Extremum seeking-based tracking for unknown systems with unknown control directions

We develop controllers which perform trajectory tracking for a class of unknown linear and nonlinear systems. In the linear case we work under the assumption that the time-varying input vector, which is otherwise unknown, satisfies a persistency of excitation condition over a sufficiently short window. In the multi-input nonlinear case the square of the unknown time-varying input matrix has a known lower bound. Our design does not guarantee perfect regulation to the desired trajectory, but ensures semiglobal uniform ultimate boundedness of the error system and prevents large overshoots even when the initial estimate of the control direction is wrong. The stability analysis is inspired by the approach in a recent work by Dürr, Stankovic, and Johansson, which combines a Lie bracket second-order averaging result of Gurvits and Li with an extension of the perturbation theory semiglobal practical stability result of Moreau and Aeyels.