Non-linear control with Lyapunov stability applied to spacecraft with flexible structures

Abstract

The method of input-output feedback linearization incorporating the Lyapunov stability analysis was applied in this study to design a stable control law for the problem of reorienting a spacecraft with flexible appendages. Only three mutually orthogonal torque actuators on the hub are required for the proposed control law to perform a desired simultaneous multi-axis reorientation. Mathematical modelling of the system gives a set of coupled ordinary and partial differential equations, which includes attitude dynamics of the spacecraft and dynamics of the flexible structures. To simplify the system equations for controller design, deformations of the flexible structures were assumed to be small and mode-shape functions were applied first. Furthermore, the set of non-linear equations governing the attitude motions was transformed into a Euler parameters representation. Through the method of feedback linearization and vector subtraction in the Euler parameters space, the dynamics of the attitude errors were formulated as a set of stable second-order ordinary differential equations with constant coefficients, which are the gains of the attitude feedback control law. Vibration control of the flexible structures in the form of adaptive damping was also derived from the procedure of Lyapunov stability analysis and becomes a part of the attitude feedback control law. The stability of the overall dynamic system can be achieved by tuning the selected control gains in the Lyapunov analysis. Attitude manoeuvre of a model spacecraft was tested using the proposed control law and the simulation results were compared for the cases with and without adaptive structural damping. This study also shows that, by selecting the adaptive damping coefficients, the optimal time and torque manoeuvre of the flexible spacecraft can be determined.