Convergence Of Error Research Articles

A Sensor Fusion Approach to Observe Quadrotor Velocity.

The growing use of Unmanned Aerial Vehicles (UAVs) raises the need to improve their autonomous navigation capabilities. Visual odometry allows for dispensing positioning systems, such as GPS, especially on indoor flights. This paper reports an effort toward UAV autonomous navigation by proposing a translational velocity observer based on inertial and visual measurements for a quadrotor. The proposed observer complementarily fuses available measurements from different domains and is synthesized following the Immersion and Invariance observer design technique. A formal Lyapunov-based observer error convergence to zero is provided. The proposed observer algorithm is evaluated using numerical simulations in the Parrot Mambo Minidrone App from Simulink-Matlab.

Fixed‐time formation control of second‐order nonlinear multi‐agent systems using the neural network dynamic sliding mode

AbstractThe problem of fixed‐time formation control for a class of second‐order nonlinear multi‐agent systems is studied. For a class of such systems, a control algorithm is proposed to maintain the connections among the agents while avoiding collisions. Furthermore, a radial basis function neural network is used in the design to precisely approximate the nonlinear function for the nonlinear terms in the model. Then, a dynamic sliding mode control method is proposed to suppress the chattering phenomenon that may arise due to the sliding mode control. A sufficient condition for the system to achieve fixed‐time formation is obtained by using different methods, such as Lyapunov stability. Finally, the effectiveness of the proposed algorithm is verified by example. Simulation experiments reveal that the proposed method has faster error convergence and better robust control than conventional algorithms.

A guaranteed fixed-step convergence approach to discrete sliding mode control of linear disturbed systems

Current methods for achieving fixed-time convergence in continuous sliding mode control are extensively studied, whereas there are limited studies on fixed-step convergence in discrete sliding mode control. The control objective of this paper aims to develop control schemes assuring that the number of convergence steps of the tracking error remains within the desired range, even as the initial value increases, and this paper proposes a guaranteed fixed-step convergence approach to form the reaching law. An upper bound on convergent steps, independent of the initial value, is established, thereby having the fixed-step convergence property. The performance assessment, including attractiveness, invariance, and convergence steps, is provided to meet the required specifications. The steady-state band results offer guidance in selecting parameters to achieve the control objective. To achieve fixed-step convergence of the tracking error, one can develop a dead-beat terminal sliding mode control scheme, where the tracking error is governed by the prescribed error dynamics. To illustrate the approach more concretely, specific designs are proposed for both switching and non-switching reaching laws aimed at ensuring fixed-step convergence. The simulation and experiment results validate the performance evaluation and demonstrate effectiveness of the proposed control schemes.

Adaptive Takagi–Sugeno (T–S) fuzzy observer based fault tolerant control for DC–DC converters with robustness against uncertainties

AbstractThis paper proposes a Takagi–Sugeno fuzzy system model based fault tolerant control scheme for DC–DC converters, which is robust against parameter uncertainties and achieves the output voltage of an ideal converter. The control involves estimating the duty cycle change in the form of a fault parameter required to track the output voltage, in the presence of several uncertain conditions including converter losses, variation in input voltage, and unknown and changing output load. An adaptive law is designed to estimate the fault parameter that guarantees state and parameter error convergence. The adaptive law is derived using the Lyapunov stability theorem and the required parameters are evaluated by solving a linear matrix inequalities optimization problem. The load resistance is estimated in parallel by using a Kalman filter and fed to the fault parameter estimation scheme. Furthermore, a fast and robust method to detect short and open circuit switch faults is also presented. The proposed technique offers a simple, yet effective method to regulate the output voltage under several faulty and uncertain conditions. The proposed technique is tested on a DC–DC boost converter simulation model and the demonstrated MATLAB/Simulink results show the effectiveness of the proposed algorithm.

Smoothness: The key factor in well-log information-assisted PINNtomo

Seismic traveltime tomography is popularly used to provide a smooth background model for subsequent depth migration or full waveform inversion (FWI). The core of the traveltime tomography process requires an eikonal solver for seismic traveltime calculation. Most of the conventional eikonal solvers, suffer from first-order convergence errors and difficulties in dealing with irregular topography, which will lead to inaccurate seismic tomography results. To solve these issues, physics-informed neural networks (PINNs) have been developed and demonstrated remarkable success in simultaneous traveltime calculation and velocity construction, which is referred to as PINNtomo. It combines the physics constraint (eikonal equation) and data constraint (traveltime fitting) to form the loss function of the networks. Though PINNtomo can manage to recover a smooth background velocity, it will fail in the inversion for a complicated high-resolution velocity model. We take advantage of the well-log velocity as a prior information to provide an additional model constraint in the loss function to improve the performance of PINNtomo. We found that the well-log information will provide high-resolution components in the inverted velocity model. However, it will destroy the lateral structure of the true model. We propose to use a smoothed version of the well-log velocity as the prior model constraint in PINNtomo. As a result, the accuracy of the inverted background velocity will be improved, which will be beneficial to the subsequent FWI.

