Extremum seeking loops with assumed functions: estimation and control

Extremum seeking (also peak-seeking) controllers are designed to operate at an a priori unknown set-point that extremizes the value of a performance function. Traditional approaches to the problem assume a time-scale separation between the gradient computation and function minimization and the system dynamics. The work here, in contrast, assumes that the performance function can be approximated by an assumed function with a finite number of parameters. These parameters axe estimated on-line and the extremum seeking controller operates based on these estimated values. To analyze our current scheme, quadratic functions or exponentials of quadratic functions are assumed as approximations to the performance function. A significant advantage of these functions is that they allow the peak-seeking control loop to be reduced to a linear system. For such a loop, the wealth of linear system analysis and synthesis tools can be employed. First, the control loop is analyzed assuming that the parameters in the function are known (full information case) and then when the parameters are estimated on line (the partial information case).