A Method to Find Non-Zero Operating Point Volterra Models for Port-Based Ordinary Differential Equations

2010 ◽  
Author(s):  
Eliot Motato ◽  
Clark Radcliffe ◽  
Jose Luis Viveros
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Transfer Functions ◽  
Operating Conditions ◽  
Operating Point ◽  
Input Output ◽  
Nonlinear Functions ◽  
Nonlinear Odes ◽  
Physical Systems ◽  
Volterra Models

Nonlinear physical systems frequently perform around constant non-zero input-output operating conditions. This local behavior can be modeled using port-based nonlinear ordinary differential equations (ODEs). An ODE local solution around an specific input-output operating point can be obtained through the Volterra transfer function (VTF) model. In a past work a procedure for obtaining MIMO Volterra models from port-based nonlinear ODEs was presented. This previous work considered only systems operating at zero input-output conditions subject to linear inputs. In this work the process for obtaining MIMO Volterra transfer functions is extended for systems operating at non-zero input-output conditions. This extension also allows systems that are nonlinear functions of their inputs and input derivatives.

scholarly journals Passive identification of multivariable stabilization system elements’ dynamics

2020 ◽  
Vol 12 (1) ◽  
pp. 32-40
Author(s):  
S. I. Osadchy ◽  
V. A. Zozulia
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Transfer Functions ◽  
Control Object ◽  
Operating Conditions ◽  
Necessary Condition ◽  
Identification Algorithm ◽  
Stabilization System ◽  
Spectral Densities ◽  
Design Work

Increasing domestic automatic control systems competitiveness requires the design work intensification based on the implementation of modern information technologies. Such implementation necessary condition is based on the control object dynamic model's improvement and real operating perturbations characteristics estimates' development with the help of the latest identification methods design and usage. Objective - to create an algorithm for structural identifying the controller transfer functions matrix, the control object's system of ordinary differential equations and the perturbations spectral densities matrix. The algorithm initial data  include stabilization system control signals and it's output signals obtained in standard operating conditions.  Method - minimization of an identification errors weighted dispersions sum by the Wiener – Kolmogorov’s method.  Equations that allow one to calculate matrices of the controller's transfer functions and of the perturbation’s spectral densities, as well as to find a control object's ordinary differential equations system that minimize the identification error’s variance have been found. Processing results of the stabilization system’s work simulation based on these equations proves their correctness and illustrates the methodology of applying the new identification algorithm.  Application of the new identification algorithm is limited by the following conditions fulfillment: signals in the stabilization circuit belong to a set of centered stationary random processes; measurement noises operate at the controller input and are independent from perturbations, the sensors transfer matrix is known.

scholarly journals Enhancing Accuracy of Runge–Kutta-Type Collocation Methods for Solving ODEs

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 174
Author(s):  
Janez Urevc ◽  
Miroslav Halilovič
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Integration ◽  
Integral Form ◽  
Collocation Methods ◽  
Nonlinear Odes ◽  
New Class ◽  
New Methods ◽  
Runge Kutta

In this paper, a new class of Runge–Kutta-type collocation methods for the numerical integration of ordinary differential equations (ODEs) is presented. Its derivation is based on the integral form of the differential equation. The approach enables enhancing the accuracy of the established collocation Runge–Kutta methods while retaining the same number of stages. We demonstrate that, with the proposed approach, the Gauss–Legendre and Lobatto IIIA methods can be derived and that their accuracy can be improved for the same number of method coefficients. We expressed the methods in the form of tables similar to Butcher tableaus. The performance of the new methods is investigated on some well-known stiff, oscillatory, and nonlinear ODEs from the literature.

