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arxiv: 2604.11183 · v1 · submitted 2026-04-13 · 🧮 math.OC

Closed-loop analysis of linear stochastic MPC with risk-averse constraints

Jonas Schie{\ss}l , Ruchuan Ou , Michael H. Baumann , Timm Faulwasser , Lars Gr\"une This is my paper

Pith reviewed 2026-05-10 15:33 UTC · model grok-4.3

classification 🧮 math.OC
keywords stochastic model predictive controlrisk-averse constraintsconditional value-at-riskrecursive feasibilityclosed-loop analysisstochastic dissipativityindirect feedbacklinear systems
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The pith

Extending indirect feedback to risk-averse constraints in linear stochastic MPC ensures recursive feasibility, closed-loop satisfaction, and near-optimal averaged performance.

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

The paper extends the indirect feedback approach for linear stochastic MPC from chance constraints to risk-averse constraints like conditional value-at-risk. This change lets the controller consider both the likelihood and the size of bad violations instead of probabilities alone. The authors prove the resulting scheme stays recursively feasible and meets the risk-averse constraints when the system runs in closed loop. They further show that stochastic dissipativity plus proper terminal conditions make the long-run average performance nearly optimal. Readers working on safe control under uncertainty would care because the method handles extreme events more directly than earlier probabilistic versions.

Core claim

We extend the indirect feedback approach for linear stochastic MPC from chance constraints to risk-averse constraints like the conditional value-at-risk. For the resulting risk-averse MPC scheme, we establish recursive feasibility and closed-loop constraint satisfaction. Furthermore, based on a stochastic dissipativity notion and suitable conditions on the terminal ingredients we show that (near)-optimality of the averaged closed-loop performance can be ensured.

What carries the argument

The indirect feedback approach adapted to risk-averse constraints such as conditional value-at-risk, which carries the proofs of recursive feasibility, closed-loop satisfaction, and performance via stochastic dissipativity.

If this is right

  • Recursive feasibility holds for every time step in the risk-averse MPC scheme.
  • Closed-loop state and input trajectories satisfy the risk-averse constraints.
  • Averaged closed-loop performance is near-optimal whenever the dissipativity condition holds.
  • The guarantees apply to linear systems driven by unbounded random disturbances.

Where Pith is reading between the lines

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

  • The same indirect-feedback structure could be tested with other tail-risk measures beyond conditional value-at-risk.
  • Practical verification of the stochastic dissipativity condition would be needed before deployment on physical plants.
  • The closed-loop analysis might extend naturally to certain classes of nonlinear stochastic systems.
  • Comparing the resulting constraint tightness with distributionally robust MPC could reveal trade-offs in conservatism.

Load-bearing premise

Suitable conditions on the terminal ingredients hold and the stochastic dissipativity notion applies to the closed-loop system.

What would settle it

A numerical simulation of the closed-loop system under the risk-averse MPC that shows repeated violations of the risk-averse constraints or averaged performance far from the optimum, even when terminal conditions are met, would falsify the claims.

Figures

Figures reproduced from arXiv: 2604.11183 by Jonas Schie{\ss}l, Lars Gr\"une, Michael H. Baumann, Ruchuan Ou, Timm Faulwasser.

Figure 1
Figure 1. Figure 1: Evolution of E[Xcl 1 (k)] (blue), VaR0.6(Xcl 1 (k)) (orange), CVaR0.6(Xcl 1 (k)) (green), and EVaR0.6(Xcl 1 (k)) (red) as well as the upper constraint bound p = 2 (dashed black). The subplots (top to bottom) correspond to the constraints (41) with ρ = E, ρ = VaR0.6, ρ = CVaR0.6, and ρ = EVaR0.6, respectively. 0 50 100 150 200 250 5 10 15 20 25 K * K̃ [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Averaged closed-loop performance of Algorithm ˜ [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
read the original abstract

Chance constraints are widely used in stochastic model predictive control (MPC) to enforce probabilistic state and input constraints in the presence of unbounded disturbances. However, they only restrict violation probabilities and do not account for the magnitude of rare but severe constraint violations. In this paper, we extend the indirect feedback approach for linear stochastic MPC from chance constraints to risk-averse constraints like the conditional value-at-risk. For the resulting risk-averse MPC scheme, we establish recursive feasibility and closed-loop constraint satisfaction. Furthermore, based on a stochastic dissipativity notion and suitable conditions on the terminal ingredients we show that (near)-optimality of the averaged closed-loop performance can be ensured.

