Koopman-Operator Control Optimization for Relative Motion in Space

Abstract

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 dynamic system of finite dimension into a linear system with an infinite number of dimensions. As a result, 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. Because the optimal control technique can be reduced to a two-point 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 modeled 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. The proposed numerical applications are analyzed to show the robustness and limitations of the novel technique.