On the convergence rate of extremum seeking control

Extremum seeking control (ESC) is an adaptive optimization method originally proposed for static systems but later extended to Hammestein/Wiener-like systems and more recently also to more general dynamic systems. In the latter case the focus has been on proving convergence and stability of solutions in the vicinity of the optimum. The proofs are in general based on combining asymptotic methods like singular perturbations and averaging, which leads to a three time-scale factorization of the problem where the control action is forced to be several orders of magnitude slower than the open-loop dynamics of the plant. This implies that the convergence rate will be impractically slow for many applications. In this paper, we employ Tikhonov theory and averaging to study the rate of convergence while employing only two time-scales. In particular, the analysis places no restrictions on the rate of the gradient estimation and therefore allows for significantly faster control compared to the conventional approach. The plant is approximated as a linear parameter varying system (LPV) which is then used to derive a global quantitative expression for the convergence rate in terms of the ESC parameters and the frequency response of the LPV plant. For Hammerstein/Wiener-like systems, the derived expression is used to show that the ESC loop behaves like a gradient descent method while it has a more complex behavior in the general case. Finally, an isothermal biochemical reactor is used to illustrate the results and some of the difficulties which might arise in the general case, such as the fact that the convergence rate can be low locally even if the gradient of the cost function is large.