Autonomous Satellite Rendezvous via Hybrid Feedback Optimization

As satellites have proliferated, interest has increased in autonomous rendezvous, proximity operations, and docking (ARPOD). A fundamental challenge in these tasks is the uncertainties when operating in space, e.g., in measurements of satellites' states, which can make future states difficult to predict. Another challenge is that satellites' onboard processors are typically much slower than their terrestrial counterparts. Therefore, to address these challenges we propose to solve an ARPOD problem with feedback optimization, which computes inputs to a system by measuring its outputs, feeding them into an optimization algorithm in the loop, and computing some number of iterations towards an optimal input. We focus on satellite rendezvous, and satellites' dynamics are modeled using the continuous-time Clohessy-Wiltshire equations, which are marginally stable. We develop an asymptotically stabilizing controller for them, and we use discrete-time gradient descent in the loop to compute inputs to them. Then, we analyze the hybrid feedback optimization system formed by the stabilized Clohessy-Wiltshire equations with gradient descent in the loop. We show that this model is well-posed and that maximal solutions are both complete and non-Zeno. Then, we show that solutions converge exponentially fast to a ball around a rendezvous point, and we bound the radius of that ball in terms of system parameters. Simulations show that this approach provides up to a 98.4\% reduction in the magnitude of disturbances across a range of simulations, which illustrates the viability of hybrid feedback optimization for autonomous satellite rendezvous.