State-space representation for fractional order controllers

Automatica ◽  
2000 ◽  
Vol 36 (7) ◽  
pp. 1017-1021 ◽  
Author(s):  
H.-F. Raynaud ◽  
A. Zergaı̈noh
Keyword(s):  
State Space ◽  
Fractional Order ◽  
Space Representation ◽  
State Space Representation

Algebraic fractional order differentiator based on the pseudo-state space representation

2019 ◽  
Vol 22 (5) ◽  
pp. 1395-1413
Author(s):  
Xing Wei ◽  
Da-Yan Liu ◽  
Driss Boutat ◽  
Yi-Ming Chen
Keyword(s):  
Integral Equation ◽  
State Space ◽  
Fractional Order ◽  
Initial Conditions ◽  
Algebraic Method ◽  
Space Representation ◽  
State Space Representation ◽  
Algebraic Formula ◽  
Fractional Order Differentiator ◽  
Fractional Order Integral

Abstract The aim of this paper is to design an algebraic fractional order differentiator for a class of commensurate fractional order linear systems modeled by the pseudo-state space representation. For this purpose, a new algebraic method is introduced by designing an operator which can transform the considered system into a fractional order integral equation by eliminating unknown initial conditions. Based on the obtained equation, the desired fractional derivative is exactly given by a new algebraic formula using a recursive way. Then, a digital fractional order differentiator is introduced in discrete noisy cases. Finally, numerical results are given to illustrate the accuracy and the robustness of the proposed method.

Equivalence of History-Function Based and Infinite-Dimensional-State Initializations for Fractional-Order Operators

2013 ◽  
Vol 8 (4) ◽  
Author(s):  
Tom T. Hartley ◽  
Carl F. Lorenzo ◽  
Jean-Claude Trigeassou ◽  
Nezha Maamri
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
State Space ◽  
Fractional Order ◽  
Laplace Transforms ◽  
Space Representation ◽  
Infinite Dimensional ◽  
State Space Representation ◽  
Fractional Order Differential Equations ◽  
Dimensional State Space

Proper initialization of fractional-order operators has been an ongoing problem, particularly in the application of Laplace transforms with correct initialization terms. In the last few years, a history-function-based initialization along with its corresponding Laplace transform has been presented. Alternatively, an infinite-dimensional state-space representation along with its corresponding Laplace transform has also been presented. The purpose of this paper is to demonstrate that these two approaches to the initialization problem for fractional-order operators are equivalent and that the associated Laplace transforms yield the correct initialization terms and can be used in the solution of fractional-order differential equations.

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