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REVIEW 4 major objections 6 minor 29 references

A Unified Alternating Optimization Framework for Joint Sensor and Actuator Configuration in LQG Systems

T0 review · 4 major / 6 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read LQG sensor and actuator design becomes a differentiable optimization problem

desk verdict Useful continuous co-design formulation and solid gradient derivations, but the Algorithm 2 convergence proof rests on an invalid spectral abscissa claim. read the letter →

arxiv 2504.18731 v1 pith:7KK3EYFY submitted 2025-04-25 eess.SY cs.SY

Nachuan Yang , Yuzhe Li , Ling Shi , Tongwen Chen This is my paper · ORCID
classification eess.SYcs.SY MSC 93E2090C2693B52
keywords LQGcontrolsensorconfigurationactuatorco-designADMMalgebraicRiccatiequationsparsitypromotionlow-rank
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper claims that the LQG performance cost is differentiable with respect to the actuator matrix $B$ and the sensor matrix $C$, with gradients given by closed-form expressions built from the algebraic Riccati solutions and a pair of Lyapunov equations. This matters because existing methods typically select sensors and actuators from a predetermined candidate set, whereas this formulation allows designing them from scratch under convex configuration costs and constraints. The paper derives KKT stationarity conditions and proposes an ADMM-based alternating algorithm whose accumulation point is claimed to be a stationary point. It then specializes the framework to sparsity, low-rank, and structure-constrained configuration costs, with explicit proximal updates for each.

What carries the argument

The central object is the pair of continuous-time algebraic Riccati equations (CAREs) for the optimal observer and controller, whose unique positive-definite solutions $X$ and $P$ exist under the stabilizability and detectability assumption. The paper perturbs $B$ and $C$ through these CAREs, rearranges trace identities, and obtains gradient formulas expressed through Lyapunov equations; this is what turns the implicit LQG cost into an explicitly differentiable objective. The optimization machinery is an ADMM splitting that separates the smooth nonconvex LQG term from the convex nonsmooth configuration cost, with auxiliary-variable updates that have closed forms for the three scenarios considered.

What would settle it

A numerical scan that approaches the boundary of the stabilizable region and finds bounded LQG cost while the optimal feedback gain grows without bound would falsify the coercivity premise behind Theorem 4.

Watch

Extended reading notes

Core claim

The central claim is that the LQG cost $J_{\mathrm{LQG}}$, despite depending on $B$ and $C$ only implicitly through two continuous-time algebraic Riccati equations, has closed-form gradients: $\partial J_{\mathrm{LQG}}/\partial B = -2P(G_1+G_2)PBR^{-1}$ and $\partial J_{\mathrm{LQG}}/\partial C = -2\Pi_v^{-1}CX(H_1+H_2)X$, where $P,X$ are the Riccati solutions and $G_1,G_2,H_1,H_2$ solve Lyapunov equations built from the optimal closed-loop and observer matrices. Theorem 3 converts these gradients into necessary first-order optimality conditions for the constrained joint configuration problem, and Theorem 4 states that the accumulation point of Algorithm 2, an ADMM-based alternating optimization, satisfies these conditions and is therefore at least a stationary point. The paper also gives closed-form proximal updates for three representative scenarios: soft thresholding for $\ell^1$ sparsity, singular value thresholding for the nuclear norm, and a mask projection for structure constraints.

Load-bearing premise

The proof that Algorithm 2 reaches a stationary point rests on two unsupported premises: that the LQG cost diverges at the boundary of the stabilizable/detectable region even when the optimal gains diverge, and that each alternating update never increases the total cost.

