Adaptive identification of systems with distributed lags

In this paper, we propose adaptive algorithms for parametric identification of discrete systems with lagged variables. For the case with lagged values of input variables, we develop adaptive algorithms and prove their convergence and the boundedness of all trajectories in the adaptive system. The convergence area of an adaptive algorithm depends on the effective perturbation. Models with multiplicative parameters are proposed to reduce the number of evaluated parameters. The procedure of choosing the vector of basic parameters of a such model is developed. The functioning of adaptive identification system is proved for this case. It is shown that if multiplicative identification is applied, the parameters of the evaluation system should be selected from the condition of minimizing the criterion that depends on the prediction error. In the case where lagged variables are interconnected, we propose a transformation which allows one to exclude this mutual influence of variables. The second part of the work is devoted to the analysis of adaptive algorithms for the systems described by equations with the lagged output variable. A linear dependence of the output of the system and the effective perturbation is assumed. Application of adaptive algorithms described in the first part of the work does not make it possible to obtain valid estimates of parameters. We proposed to estimate the existing perturbation for solving the problem. Relevant procedures are described and their functioning is proved. We also present the results of modeling that confirm the functioning of adaptive methods.