Pith. sign in

REVIEW 3 major objections 5 minor 29 references

Convergence of stochastic nonlinear systems and implications for Stochastic Model Predictive Control

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Under an ISS Lyapunov condition and a mild assumption on disturbance probabilities, the state of a stochastic nonlinear system converges with probability 1 to the minimal robust positively invariant set.

desk verdict Solid almost-sure convergence machinery for stochastic MPC, but the claimed convergence to the minimal RPI set outruns the assumptions—the set may lack interior, so the tight-ultimate-bound corollaries need an extra condition. read the letter →

arxiv 1908.00483 v2 pith:O6VSKO32 submitted 2019-08-01 math.OC

Diego Muñoz-Carpintero , Mark Cannon This is my paper · ORCID
classification math.OC
keywords stochasticconvergenceboundscontrolnonlinearstateformulationsimplications
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

Model predictive control (MPC) is a method where a controller looks ahead a few steps, solves an optimization problem, applies the first action, and repeats. When the system is hit by random disturbances, standard guarantees often only show the state stays bounded, not where it eventually goes. This paper considers the minimal robust positively invariant (RPI) set: the smallest region from which the system cannot be pushed out by any allowed disturbance. If the state reaches this region, it stays inside it forever.

The authors use input-to-state stability (ISS), a property saying the state remains bounded by the size of the initial state plus the largest disturbance, and that the influence of past disturbances shrinks with time. They combine ISS with a probabilistic condition: arbitrarily small disturbances have positive probability. Then, with positive probability, a long enough string of tiny disturbances occurs, which is enough to pull the state into the RPI set. Because such strings appear infinitely often, the Borel-Cantelli argument guarantees that eventually the state enters the minimal RPI set and remains there with probability 1.

The paper then applies this result to three stochastic MPC algorithms. For linear systems with affine disturbance feedback, it shows the closed-loop state converges almost surely to the minimal RPI set and gives a tight bound on the time-average quadratic cost. For a nonlinear constraint-tightening MPC, it proves convergence that was not previously established. The main limitation is that disturbances must have positive probability of being arbitrarily small, which excludes some distributions.

Extended reading notes

Core claim

Theorem 5: Under Assumptions 1-3, for any x0 in X, P{lim_{k to infinity} 1_Omega(x_k)=1}=1, i.e., the state converges with probability 1 to any RPI set Omega containing the origin in its interior; Corollary 7 then gives convergence to the minimal RPI set X_infty. Theorem 8 additionally bounds the time-average quadratic state cost by lss, the value associated with the linear dynamics inside the limit set.

Load-bearing premise

Assumption 1 requires P{||w||<=lambda}>0 for all lambda>0: the disturbance must have positive probability of being arbitrarily small. This is load-bearing because Lemma 4 needs the event w in W(epsilon) to have positive probability, so long runs of small disturbances can trigger the finite-time ISS decay of Proposition 3; without it, the Borel-Cantelli argument fails. The paper itself notes this excludes some disturbance distributions.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies almost sure convergence of discrete-time stochastic nonlinear systems with additive i.i.d. disturbances. Under an input-to-state stability (ISS) assumption and a condition that arbitrarily small disturbances have positive probability, Theorem 5 proves that the state converges with probability 1 to any robust positively invariant (RPI) set containing the origin in its interior. The authors then claim (Corollary 7) that this implies almost sure convergence to the minimal RPI set X∞, and, under an additional linear-dynamics-on-a-limit-set condition, they prove a bound on the asymptotic average quadratic cost (Theorem 8). These results are applied to three existing stochastic MPC formulations from [18], [19], and [12], yielding new convergence statements for those controllers. The core Borel-Cantelli / ISS argument is carefully executed and is correct for RPI sets with nonempty interior; however, the extension to the minimal RPI set is not justified and is false in degenerate cases.

