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arxiv: 2604.03490 · v1 · submitted 2026-04-03 · 📡 eess.SY · cs.SY

Conditions for Complete Decentralization of the Linear Quadratic Regulator

Addie McCurdy , Isabel Collins , Emily Jensen This is my paper

Pith reviewed 2026-05-13 18:18 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords linear quadratic regulatordecentralized controloptimal controlinterconnected systemsparameter conditionsRiccati equationblock-diagonal gain
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The pith

The optimal Linear Quadratic Regulator for interconnected systems can be completely decentralized under specific parameter conditions.

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

This paper seeks to determine when the optimal Linear Quadratic Regulator control law can be split so that each subsystem computes its actuation using only its own local information. It first derives these conditions for several simple cases such as two-subsystem setups and supplies physical interpretations through concrete examples. These basic results are then used to characterize when complete decentralization holds in larger, more complex systems. A reader would care because many real-world networks cannot rely on constant central communication or computation, so identifying when local decisions still achieve the global optimum enables practical scalable control.

Core claim

For several simple linear system configurations, explicit conditions on the dynamics and cost matrices exist such that the algebraic Riccati equation solution produces a block-diagonal optimal feedback gain. This gain structure means each subcontroller needs only its local state to implement the globally optimal policy. The paper supplies physical interpretations of these conditions via examples and leverages the simple cases to analyze decentralization in composite systems.

What carries the argument

The algebraic Riccati equation whose solution yields the LQR feedback gain, specifically the parameter conditions that force this gain to be block-diagonal with respect to the given subsystem partition.

If this is right

  • For two-subsystem systems, decentralization holds when dynamic coupling and cost cross-terms satisfy explicit equalities or ratio conditions.
  • Physical examples demonstrate that either complete decoupling or balanced symmetric coupling can make local control optimal.
  • Complex systems are analyzed by reducing them to combinations of the simple cases whose conditions are already known.
  • When the conditions hold, the decentralized policy achieves the identical closed-loop cost as the centralized policy.

Where Pith is reading between the lines

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

  • Engineers could use the conditions to tune parameters in networked systems such as vehicle formations or power networks to enable fully local optimal control.
  • The approach of building from simple cases might extend to checking decentralization in systems with mild nonlinearities or time delays, though the paper stays within linear unconstrained settings.
  • The physical interpretations suggest that symmetry in costs or couplings is often the key enabler, which could guide design in other optimal control settings.

Load-bearing premise

The systems are assumed to admit a completely decentralized optimal controller only when the dynamics and cost parameters satisfy certain characterizable conditions, with no input constraints present.

What would settle it

Take any simple two-subsystem system whose parameters satisfy the derived conditions, solve the full Riccati equation for the centralized LQR gain, and check whether the resulting gain matrix contains any nonzero entries coupling the subsystems; nonzero coupling entries would falsify the claim.

Figures

Figures reproduced from arXiv: 2604.03490 by Addie McCurdy, Emily Jensen, Isabel Collins.

Figure 1
Figure 1. Figure 1: Visualization of distributed (top) and decentralized [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: shows the closed loop H2 norm of this system in feedback with the optimal LQR controller for various choices of γ0 γ2 and q0 q2 . In this case, the parameter choices that give a completely decentralized controller (shown by the red dot in the figure) result in neither the highest nor the lowest closed loop H2 norm. Because there is not a direct tradeoff between decentraliza￾tion and performance, it may be … view at source ↗
Figure 3
Figure 3. Figure 3: Heat plot of closed loop H2 norm with varying choices of Q and a2/a0. The curve of decentralization is shown in black. 3.2 Example: aquarium population dynamics An example of a physical system described by Theorem 1 is a fish tank into which fish can be freely added and removed as depicted in [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Diagram for physical example of 2 × 2 system with bass as the predators and shrimp as the prey. 4. SPATIALLY INVARIANT SYSTEMS Spatial invariance is the property that system dynamics are invariant with respect to spatial translations. In the discrete setting with finite spatial extent, this is equivalent to all system matrices being (block) circulant. Since the DFT diagonalizes circulant matrices, the LQR … view at source ↗
Figure 5
Figure 5. Figure 5: Heat transfer across a wall A =  a0 a1 a1 a0  , B =  b0 b1 b1 b0  , Q =  q0 q1 q1 q0  , R =  r0 r1 r1 r0  . (16) Corollary 3. Let A, B, Q, R ∈ R 2×2 be defined as in (16). Then a sufficient condition for completely decentralized LQR is a0 − a1 a0 + a1 = b0 − b1 b0 + b1 and q0 − q1 q0 + q1 = r0 − r1 r0 + r1 . (17) Proof. See Appendix B.3 Note that (17) is satisfied if all matrices are diagonal (a1 =… view at source ↗
read the original abstract

An unconstrained optimal control policy is completely decentralized if computing actuation for each subsystem only requires information directly available to its own subcontroller. Parameters that admit a completely decentralized optimal controller have been characterized in a variety of systems, but attempts to physically explain the phenomenon have been limited. As a step toward a general characterization of complete decentralization, this paper presents conditions for complete decentralization of Linear Quadratic Regulators for several simple cases and physically interprets these conditions with illustrative examples. These simple cases are then leveraged to characterize complete decentralization of more complex systems.

