Convergence Of Error Research Articles

UWB RCS calculations of smooth targets via the multi-band beam summation method

We present an ultra-wide-band beam-summation (UWB-BS) algorithm for calculating the radar cross-section (RCS) of large smooth targets whose asymptotic numerical complexity is of O ( 1 ) . In this approach, the incident field is expanded into a sparse set of iso-diffracting Gaussian beams (ID-GB), tiling the phase space (PS) of beam origins and directions. The field of each ID-GB is then tracked through reflections from the scatterer until the beam emerges from the target zone and propagates to the far zone using a closed-form near-to-far (N2F) zone transformation, and finally, the beam contributions are aggregated to obtain the scattered field. The algorithm can accommodate a multi-octave frequency band by dividing it into octave-wide sub-bands and introducing a UWB-BS scheme such that the PS beam lattices in the higher sub-bands are obtained by spatial-spectral upsampling those of the lower bands. The PS summation of ID-GBs is a priori localized and involves a roughly constant relatively small number of contributing beams in all the sub-bands, leading to an O ( 1 ) (frequency-independent) asymptotic computational complexity. Furthermore, with the ID parametrization, the beam axes and their propagation and scattering parameters are frequency-independent and, therefore, need to be calculated only once and then reused for all frequencies, thus greatly reducing the numerical cost of computing the RCS at multiple frequencies. The paper describes all aspects of the algorithm, including the choice of the beam parameters for asymptotic error convergence and the utilization of localization for algorithmic efficacy.

Coupled error super-twisted sliding mode active fault-tolerant control for robotic system with actuator fault.

This paper focuses on a robotic system with actuator fault and presents a sliding mode active fault-tolerant control method based on coupled position error. The actuator fault is first detected by a fault alerter with a predetermined threshold. After the fault is successfully detected, the fault is estimated by a fault observer. Coupling the position error with the weighted position error of different joints, a non-singular fast terminal sliding mode surface is constructed, and a super-twisted algorithm is introduced to design a coupled error super-twisted sliding mode controller (CESSMC). Furthermore, the fault estimation is incorporated with the CESSMC to accomplish the coupled error super-twisted sliding mode active fault-tolerant control. The stability and the finite-time convergence of the system are theoretically proved. Simulations and experiments are conducted to verify the effectiveness of the proposed method. The proposed method can achieve the position error convergence in finite time and reduce chattering in the control input. With the tight coupling between position error and weighted position error, the fault compensation effect and trajectory tracking accuracy can be improved.

Flow dependence of forecast uncertainty in a large convection‐permitting ensemble

AbstractThe flow dependence of forecast distributions is studied using a 120‐member ICON‐D2 ensemble for two representative case studies of weak and strong synoptic forcing of convection. The bootstrapping technique is used to investigate the convergence of sampling error for a range of surface and midtroposphere variables. Convergence is generally observed for the mean and the standard deviation, but not for the 95th percentile, especially in strong forcing. Additionally, maps of uncertainty are introduced, which allow for a more detailed analysis of the spatial pattern of uncertainty and facilitate the interpretation of the sources and evolution of forecast uncertainty in different synoptic forcing conditions. It is found that convection significantly increases uncertainty in model variables and shapes the spatial uncertainty pattern, with weak forcing leading to patchy uncertainty and strong forcing resulting in a more coherent pattern due to large convective systems. Overall, the main result of this work is the strong connection between uncertainty of forecast variables and convection, with synoptic forcing being crucial in determining the spatial distribution and uncertainty evolution within 24 h.

Meshless Error Recovery Parametric Investigation in Incompressible Elastic Finite Element Analysis

The meshless displacement error-recovery parametric investigation in finite element method-based incompressible elastic analysis is presented in this study. It investigates key parameters such as interpolation schemes, patch configurations, dilation indexes, weight functions, and meshing patterns. The study evaluates error recovery effectiveness (local and global), convergence rates, and adaptive mesh improvement for triangular/quadrilateral discretization schemes. It uses meshless moving least squares (MLS) interpolation with rectangular and circular support regions and solves benchmark plate and cylinder problems. It is observed that a circular influence region, a cubic spline weight function, and regular mesh patterns yield a better performance of than an MLS-based error recovery method. The study also concludes that lower dilation index values with rectangular influence regions are preferable for regular meshes, while higher dilation index values with radial influence regions are suitable for preferable meshes to enhance MLS error recovery.

Information FOMO: The Unhealthy Fear of Missing Out on Information-A Method for Removing Misleading Data for Healthier Models.

Misleading or unnecessary data can have out-sized impacts on the health or accuracy of Machine Learning (ML) models. We present a Bayesian sequential selection method, akin to Bayesian experimental design, that identifies critically important information within a dataset while ignoring data that are either misleading or bring unnecessary complexity to the surrogate model of choice. Our method improves sample-wise error convergence and eliminates instances where more data lead to worse performance and instabilities of the surrogate model, often termed sample-wise "double descent". We find these instabilities are a result of the complexity of the underlying map and are linked to extreme events and heavy tails. Our approach has two key features. First, the selection algorithm dynamically couples the chosen model and data. Data is chosen based on its merits towards improving the selected model, rather than being compared strictly against other data. Second, a natural convergence of the method removes the need for dividing the data into training, testing, and validation sets. Instead, the selection metric inherently assesses testing and validation error through global statistics of the model. This ensures that key information is never wasted in testing or validation. The method is applied using both Gaussian process regression and deep neural network surrogate models.

