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Energy-based control of numerical errors in time-domain simulation of dynamic circuits

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I-FUNDAMENTAL THEORY AND APPLICATIONS 2001年第5期48卷 543-551页

作者： Brambilla, A D'Amore, D Politecn Milan Dipartimento Elettron & Informazione I-20133 Milan Italy

In this paper, we consider the accuracy of integration algorithms such as the implicit Euler and the trapezoidal ones, which are largely employed in the time domain circuit analysis. These algorithms require to make hypotheses on the intersample shape and on the "energy content" of the sampled waveforms. For example, the implicit Euler algorithm supposes functions to be piecewise constant. When these hypotheses are violated, some errors are introduced by the integration process into the solution waveform. We consider the energy of the sampled functions, and through energy balance equations, estimate the accuracy of the integration algorithm. Furthermore, we propose an implicit algorithm to determine an adequate integration time step during numerical time domain analysis, This algorithm is based on a global energy balance equation and not on the conventional estimation of the local truncation error. It avoids the "cut and try" mechanism used in SPICE to determine the time step that satisfies the desired error tolerance.

关键词： electrical simulation numerical algorithms numerical integration

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numerical OUTFLOW BOUNDARY-CONDITION FOR NAVIER-STOKES FLOW CALCULATIONS BY A LINE ITERATIVE METHOD

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AIAA JOURNAL 1985年第12期23卷 1847-1848页

作者： SHYY, W General Electric Research and Development Center Schenectady New York

This study investigates the use of explicit/implicit, step/linear space extrapolation as numerical outflow boundary conditions for turbulent flow calculations. The emphasis is on the interplay between the use of curvilinear coordinate systems for complex flow calculation and the semiimplicit line iterative procedure. Both a planar singlechanneled and an annular multichanneled flow configuration were used as test models. It was found that implicit linear space extrapolation needs more computing time and may not yield a unique solution. On the other hand, explicit zero-order space extrapolation appears to be a robust outflow boundary condition since it has the largest stability limit with respect to underrelaxation factors and initial conditions. © 1985 American Institute of Aeronautics and Astronautics, Inc., All rights reserved.

关键词： Computing Time Turbulent Flow Curvilinear Coordinates Turbulence Kinetic Energy numerical algorithms Mass Flux Navier Stokes Equations Gas Turbine Combustor Flow Characteristics Fluid Mechanics

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A numerical Algorithm for Optimal Control of Systems with Parameter Uncertainty

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IFAC-PapersOnLine 2016年第18期49卷 468-475页

作者： Walton, Claire Phelps, Chris Gong, Qi Kaminer, Isaac Naval Postgraduate School MontereyCA93943 United States University of California Santa Cruz Santa CruzCA95060 United States

This paper describes a numerical scheme for computing optimal solutions to a class of nonlinear optimal control problems in which parameter uncertainty may be a feature of the state dynamics or objective function. Consistency results are provided for states and controls generated by the algorithm as well as for the adjoint variables. © 2016

关键词： Parameter estimation numerical methods Optimal control systems Consistency result Non linear control Nonlinear optimal control problems numerical algorithms Objective functions Optimal controls Optimal solutions Parameter uncertainty

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Post-stabilization of invariants and application to numerical analysis of chaos for some 3-dimensional systems

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INTERNATIONAL JOURNAL OF MODERN PHYSICS C 2006年第11期17卷 1613-1628页

作者： Wu, Xin He, Ji Zhou Nanchang Univ Dept Phys Nanchang 330047 Peoples R China

This research relates to a numerical integrator with post-stabilization of several constraints for an autonomous dynamical system. A generally analytical approach shows that the total energy correction is not valid in most cases, while post-stabilization of each independent energy is. As a typical test example, we consider a non-integrable Hamiltonian system of three degrees of freedom, which can be split into two independent pieces, one 1D harmonic oscillator and another 2D non-integrable system, by using a transformation of variables. Phase portraits on Poincaxe sections about the 2D system manifest that our analysis is reasonable. In addition, a problem how to compute Lyapunov exponents in constrained systems is proposed. As a suggestion, it is best to stabilize all constraints involving each energy integral and its corresponding variation in order to avoid spurious Lyapunov exponents. Because an appropriately larger time step is acceptable in this sense, it is not expensive to use the fast Lyapunov indicators to describe the global dynamics of phase space for the 3D system, where regions of chaos and order are clearly identified.

