A stochastic MPC scheme for distributed systems with multiplicative uncertainty

Christoph Mark; Steven Liu

REVIEW 3 major objections 4 minor 24 references

A stochastic MPC scheme for distributed systems with multiplicative uncertainty

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A feasible start guarantees recursive feasibility, mean-square convergence, and chance constraint satisfaction in distributed stochastic MPC.

desk verdict A promising distributed stochastic MPC scheme whose main theorem does not cover the actual preferred feedback branch (S1), leaving the convergence guarantee unproven for the implemented algorithm. read the letter →

arxiv 1908.09337 v3 pith:VOTOLMHQ submitted 2019-08-25 math.OC

Christoph Mark , Steven Liu This is my paper · ORCID

classification math.OC MSC 93C5593A1493E2090C22

keywords distributedmodelpredictivecontrolstochasticMPCmultiplicativeuncertaintychanceconstraintsCantelliinequalityalternatingdirectionmethodofmultipliersmeansquarestabilityrecursivefeasibility

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

This paper proposes a fully distributed stochastic model predictive control (MPC) scheme for networks of linear systems whose dynamics are corrupted by multiplicative, unbounded zero-mean noise. The authors replace probabilistic state and input constraints with mean-and-covariance constraints via Cantelli's inequality, then solve the resulting optimization with consensus ADMM so no central coordinator is needed. The central claim is Theorem 19: if the distributed optimization is feasible at time zero and solved exactly, the closed loop stays feasible, the expected quadratic state norm tends to zero, and the original chance constraints hold at every time. The practical value is that networked control designers get scalability without sacrificing stability or safety certificates.

What carries the argument

The central machinery is a distributed terminal design: a block-diagonal terminal cost $V_{f,i}(z_i)=z_i^\top P_i z_i$, a structured local terminal controller $K_{N_i}$, time-varying local terminal sets $Z_{f,i}(\alpha_i(k))$ whose sizes are updated by relaxation functions $\Gamma_{N_i}$, and a block-diagonal terminal covariance bound $\hat{\Sigma}_{f,i}$. These are computed by solving distributed LMIs and a distributed linear program, and they make the terminal region invariant while satisfying tightened mean and covariance constraints. Online, the global optimization problem is rewritten in consensus form and solved in parallel by ADMM; each subsystem optimizes over its own and its neighbors' predicted trajectories and exchanges copies to reach agreement. Chance constraints are converted into deterministic constraints on means, inputs, and covariances via Cantelli's inequality, $f(p)=\sqrt{p/(1-p)}$, and then linearized with a parameter $\epsilon$ that trades conservatism against feasibility.

What would settle it

A decisive test is to run the Section 5 example with a boundary-feasible initial condition and decreasing stopping tolerances $\epsilon_c=10^{-2},10^{-4},10^{-6}$; if any closed-loop trajectory under finite $\epsilon_c$ becomes infeasible at the next step, or violates the chance constraint at the required $p_{i,x} \ge 0.7$, then the theorem's guarantees do not extend to the implemented algorithm.

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Extended reading notes

Core claim

The paper establishes that every stabilizing ingredient of stochastic MPC—terminal cost, terminal controller, terminal set scaling, and covariance bound—can be synthesized distributedly for multiplicative uncertainties, and the online distributed problem then inherits the recursive feasibility, mean-square stability, and chance constraint satisfaction of the centralized design. Specifically, Theorem 19 states: if Problem 16 is feasible at $k=0$ and Algorithm 1 converges exactly (Assumption 18), then the closed loop is recursively feasible, $\mathbb{E}(\|x(k)\|_Q^2) \to 0$ as $k \to \infty$, and the chance constraints (2) are satisfied for all $k \ge 0$. This extends centralized stochastic MPC for multiplicative unbounded uncertainty to networks and removes the need for a central node in both offline design and online operation.

Load-bearing premise

The proof assumes the distributed optimization problem is solved exactly at every step, while the implemented algorithm stops at a finite tolerance, and the paper gives no bound on how this gap affects recursive feasibility, mean-square convergence, or chance constraint satisfaction.

