Stability and chaotic dynamics of a rate gyro with feedback control under uncertain vehicle spin and acceleration

An analysis of stability and chaotic dynamics is presented by a single-axis rate gyro subjected to linear feedback control loops. This rate gyro is supposed to be mounted on a space vehicle which undergoes an uncertain angular velocity ωZ(t) around its spin axis. And simultaneously acceleration ω̇X(t) occurs with respect to the output axis. The necessary and sufficient conditions of stability for the autonomous case, whose vehicle undergoes a steady rotation, were provided by Routh–Hurwitz theory. Also, the degeneracy conditions of the non-hyperbolic point were derived and the dynamics of the resulting system on the center manifold near the double-zero degenerate point by using center manifold and normal form methods were examined. The stability of the non-linear non-autonomous system was investigated by Liapunov stability and instability theorems. As the electrical time constant is much smaller than the mechanical time constant, the singularly perturbed system can be obtained by the singular perturbation theory. The Liapunov stability of this system by studying the reduced and boundary-layer systems was also analyzed. Numerical simulations were performed to verify the analytical results. The stable regions of the autonomous system were obtained in parametric diagrams. For the non-autonomous case in which ωZ(t) oscillates near boundary of stability, periodic, quasiperiodic and chaotic motions were demonstrated by using time history, phase plane and Poincaré maps.