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REVIEW 2 major objections 5 minor 53 references

Non-asymptotic Bounds of Learning-based Linear MPC With Input Constraints and Unbounded Stochastic Noise

T0 review · 2 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read This paper claims that a certainty-equivalence MPC-deadbeat switching law with online least-squares estimation renders unknown input-constrained linear systems globally practically stable in high probability, with explicit non-asymptotic bo

desk verdict Genuinely new CE-MPC stability theorem whose main identification lemma lives in a companion paper; send it to referees, but require the missing proof before acceptance. read the letter →

arxiv 2607.13513 v1 pith:CEGXS5WU submitted 2026-07-15 eess.SY cs.SY

Changyi Lei , Seth Siriya , Dragan Nešić , Ye Pu This is my paper · ORCID
classification eess.SYcs.SY MSC 93C5593E3593D30
keywords learning-basedMPCcertainty-equivalencecontrolrecursiveleastsquaresinputconstraintssub-Gaussiannoisenon-asymptoticboundspersistentexcitationhysteresisswitching
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 tries to prove that a controller can learn and stabilize an unknown linear system at the same time, even when the input is hard-saturated and the noise is unbounded. It proposes a certainty-equivalence scheme that estimates the system matrices online with regularized least squares and switches between two feedback laws: model predictive control and a saturated deadbeat controller, with dither added for excitation. The central result is a high-probability bound: after a finite transient, the state stays inside a tube that shrinks to a noise-dependent offset, with failure probability at most 3δ. This matters because existing adaptive MPC guarantees usually require a known uncertainty set, a pre-stabilizing controller, or terminal ingredients; here none of those are assumed. If the proof is correct, it turns simultaneous learning and constrained stabilization into a quantifiable, global property rather than a regional one.

What carries the argument

The central object is the four-part controller assembled in Algorithm 1: an RLS estimator, a dither signal v uniform on [−C,C]^κm, a saturated deadbeat law π_DB, and an MPC law π_MPC, glued together by a hysteresis switching rule with width c_3. The hysteresis is load-bearing because it restores global Lipschitz continuity of the control law with respect to the estimated parameters—a property the MPC law alone lacks. The exponential stage cost ℓ(x,u)=Q(e^{|x|}−1)^2+u^⊤Ru supplies a state-independent horizon bound that makes the finite-horizon value function decrease without terminal constraints. These pieces let the authors propagate finite-time RLS error bounds through the value functions a

What would settle it

For the numerical example in Section V, compute the empirical probability that ζ^⊤[(x+w);(π_s(x+w,θ)+v)]^2 ≥ c_PE over many noise draws, sweeping x over a large grid, ζ over the unit sphere, and θ in a neighborhood of θ̂; if for some (x,ζ,θ,s) this probability falls below the asserted p_PE, Lemma 1 fails and the burn-in bound of Proposition 1 no longer follows. More directly, run long closed-loop simulations and check whether the RLS error bound (25) holds after T_burn-in; a single episode in which ∥θ̂_t−θ*∥ exceeds ϵ_t falsifies the claim.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1: under sub-Gaussian disturbances and κ-step reachability, the switching law of Algorithm 1 with a prediction horizon N satisfying condition (12) gives P(|x_t| ≤ η̃(t) + c̃) ≥ 1 − 3δ for all t ≥ t_1, where η̃ ∈ L decays and c̃ is a noise-dependent offset. The proof chain is: RLS with injected dither ensures persistent excitation (Lemma 1), so the estimation error is bounded non-asymptotically after a finite burn-in (Proposition 1); the exponential stage cost makes the finite-horizon MPC value function a Lyapunov function without terminal constraints (Lemma 3); hysteresis switching makes the closed-loop law globally Lipschitz in the parameter estimate; together t

Load-bearing premise

The whole result rests on Lemma 1's claim that the closed-loop regressor is uniformly persistently exciting for every state, estimate, and controller mode with fixed constants c_PE and p_PE—a uniform anti-concentration bound whose proof is deferred to a companion paper and is not verified here.

