Stability and Chaotic Dynamics of a Rate Gyro with Feedback Control

Abstract

An analysis is presented of a single-axis rate gyro subjected to linear feedback control mounted on a space vehicle that is spinning with uncertain angular velocity ωZ(t) about its spin of the gyro. The stability of the nonlinear nonautonomous system is investigated by Liapunov stability and instability theorems. As the electrical time constant is much smaller than the mechanical time constant, the full singularly perturbed system is obtained. We study the stability of the system by forming a Liapunov function candidate as a linear combination of the Liapunov functions for the reduced and boundary-layer systems. When the perturbation near angular velocity ωZ(t) of the space vehicle is harmonic, the feedback control system reduces to a planar system of parametrical excitation by the singular perturbation theory. Using the Melnikov technique, we can give criteria for the existence of chaos in the gyro motion. The numerical results, in which the existence of chaos is demonstrated, are presented in the form of phase plane, Poincaré maps, bifurcation diagrams and Lyapunov exponents.