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CLC number: TP13

On-line Access: 2017-12-04

Received: 2016-06-04

Revision Accepted: 2016-10-15

Crosschecked: 2017-11-01

Cited: 0

Clicked: 1498

Citations:  Bibtex RefMan EndNote GB/T7714

 ORCID:

Xue-song Chen

http://orcid.org/0000-0001-9530-0644

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Frontiers of Information Technology & Electronic Engineering  2017 Vol.18 No.10 P.1479-1487

http://doi.org/10.1631/FITEE.1601101


Galerkin approximation with Legendre polynomials for a continuous-time nonlinear optimal control problem


Author(s):  Xue-song Chen

Affiliation(s):  School of Applied Mathematics, Guangdong University of Technology, Guangzhou 510006, China

Corresponding email(s):   chenxs@gdut.edu.cn

Key Words:  Generalized Hamilton-Jacobi-Bellman equation, Nonlinear optimal control, Galerkin approximation, Legendre polynomials


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.

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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.

连续非线性最优控制问题的勒让德-伽辽金逼近方法

概要: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。

关键词:广义哈密尔顿-雅可比-贝尔曼方程;非线性最优控制;伽辽金逼近(Galerkin approximation);勒让德多项式(Legendre polynomials)

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