Proportional-integral extremum-seeking control

This paper proposes a proportional-integral extremum-seeking control technique. The technique is a generalization of the standard perturbation based techniques that provides fast transient performance of the closed-loop system to the optimum equilibrium of a measured objective function. The main contribution is that the formal development of this technique does not require the need for a time-scale separation. It is assumed that the equations describing the dynamics of the nonlinear system and the cost function to be minimized are unknown. The cost function and its first time derivative are assumed to be measured. The equilibrium of the unknown dynamics are assumed to be asymptotically stable and the cost function dynamics are assumed to be of relative order one. It is shown that the closed-loop ESC can also stabilize the steady-state optimum for an unknown unstable nonlinear control system. The stabilization result is quite general and provides a new approach to output feedback control of nonlinear systems.The effectiveness of the proposed approach is demonstrated using several simulation examples.