scholarly journals Optimization for Software Implementation of Fractional Calculus Numerical Methods in an Embedded System

Entropy ◽  
2020 ◽  
Vol 22 (5) ◽  
pp. 566
Author(s):  
Mariusz Matusiak
Keyword(s):  
Embedded System ◽  
Fractional Order ◽  
Numerical Algorithms ◽  
Computation Time ◽  
Optimization Methods ◽  
Optimization Techniques ◽  
Software Implementation ◽  
Software Optimization ◽  
Fractional Order Differential Equations ◽  
Non Locality

In this article, some practical software optimization methods for implementations of fractional order backward difference, sum, and differintegral operator based on Grünwald–Letnikov definition are presented. These numerical algorithms are of great interest in the context of the evaluation of fractional-order differential equations in embedded systems, due to their more convenient form compared to Caputo and Riemann–Liouville definitions or Laplace transforms, based on the discrete convolution operation. A well-known difficulty relates to the non-locality of the operator, implying continually increasing numbers of processed samples, which may reach the limits of available memory or lead to exceeding the desired computation time. In the study presented here, several promising software optimization techniques were analyzed and tested in the evaluation of the variable fractional-order backward difference and derivative on two different Arm® Cortex®-M architectures. Reductions in computation times of up to 75% and 87% were achieved compared to the initial implementation, depending on the type of Arm® core.

Balancing of Flexible Rotors Using Convex Optimization Techniques: Optimum Min-Max LMI Influence Coefficient Balancing

2008 ◽  
Vol 130 (2) ◽  
Author(s):  
Costin D. Untaroiu ◽  
Paul E. Allaire ◽  
William C. Foiles
Keyword(s):  
Convex Optimization ◽  
Optimization Theory ◽  
Numerical Algorithms ◽  
Optimization Methods ◽  
Industrial Applications ◽  
Optimization Techniques ◽  
Linear Matrix ◽  
Influence Coefficient ◽  
Flexible Rotors ◽  
Optimum Solution

In some industrial applications, influence coefficient balancing methods fail to find the optimum vibration reduction due to the limitations of the least-squares optimization methods. Previous min-max balancing methods have not included practical constraints often encountered in industrial balancing. In this paper, the influence coefficient balancing equations, with suitable constraints on the level of the residual vibrations and the magnitude of correction weights, are cast in linear matrix inequality (LMI) forms and solved with the numerical algorithms developed in convex optimization theory. The effectiveness and flexibility of the proposed method have been illustrated by solving two numerical balancing examples with complicated requirements. It is believed that the new methods developed in this work will help in reducing the time and cost of the original equipment manufacturer or field balancing procedures by finding an optimum solution of difficult balancing problems. The resulting method is called the optimum min-max LMI balancing method.

Research on Software Optimization Solutions of E-Commerce Site

2012 ◽  
Vol 198-199 ◽  
pp. 626-630
Author(s):  
Xin Wang
Keyword(s):  
Electronic Commerce ◽  
Dynamic Load ◽  
Performance Optimization ◽  
Optimization Methods ◽  
System Optimization ◽  
Optimization Techniques ◽  
Load Optimization ◽  
Software Optimization ◽  
Database Performance ◽  
Cluster Technology

There are generally two types of E-commerce platform optimized programs: hardware optimization and software optimization, This paper first analyzes the system optimization techniques of software optimization, Including dynamic load optimization technology and cluster technology; Then studies the database performance optimization methods from the table, connection pooling, query and several other aspects; Finally to carry on the research to optimization electronic commerce platform used the cache technology. Proposes a universal significance of E-commerce platform software optimization solutions, these studies have some references for relevant E-commerce website designers and maintainers, and provides a strategy for the corresponding E-commerce enterprises to optimize platform environments.

Sixth-Kind Chebyshev Spectral Approach for Solving Fractional Differential Equations

2019 ◽  
Vol 20 (2) ◽  
pp. 191-203 ◽  
Author(s):  
W. M. Abd-Elhameed ◽  
Y. H. Youssri
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Chebyshev Polynomials ◽  
Symmetric Functions ◽  
Numerical Algorithms ◽  
Initial Boundary ◽  
Algebraic Equations ◽  
Fractional Order Differential Equations ◽  
New Type ◽  
Connection Formulae

AbstractThe basic aim of this paper is to develop new numerical algorithms for solving some linear and nonlinear fractional-order differential equations. We have developed a new type of Chebyshev polynomials, namely, Chebyshev polynomials of sixth kind. This type of polynomials is a special class of symmetric orthogonal polynomials, involving four parameters that were constructed with the aid of the extended Sturm–Liouville theorem for symmetric functions. The proposed algorithms are basically built on reducing the fractional-order differential equations with their initial/boundary conditions to systems of algebraic equations which can be efficiently solved. The new proposed algorithms are supported by a detailed study of the convergence and error analysis of the sixth-kind Chebyshev expansion. New connection formulae between Chebyshev polynomials of the second and sixth kinds were established for this study. Some examples were displayed to illustrate the efficiency of the proposed algorithms compared to other methods in literature. The proposed algorithms have provided accurate results, even using few terms of the proposed expansion.

scholarly journals Optimization techniques for economic load dispatch problem in power plant

