奇异最优LQ控制系统的Hamilton算法
A Hamiltonian Algorithm for Singular Optimal LQ Control Systems
摘要：
本文提出了一种哈密顿算法，理论上和数值上都得到了实现Pontryagine最大值原理的奇异线性二次型最优控制问题的简化方程。该算法的灵感来自著名的Rabier-Rheinhboldt约束算法用于解决微分-代数方程。它的几何内容是充分利用实施的Hamilton扩展，这是更接近Gotay-Nester前辛约束算法用于解决奇异哈密顿系统。这样，给出了一个最优反馈以隐式形式给出的最优控制问题，得到了描述Pontryaguine最大值原理最大值原理的一阶微分条件的相容方程组.这样的方程被证明是哈密尔顿和一组第一类约束对应的控制，没有确定的，得到明确的。该算法的强度显示通过展示一个数值实现与部分反馈的控制，提供了一个部分最优反馈法的问题。数值算法的灵感来自作者以前的分析奇异线性-二次最优控制问题的一致性条件。数值算法是显式的哈密顿。它表明，只有两种可能性可以出现约化的哈密顿PMP：即简化方程是完全确定的，或者它们依赖于一个家庭的自由参数，将被称为“规范”控制的第一类约束所确定的系统。这些额外自由度的存在为寻求解决方案提供了新的可能性。通过讨论各种相关的数值实验，算法的稳定性的数值证据。
英文摘要：
A Hamiltonian algorithm, both theoretical and numerical, to obtain the reduced equations implementing Pontryagine's Maximum Principle for singular linear-quadratic optimal control problems is presented. This algorithm is inspired on the well-known Rabier-Rheinhboldt constraints algorithm used to solve differential-algebraic equations. Its geometrical content is exploited fully by implementing a Hamiltonian extension of it which is closer to Gotay-Nester presymplectic constraint algorithm used to solve singular Hamiltonian systems. Thus, given an optimal control problem whose optimal feedback is given in implicit form, a consistent set of equations is obtained describing the first order differential conditions of Pontryaguine's Maximum Principle. Such equations are shown to be Hamiltonian and the set of first class constraints corresponding to controls that are not determined, are obtained explicitly. The strength of the algorithm is shown by exhibiting a numerical implementation with partial feedback on the controls that provides a partial optimal feedback law for the problem. The numerical algorithm is inspired on a previous analysis by the authors of the consistency conditions for singular linear--quadratic optimal control problems. The numerical algorithm is explicitly Hamiltonian. It is shown that only two possibilities can arise for the reduced Hamiltonian PMP: that the reduced equations are completely determined or that they depend on a family of free parameters that will be called the "gauge" controls of the system determined by the first class constraints. The existence of these extra degrees of freedom opens new possibilities for the search of solutions. Numerical evidence of the stability of the algorithm is presented by discussing various relevant numerical experiments.
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