Structural properties of LPV to LFR transformation: minimality, input-output behavior and identifiability

TL;DR

This paper explores the properties of transforming ALPV models into LFRs, focusing on minimality, input-output behavior, and identifiability, with implications for system identification and control.

Contribution

It establishes the equivalence of minimality, input-output behavior, and identifiability between ALPV and LFR representations, providing a theoretical foundation for system analysis.

Findings

Minimal ALPV models produce minimal LFRs

Input-output behavior is preserved in the transformation

Identifiability properties are equivalent between ALPV and LFR

Abstract

In this paper, we introduce and study important properties of the transformation of Affine Linear Parameter-Varying (ALPV) state-space representations into Linear Fractional Representations (LFR). More precisely, we show that $(i)$ state minimal ALPV representations yield minimal LFRs, and vice versa, $(ii)$ the input-output behavior of the ALPV represention determines uniquely the input-output behavior of the resulting LFR, $(iii)$ structurally identifiable ALPVs yield structurally identifiable LFRs, and vice versa. We then characterize LFR models which correspond to equivalent ALPV models based on their input-output maps. As illustrated all along the paper, these results have important consequences for identification and control of systems described by LFRs.