2021 ◽  
pp. 10-17
Keyword(s):  
Computation Time ◽  
Optimization Methods ◽  
Optimization Techniques ◽  
Economic Load Dispatch ◽  
Solution Quality ◽  
Hybrid Techniques ◽  
Power Stations ◽  
Programming Techniques ◽  
Single Method ◽  
Parameter Robustness

This paper presents survey of optimization techniques used to solve the problem of economic load dispatch in power stations. Since there is no single method available for solving all economic dispatch problems efficiently, thus a number of different optimization methods have been developed to solve this problem. These techniques were divided into three types depending on the efficiency of the solution: stochastic process techniques, statistical methods, and mathematical programming techniques which divided to local optimization, and global optimization. It is found that is better to use hybrid techniques to overcome load dispatch problems so as to achieve significant improvements in computation time, convergence properties, solution quality, or parameter robustness.

scholarly journals Optimal Digital Implementation of Fractional-Order Models in a Microcontroller

Entropy ◽  
2020 ◽  
Vol 22 (3) ◽  
pp. 366
Author(s):  
Mariusz Matusiak ◽  
Marcin Bąkała ◽  
Rafał Wojciechowski
Keyword(s):  
Fractional Order ◽  
Absolute Error ◽  
Optimization Techniques ◽  
Memory Storage ◽  
Digital Implementation ◽  
Software Optimization ◽  
Brushless Dc ◽  
Fractional Differintegral ◽  
Non Local ◽  
Short Memory

The growing number of operations in implementations of the non-local fractional differentiation operator is cumbersome for real applications with strict performance and memory storage requirements. This demands use of one of the available approximation methods. In this paper, the analysis of the classic integer- (IO) and fractional-order (FO) models of the brushless DC (BLDC) micromotor mounted on a steel rotating arms, and next, the discretization and efficient implementation of the models in a microcontroller (MCU) is performed. Two different methods for the FO model are examined, including the approximation of the fractional-order operator s ν ( ν ∈ R ) using the Oustaloup Recursive filter and the numerical evaluation of the fractional differintegral operator based on the Grünwald–Letnikov definition and Short Memory Principle. The models are verified against the results of several experiments conducted on an ARM Cortex-M7-based STM32F746ZG unit. Additionally, some software optimization techniques for the Cortex-M microcontroller family are discussed. The described steps are universal and can also be easily adapted to any other microcontroller. The values for integral absolute error (IAE) and integral square error (ISE) performance indices, calculated on the basis of simulations performed in MATLAB, are used to evaluate accuracy.

scholarly journals A numerical technique for a general form of nonlinear fractional-order differential equations with the linear functional argument

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Khalid K. Ali ◽  
Mohamed A. Abd El salam ◽  
Emad M. H. Mohamed
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Fractional Order ◽  
Linear Functional ◽  
Numerical Technique ◽  
Operational Matrix ◽  
Fractional Order Differential Equations ◽  
The Given ◽  
Functional Argument ◽  
Special Case

AbstractIn this paper, a numerical technique for a general form of nonlinear fractional-order differential equations with a linear functional argument using Chebyshev series is presented. The proposed equation with its linear functional argument represents a general form of delay and advanced nonlinear fractional-order differential equations. The spectral collocation method is extended to study this problem as a discretization scheme, where the fractional derivatives are defined in the Caputo sense. The collocation method transforms the given equation and conditions to algebraic nonlinear systems of equations with unknown Chebyshev coefficients. Additionally, we present a general form of the operational matrix for derivatives. A general form of the operational matrix to derivatives includes the fractional-order derivatives and the operational matrix of an ordinary derivative as a special case. To the best of our knowledge, there is no other work discussed this point. Numerical examples are given, and the obtained results show that the proposed method is very effective and convenient.

scholarly journals Central Tunicate Swarm NFOPID-Based Load Frequency Control of the Egyptian Power System Considering New Uncontrolled Wind and Photovoltaic Farms

Energies ◽  
2021 ◽  
Vol 14 (12) ◽  
pp. 3604
Author(s):  
Hady H. Fayek ◽  
Panos Kotsampopoulos
Keyword(s):  
Power System ◽  
Fractional Order ◽  
Frequency Control ◽  
Optimization Techniques ◽  
Load Frequency Control ◽  
The Novel ◽  
Operating Condition ◽  
Load Frequency ◽  
Non Linear ◽  
Fractional Order Pid

This paper presents load frequency control of the 2021 Egyptian power system, which consists of multi-source electrical power generation, namely, a gas and steam combined cycle, and hydro, wind and photovoltaic power stations. The simulation model includes five generating units considering physical constraints such as generation rate constraints (GRC) and the speed governor dead band. It is assumed that a centralized controller is located at the national control center to regulate the frequency of the grid. Four controllers are applied in this research: PID, fractional-order PID (FOPID), non-linear PID (NPID) and non-linear fractional-order PID (NFOPID), to control the system frequency. The design of each controller is conducted based on the novel tunicate swarm algorithm at each operating condition. The novel method is compared to other widely used optimization techniques. The results show that the tunicate swarm NFOPID controller leads the Egyptian power system to a better performance than the other control schemes. This research also presents a comparison between four methods to self-tune the NFOPID controller at each operating condition.

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