A Novel General and Robust Method Based on NAOP for Solving Nonlinear Ordinary Differential Equations and Partial Differential Equations by Cellular Neural Networks

2013 ◽  
Vol 135 (3) ◽  
Author(s):  
Jean Chamberlain Chedjou ◽  
Kyandoghere Kyamakya
Keyword(s):  
Neural Networks ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Cellular Neural Networks ◽  
Nonlinear Ordinary Differential Equations ◽  
Van Der Pol ◽  
Partial Differential ◽  
Nonlinear Odes ◽  
First Order

This paper develops and validates through a series of presentable examples, a comprehensive high-precision, and ultrafast computing concept for solving nonlinear ordinary differential equations (ODEs) and partial differential equations (PDEs) with cellular neural networks (CNN). The core of this concept is a straightforward scheme that we call "nonlinear adaptive optimization (NAOP),” which is used for a precise template calculation for solving nonlinear ODEs and PDEs through CNN processors. One of the key contributions of this work is to demonstrate the possibility of transforming different types of nonlinearities displayed by various classical and well-known nonlinear equations (e.g., van der Pol-, Rayleigh-, Duffing-, Rössler-, Lorenz-, and Jerk-equations, just to name a few) unto first-order CNN elementary cells, and thereby enabling the easy derivation of corresponding CNN templates. Furthermore, in the case of PDE solving, the same concept also allows a mapping unto first-order CNN cells while considering one or even more nonlinear terms of the Taylor's series expansion generally used in the transformation of a PDE in a set of coupled nonlinear ODEs. Therefore, the concept of this paper does significantly contribute to the consolidation of CNN as a universal and ultrafast solver of nonlinear ODEs and/or PDEs. This clearly enables a CNN-based, real-time, ultraprecise, and low-cost computational engineering. As proof of concept, two examples of well-known ODEs are considered namely a second-order linear ODE and a second order nonlinear ODE of the van der Pol type. For each of these ODEs, the corresponding precise CNN templates are derived and are used to deduce the expected solutions. An implementation of the concept developed is possible even on embedded digital platforms (e.g., field programmable gate array (FPGA), digital signal processor (DSP), graphics processing unit (GPU), etc.). This opens a broad range of applications. Ongoing works (as outlook) are using NAOP for deriving precise templates for a selected set of practically interesting ODEs and PDEs equation models such as Lorenz-, Rössler-, Navier Stokes-, Schrödinger-, Maxwell-, etc.

An Experimental Determination of Differential Equations to Describe Simple Nonlinear Systems

1967 ◽  
Vol 89 (2) ◽  
pp. 393-398 ◽  
Author(s):  
L. L. Hoberock ◽  
R. H. Kohr
Keyword(s):  
Differential Equations ◽  
Continuous Functions ◽  
Nonlinear Functions ◽  
Physical Systems ◽  
Lumped Parameter ◽  
System Input ◽  
System Variables ◽  
Time Invariant ◽  
By Products

A method is presented for the determination of ordinary differential equations to describe the performance of existing lumped-parameter, time-invariant, nonlinear physical systems. It is assumed initially that the nonlinear elements can be described by products of continuous functions of system variables and these system variables themselves, which consist of the input and output of the system and their time derivatives. It is also assumed that the system input may be specified and that the output can be measured. The method yields graphical representations of unknown nonlinear functions in an assumed system differential equation. Examples illustrating the accuracy of the procedure are presented, and results obtained in the identification of two physical systems are given.

scholarly journals On the complete integrability and linearization of nonlinear ordinary differential equations. IV. Coupled second-order equations

2008 ◽  
Vol 465 (2102) ◽  
pp. 609-629 ◽  
Author(s):  
V.K Chandrasekar ◽  
M Senthilvelan ◽  
M Lakshmanan
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Nonlinear Differential Equations ◽  
Second Order ◽  
Nonlinear Ordinary Differential Equations ◽  
Integrals Of Motion ◽  
Nonlinear Odes ◽  
Second Order Equations ◽  
Integrating Factors

Coupled second-order nonlinear differential equations are of fundamental importance in dynamics. In this part of our study on the integrability and linearization of nonlinear ordinary differential equations (ODEs), we focus our attention on the method of deriving a general solution for two coupled second-order nonlinear ODEs through the extended Prelle–Singer procedure. We describe a procedure to obtain integrating factors and the required number of integrals of motion so that the general solution follows straightforwardly from these integrals. Our method tackles both isotropic and non-isotropic cases in a systematic way. In addition to the above-mentioned method, we introduce a new method of transforming coupled second-order nonlinear ODEs into uncoupled ones. We illustrate the theory with potentially important examples.

