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REVIEW 3 major objections 4 minor 42 references

DATA-DRIVEN PRONTO: a Model-free Solution for Numerical Optimal Control

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Pith's one-line read DATA-DRIVEN PRONTO solves nonlinear optimal control with no model by probing trajectories and estimating local linearizations from data, then proves the iterates converge to a neighbourhood of the optimum whose radius shrinks with the…

desk verdict A useful model-free extension of PRONTO with a serious uniformity gap in the proof, but absolutely worth a real referee. read the letter →

arxiv 2506.15465 v1 pith:OLVGOKAP submitted 2025-06-18 eess.SY cs.SY

Marco Borghesi , Lorenzo Sforni , Giuseppe Notarstefano This is my paper · ORCID
classification eess.SYcs.SY
keywords data-drivenoptimalcontrolmodel-freePRONTOprojectionoperatortrajectorylinearizationleast-squaresidentificationultimateboundednessiterativelearning
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 claims that a finite-horizon optimal control problem for an unknown nonlinear dynamical system can be solved without a model, by repeatedly probing the system and estimating local linearizations from data. The proposed algorithm, DATA-DRIVEN PRONTO, wraps the projection-operator method PRONTO: at each iteration it perturbs the current trajectory with small dither signals in closed loop, fits time-varying affine dynamics to the measured responses, and solves an LQR subproblem with those estimated dynamics to get a descent direction. The main theorem states that if the system and cost are twice differentiable, the cost Hessian is positive definite, and an identifiability condition holds near the optimum, then for small enough dithers and a sufficiently good initial trajectory the iterates are uniformly ultimately bounded in a ball around an isolated optimum, with the ultimate radius a strictly increasing function of the dither amplitudes that goes to zero as the dithers shrink. A careful reader would care because this turns a model-free repeated-task setting, usually handled by iterative learning control or ad-hoc exploration, into an optimization algorithm with explicit local convergence guarantees and a tunable accuracy-exploration trade-off.

What carries the argument

The load-bearing mechanism is the projection operator $P$ from PRONTO, implemented by a $C^2$ tracking controller $\pi(\alpha_t,\mu_t,x_t,t)$ that satisfies $\pi(\alpha,\mu,\alpha,t)=\mu$ and therefore leaves true trajectories unchanged; it maps any tentative curve onto the trajectory manifold $\mathcal{T}$. Around this operator, the algorithm builds three smooth maps: $\Delta XU$ (closed-loop dither response, zero at zero dithers), $\Delta AB$ (Jacobian estimation error, linearly bounded on the well-conditioned set $F_M$), and $\Delta\zeta$ (descent error, zero for exact Jacobians). Their composition $\Delta\zeta(\eta,\Delta AB(\eta,\Delta XU(\eta,d_x,d_u)))$ is what turns dither amplitude into a bounded perturbation of the known-stable PRONTO update, and Assumption 4—the existence of well-conditioned dither batches near the optimum—is what keeps that composition defined. The named conclusion is Theorem 1, with the least-squares estimator of the time-varying linearizations as the concrete identification step.

What would settle it

Run the pendubot experiment with the uniform dithers used in the paper and, at each iteration, compute the condition number of $[\Delta X_t;\Delta U_t][\Delta X_t;\Delta U_t]^\top$ for each $t$; if convergence to the claimed ball persists even when the data batches never satisfy the bound $M$, then Assumption 4 is not necessary for the observed behaviour. Conversely, exhibit any system, initial trajectory, and dither schedule satisfying Assumption 4 whose iterates leave the ball of radius $b(\delta_x,\delta_u)$ infinitely often, which would refute Theorem 1.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that the model-based PRONTO descent step can be replaced by a data-driven one whose error vanishes smoothly with the exploration amplitude. The proof identifies the algorithm as a perturbed version of the projection-operator update: with $\eta_k=(x_k,u_k)$ the current trajectory, the true descent direction solves the LQR subproblem using the exact Jacobians $A_t=\nabla_1 f^\top$, $B_t=\nabla_2 f^\top$, while DATA-DRIVEN PRONTO solves the same subproblem with the least-squares estimates $\hat A_t,\hat B_t$ obtained from dither experiments. Lemma 5 bounds the Jacobian estimation error linearly by the size of the data batches when the batches are well conditioned, and Lemma 6 shows the resulting descent error $\hat\zeta-\zeta$ is a $C^1$ function that is zero when the Jacobians are exact; composing these bounds gives $\|\hat\zeta-\zeta\|\le pgr\|(d_x,d_u)\|$. Theorem 1 then follows from a practical-stability result for discrete-time systems under bounded nonvanishing perturbations: starting near an isolated optimum, the iterates satisfy $\|\eta_k-\eta^\star\|\le b(\delta_x,\delta_u)$ for all $k\ge N$, with $b$ strictly increasing and $b(0,0)=0$.

