Linear Reformulation of Event-Triggered LQG Control under Unreliable Communication

Dipankar Maity; Zahra Hashemi

arxiv: 2604.06060 · v1 · submitted 2026-04-07 · 📡 eess.SY · cs.SY

Linear Reformulation of Event-Triggered LQG Control under Unreliable Communication

Zahra Hashemi , Dipankar Maity This is my paper

Pith reviewed 2026-05-10 19:37 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords event-triggered controlLQG controlpacket-erasure channelmixed-integer linear programmingerror covariancemodel predictive controltransmission scheduling

0 comments

The pith

Event-triggered LQG control over i.i.d. packet-erasure channels reduces to solving a compact mixed-integer linear program.

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

The paper shows how to choose when to transmit sensor data for linear-quadratic-Gaussian control when packets can be lost at random. It expands the estimation error covariance into sums of fixed Gramian matrices multiplied by survival factors that depend only on how many transmission attempts occur in each interval. This turns the joint problem of picking a control law and a send-or-skip schedule into an unconstrained binary optimization task that is then converted exactly into a mixed-integer linear program. The resulting formulation is small enough to solve repeatedly in a receding-horizon loop, and numerical tests on a Boeing-747 model show it uses far fewer transmissions while keeping cost low across different channel reliabilities.

Core claim

By expressing the estimation error covariance as a sum of precomputable Gramian matrices each multiplied by a survival probability factor that depends solely on the count of transmission attempts within each time interval, the co-design of control input and transmission schedule becomes an unconstrained binary optimization problem. This binary program is then exactly linearized using running counters of attempts and a one-hot encoding of the attempt numbers, resulting in a mixed-integer linear program suitable for model predictive control implementation.

What carries the argument

Closed-form expansion of error covariance as precomputable Gramian terms scaled by survival factors that depend only on transmission attempt counts per interval, then linearized exactly via running attempt counters and one-hot encoding into a compact MILP.

If this is right

The scheduler runs online in receding horizon because the MILP remains compact after linearization.
On the Boeing-747 benchmark the MPC scheduler achieves lower total cost while using substantially fewer transmissions than a one-shot baseline.
Performance holds across a range of channel success probabilities without retuning the formulation.
Certainty equivalence decouples the LQR feedback gain from the scheduling decisions under randomized deliveries.

Where Pith is reading between the lines

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

The same covariance expansion could be reused for other event-triggered problems that share i.i.d. uncertainty and quadratic costs.
The compact MILP size makes real-time embedded implementation on modest hardware feasible for systems that must trade communication energy against control accuracy.
Additional linear constraints such as hard limits on total transmissions or energy per window can be added directly without changing the overall structure.

Load-bearing premise

The packet-erasure channel is memoryless and identically distributed over time, and certainty equivalence lets the optimal control law be designed separately from the transmission decisions.

What would settle it

Run Monte-Carlo closed-loop simulations of the MILP schedule against the exact dynamic-programming optimum on a low-dimensional LQG plant with known i.i.d. erasure probability and verify that the achieved quadratic costs differ by less than a small numerical tolerance.

Figures

Figures reproduced from arXiv: 2604.06060 by Dipankar Maity, Zahra Hashemi.

**Figure 2.** Figure 2: Packet–erasure channel: state 1 = success, 0 = erasure. For each time [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** Figure 3: State-error predictions and scheduling decisions under one-shot and [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 5.** Figure 5: Comparison of one–shot and MPC schedulers versus channel success [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

read the original abstract

We consider event-triggered linear-quadratic Gaussian (LQG) control when sensor updates are transmitted over an i.i.d. packet-erasure channel. Although the optimal controller in a standard LQG setup is available in closed form, choosing when to transmit remains computationally and analytically difficult because packet drops randomize packet delivery and couple scheduling decisions with the estimation-error dynamics, making direct dynamic-programming solutions impractical. By certainty equivalence, the co-design problem becomes choosing a binary send/skip sequence that balances control performance and communication cost. We derive a closed-form expansion of the error covariance as precomputable Gramian terms scaled by a survival factor that depends only on the number of transmission attempts on each interval. This converts the problem into an unconstrained binary program that we linearize exactly via running attempt counters and a one-hot encoding, yielding a compact MILP well suited to receding-horizon implementation. On the linearized Boeing-747 benchmark, a model predictive control (MPC) scheduler lowers cost while attempting far fewer transmissions than a one-shot baseline across channel success rates.

