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arxiv: 2511.10473 · v2 · submitted 2025-11-13 · 🧮 math.OC

Riccati-ZORO: An efficient algorithm for heuristic online optimization of internal feedback laws in robust and stochastic model predictive control

Florian Messerer , Yunfan Gao , Jonathan Frey , Moritz Diehl This is my paper

Pith reviewed 2026-05-17 22:18 UTC · model grok-4.3

classification 🧮 math.OC
keywords tube model predictive controlrobust MPCstochastic MPCfeedback law optimizationellipsoidal tubesonline optimizationconstraint tightening
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The pith

Riccati-ZORO splits tube MPC into nominal trajectory and feedback subproblems to drop complexity from O(n_x^6) to O(n_x^3).

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

The paper develops an algorithm that jointly tunes the main trajectory and the internal feedback law used to shape uncertainty tubes in robust and stochastic model predictive control. It alternates between solving a standard nominal optimal control problem with fixed backoff margins and solving an unconstrained problem that adjusts the feedback gains while the trajectory is held fixed. A heuristic cost term guides the gain update by emphasizing backoff reduction near active constraints. For ellipsoidal tubes and linear state feedback, the split removes the sixth-order dependence on state dimension that appears in a direct joint formulation.

Core claim

By alternating a nominal OCP with fixed backoffs and an unconstrained tube OCP that optimizes feedback gains under a proximity-based uncertainty cost, the algorithm optimizes both the reference trajectory and the internal feedback law at the computational cost of a standard nominal problem.

What carries the argument

Alternation between a nominal OCP with fixed backoffs and an unconstrained tube OCP that optimizes linear feedback gains for ellipsoidal tubes using a heuristic cost based on nominal-trajectory proximity to constraints.

If this is right

  • Tube-based robust and stochastic MPC can be solved online with state-dimension scaling identical to that of a nominal OCP.
  • The same decomposition structure applies to both ellipsoidal robust tubes and stochastic tubes under linear feedback.
  • Two open implementations exist: one in CasADi for prototyping and one integrated into the acados solver for high-performance use.

Where Pith is reading between the lines

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

  • The approach may extend to nonlinear dynamics if the tube propagation step can be approximated by a similar Riccati-like update.
  • The heuristic cost could be replaced by a learned surrogate without changing the outer alternation structure.
  • The resulting feedback gains could be warm-started from one time step to the next to further reduce per-iteration work.

Load-bearing premise

The heuristic that reduces backoffs according to how close the nominal trajectory comes to constraints will produce feedback gains that are useful for the overall tube problem.

What would settle it

A benchmark where the closed-loop performance or feasibility margin obtained with the optimized gains is no better than the margin obtained with fixed-gain tubes on the same nominal trajectory.

Figures

Figures reproduced from arXiv: 2511.10473 by Florian Messerer, Jonathan Frey, Moritz Diehl, Yunfan Gao.

Figure 1
Figure 1. Figure 1: Schematic structure of Riccati-ZORO for k = N − 1, . . . , 0, followed, for k = 0, . . . , N − 1, by the forward Lyapunov recursion P ⋆ 0 = P¯ 0, (13a) P ⋆ k+1 = (Ak + BkK⋆ k )P ⋆ k (Ak + BkK⋆ k ) ⊤ + W˜ k. (13b) IV. RICCATI-ZORO In this section we describe the algorithm Riccati-ZORO, which is aimed at problems of structure (1), but optimizes the feedback gains based on a heuristic cost instead of exactly.… view at source ↗
Figure 2
Figure 2. Figure 2: Visualization of the constraint-adaptive uncertainty weighting [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Stationary points of different algorithms for the towing kite problem, with the objective of maximizing the average thrusting force [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Computation times for the hanging chain of masses benchmark. [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
read the original abstract

We present Riccati-ZORO, an algorithm for tube-based optimal control problems (OCP). Tube OCPs predict a tube of trajectories in order to capture predictive uncertainty. The tube induces a constraint tightening via additional backoff terms. This backoff can significantly affect the performance, and thus implicitly defines a cost of uncertainty. Optimizing the feedback law used to predict the tube can significantly reduce the backoffs, but its online computation is challenging. Riccati-ZORO jointly optimizes the nominal trajectory and uncertainty tube based on a heuristic uncertainty cost design. The algorithm alternates between two subproblems: (i) a nominal OCP with fixed backoffs, (ii) an unconstrained tube OCP, which optimizes the feedback gains for a fixed nominal trajectory. For the tube optimization, we propose a cost function informed by the proximity of the nominal trajectory to constraints, prioritizing reduction of the corresponding backoffs. These ideas are developed for ellipsoidal tubes under linear state feedback. In this case, the decomposition into the two subproblems yields a substantial reduction of the computational complexity with respect to the state dimension from $\mathcal{O}(n_x^6)$ to $\mathcal{O}(n_x^3)$, i.e., the complexity of a nominal OCP. We investigate the algorithm in numerical experiments, and provide two open-source implementations: a prototyping version in CasADi and a high-performance implementation integrated into the acados OCP solver.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper introduces Riccati-ZORO, an alternating optimization algorithm for tube-based OCPs in robust and stochastic MPC. It splits the problem into (i) a nominal OCP with fixed backoffs and (ii) an unconstrained tube OCP that optimizes linear feedback gains for ellipsoidal tubes using a heuristic cost based on proximity of the nominal trajectory to constraints. The central claim is that this decomposition reduces online complexity w.r.t. state dimension from O(n_x^6) to O(n_x^3) while providing open-source implementations in CasADi and acados together with numerical experiments.

