Scalable computation of input-normal/output-diagonal balanced realization for control-affine polynomial systems

TL;DR

This paper introduces a scalable tensor-based method for computing input-normal/output-diagonal nonlinear balancing transformations in control-affine polynomial systems, facilitating model reduction.

Contribution

It provides explicit equations and a scalable Kronecker product-based implementation for nonlinear balancing transformations, along with open-source software.

Findings

Transformation computation is as efficient as energy function calculation

Explicit algebraic structure for transformation equations

Rigorous analysis of solvability and complexity

Abstract

We present a scalable tensor-based approach to computing input-normal/output-diagonal nonlinear balancing transformations for control-affine systems with polynomial nonlinearities. This transformation is necessary to determine the states that can be truncated when forming a reduced-order model. Given a polynomial representation for the controllability and observability energy functions, we derive the explicit equations to compute a polynomial transformation to induce input-normal/output-diagonal structure in the energy functions in the transformed coordinates. The transformation is computed degree-by-degree, similar to previous Taylor-series approaches in the literature. However, unlike previous works, we provide a detailed analysis of the transformation equations in Kronecker product form to enable a scalable implementation. We derive the explicit algebraic structure for the equations, present rigorous analyses for the solvability and algorithmic complexity of those equations, and provide general purpose open-source software implementations for the proposed algorithms to stimulate broader use of nonlinear balanced truncation model. We demonstrate that with our efficient implementation, computing the nonlinear transformation is approximately as expensive as computing the energy functions.