Extremum seeking loops with quadratic 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 timescale 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 a quadratic function with a finite number of parameters. These parameters are estimated on-line and the extremum seeking controller operates based on these estimated values. A significant advantage of a quadratic function is that it allows 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 analysed 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).