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 turns event-triggered LQG scheduling over i.i.d. erasures into a compact MILP via covariance expansion and exact linearization with counters plus one-hot variables, and the Boeing benchmark shows practical gains.

read the letter

The main advance is a closed-form expansion of the expected error covariance as precomputable Gramians scaled by a survival factor tied to the number of attempts per interval, followed by an exact linearization of the resulting binary program using running attempt counters and one-hot encoding. This produces a MILP that supports receding-horizon MPC scheduling while respecting the packet-erasure channel and communication cost. Certainty equivalence lets them separate the LQG control law from the scheduling decisions, which keeps the formulation tractable. The Boeing-747 numerical example is a clear plus: the scheduler cuts total cost and transmission attempts relative to a one-shot baseline across a range of success probabilities, giving concrete evidence that the method works on a realistic plant. That combination of reformulation and demonstration is what makes the work worth attention for networked control applications. The potential weak point is whether the covariance expansion really depends only on attempt counts. Standard unrolling of the intermittent Kalman covariance weights each Gramian by the probability that the last successful reception occurred at a specific earlier time, and those probabilities are sensitive to the exact placement of attempts inside the interval, not merely their total number. The paper counters this with the running counters and one-hot variables, so the linearization is exact only if those auxiliaries fully recover the position-dependent products without relaxation or approximation. The abstract presents the result as exact, and the benchmark is consistent with that claim, but the derivation itself needs verification to confirm no hidden error enters when intervals span multiple steps. This is aimed at researchers in networked and cyber-physical control who need schedulers that are both optimal-ish and computationally feasible under losses. It deserves a serious referee because the MILP reformulation is a concrete computational contribution even if it rests on standard LQG ingredients.

Referee Report

2 major / 1 minor

Summary. The manuscript considers event-triggered LQG control over an i.i.d. packet-erasure channel. By certainty equivalence the co-design reduces to selecting a binary send/skip sequence. The central technical claim is a closed-form expansion of the error covariance as precomputable Gramian terms scaled by a survival factor that depends only on the number of transmission attempts per interval; this converts the problem to an unconstrained binary program that is then exactly linearized into a compact MILP via running attempt counters and one-hot encoding, enabling receding-horizon MPC. Numerical results on a Boeing-747 model show lower cost with substantially fewer transmissions than a one-shot baseline across success rates.

Significance. If the covariance expansion and exact linearization hold, the work supplies a practical, solver-ready formulation for joint control-communication scheduling under packet losses that avoids dynamic programming and retains the structure of standard LQG certainty equivalence. The MILP reformulation and its receding-horizon suitability constitute the primary contribution; the benchmark provides concrete evidence of transmission reduction without performance loss.

major comments (2)

[Abstract and covariance-expansion derivation] Abstract and covariance-expansion derivation: the claim that the error covariance admits an exact closed-form expansion as precomputable Gramian terms scaled by a survival factor depending only on the count of transmission attempts per interval is load-bearing for the entire reformulation. The standard unrolling of the intermittent-observation Riccati recursion gives E[P_k] as a sum over possible last-success times t, with weights equal to the product of failure probabilities on the specific send decisions between t and k. These products are position-dependent; a factor that uses only the attempt count cannot be exact unless the subsequent one-hot encoding and running counters fully restore the timing information. Please supply the explicit unrolled expression (with the auxiliary variables) and prove equivalence to the random Riccati recursion.
[Linearization via running counters and one-hot encoding] Linearization via running counters and one-hot encoding: the assertion of an exact (non-relaxed) MILP representation stands or falls on whether the auxiliary variables recover the position-dependent last-success probabilities without approximation or bound. If any inequality is introduced in the encoding of the survival factor, the subsequent claim that the MILP solves the original problem exactly is invalidated.

minor comments (1)

[Benchmark results] The Boeing-747 benchmark section would benefit from an explicit statement of the LQG weighting matrices, the horizon length used in the MPC scheduler, and the precise definition of the one-shot baseline to facilitate reproduction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review. The comments correctly identify the core technical claims whose rigor must be fully transparent. We address each major point below and will incorporate explicit derivations and proofs in the revised manuscript.

read point-by-point responses

Referee: [Abstract and covariance-expansion derivation] Abstract and covariance-expansion derivation: the claim that the error covariance admits an exact closed-form expansion as precomputable Gramian terms scaled by a survival factor depending only on the count of transmission attempts per interval is load-bearing for the entire reformulation. The standard unrolling of the intermittent-observation Riccati recursion gives E[P_k] as a sum over possible last-success times t, with weights equal to the product of failure probabilities on the specific send decisions between t and k. These products are position-dependent; a factor that uses only the attempt count cannot be exact unless the subsequent one-hot encoding and running counters fully restore the timing information. Please supply the explicit unrolled expression (with the auxiliary variables) and prove equivalence to the random Riccati.

