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中文核心期刊

哈密顿系统正则变换在时变最优控制中的应用

Time-varying optimal control via canonical transformation of hamiltonian system

  • 摘要: 利用哈密顿系统正则变换和生成函数理论求解线性时变最优控制问题,构造了新的最优控制律形式并提出了控制增益计算的保结构算法. 利用生成函数求解最优控制导出的哈密顿系统两端边值问题,并构造线性时变系统的最优控制律,由第2类生成函数所构造的最优控制律避免了末端时刻出现无穷大反馈增益. 控制系统设计中需求解生成函数满足的时变矩阵微分方程组. 根据生成函数与哈密顿系统状态转移矩阵之间的关系,从正则变换的辛矩阵描述出发,导出了求解这组微分方程组的保结构递推算法.为了保持递推计算中的辛矩阵结构,哈密顿系统状态转移矩阵的计算中利用了Magnus级数.

     

    Abstract: This paper presents a unified canonical transformationand generating function approach, including associated numerical algorithms,for linear time-varying optimal control problems with various terminalconstraints. Generating functions are employed to find the optimal controllaw by solving Hamiltonian two-point-boundary-value problems. Thetime-varying optimal control laws constructed by the second type generatingfunction do not have infinite feedback gain at terminal time, which isdifferent from other existing solutions. Motivated by practical design oftime-varying optimal control systems, a structure-preserving matrixrecursive algorithm is proposed to solve coupled time-varying matrixdifferential equations of the generating function; derivation of therecursive algorithm is based on symplectic formulation of canonicaltransformation. To preserve symplectic structure of matrices in therecursive computation, state transition matrices of the Hamiltonian systemare calculated by Magnus series. In fact, the canonical transformation andgenerating function method leads to a geometric perspective to thedesign and computation of optimal control systems.%control systems synthesis and computation.

     

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