Editorial extensions

If this is right

  • Designers can optimize sensor and actuator matrices from scratch, with no predefined candidate set, while the stabilizability and detectability constraints are enforced by the optimization itself.
  • A returned configuration $(B^*,C^*)$ can be checked for stationarity by directly evaluating the two gradient formulas and the corresponding normal-cone conditions.
  • Varying the relative weight $\gamma$ traces a performance-versus-configuration-cost frontier, as demonstrated numerically for sparsity, low-rank, and structure-constrained designs.
  • Low-rank solutions come with explicit factorizations $B=B_1B_2$ and $C=C_1C_2$, so the designed matrices translate directly into a reduced set of actuators and sensors.
  • Because the outer loop alternates and each inner ADMM update is either a smooth gradient step or a convex proximal step, the framework can be embedded in numerical solvers with modest per-iteration cost.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same Riccati-perturbation derivation is likely to produce analogous gradient formulas for discrete-time LQG or $H_2$ performance, although the paper does not state these extensions.
  • The stationarity guarantee is local, so global optimality should not be assumed; restarting Algorithm 2 from several initial pairs is a natural safeguard that the paper does not discuss.
  • If the true design problem is binary (a sensor or actuator is either installed or not), the continuous matrices returned by this framework would need a rounding or mixed-integer layer, which the paper leaves implicit.
  • The coercivity premise behind the boundary argument is directly testable: a numerical scan of near-unstabilizable pairs would reveal whether the LQG cost actually diverges when the optimal feedback gain itself blows up.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The paper studies the joint design of the actuator matrix B and sensor matrix C for a continuous-time LQG system without assuming a pre-specified set of candidate components. The objective is J(B,C)=J_LQG+γ(Φ(B)+Ψ(C)) subject to convex configuration constraints and to stabilizability/detectability of (A,B) and (A,C). The main theoretical results are analytic gradient formulas for J_LQG with respect to B and C (Theorems 1 and 2), KKT-type necessary conditions (Theorem 3), and an ADMM-based alternating optimization algorithm (Algorithm 2) that is claimed to converge to a stationary point of the problem (Theorem 4). The paper also specializes the proximal updates to three scenarios: ℓ1 sparsity promotion, nuclear-norm low-rank promotion, and support-structure constraints. Numerical experiments on the REA1 benchmark compute returned matrices and verify the corresponding stationarity residuals.

Significance. The analytic gradient formulas are the paper's clearest contribution: they are derived directly from the CARE/Lyapunov equations, contain no fitted parameters, and provide a practical stationarity check. The KKT conditions and the closed-form ADMM updates for the three scenarios are useful building blocks. The numerical verification is also a strength: the authors check the KKT residuals of the returned solutions rather than only plotting objective decrease. However, the central convergence claim of Algorithm 2 (Theorem 4) is not proven as written: its proof depends on an invalid step in Proposition 1 and on an unproven monotonicity assertion. The gradient and stationarity results of Theorems 1–3 appear sound, but the algorithmic guarantee needs substantial repair before the paper can be accepted.

