Discrete time extremum seeking using stochastic perturbations

Extremum seeking using periodic, deterministic perturbations has been an effective method for non-model based real time optimisation when only limited knowledge of the system is available. However, periodicity can naturally lead to predictability which is undesirable in some applications and unrepresentative of some biological optimisation processes such as bacterial chemotaxis. With this in mind, it is useful to investigate the benefit of employing stochastic perturbations in the context of a typical extremum seeking architecture, and to compare the approach with existing stochastic optimisation techniques. In this work, we show that the convergence towards the extremum of a static map can be guaranteed with the stochastic extremum seeking algorithm, and quantify the behaviour of the system at the extremum in terms of the extremum seeking constants and map parameters. Finally simulation results are used to demonstrate the stochastic closed loop system convergence and behaviour about the extremum. For the sake of analogy with the classical methods of stochastic approximation, stochastic extremum seeking in this paper is pursued in discrete time.