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A symplectic indirect approach for a class of nonlinear optimal control problems of differential‐algebraic systems
Author(s) -
Shi Boyang,
Peng Haijun,
Wang Xinwei,
Zhong Wanxie,
Gao Lingchong,
Fottner Johannes
Publication year - 2021
Publication title -
international journal of robust and nonlinear control
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.361
H-Index - 106
eISSN - 1099-1239
pISSN - 1049-8923
DOI - 10.1002/rnc.5414
Subject(s) - symplectic geometry , optimal control , mathematics , nonlinear system , hamiltonian system , hamiltonian (control theory) , algebraic equation , dynamical systems theory , differential algebraic equation , jacobian matrix and determinant , differential equation , computation , ordinary differential equation , mathematical optimization , mathematical analysis , algorithm , physics , quantum mechanics
Abstract Differential‐algebraic equations (DAEs) can model constrained dynamical systems and processes from practical engineering. Therefore, research on nonlinear optimal control problems of DAEs is of theoretical significance for optimal control of constrained systems, which can generate reference trajectories and control inputs for online control strategies. In terms of the numerical solution of this type of problem, research on indirect numerical methods is still insufficient and less research focuses on symplectic‐preserving methods. In this article, a symplectic indirect approach is proposed for optimal control problems subject to index‐1 DAEs. Necessary conditions of the optimal control problem constitute a Hamiltonian boundary value problem (HBVP) and there exists a symplectic structure in the Hamiltonian system. In the proposed approach, based on specified properties of generating functions, discrete equations can preserve the symplectic structure of the Hamiltonian system. In the iterative solution, the Jacobian matrices of the discrete equations are sparse and symmetric, which are very significant to save memory and improve efficiency in practical computation. In numerical examples, the proposed approach can provide highly accurate state variables and control inputs with fewer iterations. More accurate cost functional can be obtained. Problems from the chemistry process also can be solved effectively, it verifies the problem‐solving ability of the proposed approach.