⚡ electrical engineering

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

This paper establishes that the stability of minimal realizations for input-output maps in state-linear, state-affine, LPV, and linear switched systems is directly characterized by finite Hankel rank and uniform decay of responses, demonstrating that the forgetting rate of the input-output data dictates the decay rate of every minimal realization.

Original authors: Mihály Petreczky, Juan-Pablo Ortega, Florian Rossmannek, Bálint Daróczy

Published 2026-07-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Mihály Petreczky, Juan-Pablo Ortega, Florian Rossmannek, Bálint Daróczy

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to understand a mysterious black box. You can push buttons (inputs) and watch what lights up on the screen (outputs), but you cannot see the gears and wires inside. This is the world of input-output maps.

In engineering and data science, we often want to build a "model" of this black box—a simplified version with internal parts (states) that explains how the buttons lead to the lights. This is called a realization.

The big question this paper answers is: If the black box's behavior eventually "calms down" (the lights stop flashing wildly and fade to zero), does that mean the internal gears of our model will also calm down?

Here is the breakdown of their findings using simple analogies:

1. The Three Types of Black Boxes

The authors looked at four specific types of mathematical models, which they group into a few main categories:

State-Linear Systems (SLS): Think of these as a set of different machines. At any moment, you pick one machine to run. The machine changes its internal state based on a switch you flip.
State-Affine Systems (SAS): These are like the machines above, but they also have a "push" or "pull" added to them (like a constant wind blowing on a sailboat).
LPV and Switched Systems: These are variations where the rules change based on a schedule or a switch, similar to driving a car that changes its engine settings depending on the road conditions.

2. The "Echo" vs. The "Fade"

The paper introduces two main concepts to describe how these systems behave over time:

The "Echo" (Stability): This is about the internal gears. A system is "stable" if, no matter how you start it or what buttons you press, the internal parts eventually settle down and stop moving wildly. In math terms, this is called GUES (Globally Uniformly Exponential Stability). It means the system forgets its past quickly, like a ripple in a pond that disappears fast.
The "Fade" (Input-Output Decay): This is about what you see on the screen. If you stop pressing buttons, does the light on the screen eventually go dark? Or does it keep flickering forever?
- Uniform Decay: The light gets dimmer and dimmer until it's off.
- Input Forgetting: The system "forgets" what you did a long time ago. If you change a button you pressed 100 steps ago, the screen today shouldn't care.

3. The Big Discovery: The "Mirror" Effect

The authors proved a powerful rule: The outside behavior perfectly predicts the inside health.

The Rule: If you have a family of input-output maps (the black box behavior) that fades away (the lights go dark) and has a finite complexity (it's not infinitely complicated), then you can always build a model of it where the internal gears are stable.
The "Minimal" Model: Among all possible models that explain the black box, there is a "minimal" one (the simplest, most efficient version). The paper proves that if the black box fades away, this simplest model must be stable. You can't have a simple, fading black box with a chaotic, unstable internal engine.
The Speed Match: If the black box fades away at a specific speed (say, halving its brightness every second), the internal gears of the minimal model will also settle down at that exact same speed.

4. Why This Matters (According to the Paper)

The authors give two main reasons why this is useful, without promising future medical or industrial miracles:

Trust in Models: When we use computers to learn a system's behavior (identification), we often get many different models that all fit the data. Usually, we worry that a "stable" model might be a fluke and a "chaotic" one might be the truth. This paper says: If the data fades away, any simplest model you build will be stable. So, if you design a controller to stabilize the simplest model, it will stabilize the system's behavior, regardless of which specific simple model you picked.
Learning and Prediction: In machine learning, we often force our models to be stable so they don't make wild, accumulating errors over time. This paper gives you a checklist: Before you force stability, check if the real-world data actually fades away. If the data doesn't fade, no amount of forcing will make a stable model that fits the truth. If the data does fade, you are safe to build a stable model.

5. The "Reservoir" Connection

The paper mentions a specific application in Reservoir Computing (a type of AI). In this field, there is a concept called "Fading Memory" (the system forgets old inputs). The authors show that for these specific types of systems, "Fading Memory" is exactly the same thing as having a "Stable Internal Engine." If the system forgets the past, its internal gears are safe and sound.

Summary

Think of the black box as a drum.

If you hit the drum and the sound fades away quickly (Input-Output Decay), the paper proves that the drum's internal structure (the Minimal Realization) is solid and won't vibrate forever.
Furthermore, the speed at which the sound fades is exactly the speed at which the internal vibrations die out.
This holds true whether the drum is a simple linear one, a complex one with extra pushes, or one that changes its shape based on a schedule.

The paper essentially says: If the output behaves well (fades), the simplest internal explanation must also behave well.

