Extremum seeking control of the CANON process—Existence of multiple stationary solutions

• Extremum seeking control (ESC) is applied to optimization of the CANON process used in wastewater treatment.
• Simulations reveal existence of multiple stationary solutions to the ESC problem despite the objective function being convex.
• Conditions for stationarity of the ESC algorithm are presented.
• The results imply that ESC may produce multiple stationary solutions of which only one is related the conditions at the extremum point.

The paper considers the use of extremum seeking control (ESC) for real-time optimization of the completely autotrophic nitrogen removal over nitrite (CANON) process. In particular, the feasibility of employing relatively fast estimation and control, compared to the open-loop dynamics, is considered. From simulations it is found that the ESC can lock onto stationary solutions far removed from the optimum and that, for some control parameters, multiple stationary solutions may exist. To verify the simulation results we analyze a general dynamic model for ESC and show that all stationary solutions can be characterized by simple conditions on the gain and phase lag of the frequency response of the locally linearized system. In order to relate the derived conditions to conditions for optimality, we consider characteristic dynamic properties of systems displaying extremum points and show that the phase lag condition implies that a stationary solution in the vicinity of the optimum is likely to exist even with relatively fast estimation and control. However, the phase lag condition can also be fulfilled at operating points with no relation to the optimum whatsoever. Thus, the ESC can yield multiple stationary solutions and for some initial conditions lock onto solutions far removed from the optimum. A simple stability analysis is presented to verify that multiple solutions can be locally asymptotically stable. A simple bioreactor model is considered to illustrate the results and we show that the ESC provides three stationary solutions, of which two are stable and one unstable, even for relatively slow estimation schemes. We also confirm that the stationary solutions of the CANON process, found through simulations, fulfill both the phase lag condition and the stability criterion derived in the paper.