Editorial extensions

If this is right

  • Global practical stability: the high-probability bound holds for any initial state, so learning can begin while the plant is far from the origin.
  • No terminal constraints or known compact parameter set: the only requirements are κ-step reachability, a known noise-variance lower bound, and ∥A∥ ≤ 1.
  • Explicit controller tuning: Corollary 1 gives concrete sufficient conditions on N, R/Q, and u_max that satisfy the main stability condition (12).
  • Quantified failure probability: the 3δ term decomposes into estimation, block-level, and per-step events, so a user can trade off confidence against horizon and excitation levels.
  • Hard input constraints are respected at every time step by reserving C of u_max for dither and saturating the deadbeat law.

Reading between the lines

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

  • The hysteresis-switching construction suggests a general recipe: when a certainty-equivalence controller is only locally Lipschitz in the parameters, adding hysteresis between two controllers can recover global Lipschitz continuity and make finite-time identification errors propagatable through the value function—this could transfer to other adaptive MPC designs.
  • The exponential stage cost is doing heavy lifting; the same trick may yield terminal-cost-free stability bounds for other constrained nonlinear MPC problems, though it makes the optimization more nonlinear.
  • The uniform persistency-of-excitation lemma is the unverified load-bearing block; a natural next step is to compute or bound c_PE and p_PE explicitly for structured examples, since the burn-in time and all downstream bounds scale inversely with them.
  • A systematic numerical sweep over dither amplitude C, horizon N, and noise variance could map the conservatism of the bound and test whether the ensemble behavior observed in the simulation reflects the worst-case theory.
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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

2 major / 5 minor

Summary. The paper proposes a certainty-equivalence learning-based MPC scheme for unknown discrete-time LTI systems with hard input constraints and unbounded sub-Gaussian disturbances. The controller switches between a saturated deadbeat law and an MPC law, with hysteresis, and injects bounded excitation; system parameters are updated online by regularized least squares. The main result, Theorem 1, asserts a non-asymptotic high-probability practical stability bound after a finite transient, under a horizon condition (12). The proof proceeds through a uniform persistent-excitation lemma, a non-asymptotic RLS error bound, Lyapunov-like bounds for the deadbeat and MPC value functions, a convex-decay stochastic ISS argument, and a conversion from κ-step to per-step bounds. A simulation illustrates the behavior.

Significance. If the result is correct, it is a notable advance: simultaneous online identification and global stabilization of an unknown linear system under hard input constraints and unbounded noise, without pre-stabilizing controllers, terminal ingredients, or a known compact uncertainty set. The switching construction and the attempt to propagate finite-time identification errors through the MPC value function are interesting and nontrivial. However, the paper is heavily dependent on companion works: the uniform PE lemma (Lemma 1) is not proved here, the RLS bound is imported from [35], the final stochastic-ISS step imports [40, Theorem 2], and the DB Lipschitz lemma is from [47]. The algebraic core (Lemmas 2, 3, 5, 7, 8 and Proposition 2) is presented in detail, but one key inequality in Lemma 5 appears invalid. The contribution is therefore promising but not yet established in the manuscript as written.

