非线性最小二乘跟踪的对偶变量变分方法

寻广彬; 彭海军; 邬树楠; 吴志刚

doi:10.6052/0459-1879-15-399

非线性最小二乘跟踪的对偶变量变分方法

寻广彬¹,
彭海军^1,2,
邬树楠^2,3,
吴志刚^2,3,

1.
大连理工大学工程力学系, 大连 116024
2. 大连理工大学工业装备结构分析国家重点实验室, 大连 116024
3. 大连理工大学航空航天学院, 大连 116024

基金项目: 国家自然科学基金（11432010）和中央高校基本科研业务费（DUT15LK31）资助项目.

详细信息

通讯作者:
吴志刚,教授,主要研究方向:飞行器动力学与控制

中图分类号: O313;O241.2
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出版历程
- 收稿日期: 2015-11-01
- 修回日期: 2016-07-25
- 刊出日期: 2016-09-17

DUAL VARIABLE VARIATIONAL METHOD FOR NONLINEAR LEAST SQUARES SHADOWING PROBLEM

1.
Department of Engineering Mechanics, Dalian University of Technology, Dalian 116024, China
2. State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, China
3. School of Aeronautics and Astronautics, Dalian University of Technology, Dalian 116024, China

摘要

摘要: 最小二乘跟踪方法是近几年提出的一种计算动力系统跟踪轨迹的方法.基于最小二乘跟踪的灵敏度分析算法可以有效避免传统的非线性系统灵敏度分析方法中的病态初值问题，因此其在混沌系统灵敏度分析方面有着重要的应用.针对非线性的最小二乘跟踪问题，首先将其重新描述为带有约束的非线性最优控制问题，引入协态变量并将系统的哈密顿函数表示为关于状态变量和协态变量的函数.然后将目标函数的积分时间离散化，根据对偶变量变分原理，以离散区间两端的状态变量作为独立变量，用Lagrange插值多项式近似离散区间内的状态变量和协态变量，进而将非线性最优控制问题转化为求解非线性方程组问题.这种算法无需对原问题做线性化处理，避免了复杂的线性化过程以及可能因此造成的误差，同时为求解非线性最小二乘跟踪问题提供了新的思路.根据最小二乘方法可以得到两条设计参数有微小变化的状态轨迹，基于这两条状态轨迹可进一步计算出系统关于设计参数的灵敏度，范德波振子作为数值算例验证了该方法在求解最小二乘跟踪问题以及计算非线性系统灵敏度时的有效性.
- 对偶变量变分 /
- 最小二乘跟踪 /
- 非线性系统 /
- 灵敏度分析
Abstract: The least squares shadowing (LSS) method, to compute the shadowing trajectories of dynamical systems, has been presented in recent years. The ill-posed initial value problem within the conventional sensitivity analysis algorithm for nonlinear systems can be effectively avoided based on the LSS method, which therefore has significant applications in the sensitivity analysis of chaotic systems. To achieve nonlinear LSS problem, it will be firstly represented as a nonlinear optimal control problem subject to constraints. By introducing the costate variables, the Hamiltonian function is represented depending on state and costate variables. The integral time of objective function is then discretized, and the state variables at ends of time interval are taken as independent variables. Then approximate the state and costate variables in the time interval using the Lagrange polynomials. This problem is finally transformed into solving nonlinear equations via dual variable variation principle. The linearizing process is avoided for the proposed algorithm, and then the errors, caused by the complex linearizing process, are also reduced, which provides affinew solution for solving the shadowing problem. Two state trajectories with designing parameters slightly changed can be obtained by LSS problem, and then the sensitivity of the nonlinear system is calculated from the two trajectories. Van der Pol oscillator as a numerical example shows that this method is effective for the LSS problem and sensitivity analysis of nonlinear systems.
- dual variable variation /
- least squares shadowing /
- nonlinear systems /
- sensitivity analysis

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