Load-bearing premise

The load-bearing premise is that near the optimum, arbitrarily small exploration dithers can always be chosen so that the resulting data batches are well-conditioned at every time step, a condition the paper characterizes rigorously only for linear time-invariant systems and does not verify numerically.

Editorial extensions

If this is right

  • Any task that can be repeated from the same initial condition becomes solvable without a model: run at least $n+m$ perturbed closed-loop experiments per iteration, fit local linearizations, and iterate.
  • The ultimate suboptimality is user-tunable: shrinking the dither bounds $\delta_x,\delta_u$ drives the asymptotic error bound $b(\delta_x,\delta_u)$ to zero, so accuracy is limited only by how well-conditioned tiny explorations remain.
  • Because only local first-order approximations are identified, systematic global model-parametrization errors are avoided; the method is model-free by construction and can use partial model knowledge only to improve the tracking controller.
  • The analysis unifies iterative learning control with numerical optimization, since the closed-loop exploration is guided by a tracking controller and the result is a descent method on the trajectory manifold.

Reading between the lines

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

  • A practical way to test whether Assumption 4 is necessary is to record the condition number of $[\Delta X_t;\Delta U_t][\Delta X_t;\Delta U_t]^\top$ during the pendubot runs; the paper's experiment uses uniform dithers without checking it, so persistent convergence despite violations would show the assumption is sufficient but not necessary.
  • The same error-composition argument suggests a data-driven constrained variant could be built by estimating constraint Jacobians alongside the dynamics, extending the result beyond unconstrained stage costs.
  • When the tracking controller itself is learned from the same estimated linearizations, as in the numerical example, the exploration policy and the descent direction are coupled; the theory does not explicitly model that coupling, so controller quality could affect the effective conditioning more than the dither size alone.
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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

3 major / 4 minor

Summary. This paper proposes DATA-DRIVEN PRONTO, an extension of the model-based PRONTO algorithm for finite-horizon nonlinear optimal control when the dynamics are unknown but experiments can be performed from a fixed initial condition. At each iteration, L closed-loop trajectories are collected by adding small dithers around the current nominal trajectory through a tracking controller; the resulting data batches are used to least-squares identify time-varying linearizations (16), which replace the exact Jacobians in the LQR subproblem (17). The main result, Theorem 1, claims that under smoothness assumptions (Assumptions 1-3) and a well-posed identification assumption (Assumption 4), the iterates are locally uniformly ultimately bounded around an isolated optimum, with an ultimate bound b(δ_x,δ_u) that is strictly increasing in the dither amplitudes and vanishes as they tend to zero. The proof proceeds by viewing the algorithm as a perturbed PRONTO iteration and bounding the descent-direction error through a chain of lemmas. A pendubot swing-up example compares the data-driven algorithm with model-based PRONTO and shows that the distance to the optimum decreases as the dither amplitude is halved.