major comments (2)
  1. [Section IV-A, Lemma 1 and Remark 6] Lemma 1 is the sole source of the uniform PE constants c_PE, p_PE used by Proposition 1 to define T_burn-in and the RLS error bound ϵ_t. Its proof is not in this manuscript: Remark 6 explicitly sends the reader to the companion paper [46]. The lemma quantifies over all x, ζ, s, and θ with no compactness or closeness assumptions, so this is not a routine detail. Moreover, Section VI states ‘We proved the probabilistic PE condition…’, which contradicts Remark 6. Since T_burn-in, Eq. (25), Proposition 3, and Theorem 1 all collapse if the PE constants are not available, the full proof (or a self-contained statement with explicit c_PE, p_PE) must appear in this paper before the main claim can be assessed.
  2. [Section IV-B, Lemma 5, Eq. (43)] The displayed inequality E[σ(|˜g1(x)+R*(π diff)+R*¯v+Rκ(A,I)¯w|)] ≤ E[σ(|˜g1(x)|+D_MPC∥R*∥∥θ̂−θ*∥+|h|)] = σ(...) is not valid. Since σ(s)=(e^s−1)^2 is convex and increasing, an upper bound on E[σ(|Y+N|)] requires control of E[e^{2|N|}], whereas h = ln E e^{|R*¯v|} + ln E e^{|Rκ(A,I)¯w|} only controls E[e^{|N|}] (and in fact E[e^{2|N|}] ≥ (E[e^{|N|}])^2, typically strictly). For a concrete failure mode, take N∼N(0,1); then E[(e^{|N|}−1)^2] ≈ 9.9, while (e^{ln E e^{|N|}}−1)^2 ≈ 3.1. The same issue propagates to Lemma 7, Proposition 2, and Theorem 1 through the terms E_w1, E_w2, and E_w. The proof needs a corrected bound, e.g. using a bound on E[e^{2|N|}] or a different concentration argument, and the downstream conditions (12), (50) must be re-derived accordingly.
minor comments (5)
  1. [Section III-C, Remark 4] Remark 4 says the theorem gives probability at least 1−δ, but Theorem 1 states 1−3δ. This mismatch should be corrected.
  2. [Section VI] The conclusion describes ‘additive i.i.d bounded stochastic disturbances’, but Assumption 1 and the abstract allow unbounded sub-Gaussian noise. Please fix the wording.
  3. [Section IV-B, Lemmas 2 and 7] Both lemmas refer to ‘p being defined in (19)’, but p is defined in Eq. (16); Eq. (19) defines q_3^max. This typo should be corrected in the final version.
  4. [Section III-C, Theorem 1 and Corollary 1] Condition (12) and Corollary 1 are expressed in terms of the true unknown matrices R_* and A, e.g. σ_min(R_*), ∥R_*^{-1} A^κ∥, and ¯E_w. As a result, the horizon/excitation conditions are not verifiable from prior knowledge. The theorem is still a valid existence statement, but the paper should clarify this and, if possible, provide computable sufficient conditions or bounds.
  5. [Throughout] There are several typos and small errors: ‘asic algebraic manipulations’, ‘taht’, ‘PN’ in Sec. IV-B, and an undefined H_1 in the proof of Lemma 4 (the constraint matrix M is defined but H_1 is not). A careful proofreading pass is needed.

Circularity Check

1 steps flagged · score 4.0 of 10

Load-bearing same-author citations for the identification step; no definitional equivalence, but Theorem 1 rests on an unproved Lemma 1 delegated to a companion paper by the same authors.

  1. self citation load bearing [Section IV-A, Lemma 1 and Remark 6; Proposition 1; used in the proof of Proposition 3]
    "Lemma 1: 'There exist constants c_PE > 0 and p_PE > 0 such that, for all x∈R^n, ζ∈S^{n+κm-1}, s∈{MPC,DB}, and θ∈R^{n×(n+m)}, P(|ζ^⊤[(x+w);(π_s(x+w,θ)+v)]|^2 ≥ c_PE) ≥ p_PE.' Remark 6: 'The proof can be found in [46].' Proposition 1 proof: 'Proposition 1 can be proved directly using Lemma 1, Corollary 1 and Example 2 of [35].' Proposition 3: 'Firstly, by Proposition 1 ... there must exist a time step t3 ... such that (51) holds ... with probability at least 1−δ.'"

    Theorem 1's final bound is mediated by Proposition 3, which uses Proposition 1 to obtain the parameter-error threshold (51) that triggers the switching/ISS argument. Proposition 1's error bound ϵ_t and finite T_burn-in depend on the uniform PE constants c_PE,p_PE of Lemma 1. This lemma is not proved in the present paper; Remark 6 explicitly defers to [46], a same-author companion paper. Thus the load-bearing identification premise is justified only by a self-citation whose proof is not exhibited here, and the theorem's probability guarantee collapses if that lemma is not valid. This is not an equation-to-equation identity, but the claimed derivation reduces at this point to an unverified same-group citation.

full rationale

The paper is not definitionally circular: no fitted parameter is relabeled as a prediction, and no displayed equation is assumed equal to the theorem's conclusion. The Lyapunov and Convex-Decay ISS analysis in Lemmas 2, 3, 5, 7 and Proposition 2 is carried out in the text with explicit inequalities. However, the identification engine that feeds the stability theorem is imported from same-group preprints: Lemma 1 (uniform persistent excitation with explicit c_PE,p_PE) is stated without proof and Remark 6 says its proof is in [46]; Proposition 1 adapts [35] using Lemma 1; Proposition 3 then invokes Proposition 1 to obtain the threshold (51). Because T_burn-in, ϵ_t, and the triggering of the entire switching analysis depend on those constants, the central high-probability bound is only as strong as an unverified same-author lemma. This is load-bearing self-citation rather than equivalence-by-construction, so I assign a moderate score rather than a high one.

