arxiv: 2604.07069 · v1 · submitted 2026-04-08 · 📡 eess.SY · cs.LG· cs.SY· math.DS

Recognition: 2 theorem links

· Lean Theorem

Controller Design for Structured State-space Models via Contraction Theory

Muhammad Zakwan , Vaibhav Gupta , Alireza Karimi , Efe C. Balta , Giancarlo Ferrari-Trecate

Authors on Pith no claims yet

Pith reviewed 2026-05-10 18:08 UTC · model grok-4.3

classification 📡 eess.SY cs.LGcs.SYmath.DS

keywords structured state-space modelscontraction theorycontrollabilityobservabilitylinear matrix inequalitiesseparation principleoutput feedback controlnonlinear systems

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The pith

Structured state-space models permit the first controllability and observability analysis, enabling LMI-based contraction controllers and a separation principle for output feedback.

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

The paper develops an indirect data-driven approach to output feedback controller design for nonlinear systems by treating structured state-space models as surrogate models. It supplies the initial controllability and observability conditions for these models. Those conditions produce scalable linear-matrix-inequality designs that rely on contraction theory. A separation principle is also derived, so that observers and state-feedback controllers can be synthesized independently while the closed-loop system remains exponentially stable.

Core claim

The authors establish controllability and observability conditions for structured state-space models. These conditions lead to linear matrix inequality formulations for designing contracting controllers. They further prove a separation principle that permits independent synthesis of observers and state-feedback laws without compromising the exponential stability of the closed-loop system.

What carries the argument

Structured state-space models analyzed for controllability and observability using contraction theory to derive LMI-based control conditions.

If this is right

Scalable control design becomes possible for long-sequence dynamical systems via LMIs.
Observer and controller can be designed separately while preserving closed-loop exponential stability.
The method supports data-driven identification of nonlinear systems followed by controller synthesis.
Exponential stability guarantees are obtained for the surrogate model under the derived conditions.

Where Pith is reading between the lines

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

Similar controllability and separation analysis might extend to other sequence models that maintain linear complexity in sequence length.
If the surrogate matches the plant dynamics sufficiently well, the framework could support real-time nonlinear control on embedded hardware.
Robustness extensions could incorporate bounded disturbances or model mismatch directly into the LMI conditions.

Load-bearing premise

The structured state-space model must serve as an accurate surrogate for the underlying nonlinear system so that its controllability, observability, and contraction properties transfer to guarantee stability on the real plant.

What would settle it

A counterexample in which an SSM meets the derived LMI conditions and separation principle yet the corresponding real nonlinear system under the designed output-feedback controller fails to exhibit exponential stability would falsify the transfer of guarantees.

Figures

Figures reproduced from arXiv: 2604.07069 by Alireza Karimi, Efe C. Balta, Giancarlo Ferrari-Trecate, Muhammad Zakwan, Vaibhav Gupta.

**Figure 1.** Figure 1: Typical SSM layer The recurrent unit is typically a discrete-time state-space model. In this paper, we focus on an LRU [10], defined using the following discrete-time state-space equations: xk+1 = Axk + Bu¯k (1a) y¯k = Cxk + Du¯k (1b) where x ∈ R nx , u¯ ∈ R nu¯ , and y¯ ∈ R ny¯ denote the state, input, and output, respectively.2 The matrices A ∈ R nx×nx , B ∈ R nx×nu¯ , C ∈ R ny¯×nx , and D ∈ R ny¯×nu¯ ar… view at source ↗

**Figure 2.** Figure 2: A sample validation trajectory after the training [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: Controller response on the nonlinear mathematical [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

read the original abstract

This paper presents an indirect data-driven output feedback controller synthesis for nonlinear systems, leveraging Structured State-space Models (SSMs) as surrogate models. SSMs have emerged as a compelling alternative in modelling time-series data and dynamical systems. They can capture long-term dependencies while maintaining linear computational complexity with respect to the sequence length, in comparison to the quadratic complexity of Transformer-based architectures. The contributions of this work are threefold. We provide the first analysis of controllability and observability of SSMs, which leads to scalable control design via Linear Matrix Inequalities (LMIs) that leverage contraction theory. Moreover, a separation principle for SSMs is established, enabling the independent design of observers and state-feedback controllers while preserving the exponential stability of the closed-loop system. The effectiveness of the proposed framework is demonstrated through a numerical example, showcasing nonlinear system identification and the synthesis of an output feedback controller.

