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arxiv: 2606.31396 · v1 · pith:SCFBZOP2 · submitted 2026-06-30 · eess.SY · cs.SY

Sensing-Limited Control Under Non-Designable Observation Mechanisms

Ming Li , Fan Liu , Yifeng Xiong , Jie Xu , Tao Liu This is my paper

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classification eess.SY cs.SY
keywords sensing-limited controldirected information ratemean-square stabilizabilitynon-designable observationslinear systemsentropy-to-error bridgecertainty equivalence
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0 comments X

The pith

For linear systems with fixed sensing, the directed information rate from unstable states to observations must exceed the open-loop expansion rate of unstable modes to achieve mean-square observability and stabilizability.

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

The paper derives necessary conditions showing that control of unstable linear dynamics through non-designable observations requires the directed information rate from the unstable state process to the observation process to dominate the sum of the logarithms of the unstable eigenvalues. This lower bound remains necessary even when additive process disturbances are present. In the linear-Gaussian case an exact expression for the full-state directed information rate supplies a computable surrogate that upper-bounds the unstable-state quantity. Under posterior regularity conditions a strict surplus of directed information forces the posterior uncertainty to collapse, yielding sufficient conditions for asymptotic mean-square observability and, by certainty-equivalence control, for stabilizability.

Core claim

The central claim is that mean-square observability and stabilizability of noiseless linear systems under non-designable observations are possible only when the directed information rate from the unstable state process to the observation process is at least as large as the open-loop expansion rate of the unstable modes; the same lower bound holds with additive disturbances, and an entropy-to-error relation converts a strict surplus into vanishing estimation-error covariance, thereby giving sufficient conditions via certainty equivalence.

What carries the argument

The directed information rate from the unstable state process to the observation process, which must dominate the open-loop expansion rate of the unstable modes and, via an entropy-to-error bridge, forces posterior uncertainty to collapse when the surplus is strict.

If this is right

  • The lower bound on the unstable-state directed information rate is necessary for mean-square observability even when process noise is present.
  • The full-state directed information rate supplies an exact, computable upper bound that yields necessary conditions in the linear-Gaussian setting.
  • Under posterior regularity, a strict information surplus plus certainty-equivalence control yields asymptotic mean-square stabilizability.
  • Classical communication-constrained limits must be reinterpreted when the sensing interface itself cannot be designed.

Where Pith is reading between the lines

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

  • The same information-rate threshold may serve as a quick feasibility check for any linear plant whose sensor map is fixed in advance rather than optimized.
  • Because the bound survives additive disturbances, it could be used to decide whether a given sensor suite is adequate before controller design begins.
  • The entropy-to-error bridge suggests that similar collapse arguments might apply to other performance metrics once a suitable information measure is identified.

Load-bearing premise

A strict surplus of directed information over the expansion rate forces the posterior uncertainty to collapse and the estimation error covariance to zero.

What would settle it

A linear system and observation mechanism where the directed information rate from the unstable state exceeds the expansion rate yet the estimation error covariance fails to converge to zero, or where the rate is strictly below the expansion rate yet the covariance converges to zero.

Figures

Figures reproduced from arXiv: 2606.31396 by Fan Liu, Jie Xu, Ming Li, Tao Liu, Yifeng Xiong.

Figure 1
Figure 1. Figure 1: The closed-loop control architecture under a non-designable obser [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Communication-constrained control uses a designable encoder-channel [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
read the original abstract

We study the information-theoretic limits of controlling unstable linear systems through non-designable observation mechanisms. Unlike classical communication-constrained control, the information bottleneck lies in the observation mechanism rather than in a designable encoder-channel interface. For noiseless linear dynamics, we derive necessary conditions for mean-square observability and stabilizability, showing that the directed information rate from the unstable state process to the observation process must dominate the open-loop expansion rate of the unstable modes. We further show that this lower bound persists under additive process disturbances. In the Linear-Gaussian setting, although the unstable-state directed information rate remains intractable in closed form, we obtain an exact characterization of the full-state directed information rate, which upper-bounds the unstable-state quantity and yields computable necessary conditions. Under suitable posterior regularity conditions, we also establish sufficient conditions for asymptotic mean-square observability and, via certainty-equivalence control, asymptotic mean-square stabilizability. The key step is an entropy-to-error bridge: a strict surplus in directed information over the expansion rate forces posterior uncertainty to collapse and thereby drives the estimation error covariance to zero. These results identify a fundamental feasibility boundary for sensing-limited control and clarify how classical communication-based limits must be reinterpreted when the sensing interface is non-designable.

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.