关键词： celestial mechanics numerical algorithms post-stabilization chaos Lyapunov spectra fast Lyapunov indicators

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An enhanced numerical procedure for the shakedown analysis in multidimensional loading domains

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COMPUTERS & STRUCTURES 2017年第Dec.期193卷 155-171页

作者： Spiliopoulos, K. V. Panagiotou, K. D. Natl Tech Univ Athens Dept Civil Engn Inst Struct Anal & Antiseism Res Zografou Campus Athens 15780 Greece Rhein Westfal TH Aachen AICES Schinkelstr 2 D-52062 Aachen Germany Rhein Westfal TH Aachen Inst Appl Mech Mies van der Rohe Str 1 D-52074 Aachen Germany

The Residual Stress Decomposition Method for Shakedown (RSDM-S) is a new iterative direct method to estimate the shakedown load in a 2-dimensional (2D) loading domain. It may be implemented to any existing finite element code, without the need to use a mathematical programming algorithm. An improved and enhanced RSDM-S is proposed herein. A new convergence criterion is presented that makes the procedure almost double as fast. At the same time, the procedure is formulated in a 3 -dimensional (3D) polyhedral loading domain, consisting of independently varying mechanical and thermal loads. Using a cyclic loading program that follows the outline of this domain, it is shown that there is hardly any increase in the computational time when passing from a 2D to a 3D domain. Finally, keeping the efficiency, using an alternative cyclic loading program, an automation of the approach to any n-dimensional loading domain is presented. Examples of application are included. (C) 2017 Elsevier Ltd. All rights reserved.

关键词： numerical algorithms Direct methods Cyclic loading Plasticity Shakedown Residual stresses

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The Gauss-Seidel numerical procedure for Markov stochastic games

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL 2004年第10期49卷 1779-1782页

作者： Kushner, HJ Brown Univ Dept Appl Math Lefschetz Ctr Dynam Syst Providence RI 02912 USA

Consider the problem of value iteration for solving Markov stochastic games. One simply iterates backward, via a Jacobi-like procedure. The convergence of the Gauss-Seidel form of this procedure is shown for both the discounted and ergodic cost problems, under appropriate conditions, with extensions to problems where one stops when a boundary is hit or if any one of the players chooses to stop, with associated costs. Generally, the Gauss-Seidel procedure accelerates convergence.

关键词： Gauss-Seidel procedure Markov games numerical algorithms stochastic games

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A numerical algorithm for simulation of the Q-switched fiber laser using the travelling wave model

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LITHUANIAN MATHEMATICAL JOURNAL 2008年第3期48卷 270-281页

作者： Ciegis, R. Dement'ev, A. Laukaityte, I. Vilnius Gediminas Tech Univ LT-10223 Vilnius Lithuania Inst Phys LT-2600 Vilnius Lithuania

We develop a finite-difference scheme for approximation of a system of nonlinear PDEs describing the Q-switching process. We construct it by using staggered grids. The transport equations are approximated along characteristics, and quadratic nonlinear functions are linearized using a special selection of staggered grids. The stability analysis proves that a connection between time and space steps arises only due to approximation requirements in order to follow exactly the directions of characteristics. The convergence analysis of this scheme is done in two steps. First, some estimates of the uniform boundedness of the discrete solution are proved. This part of the analysis is done locally, in some neighborhood of the exact solution. Second, on the basis of the obtained estimates, the main stability inequality is proved. The second-order convergence rate with respect to the space and time coordinates follows from this stability estimate. Using the obtained convergence result, we prove that the local stability analysis in the selected neighborhood of the exact solution is sufficient.

