Optimal Regulation and Passivity Results for Axisymmetric Rigid Bodies Using Two Controls

We present a partial solution to the problem of optimal feedback reorientation of the symmetry axis of an axially-symmetric rigid body. The performance index is quadratic in the state and the control variable and the optimal reorientation maneuver requires the use of only two control torques. Because of the passivity characteristics and the cascade structure of the system we first state two optimal regulation problems for the dynamics and the kinematics subsystems, separately. In this case one is able to find explicit solutions to the associated HamiltonJacobi equations. For the complete system we present solutions for two partial cases. The first case is when there is no penalty on the control input. In this case, one can asymptotically recover the cost for the kinematics by making the dynamics sufficiently fast. The second case investigates restrictions imposed by optimality considerations on the aforementioned control law in order to avoid high gain. Introduction The optimal contro...