Analytical Modeling for the Shape Control of an Adaptive Composite Satellite Dish

A three-dimensional dynamic analysis of a satellite dish in spherical coordinate system is investigated. The method of separation of variables is employed to obtain the explicit 3D solution of the partial differential governing equation of the active composite satellite dish (ACSD). Then, the mode shape functions are expanded as a combination of periodic functions, associated Legendre functions, and spherical Bessel functions. The validation of the theoretical model is performed by comparing the developed analytical mode shapes with finite element method (FEM) mode shapes. Also, using the developed analytical model, the disturbance observer (DOB) controller is employed for the ACSD shape control. The numerical results show that, by employing the DOB controller, more accurate shape control of the satellite dish is achieved and less control energy is consumed, when piezoelectric actuator patches are placed on their optimal locations.