关键词： Q-switch nonlinear problems numerical algorithms convergence analysis staggered grids

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numerical Computation of Lyapunov Matrices for Integral Delay Systems

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IFAC-PapersOnLine 2017年第1期50卷 13342-13347页

作者： Arismendi-Valle, Héctor Melchor-Aguilar, Daniel Division of Applied Mathematics IPICYT San Luis PotosíSLP78216 Mexico

This paper focuses on the problem of computing Lyapunov matrices for integral delay systems. It is shown that these Lyapunov matrices cannot be computed by means of the existing methods for Lyapunov matrices of differential delay systems. Then, a numerical algorithm for constructing piecewise linear approximations of Lyapunov matrices for integral delay systems is proposed. An example illustrates the procedure. © 2017

关键词： Matrix algebra Approximation algorithms Lyapunov functions Piecewise linear techniques Differential delay systems Integral Delay Systems Lyapunov matrix Lyapunov Krasovskii functionals numerical algorithms numerical computations Piecewise linear approximations

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NOSNOC: A Software Package for numerical Optimal Control of Nonsmooth Systems

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IEEE CONTROL SYSTEMS LETTERS 2022年 6卷 3110-3115页

作者： Nurkanovic, Armin Diehl, Moritz Univ Freiburg Dept Microsyst Engn IMTEK D-79110 Freiburg Germany Univ Freiburg Dept Math D-79110 Freiburg Germany

This letter introduces the NOnSmooth numerical Optimal Control (NOSNOC) open-source software package. It is a modular MATLAB tool based on CasADi and IPOPT for numerically solving Optimal Control Problems (OCP) with piecewise smooth systems (PSS). The tool supports: 1) automatic reformulation of systems with state jumps into PSS (via the time-freezing reformulation [1]) and of PSS into computationally more convenient forms, 2) automatic discretization of the OCP via, e.g., the recently introduced Finite Elements with Switch Detection [2] which enables high accuracy optimal control and simulation of PSS, 3) solution methods for the resulting discrete-time OCP. The nonsmooth discrete-time OCP are solved with techniques of continuous optimization in a homotopy procedure, without the use of integer variables. This enables the treatment of a broad class of nonsmooth systems in a unified way. Two tutorial examples are given. A benchmark shows that NOSNOC provides both faster and more accurate solutions than conventional approaches, including mixed-integer formulations.

关键词： Switches Finite element analysis Standards Optimal control Optimization Open source software Matlab Software hybrid systems optimal control numerical algorithms

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A Time-Freezing Approach for numerical Optimal Control of Nonsmooth Differential Equations With State Jumps

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IEEE CONTROL SYSTEMS LETTERS 2021年第2期5卷 439-444页

作者： Nurkanovic, Armin Sartor, Tommaso Albrecht, Sebastian Diehl, Moritz Siemens Corp Technol D-81739 Munich Germany Univ Freiburg Dept Microsyst Engn IMTEK D-79110 Freiburg Germany Katholieke Univ Leuven Dept Mech Engn MECO Res Team B-3000 Leuven Belgium Univ Freiburg Dept Math D-79110 Freiburg Germany

We present a novel reformulation of nonsmooth differential equations with state jumps enabling their easier simulation and use in optimal control problems without the need for integer variables. The main idea is to introduce an auxiliary differential equation to mimic the state jump map. Thereby, a clock state is introduced which does not evolve during the runtime of the auxiliary system. The pieces of the trajectory that correspond to the parts when the clock state was evolving recover the solution of the original system with jumps. Our reformulation results in nonsmooth ordinary differential equations where the discontinuity is in the first time derivative of the trajectory, rather than in the trajectory itself. This class of systems is easier to handle both theoretically and numerically. The reformulation is suitable for partially elastic mechanical impact problems. We provide numerical examples demonstrating the ease of use of this reformulation in both simulation and optimal control. In the optimal control example, we solve a sequence of nonlinear programming problems (NLPs) in a homotopy penalization approach and recover a time-optimal trajectory with state jumps.

关键词： Differential equations Trajectory Dynamical systems Optimal control Mathematical model Clocks numerical models Optimal control numerical algorithms hybrid systems

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