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arxiv: 2603.27833 · v3 · submitted 2026-03-29 · 🧮 math.OC · cs.IT· cs.MA· cs.RO· cs.SY· eess.SY· math.IT

Separation is Optimal for LQR under Intermittent Feedback

Abdullah Y. Etcibasi , C. Emre Koksal , Eylem Ekici This is my paper

Pith reviewed 2026-05-14 21:23 UTC · model grok-4.3

classification 🧮 math.OC cs.ITcs.MAcs.ROcs.SYeess.SYmath.IT
keywords separation principlelinear quadratic regulatorintermittent feedbackcommunication constraintsoptimal schedulingthreshold policydynamic programmingsymmetric disturbances
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The pith

Separation principle holds for LQR under intermittent feedback with symmetric disturbances

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

The paper proves that the separation principle remains valid for linear quadratic regulator problems when feedback arrives only intermittently, provided the disturbances are independent, identically distributed, zero-mean, and symmetrically distributed. Under this condition the optimal controller takes the form of a discounted linear feedback law whose gains do not depend on the scheduling policy, while the optimal scheduler is a symmetric threshold rule applied to the accumulated disturbance since the last successful update. A reader would care because the result decouples two difficult design tasks: one can first pick the controller as in the classical full-information case and then separately optimize the communication schedule without losing optimality.

Core claim

We first prove that the separation principle holds for communication-constrained LQR problems under i.i.d. zero-mean disturbances with a symmetric distribution. We then solve the dynamic programming problem and show that the optimal scheduling policy is a symmetric threshold rule on the accumulated disturbance since the most recent update, while the optimal controller is a discounted linear feedback law independent of the scheduling policy.

What carries the argument

Separation principle that decouples the discounted linear feedback controller from the symmetric threshold scheduler on accumulated disturbance

If this is right

  • The controller can be designed exactly as in the classical LQR case without reference to the communication schedule.
  • Scheduling decisions reduce to checking whether the magnitude of accumulated disturbance exceeds a fixed threshold.
  • The overall optimal policy is obtained by independently solving the standard Riccati equation for the controller and a one-dimensional dynamic program for the thresholds.
  • The result holds for any symmetric zero-mean disturbance distribution, not just Gaussian.

Where Pith is reading between the lines

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

  • The same separation may simplify design in remote-control applications where bandwidth is limited but noise symmetry can be verified from data.
  • If symmetry fails, the coupled optimization problem would have to be solved jointly, potentially requiring approximate dynamic programming.
  • Explicit threshold computation for low-dimensional systems could be performed once the Riccati solution is known.
  • The structure suggests similar threshold policies might appear in other quadratic-cost problems with intermittent observations.

Load-bearing premise

The disturbances must be independent, identically distributed, zero-mean, and symmetrically distributed around zero.

What would settle it

A numerical counter-example in which the optimal feedback gain changes when the scheduler is altered, even though the disturbances remain i.i.d., zero-mean, and symmetric.

Figures

Figures reproduced from arXiv: 2603.27833 by Abdullah Y. Etcibasi, C. Emre Koksal, Eylem Ekici.

Figure 1
Figure 1. Figure 1: Block diagram of the closed-loop control system [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Trellis diagram illustrating the evolution of the value functions across time [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Running-average LQR cost versus time under Gaussian disturbances. All policies are [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Steady-state LQR cost versus the open-loop gain [PITH_FULL_IMAGE:figures/full_fig_p025_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Steady-state LQR cost versus the target communication rate [PITH_FULL_IMAGE:figures/full_fig_p025_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Steady-state LQR cost versus disturbance standard deviation [PITH_FULL_IMAGE:figures/full_fig_p026_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Steady-state LQR cost versus rs for a = 1 under Laplace disturbances. The behavior closely matches the Gaussian case: ZOH policies become unstable for rs < 0.4, and the optimal policy achieves the lowest cost across all rates. and Am = \m j=1 ( [PITH_FULL_IMAGE:figures/full_fig_p027_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Normalized steady-state cost difference relative to Gaussian disturbances versus [PITH_FULL_IMAGE:figures/full_fig_p028_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Normalized steady-state cost difference relative to Gaussian disturbances versus [PITH_FULL_IMAGE:figures/full_fig_p029_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Normalized steady-state cost difference relative to Gaussian disturbances versus [PITH_FULL_IMAGE:figures/full_fig_p030_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Normalized steady-state cost difference relative to Gaussian disturbances versus [PITH_FULL_IMAGE:figures/full_fig_p031_11.png] view at source ↗
read the original abstract

