连续非线性最优控制问题的勒让德-伽辽金逼近方法

陈学松

doi:10.1631/FITEE.1601101

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连续非线性最优控制问题的勒让德-伽辽金逼近方法

常规文章 | Updated：2022-05-19

- 连续非线性最优控制问题的勒让德-伽辽金逼近方法
- Galerkin approximation with Legendre polynomials for a continuous-time nonlinear optimal control problem
- 信息与电子工程前沿（英文） 2017年18卷第10期页码：1479-1487
- Affiliations：
  
  School of Applied Mathematics, Guangdong University of Technology, Guangzhou 510006, China
- Author bio：
  
  [ "CHEN Xue-song, E-mail: chenxs@gdut.edu.cn" ]
- Funds：
  
  Project supported by the China Postdoctoral Science Foundation (No. 2013M531842), the Natural Science Foundation of Guangdong Province, China (No. 2015A030313497), and the Science and Technology Program of Guangzhou, China (No. 2014KP000039)
- DOI：10.1631/FITEE.1601101
  中图分类号： TP13
- 收稿：2016-06-04，
  
  修回：2017-11-;1，
  
  纸质出版：2017-10
- Accepted：
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陈学松. 连续非线性最优控制问题的勒让德-伽辽金逼近方法[J]. 信息与电子工程前沿（英文）, 2017,18(10):1479-1487.

Xue-song CHEN. Galerkin approximation with Legendre polynomials for a continuous-time nonlinear optimal control problem[J]. Frontiers of Information Technology & Electronic Engineering, 2017, 18(10): 1479-1487.
陈学松. 连续非线性最优控制问题的勒让德-伽辽金逼近方法[J]. 信息与电子工程前沿（英文）, 2017,18(10):1479-1487. DOI： 10.1631/FITEE.1601101.

Xue-song CHEN. Galerkin approximation with Legendre polynomials for a continuous-time nonlinear optimal control problem[J]. Frontiers of Information Technology & Electronic Engineering, 2017, 18(10): 1479-1487. DOI： 10.1631/FITEE.1601101.

摘要

使用逼近方法获得一类连续非线性最优控制问题的近似最优解。该方法基于伽辽金逼近理论（Galerkin approximation）求解广义哈密尔顿-雅可比-贝尔曼（Hamilton-Jacobi-Bellman

GHJB）方程。勒让德-伽辽金逼近方法（Galerkin approximation with Legendre polynomials

GALP）尚未被用于求解连续非线性最优控制问题。由于勒让德多项式（Legendre polynomials）具有正交性，在计算函数内积时，该方法可以明显减少积分计算量。详细证明了此方法的稳定性和收敛性。数值算例表明，按此方法获得的控制律，能够收敛到连续非线性控制系统的最优控制。

Abstract

We investigate the use of an approximation method for obtaining near-optimal solutions to a kind of nonlinear continuous-time (CT) system. The approach derived from the Galerkin approximation is used to solve the generalized Hamilton-Jacobi-Bellman (GHJB) equations. The Galerkin approximation with Legendre polynomials (GALP) for GHJB equations has not been applied to nonlinear CT systems. The proposed GALP method solves the GHJB equations in CT systems on some well-defined region of attraction. The integrals that need to be computed are much fewer due to the orthogonal properties of Legendre polynomials

which is a significant advantage of this approach. The stabilization and convergence properties with regard to the iterative variable have been proved. Numerical examples show that the update control laws converge to the optimal control for nonlinear CT systems.

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references

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