Significance. If repaired, the paper would provide a useful and general framework for deriving almost sure convergence and tight long-run performance bounds for stochastic MPC, going beyond earlier analyses that only establish boundedness or Lyapunov-type inequalities. The proof technique based on ISS Lyapunov functions plus Borel-Cantelli is elegant and is correctly applied to interior RPI sets. The claimed stronger result of almost sure entry into the minimal RPI set, however, is not valid without additional assumptions, and this affects several corollaries and the nonlinear MPC application. The proposed fix—either adding an interiority assumption on X∞ or restating the result as convergence in distance to X∞—is local and preserves much of the paper's value. The work is not accompanied by code or machine-checked proofs, but the main mathematical arguments are presented in a self-contained and generally careful way.

major comments (3)
  1. [Section III.A, Corollary 7 and Remark 6] Corollary 7 is false as stated. Theorem 5 requires Assumption 2, which demands that the RPI set Ω contains the origin in its interior, but the minimal RPI set X∞ of Definition 2 is only the intersection of RPI sets containing the origin and need not have nonempty interior. For example, take n=1, x_{k+1}=0.5x_k+0·w_k, W=[-1,1], and let w_k be i.i.d. uniform on W so that Assumption 1 holds. Then X∞={0}, which does not contain the origin in its interior, and for x0=1 the trajectory is x_k=2^{-k}; it never equals 0, so P{lim_{k→∞} 1_{X∞}(x_k)=1}=0, contradicting (8). The proof of Corollary 7 asserts that the minimal RPI set 'also satisfies this assumption' (Assumption 2) without proof, and that assertion is false. The correct conclusion from Theorem 5 is P{lim_{k→∞} d(x_k,X∞)=0}=1, or eventual entry into any RPI set that contains the origin in its interior. This correction propagates to Remark 6, Corollaries 10 and 13, and the closing statement of Section IV.C.
  2. [Section IV.C, Proposition 14] Proposition 14 does not verify Assumption 2 for Ω=X∞. The proof states 'Assumption 2 holds because X∞ is bounded due to X∞⊆Z', but Assumption 2 also requires that Ω contains the origin in its interior, and boundedness does not imply this. In the same degenerate linear example as in the previous comment, X∞={0} is RPI and bounded but has empty interior, so Proposition 14 fails and Corollary 7 cannot be invoked for the nonlinear MPC law of (26). Please either add an explicit interiority condition on X∞ (for the nonlinear case, or on the disturbance matrix D and W in the linear cases) or restate the result as convergence in distance to X∞.
  3. [Section III.B, Theorem 8] The proof of Theorem 8 invokes the summability bound ∑_{j} P{xj∉Γ} ≤ Nf p^{-Nf} from Lemma 4, but Lemma 4 applies only to an RPI set Γ satisfying Assumption 2, i.e., containing the origin in its interior. Assumption 4 as stated only requires Γ to be an RPI set on which the dynamics are linear; it does not require 0∈int Γ. The paragraph preceding Assumption 4 says 'an RPI set containing the origin,' but the formal assumption should state this explicitly. Without that interiority condition, the derivation of (10) is incomplete because P{xj∉Γ} need not be summable for Γ with empty interior (e.g., Γ={0} in the scalar example above, where P{xj∉{0}}=1 for all j).
minor comments (5)
  1. [Lemma 4 proof] In the paragraph before equation (6), '⌊N/Nf⌋' should be '⌊k/Nf⌋'.
  2. [Section III.A, paragraph after Lemma 4] 'Borell-Cantelli' should be 'Borel-Cantelli'.
  3. [Theorem 11 proof] The proof invokes [21, Lem. 22] to conclude ISS, but the displayed inequality only treats w=0; please state the lemma or explain how it supplies the K-function bound on the disturbance term needed in (3b).
  4. [Definition 2] The definition '0 ∈ X ⊆ X' uses the symbol X for both the state space and an element of the intersection, which is confusing; the intended meaning is probably that each set in the intersection contains 0 and is a subset of the state space.
  5. [Reference [22]] The reference [22] lists '2009 European' but the conference appears to be the 2019 European Control Conference; please correct the year.
Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper's contribution is a theorem that combines standard control-theoretic assumptions (ISS, RPI sets) with a mild probabilistic condition. It introduces no free parameters or new entities; the only extra assumption specific to an application is Assumption 5 for the striped MPC, whose satisfaction is delegated to [19].

