A Direct State-Space Realization of Discrete-Time Linear Parameter-Varying Input-Output Models

TL;DR

This paper presents a novel direct state-space realization method for discrete-time linear parameter-varying input-output models that simplifies stability analysis and controller design by avoiding nonlinear dependencies.

Contribution

It introduces a direct state-space realization that is reachable and reconstructible, avoiding nonlinear dependencies, under certain conditions, and demonstrates its effectiveness through numerical examples.

Findings

The realization is reachable under coprimeness and well-posedness conditions.

It is never observable but is reconstructible with unobservable directions converging to zero.

Numerical examples validate the theoretical results in LPV and LTI cases.

Abstract

A minimal state-space (SS) realization of an identified linear parameter-varying (LPV) input-output (IO) model usually introduces dynamic and nonlinear dependency of the state-space coefficient functions, complicating stability analysis and controller synthesis. The aim of this paper is to introduce and analyze a direct SS realization of this IO model that avoids this nonlinear and dynamic dependency, at the cost of introducing a nonminimal state. It is shown that this direct SS realization 1) is reachable under a coprimeness condition on the coefficient functions of the IO model and a well-posedness condition on the model order, and 2) is never observable but that the unobservable directions converge to zero in a finite amount of steps, i.e., that the realization is reconstructible. The derived results are illustrated through numerical examples in both the LPV and LTI case.