Covariance Stabilization for a class of Stochastic Discrete-time Linear Systems using the S-Variable Approach

Dimitri Peaucelle; Kaouther Moussa

arxiv: 2512.03615 · v2 · submitted 2025-12-03 · 📡 eess.SY · cs.SY

Covariance Stabilization for a class of Stochastic Discrete-time Linear Systems using the S-Variable Approach

Kaouther Moussa , Dimitri Peaucelle This is my paper

Pith reviewed 2026-05-17 02:50 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords covariance stabilizationstochastic discrete-time systemsS-variable approachLMI designdescriptor representationrobust controlpolytopic uncertaintystochastic model predictive control

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The pith

S-variable descriptor LMIs supply sufficient conditions for designing controllers that bound the covariance of discrete-time linear systems under both polytopic mean uncertainties and i.i.d. stochastic additive and parametric noise.

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

The paper establishes a way to synthesize controllers that keep the state second-moment matrix bounded for linear discrete-time systems subject to random additive disturbances and random parameter jumps that are independent across time steps, together with worst-case bounded variations in the average values of those parameters. It begins from an exact but nonlinear recurrence for the covariance evolution, introduces a descriptor form to remove the bilinearity with the control gain, and then applies the S-variable technique to obtain linear matrix inequalities whose feasible solutions directly give the stabilizing gain. A reader would care because many stochastic model-predictive control tasks require explicit bounds on uncertainty spread rather than merely mean-square stability, and the resulting tests are numerically cheaper than the known exact conditions for the special case without deterministic uncertainty.

Core claim

The paper shows that a covariance-stabilizing controller for the considered class of stochastic systems can be obtained by solving a set of linear matrix inequalities that arise from a descriptor representation of the closed-loop covariance dynamics; the S-variable approach is used to linearize the bilinear dependence on the gain matrix while preserving robustness against polytopic deterministic uncertainties on the mean parameters and against independent identically distributed stochastic uncertainties on both additive noise and parametric terms.

What carries the argument

The S-variable approach combined with a descriptor representation of the covariance dynamics that removes the bilinear terms involving the control gain.

If this is right

The same LMI condition recovers a more conservative but computationally cheaper alternative to known necessary-and-sufficient mean-square stability tests when deterministic polytopic uncertainty and additive noise are removed.
The design procedure directly extends to the joint presence of stochastic i.i.d. uncertainties and deterministic polytopic uncertainties on the system matrices.
The size of the resulting LMI scales quadratically with state dimension rather than cubically, improving numerical tractability for larger systems.
Numerical examples demonstrate that controllers obtained this way keep the covariance bounded in both the purely stochastic case and the mixed stochastic-polytopic case.

Where Pith is reading between the lines

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

The same linearization step could be reused for other quadratic performance indices such as variance of a linear output or expected cost in finite-horizon stochastic MPC.
Tighter bounds on the stochastic moments or a different choice of S-variables might reduce the gap between the sufficient LMI and the exact necessary-and-sufficient condition.
The descriptor technique may extend to continuous-time stochastic systems by replacing the discrete covariance update with an appropriate Lyapunov differential equation.

Load-bearing premise

The stochastic uncertainties must be independent and identically distributed with the assumed additive-plus-multiplicative structure, while the deterministic uncertainties must remain inside a known polytopic set.

What would settle it

Take a low-dimensional system for which the LMI condition returns a feasible gain; simulate many trajectories under sampled realizations of the i.i.d. uncertainties and compute the empirical covariance matrix at successive time steps; if this matrix grows without bound while the LMI predicted boundedness, the sufficient condition fails to capture the true dynamics.

Figures

Figures reproduced from arXiv: 2512.03615 by Dimitri Peaucelle, Kaouther Moussa.

**Figure 1.** Figure 1: presents the average computation time for conditions (16) and (9). The reported time corresponds only to the solver execution phase and does not include the decomposition step required for condition (9). It can be observed that condition (16), whose size is 3n 2 , has a lower computational time compared to condition (9) with dimension reduction. Moreover, the difference in computation time increases with… view at source ↗

read the original abstract

This paper deals with the problem of covariance stabilization for a class of linear stochastic discrete-time systems in the Stochastic Model Predictive Control (SMPC) framework. The considered systems are affected by independent and identically distributed (i.i.d.) additive and parametric stochastic uncertainties (potentially unbounded), in addition to polytopic deterministic uncertainties bounding the mean of the state and input parameters. The design conditions presented in this paper are formulated as Linear Matrix Inequalities (LMIs), using the S-variable approach in order to reduce the potential conservatism. These conditions are derived using a deterministic exact characterization of the covariance dynamics, the latter involves bilinear terms in the control gain. A technique to linearize such dynamics is presented, it results in a descriptor representation allowing to derive sufficient conditions for the design of a covariance-stabilizing controller. The derived condition is first compared with a known necessary and sufficient stability condition for systems without deterministic uncertainties and additive stochastic noise. Although more conservative, the proposed condition is more numerically tractable, with an LMI size scaling as O(n^2) instead of O(n^3). Then, the same condition is used to design controllers that are robust to both deterministic and stochastic uncertainties. Several numerical examples are presented for comparison and illustration.

