Power-Series Approach to Moment-Matching-Based Model Reduction of MIMO Polynomial Nonlinear Systems

TL;DR

This paper introduces a power-series based method for reducing high-order MIMO polynomial nonlinear systems by matching moments, providing analytical reduced models and new theoretical insights into the reduction process.

Contribution

It develops a power-series decomposition technique for nonlinear model reduction, offering algorithms for precise order and parameter determination, and reveals fundamental bounds and conditions for nonlinear reduced models.

Findings

Lower bound for reduced model order can be less than the number of matched moments.

The ratio of input to output channels influences the lower bound.

A reduced model can have either linear state or output equations under mild conditions.

Abstract

The model reduction problem for high-order multi-input, multi-output (MIMO) polynomial nonlinear systems based on moment matching is addressed. The technique of power-series decomposition is exploited: this decomposes the solution of the nonlinear PDE characterizing the center manifold into the solutions of a series of recursively defined Sylvester equations. This approach allows yielding nonlinear reduced-order models in very much the same way as in the linear case (e.g. analytically). Algorithms are proposed for obtaining the order and the parameters of the reduced-order models with precision of degree $\kappa$. The approach also provides new insights into the nonlinear moment matching problem: first, a lower bound for the order of the reduced-order model is obtained, which, in the MIMO case, can be strictly less than the number of matched moments; second, it is revealed that the lower bound is affected by the ratio of the number of the input and output channels; third, it is shown that under mild conditions, a nonlinear reduced-order model can always be constructed with either a linear state equation or a linear output equation.