New results on global stabilization of feedforward systems via small feedback

We previously (1998) presented a general framework for stabilizing a chain of integrators perturbed by a higher-order vector field, by a nested saturation controller whose saturation levels depend on the system state. Although the previous result has been used to solve a number of global stabilization problems in the low dimensional case, it cannot straightforwardly be applied to higher dimensional nonlinear systems. To overcome this drawback, we present here, in contrast to the state-dependent nested saturation design, a Lyapunov-based small control scheme for a rather general feedforward system whose linearization is not necessarily controllable. A set of weaker growth conditions are identified under which a bottom-up, step-by-step, recursive design procedure can still be successfully used to construct a globally stabilizing controller whose norm is arbitrarily small. Our approach relies heavily on an effective coupling of the forwarding strategy for feedforward systems with a nonlinear chain of integrators, and of the synthesis technique of using arbitrarily small controls, developed for global stabilization of non-affine systems with marginally stable free dynamics