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

Abstract

A conceptual and mathematical framework is presented for optimally approximating a large-scale continuous-time-parameter nonlinear dynamical system <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S_C</tex> by a continuous-time-parameter model <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\hat{S}_C</tex> as well as a discrete-time-parameter model <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\hat{S}_D</tex> , 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S_C</tex> can be represented by a Volterra functional series <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">V_t</tex> , belonging to a Fock space <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F_{\underline{\rho}}(E)</tex> , the input-output maps for the simulators <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\hat{S}_C</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\hat{S}_D</tex> are obtained as "best approximations" in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F_{\underline{\rho}}(E)</tex> for the entire (untruncated) series <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">V_t</tex> . 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.