# Model Boundary Approximation Method as a Unifying Framework for Balanced Truncation and Singular Perturbation Approximation

## TL;DR

This paper unifies two major model reduction techniques, Balanced Truncation and Singular Perturbation, within a framework derived from the Model Boundary Approximation Method, enabling interpolation and potential extension to nonlinear systems.

## Contribution

It demonstrates that these techniques are boundary cases on a model manifold and introduces a method to interpolate between them, suggesting extensions to nonlinear systems.

## Key findings

- Balanced Truncation and Singular Perturbation are boundary points on a model manifold.
- MBAM allows interpolation between the two reduction techniques.
- Potential extension of these methods to nonlinear systems.

## Abstract

We show that two widely accepted model reduction techniques, Balanced Truncation and Balanced Singular Perturbation Approximation, can be derived as limiting approximations of a carefully constructed parameterization of Linear Time Invariant (LTI) systems by employing the Model Boundary Approximation Method (MBAM), a recent development in the Physics literature. This unifying framework of these popular model reduction techniques shows that Balanced Truncation and Balanced Singular Perturbation Approximation each correspond to a particular boundary point on a manifold, the "model manifold," which is associated with the specific choice of model parameterization and initial condition, and is embedded in a sample space of measured outputs, which can be chosen arbitrarily, provided that the number of samples exceeds the number of parameters. We also show that MBAM provides a novel way to interpolate between Balanced Truncation and Balanced Singular Perturbation Approximation, by exploring the set of approximations on the boundary of the manifold between the elements that correspond to the two model reduction techniques; this allows for alternative approximations of a given system to be found that may be better under certain conditions. The work herein suggests similar types of approximations may be obtainable in topologically similar places (i.e. on certain boundaries) on the model manifold of nonlinear systems if analogous parameterizations can be achieved, therefore extending these widely accepted model reduction techniques to nonlinear systems.