Maximum Convergence Rate Research Articles

Development of an Efficient Variable Step-Size Gradient Method Utilizing Variable Fractional Derivatives

The fractional gradient method has garnered significant attention from researchers. The common view regarding fractional-order gradient methods is that they have a faster convergence rate compared to classical gradient methods. However, through conducting theoretical convergence analysis, we have revealed that the maximum convergence rate of the fractional-order gradient method is the same as that of the classical gradient method. This discovery implies that the superiority of fractional gradients may not reside in achieving fast convergence rates compared to the classical gradient method. Building upon this discovery, a novel variable fractional-type gradient method is proposed with an emphasis on automatically adjusting the step size. Theoretical analysis confirms the convergence of the proposed method. Numerical experiments demonstrate that the proposed method can converge to the extremum point both rapidly and accurately. Additionally, the Armijo criterion is introduced to ensure that the proposed gradient methods, along with various existing gradient methods, can select the optimal step size at each iteration. The results indicate that, despite the proposed method and existing gradient methods having the same theoretical maximum convergence speed, the introduced variable step size mechanism in the proposed method consistently demonstrates superior convergence stability and performance when applied to practical problems.

Maximum Convergence Rate Control of a Switched Electrical Network with Symmetries

The purpose of the present research endeavor is to propose a novel control strategy for a DC-DC electrical converter realized as a switched circuit. The present endeavor is based on an early work by Leonard and Krishnaprasad where a prototypical DC-DC converter was modeled as a state space dynamical system and controlled by an open-loop strategy based on Lie group theory. In this work, we shall rather introduce a closed-loop control strategy based on maximum convergence rate, study some features of the novel method, and illustrate its behavior by numerical simulations. A numerical comparison of the two methods complements the paper. The numerical comparison shows how the proposed feedback control method outperforms the static one in terms of convergence rate and resiliency against parameters mismatch.

Deception attack detection based on bandwidth allocation for switched systems with quantization

This article discusses a more practical deception attack on the continuous-time switched systems, i.e. a malicious attacker tampers with the binary 01 code transmitted through the network. Assume that the switching is slow enough and the attacker’s energy is limited, we propose a coding transmission strategy based on bandwidth allocation, which selects a part of the network bandwidth for reverse data transmission. Under such strategy, partial detection of attacks and synchronous update of quantization rules are realized. Combined the increasing/decreasing rate of Lyapunov function with the success rate of attack detection, the sufficient conditions ensuring the mean square exponential stability of the system are given. Taking the maximum convergence rate as a performance indicator, the optimal bandwidth allocation scheme is provided for the specific values, which realizes a trade-off between attack detection rate and quantization error. Two-tank system is adopted to illustrate the effectiveness of the theoretical analysis.

Direct Deadbeat Control of a Buck PWM Converter

The paper explains the shortcomings of the well-known modal control strategy, which prevent its practical application to achieve the highest possible dynamic performance of switching semiconductor converters. A method for eliminating these shortcomings is suggested as applied to a self-contained modal control version with specifying zero roots of the characteristic polynomial and obtaining a nilpotent system matrix. A technique for synthesizing direct deadbeat control with a nilpotent system matrix is described in detail to implement a finite transient that terminates within a number of cycles not exceeding the order of the difference equation with which a discrete system is described. The control object is a well-known circuit of a DC-DC buck converter with an inductive-capacitive filter and a proportional input-output characteristic provided by an integral link in the feedback loop. The efficiency of the proposed deadbeat controller version is confirmed through mathematically simulating the operation of a PWM converter controlling the load current. The dynamic operation modes are studied for the case of a step change in the load resistance and input voltage relative to their nominal values for a control system with I-, PI- and deadbeat controllers. A comparison of the obtained results shows that the transients in the system equipped with a deadbeat controller terminate most quickly: within three cycles without an overshoot. This corresponds to the maximum convergence rate to the equilibrium state for a third-order system.

