Power series-based Koopman model with application to spacecraft relative motion control

This paper presents a power series-based method for efficiently computing the Koopman operator of nonlinear systems and investigates its application to model predictive control for spacecraft relative motion. Firstly, the power series function is introduced as the function basis for the Koopman operator, eliminating the necessity for complex high-dimensional symbolic integration typically of conventional methods. This significantly reduces computational time by directly extracting coefficients from symbolic polynomials. Then, a mapping relationship is established between the control inputs and the Koopman linear system, leading to the development of a bilinear Koopman model with control terms. Furthermore, we enhance the traditional model predictive controller to enable the derived Koopman bilinear control model to be applied in a linear controller, achieving rapid online planning and control of the original system. Simulations based on three-dimensional high-order relative motion equations of spacecraft show that our method requires only 1 % of the time needed for traditional Koopman matrix computation. The resultant Koopman linear model exhibits precise prediction capabilities over a wide range. The proposed Koopman model predictive control algorithm enables the spacecraft to respond to real-time environmental deviations and autonomously complete assigned tasks in various relative orbital missions despite external disturbances.