Michal Macias

Analog Modeling of Fractional Switched Order Derivative Using Different Switching Schemes

The paper presents comparison of two different switching schemes of variable order derivatives. The first one is additive-switching scheme, when change of order is caused by adding a derivative at the beginning of the system. The second one, introduced in this paper, is reductive-switching scheme, when change of order is caused by removing a derivative at the beginning of the system. For both methods numerical schemes are given and analyzed. Based on presented switching schemes results of analog modeling are presented. Results were obtained using analog approximations of integrators of orders 0.5 and 0.25. Finally, results of analog modeling were compared with numerical approach.

An alternative recursive fractional variable-order derivative definition and its analog validation

The paper presents an alternative recursive definition of fractional variable-order derivative (so-called ε-type) and its switching scheme. Nextly, an analog realization equivalent to this type of definition is introduced. The experimental results of switching fractional variable-order integrator are shown and compare to its numerical implementation.

Analog Modeling of Fractional Switched-Order Derivatives: Experimental Approach

The article presents experimental results of modeling switched-order integrators based on domino-ladder approximations of order 0.5 and 0.25. Results were obtained for increasing and decreasing the fractional order. As fractional order impedances, a half-order domino ladder impedance, and quarter-order domino ladder structure were used. The quarter-order impedance was implemented with using over 5000 discrete elements. The experimental circuits are based on switching scheme that is numerically identical to the second order type of fractional variable order derivative. Experimental results were analyzed and compared with numerical results.

Matrix Approach and Analog Modeling for Solving Fractional Variable Order Differential Equations

In the paper, a matrix approach for solving fractional variable order linear differential equations of an additive-switching type will be presented. Introduced method is based on a duality property between additive and recursive type of variable order differential definitions. Obtained solutions will be validated by comparing them with analog model results.

On a New Symmetric Fractional Variable Order Derivative

The paper presents particular definitions of symmetric fractional variable order derivatives. The \(\mathcal {AD}\) and \(\mathcal {DA}\) types of the fractional variable order derivatives and their properties have been introduced. Additionally, the switching order schemes equivalent to these types of definitions have been shown. Finally, the theoretical considerations have been validated on numerical examples.

Numerical schemes for initialized constant and variable fractional-order derivatives: matrix approach and its analog verification

This paper presents a method for including initial conditions in recursive constant and variable fractional-order derivatives. The initial conditions are assumed to be in the form of a time-constant function. The numerical scheme for solving fractional-order differential equations, given in the matrix form, is presented as well. Results obtained with the proposed numerical algorithm are compared with the analog realization of a fractional-order inertial system. For the constant-order case, analog models were built based on domino ladder approximations for 0.5 and 0.25 orders. For the variable-order case, the analog model was built based on the reductive-switching structure. Comparison with the physical systems shows the ability of the proposed method to describe the behavior of real fractional-order systems with initial conditions.

Experimental Results of Modeling Variable Order System Based on Discrete Fractional Variable Order State-Space Model

The paper presents experimental results of modelling fractional variable order system with using Discrete Fractional Variable Order State-Space Model. Experimental results were obtained based on modified multi-order switching analog realization for a case of constant parameter case, that is also introduced in this paper. During identification process two algorithms were used: direct and dual. Additionally, joint estimation results for parameter estimation were presented, in order to verification of constant parameter for proposed analog model.

On the Output-Additive Switching Strategy for a New Variable Type and Order Difference

The paper introduces definition of recursive fractional order difference for the case when type of variable order changing is varying in time. The equivalent switching strategies for this definition, which allow to better understand mechanism of type of variable order definition changing, are also given. Numerical results of comparison between given switching schemes and the definition are presented and analyzed.

Practical analog realization of multiple order switching for recursive fractional variable order derivative

In the paper, a novel practical analog modeling of multiple-switching fractional order systems based on recursive fractional variable order integrals, is introduced. Presented circuit, based on domino ladder approximation of half order impedance, allows to multiple-switching between orders 0.5 and 1. Moreover, numerical scheme for solving variable order differential equations is introduced. Experimental results of modeling are also presented and compared with numerical simulations.

Solution of fractional variable order differential equation

The aim of this paper is to introduce an approach for analytical solving fractional variable order linear differential equations. The approach is based on switching scheme that realizes variable order derivative. The obtained analytical solution is compared with numerical and analog model results.