Nonlinear Robust $H_{\infty }$ Control for Spacecraft Body-Fixed Hovering Around Noncooperative Target Via Modified $\theta -D$ Method

This paper addresses the robust control problem of spacecraft body-fixed hovering near a noncooperative target in the presence of parametric uncertainties and disturbances. A closed-form nonlinear robust $H_{\infty }$ controller constructed by the solution of Hamilton–Jacobi–Isaacs (HJI) inequality is designed, which can stabilize the closed-loop system with the $H_{\infty }$ performance. To obtain the solution of HJI, an effective numerical approximation approach called the modified $\theta \text{--}D$ method is proposed, which can obtain a numerical approximation solution of HJI by only solving one algebraic Riccati inequality and several Lyapunov equations rather than a partial differential inequality. The detailed numerical calculation processes of solving the HJI are presented, which shows that the modified $\theta \text{--}D$ method can be implemented offline with less computational burden. Finally, the effectiveness of the designed robust control scheme is demonstrated via a numerical example.