Realization Theory for LPV State-Space Representations With Affine Dependence

Abstract

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 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 KalmanHo algorithm formulated in [1]. These results thus represent the basis of systems theory for LPV-SSA representation.

Realization and identification algorithm for stochastic LPV state-space models with exogenous inputs

“…Let Ψ(w) = [Ψ y s (1w) · · · Ψ y s (n µ w)], ∀w ∈ Σ * . Then, R = ({Â σ , } σ∈Σ ,Ĝ,Ĉ), withĜ = Ĝ 1 · · ·Ĝ nµ , is a representation of Ψ and by (Petreczky et al, 2017, Theorem 1, Theorem 2) and the definition of observability and reachability for representations in (Petreczky and Vidal, 2018, Appendix C), it follows that R is observable and reachable, and hence R is minimal by Berstel and Reutenauer (1984); Sontag (1979).…”

“…Recall from Petreczky et al (2017); Cox et al (2018) that a deterministic LPV-SSA representation (with affine dependence) is a system of the form…”

“…We will refer to M S as the sub-Markov function of the deterministic LPV-SSA representation of S . From Petreczky et al (2017) it then follows that two deterministic LPV-SSA representations S 1 , S 2 have the same input-output behavior, if and only if their sub-Markov parameters are equal, i.e., M S1 = M S2 . Moreover, the sub-Markov parameters can be determined from the input-output behavior.…”

“…It also allows to formulate identification algorithms for estimating state-space representation from a finite set of observations. The realization theory for deterministic Linear Parameter-Varying State-Space with Affine dependence (LPV-SSA) representation has been developed in Tóth et al (2012); Petreczky et al (2017). The results of Tóth et al (2012); Petreczky et al (2017) were used to derive LPV-SS identification algorithm in Cox et al (2015Cox et al ( , 2018.…”

“…The realization theory for deterministic Linear Parameter-Varying State-Space with Affine dependence (LPV-SSA) representation has been developed in Tóth et al (2012); Petreczky et al (2017). The results of Tóth et al (2012); Petreczky et al (2017) were used to derive LPV-SS identification algorithm in Cox et al (2015Cox et al ( , 2018. These methods are focused mainly on ⋆ This work was partially funded by CPER Data project, which is co-financed by European Union with the financial support of European Regional Development Fund (ERDF), French State and the French Region of Hauts-de-France.…”