Optimal Control on Lie Groups: The Projection Operator Approach
Optimal Control on Lie Groups: The Projection Operator Approach
复制标题
李群的最优控制：投影算子方法
DOI：
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发表时间：
2013
影响因子：
6.8
通讯作者：
Antonio Pedro Aguiar
中科院分区：
文献类型：
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作者：
A. Saccon;J. Hauser;Antonio Pedro Aguiar
许多具有实际意义的非线性系统都是在李群或李群作用的流形上演化的。例子从飞机和水下航行器到量子力学系统。在本文中，我们开发了一个算法来解决连续时间最优控制问题的系统发展（非紧）李群。该算法推广了最初为向量空间系统开发的轨迹优化投影算子方法。概念推广系统的理论工具，如Riccati方程和线性和二次系统近似的发展。在这个发展中，两个流形之间的映射的协变导数在为所需的李群计算提供链式规则方面起着关键作用。最后给出了SO（3）上的一个最优控制问题的例子，以突出实现细节并证明该方法的有效性。
Many nonlinear systems of practical interest evolve on Lie groups or on manifolds acted upon by Lie groups. Examples range from aircraft and underwater vehicles to quantum mechanical systems. In this paper, we develop an algorithm for solving continuous-time optimal control problems for systems evolving on (noncompact) Lie groups. This algorithm generalizes the projection operator approach for trajectory optimization originally developed for systems on vector spaces. Notions for generalizing system theoretic tools such as Riccati equations and linear and quadratic system approximations are developed. In this development, the covariant derivative of a map between two manifolds plays a key role in providing a chain rule for the required Lie group computations. An example optimal control problem on SO(3) is provided to highlight implementation details and to demonstrate the effectiveness of the method.