Geometric based estimation and nonlinear PI controller for dynamic optimization problem

In this paper, the minimization of an unknown but measurable cost function of the state variables of nonlinear systems governed by uncertain dynamics is considered. An extremum-seeking algorithm is proposed to solve this uncertain dynamic optimization problem without the need for a time-scale separation. The Lie derivatives of the convex cost function with respect to nonlinear dynamics of the system are regarded as time-varying parameters. A new technique based on the concept of invariant manifolds is proposed for the adaptive estimation of the time-varying parameters. A nonlinear proportional-integral approach is then used to formulate the extremum-seeking controller. This approach is shown to avoid the need for a time-scale separation in real-time optimization problem. The effectiveness of the proposed method is illustrated with a simulation example.