Authors: We agree that an explicit unrolled expression and equivalence proof are necessary for verification. Because erasures are i.i.d., the probability of no success since last reception at t equals (1-p) raised to the exact number of transmission attempts in (t,k]; non-attempt intervals contribute nothing to the product. The one-hot variables encode the possible last-success time t while the running counters accumulate the attempt count since that t, allowing each survival factor to be evaluated exactly. We will add the full unrolled expression for E[P_k] in terms of these auxiliaries together with a lemma proving equivalence to the expectation of the random Riccati recursion in the revised manuscript. revision: yes
Referee: [Linearization via running counters and one-hot encoding] Linearization via running counters and one-hot encoding: the assertion of an exact (non-relaxed) MILP representation stands or falls on whether the auxiliary variables recover the position-dependent last-success probabilities without approximation or bound. If any inequality is introduced in the encoding of the survival factor, the subsequent claim that the MILP solves the original problem exactly is invalidated.

Authors: The linearization employs only equality constraints. The running counters are updated by exact recurrence equalities driven by the binary send decisions, and the one-hot encoding selects the precise survival factor corresponding to each possible last-success interval. No inequalities or relaxations appear in the encoding of the survival factor. Consequently the MILP is equivalent to the original binary program. We will insert a lemma establishing this exact equivalence in the revision. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation builds on standard LQG and i.i.d. assumptions without self-referential reduction

full rationale

The paper's central step is deriving a closed-form covariance expansion under i.i.d. erasures and certainty equivalence, then converting the resulting binary program to MILP via counters and one-hot encoding. No quoted equation reduces the claimed Gramian scaling to a fitted parameter or prior self-citation by construction. The approach cites standard LQG results as external support rather than load-bearing self-references that would make the expansion tautological. The derivation chain remains self-contained against external benchmarks such as the random Riccati recursion and MILP linearization techniques.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard assumptions in stochastic control plus the specific channel model; no free parameters or new entities are introduced in the abstract.

axioms (2)

domain assumption The communication channel is i.i.d. packet-erasure
Stated in abstract as the basis for modeling unreliable communication and deriving the survival factor.
domain assumption Certainty equivalence holds for the co-design problem
Invoked to separate the control law from the transmission scheduling decisions.

pith-pipeline@v0.9.0 · 5482 in / 1488 out tokens · 54575 ms · 2026-05-10T19:37:11.348426+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.

We derive a closed-form expansion of the error covariance as precomputable Gramian terms scaled by a survival factor that depends only on the number of transmission attempts on each interval... yielding a compact MILP
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat recovery and orbit embedding unclear

?

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

Σt|k = sum_τ (1-p)^{sum_{s=τ}^t θ_s} G_{t,τ}

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.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

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V . Kostina and B. Hassibi, “Rate-cost tradeoffs in control,”IEEE Transactions on Automatic Control, vol. 64, pp. 4525–4540, 2019. # attempts (a) Average number of attempts (b) Total realized cost Fig. 5. Comparison of one–shot and MPC schedulers versus channel success probabilityp. Solid lines denote the mean; shaded bands show±1standard deviation over 1...

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S. P. Boyd and L. Vandenberghe,Convex Optimization. Cambridge University Press, 2004. APPENDIXA PROOF OFTHEOREM1 Proof.For a decision at timek, define the cost-to-go under the two actions J skip k (Θ) :=E "T−1X t=k ∥et∥2 Γt +λθ t θk = 0, e s k # , J att k (Θ) :=E "T−1X t=k ∥et∥2 Γt +λθ t θk = 1, e s k # , and their optima J skip,⋆ k := min Θ J skip k (Θ),...

