Best causal mathematical models for a nonlinear system

We provide new causal mathematical models of a nonlinear system S which are specifications of a nonlinear operator P-p of degree p = 1, 2,.... The operator P-p is determined from a special orthogonalization procedure and minimization of the mean squared difference between outputs of S and P-p. As a result, these models have smallest possible associated errors in the class of such operators P-p. The causality condition is implemented through the use of specific matrices called lower trapezoidal. The associated computational work is reduced by the use of the orthogonalization procedure. We provide a strict justification of the proposed approach including theorems on an explicit representaton of the models' parameters, and theorems on the associated error representation. The possible extensions of the proposed approach and its potential applications are outlined.