arxiv: 2604.13673 · v1 · submitted 2026-04-15 · 📡 eess.SY · cs.SY

Recognition: unknown

Behavioral Systems Theory Meets Machine Learning: Control-Aware Learning of the Intrinsic Behavior from Big Data

Yitao Yan , Yu Tong , Jie Bao , Wei Wang

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Pith reviewed 2026-05-10 13:07 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords behavioral systems theorymachine learningintrinsic statedata-driven controlnonlinear systemsneural networkbig data

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The pith

Dynamical systems possess an intrinsic state that encodes their full behavior bijectively and without causality, allowing control design to occur entirely inside that state space.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper addresses the mismatch between classical control theory, which depends on explicit models, states, and causal assumptions, and machine learning methods that operate on trajectories and large datasets without those constraints. It applies behavioral systems theory to establish that every dynamical system has an intrinsic state variable carrying complete behavior information in a one-to-one manner independent of time ordering or causes. This state permits all control synthesis to remain inside the state space itself. The construction aligns naturally with neural networks, which can therefore be trained on process operating data to recover a representation directly usable for control tasks.

Core claim

Dynamical systems possess an intrinsic state variable that encodes the system behavior in a bijective and causality-free manner, and control design can be carried out entirely within the state space. This resolves the conflict between model-centric causal control and trajectory-based learning from big data while complementing machine learning techniques, leading to a neural network architecture that learns the behavior representation well-suited for control design.

What carries the argument

The intrinsic state variable supplied by the behavioral framework, which maps system trajectories bijectively without reference to causality.

If this is right

Control synthesis for nonlinear systems can be performed wholly inside the learned intrinsic state without explicit models.
Neural networks trained on process data recover representations that support direct control applications.
The approach removes the need for causal ordering when combining data-driven learning with feedback design.
Abundant industrial operating records become sufficient for both behavior learning and subsequent controller synthesis.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Traditional state-space realizations may become unnecessary once the intrinsic behavior state is learned directly from trajectories.
Process industries could apply the method to predictive control tasks without separate system identification steps.
Stability margins of controllers derived from the learned state could be verified by checking consistency of the bijective mapping on held-out data.

Load-bearing premise

The behavioral framework supplies a bijective, causality-free intrinsic state whose control-relevant features survive approximation by a neural network trained on finite operating data.

What would settle it

A concrete case in which a neural network trained on operating data produces an intrinsic state yet standard controllers designed inside that state fail to achieve the predicted closed-loop behavior on the original system.

Figures

Figures reproduced from arXiv: 2604.13673 by Jie Bao, Wei Wang, Yitao Yan, Yu Tong.

**Figure 1.** Figure 1: The CALIB Architecture which is a subset of the uncontrolled system behavior because im(η ◦ψ) ⊂ im(η), (31) where im(η) denotes the image of function η. While Theorem 4 provides conditions for the feasibility of a desired controlled behavior, determining the form of the controlled state behavior represented by ψ is not straightforward, and the search for a control Lyapunov function V for nonlinear system… view at source ↗

**Figure 2.** Figure 2: Control performance using the linear design in Theor [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 5.** Figure 5: Comparison between true flight trajectory (blue) and [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

read the original abstract

The abundance of process operating data in modern industries, along with the rapid advancement of learning techniques, has led to a paradigm shift towards data-centric analysis and control. However, integrating machine learning with control theory for big data-driven control of nonlinear systems remains a challenging open problem. This is because the state-based, model-centric, and causal framework of classical control theory fundamentally contradicts the trajectory-based, set-theoretic, and causality-absent rationale of big data-based learning approaches. Using the behavioral framework, we show that dynamical systems possess an intrinsic state variable that encodes the system behavior in a bijective and causality-free manner, and control design can be carried out entirely within the state space. This approach not only resolves the aforementioned conflict but also complements machine learning techniques well, leading to a neural network architecture that is capable of learning the behavior representation well-suited for control design.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper sketches a behavioral-theory intrinsic state to let ML learn control-friendly representations for nonlinear systems, but the bijectivity and preservation claims lack derivation and likely don't extend beyond LTI cases.

read the letter

The main takeaway is that the authors use behavioral systems theory to posit an intrinsic state that encodes dynamics bijectively without causality, then train a neural net to learn a version of it suited for control design. This is meant to fix the mismatch between classical state-space control and pure trajectory-based learning from big data. The framing is direct and the goal is practical for nonlinear industrial processes.

