Stability of input-output maps and their minimal realizations in state-linear, state-affine, LPV, and linear switched systems

Jul 16, 2026

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Stability is characterized directly from input–output data through Hankel-rank and time decay properties. An input–output family admits a stable finite-dimensional state-linear realization exactly when it has finite Hankel-rank and a uniformly decaying response; for state-linear realizable maps, the decay must be exponential. The analysis extends to state-affine, LPV, and linear switched systems using input-forgetting notions, linking forgetting to impulse-response decay (sub-Markov parameters). The decay/forgetting rate fully determines the decay rate of every minimal realization.

LanguageEnglish

CategoryResearch & Report

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Pages7

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Introduction

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Preliminaries

Stability of input-output maps and their minimal realizations in state-linear, state-affine, LPV, and linear switched systems Mihly Petreczky, Juan-Pablo Ortega, Florian Rossmannek and Blint Darczy arXiv :2607 .03849v1 [math .OC] 4 Jul 2026 Abstract—Stability is often assumed in learning and identification, yet it is rarely characterized directly from input–output data. We show that an input–output family admits a stable finite-dimensional state-linear realization iffit has finite Hankel-rank and its response decays uniformly with time; for state-linear realizable maps this decay is necessarily exponential. We extend these results to stateaffine, LPV, and linear switched systems via suitable inputforgetting notions, and relate forgetting to decay of impulse responses (sub-Markov parameters). In all cases, the decay/forgetting rate determines the decay rate of every minimal realization. Index Terms—Stability, input forgetting, minimal realization, state-affine systems, LPV systems. I. INTRODUCTION In this paper we are interested in characterizing input-output maps of stable state-linear systems (SLS) x (t + 1) = Au (t)x (t), x(0) ∈ B, y (t) = Cx(t) (1) where x (t) ∈ Rn is the state, y (t) ∈ Rny is the output and u (t) ∈ U is the input at time t, Au (t) and C are matrices of suitable sizes, and B = {Bj ∈ Rn }j∈J is a family of initial states indexed by a set J. SLSs are also known as switched linear systems (with no additive input) when U is finite or {Au }u∈U is compact. We prefer the term SLS as it is older [1] and it is not associated with assumptions on U. Stability of (1) is well-understood, without claiming completeness [2]–[6] . In particular, it is known that the asymptotic stability of (1) is equivalent to uniform global exponential stability (GUES) and both are equivalent to the joint spectral radius of {Au }u∈U being less than one. However, it is less clear how stability properties of the input-output behavior of (1) relate to the stability of (1), e.g., it is unclear if (1) is GUES if its input-output maps decay. Our first contribution is to show that a family of inputoutput maps has a GUES SLS realization if and only if it hasan SLS realization and the values of the input-output maps M. Petreczky is with Univ. Lille, CNRS, Centrale Lille, UMR 9189 CRIStAL, F-59000 Lille, France ([mihaly.petreczky@centralelille.fr](mihaly.petreczky@centralelille.fr)). J.-P. Ortega and F. Rossmannek are with SPMS, NTU, Singapore (email: juan-pablo.ortega,[florian.rossmannek@ntu.edu.sg](florian.rossmannek@ntu.edu.sg)). B. Darczy is with HUN-REN SZTAKI, Budapest, Hungary (e-mail: daroczy.balint@sztaki.hun-ren.hu) . decay to zero; in that case, decay is necessarily exponential, and all minimal realizations are GUES with this decay rate. This is relevant for two reasons. First, state-space representations (SSRs) obtained through identification or model reduction capture, at best, only the true input-output behavior, and therefore cannot be identified with the true system itself. Hence, in the absence of prior knowledge, any SSR reproducing the true input-output behavior is an equally valid candidate model. Therefore, analysis and control design for one such SSR should apply to other SSRs with the same input-output map, e.g., a controller that stabilizes one SSR (and thus its input-output behavior) should stabilize other SSRs with the same input-output behavior. Our results suggest that this might be the case if the other SSRs are minimal. Second, in learning/system identification, stability is often imposed on SSRs to guarantee robust long-term prediction and prevent error accumulation. However, this restriction is only meaningful if the true input-output behavior admits a stable realization. Our results provide conditions for this. As a second contribution we apply the results for SLSsto state-affine systems (SAS) [1], linear parameter-varying systems with affine dependence on the scheduling variable (LP

FAQs

What conditions guarantee a stable finite-dimensional state-linear realization from input–output data?

A stable realization exists exactly when the input–output family has finite Hankel-rank and the input–output response decays uniformly with time.

Why does decay of input–output maps imply exponential stability for state-linear realizations?

For state-linear realizable maps, the required decay rate is necessarily exponential, and every minimal realization is GUES with that same decay rate.

How are the results extended to state-affine, LPV, and linear switched systems?

They use appropriate input-forgetting notions: for these classes, realizability by a GUES system is equivalent to the decay of impulse responses (sub-Markov parameters), equivalently to ℓ∞-forgetting or related forgetting properties.

Jul 16, 2026

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