# Comparing global input-output behavior of frozen-equivalent LPV state-space models

## TL;DR

This paper analyzes how frozen-equivalent LPV models can behave differently under varying scheduling signals and provides an error bound relating their input-output differences to the scheduling signal's rate of change.

## Contribution

It introduces an analytic error bound for the output difference of frozen-equivalent LPV models depending on scheduling signal dynamics and basis discrepancies.

## Key findings

- Error bound depends on scheduling signal change rate
- Output difference can be minimized with slow-changing signals
- Choice of scheduling signal influences model behavior

## Abstract

It is known that in general, \emph{frozen equivalent} (Linear Parameter-Varying) LPV models, \emph{i.e.}, LPV models which have the same input-output behavior for each constant scheduling signal, might exhibit different input-output behavior for non-constant scheduling signals. In this paper, we provide an analytic error bound on the difference between the input-output behaviors of two LPV models which are frozen equivalent. This error bound turns out to be a function of both the speed of the change of the scheduling signal and the discrepancy between the coherent bases of the two LPV models. In particular, the difference between the outputs of the two models can be made arbitrarily small by choosing a scheduling signal which changes slowly enough. An illustrative example is presented to show that the choice of the scheduling signal can reduce the difference between the input-output behaviors of frozen-equivalent LPV models.