UC San Diego

The work in this thesis describes our attempts for theoretical advancements to and the experimental results of extremum seeking schemes. Extremum seeking is a form of adaptive control where an unknown but assumed concave objective function is maximized. While many black-box optimization schemes exist, each with their own particular assumptions and nomenclature, extremum seeking analysis in the controls community typically deals with an unknown objective function associated with the non-linear equilibrium profile of a dynamic process. An elegant and rigorous analysis paradigm exists for the general structure of this problem. It exploits the time scale difference in process, estimation, and learning dynamics so as to use averaging and singular perturbation theory. The slowest timescale approximates a continuous optimization algorithm. The middle timescale constructs estimates for the `slower' learning algorithm. And the fastest timescale are the process dynamics.

In reality, optimization problems are typically not unconstrained. A real process will likely have some physical limitation or a even a user imposed virtual limitation. In order to enforce input constraints to the extremum seeking setting, we must better understand the constrained optimization algorithm the "average" extremum seeking system will approximate.

We initially investigate the use of continuous-time gradient based optimization scheme for the convex programming problem. The dynamics of the optimization parameter are described by a continuous projection of the map's gradient and appear an appropriate target system for the average extremum seeking system. We then assume a general unknown quasi-concave objective function and provide that the 1-dimensional sinusoidal extremum seeking scheme is domain-limited semi-globally practically asymptotically stable about the constrained minimum. The incorporation of a projection operator prohibits the estimate system from leaving the constraint set and the perturbed system from leaving the constraint set dilated by the perturbation amplitude.

The usual working assumption is that the extremum seeking process will regulate the system to a neighborhood of the output's extremal value. However, it does not need to be the case. We present a generalization to the scalar Newton-based extremum seeking algorithm which through perturbation-induced measurements of an unknown steady-state input-to-output map, maximizes the map's higher derivatives. The Newton-based extremum seeking approach removes the dependence of the average convergence rate on the unknown Hessian of the higher derivative, an effort to improve performance over standard gradient-based extremum seeking. We prove local stability of the algorithm for general nonlinear equilibrium profiles of dynamic maps and extend this abstraction to multiplayer non-cooperative games, but limit our attention to a 2-player game for notational simplicity and length considerations.

Experimental results and a discussion of application process are shown for a negative ion surface plasma source . The converter voltage of a $H^{-}$ ion source is utilized by the Los Alamos linear accelerator is optimized in order to maximize the output intensity of the beam current.