Current motion and deformation of the Nazca Plate: new constraints from GPS measurements

SUMMARYWe use new GPS data to determine an updated Euler pole describing the present-day motion of the oceanic Nazca Plate. Our solution includes continuous GPS (cGPS) measurements at Malpelo Island offshore Colombia, two sites in the Galapagos archipelago, Easter Island and Salas y Gomez Island in the western part of the plate and Robinson Crusoe Island offshore Chile. A careful analysis of geodetic time-series reveals that (1) previous estimates using former cGPS site EISL are biased by several millimetres per year eastward due to station malfunctioning (2) north velocity component of cGPS site GLPS at Santa Cruz Island in the Galapagos is impacted by volcanic deformation at the 1–2 mm yr–1 level, probably caused by the recurrent volcanic activity of the Sierra Negra volcano. In addition, we find shortening at ∼1 mm yr–1 between Easter Island (cGPS ISPA) and Salas y Gomez Island (cGPS ILSG), consistent with the elastic deformation induced by rapid opening at the East Pacific rise. cGPS site at Robinson Crusoe Island shows ∼4–5 mm yr–1 abnormally fast East velocity induced by the visco-elastic relaxation following the Maule Mw 8.8 2010 earthquake. Using this information, we determine a new Euler pole (longitude: –90.93°E, latitude 56.19°N, 0.588 deg Myr–1) describing the present-day Nazca–South America Plate motion, using five sites (Malpelo Island, two sites in the Galapagos archipelago, Easter Island and Salas y Gomez Island). The proposed Euler pole provides a weighted root mean square (wrms) of residual velocities of 0.6 mm yr–1, slightly higher than usually observed for other major tectonic plates and accounting for the uncertainty of potential volcanic–tectonic deformation. Our model predicts a maximum convergence rate at 65.5 ± 0.8 mm yr–1 at latitude ∼30°S along the Chile trench, decreasing to 50.8 ± 0.7 mm yr–1 in northern Colombia and 64.5 ± 0.9 mm yr–1 in southern Chile (1σ confidence level). Comparison with the geological models NUVEL1A and MORVEL indicates constant decrease since 3.16 Ma of opening rate along the Nazca–Antarctic Plate boundary spreading centres at ∼1 cm yr–1 per Myr. Combined with the ITRF2014 pole for the Pacific and Antarctic plates, our derived Euler pole predicts closure at the ∼1 mm yr–1 level for Pacific–Antarctic–Nazca Plate circuit. However, combining our results with MORVEL estimates for the Cocos Plate, the non-closure of the Pacific–Cocos–Nazca Plate circuit is 9.7 ± 1.6 mm yr–1, 30 per cent lower than the 14 ± 5 mm yr–1 reported in MORVEL model, but still significant. A small (∼1.5 mm yr–1) velocity residual at Malpelo Island neither supports the hypothesis of an independent Malpelo microplate offshore Colombia nor large scale internal deformation induced by thermal contraction. Our solution rather suggests that non-closure of the Pacific–Cocos–Nazca Plate circuit arises from the determination of the Cocos Plate motion in MORVEL, an hypothesis further supported by the large discrepancy between MORVEL’s prediction and the observed GPS velocity observed at Cocos Islands (cGPS ISCO).

Investigation of Near-Field Source Localization Using Uniform Rectangular Array

In fifth-generation (5G) wireless communications, large-scale arrays pose a challenge to the accuracy of signal models based on the plane wavefront. In this paper, a novel method for 3D near-field direction of arrival (DOA) estimation is proposed based on large-scale uniform rectangular array (URA). First, the near-field signal model based on the vertical rectangular array and the delay phase shift of the received array is presented. Afterwards, the proposed method divides the complete parameters set into multiple-parameters subsets, and only estimates one of them in each iteration, leaving the others in the fixed subset. As a result, we can obtain the maximum convergence rate of the deterministic maximum likelihood (DML) algorithm. Finally, the simulation results demonstrate that the root mean square errors (RMSEs) of the proposed algorithm are closer to the Cramer-Rao lower bounds and converge faster than those of the DML algorithm, confirming its effectiveness and superiority.

