Increase the Dynamic Accuracy of a System with PID-regulator by Numerical Optimization

Many tasks of control of technical objects require the design of regulators. Correct calculation of the regulator provides high static and dynamic accuracy. Among the many methods for solving this problem, the most successful methods are based on numerical optimization. The reasons for this are that almost any complexity of the mathematical model of an object is not an obstacle for these methods; many specialized and universal mathematical software products such as MATLAB, Simulink, Mathcad, VisSim, SimInTech, and others offer algorithmic support for modeling and optimization of such systems. These software products have built-in algorithms for numerical optimization of many parameters, such as Powell, Polak-Ribiere, Fletcher-Reeves, Monte-Carlo. If the object model is known with sufficient accuracy and the objective (cost) function together with the test tasks for the system are justified, then such software most often quite successfully solve the problem of numerical calculation of regulator coefficients. However, in some cases, the desired extremum of the objective function is located in such a range of the values of the desired coefficients that do not allow the problem to be considered successfully solved, since theory and practice indicate that these coefficients in this case should be positive, and the optimization result gives negative values for some of such coefficients. In particular, the coefficient of the derivative link is sometimes negative. This paper provides a theoretical justification and model confirmation of a method that allows to exclude such a result that provides a more successful solution of the problem, and the resulting regulator provides the best dynamic accuracy along with the best static accuracy. The method is the use of Gaussian noise generator as a disturbing effect in the system when modeling its operation for the purpose of numerical optimization of the regulator. With all this, the noise level should be chosen sufficiently small; the choice of the best cost function for these purposes and optimization of the task and simulation time are also necessary. All recommendations are combined into a formal description of the proposed methodology for solving the problem.