Stabilizing and tracking unknown steady states of dynamical systems

An adaptive dynamic state feedback controller for stabilizing and tracking unknown steady states of dynamical systems is proposed. We prove that the steady state can never be stabilized if the system and controller in sum have an odd number of real positive eigenvalues. For two-dimensional systems, this topological limitation states that only an unstable focus or node can be stabilized with a stable controller, and stabilization of a saddle requires the presence of an unstable degree of freedom in a feedback loop. The use of the controller to stabilize and track saddle points (as well as unstable foci) is demonstrated both numerically and experimentally with an electrochemical Ni dissolution system.