Assumptions & free parameters 5 free parameters · 7 assumptions · 0 invented entities

The paper introduces no new physical or conceptual entity beyond a switching controller; the central claim rests on four imported same-group results (PE lemma, RLS bound, stochastic-ISS framework, DB Lipschitz lemma), user-chosen design parameters, and the structural assumptions on the plant. The most honest measure is that the main theorem is a dependent assembly of strong prior machinery rather than a self-contained derivation.

free parameters (5)
  • Prediction horizon N = 5 in simulation; theorem only requires existence under (12)
    Central stability bound is conditional on N satisfying (12), which depends on unknown true matrices R*, Rκ(A,I), and ∥R†*Aκ∥. No computable formula from known constants is supplied.
  • Excitation magnitude C = 3 in simulation
    User-chosen in (0,u_max); affects p and condition (12).
  • Cost weights Q and R = Q=1, R=0.1I in simulation
    User-chosen; Lemma 3 and the horizon condition depend on λmax(R)/Q.
  • Switching thresholds c_MPC, c_DB, c3, c4 = c_MPC=200, c_DB=100, c3=50 in simulation
    User-chosen satisfying (7); influence q2, Lemma 8, and the construction of ρ.
  • RLS regularization parameter ψ (λ in simulation) = 0.1 in simulation
    Regularization in (3) affects the error bound ϵ_t and the burn-in time T_burn-in.
assumptions (7)
  • domain assumption Disturbance sequence is i.i.d., zero-mean, σ_W²-sub-Gaussian with known lower bound on covariance eigenvalues (Assumption 1)
    Used in Lemma 1's anti-concentration argument and in [35]'s error bound; restricts the noise class.
  • domain assumption System matrices satisfy ∥A∥≤1 and (A,B) is κ-step reachable with known lower bound σ_B on σmin(Rκ(A,B)) (Assumption 2)
    Needed for the saturated deadbeat controller, for p in (16), and for global stabilizability under input saturation.
  • ad hoc to paper Uniform persistent-excitation constants c_PE>0 and p_PE>0 exist for all states, parameters, and controller modes (Lemma 1)
    Load-bearing for Proposition 1; proof delegated to companion paper [46], not contained in this manuscript.
  • domain assumption The non-asymptotic RLS error bound of [35] (Proposition 1) holds under the closed-loop controller
    Imported from the authors' prior work; not re-derived here and depends on Lemma 1.
  • domain assumption Applicability of [40, Theorem 2] (Convex-Decay Stochastic ISS) and [48, Theorem 1] (MPC without terminal constraints)
    Used to convert the Lyapunov decrease condition into the final high-probability state bound and the MPC Lyapunov decrement.
  • ad hoc to paper Exponential stage cost ℓ=Qσ(x)+u^TRu is chosen to allow a state-independent horizon (Remark 2)
    The proof of Lemma 3 relies on the exponential cost structure; a different cost may not admit the same state-independent horizon bound.
  • domain assumption Hysteresis switching construction from [41] yields a globally Lipschitz closed-loop control law with respect to parameters
    Imported from [41] and central to propagating finite-time identification errors through the control law.

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Cite this review

Pith. "Pith review of Non-asymptotic Bounds of Learning-based Linear MPC With Input Constraints and Unbounded Stochastic Noise." pith.science (2026). https://pith.science/paper/CEGXS5WU

@misc{pith2026260713513,
  author       = {Pith},
  title        = {Pith review of: Non-asymptotic Bounds of Learning-based Linear MPC With Input Constraints and Unbounded Stochastic Noise},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CEGXS5WU}},
  note         = {Machine review of arXiv:2607.13513}
}
read the original abstract

This paper studies learning-based model predictive control (MPC) for stabilizing unknown discrete-time linear systems with hard input constraints and additive unbounded sub-Gaussian disturbances. We adopt a certainty-equivalence (CE) design that combines a switching MPC control law with online regularized least-squares (RLS) parameter estimation. The resulting switching control law blends the MPC with a saturated deadbeat controller, ensuring global closed-loop stability. Building upon non-asymptotic error bound of least-squares, we derive non-asymptotic, high-probability stability bounds for the closed-loop system under the proposed switching controller. Numerical experiments illustrate and support the theoretical findings.

