Bounded extremum seeking for single-variable static map with large measurement delay via time-delay approach to averaging

In this paper, we present a time-delay approach to gradient-based bounded extremum seeking (ES) with large measurement constant delay, for an unknown single-input static quadratic map. We assume that the extremum point and the Hessian H belong to known intervals, whereas the sign of H is known. We apply a time-delay approach to the bounded ES system and arrive at the neutral type system with a nominal linear delayed system. We present the latter system as a retarded one and employ variation of constants formula for practical stability analysis. Explicit conditions in terms of simple scalar inequalities depending on tuning parameters and delay are established to guarantee the practical stability of the bounded ES control systems. Given any delay and neighborhood of the extremum point and through the solution of the constructed inequalities, we find lower bounds on the dither period that ensures the practical stability.