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Approximate Solutions to the Hamilton-Jacobi Equations for Generating Functions

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Journal of Systems Science & Complexity 2020年第2期33卷 261-288页

作者： HAO Zhiwei FUJIMOTO Kenji ZHANG Qiuhua Department of Astronautical Science and Mechanics Harbin Institute of TechnologyHarbin 150001China Department of Aeronautics and Astronautics Kyoto UniversityC3 Bldg.Kyotodaigaku-katsuraNishikyo-kuKyoto 615-8540Japan

For a nonlinear finite time optimal control problem,a systematic numerical algorithm to solve the Hamilton-Jacobi equation for a generating function is proposed in this paper.This algorithm allows one to obtain the taylor series expansion of the generating function up to any prescribed order by solving a sequence of first order ordinary differential equations recursively.Furthermore,the coefficients of the taylor series expansion of the generating function can be computed exactly under a certain technical condition.Once a generating function is found,it can be used to generate a family of optimal control for different boundary conditions.Since the generating function is computed off-line,the on-demand computational effort for different boundary conditions decreases a lot compared with the conventional method.It is useful to online optimal trajectory generation problems.Numerical examples illustrate the effectiveness of the proposed algorithm.

关键词： Generating functions Hamilton-Jacobi equations optimal control taylor series expansion two-point boundary-value problems

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A Nonlinear Optimal Control Approach for Tracked Mobile Robots

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Journal of Systems Science & Complexity 2021年第4期34卷 1279-1300页

作者： RIGATOS Gerasimos Unit of Industrial Automation IndiLstrial Systems InstituteRion Patras 26504Greece

The article proposes a nonlinear optimal(H-infinity)control approach for the model of a tracked robotic vehicle.The kinematic model of such a tracked vehicle takes into account slippage effects due to the contact of the tracks with the ground.To solve the related control problem,the dynamic model of the vehicle undergoes first approximate linearization around a temporary operating point which is updated at each iteration of the control algorithm.The linearization process relies on first-order taylor series expansion and on the computation of the Jacobian matrices of the state-space model of the vehicle.For the approximately linearized description of the tracked vehicle a stabilizing H-infinity feedback controller is designed.To compute the controller’s feedback gains an algebraic Riccati equation is solved at each time-step of the control method.The stability properties of the control scheme are proven through Lyapunov analysis.It is also demonstrated that the control method retains the advantages of linear optimal control,that is fast and accurate tracking of reference setpoints under moderate variations of the control inputs.

关键词： Approximate linearization global asymptotic stability H-infinity control Jacobian matrices Lyapunov analysis nonlinear optimal control taylor series expansion tracked robotic vehicle

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Power system transient stability simulation under uncertainty based on taylor model arithmetic

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Frontiers of Electrical and Electronic Engineering in China 2009年第2期4卷 220-226页

作者： Shouxiang WANG Zhijie ZHENG Chengshan WANG School of Electrical Engineering and Automation Tianjin UniversityTianjin 300072China

The taylor model arithmetic is introduced to deal with uncertainty.The uncertainty of model parameters is described by taylor models and each variable in functions is replaced with the taylor model(TM).Thus,time domain simulation under uncertainty is transformed to the integration of TM-based differential equations.In this paper,the taylor series method is employed to compute differential equations;moreover,power system time domain simulation under uncertainty based on taylor model method is presented.This method allows a rigorous estimation of the influence of either form of uncertainty and only needs one simulation.It is computationally fast compared with the Monte Carlo method,which is another technique for uncertainty analysis.The proposed method has been tested on the 39-bus New England system.The test results illustrate the effectiveness and practical value of the approach by comparing with the results of Monte Carlo simulation and traditional time domain simulation.

关键词： interval arithmetic power systems taylor series expansion taylor model time domain simulation transient stability uncertainty

A Nonlinear Optimal Control Method for Attitude Stabilization of Micro-Satellites

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Guidance, Navigation and Control 2022年第3期2卷 30-67页

作者： G.Rigatos M.Abbaszadeh K.Busawon L.Dala Unit of Industrial Automation Industrial Systems Institute 26504 Rion PatrasGreece Department of ECSE Rensselaer Polytechnic Institute New York 12065 USA Nonlinear Control Group University of Northumbria Newcastle NE18ST UK Department of Mechanical Engineering University of Northumbria Newcastle NE18ST UK

Attitude control and stabilization of micro-satellites is a nontrivial problem due to the highly nonlinear and multivariable structure of the satellites'state-space model.In this paper,a novel nonlinear optimal(H-infinity)control approach is developed for this control problem.The dynamic model of the satellite's attitude dynamics undergoesfirst approximate linearization around a temporary operating point which is updated at each iteration of the control algorithm.The linearization process relies on first-order taylor series expansion and on the computation of the Jacobian matrices of the state-space model of the satellite's attitude dynamics.For the approximately linearized description of the satellite's attitude a stabilizing H-infinity feedback controller is designed.To compute the controller's feedback gains,an algebraic Riccati equation is solved at each time-step of the control method.The stability properties of the control scheme are proven through Lyapunov analysis.It is also demonstrated that the control method retains the advantages of linear optimal control that is fast and accurate tracking of the reference setpoints under moderate variations of the control inputs.

