Optimal Control with Lyapunov Stability Guarantees for Space Applications

TL;DR

This work tackles the infinite-horizon nonlinear optimal control problem for space applications by splitting it into a nonlinear transfer toward a terminal set and a subsequent linear regulation phase within that set. By leveraging the infinite-horizon cost-to-go as a Control Lyapunov Function, the authors prove global asymptotic stability and demonstrate convergence of the AC-OCP cost to the IH-OCP cost as the terminal region contracts to the origin. The solution employs an iLQR-driven finite-horizon phase to reach the terminal set, followed by an LQR controller for stabilization, with the transfer time optimized for minimal cost. Numerical results on spacecraft attitude control, rendezvous, and soft-landing validate robustness and stability, highlighting the method’s practicality for complex space missions. A soft-landing case shows the need for nonlinear handling with a penalty to manage altitude constraints, underscoring the importance of careful transition planning between nonlinear and linear regimes.

Abstract

This paper investigates the infinite horizon optimal control problem (OCP) for space applications characterized by nonlinear dynamics. The proposed approach divides the problem into a finite horizon OCP with a regularized terminal cost, guiding the system towards a terminal set, and an infinite horizon linear regulation phase within this set. This strategy guarantees global asymptotic stability under specific assumptions. Our method maintains the system's fully nonlinear dynamics until it reaches the terminal set, where the system dynamics is linearized. As the terminal set converges to the origin, the difference in optimal cost incurred reduces to zero, guaranteeing an efficient and stable solution. The approach is tested through simulations on three problems: spacecraft attitude control, rendezvous maneuver, and soft landing. In spacecraft attitude control, we focus on achieving precise orientation and stabilization. For rendezvous maneuvers, we address the navigation of a chaser to meet a target spacecraft. For the soft landing problem, we ensure a controlled descent and touchdown on a planetary surface. We provide numerical results confirming the effectiveness of the proposed method in managing these nonlinear dynamics problems, offering robust solutions essential for successful space missions.

Figures (10)

• Figure 1: Schematic illustrating the strategy to solve the infinite horizon optimal control problem
• Figure 2: Change in total cost and regulation cost as a function of transfer time for attitude control.
• Figure 3: Evolution of states and control for the attitude control problem with a transfer time of 10 seconds. The dotted red line shows the transition in the above plots.
• Figure 4: Evolution of states and control for the attitude control problem with a transfer time of 80 seconds. The dotted red line shows the transition in the above plots.
• Figure 5: Change in total cost and regulation cost as a function of transfer time for rendezvous manuever.

Theorems & Definitions (3)

• Corollary 1
• Theorem 1
• Lemma 1