This study considers the problem of controller synthesis to control position stabilization of triaxial powered gyrostabi-lizer with moving base that operates under external disturbing factors. To supress the uncertain external disturbances with finite energy, H∞-optimization approach is used. The Kalman filtering and H∞-suboptimal state estimate feedback as well as the concept of mixed H2/H∞ control are applied in the case of noisy measurement channels. To solve the controller synthesis problems, we apply semidefinite programming and convex optimization. The analysis and comparison of the results of the position stabilization problem solution by various methods are presented.