KONTROL SLIDING MODE ADAPTIF DENGAN VAKSINASI DAN TREATMENT PADA MODEL PENYEBARAN TUBERKULOSIS

This research examines the design of adaptive sliding mode control in a tuberculosis (TB) disease spread model, considering uncertainties within the model. The TB model involves five state variables (susceptible individuals, vaccinated individuals, individuals with latent TB infection, individuals with active TB infection, and individuals under treatment) and three control input variables consisting of vaccination and treatment. The objective of this control design is to reduce the number of susceptible individuals, individuals with active TB infection, and the number of individuals under treatment by tracking the given reference functions. Stability and convergence of tracking errors of the controlled system are proved using the Lyapunov Stability Theorem. Numerical simulations are conducted to evaluate the performance of the designed control under various parameter uncertainty conditions. Based on the simulation results, it is shown that the adaptive sliding mode controllers guarantee the convergence of tracking errors is achieved despite the presence of uncertainty in the model.

A new approach of B-spline wavelets to solve fractional differential equations

This paper presents a groundbreaking method for solving the multi-order fractional differential (M-OFD), both linear and nonlinear, as well as fractional partial differential equations (FPDE)s. This approach involves constructing an operational matrix of fractional derivatives using linear B-spline (LB-S) wavelet functions with perfect subtlety. The new method has two crucial features. Firstly, it simplifies the problem by converting it into a set of algebraic equations, which is a significant advantage. This makes the method highly accurate and reliable. Secondly, it uses thresholding to dramatically reduce the computational workload in linear problems. This leads to lightning-fast and highly efficient problem-solving. The newly developed scheme underwent a thorough examination of its error estimates and convergence, revealing remarkable results in terms of accuracy and efficiency. The analysis provides a comprehensive understanding of the scheme’s performance, highlighting its potential as a dependable and effective method. Based on the findings, it is evident that the proposed method not only delivers exceptional precision but also operates with remarkable efficiency.

Convergence and accuracy of FDTD modelling for periodic plasmonic systems

Despite the widespread use of the finite-difference time-domain (FDTD) method for modeling plasmonic systems, there is a lack of detailed convergence and accuracy studies for periodic nanoparticle systems in which both particle radius and interparticle distance are critical parameters. Using an in-house parallelized 3D-FDTD code for which we implemented interface field averaging, convergence and accuracy were evaluated for various spherical particle radii, inter-distances, and radius-to-mesh size ratios. We found that Interface Field Averaging (IFA) FDTD improved accuracy and convergence with respect to per-component (Per-C) meshing. In the worst case of this study, the convergence error decreased from 4.9% to 2.6% only by using IFA. Accuracy was verified by benchmarking our simulation results with COMSOL Multiphysics software. Furthermore, we notice that there exists no general rule for choosing the mesh size. Careful convergence testing should therefore be carried out systematically.

Energy norm error estimates and convergence analysis for a stabilized Maxwell’s equations in conductive media

The aim of this article is to investigate the well-posedness, stability of solutions to the time-dependent Maxwell’s equations for electric field in conductive media in continuous and discrete settings, and study convergence analysis of the employed numerical scheme. The situation we consider would represent a physical problem where a subdomain is emerged in a homogeneous medium, characterized by constant dielectric permittivity and conductivity functions. It is well known that in these homogeneous regions the solution to the Maxwell’s equations also solves the wave equation, which makes computations very efficient. In this way our problem can be considered as a coupling problem, for which we derive stability and convergence analysis. A number of numerical examples validate theoretical convergence rates of the proposed stabilized explicit finite element scheme.

