Approximation theory for deterministic and stochastic nonlinear systems

Approximation of the input-output behavior of nonlinear systems characterized by fading memory properties is addressed. System structures are proposed as approximants along with approximation criteria and general conditions under which these structures satisfy the approximation criteria are given. The results fall into three general areas depending on the deterministic or stochastic nature of the system and the existence of a state-equation representation.; Realizable structures are proposed as approximants for continuous-time systems defined on deterministic signal spaces. Such structures are shown to uniformly approximate, arbitrarily well, systems with approximately finite memory on large classes of 'useful' signal spaces. Additional results are given for deterministic, discrete-time systems known to possess an absolutely summable state-equation representation. It is shown that, for such systems, a uniform approximation of the input-output response may be obtained by uniformly approximating the associated state-update and output functions in the usual state-equation interconnection.; Uniform approximation of the input-output response of a system operating on a stochastic process is shown to be considerably more complex than in the deterministic case. By defining a somewhat weaker approximation criterion, called uniform, in-probability approximation, useful results can be obtained for a large class of systems which have approximately finite memory in probability, a property defined herein. The only significant additional requirement is that the sample functions of the process are sufficiently 'smooth.' In particular, it is shown that all Gaussian processes with rational spectra satisfy this requirement.