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

SSMs get their first controllability and observability analysis plus a separation principle via contraction theory, but the transfer of closed-loop guarantees from the surrogate to the true nonlinear plant lacks supporting bounds.

read the letter

The main thing here is the controllability and observability analysis for structured state-space models, which feeds directly into LMI conditions for output-feedback design and a separation principle that keeps exponential stability when the observer and controller are designed separately. That combination is new on the stated terms and fits the efficiency advantage of SSMs over quadratic-cost alternatives for long sequences. The paper does a reasonable job laying out how contraction metrics translate into tractable LMIs and showing the pipeline on a numerical example that includes both identification and closed-loop synthesis. The separation principle is a clean addition that simplifies the workflow without extra conservatism in the model. The soft spot is the indirect route to the real plant. All formal results sit on the fitted SSM, yet there is no general bound on how large the approximation error can be before the contraction property or the LMI-derived gains stop guaranteeing stability on the original nonlinear dynamics. The single numerical case works but does not test sensitivity to identification residuals or supply incremental stability margins. This paper is for people working on scalable data-driven control who already use or want to use SSMs. A reader who needs verifiable guarantees on approximate models will want to inspect the derivations closely. I would send it to peer review to check the new analysis and to see whether the robustness gap can be closed or at least quantified.

Referee Report

2 major / 2 minor

Summary. The paper proposes an indirect data-driven output-feedback controller synthesis method for nonlinear systems that uses Structured State-space Models (SSMs) as surrogate models. It claims to deliver the first controllability and observability analysis for SSMs, which enables scalable LMI-based state-feedback and observer design via contraction theory, establishes a separation principle that preserves exponential stability, and demonstrates the approach on a numerical nonlinear system identification and control example.

Significance. If the stability transfer from the SSM surrogate to the true nonlinear plant can be made rigorous, the work would offer a computationally attractive bridge between efficient sequence modeling and contraction-theoretic control, potentially enabling scalable output-feedback design for systems where direct nonlinear analysis is intractable. The separation principle and LMI formulation are potentially valuable contributions if the underlying controllability/observability results hold.

major comments (2)

[Introduction and closed-loop stability section] The central claim that the synthesized controller stabilizes the original nonlinear plant (Introduction and Section on closed-loop stability) rests on the SSM serving as a faithful surrogate, yet no incremental stability margin, approximation-error bound, or robustness result is supplied to guarantee that the contraction metric and LMI-derived gains remain valid under realistic identification residuals. The numerical example shows only one successful case without quantifying the required model accuracy.
[Controllability and observability analysis section] § on controllability/observability analysis: the claimed 'first analysis' and resulting LMI conditions for contraction are presented without an explicit statement of the precise SSM state-space realization used (e.g., the structured matrices A, B, C and their dependence on the learned parameters), making it impossible to verify whether the controllability rank conditions or the LMI feasibility indeed imply the stated exponential stability for the surrogate.

minor comments (2)

[Abstract and Introduction] The abstract and introduction use 'scalable' and 'parameter-free' without quantifying computational complexity relative to sequence length or comparing against standard LMI solvers for the specific SSM dimension.
[Preliminaries and main results] Notation for the contraction metric and the LMI variables is introduced without a dedicated nomenclature table or consistent cross-referencing to the SSM equations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for their thorough review and valuable suggestions. We address each of the major comments in detail below and indicate the revisions we plan to make to the manuscript.

read point-by-point responses

Referee: [Introduction and closed-loop stability section] The central claim that the synthesized controller stabilizes the original nonlinear plant (Introduction and Section on closed-loop stability) rests on the SSM serving as a faithful surrogate, yet no incremental stability margin, approximation-error bound, or robustness result is supplied to guarantee that the contraction metric and LMI-derived gains remain valid under realistic identification residuals. The numerical example shows only one successful case without quantifying the required model accuracy.