Про один алгоритм ідентифікації лінійних обʼєктів на основі найменшого ексцесу

Розглянуто задачу ідентифікації параметрів лінійного обʼєкта за наявності негауссівських завад. Алгоритм ідентифікації є градієнтною процедурою мінімізації критерію найменшого середнього ексцесу. Використання такого функціоналу дозволяє отримати оцінки з робастними властивостями. Алгоритм ідентифікації — це градієнтна процедура. Визначено умови збіжності процедури, що застосовується, в середньому і середньоквадратичному за наявності негауссівських завад вимірів. Крім того, отримано оцінки для визначення оптимального значення параметра алгоритму, що забезпечують його максимальну швидкість збіжності. На основі цих оцінок визначено асимптотичні та неасимптотичні значення похибок оцінювання параметрів та похибок ідентифікації. У звʼязку з цим отримані вирази містять невідомі параметри (значення дисперсій сигналів і завад). Для їх застосування слід використовувати оцінки цих параметрів. Отримані співвідношення досить громіздкі, однак їх спрощення дозволяє провести якісний аналіз сталості. Слід зазначити, що всі отримані в роботі оцінки залежать від низки параметрів, проблема визначення яких залишається відкритою. Вони дозволяють досліднику попередньо оцінити можливості алгоритму ідентифікації та ефективність його використання при вирішенні практичних задач.

Numerical implementation of biological kinetics multi-species model in the Azov Sea

Paper covers the research of biological kinetics processes based on a multi-species model of plankton and fish interaction of the Azov Sea at low and high size considering the V erhulst and Allee effects, competition for resources, taxis, catching, spatial distribution of biogenic matter and detritus. Discrete analogue of developed model problem of water ecology, included in a software complex, were calculated using schemes of increased order of accuracy considering the partial filling of computational cells. The system of grid equations of large dimension, arising at discretization, has been solved on the basis of a two-layer variational type method – the minimum corrections method having the maximum convergence rate. Effective parallel algorithms were developed for numerical implementation of biological kinetics problem and oriented on multiprocessor computer system and NVIDIA Tesla K80 graphics accelerator with the data storage format modification of. Due to it, the reproduction processes of biogeocenose populations have been analysed in real and accelerated time.

Rapid Transitions With Robust Accelerated Delayed-Self-Reinforcement for Consensus-Based Networks

Rapid transitions are important for quick response of consensus-based, multi-agent networks to external stimuli. While high-gain can increase response speed, potential instability tends to limit the maximum possible gain, and therefore, limits the maximum convergence rate to consensus during transitions. Since the update law for multi-agent networks with symmetric graphs can be considered as the gradient of its Laplacian-potential function, Nesterov-type accelerated-gradient approaches from optimization theory, can further improve the convergence rate of such networks. An advantage of the accelerated-gradient approach is that it can be implemented using accelerated delayed-self-reinforcement (A-DSR), which does not require new information from the network nor modifications in the network connectivity. However, the accelerated-gradient approach is not directly applicable to general directed graphs since the update law is not the gradient of the Laplacian-potential function. The main contribution of this work is to extend the accelerated-gradient approach to general directed graph networks, without requiring the graph to be strongly connected. Additionally, while both the momentum term and outdated-feedback term in the accelerated-gradient approach are important in general, it is shown that the momentum term alone is sufficient to achieve balanced robustness and rapid transitions without oscillations in the dominant mode, for networks whose graph Laplacians have real spectrum. Simulation results are presented to illustrate the performance improvement with the proposed Robust A-DSR of 40% in structural robustness and 50% in convergence rate to consensus, when compared to the case without the A-DSR. Moreover, experimental results are presented that show a similar 37% faster convergence with the Robust A-DSR when compared to the case without the A-DSR.