assumptions (6)
  • domain assumption Disturbance sequence is i.i.d., zero-mean, with bounded support W and P{||w||<=lambda}>0 for all lambda>0 (Assumption 1).
    Used in Lemma 4 to guarantee the small-disturbance event W(epsilon) has positive probability, making long runs of contraction possible.
  • domain assumption There exists an RPI set Omega containing the origin in its interior (Assumption 2).
    Defines the target limit set; needed so epsilon>0 in Proposition 3.
  • domain assumption System is ISS with region of attraction X, equivalently admits an ISS-Lyapunov function (Assumption 3).
    Provides the Lyapunov decrease used in Propositions 2-3 and the boundedness bound r(x0).
  • domain assumption For linear-in-limit-set results, f(x,w)=Phi x+D w on Gamma times W with Phi Schur stable (Assumption 4).
    Enables explicit computation of X_infty and the tight average-cost bound in Theorem 8.
  • domain assumption For the striped MPC application, the minimal RPI set is contained in the region where the optimal perturbation is zero, X_infty subset of Xuc (Assumption 5).
    Needed so the closed-loop dynamics are linear on the limit set; not verified for all problem data, relies on [19].
  • standard math Borel-Cantelli lemma and ISS-Lyapunov equivalence (Theorem 1 from [24]) are used.
    Backbone of the almost-sure convergence argument.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Convergence of stochastic nonlinear systems and implications for Stochastic Model Predictive Control." pith.science (2026). https://pith.science/paper/O6VSKO32

@misc{pith2026190800483,
  author       = {Pith},
  title        = {Pith review of: Convergence of stochastic nonlinear systems and implications for Stochastic Model Predictive Control},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O6VSKO32}},
  note         = {Machine review of arXiv:1908.00483}
}
read the original abstract

The stability of stochastic Model Predictive Control (MPC) subject to additive disturbances is often demonstrated in the literature by constructing Lyapunov-like inequalities that ensure closed-loop performance bounds and boundedness of the state, but tight ultimate bounds for the state and non-conservative performance bounds are typically not determined. In this work we use an input-to-state stability property to find conditions that imply convergence with probability 1 of a disturbed nonlinear system to a minimal robust positively invariant set. We discuss implications for the convergence of the state and control laws of stochastic MPC formulations, and we prove convergence results for several existing stochastic MPC formulations for linear and nonlinear systems.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

29 extracted references · 28 canonical work pages

  1. [20]

    Con straint- tightening and stability in stochastic model predictive co ntrol,

    M. Lorenzen, F. Dabbene, R. Tempo, and F. Allgower, “Con straint- tightening and stability in stochastic model predictive co ntrol,” IEEE Transactions on Automatic Control , 2016

    work page 2016

  2. [22]

    On the convergenc e of stochastic MPC to terminal modes of operation,

    D. Mu˜ noz-Carpintero and M. Cannon, “On the convergenc e of stochastic MPC to terminal modes of operation,” in Control Conference (ECC), 2009 European, p. accepted for publication, 2019

    work page 2009

  3. [19]

    Efficient prediction strategies for disturbance compensation in sto chastic MPC,

    B. Kouvaritakis, M. Cannon, and D. Mu˜ noz-Carpintero, “Efficient prediction strategies for disturbance compensation in sto chastic MPC,” International Journal of Systems Science , vol. 44, no. 7, pp. 1344–1353, 2013

    work page 2013

  4. [12]

    A constraint-tightening approach to nonlinear model pred ictive control with chance constraints for stochastic systems,