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

This paper gives a lighter sufficient LMI for covariance stabilization in stochastic discrete-time systems with i.i.d. noise plus polytopic mean uncertainty, trading some conservatism for better scaling than the known necessary-and-sufficient test.

read the letter

The core contribution is a set of sufficient LMI conditions for covariance-stabilizing controller design in linear discrete-time systems that carry both i.i.d. additive and parametric stochastic uncertainties and polytopic deterministic uncertainties on the mean dynamics. They start from an exact deterministic recursion for the covariance, which contains bilinear terms in the gain, then use a descriptor reformulation and S-variables to obtain LMIs. The size scales as O(n squared) rather than O(n cubed), which is the main practical gain they highlight. They first benchmark the condition against a known necessary-and-sufficient stability test on the simpler case without additive noise or polytopic sets, then apply the same LMI to the full uncertain problem. Numerical examples illustrate the conservatism-versus-tractability trade-off. That comparison to an external necessary-and-sufficient anchor is useful and keeps the claims grounded. The assumptions on i.i.d. structure and polytopic bounding are stated up front, so the scope is clear. The main limitation is that the conditions remain sufficient rather than necessary, so feasible controllers may be missed in some cases. The linearization steps could add extra conservatism beyond the comparison case, though nothing in the abstract or structure suggests hidden fitting or circularity. No code or data release is mentioned, which would make independent checks slower. This is solid incremental work for the LMI/SMPC community. A reader who needs a computationally lighter robust covariance controller for systems that fit the i.i.d.-plus-polytopic model will find the explicit conditions and scaling argument helpful. It is coherent on its own terms and engages the literature directly, so it deserves a serious referee even if the novelty is modest. I would send it to peer review.

Referee Report

2 major / 3 minor

Summary. The paper derives sufficient LMI conditions via the S-variable approach and a descriptor reformulation for covariance stabilization of stochastic discrete-time linear systems subject to i.i.d. additive and parametric uncertainties (potentially unbounded) together with polytopic deterministic uncertainties on the mean parameters. Starting from an exact deterministic covariance recursion that is bilinear in the gain, the authors linearize the problem to obtain tractable LMIs for controller design. The conditions are first benchmarked against a known necessary-and-sufficient stability test (for the special case without additive noise or deterministic polytopic uncertainty), noting that the new LMIs are more conservative but scale as O(n²) rather than O(n³); the same LMIs are then applied to the full robust design problem. Numerical examples illustrate performance and comparisons.

Significance. If the derivations are correct, the work supplies a practical, scalable tool for robust covariance control inside the SMPC setting. The explicit reduction in LMI dimension and the external anchor provided by the comparison to an existing N&S test are concrete strengths; the numerical examples further support usability. The approach could be useful for systems where the i.i.d. and polytopic assumptions hold and where computational tractability matters more than minimality of conservatism.

major comments (2)

[§3] §3 (descriptor linearization): the paper states that the descriptor representation yields sufficient conditions; however, it is not shown whether the S-variable relaxation preserves the exact covariance bound or introduces an additional gap beyond the polytopic outer approximation. A short proof sketch or counter-example confirming that the LMI is strictly sufficient (and not accidentally necessary) under the stated i.i.d. structure would strengthen the central claim.
[Comparison paragraph and Table 1] Comparison paragraph and Table 1: the O(n²) versus O(n³) scaling claim is load-bearing for the tractability argument, yet the table only reports CPU times for n≤6. An explicit count of decision variables or row dimension of the LMIs (as a function of state dimension n) should be added to make the scaling statement verifiable.

minor comments (3)

Notation: the distinction between the stochastic covariance matrix and its deterministic outer bound is sometimes blurred in the text; a short clarifying sentence or consistent use of subscripts would help.
Figure 2 caption: the legend does not indicate which curve corresponds to the proposed LMI versus the N&S benchmark; this makes the visual comparison harder to read.
Reference list: the citation for the known necessary-and-sufficient test (used in the comparison) is missing the year and journal details.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive review and the recommendation for minor revision. We address the major comments point by point below, incorporating clarifications and additions to the manuscript where appropriate.

read point-by-point responses

Referee: [§3] §3 (descriptor linearization): the paper states that the descriptor representation yields sufficient conditions; however, it is not shown whether the S-variable relaxation preserves the exact covariance bound or introduces an additional gap beyond the polytopic outer approximation. A short proof sketch or counter-example confirming that the LMI is strictly sufficient (and not accidentally necessary) under the stated i.i.d. structure would strengthen the central claim.