Space-time CutFEM on overlapping meshes II: simple discontinuous mesh evolution

We present a cut finite element method for the heat equation on two overlapping meshes: a stationary background mesh and an overlapping mesh that evolves inside/“on top” of it. Here the overlapping mesh is prescribed by a simple discontinuous evolution, meaning that its location, size, and shape as functions of time are discontinuous and piecewise constant. For the discrete function space, we use continuous Galerkin in space and discontinuous Galerkin in time, with the addition of a discontinuity on the boundary between the two meshes. The finite element formulation is based on Nitsche’s method. The simple discontinuous mesh evolution results in a space-time discretization with a slabwise product structure between space and time which allows for existing analysis methodologies to be applied with only minor modifications. We follow the analysis methodology presented by Eriksson and Johnson (SIAM J Numer Anal 28(1):43–77, 1991; SIAM J Numer Anal 32(3):706–740, 1995). The greatest modification is the introduction of a Ritz-like “shift operator” that is used to obtain the discrete strong stability needed for the error analysis. The shift operator generalizes the original analysis to some methods for which the discrete subspace at one time does not lie in the space of the stiffness form at the subsequent time. The error analysis consists of an a priori error estimate that is of optimal order with respect to both time step and mesh size. We also present numerical results for a problem in one spatial dimension that verify the analytic error convergence orders.

Zero‐sum game‐based H∞ optimal finite‐time prescribed performance control for nonlinear multiagent systems with actuator faults

AbstractThis article proposes an optimal leader‐follower consensus control protocol with finite‐time prescribed performance for nonlinear multiagent systems suffering from the actuator fault based on the zero‐sum game framework. First, to guarantee the specified performance of the system within a finite time frame, a novel error transformation function (ETF) is constructed. The ETF proposed in this paper can adjust the mapping range of error transformation to achieve better error convergence performance, thereby enhancing the flexibility of the controller design. Then, to compensate for the negative impact of actuator fault, an error‐related item is separated from the optimal performance index function to design an online fault estimator, which can obtain a better error estimation accuracy. Furthermore, to lessen the need for difficult‐to‐verify persistent excitation signal in adaptive dynamic programming algorithm, both historical and real‐time data are applied to adjust the updating law of neural network weight. Moreover, it demonstrates that the signals within the closed‐loop system are bounded and the expected finite‐time prescribed performance can be ensured in the presence of faults. Finally, a numerical example of multiple single‐link robots manipulator system is executed to validate the efficacy of the developed scheme.

Convergence and error estimates of a penalization finite volume method for the compressible Navier–Stokes system

Abstract In numerical simulations a smooth domain occupied by a fluid has to be approximated by a computational domain that typically does not coincide with a physical domain. Consequently, in order to study convergence and error estimates of a numerical method domain-related discretization errors, the so-called variational crimes, need to be taken into account. In this paper we apply the penalty approach to control domain-related discretization errors. We embed the physical domain into a large enough cubed domain and study the convergence of a finite volume method for the corresponding domain-penalized problem. We show that numerical solutions of the penalized problem converge to a generalized, the so-called dissipative weak, solution of the original problem. If a strong solution exists, the dissipative weak solution emanating from the same initial data coincides with the strong solution. In this case, we apply a novel tool of the relative energy and derive the error estimates between the numerical solution and the strong solution. Extensive numerical experiments that confirm theoretical results are presented.

A practical leader–follower hybrid control scheme for wheeled mobile robots

This paper addresses the problem of tracking control of wheeled mobile robots in leader–follower formation. By applying the backstepping technique, a hybrid control scheme consisting of continuous control and intermittent control is proposed, and leader–follower formation tracking is achieved in the presence of model uncertainty and external disturbance. A relationship between the intermittent control width and control gain and the convergence of the position errors is obtained for the tracking accuracy estimation. As a special case, the concerned intermittent control can be converted to impulsive control with a sufficiently small control width, and tracking of the trajectory can be realized by directly driving the state variables of the follower at well-designed discrete instants. The main novelty of the proposed hybrid control scheme is to drive the tracking errors into a preset region via intermittent control (or impulsive control) signals and then achieve the tracking goal in the practical sense via continuous control signals. An example with the corresponding numerical simulations and experimental tests is given to illustrate the effectiveness of the proposed method.

Space-time CutFEM on overlapping meshes I: simple continuous mesh motion

We present a cut finite element method for the heat equation on two overlapping meshes: a stationary background mesh and an overlapping mesh that moves around inside/“on top” of it. Here the overlapping mesh is prescribed by a simple continuous motion, meaning that its location as a function of time is continuous and piecewise linear. For the discrete function space, we use continuous Galerkin in space and discontinuous Galerkin in time, with the addition of a discontinuity on the boundary between the two meshes. The finite element formulation is based on Nitsche’s method and also includes an integral term over the space-time boundary between the two meshes that mimics the standard discontinuous Galerkin time-jump term. The simple continuous mesh motion results in a space-time discretization for which standard analysis methodologies either fail or are unsuitable. We therefore employ what seems to be a relatively uncommon energy analysis framework for finite element methods for parabolic problems that is general and robust enough to be applicable to the current setting. The energy analysis consists of a stability estimate that is slightly stronger than the standard basic one and an a priori error estimate that is of optimal order with respect to both time step and mesh size. We also present numerical results for a problem in one spatial dimension that verify the analytic error convergence orders.

