scholarly journals Consistent Approximation of Fractional Order Operators

2021 ◽  
Author(s):  
YiHeng Wei ◽  
Yangquan Chen ◽  
Yingdong Wei ◽  
Xuefeng Zhang
Keyword(s):  
Fractional Order ◽  
Approximation Error ◽  
Approximate Model ◽  
Model Parameters ◽  
High Approximation ◽  
Numerical Approximations ◽  
Approximation Accuracy ◽  
Consistent Approximation ◽  
Approximate Models ◽  
Multiple Orders

Abstract Fractional order controllers become increasingly popular due to their versatility and superiority in various performance. However, the bottleneck in deploying these tools in practice is related to their implementation. Numerical approximations are usually employed in which the approximation of fractional differintegrator is a foundation. Generally, the following three identical equations always hold, i.e., $\frac{1}{s^\alpha}\frac{1}{s^{1-\alpha}} = \frac{1}{s}$, $s^\alpha \frac{1}{s^\alpha} = 1$ and $s^\alpha s^{1-\alpha} = s$. However, for the approximate models of fractional differintegrator $s^\alpha$, $\alpha\in(-1,0)\cup(0,1)$, there usually exist some conflicts on the mentioned equations, which might enlarge the approximation error or even cause fallacious in multiple orders occasion. To overcome the conflicts, this brief develops a piecewise approximate model and provides two procedures for designing the model parameters. The comparison with several existing methods shows that the proposed methods do not only satisfy the equalities but also achieve high approximation accuracy. From this, it is believed that this work can serve for simulation and realization of fractional order controllers more friendly.

scholarly journals Fractional-Order Logistic Differential Equation with Mittag–Leffler-Type Kernel

2021 ◽  
Vol 5 (4) ◽  
pp. 273
Author(s):  
Iván Area ◽  
Juan J. Nieto
Keyword(s):  
Differential Equation ◽  
Power Series ◽  
Differential Equations ◽  
Formal Power Series ◽  
Fractional Order ◽  
Numerical Approximations ◽  
Formal Power

In this paper, we consider the Prabhakar fractional logistic differential equation. By using appropriate limit relations, we recover some other logistic differential equations, giving representations of each solution in terms of a formal power series. Some numerical approximations are implemented by using truncated series.

scholarly journals Adaptive Neural Network Sliding Mode Control for Quad Tilt Rotor Aircraft

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
Yanchao Yin ◽  
Hongwei Niu ◽  
Xiaobao Liu
Keyword(s):  
Neural Network ◽  
Sliding Mode Control ◽  
Sliding Mode ◽  
Approximation Error ◽  
Model Parameters ◽  
Control Law ◽  
The Novel ◽  
Adaptive Neural Network ◽  
Mode Control ◽  
Attitude Tracking

A novel neural network sliding mode control based on multicommunity bidirectional drive collaborative search algorithm (M-CBDCS) is proposed to design a flight controller for performing the attitude tracking control of a quad tilt rotors aircraft (QTRA). Firstly, the attitude dynamic model of the QTRA concerning propeller tension, channel arm, and moment of inertia is formulated, and the equivalent sliding mode control law is stated. Secondly, an adaptive control algorithm is presented to eliminate the approximation error, where a radial basis function (RBF) neural network is used to online regulate the equivalent sliding mode control law, and the novel M-CBDCS algorithm is developed to uniformly update the unknown neural network weights and essential model parameters adaptively. The nonlinear approximation error is obtained and serves as a novel leakage term in the adaptations to guarantee the sliding surface convergence and eliminate the chattering phenomenon, which benefit the overall attitude control performance for QTRA. Finally, the appropriate comparisons among the novel adaptive neural network sliding mode control, the classical neural network sliding mode control, and the dynamic inverse PID control are examined, and comparative simulations are included to verify the efficacy of the proposed control method.

Design of an integer order proportional–integral/proportional–integral–derivative controller based on model parameters of a certain class of fractional order systems

2018 ◽  
Vol 233 (3) ◽  
pp. 320-334 ◽  
Author(s):  
Erhan Yumuk ◽  
Müjde Güzelkaya ◽  
İbrahim Eksin
Keyword(s):  
Fractional Order ◽  
Design Parameter ◽  
Design Methodology ◽  
Model Parameters ◽  
Proportional Integral Derivative ◽  
Level Control ◽  
Proportional Integral Derivative Controller ◽  
Integer Order ◽  
Proportional Integral ◽  
Order Pole

In this study, we deal with systems that can be represented by single fractional order pole models and propose an integer order proportional–integral/proportional–integral–derivative controller design methodology for this class. The basic principle or backbone of the design methodology of the proposed controller relies on using the inverse of the fractional model and then approximating this fractional controller transfer function by a low integer order model using Oustaloup filter. The emerging integer order controller reveals itself either in pre-filtered proportional–integral or proportional–integral–derivative form by emphasizing on the dominancy concept of pole-zero configuration. Parameters of the proposed controllers depend on the parameters of the single fractional order pole model and the only free design parameter left is the overall controller gain. This free design parameter is determined via some approximating functions relying on an optimization procedure. Simulation results show that the proposed controller exhibits either satisfactory or better results with respect to some performance indices and time domain criteria when they are compared to classical integer order proportional–integral–derivative and fractional order proportional–integral–derivative controllers. Moreover, the proposed controller is applied to real-time liquid level control system. The application results show that the proposed controller outperforms the other controllers.

