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 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.

scholarly journals Experimental Assessment of Fractional-Order PDD1/2 Control of a Brushless DC Motor with Inertial Load

Actuators ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 13
Author(s):  
Luca Bruzzone ◽  
Pietro Fanghella ◽  
Mario Baggetta
Keyword(s):  
Fractional Order ◽  
Position Control ◽  
Experimental Tests ◽  
Research Area ◽  
Digital Implementation ◽  
Integer Order ◽  
Brushless Dc ◽  
Brushless Motor ◽  
Control Schemes ◽  
Derivative Term

The application of Fractional Calculus to control mechatronic devices is a promising research area. The most common approach to Fractional-Order (FO) control design is the PIλDµ scheme, which adopts integrals and derivatives of non-integer order λ and µ. A different possible approach is to add FO terms to the PID control, instead of replacing integer order terms; for example, in the PDD1/2 scheme, the half-derivative term is added to the classical PD. In the present paper, by mainly focusing on the transitory behaviour, a comparison among PD, PDµ, and PDD1/2 control schemes is carried out, with reference to a real-world mechatronic implementation: a position-controlled rotor actuated by a DC brushless motor. While using a general non-dimensional approach, the three control schemes are first compared by continuous-time simulations, and tuning criteria are outlined. Afterwards, the effects of the discrete-time digital implementation of the controllers are investigated by both simulation and experimental tests. The results show how PDD1/2 is an effective and almost cost-free solution for improving the trajectory-tracking performance in position control of mechatronic devices, with limited computational burden and, consequently, easily implementable on most commercial motion control drives.

An enhanced whale optimized fractional order piλ controller based power quality improvement for sensorless brushless dc motor under dynamic operating conditions

2021 ◽  
pp. 1-20
Author(s):  
R Prakash ◽  
K Ayyar
Keyword(s):  
Power Quality ◽  
Fractional Order ◽  
Optimization Algorithm ◽  
Speed Control ◽  
Optimization Techniques ◽  
Operating Conditions ◽  
The Other ◽  
Brushless Dc ◽  
Whale Optimization ◽  
Dynamic Operating

This paper presents an Enhanced Whale Optimization Algorithm (EWO) approach for tuning to perfection of Fractional Order Proportional Integral and integral order Controller (FOPI λ ) is used to sensorless speed control of permanent magnet Brushless DC (PMBLDC) motor under the operating dynamic condition such as (i) speed change by set speed command signal (ii) varying load conditions, (iii) integrated conditions and (iv) controller parameters uncertainty. On the other hand, it deals with a reduced THD (Total Harmonic Distortion) under dynamic operating conditions to improve the power quality for the above control system. Here present are three optimization techniques, namely (i) Enhanced Whale Optimization (EWO), (ii) Invasive Weed Optimization (IWO), and (iii) Social Spider Optimization (SSO) for fine-tuning of the FOPI λ controller parameters with reduction of THD. The proposed optimization algorithm optimized FOPI λ controller are compared under various BLDC motor operating conditions. Based on the results of MATLAB/Simulink models, the proposed algorithms are evaluated. Here, both the simulation and the results of the experiments are validated for the proposed controller technique. It demonstrates that the effectiveness of the proposed controllers is completely validated by comparing the three intelligent optimization techniques mentioned above. The EWO optimized FOPI λ controller for speed control of sensorless PMBLDC motor clearly outperforms the other two intelligent controllers by minimizing the time domain parameters, THD, performance Indices error, convergence time, control efforts, cost function, mean and standard deviation.

scholarly journals A New No Equilibrium Fractional Order Chaotic System, Dynamical Investigation, Synchronization, and Its Digital Implementation

Inventions ◽  
2021 ◽  
Vol 6 (3) ◽  
pp. 49
Author(s):  
Zain-Aldeen S. A. Rahman ◽  
Basil H. Jasim ◽  
Yasir I. A. Al-Yasir ◽  
Raed A. Abd-Alhameed ◽  
Bilal Naji Alhasnawi
Keyword(s):  
Adaptive Control ◽  
Fractional Order ◽  
Chaotic System ◽  
Real World ◽  
Experimental Results ◽  
Chaotic Attractors ◽  
State Variables ◽  
Digital Implementation ◽  
Dynamical Behaviors ◽  
Real World Applications

In this paper, a new fractional order chaotic system without equilibrium is proposed, analytically and numerically investigated, and numerically and experimentally tested. The analytical and numerical investigations were used to describe the system’s dynamical behaviors including the system equilibria, the chaotic attractors, the bifurcation diagrams, and the Lyapunov exponents. Based on the obtained dynamical behaviors, the system can excite hidden chaotic attractors since it has no equilibrium. Then, a synchronization mechanism based on the adaptive control theory was developed between two identical new systems (master and slave). The adaptive control laws are derived based on synchronization error dynamics of the state variables for the master and slave. Consequently, the update laws of the slave parameters are obtained, where the slave parameters are assumed to be uncertain and are estimated corresponding to the master parameters by the synchronization process. Furthermore, Arduino Due boards were used to implement the proposed system in order to demonstrate its practicality in real-world applications. The simulation experimental results were obtained by MATLAB and the Arduino Due boards, respectively, with a good consistency between the simulation results and the experimental results, indicating that the new fractional order chaotic system is capable of being employed in real-world applications.

