The article deals with the problem of adaptive control of a single-axis vibratory gyroscope. In order to improve both the control quality of the vibratory gyroscope and the identifying properties of the adaption algorithm, an extension of the original system is proposed by adding additional integrators to the gyroscope inputs, i. e. enhancing of the astatism. On the other hand, smoothness of the electrostatic forces applied to the axes of a gyroscope is improved. Smooth control algorithms as well as algorithms of the sliding mode with a tuning surface for the gyroscope with integrator was designed by the speed bigradient method (SBGM). SBGM consists of three stages. At the first stage, an "ideal" virtual control for an output subsystem, which is the gyroscope, is designed. The "ideal" virtual control ensures achievement of the control goal for the output subsystem, assuming the object parameters are known. At the second stage, the unknown parameters are replaced with the tunable ones, and adaptation algorithm is designed. At the third stage, a deviation of the manifold, that is a difference between the input subsystem, which is an integrator of the output and virtual control, is selected. Control law ensuring the convergence of the system trajectories to the manifold is designed. The relevance of adding of the additional integrators to the inputs of the vibratory gyroscope; analysis of the robust properties of designed adaptive control algorithms with respect to the additive disturbances; comparative analysis of the convergence, accuracy, and presence of the identifying properties of the designed algorithms are presented. The theoretical results are proved by a computer simulation in MATLAB.