INTEGRATED UNSUPERVISED AND ERROR-DRIVEN LEARNING FOR INCREMENTAL PROCESS IDENTIFICATION WITH NEURAL SINGLETON MODELS

2004 ◽  
Vol 04 (03) ◽  
pp. 267-282
Author(s):  
GANCHO VACHKOV
Keyword(s):  
Learning Algorithm ◽  
Approximation Error ◽  
Integrated Learning ◽  
Identification Procedure ◽  
Approximation Accuracy ◽  
Feedback Information ◽  
Incremental Model ◽  
Comparison Results ◽  
Algorithm Comparison ◽  
Data Density

In the paper a special incremental identification procedure for learning a Neural Singleton (NS) Model, as a simplification of the RBFNN model is proposed. At each epoch of this incremental procedure a special integrated learning algorithm for the neuron centers and then a supervised learning for the singletons are performed. As a result, the approximation error of the NS model is gradually decreased with the increment of the neurons at each learning epoch. The newly proposed integrated unsupervised and Error-Driven learning algorithm is the main part of the incremental identification procedure. It is a modified version of the Neural-Gas learning algorithm, which uses the normalised evaluation error as feedback information. Thus the integrated learning algorithm is able to drive the neurons to the "more interesting" areas in the input space, where bigger evaluation errors exist, instead of just locating them in the areas of higher data density. The final result is an incremental growing type of NS model that gradually improves its approximation accuracy at each learning epoch. A detailed test example is shown in the paper in order to evaluate the performance and to show the important features of the whole incremental identification procedure, including the integrated learning algorithm. Comparison results with the conventional (one-epoch and non-incremental) model are also given.

Fast 2-D ray+Born migration/inversion in complex media

Geophysics ◽  
1999 ◽  
Vol 64 (1) ◽  
pp. 162-181 ◽  
Author(s):  
Philippe Thierry ◽  
Stéphane Operto ◽  
Gilles Lambaré
Keyword(s):  
Numerical Approximation ◽  
Reference Model ◽  
Real Data ◽  
Model Parameters ◽  
Complex Media ◽  
Inversion Algorithm ◽  
Numerical Approximations ◽  
Data Set ◽  
Cpu Time ◽  
Computational Evaluation

In this paper, we evaluate the capacity of a fast 2-D ray+Born migration/inversion algorithm to recover the true amplitude of the model parameters in 2-D complex media. The method is based on a quasi‐Newtonian linearized inversion of the scattered wavefield. Asymptotic Green’s functions are computed in a smooth reference model with a dynamic ray tracing based on the wavefront construction method. The model is described by velocity perturbations associated with diffractor points. Both the first traveltime and the strongest arrivals can be inverted. The algorithm is implemented with several numerical approximations such as interpolations and aperture limitation around common midpoints to speed the algorithm. Both theoritical and numerical aspects of the algorithm are assessed with three synthetic and real data examples including the 2-D Marmousi example. Comparison between logs extracted from the exact Marmousi perturbation model and the computed images shows that the amplitude of the velocity perturbations are recovered accurately in the regions of the model where the ray field is single valued. In the presence of caustics, neither the first traveltime nor the most energetic arrival inversion allow for a full recovery of the amplitudes although the latter improves the results. We conclude that all the arrivals associated with multipathing through transmission caustics must be taken into account if the true amplitude of the perturbations is to be found. Only 22 minutes of CPU time is required to migrate the full 2-D Marmousi data set on a Sun SPARC 20 workstation. The amplitude loss induced by the numerical approximations on the first traveltime and the most energetic migrated images are evaluated quantitatively and do not exceed 8% of the energy of the image computed without numerical approximation. Computational evaluation shows that extension to a 3-D ray+Born migration/inversion algorithm is realistic.