Technical Summary: Stability of Input-Output Maps and Their Minimal Realizations

Problem Statement
The paper addresses a fundamental gap in system identification and learning: while stability is frequently assumed for state-space representations (SSRs) to ensure robust long-term prediction and prevent error accumulation, it is rarely characterized directly from input-output data. Specifically, it is unclear whether the decay of an input-output map implies the stability (specifically Global Uniform Exponential Stability, or GUES) of the underlying system realization. This is critical because SSRs obtained via identification or model reduction capture only the input-output behavior; without prior knowledge, any SSR reproducing this behavior is a valid candidate. The authors investigate the conditions under which a family of input-output maps admits a stable finite-dimensional realization and how the decay rates of these maps relate to the stability of minimal realizations across four specific system classes: State-Linear Systems (SLS), State-Affine Systems (SAS), Linear Parameter-Varying systems with affine dependence (LPV-SSA), and Linear Switched Systems (LSS).

Methodology
The authors employ realization theory combined with known stability results for SSRs. The core methodological approach involves:

Reduction to State-Linear Systems: For SAS, LPV-SSA, and LSS, the authors utilize established transformations to associate these systems with State-Linear Systems (SLS). For SAS, this involves defining a family of response maps $\Psi_H$ derived from the original map $H$ . For LPV-SSA and LSS, this involves mapping the system to an SLS based on sub-Markov parameters (impulse responses).
Hankel-Rank Analysis: The existence of a finite-dimensional realization is tied to the finiteness of the Hankel-rank of the input-output family.
Decay and Forgetting Definitions: The authors define specific decay properties for input-output maps:
- Uniform Decay (UD) / Uniform Exponential Decay (UED): For SLS, defined by the uniform convergence of response maps to zero.
- Input Forgetting (UF) / Uniform Exponential Forgetting (UEF): For SAS, defined by the influence of past inputs on current outputs fading over time.
- Impulse Response Decay (IRR-UD/IRR-UED): For LPV-SSA/LSS, defined by the decay of sub-Markov parameters.
- $\ell_\infty$ -Forgetting: A property combining BIBO stability with the uniform decay of the effect of past inputs.
Minimal Realization Analysis: The paper analyzes minimal realizations (those that are both span-reachable and observable) to determine if stability properties of the input-output map are inherited by the state-space dynamics.

Key Contributions and Results

State-Linear Systems (SLS):
The paper establishes that a family of input-output maps admits a GUES SLS realization if and only if the family has a finite Hankel-rank and the maps decay uniformly to zero. Crucially, if the maps decay uniformly, they necessarily decay exponentially. Furthermore, if the input-output family is uniformly exponentially decaying (UED) with rate $\beta$ , then all minimal SLS realizations are GUES with the same rate $\beta$ . This implies that stability is an intrinsic property of the minimal realization if the input-output behavior decays.
State-Affine Systems (SAS):
By reducing SAS to SLS, the authors show that a response map $H$ admits a GUES SAS realization if and only if $H$ is uniformly forgetting (UF) and the associated family $\Psi_H$ has a finite Hankel-rank. The paper proves that for SAS, UF is equivalent to Uniform Exponential Forgetting (UEF). Consequently, if a map is UEF with rate $\beta$ , all minimal SAS realizations are GUES with rate $\beta$ . This result connects the "fading memory" property in reservoir computing directly to the stability of minimal finite-dimensional realizations.
LPV-SSA and Linear Switched Systems (LSS):
The results are extended to LPV-SSA and LSS via their impulse response representations (IRR). The existence of a GUES realization is equivalent to the existence of an IRR that decays uniformly (IRR-UD), which implies exponential decay (IRR-UED). The paper demonstrates that for these systems, $\ell_\infty$ -forgetting (BIBO stability plus uniform input forgetting) is equivalent to the existence of a GUES realization. As with SLS and SAS, the decay rate of the impulse response dictates the decay rate of every minimal realization.
Minimal Realizations:
A central finding across all classes is that the decay/forgetting rate of the input-output map is not just a necessary condition for stability but determines the decay rate of every minimal realization. Minimal realizations are necessarily GUES with this specific rate.

Significance and Claims
The paper claims significance primarily for system identification and learning theory. It argues that since identified SSRs are often non-unique (representing the same input-output behavior), stability analysis and control design should ideally apply to all SSRs sharing that behavior. The results suggest that if the input-output map decays, then minimal realizations will inherently possess the same stability properties (GUES) and decay rates. This provides a theoretical justification for imposing stability constraints on identified models: if the true input-output behavior admits a stable realization, the minimal realization will be stable.

Additionally, for SAS, the work bridges the gap between "fading memory" properties studied in reservoir computing and the stability of minimal finite-dimensional realizations, suggesting that fading memory maps on infinite sequences correspond to those realizable by GUES SASs.

The authors remain modest regarding the scope, noting that while internal stability, BIBO stability, and $\ell_p$ -gains are well-understood for linear systems, the connections for structured nonlinear classes (like SLS/SAS/LPV-SSA/LSS) are subtler. Their contribution is specifically establishing the equivalence between the decay of input-output maps and the stability of their minimal realizations, rather than providing a complete characterization of all possible stability notions for these systems. Future work is suggested to explore connections with Lyapunov functions (quadratic, polyhedral, SOS) and echo-state properties.