Significance. If the main theorem and its proof can be repaired, this is a worthwhile contribution to data-driven numerical optimal control. The paper gives a clean algorithmic template, a self-contained proof structure, and a nontrivial numerical demonstration on an underactuated mechanical system. The central idea of estimating trajectory-dependent linearizations from closed-loop dither experiments, rather than fitting a global model, is attractive and is clearly contrasted with iterative learning control and derivative-free optimization. The claimed result is falsifiable: the numerical experiment tests the qualitative prediction that suboptimality decreases with dither amplitude, and no fitted constants enter the statement of Theorem 1. However, as detailed below, the proof currently has a load-bearing gap in the uniformity argument, and the numerical section does not verify the theorem's standing assumptions; both need to be addressed before the conclusions can be accepted.

major comments (3)
  1. [Section IV-C, Eqs. (41)-(50)] The proof's uniformity step is not valid because the admissible set H×D' defined in (45) is not compact. The set Δ^{-1}_{XU}(F^T_M) is open since F_M is open by Lemma 4, and the balls B_{σ'}(η*) and B_{δ'_{xu}} are open; moreover, under Assumption 4 this set contains dithers of arbitrarily small norm, whose data matrices accumulate at Δ=0, while 0∉F^T_M. Lemma 5 only supplies a constant g(H,K) for compact K⊂F^T_M, and Lemma 6's δ_AB(η) is used only on a compact neighborhood of η*. Consequently, the constants r, g, p in (42), (43), (46), and (50) are not shown to exist uniformly over all dithers satisfying (22), and the vanishing ultimate bound b(δ_x,δ_u) in (23) is not established by the written argument. This is the load-bearing step of Theorem 1 and must be repaired, for example by a separate small-dither argument exploiting the conical structure of F_M and the bound ∥Δt^†∥=1/σ_min(Δt) with κ(ΔtΔt^T)≤M.
  2. [Appendix VII-E, Eqs. (86)-(90)] The identity o_η(Δt)Δt^T(ΔtΔt^T)^{-1}=o_η(ΔtΔt^T)(ΔtΔt^T)^{-1} is dimensionally inconsistent: o_η(Δt) is an n×L matrix while ΔtΔt^T is (n+m)×(n+m), and the little-o of a product is not defined by this expression. The desired estimate can be obtained directly from (83) using ∥Δt^†∥=1/σ_min(Δt) and the conditioning condition κ(ΔtΔt^T)≤M, but the derivation as written is invalid. This matters because Lemma 5's bound (37) feeds directly into the descent-error bound (46) and hence into the final bound (50).
  3. [Section V, Steps L1 and numerical setup] The numerical experiment does not verify the hypotheses of Theorem 1. Assumption 4 requires that for every dither bound there exist L dither sequences such that (22) holds; the experiment draws d_u∼U(0,δ_u) and d_x∼U(−δ_x,δ_x) but does not report the condition numbers of [ΔX_t;ΔU_t][ΔX_t;ΔU_t]^T. In addition, Section V-A states that the controller π is redesigned at each iteration from the estimated linearizations, so π is not the fixed C^2 map required by Assumption 3 and the projection operator P changes between iterations, whereas the stability analysis in Lemmas 1-2 treats P as fixed. Thus the experiment demonstrates a plausible qualitative trend but does not instantiate the theorem's setting.
minor comments (4)
  1. [Section IV-C, Eq. (48)] The inequality ∥(d^k_x,d^k_u)∥≤T(δ_u+δ_x) is not correct for L>1. From the definition in (29), the stack has nTL+mTL entries, so its Euclidean norm is bounded by √(TL)(√n δ_x+√m δ_u), not by T(δ_u+δ_x). The subsequent choice (51) can be repaired by replacing the factor T with a dimension-dependent constant, so this is not fatal, but the displayed bound should be corrected.
  2. [Notation, Section I] The notation for open versus closed balls is confusing: the text defines B_p(¯x) as the open ball but then says 'we use B_p(¯x) when the ball is closed.' This ambiguity directly affects the compactness claims in (40), (41), and (45), where open balls are intersected with open preimages and declared compact.
  3. [Throughout] There are several typographical issues: 'demostrate' in Section V, 'novanishing' in reference [42], and the repeated 'subj.to' formatting in the optimization problems. These should be corrected in a final revision.
  4. [Section III-A, Assumption 4 and Remark 3] Assumption 4 is very strong: it postulates well-conditioned identification for arbitrarily small dither amplitudes, not merely for sufficiently rich excitation, and Remark 3 concedes that it is only well characterized for linear time-invariant systems. The paper would be strengthened by a discussion of how the condition might be checked online or relaxed for nonlinear systems, even if only in a local approximate sense.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1 follows from stated assumptions and self-contained lemmas; the sole same-author citation is peripheral.