Editorial extensions

If this is right

Under the theorem's conditions, closed-loop trajectories of every subsystem satisfy their chance constraints at all times $k \ge 0$, even though the multiplicative noise has unbounded support.
The terminal ingredients can be computed without a central node, so the method scales to larger networks where centralized synthesis or centralized initialization is intractable.
With exact ADMM convergence, the closed loop is mean-square stable: $\mathbb{E}(\|x(k)\|_Q^2) \to 0$ as $k \to \infty$.
The linearization parameter $\epsilon \in (0,1]$ gives the designer a dial for trading conservatism of the approximated chance constraints against feasibility and closed-loop cost.
The two initialization strategies (feedback and prediction) let the controller maintain recursive feasibility even when resetting the mean and covariance to the measured state would be infeasible.

Reading between the lines

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

Because Theorem 19 assumes exact convergence of the ADMM solver ($\epsilon_c=0$) whereas Algorithm 2 stops at a finite tolerance, the guarantees for the implemented controller are unquantified; a natural extension is to derive a bound on how suboptimality degrades recursive feasibility, mean-square convergence, and chance constraint satisfaction.
The numerical table suggests a trade-off: looser ADMM tolerances act as extra constraint tightening and reduce observed violations, while tighter tolerances operate closer to the probabilistic limit and show more violations; using the tolerance as an intentional tuning knob is a testable idea the paper leaves open.
As the paper notes in Remark 5, replacing the distribution-free Cantelli bound with the exact quantile of the disturbance should reduce conservatism; a quantitative comparison of feasible volume and closed-loop cost is a straightforward extension.
The block-diagonal upper bound on each neighbor covariance is a deliberate source of conservatism; relaxing it would be a step toward the coupling chance constraints the conclusion names as future work.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Christoph,

I read the Mark-Liu paper. The headline: it does something new relative to [9] — a fully distributed DSMPC for multiplicative uncertainty with distributed synthesis of terminal ingredients and an ADMM-based online solver. The LMI framework in Section 3 is a real extension of Conte et al. and looks carefully done. The numerical example is honest, with Monte-Carlo violations and cost comparisons at different tolerances. That part is the paper's strength.

The problem is Theorem 19. The proof in Appendix C constructs a feasible candidate for time k+1 by shifting the previous optimal solution, i.e. using the S2 initialization (prediction). But Algorithm 2, when every subsystem passes the S1 feasibility check, solves Problem 16 with S1 (feedback, zero covariance). The shifted candidate is not feasible for that problem unless the measured state happens to equal the predicted mean and the predicted covariance is zero, which is generally false. The proof never supplies a feasible candidate or a cost-decrease argument for the S1 branch. So the recursive feasibility claim is actually fine because the S1 check guarantees feasibility by construction, but the mean-square convergence argument only works for a hypothetical run that always uses S2. Since the algorithm prefers S1 whenever possible, the stated guarantee does not follow for the implemented controller. This is not a minor glitch; it's a missing half of the theorem.

There's a second, more conventional gap: Assumption 18 assumes exact ADMM convergence, while Algorithm 1 uses a finite tolerance εc. The paper doesn't quantify how suboptimality affects feasibility or the Lyapunov decrease. For MPC papers this is often waved away, but here it compounds the first issue: even with εc=0, Theorem 19 is not established for S1.

I'd send it to review if I were the editor — the core idea deserves a chance to be fixed. The authors need to either prove the S1 branch enjoys a similar descent property, or restrict the algorithm to S2 and accept the loss of feedback, or add a safeguard that guarantees the cost decrease. The distributed synthesis part is solid on its own and could be published separately. For now, cite it for the synthesis method, not for the closed-loop guarantees.

Best.

Referee Report

3 major / 4 minor

Summary. The paper proposes a distributed stochastic model predictive control (DSMPC) scheme for networks of linear systems with multiplicative, unbounded zero-mean noise and local chance constraints. Chance constraints are converted, via Cantelli's inequality and a linearization, into constraints on the predicted mean and an upper bound on the predicted covariance. The terminal ingredients, including a structured terminal controller, terminal cost, relaxation functions, and terminal covariance bound, are synthesized through distributed LMIs and a distributed LP. The online algorithm solves the global MPC problem with consensus ADMM, using two initialization strategies: S1, which initializes the mean at the measured state and the covariance at zero, and S2, which uses the shifted optimal prediction from the previous step. The main result, Theorem 19, claims recursive feasibility, mean-square convergence of the state to zero, and satisfaction of the chance constraints at all times, assuming exact solution of the distributed optimization. A three-subsystem numerical example illustrates the closed-loop behavior and compares the distributed design with a centralized one.

