Feedback Stabilization via Center Manifold Reduction with Application to Tethered Satellites

Abstract

Center manifold reduction has recently been introduced as a tool for design of stabilizing control laws for nonlinear systems in critical cases. In this dissertation, the center manifold approach is elaborated for general such nonlinear systems in several critical cases of interest, and the results are applied to the control of tethered satellite systems (TSS). In addition, to address stability questions for satellite deployment via TSS, we obtain new results in finite-time stability theory. The critical cases considered in the general feedback stabilization studies include the cases in which the system linearization possesses a simple zero eigenvalue (of multiplicity one or two), a pair of simple pure imaginary eigenvalues, one zero eigenvalues along with a pair of simple pure imaginary eigenvalues, and two pairs of simple pure imaginary eigenvalues. The calculations involve center manifold reduction, normal form transformations, and Liapunov function construction for critical systems. These calculations are explicit. The tethered satellite systems considered here consist of a satellite and subsatellite connected by a tether, in orbit around the Earth. The Lagrangian formulation of dynamics is used to obtain a nonlinear system of ordinary differential equations for TSS dynamics. For simplicity, a rigid, massless tether is assumed. Linear analysis reveals the presence of critical eigenvalues in the station- keeping mode of operation. This renders useful results on stabilization in critical cases to this application. The control variable assumed is tether tension feedback. Besides the design of stabilizing station-keeping controllers, stability of deployment and instability of retrieval are also shown for a constant angle deployment/retrieval scheme.