Synthesis of Robust PID Controllers by Double Optimization Method

Abstract

The design of adaptive controllers allows to solve the problem of control of the object with non-stationary parameters. However, if the parameters of the object do not change too much or if only a certain interval of their change is known, it may turn out that an adaptive controller is not required, since the problem can be solved with the help of a robust controller. The robust controller allows to provide an acceptable quality of control even if the parameters of the mathematical model of the object change in some predetermined interval. A method of designing such controllers is known as the method of numerical optimization of the controllers used in the ensemble of systems in which the models of objects are different and the models of controllers are identical. The ensemble uses object models with extreme parameter values. The disadvantages of this method are too many systems that need to be modeled and optimized at the same time if there are several parameters to be changed. In addition, the worst combination of model parameters may not be boundary, but middle, in this case this method is not applicable. This article offers and analyzes an alternative method of designing a robust controller on a numerical example. The essence of this method is the numerical optimization of the regulator for the model with the worst combination of the values of all modifiable parameters. The search for the worst combination of parameters is also carried out using the method of numerical optimization. In this case, a combination of model parameters is found in which the best relation of regulator coefficients gives the worst result of the system. The problem is solved in several optimization cycles with alternating cost functions. The utility of the method is illustrated numerically by an example of a third order dynamic object with a series linked delay element.