Figures

Figures reproduced from arXiv: 2607.13513 by the authors.

Figure 1
Figure 1. States and estimation errors ensemble VI. CONCLUSION In this paper, we propose an excited certainty equivalence Model Predictive Control scheme for adaptive control of discrete-time linear systems subject to additive i.i.d bounded stochastic disturbances with input constraints, assuming that the system is at least marginally stable. We proved the probabilistic PE condition for any process disturbance that obeys the … view at source ↗

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Reference graph

Works this paper leans on

53 extracted references · 5 linked inside Pith

  1. [46]

    Learning-based homothetic tube mpc with non-asymptotic guarantees,

    C. Lei, S. Siriya, D. Ne ˇsi´c, and Y . Pu, “Learning-based homothetic tube mpc with non-asymptotic guarantees,” 2026. [Online]. Available: https://arxiv.org/abs/2607.12343

    arXiv 2026

  2. [35]

    Non-asymptotic bounds for closed-loop identification of unstable nonlinear stochastic systems,

    S. Siriya, J. Zhu, D. Ne ˇsi´c, and Y . Pu, “Non-asymptotic bounds for closed-loop identification of unstable nonlinear stochastic systems,” arXiv preprint arXiv:2412.04157, 2024

    arXiv 2024

  3. [40]

    A framework for adap- tive stabilisation of nonlinear stochastic systems,

    S. Siriya, J. Zhu, D. Ne ˇsi´c, and Y . Pu, “A framework for adap- tive stabilisation of nonlinear stochastic systems,”arXiv preprint arXiv:2511.17436, 2025

    arXiv 2025

  4. [47]

    Stability bounds for learning- based adaptive control of discrete-time multi-dimensional stochastic linear systems with input constraints,

    S. Siriya, J. Zhu, D. Ne ˇsi´c, and Y . Pu, “Stability bounds for learning- based adaptive control of discrete-time multi-dimensional stochastic linear systems with input constraints,” in2023 62nd IEEE Conference on Decision and Control (CDC). IEEE, 2023, pp. 3802–3807

    2023

  5. [1]

    Borrelli, A

    F. Borrelli, A. Bemporad, and M. Morari,Predictive control for linear and hybrid systems. Cambridge University Press, 2017

    2017

  6. [2]

    Rawlings, D

    J. Rawlings, D. Mayne, and M. Diehl,Model Predictive Control: The- ory, Computation, and Design. Nob Hill Publishing, 2017. [Online]. Available: https://books.google.com.au/books?id=MrJctAEACAAJ

    2017

  7. [3]

    Robust model predic- tive control of constrained linear systems with bounded disturbances,

    D. Q. Mayne, M. M. Seron, and S. V . Rakovi ´c, “Robust model predic- tive control of constrained linear systems with bounded disturbances,” Automatica, vol. 41, no. 2, pp. 219–224, 2005

    2005

  8. [4]

    Learning-based model predictive control: Toward safe learning in control,

    L. Hewing, K. P. Wabersich, M. Menner, and M. N. Zeilinger, “Learning-based model predictive control: Toward safe learning in control,”Annual Review of Control, Robotics, and Autonomous Sys- tems, vol. 3, no. 1, pp. 269–296, 2020

    2020

Show all 53 references
  1. [5]

    Learning- based model predictive control for safe exploration,

    T. Koller, F. Berkenkamp, M. Turchetta, and A. Krause, “Learning- based model predictive control for safe exploration,” in2018 IEEE conference on decision and control (CDC). IEEE, 2018, pp. 6059– 6066

    2018

  2. [6]

    Cautious model predictive control using gaussian process regression,

    L. Hewing, J. Kabzan, and M. N. Zeilinger, “Cautious model predictive control using gaussian process regression,”IEEE Transactions on Control Systems Technology, vol. 28, no. 6, pp. 2736–2743, 2019

    2019

  3. [7]

    Kpc: Learning-based model predictive control with deterministic guaran- tees,

    E. T. Maddalena, P. Scharnhorst, Y . Jiang, and C. N. Jones, “Kpc: Learning-based model predictive control with deterministic guaran- tees,” inLearning for Dynamics and Control. PMLR, 2021, pp. 1015–1026