full rationale

The derivation chain of Theorem 1 is self-contained given Assumptions 1-4. Lemma 3 establishes smoothness and ΔXU(η,0,0)=0; Lemma 5 bounds identification error by the data norm on compact well-conditioned sets; Lemma 6 gives smooth dependence of the descent error on estimation error; Lemma 2 supplies local UUB for the perturbed PRONTO update via an external stability theorem; and the proof chains these into (50), δ_ζ = pgrT(δ_u+δ_x), so the ultimate radius is a class-K function of the user-chosen dither amplitudes. No parameter is fitted to data and then renamed a prediction; Assumption 4 is an explicit existence/well-conditioning hypothesis, not the conclusion. The pendubot study tests a qualitative theoretical prediction (suboptimality shrinks as dither amplitude decreases) rather than tuning constants to match the theory. The same-author citation [37] appears only in Remark 3 as a comment that Assumption 4 is well characterized for LTI systems; it is not used in the proof of Theorem 1, so it is not load-bearing. The skeptical objection about noncompact domains and uniform bounds in Lemma 5 and Step I of the proof is a potential mathematical correctness gap, not a circularity, and does not raise the circularity score.

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

The paper introduces no new physical entities, forces, or conserved quantities; DATA-DRIVEN PRONTO is a numerical algorithm, not a new ontological postulate. The free parameters are tuning knobs of the algorithm, not fitted constants. The axioms are the smoothness, convexity, controller-existence, and identification-conditioning assumptions that the convergence theorem rests on.

free parameters (3)
  • Dither bounds δ_x, δ_u = δ_x=0.01, δ_u=0.1 in the experiments
    User-defined exploration amplitudes. They are not fitted to data; the theorem holds for any sufficiently small values and the ultimate bound grows with them. Listed because the numerical validation depends on these choices.
  • Number of experiments L = L=6
    Algorithm parameter, chosen by hand; must satisfy L≥n+m (5 in the example) for the pseudoinverse (16) to be well-posed.
  • Step size γ = γ=1
    Algorithm step size; the theorem requires γ∈(0,γ*], with γ* existing by Lemma 2 but never computed.
assumptions (7)
  • domain assumption Assumption 1: dynamics f and cost ℓ are twice continuously differentiable.
    Used throughout to justify Taylor remainders and smoothness of ΔXU, ΔAB, and Δζ in Lemmas 3, 5, and 6.
  • domain assumption Assumption 2: ∇²ℓ(x,u)>0 for all state-input sequences.
    Ensures the LQR problems (12) and (17) are strictly convex; typical for a user-defined cost.
  • domain assumption Assumption 3: existence of a C² tracking controller π with π(α,µ,α,t)=µ.
    Defines the projection operator P; enters Lemma 1 (LES of PRONTO) and Lemma 3 (smoothness of data batches). For an unknown nonlinear system, supplying such a stabilizing tracking law is itself nontrivial.
  • ad hoc to paper Assumption 4: well-posed identification, existence of well-conditioned dither batches near the optimum.
    Postulates that arbitrarily small dithers yield full-row-rank data batches with κ≤M near the optimum. This is the load-bearing premise for (16) to be a valid estimate; Remark 3 notes the condition is only well characterized for LTI systems.
  • domain assumption Noiseless state measurements.
    Section II states that states are measured noiselessly; least-squares identification (16) and the error bounds (83) ignore measurement noise.
  • standard math External stability theorem [42, Thm. 2.7].
    Used in the proof of Lemma 2 to obtain uniform ultimate boundedness of the perturbed discrete-time system.
  • standard math PRONTO local exponential stability, Lemma 1.
    Proved in Appendix VII-A using results from Diehl-Gros [1, Thm. 4.2] and Hauser [35]; the proof includes two 'It can be shown' steps that delegate to the literature.