major comments (4)
  1. [III.A, Proposition 1, Eq. (28)] The proof of Proposition 1 is invalid. Equation (28) asserts σ(A+B_iK_i^*)→σ(A+BK^*)=0 as B_i→B∈∂S, but K_i^*=−R^{-1}B_i^TP_i depends on the CARE solution P_i, and P_i may diverge as stabilizability is lost. For example, with scalar A=1, B_i=ε_i→0, and Q=R=1, the CARE solution is P_i=(1+√(1+ε_i^2))/ε_i^2, so A+B_iK_i^*=1−ε_i^2P_i=−√(1+ε_i^2)→−1, not 0. Consequently the Lyapunov lower bound (30) does not force tr(P_i)→∞. Because Proposition 1 is the basis for the coercivity used to keep iterates in S and to ensure accumulation points in Theorem 4, Theorem 4 is not established by the argument given.
  2. [III.A, Remark 4, Eq. (36)] Remark 4 asserts the monotone decrease J(B_{h+1},C_h)≤J(B_h,C_h), but this does not follow from the ADMM inner loop as implemented. Algorithm 2 stops the inner ADMM at fixed positive tolerances ǫpri and ǫdual, so B_{h+1}=M_k is not an exact minimizer of (32); for a nonconvex smooth term, the ADMM stationarity relations (38)–(39) do not imply a decrease of the original objective J. The cited reference [24] concerns convergence of ADMM to a stationary point of the augmented or original problem under additional conditions; it does not supply the outer monotonicity (36). Without (36), the outer iterates need not converge, so the existence of the accumulation point (B^*,C^*) assumed in Theorem 4 is unjustified.
  3. [III.A, Theorem 4 proof, Eqs. (38)–(44)] The proof of Theorem 4 has a double-limit gap. It derives stationarity for the inner iterate B_{k+1} while the outer variable is fixed at C_h, but then writes C^* and passes k→∞ and h→∞ simultaneously without justifying the interchange. It also replaces ∂J(B_{k+1},C^*)/∂B by ∂J(M_{k+1},C^*)/∂B on the strength of ‖B_{k+1}−M_{k+1}‖≤ǫpri; this requires a Lipschitz estimate for the gradient and a limit argument as ǫpri→0, and similarly for ρ(M_{k+1}−M_k)→0 as ǫdual→0. The tolerances are fixed positive constants in Algorithm 2, so the claimed limit is not immediate. As written, the conclusion that (B^*,C^*) satisfies (44)–(45) does not follow.
  4. [III.A, Theorem 3] The second, more specific part of Theorem 3 represents the normal cone N_{Ω_B}(B) as the set of all linear combinations of gradients of the equality and inequality descriptions with μ^j_B≥0. This representation requires a constraint qualification (for example, a Slater-type condition for the convex description); the statement and proof do not mention any such qualification. Without it, the displayed KKT conditions can fail at nonregular feasible points. The three concrete scenarios studied later have simple constraint sets where the issue is benign, but the general theorem as stated is missing an assumption.
minor comments (6)
  1. [II.A / Theorem 2] In Theorem 2, 'Given a constant actuator matrix B∈D, for any sensor matrix C∈S' should be 'B∈S and C∈D'; the symbols appear swapped.
  2. [III.A, Theorem 1 proof] In the increment of the CARE in the proof of Theorem 1, the term '∆P A^T' should read '∆P A'.
  3. [III.D, Lemma 5 proof] The proof of Lemma 5 begins 'Given that Φ(B)=0', but in this scenario Φ(M)=‖M‖_F^2; the zero-cost claim is a typo.
  4. [IV.C] The displayed matrices B^* and C^* for the structure-constrained scenario appear to have misaligned rows and columns: B^* is printed with a single 0 in the first row, and C^* appears to have five rows. Please correct the typesetting.
  5. [III.A, Algorithm 1] Algorithm 1 does not specify a stopping criterion for the inner gradient descent, although Theorem 4's proof treats B_{k+1} as an exact stationary point of (37); the dependence on the inner tolerance should be stated.
  6. [III.A, Remark 4] The choice of ρ is only described as 'sufficiently large'; the simulations fix ρ=1 with no sensitivity study, so the practical reliability of the convexification argument is not demonstrated.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the gradient and KKT derivations are self-contained, and the stationarity verification is a post-hoc consistency check, not an input-equivalent prediction.

full rationale

The paper's central results — the LQG gradient formulas in Theorems 1 and 2 — are derived by increment analysis of the dual algebraic Riccati equations (5)–(6) and by solving auxiliary Lyapunov equations. No fitted constants appear in this derivation, no target value is assumed, and no load-bearing self-citation is used. Theorem 3 obtains KKT conditions by applying standard variational-inequality theory [20] to these gradients, which is a direct mathematical consequence rather than a renaming of a known result. Theorem 4's proof takes the ADMM iterates and checks that, in the limit of vanishing residuals, they satisfy the same first-order conditions; the simulation sections verify the returned matrices against Corollaries 1–3. This is a consistency check of the numerical output, not a 'prediction' forced by construction. The concerns noted by the skeptical reader — Proposition 1's coercivity proof relying on the questionable limit sigma(A + B_i K_i^*) -> 0 in Eq. (28), and the asserted but unproven monotone decrease in Remark 4 — are correctness or rigor gaps in the convergence argument, not circular reductions: they do not make the gradient formulas, KKT conditions, or stationarity checks equivalent to their inputs by definition. Since no step identified reduces to its own inputs and no self-citation chain carries the derivation, the circularity score is 0.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

