Bifurcation in Singularity Parameterized ODEs

In this paper we will study differential algebraic equations (DAEs) through studying singularly perturbed ODEs. That's the ODEs will be transformed to an DAEs when the perturbed parameter approach to 0. This will permit us to apply the classicalbifurcation theory of ODEs for the new system (DAEs). So we will show by giving theorems, sufficient conditions for fold, pitchfork and transcritical bifurcation to be occurred in (DAEs). An illustrative example is given.

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Seminonlinear boundary value problems for nondegenerate differential-algebraic system

V N Karazin Kharkiv National University Ser Mathematics Applied Mathematics and Mechanics ◽

10.26565/2221-5646-2019-89-02 ◽

2019 ◽

Cited By ~ 1

Keyword(s):

Boundary Value Problem ◽

Boundary Value Problems ◽

Algebraic System ◽

Boundary Value ◽

Sufficient Conditions ◽

Differential Algebraic Equations ◽

Problem Solution ◽

Algebraic Equations ◽

Arbitrary Continuous Function ◽

Differential Algebraic System

In the article we obtained sufficient conditions of the existence of the nonlinear Noetherian boundary value problem solution for the system of differential-algebraic equations which are widely used in mechanics, economics, electrical engineering, and control theory. We studied the case of the nondegenerate system of differential algebraic equations, namely: the differential algebraic system that is solvable relatively to the derivative. In this case, the nonlinear system of differential algebraic equations is reduced to the system of ordinary differential equations with an arbitrary continuous function. The studied nonlinear differential-algebraic boundary-value problem in the article generalizes the numerous statements of the non-linear non-Gath boundary value problems considered in the monographs of А.М. Samoilenko, E.A. Grebenikov, Yu.A. Ryabov, A.A. Boichuk and S.M. Chuiko, and the obtained results can be carried over matrix boundary value problems for differential-algebraic systems. The obtained results in the article of the study of differential-algebraic boundary value problems, in contrast to the works of S. Kempbell, V.F. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and A.A. Boychuk, do not involve the use of the central canonical form, as well as perfect pairs and triples of matrices. To construct solutions of the considered boundary value problem, we proposed the iterative scheme using the method of simple iterations. The proposed solvability conditions and the scheme for finding solutions of the nonlinear Noetherian differential-algebraic boundary value problem, were illustrated with an example. To assess the accuracy of the found approximations to the solution of the nonlinear differential-algebraic boundary value problem, we found the residuals of the obtained approximations in the original equation. We also note that obtained approximations to the solution of the nonlinear differential-algebraic boundary value problem exactly satisfy the boundary condition.

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On the connection between nonlinear differential-algebraic equations and singularly perturbed control systems in nonstandard form

IEEE Transactions on Automatic Control ◽

10.1109/9.284898 ◽

1994 ◽

Vol 39 (5) ◽

pp. 1079-1084 ◽

Cited By ~ 14

Author(s):

H. Krishnan ◽

N.H. McClamroch

Keyword(s):

Control Systems ◽

Differential Algebraic Equations ◽

Singularly Perturbed ◽

Algebraic Equations

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On the connection between nonlinear differential-algebraic equations and singularly perturbed control systems in nonstandard form

Proceedings of 32nd IEEE Conference on Decision and Control ◽

10.1109/cdc.1993.325019 ◽

2002 ◽

Author(s):

H. Krishnan ◽

H. McClamroch

Keyword(s):

Control Systems ◽

Differential Algebraic Equations ◽

Singularly Perturbed ◽

Algebraic Equations

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Non-standard singularly perturbed control systems and differential-algebraic equations

International Journal of Control ◽

10.1080/00207179208934281 ◽

1992 ◽

Vol 55 (5) ◽

pp. 1239-1253 ◽

Cited By ~ 3

Author(s):

N. HARRIS McCLAMROCH ◽

HARIHARAN KRISHNAN

Keyword(s):

Control Systems ◽

Differential Algebraic Equations ◽

Singularly Perturbed ◽

Algebraic Equations

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Control of Closed Kinematic Chains Using A Singularly Perturbed Dynamics Model

Journal of Dynamic Systems Measurement and Control ◽

10.1115/1.2171440 ◽

2005 ◽

Vol 128 (1) ◽

pp. 142-151 ◽

Cited By ~ 17

Author(s):

Zhiyong Wang ◽

Fathi H. Ghorbel

Keyword(s):

Kinematic Chain ◽

Perturbation Parameter ◽

Parallel Robot ◽

Differential Algebraic Equations ◽

Singularly Perturbed ◽

Algebraic Equations ◽

Kinematic Chains ◽

Novel Approach ◽

Lyapunov Function Method ◽

Perturbed Model

In this paper, we propose a novel approach to the control of closed kinematic chains (CKCs). This method is based on a recently developed singularly perturbed model for CKCs. Conventionally, the dynamics of CKCs are described by differential-algebraic equations (DAEs). Our approach transfers the control of the original DAE system to the control of an artificially created singularly perturbed system in which the slow dynamics corresponds to the original DAE when the perturbation parameter tends to zero. Compared to control schemes that rely on solving nonlinear algebraic constraint equations, the proposed method uses an ordinary differential equation (ODE) solver to obtain the dependent coordinates, hence, eliminates the need for Newton-type iterations and is amenable to real-time implementation. The composite Lyapunov function method is used to show that the closed-loop system, when controlled by typical open kinematic chain schemes, achieves asymptotic trajectory tracking. Simulations and experimental results on a parallel robot, the Rice planar Delta robot, are also presented to illustrate the efficacy of our method.

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