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.

The Dual State Variable Formulation as an Alternative to Perturbation Methods for the Analysis of Non-Linear Oscillators

2006 ◽  
Author(s):  
Alvin Post ◽  
Willem Stuiver
Keyword(s):  
State Space ◽  
Perturbation Methods ◽  
Duffing Oscillator ◽  
Van Der Pol Oscillator ◽  
Space Representation ◽  
Clear Understanding ◽  
Van Der Pol ◽  
State Space Representation ◽  
State Variable ◽  
Dimensional State Space

The "dual" state variable (DSV) formulation is a new way to represent ordinary differential equations. It is based on a framework that is consistent with the analysis of linear systems, and it allows the state space representation of a system to exhibit considerable symmetry. Its use in modeling requires a clear understanding of its unique four-dimensional state space, but it can be computationally simple. The DSV formulation has been successfully applied to model the nonlinear pendulum, the Duffing oscillator, and the van der Pol oscillator, with results that are superior to those of perturbation methods. An introduction to the DSV formulation and a framework for its systematic application as a modeling tool for nonlinear oscillators are presented.

scholarly journals State-space representation of a bucket-type rainfall-runoff model: a case study with State-Space GR4 (version 1.0)

2017 ◽  
Author(s):  
Léonard Santos ◽  
Guillaume Thirel ◽  
Charles Perrin
Keyword(s):  
Water Balance ◽  
Differential Equations ◽  
State Space ◽  
Operator Splitting ◽  
Space Representation ◽  
Rainfall Runoff ◽  
State Space Representation ◽  
Rainfall Runoff Model ◽  
Unit Hydrographs ◽  
Runoff Model

Abstract. In many conceptual rainfall-runoff models, the water balance differential equations are not explicitly formulated. These differential equations are solved sequentially by splitting the equations into terms that can be solved analytically with a technique called "operator splitting". As a result, only the resolutions of the split equations are used to present the different models. This article provides a methodology to make the governing water balance equations of a bucket-type rainfall-runoff model explicit. This is done by setting up a comprehensive state-space representation of the model. By representing it in this way, the operator splitting, which complexifies the structural analysis of the model, is removed. In this state-space representation, the lag functions (unit hydrographs), which are frequent in this type of model and make the resolution of the representation difficult, are replaced by a so-called "Nash cascade". This substitution also improves the lag parameter consistency across time steps. To illustrate this methodology, the GR4J model is taken as an example. The flow time series simulated by the new representation of the model are very similar to those simulated by the classic model. The state-space representation provides a more time-consistent model with time-independent parameters.

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.

scholarly journals Visualizing the geometry of state space in plane Couette flow

2008 ◽  
Vol 611 ◽  
pp. 107-130 ◽  
Author(s):  
J. F. GIBSON ◽  
J. HALCROW ◽  
P. CVITANOVIĆ
Keyword(s):  
State Space ◽  
Couette Flow ◽  
Invariant Manifolds ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Space Representation ◽  
Plane Couette Flow ◽  
Periodic Cell ◽  
State Space Representation ◽  
Dimensional State Space

Motivated by recent experimental and numerical studies of coherent structures in wall-bounded shear flows, we initiate a systematic exploration of the hierarchy of unstable invariant solutions of the Navier–Stokes equations. We construct a dynamical 105-dimensional state-space representation of plane Couette flow at Reynolds number Re = 400 in a small periodic cell and offer a new method of visualizing invariant manifolds embedded in such high dimensions. We compute a new equilibrium solution of plane Couette flow and the leading eigenvalues and eigenfunctions of known equilibria at this Re and cell size. What emerges from global continuations of their unstable manifolds is a surprisingly elegant dynamical-systems visualization of moderate-Re turbulence. The invariant manifolds partially tessellate the region of state space explored by transiently turbulent dynamics with a rigid web of symmetry-induced heteroclinic connections.

scholarly journals Modeling and simulation aspects of AC machines

2016 ◽  
Vol 65 (2) ◽  
pp. 315-326 ◽  
Author(s):  
Michael Popp ◽  
Patrick Laza ◽  
Wolfgang Mathis
Keyword(s):  
Complex System ◽  
Differential Equations ◽  
Modeling And Simulation ◽  
Ordinary Differential Equations ◽  
State Space ◽  
Space Representation ◽  
Ac Machines ◽  
State Space Representation ◽  
Drive Systems ◽  
Algebraic Constraints

Abstract In the field of power and drive systems, electrical AC machines are mostly modeled using a set of explicit ordinary differential equations in a state space representation. It is shown, that by using other equation types for simulation, algebraic constraints arising from aggregating several machines to a more complex system can directly be considered. The effects of different model variants on numerical ODE/DAE solvers are investigated in the focus of this work in order perform efficient simulations of larger systems possessing electrical AC machines.

The Error Incurred in Using the Caputo-Derivative Laplace-Transform

2009 ◽  
Author(s):  
Tom T. Hartley ◽  
Carl F. Lorenzo
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Caputo Derivative ◽  
Fractional Order Differential Equations ◽  
Definition Of

This paper considers the initialization of fractional-order differential equations. The initialization responses obtained using the Caputo derivative are compared with the exact initialization responses from the Riemann-Liouville definition of the fractional derivative. The error incurred in using the Caputo derivative for initialization problems in fractionalorder differential equations is presented.

Construction of Differential Equations from Experimental Data

1987 ◽  
Vol 42 (8) ◽  
pp. 797-802 ◽  
Author(s):  
J. Cremers ◽  
A. Hübler
Keyword(s):  
Experimental Data ◽  
Differential Equations ◽  
State Space ◽  
Dynamic Systems ◽  
Degrees Of Freedom ◽  
Time Sequence ◽  
Space Representation ◽  
State Space Representation ◽  
Concise Description

A new algorithm to determine the number of degrees of freedom of dynamic systems is presented. To obtain a concise description of an observed chaotic time sequence, an approximation of the flow in a state space representation by series is shown to be useful.

Sign in / Sign up

Export Citation Format

Share Document