Multiple stationary solutions to the extremum seeking control problem

Extremum seeking control was originally proposed for adaptive optimization of static systems and later extended to Hammerstein and Wiener systems. More recently, stability and convergence results were presented also for general type dynamic systems with a focus on the local behavior around the optimum and under assumptions of relatively slow gradient estimation and control. In this paper we derive properties characterizing any stationary solution of the extremum seeking control scheme, i.e., we do not restrict ourselves to solutions close to optimum and allow for any frequency in the sinusoidal perturbation based gradient estimation scheme. By considering the linear properties around a stationary solution of the system, we show that stationary solutions are characterized by either a zero gradient or a phase lag condition. The former condition is satisfied at the optimum only for systems in which the zero gradient at the optimum is due to a static nonlinearity. The phase lag condition is shown to be satisfied close to the optimum for low frequency excitations, but can also be satisfied at solutions arbitrarily far from the optimum. The results imply that the extremum seeking control scheme applied to general type dynamic systems can have multiple stable stationary solutions of which some are sub-optimal and potentially far removed from the optimum. For illustration we consider extremum seeking control of a tubular bioreactor, displaying a maximum yield, and show that the closed-loop has two saddle-node bifurcations resulting in a total of three possible stationary solutions for some perturbation frequencies. A stable sub-optimal solution, with a yield less than 10% of the optimal yield, exists even with relatively slow gradient estimation.