Extremum seeking control of the CANON process - existence of sub-optimal stationary solutions

The paper considers extremum seeking control (ESC) for on-line optimization of the CANON process, a new and potentially highly effective process for ammonium removal from wastewater. For gradient estimation we employ the classical method based on periodic excitation. From simulations we find that the ESC scheme can lock onto sub-optimal stationary solutions, far removed from the optimal solution, and that the ESC may have multiple stationary solutions for given controller parameters. The cause of this is investigated through analysis of a general dynamic model. Based on the analysis, it is shown that for systems for which the optimum corresponds to the input-output transfer-function having a transmission zero at the origin, there will in general exist a number of stationary solutions to the ESC with periodic excitation. The solutions are characterized by the phase lag of the system, rather than a zero gradient of the objective function, and are hence in general not related to the optimality conditions. For systems that can be described by Hammerstein or Wiener models, as typically considered in ESC, the solution will in general correspond to the zero gradient condition fulfilled at the optimum. As shown, the CANON process can not be described by Hammerstein or Wiener models, and this then explains the observed existence of sub-optimal stationary solutions.