    2021

  4. [8]

    Learning-based non- linear model predictive control,

    D. Limon, J. Calliess, and J. M. Maciejowski, “Learning-based non- linear model predictive control,”IFAC-PapersOnLine, vol. 50, no. 1, pp. 7769–7776, 2017

    2017

  5. [9]

    Gaussian processes for dynamics learning in model predictive control,

    A. Scampicchio, E. Arcari, A. Lahr, and M. N. Zeilinger, “Gaussian processes for dynamics learning in model predictive control,”Annual Reviews in Control, vol. 60, p. 101034, 2025

    2025

  6. [10]

    Data-enabled predictive control: In the shallows of the deepc,

    J. Coulson, J. Lygeros, and F. D ¨orfler, “Data-enabled predictive control: In the shallows of the deepc,” in2019 18th European Control Conference (ECC). IEEE, 2019, pp. 307–312

    2019

  7. [11]

    Data-driven tracking mpc for changing setpoints,

    J. Berberich, J. K ¨ohler, M. A. M ¨uller, and F. Allg ¨ower, “Data-driven tracking mpc for changing setpoints,”IFAC-PapersOnLine, vol. 53, no. 2, pp. 6923–6930, 2020

    2020

  8. [12]

    On the design of terminal ingredients for data-driven mpc,

    ——, “On the design of terminal ingredients for data-driven mpc,” IFAC-PapersOnLine, vol. 54, no. 6, pp. 257–263, 2021

    2021

  9. [13]

    Robust stability analysis of a simple data-driven model predictive control approach,

    J. Bongard, J. Berberich, J. K ¨ohler, and F. Allg¨ower, “Robust stability analysis of a simple data-driven model predictive control approach,” IEEE Transactions on Automatic Control, vol. 68, no. 5, pp. 2625– 2637, 2022

    2022

  10. [14]

    A novel constraint-tightening approach for robust data-driven predictive con- trol,

    C. Kl ¨oppelt, J. Berberich, F. Allg ¨ower, and M. A. M ¨uller, “A novel constraint-tightening approach for robust data-driven predictive con- trol,”International Journal of Robust and Nonlinear Control, vol. 35, no. 7, pp. 2566–2587, 2025

    2025

  11. [15]

    Noise handling in data- driven predictive control: A strategy based on dynamic mode decom- position,

    A. Sassella, V . Breschi, and S. Formentin, “Noise handling in data- driven predictive control: A strategy based on dynamic mode decom- position,” inLearning for Dynamics and Control Conference. PMLR, 2022, pp. 74–85

    2022

  12. [16]

    Indirect adaptive model predictive control for linear systems with polytopic uncertainty,

    S. Di Cairano, “Indirect adaptive model predictive control for linear systems with polytopic uncertainty,” in2016 American Control Con- ference (ACC). IEEE, 2016, pp. 3570–3575

    2016

  13. [17]

    Adaptive receding horizon control for constrained mimo systems,

    M. Tanaskovic, L. Fagiano, R. Smith, and M. Morari, “Adaptive receding horizon control for constrained mimo systems,”Automatica, vol. 50, no. 12, pp. 3019–3029, 2014

    2014

  14. [18]

    Linear robust adaptive model predictive control: Computational com- plexity and conservatism,

    J. K ¨ohler, E. Andina, R. Soloperto, M. A. M ¨uller, and F. Allg ¨ower, “Linear robust adaptive model predictive control: Computational com- plexity and conservatism,” in2019 IEEE 58th Conference on Decision and Control (CDC). IEEE, 2019, pp. 1383–1388

    2019

  15. [19]

    Robust mpc with recur- sive model update,

    M. Lorenzen, M. Cannon, and F. Allg ¨ower, “Robust mpc with recur- sive model update,”Automatica, vol. 103, pp. 461–471, 2019

    2019

  16. [20]

    Adaptive model predictive control for a class of constrained linear systems with parametric uncertainties,

    K. Zhang and Y . Shi, “Adaptive model predictive control for a class of constrained linear systems with parametric uncertainties,”Automatica, vol. 117, p. 108974, 2020

    2020

  17. [21]

    A robust adaptive model predictive control framework for nonlinear uncertain systems,

