BEYOND VOLTERRA AND WIENER: OPTIMAL MODELING OF NONLINEAR DYNAMICAL SYSTEMS IN A NEURAL SPACE FOR APPLICATIONS IN COMPUTATIONAL INTELLIGENCE
Rui J. P. de Figueiredo · 2003
Nonlinear dynamical systems are playing a major role in a number of applications of computational intelligence. In order to maintain the current growth of the technologies supporting these applications into this new century, it is essential to develop rigorous, accurate, efficient, and insightful models for describing nonlinear dynamical systems’ behavior, including adaptation, learning, and evolution, based on input–output observations or input–output specifications. If based on observations, the modeling process is called system identification, and if based on specifications, it is called system realization or design. In this chapter, we present optimal solutions to both of the preceding problems in the setting of a Neural Space N introduced by the author in 1990 [1,2]. N is a separable Hilbert space of nonlinear maps, f, that map a given vector x from a data space, X, which itself is a separable Hilbert or Euclidean space, to an m-vector y of m scalar outputs yj 1⁄4 fjðxÞ, j 1⁄4 1; . . . ;m, and fj are bounded analytic functionals on X expressible as Volterra functional series on X [3]. The fj belong to an appropriately constructed reproducing kernel Hilbert space, F, also introduced by de Figueiredo et al. in [4] in 1980, as a generalization of the symmetric Fock space. Details on this formulation as well as applications have been presented and discussed elsewhere [5–24].