Hyperasymptotics for nonlinear ODEs I. A Riccati equation

2005 ◽  
Vol 461 (2060) ◽  
pp. 2503-2520 ◽  
Author(s):  
A.B Olde Daalhuis
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Riccati Equation ◽  
Asymptotic Expansions ◽  
Riemann Sheet ◽  
Nonlinear Ordinary Differential Equations ◽  
Numerical Illustration ◽  
Nonlinear Odes ◽  
Borel Transform ◽  
Sheet Structure

We illustrate how one can obtain hyperasymptotic expansions for solutions of nonlinear ordinary differential equations. The example is a Riccati equation. The main tools that we need are transseries expansions and the Riemann sheet structure of the Borel transform of the divergent asymptotic expansions. Hyperasymptotic expansions determine the solutions uniquely. A numerical illustration is included.

scholarly journals Quantum homotopy perturbation method for nonlinear dissipative ordinary differential equations

2021 ◽  
Author(s):  
Cheng Xue ◽  
Wu Yu-Chun ◽  
GuoPing Guo
Keyword(s):  
Differential Equations ◽  
Perturbation Method ◽  
Ordinary Differential Equations ◽  
Homotopy Perturbation Method ◽  
Quantum Algorithms ◽  
Nonlinear Differential Equations ◽  
Success Probability ◽  
Nonlinear Odes ◽  
Homotopy Perturbation ◽  
Linear Odes

Abstract While quantum computing provides an exponential advantage in solving linear differential equations, there are relatively few quantum algorithms for solving nonlinear differential equations. In our work, based on the homotopy perturbation method, we propose a quantum algorithm for solving n-dimensional nonlinear dissipative ordinary differential equations (ODEs). Our algorithm first converts the original nonlinear ODEs into other nonlinear ODEs which can be embedded into finite-dimensional linear ODEs. Then we solve the embedded linear ODEs with quantum linear ODEs algorithm and obtain a state ε-close to the normalized exact solution of the original nonlinear ODEs with success probability Ω(1). The complexity of our algorithm is O(gηTpoly(log(nT/ε))), where η, g measure the decay of the solution. Our algorithm provides exponential improvement over the best classical algorithms or previous quantum algorithms in n or ε.

scholarly journals Explicitly solvable systems of two autonomous first-order Ordinary Differential Equations with homogeneous quadratic right-hand sides

2021 ◽  
Author(s):  
Francesco Calogero ◽  
Farrin Payandeh
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
A Priori ◽  
Nonlinear Odes ◽  
First Order ◽  
Right Hand ◽  
Algebraic Constraints ◽  
Special Case

After tersely reviewing the various meanings that can be given to the property of a system of nonlinear ODEs to be solvable, we identify a special case of the system of two first-order ODEs with homogeneous quadratic right-hand sides which is explicitly solvable. It is identified by 2 explicit algebraic constraints on the 6 a priori arbitrary parameters that characterize this system. Simple extensions of this model to cases with nonhomogeneous quadratic right-hand sides are also identified, including isochronous cases

scholarly journals On the complete integrability and linearization of nonlinear ordinary differential equations. III. Coupled first-order equations

2008 ◽  
Vol 465 (2102) ◽  
pp. 585-608 ◽  
Author(s):  
V.K Chandrasekar ◽  
M Senthilvelan ◽  
M Lakshmanan
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Autonomous Systems ◽  
Three Dimensional ◽  
Complete Integrability ◽  
Nonlinear Ordinary Differential Equations ◽  
Nonlinear Odes ◽  
First Order ◽  
Open Questions ◽  
Motion Time

Continuing our study on the complete integrability of nonlinear ordinary differential equations (ODEs), in this paper we consider the integrability of a system of coupled first-order nonlinear ODEs of both autonomous and non-autonomous types. For this purpose, we modify the original Prelle–Singer (PS) procedure so as to apply it to both autonomous and non-autonomous systems of coupled first-order ODEs. We briefly explain the method of finding integrals of motion (time-independent as well as time-dependent integrals) for two and three coupled first-order ODEs by extending the PS method. From this we try to answer some of the open questions in the original PS method. We also identify integrable cases for the two-dimensional Lotka–Volterra system and three-dimensional Rössler system as well as other examples including non-autonomous systems in a straightforward way using this procedure. Finally, we develop a linearization procedure for coupled first-order ODEs.

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