On solution multiplicity and convergence rate in extremum seeking control With applications to the CANON process

Abstract: Extremum seeking control (ESC) is a classical adaptive control method aimed at locating and tracking optimal operating conditions in complex non-linear plants. Early results on ESC were restricted to plants that could bedescribed by Wiener or Hammerstein models. However, recent results haveshown that ESC will possess a stationary solution close to the optimum also for more general dynamical systems, provided the gradient estimation and feedback is sufficiently slow relative to the process dynamics. This thesis addresses the uniqueness of this solution and the achievable rate of convergence.The motivation for the work stems from the need to optimize a complex biofilm reactor, the CANON process, which if operated near a narrow optimum may significantly lower the cost of ammonium removal in wastewater treatment. Simulations of ESC applied to the CANON process reveal that, depending on initial conditions and tuning parameters, the ESC loop may converge to stationary solutions far removed from the optimum and that multiple stationary solutions may exist. Analysis of a general model for the ESC loop shows that the stationary solutions are characterized either by a gain condition or a phase lag condition on the locally linearized system, the latter indicating that the ESC loop can act as a phase-lock loop. The phase lag condition is shown to be satisfied close to the optimum, but can be fulfilled also at operating points with no relation to the optimality criterion whatsoever and this serves to explain the observed solution multiplicity. Bifurcation theory is employed to further analyze the stationary solutions of the ESC loop and conditions for existence of saddle-node bifurcations are derived. A saddle node bifurcation implies a hard loss of stability and the existence of multiple stationary solutions. It is also demonstrated, using examples, that the ESC loop may undergo other types of bifurcations, including period doubling bifurcations into chaos. For the considered example, the resulting chaotic solution is significantly closer to optimum than the underlying nominal limit cycle. Previous results on ESC applied to general dynamic systems have relied on the use of asymptotic methods, such as singular perturbations and averaging. This has resulted in a three time-scale separation of the problem, in which the gradient estimation and control have been forced to be significantly slower than the open-loop process dynamics. For most processes, including the CANON process studied in this thesis, this renders ESC of little practical use and we therefore consider relaxing some of the restrictive assumptions. Inparticular, we allow for any gradient estimation rate and significantly faster gradient feedback as compared to previous studies. Using a linear parameter varying (LPV) description of the plant, quantitative expressions for the convergence rate in terms of the ESC tuning parameters and plant properties are derived.