On Local Feedback Stabilization of Nonlinear Systems in Critical Cases

Abstract

Recently, it has been shown tha t if a so-called stability characteristic matrix of a nonlinear system having multiple critical eigenvalues is relaxed negative definite, the equilibrium point is locally asymptotically stable. A c x c real symmetric matrix A is relaxed negative definite if z 7 A z < 0 for any nonzero z E R;,+ C IR'. Based on this direct stability criterion, the task of local feedback stabilization problems for nonlinear control systems in the corresponding settings is reduced to constructing feedback laws such that the closed-loop stability characteristic matrix be relaxed negative definite. Moreover, it is shown tha t , depending on the number of controllable modes of the linearized system, there may exist a variety of design options for achieving stabilization. Application to the stabilization of the tethered satelite systems during station-keeping is also discussed. -