Non-Uniqueness of Stationary Solutions in Extremum Seeking Control

Abstract: Extremum seeking control (ESC) is a classical adaptive control method for steady-state optimization, purely based on output feedback. It is well known that the extremum seeking control loop, under certain mild conditions on the controller, has a stable stationary periodic solution in the vicinity of an extremum point of the steady-state input-output map of the plant. However, this is a local result only and this paper investigates whether this solution is necessarily unique given that the underlying optimization problem is convex. We first derive a necessary condition that any stationary solution of the ESC loop must satisfy. For plants in which the extremum point is due to a purely static nonlinearity, such as in Hammerstein or Wiener plants, the condition involves the steady-state gradient. However, for more general plants the necessary condition involves the phase lag of the locally linearized plant, indicating the possible existence of solutions without any relationship to optimality. Combining the derived necessary condition with the existence of a local solution close to the optimum, we employ bifurcation theory to trace out branches of stationary solutions. We derive conditions on when these branches bifurcate, resulting in multiple stationary solutions. The results show that cyclic fold bifurcations may exist, resulting in the existence of multiple stationary period one solutions, of which only one is related to the optimality conditions. We illustrate the results through an example in which the conversion in a chemical reactor is optimized using ESC. We show that at least five stationary solutions may exist simultaneously for realistic control parameters, and that several of these solutions are stable. One consequence of the non-uniqueness is that one in general needs to start close to the optimum to ensure convergence to the near-optimal solution.