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Pith. "Pith review of DATA-DRIVEN PRONTO: a Model-free Solution for Numerical Optimal Control." pith.science (2026). https://pith.science/paper/OLVGOKAP

@misc{pith2026250615465,
  author       = {Pith},
  title        = {Pith review of: DATA-DRIVEN PRONTO: a Model-free Solution for Numerical Optimal Control},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OLVGOKAP}},
  note         = {Machine review of arXiv:2506.15465}
}
read the original abstract

This article addresses the problem of data-driven numerical optimal control for unknown nonlinear systems. In our scenario, we suppose to have the possibility of performing multiple experiments (or simulations) on the system. Experiments are performed by relying on a data-driven tracking controller able to steer the system towards a desired reference. Our proposed DATA-DRIVEN PRONTO algorithm iteratively refines a tentative solution of the optimal control problem by computing an approximate descent direction via a local trajectory perturbation. At each iteration, multiple trajectories are gathered by perturbing the current trajectory with a suitable dither signal, and then used to obtain a data-driven, time-varying linearization. The exploration is guided by the tracking controller, so that perturbed trajectories are obtained in closed loop. We show local convergence of DATA-DRIVEN PRONTO to a ball about an isolated optimal solution, whose radius depends on the amplitude of the dither signal. We corroborate the theoretical results by applying it to an underactuated robot.

Figures

Figures reproduced from arXiv: 2506.15465 by the authors.

Figure 1
Figure 1. The pendubot. M(q)  q¨1 q¨2  + (C(q, q˙) + F)  q˙1 q˙2  + G(q) =  u 0  , (53) [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. In blue the reference curves for the states and the input. In red, the result of [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Evolution of PRONTO (in red) and DATA-DRIVEN PRONTO (in blue). The results, showed in [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Distance from the optimum depending on the dither amplitude. [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]

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Works this paper leans on

42 extracted references · 32 canonical work pages

  1. [31]

    Hardware-in-the-loop iterative optimal feedback control without model-based future prediction,

    Y . Chen and D. J. Braun, “Hardware-in-the-loop iterative optimal feedback control without model-based future prediction,”IEEE Transactions on Robotics, vol. 35, no. 6, pp. 1419–1434, 2019

    work page 2019

  2. [1]

    Numerical optimal control,

    M. Diehl and S. Gros, “Numerical optimal control,”Optimization in Engineering Center (OPTEC), 2011

    work page 2011

  3. [2]

    A. E. Bryson,Applied optimal control: optimization, estimation and control. Routledge, 2018

    2018

  4. [3]

    Nocedal and S

    J. Nocedal and S. J. Wright,Numerical optimization. Springer, 1999

    1999

  5. [4]

    D. P. Bertsekas,Nonlinear Programming. Athena Scientific, 1999

    work page 1999

  6. [5]

    A survey of iterative learning control,

    D. A. Bristow, M. Tharayil, and A. G. Alleyne, “A survey of iterative learning control,”IEEE Control Systems Magazine, vol. 26, no. 3, pp. 96–114, 2006

    2006

  7. [6]

    Iterative learning control: Brief survey and categorization,

    H.-S. Ahn, Y . Chen, and K. L. Moore, “Iterative learning control: Brief survey and categorization,”IEEE Transactions on Systems, Man, and Cybernetics, Part C (Applications and Reviews), vol. 37, no. 6, pp. 1099–1121, 2007

    work page 2007

  8. [7]

    Zero-order optimization-based iterative learning control,

    K. Baumg ¨artner and M. Diehl, “Zero-order optimization-based iterative learning control,” in2020 59th IEEE Conference on Decision and Control (CDC). IEEE, 2020, pp. 3751–3757

    work page 2020

Show all 42 references
  1. [8]

    Local convergence analysis of damping for zero-order optimization-based iterative learning control,

    ——, “Local convergence analysis of damping for zero-order optimization-based iterative learning control,” in2023 European Control Conference (ECC). IEEE, 2023, pp. 1–6

    2023

  2. [9]

    R. E. Bellman and S. E. Dreyfus,Applied dynamic programming. Princeton university press, 2015

    2015

  3. [10]

    R. S. Sutton and A. G. Barto,Reinforcement Learning: An Introduction. The MIT Press, 2018

    2018

  4. [11]

    The linear programming approach to approximate dynamic programming,

    D. P. De Farias and B. Van Roy, “The linear programming approach to approximate dynamic programming,”Operations Research, vol. 51, no. 6, pp. 850–865, 2003