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.

scholarly journals Approximate analytical fractional view of convection–diffusion equations

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 897-905
Author(s):  
Hassan Khan ◽  
Saima Mustafa ◽  
Izaz Ali ◽  
Poom Kumam ◽  
Dumitru Baleanu ◽  
...  
Keyword(s):  
Fractional Order ◽  
Variational Iteration Method ◽  
Absolute Error ◽  
Form Solution ◽  
Diffusion Equations ◽  
Nonlinear Convection ◽  
Convection Diffusion ◽  
Suggested Technique ◽  
First Time ◽  
Modified Variational Iteration Method

Abstract In this article, a modified variational iteration method along with Laplace transformation is used for obtaining the solution of fractional-order nonlinear convection–diffusion equations (CDEs). The proposed technique is applied for the first time to solve fractional-order nonlinear CDEs and attain a series-form solution with the quick rate of convergence. Tabular and graphical representations are presented to confirm the reliability of the suggested technique. The solutions are calculated for fractional as well as for integer orders of the problems. The solution graphs of the solutions at various fractional derivatives are plotted. The accuracy is measured in terms of absolute error. The higher degree of accuracy is observed from the table and figures. It is further investigated that fractional solutions have the convergence behavior toward the solution at integer order. The applicability of the present technique is verified by illustrative examples. The simple and effective procedure of the current technique supports its implementation to solve other nonlinear fractional problems in different areas of applied science.

scholarly journals Optimal Fractional PID Controller for Buck Converter Using Cohort Intelligent Algorithm

2021 ◽  
Vol 4 (3) ◽  
pp. 50
Author(s):  
Preeti Warrier ◽  
Pritesh Shah
Keyword(s):  
Fractional Order ◽  
Pid Controller ◽  
Power Converters ◽  
Buck Converter ◽  
Optimization Techniques ◽  
Superior Performance ◽  
Computational Time ◽  
Optimization Approach ◽  
Response Characteristics ◽  
Fractional Order Controller

The control of power converters is difficult due to their non-linear nature and, hence, the quest for smart and efficient controllers is continuous and ongoing. Fractional-order controllers have demonstrated superior performance in power electronic systems in recent years. However, it is a challenge to attain optimal parameters of the fractional-order controller for such types of systems. This article describes the optimal design of a fractional order PID (FOPID) controller for a buck converter using the cohort intelligence (CI) optimization approach. The CI is an artificial intelligence-based socio-inspired meta-heuristic algorithm, which has been inspired by the behavior of a group of candidates called a cohort. The FOPID controller parameters are designed for the minimization of various performance indices, with more emphasis on the integral squared error (ISE) performance index. The FOPID controller shows faster transient and dynamic response characteristics in comparison to the conventional PID controller. Comparison of the proposed method with different optimization techniques like the GA, PSO, ABC, and SA shows good results in lesser computational time. Hence the CI method can be effectively used for the optimal tuning of FOPID controllers, as it gives comparable results to other optimization algorithms at a much faster rate. Such controllers can be optimized for multiple objectives and used in the control of various power converters giving rise to more efficient systems catering to the Industry 4.0 standards.

PSO and GA tuned conventional and fractional order PID controllers for quadrotor control

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Hasan Saribas ◽  
Sinem Kahvecioglu
Keyword(s):  
Fractional Order ◽  
Pid Controller ◽  
Controller Design ◽  
Optimization Problems ◽  
Parameter Tuning ◽  
Absolute Error ◽  
Cost Functions ◽  
Pid Controllers ◽  
Content Type ◽  
Quadrotor Control

Purpose This study aims to compare the performance of the conventional and fractional order proportional-integral-derivative (PID and FOPID) controllers tuned with a particle swarm optimization (PSO) and genetic algorithm (GA) for quadrotor control. Design/methodology/approach In this study, the gains of the controllers were tuned using PSO and GA, which are included in the heuristic optimization methods. The tuning processes of the controller’s gains were formulated as optimization problems. While generating the objective functions (cost functions), four different decision criteria were considered separately: integrated summation error (ISE), integrated absolute error, integrated time absolute error and integrated time summation error (ITSE). Findings According to the simulation results and comparison tables that were created, FOPID controllers tuned with PSO performed better performances than PID controllers. In addition, the ITSE criterion returned better results in control of all axes except for altitude control when compared to the other cost functions. In the control of altitude with the PID controller, the ISE criterion showed better performance. Originality/value While a conventional PID controller has three parameters (Kp, Ki, Kd) that need to be tuned, FOPID controllers have two additional parameters (µ). The inclusion of these two extra parameters means more flexibility in the controller design but much more complexity for parameter tuning. This study reveals the potential and effectiveness of PSO and GA in tuning the controller despite the increased number of parameters and complexity.

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