A Small-Gain Approach To Extremum Seeking For Nonlinear Uncertain Systems

Extremum seeking can search the optimal point of a objective function with the gradient unknown by means of oscillation control,which is one of the most important optimization methods for uncertain systems.In the past two decades,the theoretical research on extremum seeking control has received extensive attention and obtained a series of theoretical results.Among them,extending the traditional static system optimization to dynamic system optimization is one of the hot spots of extremum seeking.However,the existing results require either that the desired optimization system be asymptotically stable or practically stable atany given operating point,or that the system have specific dynamics.There are several representative systems in nonlinear control,such as strict feedback type and output feedback type,which have been studied most widely.However,the result of extremum seeking of these systems is rare.One of the main challenges is the stability synthsesis of optimization and control.The main contribution of this paper is that we proposed an extremum seeking control system structure based on small-gain,and solved the extremum seeking control problem of nonlinear uncertain systems with strict feedback or output feedback.The specific work is as follows:(1)This paper gives the small-gain method of extremum seeking control for nonlinear uncertain systems,which transform the extremum seking problem of complex nonlinear systems into two relatively independent problems,optimization and control.Gains are used to describe the interaction between the two parts,and the small-gain theorem is used to ensure the stability of the two parts after synthsesis.In order to describe the gains between optimization and control,differential input to output stability is introduced for the first time.(2)This paper gives the extremum seeking controller design method of nonlinear uncertain systems with strict feedback or output feedback,and ensures the differential input to the output stability of the controlled system,respectively by static state feedback and observer based dynamic state feedback.And the closed-loop system satisfies the condition of smallgain at the same time,which ensures that the extremum seeking system can realize the output optimization of the nonlinear uncertain system with the gradient unknown.The relevant results are verified by numerical simulations.The simulation results show that the proposed extremum seeking control structure based on small-gain can effectively overcome the contradiction between optimal target and nonlinear uncertain dynamics,and effectively solves the extremum seeking control problem of strict feedback and output feedback nonlinear uncertain systems.