    J. K ¨ohler, P. K ¨otting, R. Soloperto, F. Allg ¨ower, and M. A. M ¨uller, “A robust adaptive model predictive control framework for nonlinear uncertain systems,”International Journal of Robust and Nonlinear Control, vol. 31, no. 18, pp. 8725–8749, 2021

    2021

  18. [22]

    Certainty-equivalent adaptive mpc for uncertain nonlinear systems,

    J. K ¨ohler, “Certainty-equivalent adaptive mpc for uncertain nonlinear systems,”arXiv preprint arXiv:2603.17843, 2026

    2026 arXiv

  19. [23]

    Robust adaptive mpc using control contraction metrics,

    A. Sasfi, M. N. Zeilinger, and J. K ¨ohler, “Robust adaptive mpc using control contraction metrics,”Automatica, vol. 155, p. 111169, 2023

    2023

  20. [24]

    Dual adaptive mpc using an exact set-membership reformulation,

    A. Parsi, D. Liu, A. Iannelli, and R. S. Smith, “Dual adaptive mpc using an exact set-membership reformulation,”IFAC-PapersOnLine, vol. 56, no. 2, pp. 8457–8463, 2023

    2023

  21. [25]

    Robust tube-based model predictive control with koopman operators,

    X. Zhang, W. Pan, R. Scattolini, S. Yu, and X. Xu, “Robust tube-based model predictive control with koopman operators,”Automatica, vol. 137, p. 110114, 2022

    2022

  22. [26]

    Data-driven mpc with terminal conditions in the koopman frame- work,

    K. Worthmann, R. Str ¨asser, M. Schaller, J. Berberich, and F. Allg¨ower, “Data-driven mpc with terminal conditions in the koopman frame- work,”arXiv preprint arXiv:2408.12457, 2024

    2024 arXiv

  23. [27]

    Stability of data-driven koopman mpc with terminal conditions,

    I. Schimperna, L. Bold, J. K ¨ohler, K. Worthmann, and L. Magni, “Stability of data-driven koopman mpc with terminal conditions,” arXiv preprint arXiv:2511.21248, 2025

    2025 arXiv

  24. [28]

    Ljung,System identification toolbox: User’s guide

    L. Ljung,System identification toolbox: User’s guide. MathWorks Incorporated Natick, MA, USA, 1995

    1995

  25. [29]

    Rates of convergence for empirical processes of stationary mixing sequences,

    B. Yu, “Rates of convergence for empirical processes of stationary mixing sequences,”The Annals of Probability, pp. 94–116, 1994

    1994

  26. [30]

    Enlarging the domain of attraction of mpc controllers,

    D. Limon, T. Alamo, and E. F. Camacho, “Enlarging the domain of attraction of mpc controllers,”Automatica, vol. 41, no. 4, pp. 629–635, 2005

    2005

  27. [31]

    Data-driven mpc with stability guarantees using extended dynamic mode decomposi- tion,

    L. Bold, L. Gr ¨une, M. Schaller, and K. Worthmann, “Data-driven mpc with stability guarantees using extended dynamic mode decomposi- tion,”IEEE Transactions on Automatic Control, 2024

    2024

  28. [32]

    Learning without mixing: Towards a sharp analysis of linear system identification,

    M. Simchowitz, H. Mania, S. Tu, M. I. Jordan, and B. Recht, “Learning without mixing: Towards a sharp analysis of linear system identification,” inConference On Learning Theory. PMLR, 2018, pp. 439–473

    2018

  29. [33]

    Finite time identification in unstable linear systems,

    M. K. S. Faradonbeh, A. Tewari, and G. Michailidis, “Finite time identification in unstable linear systems,”Automatica, vol. 96, pp. 342– 353, 2018

    2018

  30. [34]

    Near optimal finite time identification of arbitrary linear dynamical systems,

    T. Sarkar and A. Rakhlin, “Near optimal finite time identification of arbitrary linear dynamical systems,” inInternational Conference on Machine Learning. PMLR, 2019, pp. 5610–5618

    2019

  31. [36]

    Non-asymptotic identification of lti systems from a single trajectory,

    S. Oymak and N. Ozay, “Non-asymptotic identification of lti systems from a single trajectory,” in2019 American control conference (ACC). IEEE, 2019, pp. 5655–5661

    2019

  32. [37]

