Computational Scheme of a Center Manifold for Neutral Functional Differential Equations

Systems & Control Letters

Morărescu

Niculescu

2015

International audienceThe paper considers the problem of stabilization of systems possessing a multiple zero eigenvalue at the origin. The controller that we propose, uses multiple delayed measurements instead of derivative terms. Doing so, we increase the performances of the closed loop in presence of system uncertainties and/or noisy measurements. The problem formulation and the analysis is presented through a classical engineering problem which is the stabilization of an inverted pendulum on a cart moving horizontally. On one hand, we perform a nonlinear analysis of the center dynamics described by a three dimensional system of ordinary differential equations with a codimension-three triple zero bifurcation. On the other hand, we present the complementary stability analysis of the corresponding linear time invariant system with two delays describing the behavior around the equilibrium. The aim of this analysis is to characterize the possible local bifurcations. Finally, the proposed control scheme is numerically illustrated and discussed

Section: Normal Form Of the Central Dynamicsmentioning

confidence: 99%

Inverted pendulum stabilization: Characterization of codimension-three triple zero bifurcation via multiple delayed proportional gains

Systems & Control Letters

Morărescu

Niculescu

2015

“…In the third section, entitled Control of drilling vibrations, we present two controllers allowing to a stable drilling process. The methodological scheme described in Ait Babram et al [2001], Campbell [2009] is extended to the study of the parametrized model of neutral type. For the sake of self-containment, we report in the Appendix a table for the numerical settings for the parameters used in (1)-(2).…”

Section:  mentioning

confidence: 99%

Control of Drilling Vibrations: A Time-Delay System-Based Approach

IFAC Proceedings Volumes

Çela²,

Mounier

et al. 2013

The main purpose of this study is the control of both axial and torsional vibrations occurring along a rotary oilwell drilling system. This work completes a previous author's paper (Boussaada et al. [2012a]) which presents the description of the qualitative dynamical response of a rotary drilling system with a drag bit, using a model that takes into consideration the axial and the torsional vibration modes of the bit. The studied model, based on the interface bit-rock, contains a couple of wave equations with boundary conditions consisting of the angular speed and the axial speed at the top additionally to the angular and axial acceleration at the bit whose contain a realistic frictional torque. Our analysis is based on the center manifold theorem and normal forms theory whose allow us to simplify the model. By this way we design two control laws allowing to suppress the undesired vibrations guaranteeing a regular drilling process. * This work is published in the 11th IFAC Workshop on Time Delay Systems June 22-24, 2012 Grenoble, FRANCE

“…Definition 3: Given a C 1 map h from R 2 into Q. The graph of h is said to be a local manifold associated to (24) if h(0) = Dh(0) = 0 and there exists a neighborhood V of 0 ∈ R 2 such that for each ξ ∈ V , there exists δ = δ(ξ) > 0 and the solution x of (24) with initial data Φξ+h(ξ) exists on the interval ]−δ−r, δ[ and it is given by x t = Φy(t)+h(y(t)) for t ∈ [0, δ[ where y(t) is the unique solution of the ODĖ y = By + Ψ(0)F (Φy + h(y)) y(0) = ξ (30) For some particular cases of the matrix B analytic characterization of the center manifold was the subject of few studies, for instance in the case of Bogdanov-Takens singularity one can find the characterization of the function h in [1]. Note also that in the above definition it assumed that y ∈ R 2 , but it stills correct for y ∈ R m for instance we quote the case of a Double Hopf Bifurcation, the most important is to make a decomposition which separate between the eigenspace of all imaginary eigenvalues (real part equal to zero) and the space of the remaining spectral values which are assumed to have negative real parts.…”

Section: E Center Manifoldmentioning

confidence: 99%

“…where in equation (1) H is the brake motor control and α∂ t U (t, 0) represents a friction force of viscous type. For equation (2), the right hand side of the second equation designate the difference between the motor speed and rotational speed of the first pipe.…”

Section: Introductionmentioning

confidence: 99%

“…In the third section, entitled Dynamics Analysis, we establish linear stability analysis and apply bifurcation results for Bogdanov-Takens singularity. The methodological scheme described in [1], [4] is extended to the study of the parametrized model of neutral type. For the sake of selfcontainment, we report in the Appendix, a table for the numerical settings for the parameters used in (1)-(2), then we present elements of the qualitative theory of differential equations, entitled Model reduction, which comprises the outlines of a methodology enabling to approximate a system of NDDE by a system of ordinary differential equations (Center Variety) and then to study local bifurcations (Normal Forms).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Analysis of drilling vibrations: A time-delay system approach

2012 20th Mediterranean Conference on Control &Amp; Automation (MED)

Mounier

Niculescu

et al. 2012

The main purpose of this study is the description of the qualitative dynamical response of a rotary drilling system with a drag bit, using a model that takes into consideration the axial and the torsional vibration modes of the bit. The studied model, based on the interface bit-rock, contains a couple of wave equations with boundary conditions consisting of the angular speed and the axial speed at the top additionally to the angular and axial acceleration at the bit whose contain a realistic frictional torque. Our analysis is based on the center manifold Theorem and Normal forms theory whose allow us to simplify the model.