Controlling the Primary Mirror in a Space-Based Telescope Utilizing an Eigenstructure Assignment Technique

The feasibility of using a sparse array of mirrors in a satellite system is being evaluated. This study focuses on the control laws necessary for achieving the minimization of the effect of vibration which happens at the primary mirror of the satellite, by using the eigenstructure assignment technique. White Gaussian noise was assumed as the external input. AFRL has developed a 79 state model with 9 control inputs and 9 outputs, 3 sparse array mirrors, and a base. To extend the theory, a simplified model with 2 mirrors and a base which has 12 states, 4 control inputs and 8 outputs was developed. The system is not completely controllable nor observable because there are 4 rigid body modes. The control law starts from realizing the states which are controllable or uncontrollable. Then only for the controllable states, the eigenstructure assignment technique is applied. The closed-loop eigenstructure was determined by analyzing the open-loop modes. To find the best value for the eigenstructure, Newton's line search method was applied. The relative motions of the primary and secondary mirrors were described by wave reflection dynamics. Results show that this technique achieves a reasonable amount of control use for the objective and relates these result to those obtained from linear quadratic regulator (LQR) technique.