreover, a finite-time FAS tracking controller based on a finite-time observer is proposed for a HOFAS model with process faults. The proposed observer can yield zero-value convergence of state estimation error and fault estimation error in a finite time, and the proposed fault-tolerant controller can yield zero-value convergence of tracking error in a finite time. The main results are proved theoretically and illustrated experimentally.

A fourth-order kernel for improving numerical accuracy and stability in Eulerian SPH for fluids and total Lagrangian SPH for solids

The error of smoothed particle hydrodynamics (SPH) using a kernel for particle-based approximation mainly arises from smoothing and integration errors. The choice of the kernel significantly impacts numerical accuracy, stability and computational efficiency. Currently, the most popular kernels, such as B-spline, truncated Gaussian (for compact support), and Wendland kernels, have 2nd-order smoothing errors, and the Wendland kernel has become mainstream in the SPH community due to its stability and accuracy. Since the particle distribution after relaxation can achieve fast convergence of integration error with respect to support radius, it is logical to choose kernels with higher-order smoothing error to improve the numerical accuracy. In this paper, the error of the 4th-order Laguerre-Wendland kernel proposed by Litvinov et al. [1] is revisited, and another 4th-order truncated Laguerre-Gauss kernel is further analyzed and considered to replace the widely used Wendland kernel. The proposed kernel has the following three properties: One is that it avoids the pair-instability problem during the relaxation process, unlike the original truncated Gaussian kernel, and achieves much less relaxation residue than the Wendland and Laguerre-Wendland kernels; One is the truncated compact support size is the same as the non-truncated one of the Wendland kernel, which leads to both kernels' comparably computational efficiency; Another is that the truncation error of this kernel is much less than that of the Wendland kernel. Furthermore, a comprehensive set of 2D and 3D benchmark cases on Eulerian SPH for fluid dynamics and total Lagrangian SPH for solid dynamics validate the considerably improved numerical accuracy of the truncated Laguerre-Gauss kernel without introducing extra computational effort.

Data-driven integral sliding mode predictive control with optimal disturbance observer

In this paper, a novel data-driven integral sliding mode predictive control algorithm based on an optimal disturbance observer (DDISMPC-ODO) is proposed for a class of nonlinear discrete-time systems (NDTS) subject to external disturbances. The designed optimal disturbance observer realizes the precise observation of the lumped disturbance, thus ameliorating the accuracy of the controller and weakening problems with chattering. In this work, a robust pseudo-partial derivative (PPD) estimation algorithm is introduced, which not only improves the system performance, but also facilitates theoretical proof of parameter estimation and tracking accuracy. The convergence of the PPD estimation error and disturbance observation error is proved. It is also proved that the accuracy of the disturbance observation error can converge to O(T3) and then the magnitude of the sliding variable and the tracking error are also reduced to O(T3) respectively. Finally, the effectiveness of the proposed method is demonstrated by a simulation example and an experiment.

Adaptive Fuzzy Tracking Control With Global Prescribed-Time Prescribed Performance for Uncertain Strict-Feedback Nonlinear Systems.

For strict-feedback systems with mismatched uncertainties, adaptive fuzzy control techniques are developed to provide global prescribed performance with prescribed-time convergence. First, a class of prescribed-time prescribed performance functions are designed to quantify the performance constraints of the tracking error. Additionally, a novel error transformation function is provided to eliminate the initial value limitations and resolve the singularity issue in previous research. To ensure the convergence of the tracking error into a prescribed bounded region within a prescribed time and satisfactory transient performance, controllers with or without approximating structures are established. Notably, the settling time and initial condition of the prescribed performance function are completely independent of the initial tracking error and system parameters, thereby improving upon existing results. Furthermore, the disadvantage of the semi-global boundedness of tracking error induced by dynamic surface control can be eliminated through the use of a novel Lyapunov-like energy function. Finally, the effectiveness of the proposed strategies is validated through numerical simulations performed on practical examples.