CHAOS IN FRACTIONAL-ORDER POPULATION MODEL

2012 ◽  
Vol 22 (04) ◽  
pp. 1250072 ◽  
Author(s):  
IVO PETRÁŠ
Keyword(s):  
Stability Analysis ◽  
Numerical Solution ◽  
Fractional Order ◽  
Population Model ◽  
Model Parameters ◽  
New Model

This paper presents a fractional-order population model which consists of the two-predators and one-prey scheme. For this new model, the numerical solution is derived and the simulations are performed for various sets of model parameters together with stability analysis for commensurate and incomensurate orders of the fractional-order population model. The results obtained via the simulations show that chaos can be observed in such population model.

scholarly journals Approximate Models of Microbiological Processes in a Biofilm Formed on Fine Spherical Particles

Processes ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 48
Author(s):  
Szymon Skoneczny ◽  
Monika Cioch-Skoneczny
Keyword(s):  
Method Of Lines ◽  
Approximate Model ◽  
Spherical Particles ◽  
Biofilm Growth ◽  
Biofilm Reactors ◽  
Microbiological Processes ◽  
Approximate Models ◽  
The Method Of Lines ◽  
Homotopy Analysis ◽  
Air Lift

This paper concerns the dynamical modeling of the microbiological processes that occur in the biofilms that are formed on fine inert particles. Such biofilm forms e.g. in fluidized-bed bio-reactors, expanded bed biofilm reactors and biofilm air-lift suspension reactors. An approximate model that is based on the Laplace–Carson transform and a family of approximate models that are based on the concept of the pseudo-stationary substrate concentration profile in the biofilm were proposed. The applicability of the models to the microbiological processes was evaluated following Monod or Haldane kinetics in the conditions of dynamical biofilm growth. The use of approximate models significantly simplifies the computations compared to the exact one. Moreover, the stiffness that was present in the exact model, which was solved numerically by the method of lines, was eliminated. Good accuracy was obtained even for large internal mass transfer resistances in the biofilm. It was shown that significantly higher accuracy was obtained using one of the proposed models than that which was obtained using the previously published approximate model that was derived using the homotopy analysis method.

scholarly journals On Negative Binomial Approximation to k-Runs

2008 ◽  
Vol 45 (2) ◽  
pp. 456-471 ◽  
Author(s):  
Xiaoxin Wang ◽  
Aihua Xia
Keyword(s):  
Negative Binomial ◽  
Practical Interest ◽  
Poisson Approximation ◽  
Approximate Model ◽  
Approximation Accuracy ◽  
Compound Poisson Approximation ◽  
Variation Distance ◽  
Binomial Approximation ◽  
Negative Binomial Approximation ◽  
Accuracy Of Approximation

The distributions of the run occurrences for a sequence of independent and identically distributed (i.i.d.) experiments are usually obtained by combinatorial methods (see Balakrishnan and Koutras (2002, Chapter 5)) and the resulting formulae are often very tedious, while the distributions for non i.i.d. experiments are generally intractable. It is therefore of practical interest to find a suitable approximate model with reasonable approximation accuracy. In this paper we demonstrate that the negative binomial distribution is the most suitable approximate model for the number of k-runs: it outperforms the Poisson approximation, the general compound Poisson approximation as observed in Eichelsbacher and Roos (1999), and the translated Poisson approximation in Rollin (2005). In particular, its accuracy of approximation in terms of the total variation distance improves when the number of experiments increases, in the same way as the normal approximation improves in the Berry-Esseen theorem.

scholarly journals A New Double Truncated Generalized Gamma Model with Some Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-27
Author(s):  
Awad A. Bakery ◽  
Wael Zakaria ◽  
OM Kalthum S. K. Mohamed
Keyword(s):  
Likelihood Function ◽  
Survival Function ◽  
Gaussian Model ◽  
Simulated Data ◽  
Weibull Model ◽  
Approximate Model ◽  
Model Parameters ◽  
Data Sets ◽  
Gamma Model ◽  
Generalized Gamma

The generalized Gamma model has been applied in a variety of research fields, including reliability engineering and lifetime analysis. Indeed, we know that, from the above, it is unbounded. Data have a bounded service area in a variety of applications. A new five-parameter bounded generalized Gamma model, the bounded Weibull model with four parameters, the bounded Gamma model with four parameters, the bounded generalized Gaussian model with three parameters, the bounded exponential model with three parameters, and the bounded Rayleigh model with two parameters, is presented in this paper as a special case. This approach to the problem, which utilizes a bounded support area, allows for a great deal of versatility in fitting various shapes of observed data. Numerous properties of the proposed distribution have been deduced, including explicit expressions for the moments, quantiles, mode, moment generating function, mean variance, mean residual lifespan, and entropies, skewness, kurtosis, hazard function, survival function, r   th order statistic, and median distributions. The delivery has hazard frequencies that are monotonically increasing or declining, bathtub-shaped, or upside-down bathtub-shaped. We use the Newton Raphson approach to approximate model parameters that increase the log-likelihood function and some of the parameters have a closed iterative structure. Six actual data sets and six simulated data sets were tested to demonstrate how the proposed model works in reality. We illustrate why the Model is more stable and less affected by sample size. Additionally, the suggested model for wavelet histogram fitting of images and sounds is very accurate.

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