Realization Theory for LPV State-Space Representations with Affine Dependence

TL;DR

This paper develops a realization theory for affine LPV state-space models, establishing conditions for minimality, uniqueness, and the connection to Hankel matrix rank, thus laying a theoretical foundation for LPV systems.

Contribution

It introduces a Kalman-style realization framework for affine LPV state-space models, including minimality, isomorphism, and partial realization theory.

Findings

Minimal LPV-SSA models are observable and span-reachable.

Two minimal LPV-SSA models of the same system are linearly isomorphic.

The Hankel matrix rank determines the minimal model dimension.

Abstract

In this paper we present a Kalman-style realization theory for linear parameter-varying state-space representations whose matrices depend on the scheduling variables in an affine way (abbreviated as LPV-SSA representations). We deal both with the discrete-time and the continuous-time cases. We show that such a LPV-SSA representation is a minimal (in the sense of having the least number of state-variables) representation of its input-output function, if and only if it is observable and span-reachable. We show that any two minimal LPV-SSA representations of the same input-output function are related by a linear isomorphism, and the isomorphism does not depend on the scheduling variable.We show that an input-output function can be represented by a LPV-SSA representation if and only if the Hankel-matrix of the input-output function has a finite rank. In fact, the rank of the Hankel-matrix gives the dimension of a minimal LPV-SSA representation. Moreover, we can formulate a counterpart of partial realization theory for LPV-SSA representation and prove correctness of the Kalman-Ho algorithm. These results thus represent the basis of systems theory for LPV-SSA representation.