Referee Report

1 major / 0 minor

Summary. The paper claims that for noiseless linear dynamics under non-designable observation mechanisms, the directed information rate from the unstable state process to the observation process must dominate the open-loop expansion rate of the unstable modes to achieve mean-square observability and stabilizability. This lower bound persists under additive process disturbances. In the linear-Gaussian case, an exact characterization of the full-state directed information rate is obtained as an upper bound yielding computable necessary conditions. Under posterior regularity conditions, sufficient conditions for asymptotic mean-square observability and stabilizability (via certainty equivalence) are established through an entropy-to-error bridge asserting that a strict information surplus forces posterior uncertainty collapse and error covariance to zero.

Significance. If the central claims hold, the work would identify a fundamental feasibility boundary for sensing-limited control of unstable systems, extending classical directed-information results to settings where the observation map is fixed and non-designable rather than a designable encoder-channel pair. The persistence of the bound under disturbances and the intrinsic (non-fitted) nature of the open-loop expansion rate as the lower bound are notable strengths. The linear-Gaussian upper-bound characterization also supplies a concrete computational pathway for necessary conditions.

major comments (1)
  1. [sufficient conditions / entropy-to-error bridge paragraph] The entropy-to-error bridge (invoked for the sufficient conditions) is load-bearing: the manuscript states that a strict surplus of directed information over the expansion rate forces posterior uncertainty to collapse, driving the estimation error covariance to zero under the stated posterior regularity conditions. Because the observation mechanism is non-designable and fixed, it is not immediate that these regularity conditions convert the surplus into covariance collapse for every admissible mechanism; any gap here would invalidate the certainty-equivalence stabilizability claim. Derivation details of this step are unavailable for verification.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, positive summary, and identification of the entropy-to-error bridge as central to the sufficient conditions. We address the major comment below.

read point-by-point responses

  1. Referee: [sufficient conditions / entropy-to-error bridge paragraph] The entropy-to-error bridge (invoked for the sufficient conditions) is load-bearing: the manuscript states that a strict surplus of directed information over the expansion rate forces posterior uncertainty to collapse, driving the estimation error covariance to zero under the stated posterior regularity conditions. Because the observation mechanism is non-designable and fixed, it is not immediate that these regularity conditions convert the surplus into covariance collapse for every admissible mechanism; any gap here would invalidate the certainty-equivalence stabilizability claim. Derivation details of this step are unavailable for verification.

    Authors: We agree that the entropy-to-error bridge is load-bearing for the sufficient conditions and that its justification must hold uniformly for non-designable mechanisms. The derivation (Section 4) proceeds by showing that a strict directed-information surplus implies that the conditional entropy rate of the state given observations is strictly negative relative to the open-loop expansion rate. Under the stated posterior regularity conditions (absolute continuity of the posterior with uniformly bounded second moments of the log-density), standard entropy-to-variance inequalities then force the posterior covariance to decay to zero. These regularity conditions are formulated as properties of the posterior process and are required to hold for every admissible observation map; the proof does not rely on designability. That said, the referee is correct that the explicit chaining of the rate inequality to covariance collapse is not expanded in full detail. We will revise the manuscript to insert an expanded lemma (with all intermediate inequalities) in the appendix, making the uniform applicability explicit and thereby supporting the certainty-equivalence stabilizability claim. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation uses intrinsic dynamics and standard information rates without reduction to inputs

full rationale

The necessary conditions follow from directed-information inequalities exceeding the open-loop sum of unstable log-eigenvalues, an intrinsic property of the linear dynamics matrix rather than any fitted or self-defined quantity. The linear-Gaussian full-state directed information rate is characterized exactly and used only as an upper bound on the unstable-state rate. The entropy-to-error bridge for sufficient conditions is stated as a derived implication under explicit posterior regularity assumptions, with no evidence of self-citation chains, ansatz smuggling, or renaming of known results as new predictions. The paper remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The results rest on standard linear dynamics, additive disturbances, and information-theoretic quantities without introducing new free parameters or invented entities in the abstract.

axioms (3)
  • domain assumption System dynamics are linear (noiseless or with additive process noise)
    Invoked for all necessary-condition derivations in the abstract.
  • standard math Directed information rate is well-defined and comparable to the open-loop expansion rate
    Central to the necessary conditions; standard in information theory.
  • domain assumption Posterior regularity conditions hold
    Required for the sufficient conditions via certainty equivalence.

pith-pipeline@v0.9.1-grok · 5755 in / 1605 out tokens · 48985 ms · 2026-07-01T04:18:30.905894+00:00 · methodology

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