Extremum Seeking Control for Low Order System via Sliding Mode

The problem of the automatic extremum seeking control for a second-order dynamic SISO plant with incomplete information about its parameters is discussed. The plant model is a serial connection of the dynamic part and the static map with a previously unknown minimum (maximum) value. It is shown that a controller based on the method of sliding modes provides a search for an extremum with a given quality. It is necessary to define the internal variables of the dynamic part and the gradient of the static characteristic for the system to function. A special low-pass filter is used to evaluate the internal variables of the dynamic part. A feature of the system is the processes are different in time scale due to the practical implementation of the controller. The main one is a slow transient process to the extremum point, while the fast process occurs in the internal loop of the system and has the form of self-oscillations. Their parameters can be selected in a certain way. This makes it possible to obtain an estimate of the gradient, similar to the method of synchronous detection without using an additional generator of search oscillations. An analysis of the closed-loop system and recommendations for calculating the parameters of the controller are presented in this paper. The simulation in MatLab illustrates the properties of the extremum seeking controller based on sliding modes.