    Non-asymptotic identification of linear dynami- cal systems using multiple trajectories,

    Y . Zheng and N. Li, “Non-asymptotic identification of linear dynami- cal systems using multiple trajectories,”IEEE Control Systems Letters, vol. 5, no. 5, pp. 1693–1698, 2020

    2020

  33. [38]

    Active learning for identification of linear dynamical systems,

    A. Wagenmaker and K. Jamieson, “Active learning for identification of linear dynamical systems,” inConference on Learning Theory. PMLR, 2020, pp. 3487–3582

    2020

  34. [39]

    On the sample complexity of the linear quadratic regulator,

    S. Dean, H. Mania, N. Matni, B. Recht, and S. Tu, “On the sample complexity of the linear quadratic regulator,”Foundations of Compu- tational Mathematics, vol. 20, no. 4, pp. 633–679, 2020

    2020

  35. [41]

    A framework for stabilization of nonlin- ear sampled-data systems based on their approximate discrete-time models,

    D. Nesic and A. R. Teel, “A framework for stabilization of nonlin- ear sampled-data systems based on their approximate discrete-time models,”IEEE Transactions on automatic control, vol. 49, no. 7, pp. 1103–1122, 2004

    2004

  36. [42]

    Nmpc without terminal constraints,

    L. Gr ¨une, “Nmpc without terminal constraints,”IFAC Proceedings Volumes, vol. 45, no. 17, pp. 1–13, 2012

    2012

  37. [43]

    Vershynin,High-Dimensional Probability: An Introduction with Applications in Data Science, 2nd ed

    R. Vershynin,High-Dimensional Probability: An Introduction with Applications in Data Science, 2nd ed. Cambridge University Press, 2025

    2025

  38. [44]

    Non-asymptotic system identification for linear systems with nonlinear policies,

    Y . Li, T. Zhang, S. Das, J. Shamma, and N. Li, “Non-asymptotic system identification for linear systems with nonlinear policies,”IFAC- PapersOnLine, vol. 56, no. 2, pp. 1672–1679, 2023

    2023

  39. [45]

    On mean square boundedness of stochastic linear systems with bounded con- trols,

    D. Chatterjee, F. Ramponi, P. Hokayem, and J. Lygeros, “On mean square boundedness of stochastic linear systems with bounded con- trols,”Systems & Control Letters, vol. 61, no. 2, pp. 375–380, 2012

    2012

  40. [48]

    Model predictive control: for want of a local control lyapunov function, all is not lost,

    G. Grimm, M. J. Messina, S. E. Tuna, and A. R. Teel, “Model predictive control: for want of a local control lyapunov function, all is not lost,”IEEE Transactions on Automatic Control, vol. 50, no. 5, pp. 546–558, 2005

    2005

  41. [49]

    Formulas relating kl sta- bility estimates of discrete-time and sampled-data nonlinear systems,

    D. Ne ˇsi´c, A. R. Teel, and E. D. Sontag, “Formulas relating kl sta- bility estimates of discrete-time and sampled-data nonlinear systems,” Systems & Control Letters, vol. 38, no. 1, pp. 49–60, 1999

    1999

  42. [50]

    A smooth lyapunov function from a class-estimateinvolving two positive semidefinite functions,

    A. R. Teel and L. Praly, “A smooth lyapunov function from a class-estimateinvolving two positive semidefinite functions,”ESAIM: Control, Optimisation and Calculus of Variations, vol. 5, pp. 313–367, 2000

    2000

  43. [51]

    C. D. Aliprantis and K. C. Border,Infinite dimensional analysis: a hitchhiker’s guide. Springer, 2006

    2006

  44. [52]

    Lipschitz continuity for constrained processes,

    W. W. Hager, “Lipschitz continuity for constrained processes,”SIAM Journal on Control and Optimization, vol. 17, no. 3, pp. 321–338, 1979

    1979

  45. [53]

    M. W. Hirsch, S. Smale, and R. L. Devaney,Differential equations, dynamical systems, and an introduction to chaos. Academic press, 2013. APPENDIX PROOF OFCOROLLARY1 From (22), one has Q+λmax(R)∥R† ∗Aκ∥2 Q ≤1 + ξ 2 .More- over, from (21),p 2 = exp (−2σ min(R∗)(umax −C))≤ ξ 2(1+...

    2013

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.