Using Sliding Mode in Nonlinear Extremum Seeking System

In this paper, the control problem for SISO nonlinear system with unknown setpoint that is defined as the minimum (maximum) of some quality function is considered. The plant model is a serial connection of a nonlinear dynamic part and a static map with an extremum. Using the time-scale separation idea, the closed-loop system can be represented as two independent subsystems with different processes rates as it was shown in the previous papers. It is recommended to organize the sliding mode in the system inner loop in order to reduce the influence of the time-varying dynamic part parameters and uncontrolled disturbances. It was shown that the relay controller provides an additional effect in the form of the self-oscillations which parameters can be selected in a certain way. This self-oscillation signal can be used for a gradient estimate similar to a classic synchronous detection method. Therefore, the gradient estimator can be realized without using a special oscillations generator. In order to the extremum seeking in the outer loop it is enough to form a typical integral controller with information about the gradient. A detailed calculation procedure for the system with controlled oscillations is presented in this paper. It consists of recommendations on the choice of controller parameters as well as the amplitude and frequency of the search self-oscillations. A simulation example in MatLab is presented to illustrate the theoretical results.