Fractional-Order Numerical Scheme with Symmetric Structure for Fractional Differential Equations with Step-Size Control

Numerical methods for second order singular pertubation problems

A variable step size Least Mean Square (LMS) algorithm based on the difference of the error signal (ESD-VSS) is proposed.The convergence and steady state behavior of the ESD-VSS is analyzed and the key parameters are given.Simulations are conducted and the results show that, compared with the Modified Variable Step Size (MVSS) algorithm, modified LMS algorithm and the standard LMS algorithm, the ESD-VSS provides faster convergence speed and smaller steady state misadjustment.Index Terms -LMS algorithm, variable step size, adaptive filter, MSE I . IntroductionThe LMS algorithm proposed by Widrow and Hoff has been widely used in channel equalization [1] and system identification [2], due to its robustness and simplicity.However the tradeoff between the convergence speed and the steady state misadjustment is a challenging problem.To achieve faster convergence speed and small misadjustment, a number of modified algorithms have been proposed.Ref. [3] utilized the self-correlation to control the step size, and suppress the interference of irrelevant noise effectively, but the convergence speed and the tracking capability are still expected to be improved.Ref. [4] gave a variable step size LMS algorithm using estimate signal error and thus it is of relatively faster tracking speed and smaller steady state error.However, the computational complexity of the algorithm is larger due to the tangent and exponent arithmetic, which makes it difficult to implement.

We introduce a variable step size method for the numerical approximation of pathwise solutions to stochastic differential equations (SDEs). The method, which is dependent on a representation of Brownian paths as binary trees, involves estimation of local errors and of their contribution to the global error. We advocate controlling the variance of the one-step errors, conditional on knowledge of the Brownian path, in such a way that after propagation along the trajectory the error over each step will provide an equal contribution to the variance of the global error. Discretization schemes can be chosen that reduce the mean local error so that it is negligible beside the standard deviation. We show that to obtain convergence of variable step size methods for SDEs, in general it is not sufficient to evaluate the Brownian path only at the points in time where one tries to approximate the solution. We prove that convergence of such methods is guaranteed if the Lévy area is approximated well enough by further subdivision of the Brownian path and the discretization scheme employed uses appropriately both the approximate Lévy areas and increments of the Brownian path.

Summary form only given. This paper presents a Variable Step Size (VSS) technique that is integrated with a Stop-And-Go (SAG) approach for blind adaptive equalization of Quadrature Amplitude Modulation (QAM) signals. The proposed technique is applied to the Constant-Modulus Algorithm (CMA) and the Multi-Modulus Algorithm (MMA). The SAG approach allows the VSS technique to either vary the step size or to hold the previous one depending on a specific error criterion. This leads to the reduction of the amount of computations involved Simulation results show that algorithms with the VSS technique result in less steady-state Mean-Squared Errors (MSEs) than those obtained for algorithms without the VSS technique for the same speed of convergence.

To attain good performance in an acoustic echo cancellation system, it is important to have a variable step size (VSS) algorithm as part of an adaptive filter. In this paper, we are concerned with the development of a VSS algorithm for a recently proposed subband affine projection (SAP) adaptive filter. Two popular VSS algorithms in the literature are the methods of delayed coefficients (DC) and variable regularization (VR). However, the merits and demerits of them are mutually exclusive. We propose a VSS algorithm* that is a hybrid of both methods and combines their advantages. An extensive study of the new algorithm in different scenarios like the presence double-talk (DT) during the transient phase of the adaptive filter, DT during steady state, and varying DT power is conducted and reasoning is given to support the observed behavior. The importance of the method of VR as part of a VSS algorithm is emphasized.

In conventional adaptive variable step size (VSS) maximum power point tracking (MPPT) algorithms, a scaling factor is utilized to determine the required perturbation step. However, the performance of the adaptive VSS MPPT algorithm is essentially decided by the choice of scaling factor. In this paper, a neural network assisted variable step size (VSS) incremental conductance (IncCond) MPPT method is proposed. The proposed method utilizes a neural network to obtain an optimal scaling factor that should be used in current irradiance level for the VSS IncCond MPPT method. Only two operating points on the characteristic curve are needed to acquire the optimal scaling factor. Hence, expensive irradiance and temperature sensors are not required. By adopting a proper scaling factor, the performance of the conventional VSS IncCond method can be improved, especially under rapid varying irradiance conditions. To validate the studied algorithm, a 400 W prototyping circuit is built and experiments are carried out accordingly. Comparing with perturb and observe (P&O), α-P&O, golden section and conventional VSS IncCond MPPT methods, the proposed method can improve the tracking loss by 95.58%, 42.51%, 93.66%, and 66.14% under EN50530 testing condition, respectively.

System identification has wide applications in adaptive echo/feedback cancellation, active noise control, adaptive channel equalization, etc. When the system is sparse, recently, a μ-law memorised improved proportionate affine projection algorithm (MMIPAPA) has been proposed to improve the misalignment performance remarkably. However, the MMIPAPA with constant step-size has the conflicting requirement of fast convergence rate and low steady-state error. To solve this problem, a variable step-size version of the MMIPAPA (VSS-MMIPAPA) has been extended by setting each component of the a posterior error energy vector equal the system noise energy. Furthermore, through an alternative method to compute the variable step-size, when the estimated component of the a prior error energy vector is smaller than the system noise energy, further lower steady-state misalignment is achieved. This method leads to an improved VSS-MMIPAPA (IVSS-MMIPAPA). The computational complexity of the IVSS-MMIPAPA is O(P2L), which may ...