Studying the properties of a robust algorithm for identifying linear objects, which minimizes a combined functional

This paper addresses the task of identifying the parameters of a linear object in the presence of non-Gaussian interference. The identification algorithm is a gradient procedure for minimizing the combined functional. The combined functional, in turn, consists of the fourth-degree functional and a modular functional, whose weights are set using a mixing parameter. Such a combination of functionals makes it possible to obtain estimates that demonstrate robust properties. We have determined the conditions for the convergence of the applied procedure in the mean and root-mean-square measurements in the presence of non-Gaussian interference. In addition, expressions have been obtained to determine the optimal values of the algorithm's parameters, which ensure its maximum convergence rate. Based on the estimates obtained, the asymptomatic and non-asymptotic values of errors in estimating the parameters and identification errors. Because the resulting expressions contain a series of unknown parameters (the values of signal and interference variances), their practical application requires that the estimates of these parameters should be used. We have investigated the issue of stability of the steady identification process and determined the conditions for this stability. It has been shown that determining these conditions necessitates solving the third-degree equations, whose coefficients depend on the specificity of the problem to be solved. The resulting ratios are rather cumbersome but their simplification allows for a qualitative analysis of stability issues. It should be noted that all the estimates reported in this work depend on the choice of a mixing parameter, the task of determining which remains to be explored. The estimates obtained in this paper allow the researcher to pre-evaluate the capabilities of the identification algorithm and the effectiveness of its use in solving practical problems.

Consensus of Multiagent Systems With Time-Varying Input Delay via Truncated Predictor Feedback

This article investigates the consensus tracking of exponentially unstable multiagent systems with time-varying input delay. The truncated predictor feedback approach is utilized for designing delay-dependent state and output feedback protocols. And the explicit conditions that can realize consensus tracking are established in terms of parametric Lyapunov equations and scalar inequalities. Besides, the protocol design algorithms under the maximum allowable delay and the maximum convergence rate are, respectively, provided. Compared with the existing results, the salient characteristic of the current results is to reveal the quantitative relationship among the time delay, unstable plant, network topologies, and the convergence rate. Specifically, for achieving the consensus tracking, the synchronization force from network connectivity needs to dominate the anti-synchronization force from unstable open-loop poles and input delays; the maximum allowable input delay is inversely proportional to the sum of the unstable open-loop poles; the convergence rate becomes smaller as the input delay bounds or/and the sum of the unstable poles in plant increase. Numerical simulations confirm the effectiveness of the proposed theoretical design.

Finite-difference schemes and iterative methods for multidimensional elliptic equations with mixed derivatives

Finite difference schemes and iterative methods of solving anisotropic diffusion problems governing multidimensional elliptic PDE with mixed derivatives are considered. By the example of the test problem with discontinuous coefficients, it is shown that the spectral characteristics of the finite difference problem and the efficiency of their preconditioning depend on the mixed derivatives approximation method. On the basis of the comparative numerical analysis, the most adequate approximation formulas for the mixed derivatives providing a maximum convergence rate of the bi-conjugate gradients method with the incomplete LU factorization and the Fourier – Jacobi preconditioners are discovered. It is shown that the monotonicity of the finite difference scheme does not guarantee advantages at their iterative implementation. Moreover, the grid maximum principle is not provided under the conditions of essential anisotropy.

A relaxed gradient based algorithm for solving generalized coupled Sylvester matrix equations

The present work proposes a relaxed gradient based iterative (RGI) algorithm to find the solutions of coupled Sylvester matrix equations AX+YB=C,DX+YE=F. It is proved that the proposed iterative method can obtain the solutions of the coupled Sylvester matrix equations for any initial matrices X0 and Y0. Next the RGI algorithm is extended to the generalized coupled Sylvester matrix equations of the form Ai1X1Bi1+Ai2X2Bi2+⋯+AipXpBip=Ci,(i=1,2,…,p). Then, we compare their convergence rate and find RGI is faster than GI, which has maximum convergence rate, under an appropriative positive number ω and the same convergence factor µ1 and µ2. Finally, a numerical example is included to demonstrate that the introduced iterative algorithm is more efficient than the gradient based iterative (GI) algorithm of (Ding and Chen 2006) in speed, elapsed time and iterative steps.

