Best Approximation of Dynamical Systems: A Case of Causal Modelling with Memory

Abstract:
The problem of modelling of nonlinear dynamical system, 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 subject 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 modelling which have been 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 which we call  the lower p-band matrices. The degree 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.