关键词： Micro-satellites attitude control nonlinear optimal control H-infinity control differentialfiatness properties taylor series expansion Jacobian matrices Lyapunov analysis global asymptotic stability

An Alternative Approach to the Solution of Multi-Objective Geometric Programming Problems

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Open Journal of Optimization 2017年第1期6卷 11-25页

作者： Ersoy Oz Nuran Guzel Selcuk Alp Yildiz Technical University IstanbulTurkey

The aim of this study is to present an alternative approach for solving the multi-objective posynomial geometric programming problems. The proposed approach minimizes the weighted objective function comes from multi-objective geometric programming problem subject to constraints which constructed by using Kuhn-Tucker Conditions. A new nonlinear problem formed by this approach is solved iteratively. The solution of this approach gives the Pareto optimal solution for the multi-objective posynomial geometric programming problem. To demonstrate the performance of this approach, a problem which was solved with a weighted mean method by Ojha and Biswal (2010) is used. The comparison of solutions between two methods shows that similar results are obtained. In this manner, the proposed approach can be used as an alternative of weighted mean method.

关键词： Multi Objective Geometric Programming Kuhn-Tucker Conditions taylor series expansion Numerical Method Weighted Mean Method

On Simplified Models for Dynamics of Pointlike Objects

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Journal of Applied Mathematics and Physics 2021年第8期9卷 1853-1869页

作者： Marijan Ribarič Luka Š uš terš ič Jož ef Stefan Institute Ljubljana Slovenia

Motivation: We study the asymptotic-type dynamics of various real pointlike objects that one models by a variety of differential equations. Their response to an external force one defines solely by the trajectory of a single point. Its velocity eventually stops changing after cessation of the external force. The response of their acceleration to the long-term external force is slow and possibly nonlinear. Objective: Our objective is to present technique for making simplified models for the long-term dynamics of pointlike objects whose motion interacts with the surroundings. In the asymptotic-type long-term dynamics, the time variable t ∈ (tm, +∞) and tm > 0 is large, say ! Method: We apply taylor series expansion to differential equations to model the acceleration of pointlike object whose response to the long-term external force is not instantaneous and possibly nonlinear. Results: We make simplified models for the long-term dynamics of pointlike objects by taylor polynomials in time derivatives of the external force. Application: We interpret the relativistic Lorentz-Abraham-Dirac equation as an equation for modeling the long-term dynamics, where t ≥ tm ≫ 0. This interpretation resolves the conceptual and usage controversy surrounding its troublesome application to determine the trajectory of a radiating charged particle, thus contributing to the development of more adequate modeling of physical phenomena.

关键词： Point Mass Long-Term Dynamics Harmonic Oscillator Drag Force Lorentz-Abraham-Dirac Equation taylor series expansion

基于函数值Pad逼近的[1/n]阶迭代算法

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高等学校计算数学学报 2015年第3期37卷 234-243页

作者：郭巧檀结庆合肥工业大学数学学院

1引言一般的,我们在求解非线性方程的根时,利用最多的是迭代法,其迭代效果也各不一样[1-4].通常,我们在构造非线性方程求根的迭代方法有Newton迭代算法、Halley迭代算法和割线法等,而Newton迭代格式构造简单且收敛速度较快,又被认为是... 详细信息

1引言一般的,我们在求解非线性方程的根时,利用最多的是迭代法,其迭代效果也各不一样[1-4].通常,我们在构造非线性方程求根的迭代方法有Newton迭代算法、Halley迭代算法和割线法等,而Newton迭代格式构造简单且收敛速度较快,又被认为是求解一般非线性方程根的最常用方法.在:Newton迭代公式的推导过程中,利用最多的是泰勒展开式法、切线法、积分法[5].本文基于函数值Pad6逼近的行列式表示[6-7],构造出[1/0]、[1/1]、[1/2]阶Pade逼近

关键词： Pade approximation taylor series expansion Convergence Iterative algorithm

An Improved Inverse Distance Weighted Method for Interpolating ZTD Based on Regional GPS Network

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An Improved Inverse Distance Weighted Method for Interpolati...