Extremum Seeking for Linear Time-Varying Systems with Unknown Control Directions

We consider bounded extremum seeking controls for time-varying linear systems with uncertain coefficient matrices and measurement uncertainty. Using a new change of variables, Lyapunov functions, and a comparison principle, we provide practical exponential stability bounds for the states of the closed loop systems that hold for all nonnegative times. For the first time for linear time-varying systems with unknown control directions, we consider bounded extremum seeking controls in the presence of uncertain time-varying input delays with small time-varying delay uncertainties, and we provide reduction model controllers to compensate for the constant part of the delays.