Adaptive stabilization of non-linear systems at unknown equilibrium points: An invariant manifold and uniform δ-persistent excitation approach

The adaptive stabilization of a class of continuous-time non-linear systems (not necessarily chaotic) at the unknown equilibrium points is treated in this paper. The controller is designed based on the invariant manifold theory and on the concept of uniform δ-persistent excitation (Uδ-PE). Firstly, the case when there are no constraints on the parameter estimates is presented and then the case when the parameter estimates are confined to a certain region is discussed. In the last case the proposed alternative does not require knowledge of bounds but new estimates have to be introduced. Finally, the behaviour of the proposed scheme is verified through simulations on the Lorenz system for both chaotic and non-chaotic cases.