No new physical entities are introduced. The central dependencies are standard LQG assumptions, namely stabilizability and detectability and positive definite weights, plus convexity of configuration costs; the proof of the convergence theorem additionally assumes coercivity and monotone decrease that are not fully established.

free parameters (2)
  • Relative cost weight gamma = 0.01 in the main runs; varied in tradeoff curves
    User-supplied trade-off between LQG performance and configuration cost; it defines the problem rather than being fitted to data.
  • ADMM penalty parameter rho = 1 in all simulations
    Augmented Lagrangian penalty; Remark 4 claims convergence for a sufficiently large rho but no automatic tuning or verification of the threshold is provided.
assumptions (5)
  • domain assumption Assumption 1: (A,B) stabilizable and (A,C) detectable for the configured matrices.
    Invoked throughout Section II to guarantee unique positive definite CARE solutions and finite LQG cost.
  • domain assumption Noise covariance Pi_w and weight Q are strictly positive definite.
    Assumed to facilitate theory; the paper notes extension to semidefinite cases.
  • standard math The LQG optimal controller and observer are given by the two decoupled CAREs (5) and (6), i.e., the separation principle.
    Used in Sections II and III to define JLQG and to compute gradients.
  • domain assumption Configuration costs Phi and Psi and constraint sets Omega_B and Omega_C are convex.
    Stated in equation (10) and used by the ADMM M and N updates and the KKT characterization.
  • ad hoc to paper Constraint qualification holds for the explicit equality and inequality description of Omega_B and Omega_C in Theorem 3.
    The normal cone representation and KKT multipliers are used without verifying Slater or LICQ-type conditions for the problem instances.

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Pith. "Pith review of A Unified Alternating Optimization Framework for Joint Sensor and Actuator Configuration in LQG Systems." pith.science (2026). https://pith.science/paper/7KK3EYFY

@misc{pith2026250418731,
  author       = {Pith},
  title        = {Pith review of: A Unified Alternating Optimization Framework for Joint Sensor and Actuator Configuration in LQG Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7KK3EYFY}},
  note         = {Machine review of arXiv:2504.18731}
}
read the original abstract

This paper fills a gap in the literature by considering a joint sensor and actuator configuration problem under the linear quadratic Gaussian (LQG) performance without assuming a predefined set of candidate components. Different from the existing research, which primarily focuses on selecting or placing sensors and actuators from a fixed group, we consider a more flexible formulation where these components must be designed from scratch, subject to general-form configuration costs and constraints. To address this challenge, we first analytically characterize the gradients of the LQG performance with respect to the sensor and actuator matrices using algebraic Riccati equations. Subsequently, we derive first-order optimality conditions based on the Karush-Kuhn-Tucker (KKT) analysis and develop a unified alternating direction method of multipliers (ADMM)-based alternating optimization framework to address the general-form sensor and actuator configuration problem. Furthermore, we investigate three representative scenarios: sparsity promoting configuration, low-rank promoting configuration, and structure-constrained configuration. For each scenario, we provide in-depth analysis and develop tailored computational schemes. The proposed framework ensures numerical efficiency and adaptability to various design constraints and configuration costs, making it well-suited for integration into numerical solvers.

Figures

Figures reproduced from arXiv: 2504.18731 by the authors.

Figure 1
Figure 1. Convergence process of Algorithm 2 in Problem S1. [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 4
Figure 4. Convergence process of Algorithm 2 in Problem S2. [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 3
Figure 3. The configuration costs defined using the [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (3 more)
Figure 7
Figure 7. Figure 7: Convergence process of Algorithm 2 in Problem S3. [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 6
Figure 6. Figure 6: The configuration costs defined using the nuclear norm [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 8
Figure 8. Figure 8: The LQG performance JLQG versus the configuration cost JSAC under different relative weights γ. It is easy to verify that both A + B∗K∗ and A + L ∗C ∗ are Hurwitz matrices. Therefore, (A, B) is stabilizable and (A, C) is detectable. The associated LQG performance and c…

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