The stability of variable step-size LMS algorithms

1999 ◽  
Vol 47 (12) ◽  
pp. 3277-3288 ◽  
Author(s):  
S.B. Gelfand ◽  
Yongbin Wei ◽  
J.V. Krogmeier
Keyword(s):  
Variable Step Size ◽  
Step Size ◽  
Variable Step ◽  
The Stability

Variable-step-size second-order-derivative multistep method for solving first-order ordinary differential equations in system simulation

2020 ◽  
Vol 11 (01) ◽  
pp. 2050001
Author(s):  
Lei Zhang ◽  
Chaofeng Zhang ◽  
Mengya Liu
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Multistep Method ◽  
Second Order ◽  
Order Derivative ◽  
Variable Step Size ◽  
Variable Order ◽  
Step Size ◽  
Variable Step ◽  
The Stability

According to the relationship between truncation error and step size of two implicit second-order-derivative multistep formulas based on Hermite interpolation polynomial, a variable-order and variable-step-size numerical method for solving differential equations is designed. The stability properties of the formulas are discussed and the stability regions are analyzed. The deduced methods are applied to a simulation problem. The results show that the numerical method can satisfy calculation accuracy, reduce the number of calculation steps and accelerate calculation speed.

scholarly journals Variable Step Size Adams Methods for BSDEs

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Qiang Han
Keyword(s):  
High Order ◽  
Variable Step Size ◽  
Necessary And Sufficient Condition ◽  
Step Size ◽  
Truncation Errors ◽  
Adams Methods ◽  
Variable Step ◽  
High Order Convergence ◽  
The Stability ◽  
Necessary And Sufficient

For backward stochastic differential equations (BSDEs), we construct variable step size Adams methods by means of Itô–Taylor expansion, and these schemes are nonlinear multistep schemes. It is deduced that the conditions of local truncation errors with respect to Y and Z reach high order. The coefficients in the numerical methods are inferred and bounded under appropriate conditions. A necessary and sufficient condition is given to judge the stability of our numerical schemes. Moreover, the high-order convergence of the schemes is rigorously proved. The numerical illustrations are provided.

A Class of Methods with Optimal Stability Properties for the Numerical Solution of IVPs: Construction and Implementation

2017 ◽  
Vol 14 (01) ◽  
pp. 1750007
Author(s):  
Masoumeh Hosseini Nasab ◽  
Gholamreza Hojjati ◽  
Ali Abdi
Keyword(s):  
Numerical Solution ◽  
Numerical Experiments ◽  
Point Of View ◽  
Variable Step Size ◽  
Second Derivative ◽  
Step Size ◽  
Stability Properties ◽  
Variable Step ◽  
The Stability ◽  
Size Mode

Considering the methods with future points technique from second derivative general linear methods (SGLMs) point of view, makes it possible to improve their stability properties. In this paper, we extend the stability regions of a modified version of E2BD formulas to optimal one and show its effectiveness by numerical verifications. Also, implementation issues, with numerical experiments, of these methods are investigated in a variable step-size mode.

scholarly journals A Variable Step-Size Exponentially Fitted Explicit Hybrid Method for Solving Oscillatory Problems

2011 ◽  
Vol 2011 ◽  
pp. 1-17 ◽  
Author(s):  
F. Samat ◽  
F. Ismail ◽  
M. B. Suleiman
Keyword(s):  
Hybrid Method ◽  
Numerical Experiments ◽  
Variable Step Size ◽  
Step Size ◽  
Linear Combinations ◽  
Variable Step ◽  
Three Stages ◽  
Last Stage ◽  
Exponentially Fitted ◽  
The Stability

An exponentially fitted explicit hybrid method for solving oscillatory problems is obtained. This method has four stages. The first three stages of the method integrate exactly differential systems whose solutions can be expressed as linear combinations of{1,x,exp(μx),exp(−μx)},μ∈C, while the last stage of this method integrates exactly systems whose solutions are linear combinations of{1,x,x2,x3,x4,exp(μx),exp(−μx)}. This method is implemented in variable step-size code basing on an embedding approach. The stability analysis is given. Numerical experiments that have been carried out show the efficiency of our method.

scholarly journals An Accurate Block Solver for Stiff Initial Value Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
H. Musa ◽  
M. B. Suleiman ◽  
F. Ismail ◽  
N. Senu ◽  
Z. B. Ibrahim
Keyword(s):  
Initial Value Problems ◽  
Variable Step Size ◽  
Initial Value ◽  
Step Size ◽  
Differentiation Formula ◽  
Stiff Initial Value Problems ◽  
Variable Step ◽  
Compute Solution ◽  
The Stability ◽  
Size Mode

New implicit block formulae that compute solution of stiff initial value problems at two points simultaneously are derived and implemented in a variable step size mode. The strategy for changing the step size for optimum performance involves halving, increasing by a multiple of 1.7, or maintaining the current step size. The stability analysis of the methods indicates their suitability for solving stiff problems. Numerical results are given and compared with some existing backward differentiation formula algorithms. The results indicate an improvement in terms of accuracy.

Adaptive feedforward control for crosswind landing with variable step size

2021 ◽  
pp. 095441002110054
Author(s):  
Qi Bian ◽  
Menghan Zhang ◽  
Jian Ma
Keyword(s):  
Control System ◽  
Flight Control ◽  
Feedforward Control ◽  
Variable Step Size ◽  
Test Bed ◽  
Step Size ◽  
Aircraft Landing ◽  
Variable Step ◽  
The Stability ◽  
Crosswind Landing

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.

A Variable Step-Size for Sparse Nonlinear Adaptive Filters

2021 ◽  
Author(s):  
Alberto Carini ◽  
Markus V. S. Lima ◽  
Hamed Yazdanpanah ◽  
Simone Orcioni ◽  
Stefania Cecchi
Keyword(s):  
Adaptive Filters ◽  
Variable Step Size ◽  
Step Size ◽  
Variable Step

Active control for vehicle interior noise using the improved iterative variable step-size and variable tap-length LMS algorithms

2019 ◽  
Vol 67 (6) ◽  
pp. 405-414 ◽  
Author(s):  
Ningning Liu ◽  
Yuedong Sun ◽  
Yansong Wang ◽  
Hui Guo ◽  
Bin Gao ◽  
...  
Keyword(s):  
Steady State ◽  
Noise Control ◽  
Convergence Speed ◽  
Interior Noise ◽  
Lms Algorithm ◽  
Variable Step Size ◽  
Step Size ◽  
Vehicle Interior ◽  
Vehicle Interior Noise ◽  
Variable Step

Active noise control (ANC) is used to reduce undesirable noise, particularly at low frequencies. There are many algorithms based on the least mean square (LMS) algorithm, such as the filtered-x LMS (FxLMS) algorithm, which have been widely used for ANC systems. However, the LMS algorithm cannot balance convergence speed and steady-state error due to the fixed step size and tap length. Accordingly, in this article, two improved LMS algorithms, namely, the iterative variable step-size LMS (IVS-LMS) and the variable tap-length LMS (VT-LMS), are proposed for active vehicle interior noise control. The interior noises of a sample vehicle are measured and thereby their frequency characteristics. Results show that the sound energy of noise is concentrated within a low-frequency range below 1000 Hz. The classical LMS, IVS-LMS and VT-LMS algorithms are applied to the measured noise signals. Results further suggest that the IVS-LMS and VT-LMS algorithms can better improve algorithmic performance for convergence speed and steady-state error compared with the classical LMS. The proposed algorithms could potentially be incorporated into other LMS-based algorithms (like the FxLMS) used in ANC systems for improving the ride comfort of a vehicle.

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