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 paper extends the indirect feedback approach for linear stochastic MPC from chance constraints to risk-averse constraints such as conditional value-at-risk (CVaR). It establishes recursive feasibility and closed-loop constraint satisfaction for the resulting risk-averse MPC scheme. Using a stochastic dissipativity notion together with suitable terminal ingredients, it further claims to ensure (near)-optimality of the averaged closed-loop performance.

Significance. If the results hold, the work provides a meaningful extension of stochastic MPC theory by moving beyond probability-based constraints to risk measures that penalize the magnitude of violations. This is relevant for safety-critical applications. The use of dissipativity-based arguments for closed-loop performance is a standard technique in MPC; if the extension is carried through rigorously with explicit verification, it strengthens the theoretical toolkit for risk-averse stochastic control.

major comments (1)
  1. [Abstract and performance analysis section] Abstract and performance analysis section: The (near)-optimality result is conditional on a stochastic dissipativity notion holding for the closed-loop system under the risk-averse constraints and the chosen terminal ingredients. Because replacing chance constraints with CVaR changes the effective tightening and the distribution of closed-loop trajectories, the dissipativity inequality must be re-verified explicitly for the new scheme; the manuscript should contain a dedicated lemma or proposition showing that the same terminal cost and set still yield a uniform dissipation rate (or quantifying any degradation).
minor comments (2)
  1. The abstract would be clearer if it briefly indicated the precise form of the risk-averse constraint (e.g., CVaR at a fixed level) and the main assumptions on the disturbance distribution.
  2. Notation for the risk measure and the associated tightened constraints should be introduced consistently in the problem formulation section to avoid ambiguity when comparing to the chance-constrained predecessor.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive review and the positive assessment of the paper's relevance. We address the major comment below and will revise the manuscript to strengthen the performance analysis section.

read point-by-point responses

  1. Referee: [Abstract and performance analysis section] Abstract and performance analysis section: The (near)-optimality result is conditional on a stochastic dissipativity notion holding for the closed-loop system under the risk-averse constraints and the chosen terminal ingredients. Because replacing chance constraints with CVaR changes the effective tightening and the distribution of closed-loop trajectories, the dissipativity inequality must be re-verified explicitly for the new scheme; the manuscript should contain a dedicated lemma or proposition showing that the same terminal cost and set still yield a uniform dissipation rate (or quantifying any degradation).

    Authors: We agree that the change from chance constraints to CVaR requires explicit verification of the dissipativity property for the closed-loop system, as the constraint tightening and resulting trajectory distribution are affected. In the revised manuscript we will add a dedicated lemma in the performance analysis section. The lemma will prove that the same terminal cost and terminal set (chosen to satisfy the standard invariance and dissipativity conditions for the nominal linear system) continue to yield a uniform dissipation rate for the risk-averse MPC closed loop. The argument relies on the fact that the dissipativity inequality is formulated with respect to the expected stage cost and the nominal dynamics inside the terminal set; these quantities are independent of the particular risk measure used to tighten the constraints. We will also note that no degradation of the rate occurs under the maintained assumptions on the risk level. revision: yes

Circularity Check

0 steps flagged

No significant circularity; claims rest on explicit assumptions and direct proofs

full rationale

The paper directly establishes recursive feasibility and closed-loop constraint satisfaction for the risk-averse MPC extension. The averaged performance result is explicitly conditioned on a stochastic dissipativity notion plus suitable terminal-ingredient conditions, which are stated as assumptions rather than derived internally. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations that collapse the central claims to their own inputs appear in the derivation. The chain is self-contained against the stated external benchmarks and assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are stated. The performance result rests on an unspecified stochastic dissipativity notion and terminal conditions.

pith-pipeline@v0.9.0 · 5419 in / 1048 out tokens · 32713 ms · 2026-05-10T15:33:49.957289+00:00 · methodology

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Reference graph

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