Cooperative learning‐based practical formation‐containment control with prescribed performance for heterogeneous clusters of UAV/USV

SummaryIn this paper, a new approach for formation‐containment control with prescribed performances is introduced for heterogeneous autonomous vehicles involving a cluster of leader unmanned aerial vehicles (UAVs) and follower unmanned surface vessels (USVs). We introduce a two‐layer distributed control system: The upper layer focuses on guiding the UAVs to form a scalable lattice while synchronizing their movement along a predefined path, and the second layer guides the USVs to enter the convex hull formed by the UAVs, ensuring collision‐free operation with static/dynamic objects. To prevent collisions and ensure lattice formation, a set of well‐defined bump functions are utilized in the design of the backstepping control algorithm. Managing virtual controls, we incorporate a nonlinear dynamic surface control (NDSC), while a universal barrier function enhances the convergence of formation tracking errors. Furthermore, each USV employs a cooperative adaptive learning neural network to handle uncertainties in heterogeneous vehicle models. Utilizing the Lyapunov theorem, the stability of the formation‐containment of UAV/USV is achieved, and all signals in the formation‐containment systems are semiglobal uniform ultimate bounded (SGUUB). A simulation example showcases the effectiveness of our proposed approach, highlighting contributions in collision avoidance, synchronization speed, and adaptive learning. Our work advances the heterogeneous formation‐containment literature towards more realistic scenarios with safety‐critical considerations amidst multiple layers of uncertainties and unknown parameters.

Thermal and Computational Analysis of MHD Dissipative Flow of Eyring-Powell Fluid: Non-Similar Approach via Overlapping Grid-Based Spectral Collocation Scheme

The aim of the present study is to numerically investigate the non-similar flow and heat transfer in a dissipative Eyring-Powell fluid (EPF) over a stretching surface. A constant magnetic field is applied perpendicular to the stretched surface to explore the impact of the Lorentz force. Both viscous and magnetic dissipation are considered to comprehensively examine their effects on heat transfer. The problem in hand does not admit self-similar solutions as the non-Newtonian fluid parameter varies with the spatial variable along the stream-wise direction. Consequently, the set of nonlinear partial differential equations, modeling the flow problem is nondimensionalized primarily by employing a pseudo-similarity variable and stream-wise coordinate. The non-dimensional set of nonlinear partial differential equations is solved by a newly developed and efficient “overlapping multi-domain bivariate spectral local linearization method (OMD-BSLLM)”. The current study includes residual error analysis and convergence tests to demonstrate the accuracy of the numerical method applied to the current mathematical model. Graphs show fluid flow and heat transfer results for different flow parameters, while tables display skin friction and Nusselt number values. The results indicate that the non-Newtonian fluid parameter enhances both the velocity profile and temperature distribution. The fluid decelerates with increasing the dimensionless stream wise coordinate and Hartmann number. Viscous dissipation and dimensionless stream-wise coordinate enhances the temperature profile.

Power quality optimization technology for photovoltaic nanoscale electronic grid-connected systems based on FNN algorithm

With the continuous maturity and widespread application of photovoltaic power generation technology, the power quality of photovoltaic grid-connected systems has become a significant factor affecting the power grid’s stability and the lifespan of equipment. This research aims to improve the power quality in photovoltaic nanoscale electronic grid-connected systems. Through in-depth analysis of harmonic problems in grid-connected systems, a harmonic prediction and governance means using a fuzzy neural network algorithm was put forward. To improve the accuracy and adaptability of predictions, the long short-term memory network was combined with a fuzzy neural network to form an improved deep learning model. Comparative experiments confirmed that the research prediction model exhibited advantages in training convergence, prediction accuracy, and various error evaluation indicators, with an accuracy of up to 96.24% and an R2 value as high as 0.9998. It effectively reduced harmonic content by 2.4% and significantly improved the harmonic compensation effect. Based on the above-mentioned model, dynamic compensation governance for harmonics was implemented. By predicting the harmonic content in grid-connected electricity in real-time and taking corresponding measures for compensation, the adverse effects of harmonics on the entire grid-connected system were effectively reduced. The deep learning-based power quality optimization technology developed in this research can accurately predict harmonic problems in photovoltaic nanoscale electronic grid-connected systems. Moreover, it can provide efficient compensation governance solutions and help improve the entire power grid’s energy efficiency and operational stability.

