Characterizing all locally exponentially stabilizing controllers as a linear feedback plus learnable nonlinear Youla dynamics

Luca Furieri

arxiv: 2601.02244 · v2 · pith:6NMBU37Bnew · submitted 2026-01-05 · 📡 eess.SY · cs.SY

Characterizing all locally exponentially stabilizing controllers as a linear feedback plus learnable nonlinear Youla dynamics

Luca Furieri This is my paper

Pith reviewed 2026-05-21 16:09 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords nonlinear controllocal exponential stabilityYoula parametrizationstate-feedbackneural ODEsrecurrent neural networksinput-affine systemscontroller parametrization

0 comments

The pith

All locally exponentially stabilizing controllers for nonlinear systems decompose into a fixed linear state feedback plus the output of stable internal dynamics.

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

The paper establishes that any dynamic state-feedback controller making an equilibrium locally exponentially stable can be written exactly as the sum of a linear gain Kx on the plant state and the output of some internal controller dynamics that are themselves locally exponentially stable. The linear gain is chosen to stabilize only the plant's linearization at the equilibrium, while the internal part carries all the remaining nonlinear action. This gives a complete state-space parametrization of stabilizing controllers, so that the nonlinear component can be freely designed or learned without destroying the local exponential stability guarantee. A reader would care because it separates the stability task (handled by linear theory) from performance optimization (handled by the learnable part), opening a direct route to training recurrent networks or neural ODEs inside the controller.

Core claim

We derive a state-space characterization of all dynamic state-feedback controllers that make an equilibrium of a nonlinear input-affine continuous-time system locally exponentially stable. Specifically, any controller obtained as the sum of a linear state-feedback u=Kx, with K stabilizing the linearized system, and the output of internal locally exponentially stable controller dynamics is itself locally exponentially stabilizing. Conversely, every dynamic state-feedback controller that locally exponentially stabilizes the equilibrium admits such a decomposition. The result can be viewed as a state-space nonlinear Youla-type parametrization specialized to local, rather than global, and to exp

What carries the argument

The nonlinear Youla-type parametrization that decomposes every locally exponentially stabilizing dynamic state-feedback controller into linear feedback plus locally exponentially stable internal dynamics.

If this is right

The internal locally exponentially stable dynamics can be realized by stable recurrent neural networks.
These dynamics can be trained as neural ODEs to optimize closed-loop performance while the linear feedback guarantees local exponential stability.
Any existing linear stabilizer can be kept fixed and the remaining controller freedom used entirely for learning or optimization.

Where Pith is reading between the lines

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

This split may let engineers start with a classical linear design and then add data-driven nonlinear compensation without re-proving stability.
The same decomposition could be tested numerically on benchmark nonlinear plants to verify that learned networks remain inside the stable class.
It suggests a practical workflow: first compute K from the linearization, then optimize only the internal dynamics on closed-loop trajectories.

Load-bearing premise

The plant is a nonlinear input-affine continuous-time system whose linearization at the equilibrium admits a stabilizing state-feedback gain K.

What would settle it

A concrete dynamic state-feedback controller that renders the closed-loop equilibrium locally exponentially stable yet cannot be written as any linear gain Kx plus the output of locally exponentially stable internal dynamics.

read the original abstract

We derive a state-space characterization of all dynamic state-feedback controllers that make an equilibrium of a nonlinear input-affine continuous-time system locally exponentially stable. Specifically, any controller obtained as the sum of a linear state-feedback $u=Kx$, with $K$ stabilizing the linearized system, and the output of internal locally exponentially stable controller dynamics is itself locally exponentially stabilizing. Conversely, every dynamic state-feedback controller that locally exponentially stabilizes the equilibrium admits such a decomposition. The result can be viewed as a state-space nonlinear Youla-type parametrization specialized to local, rather than global, and exponential, rather than asymptotic, closed-loop stability. The residual locally exponentially stable controller dynamics can be implemented with stable recurrent neural networks and trained as neural ODEs to achieve high closed-loop performance in nonlinear control tasks.

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 gives a state-space Youla-style split for locally exponentially stabilizing controllers on nonlinear input-affine systems, but the forward direction looks vulnerable to direct feedthrough in the added dynamics.

read the letter

The central claim is that every locally exponentially stabilizing dynamic state-feedback controller for these plants can be decomposed into a fixed linear gain K (from the linearization) plus the output of some internal locally exponentially stable dynamics, and that any such sum is itself stabilizing. That is the new piece relative to the linear Youla results cited in the abstract. It is useful because it lets the residual dynamics be realized with stable recurrent networks or neural ODEs while the linear part carries the stability guarantee. That separation is the practical hook for learning-based control work. The assumptions are standard and clearly stated: input-affine continuous-time plant whose linearization is stabilizable by state feedback. The if-and-only-if framing is stated cleanly in the abstract. The soft spot is exactly the one flagged in the stress-test note. If the internal dynamics have a nonzero linear direct term from x to their output, the closed-loop linearization becomes A + B(K + D) rather than A + BK. Nothing in the abstract restricts the dynamics to be strictly proper at first order or shows how to absorb any D into K without losing the parametrization. If the full proof uses higher-order terms or a specific neighborhood argument to sidestep this, that needs to be explicit; otherwise the forward implication does not go through for general locally exponentially stable internal dynamics. The citation pattern looks light but appropriate for an initial characterization result. This is aimed at nonlinear control theorists and at people building learning-based controllers who need stability by construction. It is coherent enough on its own terms to deserve referee time, even if the linearization detail requires revision.