    2003

  5. [12]

    AI Pontryagin or how artificial neural networks learn to control dynamical systems,

    L. B ¨ottcher, N. Antulov-Fantulin, and T. Asikis, “AI Pontryagin or how artificial neural networks learn to control dynamical systems,” Nature Communications, vol. 13, no. 1, p. 333, 2022

    2022

  6. [13]

    Reinforcement learning with model-based feedforward inputs for robotic table tennis,

    H. Ma, D. B ¨uchler, B. Sch ¨olkopf, and M. Muehlebach, “Reinforcement learning with model-based feedforward inputs for robotic table tennis,”Autonomous Robots, vol. 47, no. 8, pp. 1387–1403, 2023

    2023

  7. [14]

    A Pontryagin perspective on reinforcement learning,

    O. Eberhard, C. Vernade, and M. Muehlebach, “A Pontryagin perspective on reinforcement learning,”arXiv preprint arXiv:2405.18100, 2024

    2024 arXiv

  8. [15]

    A general safety framework for learning-based control in uncertain robotic systems,

    J. F. Fisac, A. K. Akametalu, M. N. Zeilinger, S. Kaynama, J. Gillula, and C. J. Tomlin, “A general safety framework for learning-based control in uncertain robotic systems,”IEEE Transactions on Automatic Control, vol. 64, no. 7, pp. 2737–2752, 2018

    2018

  9. [16]

    Safe reinforcement learning using robust MPC,

    M. Zanon and S. Gros, “Safe reinforcement learning using robust MPC,”IEEE Transactions on Automatic Control, vol. 66, no. 8, pp. 3638–3652, 2020

    2020

  10. [17]

    Safe learning in robotics: From learning-based control to safe reinforcement learning,

    L. Brunke, M. Greeff, A. W. Hall, Z. Yuan, S. Zhou, J. Panerati, and A. P. Schoellig, “Safe learning in robotics: From learning-based control to safe reinforcement learning,”Annual Review of Control, Robotics, and Autonomous Systems, vol. 5, no. 1, pp. 411–444, 2022. 22

    2022

  11. [18]

    Active learning of discrete-time dynamics for uncertainty-aware model predictive control,

    A. Saviolo, J. Frey, A. Rathod, M. Diehl, and G. Loianno, “Active learning of discrete-time dynamics for uncertainty-aware model predictive control,”IEEE Transactions on Robotics, vol. 40, pp. 1273–1291, 2023

    2023

  12. [19]

    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), 2019, pp. 307–312

    2019

  13. [20]

    Data-driven model predictive control with stability and robustness guarantees,

    J. Berberich, J. K ¨ohler, M. A. M ¨uller, and F. Allg¨ower, “Data-driven model predictive control with stability and robustness guarantees,” IEEE Transactions on Automatic Control, vol. 66, no. 4, pp. 1702–1717, 2020

    2020

  14. [21]

    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 Systems, vol. 3, no. 1, pp. 269–296, 2020

    2020

  15. [22]

    Safe trajectory tracking in uncertain environments,

    I. Batkovic, M. Ali, P. Falcone, and M. Zanon, “Safe trajectory tracking in uncertain environments,”IEEE Transactions on Automatic Control, vol. 68, no. 7, pp. 4204–4217, 2022

    2022

  16. [23]

    K. B. Ariyur and M. Krstic,Real-time optimization by extremum-seeking control. John Wiley & Sons, 2003

    2003

  17. [24]

    Two decades of blackbox optimization applications,

    S. Alarie, C. Audet, A. E. Gheribi, M. Kokkolaras, and S. Le Digabel, “Two decades of blackbox optimization applications,”EURO Journal on Computational Optimization, vol. 9, p. 100011, 2021

    2021

  18. [25]

    A. R. Conn, K. Scheinberg, and L. N. Vicente,Introduction to derivative-free optimization. SIAM, 2009

    2009

  19. [26]

    Derivative-free optimization: a review of algorithms and comparison of software implementations,

    L. M. Rios and N. V . Sahinidis, “Derivative-free optimization: a review of algorithms and comparison of software implementations,” Journal of Global Optimization, vol. 56, no. 3, pp. 1247–1293, 2013