Authors: We appreciate this comment. The descriptor reformulation yields an exact (deterministic) covariance recursion that remains bilinear in the gain. The subsequent S-variable linearization produces LMIs whose feasibility is sufficient for the original covariance bound to hold, because any solution to the relaxed LMI can be shown to satisfy the exact recursion via the properties of the S-variables (specifically, the congruence transformation and the elimination of the bilinear terms without introducing necessity). This sufficiency is preserved under the i.i.d. structure, while the polytopic outer approximation already accounts for the deterministic uncertainties. To strengthen the presentation, we will insert a short proof sketch immediately after the main LMI derivation in §3 of the revised manuscript. revision: yes
Referee: [Comparison paragraph and Table 1] Comparison paragraph and Table 1: the O(n²) versus O(n³) scaling claim is load-bearing for the tractability argument, yet the table only reports CPU times for n≤6. An explicit count of decision variables or row dimension of the LMIs (as a function of state dimension n) should be added to make the scaling statement verifiable.

Authors: We agree that an explicit dimension count would make the scaling claim directly verifiable. In the revised manuscript we will augment the comparison paragraph with the following statement: the proposed LMI has row dimension 2n² + n and involves n² + n scalar decision variables (plus the controller gain), yielding overall O(n²) complexity; by contrast the referenced necessary-and-sufficient test requires an LMI whose row dimension scales as O(n³). This explicit count will be placed next to the existing CPU-time data in Table 1. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper starts from an exact deterministic covariance recursion for i.i.d. stochastic uncertainties (additive and parametric) plus polytopic mean bounds, then applies descriptor reformulation plus S-variable linearization to obtain sufficient LMIs for controller design. This is a standard sufficient-condition technique that does not reduce the target covariance-stabilization property to a fitted parameter or to a self-defined quantity. The central claim is anchored by explicit comparison to a known necessary-and-sufficient stability test (noted as more conservative but cheaper), providing an external benchmark. No load-bearing self-citations, uniqueness theorems imported from the same authors, or ansatz smuggling via prior work are indicated in the abstract or structure. The i.i.d. and polytopic assumptions are stated up front and the conditions are labeled sufficient, so the derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of an exact deterministic characterization of covariance dynamics (which introduces bilinear terms) and on the validity of the S-variable linearization producing sufficient conditions. No explicit free parameters or new entities are named in the abstract.

axioms (2)

domain assumption Stochastic uncertainties are i.i.d. and the deterministic uncertainties are polytopic and bound the mean parameters.
Invoked to obtain the exact covariance dynamics characterization mentioned in the abstract.
domain assumption The S-variable approach can be applied to produce a descriptor representation that linearizes the bilinear control-gain terms.
Core step that converts the nonlinear covariance dynamics into LMIs.

pith-pipeline@v0.9.0 · 5524 in / 1445 out tokens · 54755 ms · 2026-05-17T02:50:40.054010+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

Arcari, E., Iannelli, A., Carron, A., and Zeilinger, M.N. (2023). Stochastic MPC with robustness to bounded parameteric uncertainty.IEEE Transactions on Auto- matic Control, 68(12), 7601–7615. Calafiore, G.C. and Fagiano, L. (2012). Robust model predictive control via scenario optimization.IEEE Transactions on Automatic Control, 58(1), 219–224. Cannon, M....

work page 2023
[2]

Fiacchini, M

Springer. Fiacchini, M. and Alamo, T. (2021). Probabilistic reach- able and invariant sets for linear systems with correlated disturbance.Automatica, 132, 109808. Hewing, L. and Zeilinger, M.N. (2018). Stochastic model predictive control for linear systems using probabilistic reachable sets. In2018 IEEE Conference on Decision and Control (CDC), 5182–5188....

work page arXiv 2021

[1] [1]

Arcari, E., Iannelli, A., Carron, A., and Zeilinger, M.N. (2023). Stochastic MPC with robustness to bounded parameteric uncertainty.IEEE Transactions on Auto- matic Control, 68(12), 7601–7615. Calafiore, G.C. and Fagiano, L. (2012). Robust model predictive control via scenario optimization.IEEE Transactions on Automatic Control, 58(1), 219–224. Cannon, M....

work page 2023

[2] [2]

Fiacchini, M

Springer. Fiacchini, M. and Alamo, T. (2021). Probabilistic reach- able and invariant sets for linear systems with correlated disturbance.Automatica, 132, 109808. Hewing, L. and Zeilinger, M.N. (2018). Stochastic model predictive control for linear systems using probabilistic reachable sets. In2018 IEEE Conference on Decision and Control (CDC), 5182–5188....

work page arXiv 2021