Convergence Analysis on Multi-AUV Systems With Leader-Follower Architecture

In multi-autonomous underwater vehicle (multi-AUV) systems, the convergence rate is characterized by the pace of consistency of the key state information for each member. The topology with leader–follower architecture is designed as a combination of an undirected graph between followers and a digraph between leaders and followers. An overview of influences on convergence rate of the second-order consensus algorithm is elaborated in three aspects, along with the main contributions in this paper. Specifically, the explicit expression of the maximum convergence rate is established based on the root locus method, and then, the effects of control parameters on the convergence rate are analyzed. Moreover, the influences of network topologies on the convergence rate are investigated from the view of adjusting the existing connectivity, changing the weights on links, and utilizing hierarchical structure. The combination of consensus and filtering algorithm is also an approach to enhance the capacity of multi-AUV systems. In order to eliminate the accumulated errors in the process of dead reckoning, a collaborative navigation model is presented, and then, a localization approach based on consensus-unscented particle filter algorithm is proposed. Simulations results are provided to verify location performance under the assumption of Gaussian white noise in the systems. In addition, the influences of the topologies on positioning accuracy are explored.

Global exponential convergence of generalized chaotic systems with multiple time-varying and finite distributed delays

Under some simple conditions, the convergence of a generalized chaotic system about its all variables is derived by only considering the convergence of its partial variables. Furthermore, based on some inequality techniques and employing the Lyapunov method, some novel sufficient criteria are derived to ensure the state variables of the discussed mixed delay system to converge, globally exponentially to a ball in the state space with a pre-specified convergence rate. Meanwhile, the ultimate bounds of the generalized chaotic system about its all variables are induced by the ultimate bounds of the system about its partial variables. Moreover, the maximum convergence rates about partial variables are also given. The methods are simple and valid for the convergence analysis of systems with time-varying and finite distributed delays. Here, the existence and uniqueness of the equilibrium point needs not to be considered. These simple conditions here are easy to be verified in engineering applications. Finally, some illustrated examples are given to show the effectiveness and usefulness of the results.

Non-smooth Feedback Control for a Class of Constraint Systems

Beginning from the second-order input constraint systems, the paper pays attention to study the control prob- lem for a class of the constraint systems with the nalyzes the stability of the second-order input constraint systems under the condition of linear state feedback. On the basis of that, it gives the non-smooth feedback control law of the third-order state constraint systems. At the same time, it sets up the relationship between system's steady state and its initial condi- tions. Finally, it has verified the validity of the conclusion by simulation. In the daily life, systems usually work under the condi- tion of some constraints or limits which is often showed by one link from systems that has been affected by a certain degree of the characteristics of saturation nonlinearity. Influ- enced by saturation nonlinearity, this kind of systems will make the analysis and design of the systems difficult because it is not only deferent from normal one in system perfor- mance, but also difficult to adopt common means to judge its stability, especially for the situation that the change of some system state affected by the saturation nonlinearity, under which it is very difficult to use existing methods to analyze the stability of the systems. Therefore, judging from the re- searches on saturation nonlinearity control problem, there is not an effective theoretical framework to solve the stability of the systems and the design. However, although saturation nonlinearity on independent variable is continuous, it is not continuous for the derivative of variable. So that is a kind of non- smooth nonlinearity which has the system with satura- tion nonlinear. What's more, in essence, it is a class of non- smooth system, whose control problem can be considered from non-smooth control theories and design methods. Just from the perspective of non-smooth feedback design view, this paper studies that the change of system state has the con- trol problem of saturation non-linear constraint systems. In fact, as for the control problem of the system with sat- uration nonlinearity, there has been a lot of research achievements published. Reference (1) gives the sufficient and necessary condition of system stability for the reliability problem of a class of linear saturation nonlinear system. That condition shows that, in the case that the state of linearity system is completely affected by saturation nonlinearity, the system would make the global asymptotic stability come true if the system matrix met certain conditions. Especially for the second-order system, it is required that System Matrix A belongs to Hurwitz's with a line which has diagonally domi- nant characteristic. Reference (2), based on Reference (1), further discusses the convergence problem for the second- order system. In regard to the study of the second-order sys- tem, it can be found that Reference (3) studies detailed the track of a class of the second-order system under the condi- tion of saturation constraints (4). And input limited system is discussed in Reference (4) where there are the design meth- ods that linear system gets the maximum convergence rate of elliptic invariant sets in the sate of saturation constraints. These research results generally focus on a class of systems whose saturation nonlinearity has an effect on all the system states, and the restricted states of many systems are usually limited in practice (5-7). T hus, the conclusion above can not be used directly. This paper studies a class of systems which is in the lim- ited state. The systems have three states, among which there is a kind of state-saturation nonlinearity-which is often seen under the condition of actuator torque limited. Firstly, the paper starts with the feedback designs with the state of the second-order input constraint systems and draws the conclu- sion of the stability of the systems that are in the status of linearity. And then, the linear state feedback controller gotten by the second-order system directly extends to the target sys- tem and gets non-smooth feedback controller. At the same time, it further discusses the relationship between system sta- bility and initial conditions and establishes the relationship between the two. Finally, it has verified the conclusion under different initial conditions by numerical simulation.

