On nonparametric control of a dynamic system

The paper considers the problem of dual control of an inertia-free object whose input is affected by a control vari- able and an observable but uncontrollable variable. The idea of dual control belongs to A. Feldbaum and was devel- oped on the basis of Bayesian approach. In this case, the probability densities of interference, as well as input output variables are known. In particular, the case of Gaussian probability densities was investigated. As a result, algorithms of dual control with the simplest objects of the inertia-free class were obtained. In the case of combined control systems, these studies were carried out by Feldbaum’s followers. Further development of dual control theory was described by Y. Tsypkin. The probability density of the interference was unknown, but the stage of selecting the structure of the control device and the equation of the describing object were necessary. In Tsypkin’s works, the corresponding parametric algorithms for dual control are given. In this case, a simultaneous estimation of the coefficient of the model and the regulator was made on the basis of the method of sto- chastic approximations. Later on nonparametric dual control algorithms were proposed. It is this way of control design that is discussed in the current paper. The parametric model of the object and the parametric structure of the controller were unknown in the problem statement. Nevertheless, it was known from a priori information that the characteristic of the object was one-to-one in control. Below we consider nonparametric control algorithms that combine the processes of simultaneous control and the study of the object with an accuracy of the structure unknown to the parameters, but a priori with a given number of delayed elements of the output variable of the object. In other words, the depth of the memory of the managed object was determined. In this case, a nonparametric algorithm for dual control can function in conditions of passive and active accumulation of information. The technique for representing a one-dimensional inertial-free dynamic system to a multidimensional static one is presented in the paper. Some results of numerical investigation of nonparametric algorithms of dual control are also presented.