Efficient learning with projected histograms

High dimensional learning is a perennial problem due to challenges posed by the “curse of dimensionality”; learning typically demands more computing resources as well as more training data. In differentially private (DP) settings, this is further exacerbated by noise that needs adding to each dimension to achieve the required privacy. In this paper, we present a surprisingly simple approach to address all of these concerns at once, based on histograms constructed on a low-dimensional random projection (RP) of the data. Our approach exploits RP to take advantage of hidden low-dimensional structures in the data, yielding both computational efficiency, and improved error convergence with respect to the sample size—whereby less training data suffice for learning. We also propose a variant for efficient differentially private (DP) classification that further exploits the data-oblivious nature of both the histogram construction and the RP based dimensionality reduction, resulting in an efficient management of the privacy budget. We present a detailed and rigorous theoretical analysis of generalisation of our algorithms in several settings, showing that our approach is able to exploit low-dimensional structure of the data, ameliorates the ill-effects of noise required for privacy, and has good generalisation under minimal conditions. We also corroborate our findings experimentally, and demonstrate that our algorithms achieve competitive classification accuracy in both non-private and private settings.

An ultraweak-local discontinuous Galerkin method for nonlinear biharmonic Schrödinger equations

This paper proposes and analyzes a fully discrete scheme for nonlinear biharmonic Schrödinger equations. We first write the single equation into a system of problems with second-order spatial derivatives and then discretize the space variable with an ultraweak discontinuous Galerkin scheme and the time variable with the Crank–Nicolson method. The proposed scheme proves to be computationally more efficient compared to the local discontinuous Galerkin method in terms of the number of equations needed to be solved at each single time step, and it is unconditionally stable without imposing any penalty terms. It also achieves optimal error convergence in L2 norm both in the solution and in the auxiliary variable with general nonlinear terms. We also prove several physically relevant properties of the discrete schemes, such as the conservation of mass and the Hamiltonian for the nonlinear biharmonic Schrödinger equations. Several numerical studies demonstrate and support our theoretical results.

Dual-channel event-triggered model-free adaptive iterative learning control under DoS attacks

This paper investigates a dual-channel event-triggered model-free adaptive iterative learning control problem for unknown nonlinear systems under periodic denial-of-service (DoS) attacks. Firstly, periodic DoS attacks on the measured output of the system are modelled using the Bernoulli distribution. Then, to minimise the utilisation of system bandwidth resources, a dual-channel event-triggered mechanism is designed for the sensor-to-controller and controller-to-actuator channels. Subsequently, a dual-channel event-triggered model-free adaptive iterative learning control algorithm is introduced. The convergence of the tracking error is proven in terms of mathematical expectation using Lyapunov stability theory. Finally, the effectiveness of the proposed algorithm is verified through two simulation cases.

Velocity-constrained distributed formation tracking of fixed-wing UAVs with multiple leaders

This paper addresses the safety-critical formation tracking of fixed-wing unmanned aerial vehicles, where both the velocity constraint and multiple leaders are taken into consideration. A hierarchical control strategy, consisting of a distributed observer and a formation controller, is elaborately constructed to achieve the desired configuration with the velocity in a safe range. First, a novel distributed observer is designed to estimate the formation error, which promises the practical fixed-time convergence of the estimation error despite unknown inputs of leaders and disturbances. Then, the velocity constraint is enforced through the forward invariance of a safety set, for which a robust control barrier function is carefully constructed considering the effect of disturbances. Further, a local formation controller is established by embedding the robust control barrier function into quadratic programming. Compared with existing results, the proposed controller is able to achieve the formation configuration while satisfying velocity safety, even in the presence of disturbances. Simulations are conducted to demonstrate the proposed method's capability in guaranteeing velocity constraint and stabilizing formation errors.

Adaptive Finite-Time-Based Neural Optimal Control of Time-Delayed Wheeled Mobile Robotics Systems.

For nonlinear systems with uncertain state time delays, an adaptive neural optimal tracking control method based on finite time is designed. With the help of the appropriate LKFs, the time-delay problem is handled. A novel nonquadratic Hamilton-Jacobi-Bellman (HJB) function is defined, where finite time is selected as the upper limit of integration. This function contains information on the state time delay, while also maintaining the basic information. To meet specific requirements, the integral reinforcement learning method is employed to solve the ideal HJB function. Then, a tracking controller is designed to ensure finite-time convergence and optimization of the controlled system. This involves the evaluation and execution of gradient descent updates of neural network weights based on a reinforcement learning architecture. The semi-global practical finite-time stability of the controlled system and the finite-time convergence of the tracking error are guaranteed.

