In this paper, we consider nonlinear control of a symmetric spacecraft about its axis of symmetry with two control torques. Using a computationally efficient ℋ control design procedure, attitude regulation and trajectory tracking problems of the axi-symmetric spacecraft were solved. Resorting to higher order Lyapunov functions, the employed nonlinear ℋ control approach reformulates the difficult Hamilton-Jacobian-Isaacs (HJI) inequalities as semi-definite optimization conditions. Sum-of-squares (SOS) programming techniques are then applied to obtain computationally tractable solutions, from which nonlinear control laws will be constructed. The proposed nonlinear ℋ designs will be able to exploit the most suitable forms of Lyapunov function for spacecraft control and the resulting controllers will perform better than existing nonlinear control laws.

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