Soft sensor modeling based on self-organizing fuzzy neural network with clustering, merging, and splitting scheme

Abstract As a pivotal role in the control, optimization, and monitoring of contemporary industrial processes, soft sensors are frequently employed in the prediction of key quality variables. To achieve accurate prediction of key quality variables in industrial processes, a soft sensor modeling method based on the self-organizing fuzzy neural network with the clustering, merging, and splitting scheme (SOFNN-CMS) is proposed. First, the supervised fuzzy C-means clustering algorithm is proposed to identify the appropriate initial center and width of the fuzzy neural network, obtaining appropriate initial fuzzy rules. Then, a neuron merging and splitting strategy is designed to adjust the structure of the fuzzy neural network, by merging and splitting the hidden neurons according to the distance of clusters, increasing the adaptability of the fuzzy neural network. Besides, to accelerate the convergence of estimation errors, an improved Levenberg Marquardt algorithm is utilized to update neural network parameters in the training phase, realizing the soft sensor modeling of key quality variables. The effectiveness of the proposed SOFNN-CMS neural network is demonstrated on two benchmark problems and an industrial debutanizer column. Finally, the experiments showcase that the proposed SOFNN-CMS neural network can obtain better soft sensor modeling performance with a compact structure.

Application of Chelyshkov polynomials in solving stochastic model with fractional Brownian motion

Abstract This paper introduces a computational spectral collocation approach utilizing orthonormal Chelyshkov polynomials to estimate solutions for both linear and nonlinear stochastic differential equations containing fractional Brownian motion characterized by the Hurst parameter H. The method employs Newton–Cotes nodes to transform these integral equations into manageable linear and nonlinear algebraic forms. The reliability of the proposed method is established through theorems on error estimation and convergence analysis. Moreover, some examples are given to demonstrate the viability, credibility, coherence, and dependability of the approach. Also, a comparative evaluation is conducted to contrast the results obtained from the suggested approach with those produced by the spectral collocation method utilizing orthonormal Bernoulli polynomials. Furthermore, a 95 % confidence interval is constructed for the error mean, revealing that the approximations maintain high accuracy.

Cooperative Load Transportation with Quadrotors using Adaptive RISE Control

The problem of transporting cable-suspended payloads using two quadrotors is analyzed in this work. The system kinematics are treated using a virtual structure formation controller, which generates the acceleration references commanded to the aerial vehicles. The disturbances caused by the payload are treated as unmodeled disturbances and a novel robust integral of the sign of the error (RISE) controller is proposed, guaranteeing the asymptotic convergence of the tracking errors. A model reference adaptive control is also incorporated into the RISE controller, combining the advantages of adaptive and robust control. The proposal is validated by numerous experiments using two quadrotors to transport a bar-shaped payload in adverse conditions. The results allow the conclusion that the proposed system is capable of performing transportation and orientation tasks with the payload, subject to translational accelerations up to 3.5 m/s2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^2$$\\end{document}.

Decentralized learning control for high-speed trains with unknown time-varying speed delays

Treating the multi-point-mass dynamic model of high-speed trains as an interconnected system, this study proposes a decentralized iterative learning control scheme for high-speed trains to achieve the trajectory tracking goal. By making reasonable estimates of the interaction term and compensating for it, the proposed control scheme utilizes only local information from each carriage and does not need any inter-carriage information exchange. The zero-error tracking of the desired trajectory is guaranteed even in a restricted communication environment. Considering unknown time-varying speed delays in the actual high-speed train operations, a modified decentralized iterative learning control scheme is also provided to address the negative impact of speed delays. The convergence of tracking errors is strictly proven by constructing appropriate composite energy functions. Numerical simulations further verify the theoretical results.

Fixed-time control of teleoperation systems based on adaptive event-triggered communication and force estimators

Abstract A fixed-time control strategy based on adaptive event-triggered communication and force estimators is proposed for a class of teleoperation systems with time-varying delays and limited bandwidth. Two force estimators are designed to estimate the operator force and environment force instead of force sensors. With the position, velocity, force estimate signals, and triggering error, an adaptive event-triggered scheme is designed, which automatically adjusts the triggering thresholds to reduce the access frequency of the communication network. With the state information transmitted at the moment of event triggering while considering the time-varying delays, a fixed-time sliding mode controller is designed to achieve the position and force tracking. The stability of the system and the convergence of tracking error within a fixed time are mathematically proved. Experimental results indicate that the control strategy can significantly reduce the information transmission, enhance the bandwidth utilization, and ensure the convergence of tracking error within a fixed time for teleoperation systems.

Actuators with Two Double Gimbal Magnetically Suspended Control Moment Gyros for the Attitude Control of the Satellites.