A Robust Hybrid Iterative Learning Formation Strategy for Multi-Unmanned Aerial Vehicle Systems with Multi-Operating Modes

This paper investigates the formation control problem of multi-unmanned aerial vehicle (UAV) systems with multi-operating modes. While mode switching enhances the flexibility of multi-UAV systems, it also introduces dynamic model switching behaviors in UAVs. Moreover, obtaining an accurate dynamic model for a multi-UAV system is challenging in practice. In addition, communication link failures and time-varying unknown disturbances are inevitable in multi-UAV systems. Hence, to overcome the adverse effects of the above challenges, a hybrid iterative learning formation control strategy is proposed in this paper. The proposed controller does not rely on precise modeling and exhibits its learning ability by utilizing historical input–output data to update the current control input. Furthermore, two convergence theorems are proven to guarantee the convergence of state, disturbance estimation, and formation tracking errors. Finally, three simulation examples are conducted for a multi-UAV system consisting of four quadrotor UAVs under multi-operating modes, switching topologies, and external disturbances. The results of the simulations show the strategy’s effectiveness and superiority in achieving the desired formation control objectives.

Optimal Asymptotic Tracking Control for Nonzero-Sum Differential Game Systems with Unknown Drift Dynamics via Integral Reinforcement Learning

This paper employs an integral reinforcement learning (IRL) method to investigate the optimal tracking control problem (OTCP) for nonlinear nonzero-sum (NZS) differential game systems with unknown drift dynamics. Unlike existing methods, which can only bound the tracking error, the proposed approach ensures that the tracking error asymptotically converges to zero. This study begins by constructing an augmented system using the tracking error and reference signal, transforming the original OTCP into solving the coupled Hamilton–Jacobi (HJ) equation of the augmented system. Because the HJ equation contains unknown drift dynamics and cannot be directly solved, the IRL method is utilized to convert the HJ equation into an equivalent equation without unknown drift dynamics. To solve this equation, a critic neural network (NN) is employed to approximate the complex value function based on the tracking error and reference information data. For the unknown NN weights, the least squares (LS) method is used to design an estimation law, and the convergence of the weight estimation error is subsequently proven. The approximate solution of optimal control converges to the Nash equilibrium, and the tracking error asymptotically converges to zero in the closed system. Finally, we validate the effectiveness of the proposed method in this paper based on MATLAB using the ode45 method and least squares method to execute Algorithm 2.

Antisaturation fixed-time backstepping fuzzy integral sliding mode control for automatic landing of fixed-wing unmanned aerial vehicles

This paper proposes a robust adaptive fixed-time control for automatic landing systems (ALS) of small-scale fixed-wing UAVs exposed to wind disturbances, parametric uncertainties, and input saturation. The control structure consists of three subsystems, i.e., the guidance system, the attitude control system, and the thrust control system. Considering the landing geometry, the guidance system computes the desired Euler angles to stabilize the dynamics of altitude deviation, lateral deviation, and lateral speed. An auxiliary system is designed to compensate for the effect of input saturation in the attitude control system. The fixed-time method, the backstepping control (BSC), the fuzzy logic control (FLC), and the integral sliding mode control (ISMC) techniques are adopted to synthesize the control law. All three functional parts of the proposed ALS are equipped with a fixed-time fuzzy sliding mode disturbance observer (FSMDO). FLC computes the gains of signum functions adaptively to attenuate the chattering. Fixed-time convergence of the system errors to an arbitrary residual set is proved by comprehensive stability analysis. Finally, the comparative simulation results verify the better dynamic response and stronger robustness under the proposed ALS.

Distributed state constraint containment control of multi-AUV formation with unknown control direction

The control problem of multi-autonomous underwater vehicle (multi-AUV) formation systems is a hot issue in current research. The research goal is to make the formation have a more comprehensive ability to deal with emergencies under higher control accuracy. In this paper, input saturation, disturbances, and system uncertainties are considered. To achieve the convergence of system containment errors, a state constraint containment control method is proposed for multi-AUV systems with unknown control direction. Combined with unknown control direction, new auxiliary variables are constructed to compensate the input saturation. On this basis, the follower reference trajectories and errors are defined. To constrain some system states, the tan-type barrier Lyapunov function (TBLF) is proposed. The estimated values of system uncertainties and disturbances can be obtained by neural network. Compared to other works, the amount of data generated by neural network is further reduced by the optimized adaptive law in this study. The unknown control direction problem is handled based on Nussbaum function, and its limitation range is expanded. The designed control method ensures the stable operation of formation systems. The containment errors can converge to a small neighborhood of zero. By comparing the numerical simulation results with other work, the advantages of the control algorithm proposed in this paper has been verified.