In this work, we first prove that the separation principle holds for communication-constrained LQR problems under i.i.d. zero-mean disturbances with a symmetric distribution. We then solve the dynamic programming problem and show that the optimal scheduling policy is a symmetric threshold rule on the accumulated disturbance since the most recent update, while the optimal controller is a discounted linear feedback law independent of the scheduling policy.

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 / 2 minor

Summary. The manuscript proves that the separation principle holds for communication-constrained LQR problems under i.i.d. zero-mean disturbances with symmetric distribution. It solves the dynamic programming problem to establish that the optimal scheduling policy is a symmetric threshold rule on the accumulated disturbance since the most recent update, while the optimal controller reduces to a discounted linear feedback law that is independent of the scheduling policy.

Significance. If the derivations hold under the stated conditions, the result supplies a rigorous structural characterization of optimal policies for intermittent-feedback LQR, confirming separation and yielding an explicit threshold scheduler together with a controller that decouples from scheduling decisions. This supplies a clean, symmetry-driven benchmark for networked control design and could simplify analysis of communication-constrained systems.

major comments (1)
  1. [Dynamic Programming Solution] The dynamic-programming recursion establishing controller independence from scheduling (abstract and corresponding DP section) relies on symmetry to separate the value function; the manuscript should explicitly verify that the Bellman operator preserves this separation for the given disturbance class, including a short inductive step or explicit form of the cost-to-go.
minor comments (2)
  1. Clarify in the introduction whether the threshold is parameter-free or depends on the LQR cost matrices and disturbance variance; the current abstract leaves this implicit.
  2. Add a brief remark on how the i.i.d. zero-mean symmetric assumption can be relaxed or checked in practice, to aid readers applying the result.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment and the recommendation for minor revision. We address the single major comment below.

read point-by-point responses

  1. Referee: The dynamic-programming recursion establishing controller independence from scheduling (abstract and corresponding DP section) relies on symmetry to separate the value function; the manuscript should explicitly verify that the Bellman operator preserves this separation for the given disturbance class, including a short inductive step or explicit form of the cost-to-go.

    Authors: We agree that making the preservation of the separation property under the Bellman operator fully explicit will improve clarity. In the revised version we will insert a short inductive argument in the dynamic-programming section. The argument shows that if the cost-to-go at stage k+1 separates into a term that depends only on the scheduling state and a quadratic term that depends only on the controller state, then the same additive separation is preserved at stage k for any i.i.d. zero-mean symmetric disturbance. We will also record the explicit form of the cost-to-go that results once the optimal threshold scheduler and discounted linear feedback are substituted. revision: yes

Circularity Check

0 steps flagged

Derivation proceeds directly from dynamic programming on stated assumptions

full rationale

The paper states the separation principle and optimal threshold scheduler follow from solving the dynamic programming recursion under i.i.d. zero-mean symmetric disturbances. The controller is shown to reduce to a discounted linear feedback law independent of the scheduler. No equation or claim reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the symmetry condition is an explicit input assumption used to establish the threshold structure, not derived from the result itself. The derivation chain is therefore self-contained against the given model.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on the disturbance symmetry assumption to obtain both separation and the threshold policy; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Disturbances are i.i.d. zero-mean with symmetric distribution
    Invoked in the abstract to prove separation and derive the symmetric threshold scheduler.

pith-pipeline@v0.9.0 · 5375 in / 1144 out tokens · 35734 ms · 2026-05-14T21:23:11.909514+00:00 · methodology

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