Parametric Optimization of the PID Controller with Restriction Based on the Method of Conjugate Polak—Polyak—Ribier Gradients

In automatic control systems (ASR), industrial processes of various types with a delay with a limit on the amount of overregulation, a PID controller with a real differentiating link (hereinafter referred to as the PID controller) is widely used. As is known, a sign of the presence of a large delay in the object of regulation is the ratio τob/Tob ≥ 1, where τob is the value of the delay time and Tob is maximum time constant of the object of control. In the presence of a large delay and limitation in the ASR, the parametric synthesis of the PID controller by well-known frequency methods becomes difficult, which leads to interest in the development of numerical searchless algorithms for parametric optimization based on the use of sensitivity functions to determine the gradient of the optimality criterion. In this paper, an APO algorithm is formed that calculates, based on the minimum of the integral quadratic criterion, the values of the adjustable parameters of the PID controller in the specified ASR. In order to ensure that there is no re-regulation of the resulting transient process, the authors of this article propose to introduce a restriction on the regulatory effect into the automatic system at the optimization stage, which, in turn, is taken into account by adding a penalty function to the integral criterion. The proposed algorithm is based on the method of conjugate Polak—Polyak-Ribier gradients with its known advantages. The components of the gradient vector of the optimization criterion are calculated using such sensitivity functions that allow you to obtain all the components of this vector without trial search variations of the configurable parameters. To calculate the value of the optimization step, the authors implemented an appropriate algorithm based on a gradient procedure using the sensitivity function of the output coordinate of the ASR to the step value. The convergence of the generated APO algorithm was verified using a numerical procedure based on the zone of attraction of record values of the optimization criterion, which is determined by a positive-definite Hesse matrix of the integral quadratic criterion based on the difference between the averaged and the tested transients.