Convergence of extremum seeking control for state-dependent and bilinear objectives

Extremum seeking control (ESC) is a model-free and adaptive scheme for optimizing steady-state performance of dynamic systems in real time, particularly when system models are unavailable or uncertain. Classical ESC relies on exciting the system with a sinusoidal dithering signal, thereby ensuring convergence to the optimum, provided that the dither excitation frequency remains well below the dominant system dynamics. However, this time-scale separation constraint limits convergence speed and robustness, especially in systems whose dominant dynamics begin at low frequencies. Enhanced ESC methods, such as proportional-integral schemes, phasor-based estimation techniques, and frequency-domain approaches, address the time-scale separation limitation. However, the understanding of how objective definitions with identical steady-state optima affect convergence remains limited. This paper analyzes ESC closed-loop convergence in dynamical systems for state-dependent and bilinear (state-input-dependent) objectives using a Wiener-type model framework. Although both objectives share equal steady-state optima, they exhibit fundamentally different convergence behavior when the dither frequency is chosen beyond the system’s dominant dynamics (fast time scale). New insights are obtained through sequential linearization combined with frequency-domain examinations and signal-based analyses. Furthermore, improvements to the classical ESC scheme for dynamical bilinear objectives are proposed to relax time-scale separation constraints, enabling higher dither frequencies and leading to consistent and accelerated convergence.