    T. L. Santos, A. D. Bonzanini, T. A. N. Heirung, and A. Mes bah, “A constraint-tightening approach to nonlinear model pred ictive control with chance constraints for stochastic systems,” in American Control Conference (ACC), pp. 1641–1647, 2019

    work page 2019

  5. [18]

    Input-to-state stabi lity of robust receding horizon control with an expected value cost,

    P . J. Goulart and E. C. Kerrigan, “Input-to-state stabi lity of robust receding horizon control with an expected value cost,” Automatica, vol. 44, no. 4, pp. 1171–1174, 2008

    work page 2008

  6. [1]

    Kouvaritakis and M

    B. Kouvaritakis and M. Cannon, Model Predictive Control: Classical, Robust and Stochastic . Advanced Textbooks in Control and Signal Processing, Springer, 2015

    work page 2015

  7. [2]

    Stochastic model predictive control: An ove rview and perspectives for future research,

    A. Mesbah, “Stochastic model predictive control: An ove rview and perspectives for future research,” IEEE Control Systems, vol. 36, pp. 30– 44, Dec 2016

    work page 2016

  8. [3]

    Model predictive control: Recent developm ents and future promise,

    D. Q. Mayne, “Model predictive control: Recent developm ents and future promise,” Automatica, vol. 50, no. 12, pp. 2967 – 2986, 2014

    work page 2014

Show all 29 references
  1. [4]

    Model predictive control for systems with stochastic multiplicative uncertainty an d probabilistic constraints,

    M. Cannon, B. Kouvaritakis, and X. Wu, “Model predictive control for systems with stochastic multiplicative uncertainty an d probabilistic constraints,” Automatica, vol. 45, no. 1, pp. 167 – 172, 2009

    2009

  2. [5]

    Stabilizing model predi ctive control of stochastic constrained linear systems,

    D. Bernardini and A. Bemporad, “Stabilizing model predi ctive control of stochastic constrained linear systems,” IEEE Transactions on Automatic Control, vol. 57, no. 6, pp. 1468–1480, 2012

    2012

  3. [6]

    Model predictive control o f linear systems with multiplicative unbounded uncertainty and chance cons traints,

    M. Farina and R. Scattolini, “Model predictive control o f linear systems with multiplicative unbounded uncertainty and chance cons traints,” Automatica, vol. 70, pp. 258 – 265, 2016

    2016

  4. [7]

    Scenario-based approach to stochastic linear predictive control,

    J. Matusko and F. Borrelli, “Scenario-based approach to stochastic linear predictive control,” in Decision and Control (CDC), 2012 IEEE 51st Annual Conference on , pp. 5194–5199, Dec 2012

    2012

  5. [8]

    Nonlinear stochastic model predictive control via regularized polynomial chaos expansions,

    L. Fagiano and M. Khammash, “Nonlinear stochastic model predictive control via regularized polynomial chaos expansions,” in 2012 IEEE 51st IEEE Conference on Decision and Control (CDC) , pp. 142–147, 2012

    2012

  6. [9]

    The sc enario approach for stochastic model predictive control with boun ds on closed- loop constraint violations,

    G. Schildbach, L. Fagiano, C. Frei, and M. Morari, “The sc enario approach for stochastic model predictive control with boun ds on closed- loop constraint violations,” Automatica, vol. 50, no. 12, pp. 3009 – 3018, 2014

    2014

  7. [10]

    Stability for r eceding- horizon stochastic model predictive control,

    J. A. Paulson, S. Streif, and A. Mesbah, “Stability for r eceding- horizon stochastic model predictive control,” in 2015 American Control Conference (ACC), pp. 937–943, July 2015

    2015

  8. [11]

    Output-feedback mo del predictive control for stochastic systems with multiplica tive and additive uncertainty,

    J. Li, D. Li, Y . Xi, Y . Xu, and Z. Gan, “Output-feedback mo del predictive control for stochastic systems with multiplica tive and additive uncertainty,” International Journal of Robust and Nonlinear Control , vol. 28, no. 1, pp. 86–102, 2018

