An Adaptive Max-Plus Eigenvector Method for Continuous Time Optimal Control Problems
An Adaptive Max-Plus Eigenvector Method for Continuous Time Optimal Control Problems
复制标题
连续时间最优控制问题的自适应最大加特征向量法
DOI：
10.1007/978-3-030-01959-4_10
复制
发表时间：
2018
期刊：
影响因子：
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通讯作者：
P. Dower
中科院分区：
文献类型：
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作者：
P. Dower
提出了一种自适应最大加特征向量方法来近似求解连续时间非线性最优控制问题。在该方法的每个步骤中，给定一组二次基函数，应用标准最大加特征向量方法来产生感兴趣的值函数的近似值。使用该近似，根据每个基函数在逼近价值函数时活跃的位置，对由哈密顿量定义的回代误差的近似水平集进行细分。获得的多面体及其顶点根据此回代误差进行排序，从而可以识别“最坏情况”的基函数。这些基函数的位置随后被演化以产生新的基函数，从而减少这种最坏情况。在值函数近似中不活动的基函数被修剪，并且重复上述步骤。提供与最大加线性、动态规划和半凸对偶相关的基础代数性质作为开发的基础，并通过示例说明了所提出方法的实用性。
An adaptive max-plus eigenvector method is proposed for approximating the solution of continuous time nonlinear optimal control problems. At each step of the method, given a set of quadratic basis functions, a standard max-plus eigenvector method is applied to yield an approximation of the value function of interest. Using this approximation, an approximate level set of the back substitution error defined by the Hamiltonian is tessellated according to where each basis function is active in approximating the value function. The polytopes obtained, and their vertices, are sorted according to this back substitution error, allowing “worst-case” basis functions to be identified. The locations of these basis functions are subsequently evolved to yield new basis functions that reduce this worst-case. Basis functions that are inactive in the value function approximation are pruned, and the aforementioned steps repeated. Underlying algebraic properties associated with max-plus linearity, dynamic programming, and semiconvex duality are provided as a foundation for the development, and the utility of the proposed method is illustrated by example.