Duality of variable fractional order difference operators and its application in identification

Circuits Syst Signal Process

et al. 2016

The paper is devoted to variable order estimation process when measurements are obtained in two different ways: directly and by lossy network. Since the problem of fractional order estimation is highly nonlinear, dual estimation algorithm based on Unscented Fractional Order Kalman filter has been used. In dual estimation process, state variable and order estimation have been divided into two sub-processes. For estimation state variables and variable fractional order, the Fractional Kalman filter and the Unscented Fractional Kalman filter have been used, respectively. The order estimation algorithms were applied to numerical examples and to real fractional variable order inertial system realized as an analog circuit.

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Section: Discrete Variable Fractional Order State-space Systemmentioning

confidence: 99%

“…The solution of the system given by the A-type definition has the structure of D-type definition [32], namely…”

Section: Discrete Variable Fractional Order State-space Systemmentioning

confidence: 99%

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Dual Estimation of Fractional Variable Order Based on the Unscented Fractional Order Kalman Filter for Direct and Networked Measurements

Sierociuk

Circuits Syst Signal Process

et al. 2016

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“….4, are already defined in [18 -type definition. Moreover, these definitions have mutual duality properties described in [24], which can be adapted to solve the fractional variable order differential equations (see [21]). …”

Section: Definitions Of Variable Order Operatorsmentioning

confidence: 99%

“…In [23], the heat transess in specific grid-holes media whose geometry is in time was modeled by a new D-type definition. er, these definitions posses mutual duality properties d in [24], which can be adapt to solve the fractional order differential equations (see [21]). …”

Section: Time-variant Control Systemmentioning

confidence: 99%

Numerical solution of fractional variable order linear control system in state-space form

Bulletin of the Polish Academy of Sciences Technical Sciences

2017

Abstract. The aim of this paper is to introduce a matrix approach for approximate solving of non-commensurate fractional variable order linear control systems in state-space form. The approach is based on switching schemes that realize variable order derivatives. The obtained numerical solution is compared with simulation and analog model results. In our paper, a method of finding a numerical solution of fractional variable order control system in a state-space form is introduced, both for time-invariant and time-variant case. Moreover, the obtained results are also valid for system of differential equations with different types of variable order derivatives. To validate our approach the fractional variable order state-space system was physically built and the experimental results were compared with numerical implementations.The paper is organized as follows. At the beginning, in Section 2, the few types of fractional variable order derivatives are recalled, together with their discrete approximations and matrix forms. In Section 3 the solution of linear control system in statespace form for time-variant and time-invariant non-commensurate fractional variable order system is presented. An analog model of particular type of fractional variable order state-space system is introduced in Section 4. The experimental and numerical results are presented in Section 5. Finally, Section 6 summarizes the main results. Fractional variable order operatorsBelow, we recall the already known different types of fractional constant and variable order derivatives and differences. In our paper, a method of finding a numerical solution of actional variable order control system in a state-space form introduced, both for time-invariant as well as time-variant der state-space system was physically build and the experimental results were compared with numerical implementations. The paper is organized as follows. At the beginning, in Sect. 2, the few types of fractional variable order derivatives are recalled, together with their discrete approximations and matrix forms. In Sect. 3 the solution of linear control system in state-space form for time-variant and time-invariant noncommensurate fractional variable order system is presented. An analog model of particular type of fractional variable order state-space system is introduced in Sect. 4. The experimental and numerical results are collected in Sect. 5. Finally, Sect. 6 summarizes the main results. Fractional variable order operatorsBelow, we recall the already known different types of fractional constant and variable order derivatives and differences. Definitions of variable order operatorsThe following fractional constant order difference of Grünwald-Letnikov type will be used as a base of generalization onto variable orderwhere α ∈ R, l = 0, . . . , k, and h > 0 is a sample time. We will consider the following four types of fractional variable order derivatives and their discrete approximations (differences). We admit the order is changing in time, i.e., α(t) ∈ R for t > 0; and in ...

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

Lecture Notes in Electrical Engineering

Sierociuk

2015