哈密顿系统正则变换在时变最优控制中的应用

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.

TIME-VARYING OPTIMAL CONTROL VIA CANONICAL TRANSFORMATION OF HAMILTONIAN SYSTEM

This paper presents a unified canonical transformation and generating function approach, including associated numerical algorithms, for linear time-varying optimal control problems with various terminal constraints. Generating functions are employed to find the optimal control law by solving Hamiltonian two-point-boundary-value problems. The time-varying optimal control laws constructed by the second type generating function do not have infinite feedback gain at terminal time, which is different from other existing solutions. Motivated by practical design of time-varying optimal control systems, a structure-preserving matrix recursive algorithm is proposed to solve coupled time-varying matrix differential equations of the generating function; derivation of the recursive algorithm is based on symplectic formulation of canonical transformation. To preserve symplectic structure of matrices in the recursive computation, state transition matrices of the Hamiltonian system are calculated by Magnus series. In fact, the canonical transformation and generating function method leads to a geometric perspective to the design and computation of optimal control systems.