Optimal modeling of nonlinear systems

Method of variable injections

Abstract

Our work addresses a development and justification of the new approach to the modeling of nonlinear systems. Let $\f$ 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 $\f$ so that an associated error is minimized. To this end, the model $\ttt_p$ is constructed as a sum of $p+1$ particular parts, in the form $\ttt_p (\y) = \sum_{j=0}^{p}G_j H_j Q_j(\vv_j)$ where $G_j$ and $ H_j$, for $j=0,\ldots, p$, are matrices to be determined, and $\vv_j$, for $j=1,\ldots,p$, is a special random vector called the injection. We denote $\vv_0=\y$. Injections $\vv_1,\ldots, \vv_p$ are aimed to diminish the error associated with the proposed model $\ttt_p$. Further, $Q_j$ is a special transform aimed to facilitate the numerical realization of model $\ttt_p$. 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 $\vv_1,\ldots, \vv_p$ is considered. The proposed method has several degrees of freedom to diminish the associated error. They are `degree' $p$ of $\ttt_p$, choice of matrices $G_0, H_0, \ldots,$ $G_p, H_p$, dimensions of matrices $G_0, H_0, \ldots,$ $G_p, H_p$ and injections $\vv_1,\ldots, \vv_p$, respectively. Four numerical examples are provided. At the end, the open problem is formulated.