Optimal modeling of nonlinear systems: Method of variable injections

Our work addresses a development and justification of the new approach to the modeling of nonlinear systems. Let (Formula presented) be an unknown input-output map of the system with a random input and output y and x, respectively. It is assumed that y and x are available and covariance matrices formed from y and x are known. We determine a model of (Formula presented) so that an associated error is minimized. To this end, the model (Formula presented) is constructed as a sum of p + 1 particular parts, in the form (Formula presented) where G_j and H_j, for j = 0,.., p, are matrices to be determined, and v_j, for j = 1,.., p, is a special random vector called the injection. We denote v₀ = y. Further, Q_j is a special transform aimed to facilitate the numerical realization of model (Formula presented). It is determined in the way allowing us to optimally determine G_j and H_j as a solution of p + 1 separate error minimization problems which are simpler than the original minimization problem. The empirical determination of injections v₁,.., vp is considered. The proposed method has several degrees of freedom to diminish the associated error. They are ‘degree’ p of Tp, choice of matrices G₀, H₀,.., Gp, Hp, dimensions of matrices G₀, H₀,.., Gp, Hp and injections v₁,.., vp, respectively. In particular, it is shown that a variation of the injections in their dimensionality and special forms allow us to increase accuracy of the proposed model (Formula presented). The proposed approach differs from known techniques by its ingredients mentioned above. Four numerical examples are provided. At the end, the open problem is formulated.