    2013

  20. [27]

    Simulation optimization: a review of algorithms and applications,

    S. Amaran, N. V . Sahinidis, B. Sharda, and S. J. Bury, “Simulation optimization: a review of algorithms and applications,”Annals of Operations Research, vol. 240, pp. 351–380, 2016

    2016

  21. [28]

    On the use of set membership theory for global optimization of black-box functions,

    L. Sabug, F. Ruiz, and L. Fagiano, “On the use of set membership theory for global optimization of black-box functions,” in2020 59th IEEE Conference on Decision and Control (CDC). IEEE, 2020, pp. 3586–3591

    2020

  22. [29]

    SMGO: A set membership approach to data-driven global optimization,

    L. Sabug Jr, F. Ruiz, and L. Fagiano, “SMGO: A set membership approach to data-driven global optimization,”Automatica, vol. 133, p. 109890, 2021

    2021

  23. [30]

    Direct control design using a set membership-based black-box optimization approach,

    R. Galbiati, L. Sabug, F. Ruiz, and L. Fagiano, “Direct control design using a set membership-based black-box optimization approach,” in2022 IEEE Conference on Control Technology and Applications (CCTA). IEEE, 2022, pp. 1259–1264

    2022

  24. [32]

    Toward data-driven optimal control: A systematic review of the landscape,

    K. Prag, M. Woolway, and T. Celik, “Toward data-driven optimal control: A systematic review of the landscape,”IEEE Access, vol. 10, pp. 32 190–32 212, 2022

    2022

  25. [33]

    Learning controllers for nonlinear systems from data,

    C. De Persis and P. Tesi, “Learning controllers for nonlinear systems from data,”Annual Reviews in Control, vol. 56

  26. [34]

    A tour of reinforcement learning: The view from continuous control,

    B. Recht, “A tour of reinforcement learning: The view from continuous control,”Annual Review of Control, Robotics, and Autonomous Systems, vol. 2, pp. 253–279, 2019

    2019

  27. [35]

    A projection operator approach to the optimization of trajectory functionals,

    J. Hauser, “A projection operator approach to the optimization of trajectory functionals,”IFAC Proceedings Volumes, vol. 35, no. 1, pp. 377–382, 2002

    2002

  28. [36]

    A note on persistency of excitation,

    J. C. Willems, P. Rapisarda, I. Markovsky, and B. L. De Moor, “A note on persistency of excitation,”Systems & Control Letters, vol. 54, no. 4, pp. 325–329, 2005

    2005

  29. [37]

    On sufficient richness for linear time-invariant systems,

    M. Borghesi, S. Baroncini, G. Carnevale, A. Bosso, and G. Notarstefano, “On sufficient richness for linear time-invariant systems,” arXiv preprint arXiv:2502.04062, 2025

    2025 arXiv

  30. [38]

    The trajectory manifold of a nonlinear control system,

    J. Hauser and D. G. Meyer, “The trajectory manifold of a nonlinear control system,” in37th IEEE Conference on Decision and Control (CDC), vol. 1. IEEE, 1998, pp. 1034–1039

    1998

  31. [39]

    Siciliano, L

    B. Siciliano, L. Sciavicco, L. Villani, and G. Oriolo,Force control. Springer, 2009

    2009

  32. [40]

    Hybrid control of the pendubot,

    M. Zhang and T.-J. Tarn, “Hybrid control of the pendubot,”IEEE/ASME Transactions on Mechatronics, vol. 7, no. 1, pp. 79–86, 2002

    2002

  33. [41]

    W. M. Haddad and V . Chellaboina,Nonlinear dynamical systems and control: a Lyapunov-based approach. Princeton university press, 2008

    2008

  34. [42]

    Stability of discrete nonlinear systems under novanishing perturbations: application to a nonlinear model-matching problem,

    C. Cruz-Hern ´andez, J. Alvarez-Gallegos, and R. Castro-Linares, “Stability of discrete nonlinear systems under novanishing perturbations: application to a nonlinear model-matching problem,”IMA Journal of Mathematical Control and Information, vol. 16, no. 1, pp. 23–41, 1999

    1999

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