The paper proposes a novel automatic control system for the attitude of the mini-satellites equipped with an actuator having N = 2 DGMSCMGs (Double Gimbal Magnetically Suspended Control Moment Gyros) in parallel and orthogonal configuration, as well as a DGMSCMG-type sensor for the measurement of the satellite absolute angular rate. The proportional-derivative controller, designed based on the Lyapunov-functions theory, elaborates the control law according to which the angular rates applied to the servo systems for the actuation of the DGMSCMGs gyroscopic gimbals are computed. The gimbal's angular rates create gyroscopic couples acting on the satellite in order to control its attitude with respect to the local orbital frame. The new proposed control architecture was software implemented and validated, and the analysis of the obtained results proved the cancellation of the convergence errors and excellent angular rate precision.

LMI-based state observer design for supercavitating vehicles with time delay

Due to the existence of the planing force, the control of the supercavitating vehicle is difficult, and the vertical velocity which has a great influence on the planing force cannot be directly measured. Therefore, considering the time-delay effect of the planing force and the external disturbance, the state observer and controller based on linear matrix inequality (LMI) have been designed. By decomposing the planing force into two parts of time delay and non-time delay, and a longitudinal plane Linear Parameter Varying (LPV) time-delay model for supercavitating vehicles is derived. In addition, the controller and state observer for the supercavitating vehicle system with time-delay characteristics and external disturbances are simultaneously designed. The gains of the controller and observer are obtained by combining the theory of linear matrix inequalities and convex polytopes, ensuring the convergence of estimation error. At the same time, taking into account the possibility of exceeding the physical limits of the actuator when the input value is too large, the compensator has been added. Finally, the simulation results demonstrate that the designed controller exhibits good stability and tracking performance.

Research on the Magnetorheological Finishing Technology of a High-Steepness Optical Element Based on the Virtual-Axis and Spiral Scanning Path.

Magnetorheological finishing (MRF) of aspherical optical elements usually requires the coordination between the translational axes and the oscillating axes of the machine tool to realize the processing. For aspheric optical elements whose steepness exceeds the machining stroke of the equipment, there is still no better method to achieve high-precision and high-efficiency error convergence. To solve this problem, an MRF method combining virtual-axis technology and a spiral scanning path is proposed in this paper. Firstly, the distribution law of the magnetic induction intensity inside the polishing wheel is analyzed by simulation, the stability of the removal efficiency of the removal function within the ±7∘ angle of the normal angle of the polishing wheel is determined, and MRF is expanded from traditional single-point processing to circular arc segment processing. Secondly, the spiral scanning path is proposed for aspherical rotational symmetric optical elements, which can reduce the requirements of the number of machine tool axes and the dynamic performance of machine tools. Finally, an aspherical fused silica optical element with a curvature radius of 400 mm, K value of -1, and aperture of 100 mm is processed. The PV value of this optical element converges from 189.2 nm to 24.85 nm, and the RMS value converges from 24.85 nm to 5.74 nm. The experimental results show that the proposed combined process has the ability to modify curved optical elements and can be applied to ultra-precision machining of high-steepness optical elements.

Function recovery on manifolds using scattered data

We consider the task of recovering a Sobolev function on a connected compact Riemannian manifold M when given a sample on a finite point set. We prove that the quality of the sample is given by the Lγ(M)-average of the geodesic distance to the point set and determine the value of γ∈(0,∞]. This extends our findings on bounded convex domains [IMA J. Numer. Anal., 2024]. As a byproduct, we prove the optimal rate of convergence of the nth minimal worst case error for Lq(M)-approximation for all 1≤q≤∞.Further, a limit theorem for moments of the average distance to a set consisting of i.i.d. uniform points is proven. This yields that a random sample is asymptotically as good as an optimal sample in precisely those cases with γ<∞. In particular, we obtain that cubature formulas with random nodes are asymptotically as good as optimal cubature formulas if the weights are chosen correctly. This closes a logarithmic gap left open by Ehler, Gräf and Oates [Stat. Comput., 29:1203-1214, 2019].

Design and application of finite‐time tracking control for autonomous ground vehicle affected by external disturbances

AbstractPath tracking plays a critical role in autonomous driving for autonomous ground vehicle (AGV). However, AGV faces challenges in accurate tracking and chatter reduction due to external disturbances, making it difficult to meet the tracking performance requirements. Currently, sliding mode control (SMC) and disturbances observer are primarily employed for disturbance estimation. However, ensuring finite‐time robust control remains a significant challenge. To ensure rapid convergence of tracking errors and effective disturbance rejection, this paper proposed a novel non‐singular fast terminal sliding mode (NFTSM) control scheme based on finite‐time disturbance observation (FDO). First, a novel NFTSM controller based on AGV dynamic model is developed to achieve fast convergence of tracking errors. Then, to mitigate disturbances effects and suppress chatter, an innovative FDO method is employed. Finally, based on FDO, the NFTSM‐FDO establishes a control scheme that enhances disturbances suppression and accelerates convergence. The simulation and experimental results demonstrate the innovation of the proposed method. Compared with other SMC methods, the results validate the effectiveness and advantages of the proposed approach, exhibiting fast convergence and superior tracking performance.

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.