空间相对运动的Koopman算子控制Optimization
A Koopman-Operator Control Optimization for Relative Motion in Space
摘要：
提出了一种基于Koopman算子框架的高阶最优控制策略。新技术利用Koopman表示的运动方程的解决方案，开发一种能量最优的逆控制方法。算子理论可以将有限维的非线性动力系统转化为无穷维的线性系。因此，任何非线性动力学的状态被表示为高阶正交多项式的线性组合，这创建了解的状态转移多项式映射。由于最优控制技术可以归结为两点边值问题，因此利用Koopman映射在时间上连接状态变量和控制变量，通过映射反演和多项式求值得到最优值。新技术应用于空间交会应用，其中两颗卫星之间的相对运动建模与系统的拉格朗日高阶多项式级数展开，使得Clohessy-Wiltshire方程表示高阶模型的线性截断的减少。
英文摘要：
A high order optimal control strategy implemented in the Koopman operator framework is proposed in this work. The new technique exploits the Koopman representation of the solution of the equations of motion to develop an energy optimal inverse control methodology. The operator theory can reformulate a nonlinear dynamical system of finite dimension into a linear system with an infinite number of dimensions. As a results, the state of any nonlinear dynamics is represented as a linear combination of high-order orthogonal polynomials, which creates the state transition polynomial map of the solution. Since the optimal control technique can be reduced to a two-points boundary value problem, the Koopman map is used to connect the state and control variables in time, such that optimal values are obtained through map inversion and polynomial evaluation. The new technique is applied to rendezvous applications in space, where the relative motion between two satellites is modelled with a high-order polynomial series expansion of the Lagrangian of the system, such that the Clohessy-Wiltshire equations represent the reduction of the high-order model to a linear truncation.
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