    2018

  9. [13]

    Probabilistic c onstrained MPC for multiplicative and additive stochastic uncertaint y,

    M. Cannon, B. Kouvaritakis, and X. Wu, “Probabilistic c onstrained MPC for multiplicative and additive stochastic uncertaint y,” IEEE Trans. Automatic Control, vol. 54, no. 7, pp. 1626–1632, 2009

    2009

  10. [14]

    Probabilistic t ubes in linear stochastic model predictive control,

    M. Cannon, B. Kouvaritakis, and D. Ng, “Probabilistic t ubes in linear stochastic model predictive control,” Systems & Control Letters , vol. 58, no. 10, pp. 747–753, 2009

    2009

  11. [15]

    Explicit use of probabilistic distributions in linear predictive contr ol,

    B. Kouvaritakis, M. Cannon, S. V . Rakovi´ c, and Q. Cheng , “Explicit use of probabilistic distributions in linear predictive contr ol,” Automatica, vol. 46, no. 10, pp. 1719–1724, 2010

    2010

  12. [16]

    On stability and perform ance of stochastic predictive control techniques,

    D. Chatterjee and J. Lygeros, “On stability and perform ance of stochastic predictive control techniques,” IEEE Transactions on Automatic Control, vol. 60, pp. 509–514, Feb 2015

    2015

  13. [17]

    Dropout feed- back parametrized policies for stochastic predictive cont roller,

    P . K. Mishra, D. E. Quevedo, and D. Chatterjee, “Dropout feed- back parametrized policies for stochastic predictive cont roller,” IF AC- PapersOnLine, vol. 49, no. 18, pp. 59 – 64, 2016. 10th IFAC Symposium on Nonlinear Control Systems NOLCOS 2016

    2016

  14. [21]

    Op timization over state feedback policies for robust control with constraint s,

    P . J. Goulart, E. C. Kerrigan, and J. M. Maciejowski, “Op timization over state feedback policies for robust control with constraint s,” Automatica, vol. 42, no. 4, pp. 523–533, 2006

    2006

  15. [23]

    S. Meyn, R. Tweedie, and P . Glynn, Markov Chains and Stochastic Stability. Cambridge Mathematical Library, CUP , 2009

    2009

  16. [24]

    Input-to-state stability for discrete-time nonlinear systems,

    Z.-P . Jiang and Y . Wang, “Input-to-state stability for discrete-time nonlinear systems,” Automatica, vol. 37, no. 6, pp. 857 – 869, 2001

    2001

  17. [25]

    Blanchini and S

    F. Blanchini and S. Miani, Set-theoretic methods in control . Birkh¨ auser, Boston, 2008

    2008

  18. [26]

    Constrained linear syste m under disturbance feedback: Convergence with probability one,

    C. Wang, C. J. Ong, and M. Sim, “Constrained linear syste m under disturbance feedback: Convergence with probability one,” in 2008 47th IEEE Conference on Decision and Control , pp. 2820–2825, Dec 2008

    2008

  19. [27]

    T he explicit linear quadratic regulator for constrained systems,

    A. Bemporad, M. Morari, V . Dua, and E. Pistikopoulos, “T he explicit linear quadratic regulator for constrained systems,” Automatica, vol. 38, no. 1, pp. 3–20, 2002

    2002

  20. [28]

    Stochastic model predicti ve control: Output feedback, duality and guaranteed performa nce,

    M. A. Sehr and R. R. Bitmead, “Stochastic model predicti ve control: Output feedback, duality and guaranteed performa nce,” arXiv:1706.00733, 2017

    2017 arXiv

  21. [29]

    Stochastic mo del predictive control without terminal constraints,

    M. Lorenzen, M. A. Mller, and F. Allgwer, “Stochastic mo del predictive control without terminal constraints,” International Journal of Robust and Nonlinear Control , vol. 29, no. 15, pp. 4987–5001, 2019

    2019

Pith tools

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