ASYMPTOTICALLY STABILIZING FEEDBACK CONTROLS

The problem considered is the existence and construction of an asymptotically stabilizing feedback control for a rest solution of an n-dimensional, single-input, affine control system. The method is to construct high order, homogeneous approximations of the vector fields defining the control system which then describe an approximating control system such that an asymptotically stabilizing control of the latter system is a local asymptotically stabilizing control of the original system. Unlike linear approximations, there are many possible nonlinear homogeneous approximations. These are constructed from Lie algebra filtrations obtained by assigning various weights to the defining vector fields. One seeks a weighting, hence an approximating system, which satisfies the various known necessary conditions for the existence of an asymptotically stabilizing feedback control and for which such a control can be constructed as the solution to a homogeneous Hamilton-Jacobi-Bellman equation associated with an optimization (regulator) problem. © 1991.