The methods of dynamical reconstruction of an input in a system of ordinary differential equations

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Vyacheslav I. Maksimov
Keyword(s):  
Feedback Control ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Reconstruction Algorithm ◽  
System A ◽  
Ill Posed ◽  
Computational Errors ◽  
Combination Of Methods ◽  
Input Reconstruction ◽  
Dynamical Reconstruction

AbstractIn the paper, for systems described by ordinary differential equations a review of algorithms of dynamical input reconstruction by results of inaccurate observations of its solutions is given. The problem under discussion is referred to the class of dynamical inverse problems. The proposed algorithms are stable with respect to informational noises and computational errors. They are based on the combination of methods of the theory of ill-posed problems and the theory of feedback control. The essence of the methodology underlying the algorithms suggested in the paper consists in the representation of a reconstruction algorithm in the form of a feedback control algorithm for a certain artificial dynamical system, a model; such an algorithm, whose output is the realization of the control in the model, is dynamical by its definition.

On dynamical input reconstruction in a distributed second order equation

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Yury S. Osipov ◽  
Vyacheslav I. Maksimov
Keyword(s):  
Differential Equation ◽  
Feedback Control ◽  
Control Theory ◽  
Nonlinear Differential Equation ◽  
Order Equation ◽  
Second Order ◽  
Ill Posed ◽  
Computational Errors ◽  
Input Reconstruction ◽  
Feedback Control Theory

Abstract A second order nonlinear differential equation is considered. An algorithm for reconstructing an input from inaccurate measurements of the solution at discrete times is designed. The algorithm based on the constructions of feedback control theory and theory of ill-posed problems is stable with respect to informational noises and computational errors.

scholarly journals Diagonally Implicit Runge–Kutta Type Method for Directly Solving Special Fourth-Order Ordinary Differential Equations with Ill-Posed Problem of a Beam on Elastic Foundation

Algorithms ◽  
2018 ◽  
Vol 12 (1) ◽  
pp. 10 ◽  
Author(s):  
Nizam Ghawadri ◽  
Norazak Senu ◽  
Firas Adel Fawzi ◽  
Fudziah Ismail ◽  
Zarina Ibrahim
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Elastic Foundation ◽  
Fourth Order ◽  
Sixth Order ◽  
Test Problems ◽  
Type Method ◽  
Ill Posed ◽  
Runge Kutta ◽  
Fifth Order

In this study, fifth-order and sixth-order diagonally implicit Runge–Kutta type (DIRKT) techniques for solving fourth-order ordinary differential equations (ODEs) are derived which are denoted as DIRKT5 and DIRKT6, respectively. The first method has three and the another one has four identical nonzero diagonal elements. A set of test problems are applied to validate the methods and numerical results showed that the proposed methods are more efficient in terms of accuracy and number of function evaluations compared to the existing implicit Runge–Kutta (RK) methods.

scholarly journals On attaining the prescribed quality of a controlled fourth order system

2014 ◽  
Vol 24 (1) ◽  
pp. 75-85 ◽  
Author(s):  
Vladimir Kapustyan ◽  
Vyacheslav Maksimov
Keyword(s):  
Differential Equations ◽  
Fourth Order ◽  
Nonlinear Differential Equations ◽  
Solution Algorithm ◽  
Order System ◽  
Control Problems ◽  
Control Methods ◽  
Computational Errors ◽  
Input Reconstruction

Abstract In this paper, we discuss a method of auxiliary controlled models and its application to solving some robust control problems for a system described by differential equations. As an illustration, a system of nonlinear differential equations of the fourth order is used. A solution algorithm, which is stable with respect to informational noise and computational errors, is presented. The algorithm is based on a combination of online state/input reconstruction and feedback control methods.

On dynamical reconstruction of an input in a linear system under measuring a part of coordinates

2018 ◽  
Vol 26 (3) ◽  
pp. 395-410 ◽  
Author(s):  
Vyacheslav I. Maksimov
Keyword(s):  
Linear System ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Unknown Input ◽  
Phase Coordinates ◽  
Guaranteed Control ◽  
Dynamical Reconstruction

AbstractThe problem of reconstructing an unknown input under measuring a part of phase coordinates of a system of ordinary differential equations is considered. We propose a solving algorithm that is stable to perturbations and is based on the combination of ideas from the theory of dynamical inversion and the theory of guaranteed control. The algorithm consists of two blocks: the block of dynamical reconstruction of unmeasured coordinates and the block of dynamical reconstruction of an input.

scholarly journals On dynamical identification of control in a system with time delay

2012 ◽  
Vol 22 (1) ◽  
pp. 5-15 ◽  
Author(s):  
Vyacheslav Maksimov
Keyword(s):  
Time Delay ◽  
Dynamical Systems ◽  
Feedback Control ◽  
Differential Equations ◽  
Computational Errors ◽  
System With Time Delay

On dynamical identification of control in a system with time delayThe problem of identification of a control through results of observations of phase states of dynamical systems described by differential equations with time delay is discussed. The paper presents an algorithm based on the method of feedback control with a model. The algorithm is stable with respect to informational noises and computational errors.

Existence of periodic orbits of autonomous ordinary differential equations

1980 ◽  
Vol 85 (1-2) ◽  
pp. 153-172 ◽  
Author(s):  
Russell A. Smith
Keyword(s):  
Feedback Control ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Periodic Orbits ◽  
Autonomous Systems ◽  
Higher Order ◽  
Control Equation ◽  
Recurrent Orbits

SynopsisThe Poincaré-Bendixson theorem, concerning the existence of periodic orbits of plane autonomous systems, is extended to higher order systems under certain conditions. Under similar conditions, a complementary theorem on the existence of recurrent orbits is also proved. For the feedback control equation, these conditions are reduced to a form which can be easily verified in practice.

scholarly journals On reconstructing unknown characteristics of a nonlinear system of differential equations

2015 ◽  
Vol 25 (2) ◽  
pp. 163-176
Author(s):  
Alexander Kuklin ◽  
Vyacheslav Maksimov ◽  
Natalia Nikulina
Keyword(s):  
Nonlinear System ◽  
Differential Equations ◽  
Nonlinear Equations ◽  
Diffusion Of Innovations ◽  
System Of Differential Equations ◽  
Adaptive Controls ◽  
Computational Errors ◽  
Dynamical Reconstruction

Abstract Problems of dynamical reconstruction of unknown characteristics for nonlinear equations described the process of diffusion of innovations through results of observations of phase states are considered. Solving algorithms, which are stable with respect to informational noises and computational errors, are designed. The algorithms are based on the principle of auxiliary models with adaptive controls.

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