The subject of variable step size maximum power point tracking (MPPT) algorithms has been addressed in the literature. However, most of the addressed algorithms tune the variable step size according to two variables: the photovoltaic (PV) array voltage (<TEX>$V_{PV}$</TEX>) and the PV array current (<TEX>$I_{PV}$</TEX>). Therefore, both the PV array current and voltage have to be measured. Recently, maximum power point trackers that arc based on a single variable (<TEX>$I_{PV}$</TEX> or <TEX>$V_{PV}$</TEX>) have received a great deal of attention due to their simplicity and ease of implementation, when compared to other tracking techniques. In this paper, two methods have been proposed to design a variable step size MPPT algorithm using only a single current sensor for stand-alone battery storage PV systems. These methods utilize only the relationship between the PV array measured current and the converter duty cycle (D) to automatically adapt the step change in the duty cycle to reach the maximum power point (MPP) of the PV array. Detailed analyses and flowcharts of the proposed methods are included. Moreover, a comparison has been made between the proposed methods to investigate their performance in the transient and steady states. Finally, experimental results with field programmable gate arrays (FPGAs) are presented to verify the performance of the proposed methods.

Variable step size does not harm second-order integrators for Hamiltonian systems

In this article, a modified Steiglitz–McBride (SM) structure with variable step size is proposed for aircraft feedforward control to reduce lateral landing deviation in the presence of crosswind disturbance. Currently, most of the aircraft landing studies focus on the trajectory tracking problems in the longitudinal plane and seldom focus on lateral disturbances. This article develops a modified SM structure–based feedforward control system to online estimate both the primary path and the control path (secondary path), where a fuzzy logic–based variable step size strategy is implemented to ensure fast convergence rate and strong robustness under complex crosswind scenarios. As a consequence, the lateral deviation during crosswind landing could be largely reduced as fast as possible while the stability of the flight control system is maintained. A Boeing 747 model is used as a test bed, and the simulations are carried out on two different crosswind conditions to demonstrate the feasibility and effectiveness of the proposed method.

To address the challenge of local optimization and step size factor parameter setting in the full attraction model of the firefly algorithm (FA), this paper introduces level-based attraction and variable step size to FA. In the level-based attraction model, fireflies are firstly grouped in different levels according to the brightness, and each firefly randomly selects two fireflies from a higher level to learn from. By using the variable step size strategy, the searching step size decreases with the number of iterations. Herein, the level-based attracting model can increase the diversity of individual learning and improve the ability to jump out of local optimization. Meanwhile, dynamic adjustment of step size can balance the detection and development ability of the algorithm and improve the optimization accuracy of the algorithm. To evaluate the effectiveness of the proposed algorithm, comparisons are drawn against optimized FA, particle swarm optimization algorithm, artificial bee colony algorithm and differential evolution algorithm on a variety of test function sets.

Aiming at the low computational efficiency of the fixed-step solution of the standard parabolic equation (SPE) in the study of complex environmental radio wave propagation, a variable step size technology for the standard parabolic equation is proposed. Based on the relationship between the factors such as step size and frequency with the calculation error in the SSFT solution, the basic criteria for the SPE variable step size solution are proposed; secondly, the complex environment of SPE application is graded, and the corresponding step size is selected for different complex environment levels under the condition of satisfying the error requirement. Finally, the method is used to simulate the radio wave characteristics of typical complex environment. The calculation results verify the reliability and efficiency of the proposed method. It shows that the proposed method can greatly improve the computational efficiency of parabolic equations under the condition of ensuring the accuracy of calculation.

An adaptive system is a well-known method for removing noise from noise-corrupted speech. The problem arises in many situations such as airplane cockpits and automobiles. The variable step size normalized least mean squares (VSS-NLMS) algorithm incorporating a variable step size parameter whose values are based on the ratio of signal-to-noise power has very good convergence speed and low steady-state misadjustment. This paper investigates the rate of convergence by the wavelet based LMS (WLMS) algorithm to the noise speech problem. Simulation results are presented to compare the performance of the variable step size NLMS algorithm, the wavelet based LMS Algorithm, and other commonly used fixed step size LMS algorithms.

A variable step size total least squares affine-projection-like algorithm: Formula derivation and performance analysis

This work presents a low-complexity high-performance motion estimation method using a one-dimensional (1-D) gradient descent search. The proposed method consists of initial-point determination, initial-direction determination, a gradient descent search using variable step sizes and conjugate directions, and convergence checking. For each block, one of the original and predictive points with the best block-matching performance is the initial point where the predictive ones are motion vectors of its neighboring four searched blocks. In addition, the searching path is optimized by determining the initial direction based on the vector from the original point to the initial point. After determining the initial point and direction, the gradient descent search using variable step sizes and conjugate directions is performed until the searched point attains a better performance than its neighboring points. Moreover, variable step sizes for moving and slowly moving, or stationary blocks, are well addressed to improve the convergence performance of a gradient descent search. Simulation results demonstrate that our method yields a superior performance in terms of computational complexity and picture quality as compared to the three-step search, 1-D full search, block-based gradient descent search, and one-at-a-time search methods.