Authors: We thank the referee for highlighting this important aspect. Our manuscript focuses on providing controllability and observability analysis for SSMs to facilitate scalable LMI-based design using contraction theory, and establishing a separation principle. The exponential stability is rigorously shown for the closed-loop system under the SSM dynamics. Regarding the transfer of stability to the true nonlinear plant, the paper assumes that the SSM serves as an accurate surrogate model obtained via system identification, which is a standard assumption in indirect data-driven control approaches. The numerical example illustrates the practical application but does not include a comprehensive sensitivity analysis. We agree that adding a discussion on potential robustness margins and the limitations due to identification errors would be beneficial. In the revised manuscript, we will include a new subsection or paragraph in the closed-loop stability section discussing the assumptions on model fidelity and suggesting future directions for incorporating approximation error bounds. revision: partial
Referee: [Controllability and observability analysis section] § on controllability/observability analysis: the claimed 'first analysis' and resulting LMI conditions for contraction are presented without an explicit statement of the precise SSM state-space realization used (e.g., the structured matrices A, B, C and their dependence on the learned parameters), making it impossible to verify whether the controllability rank conditions or the LMI feasibility indeed imply the stated exponential stability for the surrogate.

Authors: We apologize for the lack of clarity in this section. The SSM considered in our analysis follows the standard formulation used in structured state-space models for time-series modeling, typically involving a state matrix A that is structured (e.g., diagonal with learned parameters) and input/output matrices B and C that may also have specific structures depending on the parameterization. We will revise the controllability and observability analysis section to explicitly define the state-space realization, including the forms of A, B, and C in terms of the model parameters, and show how the rank conditions and LMI feasibility lead to the contraction properties and exponential stability. revision: yes

Circularity Check

0 steps flagged

No circularity: original controllability/observability analysis and separation principle for SSMs

full rationale

The paper derives controllability and observability properties for Structured State-space Models from first principles, then applies contraction theory to obtain LMI-based controller synthesis and establishes a separation principle for output-feedback design. These steps are presented as novel contributions without any reduction to self-definitional inputs, fitted parameters renamed as predictions, or load-bearing self-citations. The abstract and described contributions show independent mathematical analysis that does not tautologically reproduce its assumptions or data. The numerical example serves only for illustration, not as the source of the core claims. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Limited information available from abstract only. The framework rests on standard properties of contraction theory for exponential stability and the assumption that SSMs can serve as faithful surrogate models for nonlinear dynamics. No explicit free parameters or invented entities are described.

axioms (2)

domain assumption Contraction theory supplies sufficient conditions for exponential stability of nonlinear systems that can be encoded as LMIs.
The paper uses contraction theory to obtain LMI conditions for controller design, invoking its incremental stability properties without deriving them.
domain assumption The SSM surrogate preserves the controllability, observability, and contraction properties of the true nonlinear system.
The control design is performed on the SSM and claimed to work for the original system; this transfer is not justified in the abstract.

pith-pipeline@v0.9.0 · 5473 in / 1412 out tokens · 54524 ms · 2026-05-10T18:08:18.818882+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

A Nonlinear Separation Principle via Contraction Theory: Applications to Neural Networks, Control, and Learning
eess.SY 2026-04 unverdicted novelty 5.0

A contraction-theory separation principle yields global exponential stability for controller-observer pairs and sharp LMI certificates for contractive RNNs, enabling stable output tracking and implicit neural network design.

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