TL;DR: In this article, a mixed-variable symplectic integrator is proposed to solve the problem of the potential energy term for the pair undergoing the encounter becoming comparable to the terms representing the unperturbed motion in the Hamiltonian, which can be overcome using a hybrid method in which the close encounter term is integrated using a conventional integrator, whilst the remaining terms are solved symplectically.

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Abstract: Mixed-variable symplectic integrators exhibit no long-term accumulation of energy error, beyond that owing to round-off, and they are substantially faster than conventional N-body algorithms. This makes them the integrator of choice for many problems in Solar system astronomy. However, in their original formulation, they become inaccurate whenever two bodies approach one another closely. This occurs because the potential energy term for the pair undergoing the encounter becomes comparable to the terms representing the unperturbed motion in the Hamiltonian. The problem can be overcome using a hybrid method, in which the close encounter term is integrated using a conventional integrator, whilst the remaining terms are solved symplectically. In addition, using a simple separable potential technique, the hybrid scheme can be made symplectic even though it incorporates a non-symplectic component.

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2,201 citations

Journal Article•10.1109/9.668834•

Continuous finite-time stabilization of the translational and rotational double integrators

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Sanjay P. Bhat¹, Dennis S. Bernstein²•Institutions (2)

Indian Institutes of Technology¹, University of Michigan²

01 May 1998-IEEE Transactions on Automatic Control

TL;DR: A class of bounded continuous time-invariant finite-time stabilizing feedback laws is given for the double integrator because Lyapunov theory is used to prove finite- time convergence.

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Abstract: A class of bounded continuous time-invariant finite-time stabilizing feedback laws is given for the double integrator. Lyapunov theory is used to prove finite-time convergence. For the rotational double integrator, these controllers are modified to obtain finite-time-stabilizing feedback that avoid "unwinding".

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1,664 citations

Proceedings Article•10.1109/PESC.2006.1711988•

A New Single-Phase PLL Structure Based on Second Order Generalized Integrator

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Mihai Ciobotaru¹, Remus Teodorescu¹, Frede Blaabjerg¹•Institutions (1)

Aalborg University¹

18 Jun 2006

TL;DR: In this article, a phase-locked-loop (PLL) method for single-phase systems was proposed to detect the phase angle, amplitude and frequency of the utility voltage.

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Abstract: Phase, amplitude and frequency of the utility voltage are critical information for the operation of the grid-connected inverter systems. In such applications, an accurate and fast detection of the phase angle, amplotude and frequency of the utility voltage is essential to assure the correct generation of the reference signals and to cope with the new upcoming standards. This paper presents a new phase-locked-loop (PLL) method for single-phase systems. The novelty consists in generating the orthogonal voltage system using a structure based on second order generalized integrator (SOGI). The proposed structure has the following advantages: — it has a simple implementation; — the generated orthogonal system is filtered without delay by the same structure due to its resonance at the fundamental frequency, — the proposed structure is not affected by the frequency changes. The solutions for the discrete implementation of the new proposed structure are also presented. Experimental results validate the effectiveness of the proposed method.

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1,259 citations

Journal Article•10.1016/J.NEUNET.2007.04.016•

2007 Special Issue: Optimization and applications of echo state networks with leaky- integrator neurons

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Herbert Jaeger¹, Mantas Lukoševičius¹, Dan Popovici¹, Udo Siewert²•Institutions (2)

Jacobs University Bremen¹, Residence Inn by Marriott²

01 Apr 2007-Neural Networks

TL;DR: Stability conditions are presented, a stochastic gradient descent method is introduced and a usefulness of leaky-integrator ESNs are demonstrated for learning very slow dynamic systems and replaying the learnt system at different speeds.

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955 citations

Journal Article•10.1063/1.1724823•

Accurate reaction paths using a Hessian based predictor–corrector integrator

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Hrant P. Hratchian¹, H. Bernhard Schlegel•Institutions (1)

Wayne State University¹

11 May 2004-Journal of Chemical Physics

TL;DR: A new integrator for the steepest descent pathway is presented, a Hessian based predictor-corrector algorithm that affords pathways comparable to the previous fourth order method at the cost of a second order approach.

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Abstract: Central to the theoretical description of a chemical reaction is the reaction pathway. The intrinsic reaction coordinate is defined as the steepest descent path in mass weighted Cartesian coordinates that connects the transition state to reactants and products. In this work, a new integrator for the steepest descent pathway is presented. This method is a Hessian based predictor-corrector algorithm that affords pathways comparable to our previous fourth order method at the cost of a second order approach. The proposed integrator is tested on an analytic surface, four moderately sized chemical reactions, and one larger organometallic system.

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877 citations