A universal extremum seeking-based stabilizer for unknown LTV systems with unknown control directions

We solve the problem of model-independent semiglobal exponential practical stabilization for any linear time-varying single-input system under the assumption that the time-varying input vector, which is otherwise unknown, satisfies a persistency of excitation condition over a sufficiently short window. We employ an extremum seeking algorithm, with the square of the norm of the state used as the cost function. 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 a perturbation theory semiglobal practical stability result of Moreau and Aeyels. As an alternative to the classical Nussbaum gain type of a design by Mudgett and Morse, when applied to time-invariant systems, our design gives up globality and perfect regulation to the origin but ensures exponential practical stabilization and prevents large overshoots that characterize the Nussbaum gain design when the initial estimate of the control direction is of the wrong sign. Furthermore, while the Nussbaum gain design leads to instability when the control direction periodically changes sign, our design guarantees stability.