Toward Best Approximation of Nonlinear Systems: A Case of Models with Memory

Abstract

The problem of nonlinear dynamical system modeling, considered in this paper, is motivated by restrictions arising in real-world tasks. The restrictions are that first, a system input cannot be entirely observed for one trial. Second, the system model must be subjected to the causality principle. Third, the input is corrupted by noise so that no relationship between the reference input and noise is known. Fourth, the model should have some degrees of freedom so that the associated accuracy can be regulated by a variation of these freedom degrees. We propose and justify new procedures for the nonlinear system modeling that are initialized by these motivations. The models are nonlinear and given by so called r-degree operators that can be reduced to a matrix form presentation. To satisfy the restrictions above, the matrices have special structures that we call the lower p-band matrices. The degree r of the models is the required degree of freedom. The rigorous analysis of errors associated with the presented techniques is given. Numerical experiments with real data demonstrate the efficiency of the proposed approach.

Keywords:

• Best operator approximation
• Nonlinear system with given memory

AMS Subject Classification:

• 93A30
• 47N70
• 37M99
• 65D15

Notes

¹The method [51] cannot be exploited for k = p + 1,…,m because of the memory condition (2.1).

²The data can be found at http://sipi.usc.edu/services/database/Database.html.