Multiple Solutions for a Class of Continuous Nonlinear Systems

1986 ◽  
Vol 108 (2) ◽  
pp. 96-105 ◽  
Author(s):  
A. J. Fish
Keyword(s):  
Nonlinear System ◽  
Nonlinear Systems ◽  
State Space ◽  
Unique Solution ◽  
Multiple Solutions ◽  
Initial Conditions ◽  
Algebraic Theory ◽  
Space Representation ◽  
State Space Representation ◽  
Nonlinear State

A nonlinear state space representation of a nonlinear system implies that the system has a unique solution for a given set of initial conditions and system controls. But not all nonlinear systems have unique solutions for a given set of initial conditions and system controls. This paper presents an algebraic theory for solving continuous nonlinear systems equations that have multiple solutions.

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.

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