Input-to-state stabilization for a class of normal form systems using extremum seeking

We present a new input-to-state-stability analysis for a class of normal form systems with unknown control directions using a recently introduced global extremum seeking algorithm in the presence of disturbances. Our new state transformation and Lyapunov analysis based on a quadratic Lyapunov function for the averaged system are a departure from previous averaging-based approaches, and cover general classes of nonlinear systems including generalized chains of integrators, providing less conservative bounds on the dither frequency compared with earlier results. Our numerical examples illustrate the effectiveness of our analysis. We also prove a robustness property for the algorithm under small time-varying delays.