Extremum Seeking System with PID controller for Second-Order Non-Stationary Plant

In this paper, an extremum seeking problem for a dynamic SISO plant is considered. Its dynamic part is represented by a linear second-order model with non-stationary parameters and a quality function has a previously unknown minimum (maximum). Since the plant contains two different components, it is possible to organize cascade control and calculate individual circuits independently of each other. In the inner circuit it is proposed to use the PID controller and determine its parameters taking into account the method of large coefficients. It is shown that in this case the processes in the internal cascade correspond to the second-order transfer function with constant parameters. The desired root distribution is ensured in the external cascade of the system by using the modal approach. Extremum seeking is carried out in the external loop, where the control law contains the gradient estimate obtained by the synchronous detection method. The calculations of the PID controller parameters for the internal circuit and the P controller for the external one are presented in the paper. Numerical simulation of an example in MatLab illustrates some theoretical results.