Significance. If Theorem 19 were fully established, the paper would make a useful contribution: it provides a fully distributed, scalable stochastic MPC formulation for multiplicative uncertainty, avoiding a central node in both design and online computation. The chance-constraint bounds are derived from a distribution-independent inequality rather than tuned to the example, and the terminal design is genuinely distributed. The numerical study gives evidence of practical behavior over many Monte-Carlo runs. However, the theoretical guarantees as stated are not proven for the algorithm actually implemented: the proof covers only the shifted S2 branch, and the exact-ADMM assumption is not satisfied by the finite-tolerance stopping rule. These gaps are load-bearing for the central claims.

major comments (3)

[Theorem 19, Algorithm 2, Appendix C] The proof of Theorem 19 constructs the feasible candidate at time k+1 by shifting the previous optimal solution, i.e. by setting the initial mean and covariance to (z*(1|k), Σ*(1|k)). This is exactly strategy S2 in constraint (23). However, Algorithm 2 (lines 4-8) solves Problem 16 with strategy S1, i.e. with (z0,Σ0)=(x(k+1),0), whenever the S1 feasibility check succeeds. The shifted candidate is not generally feasible for the S1 problem because it requires z0=z*(1|k) and Σ0=Σ*(1|k), not (x(k+1),0). No alternative feasible candidate is supplied for the S1 branch, so the cost-decrease inequality J*(k+1) ≤ J_m(1|k)+J_v(1|k) is not established for the branch that Algorithm 2 prefers. Consequently, the mean-square convergence argument applies only to a hypothetical run that always uses S2. This gap is independent of the ADMM tolerance issue.
[Assumption 18, Algorithm 1] Theorem 19 assumes that Problem 16 is solved exactly, i.e. that Algorithm 1 converges with ε_c=0. The implemented Algorithm 2 uses the finite stopping criterion ||E_i ξ − y_i||_∞ ≤ ε_c, and ADMM convergence is asymptotic, so ε_c=0 is not reached in finitely many iterations. The manuscript does not quantify how a nonzero ADMM residue affects recursive feasibility, mean-square convergence, or chance constraint satisfaction. The theorem therefore guarantees properties of a limit that the online algorithm never attains. Please either state Theorem 19 explicitly for the idealized exact solver and treat the numerical scheme as an approximation, or provide a suboptimality bound that preserves the closed-loop properties.
[Problem 16, Eq. (7)] The covariance propagation via (8)-(9) is valid only if Σ̂_{N_i}(t|k) is an upper bound on the true covariance Σ_{N_i}(t|k), as stated in (7). However, Problem 16 lists only (4a), (6), and (9) as constraints and does not include (7). If Σ̂ is a decision variable without the explicit constraint Σ̂_{N_i}(t|k) ⬣ Σ_{N_i}(t|k), the solver could choose Σ̂ too small, making the predicted covariance Σ_i(t+1|k) an underestimate of the true covariance and invalidating the chance constraint guarantee. Please add (7) to the problem statement, specify whether Σ̂ is a decision variable or a precomputed bound, and show that the constraint is preserved in the ADMM local problems.

minor comments (4)

[Eqs. (5b), (6b)] In (5b) and (6b) the subscript r should be s, consistent with the input constraint index in (2b).
[Section 3.3] The text says the synthesis is presented 'such that Assumption 4 is satisfied', but the relevant condition is Assumption 9 (the LMI (10)); please correct the cross-reference.
[Appendix C] The claim that recursive feasibility and constraint satisfaction follow directly from [8, Lem. 8] and [8, Lem. 9] should be expanded, because [8] is a deterministic setting and the terminal covariance condition (17) has no counterpart there; a short argument showing that the stochastic terminal constraints are invariant under the α_i update would make the proof self-contained.
[Algorithm 2, numerical example] The simulations do not report how often the S1 branch versus the S2 branch is selected, nor do they test the Lyapunov decrease on S1 steps; such a report would help assess whether the missing S1 analysis is a purely theoretical artifact or a practical issue.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity: the central derivation rests on external invariance lemmas and Cantelli bounds, not on fitted quantities; the only self-citation is contextual.