Event-triggered finite-time formation tracking of multi-agent systems with mismatched disturbances under switching topologies

The formation tracking of the leader–follower multi-agent systems (MASs) under switching topologies is investigated. The considered system is exposed to both the mismatched and matched disturbances in the dynamics of the leader and followers, which places higher requirements for the robustness of the control protocol. In the presence of disturbances and leader’s unknown control input, an innovative distributed observer embedded with robust terms is designed firstly to estimate leader’s states in finite time. Taking account of the switching topologies, a novel analysis scheme that divides the convergence process into two stages is proposed to establish the finite-time (FT) convergence of estimation errors. Then, by virtue of a constructed auxiliary variable, a FT controller with an event-triggered mechanism is put forward, in which multiple robust feedback terms are designed wisely to suppress the mismatched and matched disturbances effectively. As a result, the FT formation tracking can be achieved with saved resources, despite perturbed environments and switching topologies. Simulation examples are presented to confirm the effectiveness of the proposed algorithm.

Feasibility of a Tungsten-181 HDR brachytherapy source: Analysis of isotope production and source dosimetry

BackgroundInterest in Intensity Modulated Brachytherapy (IMBT) for High Dose Rate Brachytherapy (HDR) treatments has steadily increased in recent years. However, intensity modulation is not best optimized for currently used HDR sources since they emit high energy photons. To that end, the focus on IMBT has moved to middle energy sources, such as Ytterbium-169; yet even Yb-169 emits some high energy photons at a low yield. We present an alternative isotope, Tungsten-181 (T1/2 = 121 days) that is interesting due to its complete lack of high energy photon emissions. (Eavg = 58.9 keV, Emed = 57.5 keV) making it potentially favorable as high dose rate brachytherapy source from both a medical physics and health physics perspective. PurposeThe purpose of this study was to determine the feasibility of using W-181 as an HDR brachytherapy source; in this study we focused on W-181's production, dosimetric properties, and intensity modulation capabilities. MethodsWe determined the isotope production kinetics, its Dose Rate Constant, Radial Dose Function, photon self-absorption, and the shielding intensity modulation capabilities for a W-181 pellet source geometry using the MCNP6.2 Computer Simulations Code. All simulations were performed using a personal computer running an AMD Ryzen 5 3600 6-Core Processor 3.59 GHz. The number of histories run for each study were selected to produce relative simulation convergence errors in the MCNP tally output of less than 2%.Dosimetric calculations were made using the MCNP6.2 computer simulations code and activation analyses were determined mathematically using a Catenary kinetics analysis (also known as a Bateman Analysis) of W-181 and Tantlum-182 production from the neutron activation of a pure Tungsten-180 stable target. Since W-181 emits middle energy photons and has a high density, we also assessed the effects of photon self-absorption within a tungsten pellet. ResultsFrom our analysis, we determined that a 3.5 mm long and 0.6 mm in diameter is feasible for clinical applications. Our activation analyses found that these pellets can achieve W-181 activities up to 7.9Ci and 13Ci using neutron fluence rates of 4E14 cm−2 s−1 and 1E15 cm−2 s−1 respectively, which then would provide a dose rate of 1.84 ± 0.01 cG y/Ci/min at a depth of 1 cm from the source. Using our resulting Monte Carlo simulated Dose Rate Constant of 1.24 ± 0.02 cGy h−1∙U−1, a W-181 source in this geometry would require a source activity upwards of 10Ci for use in HDR treatments. In the intensity modulation analysis, only 0.1 mm of gold shielding was found to reduce a pellet's absorbed dose by over 50% while 0.3 mm of gold shielding, which is thin enough to theoretically fit between an HDR pellet and the inner catheter wall, was found to reduce the pellet's absorbed dose by over 85%. ConclusionsWhile W-181 has a lower specific activity than Ir-192 and Yb-169, it shows great promise as an isotope for use in Intensity Modulated Brachytherapy due to its easily shielded photons. We therefore expect that W-181 may lend itself best for use as a multi-pellet configuration in IMBT.

