A best approximation framework and implementation for simulation of large-scale nonlinear systems

A conceptual and mathematical framework is presented for optimally approximating a large-scale continuous-time-parameter nonlinear dynamical system
S_C
by a continuous-time-parameter model
\hat{S}_C
as well as a discrete-time-parameter model
\hat{S}_D
, which can be readily simulated respectively on an analog and on a digital computer. A reproducing kernel Hilbert space approach in appropriate weighted Fock spaces is used in the problem formulation and solution. Assuming that the input-output map of the system
S_C
can be represented by a Volterra functional series
V_t
, belonging to a Fock space
F_{\underline{\rho}}(E)
, the input-output maps for the simulators
\hat{S}_C
and
\hat{S}_D
are obtained as "best approximations" in
F_{\underline{\rho}}(E)
for the entire (untruncated) series
V_t
. Each of these models has the following features: (a) It is adaptive because it is based on a set of test input-output pairs which can be incorporated in the system by on-line multiplexing, (b) it is optimal in the sense of being a projection in a Hilbert space of nonlinear operators, (c) it is easily implementable by means of a set of interconnected linear dynamical systems and zero-memory nonlinear functions of single variables, and (d) unlike polynomic (truncated Volterra series) approximations, it constitutes a global approximation and thus is valid under both small- and large-signal operating conditions.