full rationale

The derivation chain does not reduce to its own inputs. The chance-constraint tightening comes from Cantelli's inequality (Eq. (5)) plus a standard linearization with the explicit design parameter epsilon, not from fitted data. The terminal-cost and terminal-set synthesis in Problems (20)-(21) is an offline design: feasibility of (21) defines alpha, and Lemma 15 proves equivalence with Definition 11 rather than assuming the desired conclusion. The proof of Theorem 19 in Appendix C builds shifted candidates from the previous optimal solution and invokes the external distributed-invariance results [8, Lem. 8] and [8, Lem. 9]; those results are not authored by the present authors and are not stated in terms of Theorem 19's conclusion. The reliance on [10], [12], and [8] is normal prior work, and no 'uniqueness theorem' by the present authors is invoked to force a choice. The only self-citation, [17] in Section 1.1, is a contextual reference to the authors' earlier non-iterative DSMPC concept and is never used as a premise for the main theorem. The reviewer-identified weaknesses -- Assumption 18 requiring exact ADMM convergence while Algorithm 2 uses a finite stopping tolerance, and Appendix C providing a feasible candidate only for the S2 branch while Algorithm 2 may select S1 -- are genuine rigor gaps, but they are not circularity: Theorem 19 does not assume its conclusion under Algorithm 2; it gives an incomplete proof of that implication. A proof covering only one branch is a non-circular failure of implication, not an equivalence of output to input.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The central claim depends on standard stochastic MPC assumptions (existence of stabilizing structured feedback, uncorrelated noise, undirected graph), on the Cantelli approximation and its linearization, and on an idealized exact-convergence assumption for the distributed solver. No invented physical entities are introduced.

free parameters (3)

epsilon (ǫ) = 0.5 in the numerical example
Linearization parameter in the chance constraint approximation (6); chosen by the designer, controls the trade-off between conservatism and feasibility. The theoretical results hold for any ǫ in (0,1], but the specific value affects the tightened constraints.
rho (ρ) = 10 in the numerical example
Penalty parameter of the ADMM algorithm; affects convergence speed but not the exact optimal solution. Not load-bearing for the guarantees, but required for implementation.
epsilon_c (ǫc) = 10^-2, 10^-4, 10^-6 in simulations
Stopping tolerance of the distributed ADMM; the theoretical theorem assumes ǫc=0 (Assumption 18), while the algorithm uses finite values, so this parameter is load-bearing for the practical guarantee.

assumptions (7)

domain assumption Assumption 1: disturbances w_l and w_p are uncorrelated for l != p, zero mean, unit variance, unbounded support.
Standard in stochastic MPC; needed for the covariance propagation (8) and Cantelli bound.
domain assumption Assumption 3: the coupling graph is undirected.
Required for symmetric communication in distributed ADMM and for the neighborhood definition.
domain assumption Assumption 4: (A,B) is stabilizable with a structured linear feedback K(x) = col(K_Ni x_Ni).
Necessary for the existence of a stabilizing terminal controller, which is central to the terminal design.
ad hoc to paper Assumption 9: there exist block-diagonal P and structured K such that the LMI (10) holds.
The paper provides a distributed SDP (Problem 20) to find such P and K, but this may be infeasible for a given system; the existence is assumed for the main theorem.
ad hoc to paper Assumption 18: the distributed ADMM (Algorithm 1) converges exactly to the global optimum (ǫc = 0).
Used in the proof of Theorem 19; not satisfied by the implemented Algorithm 2 which uses finite ǫc. This is a load-bearing idealization.
standard math The Cantelli inequality and the linearization (6) are used without proof, citing [10].
Standard results in chance-constrained MPC; the linearization is conservative, so it does not weaken the constraint satisfaction claim.
domain assumption The block-diagonal upper bound Sigma_Ni <= Sigma_hat_Ni in (7) is introduced to enable distributed optimization.
Adds conservatism but is needed for the distributed SDP framework; assumed a valid upper bound holds.