Backstepping-Based Inverse Optimal Attitude Control of Quadrotor

Abstract Input saturation must be taken into account for applying rapid reorientation in the large angle manoeuvre of a quadrotor. In this paper, a backstepping-based inverse optimal attitude controller (BIOAC) is derived which has the property of a maximum convergence rate in the sense of a control Lyapunov function (CLF) under input torque limitation. In the controller, a backstepping technique is used for handling the complexity introducing by the unit quaternion representation of the attitude of a quadrotor with four parameters. Moreover, the inverse optimal approach is employed to circumvent the difficulty of solving the Hamilton-Jacobi-Bellman (HJB) equation. The performance of BIOAC is compared with a PD controller in which the input torque limitation is not considered under the same unit quaternion representation using numerical simulation while the results show that BIOAC gains faster convergence with less control effort. Next, BIOAC is realized on a test bed and the effectiveness of the control law is verified by experimental studies.

Generalized method of supplementary strain in the problems of beam torsion

Generalized supplementary strain method efficiency is shown by beam elastoplastic torsion problem solving using BEM formulation. Dependence of convergence rate on modified shear modulus and on the twist angle was studied. Optimal choice of plasticity parameter leads to maximum convergence rate for this method.

Shape functions associated with super-convergent mass matrix

This article presents a novel approach in obtaining shape functions associated with super convergent element formulations with and continuity. In structural dynamics super-convergent element models are obtained by eigen-value convergence analysis, or minimizing the discretization errors leading to maximum convergence rates in their eigen-solutions. The element formulations developed by these inverse strategies are obtained without the need to define any shape functions. This article proposes a methodology to define the inverse model associated shape functions. The method employs trigonometric functions to define shape functions corresponding to the super convergent element formulations. Using the proposed strategy, the shape functions of super-convergent rod and beam elements are obtained. The presented rod and beam shape functions corresponding to the inverse formulations are just examples on how the presented methodology can be implemented.

Shape functions of superconvergent finite element models

In structural dynamics superconvergent element models are obtained by eigen-value convergence analysis, or minimizing the discretization errors leading to maximum convergence rates in their eigen-solutions. The element formulations developed by these inverse strategies are obtained in local coordinates. As no shape functions are employed in their development transforming them to global coordinates is a challenge and prevents their use in practical finite element models. To remove this obstacle a new method is proposed to obtain shape functions for superconvergent element models attained directly from the eigen-value convergence analysis or discretization error analysis. The method employs series of trigonometric functions to obtain shape functions corresponding to the superconvergent element formulations. Using the proposed strategy, the shape functions for superconvergent rod, beam and transverse vibration membrane are obtained. It is shown transformation of the superconvergent element formulation to the global coordinates using the obtained shape functions does not affect the eigen-value convergence rates.