Novel quantized iterative learning control based on spherical polar coordinates

AbstractThis study investigates the iterative learning control method for discrete‐time systems with data quantization, and it employs a quantizer based on spherical polar coordinates. The quantizer uses spherical polar coordinates to transform an unquantized signal into one with a predetermined quantization level. The quantization capability is dependent on the size of the support sphere, which is dynamically adjustable. In the scenarios of system output quantization, general quantization, and tracking error quantization, the learning control schemes are developed with consideration of the smallest sphere quantization features. The radii of the support spheres are designed independently based on the characteristics of the quantized signals (system output, control input, tracking error). The findings suggest that the system can achieve a bounded error convergence in the scenarios of system output and general quantization. In the context of quantization in tracking error, it is achievable to attain tracking performance with zero error. The example of a permanent magnet synchronous motor is provided to illustrate the findings of the study.

Fault-tolerant control for Markov jump nonlinear systems based on an observer and finite-time fast integral terminal sliding mode control

This paper develops a novel fault-tolerant control based on an observer for underactuated robot manipulators with unknown external disturbances, uncertainties, and actuator faults. First, unlike the traditional approach of treating the robot system as a parameter time-varying system, the underactuated robot manipulator with different drive modes can be considered as one of the classical Markov jump nonlinear systems (MJNSs). Second, an adaptive disturbance observer is designed to estimate the state and disturbances of the system. Finally, based on the observation results, a nonsingular fast integral terminal sliding mode controller (NFITSMC) is utilized to implement fault-tolerant control of the system. Compared with traditional observer and terminal sliding mode controller, the adoption of the novel controller and observer can improve response time and reduce chattering. Especially, in order to eliminate chattering, the integral term is introduced into the nonsingular fast terminal sliding mode controller. Structuring Lyapunov–Krasovskii functional (LKF) and based on linear matrix inequalities (LMIs) techniques, the convergence of control strategy and tracking errors is proved. The simulation results show that the actuator faults can be observed successfully and the error system is finite-time stable.

Strategic Analysis of Feature Selection Methods for Enhanced Dental Therapy Recognition in Machine Learning Applications

The popularity of smart clinics has increased significantly as a result of technical developments in fields like computer vision. At the heart of such systems is the ability to recognize objects and activities as well as perceive the environment as a whole. This is crucial for both eco-independent systems and human-machine interaction, especially in settings with constrained workspaces, like dental care. Our study delves into an extensive analysis of multiple machine learning models designed to robustly predict dental treatments. These models encompass Lazy Instance-based Learning, Sequential Minimal Optimization, Hoeffding Tree, and Random Tree. Leveraging object-oriented input sourced from gaze-guided wearable cameras, we scrutinize intricate attributes such as material properties, patients' dental conditions, and the array of instruments in use. Notably, we exploit the insight that identifying visual cues during an activity holds the potential to address the specific therapy identification challenge. Utilizing a dental data set that we gathered in the real world, we conducted our experiments and discovered that combining multiple criteria enhances accuracy in comparison to using each one alone. We did see, nevertheless, that in certain circumstances employing the symptoms alone produced superior outcomes. Additionally, symptoms demonstrated to have lesser error than combination in terms of RMS error convergence. Finally, we observed that the machine learning models' build and test durations increased as a result of the combined method. This demonstrates that adding additional parameters does not necessarily result in better outcomes in machine learning applications generally and in medical/dental applications in particular. Instead, it relies on the machine learning tool used, the settings taken into account and the input data provided. The versatility of our approach extends beyond dental contexts. It has been systematically validated across diverse domains, including the recognition of kitchen activities within smart home environments. This methodology holds relevance for various outdoor scenarios where the focal point of attention guides ongoing activities.

Adaptive Fuzzy Fixed-Time Control for Nonlinear Systems with Unmodeled Dynamics

This article concentrates on the problem of fixed-time tracking control for a certain class of nonlinear systems with unmodeled dynamics. Unmodeled dynamics are prevalent in practical engineering systems, such as axially symmetric systems like robotic arms, spacecraft, and missiles. In this paper, the fuzzy-logic systems (FLSs) are implemented to address the challenge of accurately approximating the unknown nonlinear terms that arise during the derived control algorithm process. By employing fixed-time command filters (FTCF), the “explosion of complexity” issues encountered in traditional backstepping methods will be effectively resolved. Moreover, error compensation mechanisms are derived to effectively mitigate the filtering errors that may arise from the FTCFs. The computational burden associated with FLSs is reduced through the utilization of the weight vector estimation method based on the maximal norm and an adaptive approach. A fixed-time adaptive fuzzy tracking controller is developed within the backstepping control framework to ensure the boundedness of all signals and achieve fixed-time convergence of the tracking error for the controlled system. Illustrative examples are conducted to illustrate the viability of the derived controller.