A Time-Delay Approach to Extremum Seeking for Multi-variable Static Map with Measurement Bias

For n-dimensional static quadratic maps with a time-varying bias, we propose a time-delay gradient-based extremum seeking (ES) method and provide a simplified and less conservative practical stability analysis under essentially relaxed assumption on the unknown Hessian. We present the detailed new analysis in the discrete-time, where the time-delay approach to ES has not been studied before, and extend it to the continuous-time. As in the recently introduced results on the time-delay approach (in the context of two-dimensional mappings defined over continuous-time systems), the original system is reformulated as a time-delay model, specifically a neutral-type system following Hale’s formulation in the continuous-time setting. This reformulated system represents an \(\textrm{O}(\varepsilon )\) -order perturbation of its averaged linear ODE counterpart, where \(\varepsilon \) denotes the averaging period. A transformation is further proposed to recast the neutral-type system into a linear ODE, wherein the \(\textrm{O}(\varepsilon )\) terms are treated as disturbance-like components incorporating distributed delays with duration proportional to the small parameter \(\varepsilon \) . The use of the variation of constants formula enables a practical stability analysis that replaces the conventional Lyapunov-Krasovskii (L-K) framework, yielding less restrictive conditions and superior performance. Numerical examples illustrate the efficiency of the new approach.