Modeling of Nonlinear Systems: Method of Optimal Injections

In this paper, a nonlinear system is interpreted as an operator F transforming random vectors. It is assumed that the operator is unknown and the random vectors are available. It is required to find a model of the system represented by a best constructive operator F approximation. While the theory of operator approximation with any given accuracy has been well elaborated, the theory of best constrained constructive operator approximation is not so well developed. Despite increasing demands from various applications, this subject is minimally tractable because of intrinsic difficulties with associated approximation techniques. This paper concerns the best constrained approximation of a nonlinear operator in probability spaces. The main conceptual novelty of the proposed approach is that, unlike the known techniques, it targets a constructive optimal determination of all 3p+2 ingredients of the approximating operator where p is a nonnegative integer. The solution to the associated problem is represented by a combination of new best approximation techniques with a special iterative procedure. The proposed approximating model of the system has several degrees of freedom to minimize the associated error. In particular, one of the specific features of the developed approximating technique is special random vectors called injections. It is shown that the desired injection is determined from the solution of a special Fredholm integral equation of the second kind. Its solution is called the optimal injection. The determination of optimal injections in this way allows us to further minimize the associated error.