Referee Report

1 major / 2 minor

Summary. The manuscript claims an if-and-only-if state-space characterization of all dynamic state-feedback controllers that render an equilibrium of a nonlinear input-affine continuous-time system locally exponentially stable. Any controller formed as the sum of a linear state-feedback gain K (stabilizing the linearization) plus the output of arbitrary internal locally exponentially stable (LES) dynamics is asserted to be locally exponentially stabilizing; conversely, every locally exponentially stabilizing dynamic controller admits such a decomposition. The result is presented as a local-exponential specialization of nonlinear Youla parametrization, with the residual LES dynamics suitable for implementation and training via stable recurrent neural networks or neural ODEs.

Significance. If the central characterization is correct, the result supplies a complete parametrization separating a fixed linear stabilizer from a freely learnable but internally LES nonlinear component. This would enable systematic incorporation of data-driven or neural components into nonlinear controllers while retaining local exponential stability guarantees, extending classical Youla theory beyond the linear or global-asymptotic cases. The explicit link to neural-ODE training is a concrete practical contribution.

major comments (1)

[§3, Theorem 3.1] §3, Theorem 3.1 (forward implication): the claim that u = Kx + y_c where the internal dynamics are any LES system yields local exponential stability does not hold without further restrictions. The closed-loop linearization at the equilibrium is A + B(K + D), where D is the linear direct-feedthrough term from the state to the output map of the internal dynamics. Only A + BK is assumed Hurwitz; a nonzero D can move eigenvalues into the right half-plane. The manuscript must either (i) restrict the internal dynamics to be strictly proper in their linearization or (ii) prove that LES forces D = 0. The current statement and proof sketch do not address this case.

minor comments (2)

[§2] Notation for the internal dynamics (x_c, y_c, etc.) is introduced without an explicit state-space realization in the theorem statement; adding the precise equations would improve readability.
[Abstract] The abstract mentions 'learnable nonlinear Youla dynamics' but does not state the precise stability class (local exponential) required of the internal system; this should be clarified for consistency with the theorem.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying this important technical detail in the forward direction of Theorem 3.1. The observation is correct and requires a clarification in the manuscript. We address the point below and will revise accordingly.

read point-by-point responses

Referee: [§3, Theorem 3.1] §3, Theorem 3.1 (forward implication): the claim that u = Kx + y_c where the internal dynamics are any LES system yields local exponential stability does not hold without further restrictions. The closed-loop linearization at the equilibrium is A + B(K + D), where D is the linear direct-feedthrough term from the state to the output map of the internal dynamics. Only A + BK is assumed Hurwitz; a nonzero D can move eigenvalues into the right half-plane. The manuscript must either (i) restrict the internal dynamics to be strictly proper in their linearization or (ii) prove that LES forces D = 0. The current statement and proof sketch do not address this case.

Authors: We agree that the forward implication as currently stated is incomplete without addressing the direct-feedthrough term D. A nonzero D in the linearization of the internal dynamics can indeed alter the closed-loop matrix to A + B(K + D), which may not be Hurwitz. We will revise the theorem statement, assumptions, and proof to explicitly require that the internal dynamics are strictly proper in their linearization (i.e., the output map depends only on the internal state and not directly on x, forcing D = 0). This is a natural and standard restriction in dynamic controller parametrizations and does not diminish the practical utility for learning nonlinear components via stable RNNs or neural ODEs. With this addition the forward direction holds by the same linearization and Lyapunov arguments already sketched. We will also update the abstract and introduction for consistency. Option (ii) does not hold in general, as LES of the unforced internal dynamics does not constrain the input-to-output linearization term D. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained from stability definitions; no reduction to inputs by construction

full rationale

The paper states a bidirectional characterization: any linear K plus LES internal dynamics yields local exponential stability, and conversely every stabilizing dynamic controller admits such a decomposition. This follows directly from the definitions of local exponential stability, the input-affine plant, and the linearization at equilibrium being Hurwitz under K. No equations reduce a 'prediction' to a fitted parameter, no self-definitional loop appears, and the Youla reference is to a standard concept specialized here rather than smuggled via self-citation. The central claim retains independent mathematical content from the system class and stability notions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard domain assumption that the plant is input-affine and that local exponential stability of the linearization plus internal dynamics implies the same for the closed loop; no free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption The plant is a nonlinear input-affine continuous-time system.
Explicitly stated as the system class for which the characterization holds.
domain assumption K stabilizes the linearized system at the equilibrium.
Required for the linear component of the decomposition to contribute to local exponential stability.

pith-pipeline@v0.9.0 · 5659 in / 1294 out tokens · 110278 ms · 2026-05-21T16:09:18.935295+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

any controller obtained as the sum of a linear state-feedback u=Kx, with K stabilizing the linearized system, and the output of internal locally exponentially stable controller dynamics
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The result can be viewed as a state-space nonlinear Youla-type parametrization specialized to local, rather than global, and exponential, rather than asymptotic, closed-loop stability.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.