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An introduction to event-triggered and self-triggered control,

W. P. M. H. Heemels, K. H. Johansson, and P. Tabuada, “An introduction to event-triggered and self-triggered control,” in2012 IEEE 51st Con- ference on Decision and Control (CDC), pp. 3270–3285, IEEE, 2012

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Optimal event-triggered control of nonde- terministic linear systems,

D. Maity and J. S. Baras, “Optimal event-triggered control of nonde- terministic linear systems,”IEEE Transactions on Automatic Control, vol. 65, no. 2, pp. 604–619, 2019

work page 2019

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Rate-cost tradeoffs in control,

V . Kostina and B. Hassibi, “Rate-cost tradeoffs in control,”IEEE Transactions on Automatic Control, vol. 64, pp. 4525–4540, 2019. # attempts (a) Average number of attempts (b) Total realized cost Fig. 5. Comparison of one–shot and MPC schedulers versus channel success probabilityp. Solid lines denote the mean; shaded bands show±1standard deviation over 1...

work page 2019

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On LQG joint optimal scheduling and control under communication constraints,

A. Molin and S. Hirche, “On LQG joint optimal scheduling and control under communication constraints,” inProceedings of the 48th IEEE Conference on Decision and Control (CDC), pp. 5832–5838, IEEE, 2009

work page 2009

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Where to fly, what to send: Communication-aware aerial support for ground robots,

H. Suthar and D. Maity, “Where to fly, what to send: Communication-aware aerial support for ground robots,”arXiv preprint arXiv:2512.06207, 2025

work page arXiv 2025

[6] [6]

On the optimality of certainty equivalence for event-triggered control systems,

A. Molin and S. Hirche, “On the optimality of certainty equivalence for event-triggered control systems,”IEEE Transactions on Automatic Control, vol. 58, no. 2, pp. 470–474, 2012

work page 2012

[7] [7]

Multiagent consensus subject to communica- tion and privacy constraints,

D. Maity and P. Tsiotras, “Multiagent consensus subject to communica- tion and privacy constraints,”IEEE Transactions on Control of Network Systems, vol. 9, no. 2, pp. 943–955, 2021

work page 2021

[8] [8]

Optimal LQG control of networked systems under traffic-correlated delay and dropout,

D. Maity, M. H. Mamduhi, S. Hirche, and K. H. Johansson, “Optimal LQG control of networked systems under traffic-correlated delay and dropout,”IEEE Control Systems Letters, vol. 6, pp. 1280–1285, 2021

work page 2021

[9] [9]

Network-aware optimal sampling for stochastic control systems over dynamic networks,

M. H. Mamduhi and D. Maity, “Network-aware optimal sampling for stochastic control systems over dynamic networks,”IEEE Control Systems Letters, vol. 9, pp. 1808–1813, 2025

work page 2025

[10] [10]

Adaptive sampling for linear state estimation,

M. Rabi, G. V . Moustakides, and J. S. Baras, “Adaptive sampling for linear state estimation,”SIAM Journal on Control and Optimization, vol. 50, no. 2, pp. 672–702, 2012

work page 2012

[11] [11]

Remote state estimation with com- munication costs for first-order LTI systems,

G. M. Lipsa and N. C. Martins, “Remote state estimation with com- munication costs for first-order LTI systems,”IEEE Transactions on Automatic Control, vol. 56, no. 9, pp. 2013–2025, 2011

work page 2013

[12] [12]

Value of information in feedback control: Quantification,

T. Soleymani, J. S. Baras, and S. Hirche, “Value of information in feedback control: Quantification,”IEEE Transactions on Automatic Control, vol. 67, no. 7, pp. 3730–3737, 2021

work page 2021

[13] [13]

Thelander Andr ´en,On LQG-Optimal Event-Based Sampling

M. Thelander Andr ´en,On LQG-Optimal Event-Based Sampling. PhD thesis, Lund University, 2020

work page 2020

[14] [14]

A linear programming framework for optimal event-triggered lqg control,

Z. Hashemi and D. Maity, “A linear programming framework for optimal event-triggered lqg control,”IEEE Control Systems Letters, vol. 9, pp. 2783–2788, 2025

work page 2025

[15] [15]

T−1X t=k ∥et∥2 Γt +λθ t θk = 0, e s k # , J att k (Θ) :=E

S. P. Boyd and L. Vandenberghe,Convex Optimization. Cambridge University Press, 2004. APPENDIXA PROOF OFTHEOREM1 Proof.For a decision at timek, define the cost-to-go under the two actions J skip k (Θ) :=E "T−1X t=k ∥et∥2 Γt +λθ t θk = 0, e s k # , J att k (Θ) :=E "T−1X t=k ∥et∥2 Γt +λθ t θk = 1, e s k # , and their optima J skip,⋆ k := min Θ J skip k (Θ),...

work page 2004