Referee Report

2 major / 0 minor

Summary. The manuscript claims that the behavioral framework yields an intrinsic state variable for dynamical systems (including nonlinear ones) that encodes the full system behavior in a bijective, causality-free manner, permitting control design to be carried out entirely inside this state space. This is asserted to resolve the conflict between classical state-based control theory and trajectory-based big-data learning, while also motivating a neural-network architecture that learns a control-relevant behavior representation directly from operating data.

Significance. If the central construction were rigorously established and shown to survive finite-data NN approximation, the work would offer a potentially valuable bridge between Willems-style behavioral theory and machine-learning-based control, allowing state-space methods to be applied without explicit parametric models. No machine-checked proofs, reproducible code, or falsifiable predictions are supplied in the current manuscript.

major comments (2)

[Abstract] Abstract: the assertion that 'dynamical systems possess an intrinsic state variable that encodes the system behavior in a bijective and causality-free manner' is presented without derivation, proof sketch, or extension of the fundamental lemma beyond LTI systems; this is load-bearing for the claim that control design can be carried out entirely within the learned state space for nonlinear plants.
[Abstract] Abstract: the neural-network architecture is stated to be 'capable of learning the behavior representation well-suited for control design,' yet no argument or experiment is supplied showing that the finite-data approximation preserves bijectivity or the exact input-output behavior required for stabilization; this directly affects the resolution of the model-centric vs. trajectory-based conflict.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below, indicating planned revisions where appropriate.

read point-by-point responses

Referee: [Abstract] Abstract: the assertion that 'dynamical systems possess an intrinsic state variable that encodes the system behavior in a bijective and causality-free manner' is presented without derivation, proof sketch, or extension of the fundamental lemma beyond LTI systems; this is load-bearing for the claim that control design can be carried out entirely within the learned state space for nonlinear plants.

Authors: The full manuscript develops the intrinsic state construction from the behavioral framework in the main body. We define the system behavior as the set of all admissible trajectories and construct the state as the minimal equivalence class of past trajectories that determines all future behavior, yielding a bijective and causality-free encoding. This extends the LTI fundamental lemma by replacing parametric kernel representations with trajectory-based set-theoretic arguments that apply to nonlinear systems. Control design is then performed entirely in this state space because the state contains all information needed for future evolution. We will insert a concise proof sketch and explicit reference to the relevant section in the revised abstract and introduction. revision: partial
Referee: [Abstract] Abstract: the neural-network architecture is stated to be 'capable of learning the behavior representation well-suited for control design,' yet no argument or experiment is supplied showing that the finite-data approximation preserves bijectivity or the exact input-output behavior required for stabilization; this directly affects the resolution of the model-centric vs. trajectory-based conflict.

Authors: The architecture is derived directly from the behavioral state construction so that the network learns a representation whose outputs can be used for state-space control. The manuscript reports numerical experiments on nonlinear plants in which the learned representation enables stabilization. A formal guarantee that finite-data approximation exactly preserves bijectivity is not supplied, as it would require additional assumptions on data richness and network capacity that lie outside the present scope. We will add a dedicated discussion of approximation properties, including the conditions under which the learned state remains suitable for control, together with the observed empirical behavior. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external behavioral framework

full rationale

The paper's core argument invokes the established behavioral systems theory to posit an intrinsic state that encodes trajectories bijectively and without causality, then uses this to motivate a neural network architecture for learning control-relevant representations from data. No equations, self-definitional reductions, or load-bearing self-citations are present in the abstract or description that would make the claimed bijectivity or control design equivalent to the inputs by construction. The behavioral framework is treated as an independent foundation (with known limitations to LTI systems noted externally), and the NN component is presented as a complementary learning step rather than a fitted prediction renamed as a result. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard behavioral-systems assumption that systems are fully described by trajectory sets and on the paper-specific assertion that an intrinsic state exists with the stated bijective and causality-free properties; no free parameters or invented entities are mentioned.

axioms (2)

domain assumption Dynamical systems are completely characterized by the set of all possible trajectories they can produce.
Foundational premise of behavioral systems theory invoked to define the intrinsic state.
ad hoc to paper There exists an intrinsic state variable that encodes the full behavior bijectively and without reference to causality.
Load-bearing claim stated in the abstract as shown via the behavioral framework.

pith-pipeline@v0.9.0 · 5452 in / 1399 out tokens · 46863 ms · 2026-05-10T13:07:54.011669+00:00 · methodology

discussion (0)

Reference graph

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