Extremum seeking loops with assumed functions

Extremum seeking (also peak-seeking) controllers are designed to operate at an unknown set-point that extremizes the value of a performance function. This performance function is approximated by an assumed function with a finite number of parameters. These parameters, which are estimated online, are assumed to change slowly compared to the plant and compensator dynamics. Philosophically, the approach of assuming a function is in contrast with traditional approaches that use time scale separation between gradient computation and function minimization and the system dynamics. To analyze our current scheme, quadratic functions or exponentials of quadratic functions are assumed as approximations to the performance function. This allows the peak-seeking control loop to be reduced to a linear system. For this loop, compensators can be designed and robust performance and stability analysis of the loop due to parameter uncertainty in the assumed performance functions can be obtained