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

2 major / 2 minor

Summary. The paper claims to derive conditions under which an unconstrained LQR optimal controller is completely decentralized (each subcontroller uses only locally available information) for several simple linear systems, supplies physical interpretations via examples, and then extends those conditions to characterize decentralization in more complex systems.

Significance. If the derivations and extensions hold, the work supplies a concrete, interpretable route toward general conditions for optimal decentralization in LQR problems. The emphasis on physical interpretation of the parameter conditions distinguishes the contribution from purely algebraic treatments and could inform controller design in networked systems such as vehicle platoons or power networks.

major comments (2)
  1. [Introduction and extension section] The extension step from the simple cases to more complex systems is described only at a high level in the abstract and introduction; a precise statement of the composition rule or inductive argument (e.g., in the section that follows the simple-case derivations) is needed to confirm that the conditions remain necessary and sufficient rather than merely sufficient.
  2. [Simple-case derivations] The physical-interpretation examples for the simple cases rely on specific choices of Q and R matrices; it is unclear whether the same interpretations survive when the cost matrices are allowed to have off-diagonal blocks that couple the subsystems, which would affect the load-bearing claim that the conditions characterize complete decentralization in general.
minor comments (2)
  1. Notation for the decentralized information pattern should be defined once and used consistently; the current text occasionally switches between “local state” and “directly available information” without explicit equivalence.
  2. The illustrative examples would benefit from a side-by-side comparison table of the centralized versus decentralized closed-loop eigenvalues or costs to make the “no performance loss” claim immediately visible.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment and constructive comments. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Introduction and extension section] The extension step from the simple cases to more complex systems is described only at a high level in the abstract and introduction; a precise statement of the composition rule or inductive argument (e.g., in the section that follows the simple-case derivations) is needed to confirm that the conditions remain necessary and sufficient rather than merely sufficient.

    Authors: We agree that a more precise statement is warranted. In the revised manuscript we will insert a dedicated subsection immediately after the simple-case derivations. This subsection will state the composition rule explicitly and supply an inductive argument establishing that the decentralization conditions are both necessary and sufficient for the composed system. revision: yes

  2. Referee: [Simple-case derivations] The physical-interpretation examples for the simple cases rely on specific choices of Q and R matrices; it is unclear whether the same interpretations survive when the cost matrices are allowed to have off-diagonal blocks that couple the subsystems, which would affect the load-bearing claim that the conditions characterize complete decentralization in general.

    Authors: The algebraic derivations of the decentralization conditions are performed for arbitrary (possibly block-coupled) Q and R; the examples employ diagonal costs solely for expository clarity. We will add a short clarifying paragraph after the examples that explains how the same conditions apply when off-diagonal blocks are present and how the physical interpretations generalize via effective local costs. This addition will reinforce the generality of the claim. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper derives conditions for complete decentralization of LQR controllers by starting from simple cases, providing physical interpretations via examples, and extending the characterization to more complex systems. No load-bearing steps reduce by construction to fitted parameters, self-definitions, or self-citation chains; the argument relies on standard unconstrained LQR assumptions and explicit case-by-case analysis that remains independent of the target results. The structure is self-contained against external benchmarks with no renaming of known results or smuggled ansatzes.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only abstract available, so ledger is minimal. Paper likely relies on standard LQR assumptions such as linear dynamics and quadratic costs without introducing new entities.

pith-pipeline@v0.9.0 · 5376 in / 961 out tokens · 57646 ms · 2026-05-13T18:18:42.398031+00:00 · methodology

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Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

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    Arbelaiz, J., Bamieh, B., Hosoi, A.E., and Jadbabaie, A. (2020). Distributed Kalman filtering for spatially- invariant diffusion processes: The effect of noise on com- munication requirements. In2020 59th IEEE Confer- ence on Decision and Control (CDC), 622–627. IEEE. Arbelaiz, J., Bamieh, B., Hosoi, A.E., and Jadbabaie, A. (2021). Optimal structured cont...

    work page doi:10.1109/cdc45484.2021.9683472 2020

  2. [2]

    Jensen, E., Epperlein, J.P., and Bamieh, B. (2020). Local- ization of the LQR feedback kernel in spatially-invariant problems over Sobolev spaces. In2020 59th IEEE Conference on Decision and Control (CDC), 1204–1209. IEEE. Lessard, L. and Lall, S. (2016). Convexity of de- centralized controller synthesis.IEEE Transactions on Automatic Control, 61(10), 312...

    work page doi:10.1109/tac.2015.2504179 2020

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    (1987).Modeling of Heat Transfer in Buildings

    Seem, J.E. (1987).Modeling of Heat Transfer in Buildings. Ph.d. dissertation, Massachusetts Institute of Technol- ogy, Cambridge, MA. Appendix A. LEMMAS We use the following well-established fact: Lemma 4.Two quadratics: x2 +β 1x+γ 1 = 0 x2 +β 2x+γ 2 = 0 have 2 roots in common if and only if 1 = β1 β2 = γ1 γ2 . They have exactly one root in common (at val...

    work page 1987