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Cite this review

Pith. "Pith review of A stochastic MPC scheme for distributed systems with multiplicative uncertainty." pith.science (2026). https://pith.science/paper/VOTOLMHQ

@misc{pith2026190809337,
  author       = {Pith},
  title        = {Pith review of: A stochastic MPC scheme for distributed systems with multiplicative uncertainty},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VOTOLMHQ}},
  note         = {Machine review of arXiv:1908.09337}
}

read the original abstract

This paper presents a Distributed Stochastic Model Predictive Control algorithm for networks of linear systems with multiplicative uncertainties and local chance constraints on the states and control inputs. The chance constraints are approximated via Cantelli's inequality by means of expected value and covariance. The cooperative control algorithm is based on the distributed Alternating Direction Method of Multipliers, which renders the controller fully distributedly implementable, recursively feasible and ensures point-wise convergence of the states. The aforementioned properties are guaranteed through a properly selected distributed invariant set and distributed terminal constraints for the mean and covariance. The paper closes with an example highlighting the chance constraint satisfaction, numerical properties and scalability of our approach.

Figures

Figures reproduced from arXiv: 1908.09337 by the authors.

**Figure 1.** Coupling graph matrices Cii = [ 0.01 0.02 0.02 0.03 ] , Cij = [ 0.002 0.02 0 0.02 ] , ∀j 6= i, ∀i ∈ M. The input and uncertain input matrices are given by Bi = [ 0 1 ] , Di = [ 0 0.001 ] , ∀i ∈ M. The disturbance is normally distributed with w ∼ N (0, 1), the weighting matrices are set to Qi = 10I, Ri = 5, ∀i ∈ M and the prediction horizon is N = 15. We consider for each subsystem a single chance constraint Hx 1… view at source ↗

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [9]

Distributed stochastic mpc of linear systems with paramete r uncertainty and disturbances

Li Dai, Yuanqing Xia, Yulong Gao, and Mark Cannon. Distributed stochastic mpc of linear systems with paramete r uncertainty and disturbances. In Proc. 2016 35th Chinese Control Conference (CCC) , pages 4312–4317. IEEE, 2016

work page 2016

[2] [1]

Infeasibility detection in the alternating direction method of multipliers for convex optimization

Goran Banjac, Paul Goulart, Bartolomeo Stellato, and Stephen Boyd. Infeasibility detection in the alternating direction method of multipliers for convex optimization. In Proc. 2018 UKACC 12th International Conference on Control (CONTROL) , pages 340–340. IEEE, 2018

work page 2018

[3] [2]

Scenario-bas ed model predictive control of stochastic constrained linear systems

Daniele Bernardini and Alberto Bemporad. Scenario-bas ed model predictive control of stochastic constrained linear systems. In Proc. 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference, pages 6333–6338. IEEE, 2009

work page 2009

[4] [3]

A probabilistic particle-control approximation of chance-constrained stochastic predicti ve control

Lars Blackmore, Masahiro Ono, Askar Bektassov, and Brian C Williams. A probabilistic particle-control approximation of chance-constrained stochastic predicti ve control. IEEE transactions on Robotics, 26(3):502–517, 2010

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[5] [4]

Linear matrix inequalities in system and control theory , volume 15

Stephen Boyd, Laurent El Ghaoui, Eric Feron, and Venkataramanan Balakrishnan. Linear matrix inequalities in system and control theory , volume 15. Siam, 1994

work page 1994

[6] [5]

Distributed optimization and statistical learning via the alternating direction method of multipliers

Stephen Boyd, Neal Parikh, Eric Chu, Borja Peleato, Jonathan Eckstein, et al. Distributed optimization and statistical learning via the alternating direction method of multipliers. Foundations and Trends® in Machine learning , 3(1):1–122, 2011

work page 2011

[7] [6]

Stochastic tubes in model predictive control with probabilistic constraints

Mark Cannon, Basil Kouvaritakis, Saˇ sa V Rakovic, and Qifeng Cheng. Stochastic tubes in model predictive control with probabilistic constraints. IEEE Transactions on Automatic Control, 56(1):194–200, 2010

work page 2010

[8] [7]

Distributed model predictive control: A tutorial review and future research directions

Panagiotis D Christoﬁdes, Riccardo Scattolini, David M unoz de la Pena, and Jinfeng Liu. Distributed model predictive control: A tutorial review and future research directions. Computers & Chemical Engineering , 51:21–41, 2013

work page 2013