H ∞ feedback for attitude control of liquid-filled spacecraft

Abstract

An H ∞ , that is, Hardy infinity, feedback controller for the attitude control problem of a liquid-filled spacecraft is presented. The spacecraft is modeled as a main rigid body filled with an ideal liquid in uniform vortex motion and three control torques and disturbances. Quaternions are used to describe the evolution of liquid-filled spacecraft orientation to eliminate the singularities due to Eular angle representations for the kinematics motions. The H ∞ feedback controller is formulated by solving the Hamilton-Jacobi-Issacs inequality associated with the H ∞ suboptimal control problem on state manifolds according to Van der Schaft's theory(Van der Schaft, A. J., "On a State Space Approach to Non-Linear H ∞ Control," <i>Systems and Control Letters</i>, Vol. 16, No 1, 1991, pp. 1-8 and Van der Schaft, A. J., "<I>L</i>₂-Gain Analysis of Non-Linear System and Non-Linear State Feedback <I>H</i> ∞ Control," <I>IEEE Transactions on Automatic Control</I>, Vol. 37, No. 6, 1992, pp. 770-784). The orientation and angular velocities of the liquid-filled spacecraft are stabilized by appropriately choosing the feedback coefficients. The determination of the coefficients is given explicitly. The numerical simulations show that the designed feedback laws can be effectively applied to stabilize the attitude of liquid-filled spacecraft with energy dissipation and external disturbances.