LQ终端控制的生成函数法与Riccati变换法的等价性

Equivalence between generating function method and Riccati transformation method for LQ terminal control

中文关键词  LQ软终端控制  LQ硬终端控制  生成函数  Riccati变换法  哈密顿系统

英文关键词  LQ soft terminal control  LQ hard terminal control  generating function  Riccati transformation  Hamiltonian system

中文摘要

      通过定义线性哈密顿系统新形式的第3类生成函数, 建立了求解线性二次终端控制问题的生成函数方法与Riccati变换方法的直接联系, 并证明两种方法导出的最优控制律是等价的. 生成函数方法的优势在于可以灵活地处理边界约束条件. 本文根据Riccati变换方法沿时间逆向求解问题的特点, 定义了适于逆向正则变换的第3类生成函数, 用于求解哈密顿两点边值问题, 可得到用生成函数表示的最优控制律. 并通过验证生成函数方法所得最优控制律的各部分与Riccati变换法所得的结果相对应, 证明了两种方法所得控制律的等价性.

英文摘要

      By defining the third-kind generating function(GF) for a linear Hamiltonian system, this paper relates the generating function approach to the Riccati transformation method for LQ terminal control problems; and proves the equivalence of optimal terminal control laws derived by these two different methods. Since the generating function approach is adaptive to different types of boundary constraints, it provides a substantial advantage over the classical Riccati transformation method. Firstly, considering the backward sweeping character of the Riccati transformation method, we formulate the third-kind generating function in accordance with the backward canonical transform of the linear Hamiltonian system. Next, solving a Hamiltonian two-point-boundary-value problem, we obtain the new optimal control law in term of the generating function. Finally, comparing all terms of each new control law with the conventional control law derived by the Riccati transformation method, we verify the equivalence of these two different solution strategies.