Adaptive Control via Extremum Seeking: Global Stabilization and Consistency of Parameter Estimates

We propose an extremum seeking (ES) based algorithm for reference tracking and stabilization of unstable discrete-time systems with unknown control directions. The probing signal is a martingale difference sequence (m.d.s.) with a vanishing variance. It is proved that almost surely (a.s.) the input and output signals are uniformly bounded, and the tracking error converges to zero. It is also shown that (a.s.) the parameter estimates converge toward their true values. This shows that the proposed algorithm can also perform identification of open-loop unstable systems with unknown control directions.