Significance. If the O(n_x^3) scaling is realized, the method would make online feedback-law optimization practical for higher-dimensional tube MPC, potentially reducing conservatism from uncertainty backoffs. The explicit labeling of the cost as heuristic, the provision of reproducible code, and the numerical validation are positive elements that support applicability in the field.

major comments (1)
  1. [§4.2] §4.2 (tube subproblem formulation): the manuscript states that the unconstrained tube OCP is solved via Riccati recursion after incorporating the proximity-based heuristic into the quadratic cost. However, the exact algebraic expression for the heuristic term (distance/proximity metric and its weighting) and the resulting gradient/Hessian assembly must be shown explicitly to confirm that no O(n_x^4) or higher operations arise when the number of active constraints varies; otherwise the claimed reduction to nominal-OCP complexity is not guaranteed.
minor comments (2)
  1. [Abstract] The abstract and introduction would benefit from a one-sentence statement of the precise form of the heuristic cost (e.g., whether it is a weighted sum of squared distances or a different metric) to allow readers to immediately assess the complexity claim.
  2. [§5] Numerical results section: include a dedicated plot or table isolating wall-clock time of the tube subproblem versus n_x (with fixed constraint count) to directly support the O(n_x^3) scaling independent of the nominal OCP solve.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive review and the recommendation for minor revision. The single major comment is addressed below with a commitment to expand the relevant section.

read point-by-point responses

  1. Referee: [§4.2] §4.2 (tube subproblem formulation): the manuscript states that the unconstrained tube OCP is solved via Riccati recursion after incorporating the proximity-based heuristic into the quadratic cost. However, the exact algebraic expression for the heuristic term (distance/proximity metric and its weighting) and the resulting gradient/Hessian assembly must be shown explicitly to confirm that no O(n_x^4) or higher operations arise when the number of active constraints varies; otherwise the claimed reduction to nominal-OCP complexity is not guaranteed.

    Authors: We agree that the current exposition in §4.2 would benefit from greater algebraic detail. In the revised manuscript we will insert the explicit form of the heuristic term: a quadratic penalty whose distance metric is the signed Euclidean distance from the nominal state to each active constraint hyperplane, scaled by a positive weighting matrix that is diagonal and state-dependent only through the fixed nominal trajectory. Because this term is quadratic in the decision variables (the feedback gains) and is added directly to the existing quadratic cost, its gradient is linear and its Hessian is constant with respect to the gains. Assembly of the total gradient and Hessian therefore requires only a fixed number of matrix-vector products whose leading term remains O(n_x^3) per Riccati step; no combinatorial enumeration of active sets or higher-order tensor operations is introduced, regardless of how many constraints are active at the nominal trajectory. The revised text will include the precise expressions and a short complexity count confirming that the overall per-iteration cost stays O(n_x^3). revision: yes

Circularity Check

0 steps flagged

No significant circularity; complexity reduction follows from standard Riccati properties

full rationale

The paper's derivation of the O(n_x^3) scaling rests on splitting the tube OCP into a nominal problem with fixed backoffs and an unconstrained tube subproblem solved via Riccati recursion for ellipsoidal tubes under linear feedback. This reduction is a direct consequence of the known cubic complexity of Riccati-based solvers for unconstrained linear-quadratic problems once the backoffs are fixed and the tube problem is rendered unconstrained; the proximity-based heuristic cost is introduced explicitly as a design choice to prioritize backoff reduction and does not enter the complexity analysis as a fitted or self-referential quantity. No load-bearing self-citations, self-definitional steps, or ansatz smuggling appear in the provided derivation chain, and the central claim remains independent of any internal fitting or renaming of results.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The approach rests on the domain assumption that ellipsoidal tubes under linear state feedback are an adequate uncertainty model and on a heuristic cost whose parameters are not derived from first principles.

free parameters (1)
  • heuristic cost weights
    The cost function that prioritizes backoff reduction near constraints is described as heuristic and therefore likely contains tunable weights chosen by the designer.
axioms (1)
  • domain assumption Ellipsoidal tubes under linear state feedback suffice to capture predictive uncertainty for the target class of problems
    Invoked when the complexity reduction to O(n_x^3) is claimed.

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Works this paper leans on

40 extracted references · 40 canonical work pages

  1. [1]

    J. B. Rawlings, D. Q. Mayne, and M. M. Diehl,Model Predictive Control: Theory, Computation, and Design, 2nd ed. Nob Hill, 2017

    work page 2017

  2. [2]

    Kouvaritakis and M

    B. Kouvaritakis and M. Cannon,Model Predictive Control: Classical, Robust and Stochastic, 1st ed. Cham Heidelberg New York Dordrecht London: Springer, Dec. 2015

    work page 2015

  3. [3]

    Min-max feedback model predictive control for constrained linear systems,

    P. O. M. Scokaert and D. Q. Mayne, “Min-max feedback model predictive control for constrained linear systems,”IEEE Transactions on Automatic Control, vol. 43, pp. 1136–1142, 1998

    work page 1998

  4. [4]

    The scenario approach to robust control design,

    G. C. Calafiore and M. C. Campi, “The scenario approach to robust control design,”IEEE Trans. Automat. Control, 2006

    work page 2006

  5. [5]

    Handling uncertainty in economic nonlinear model predictive control: A comparative case study,

    S. Lucia, J. A. E. Andersson, H. Brandt, M. Diehl, and S. Engell, “Handling uncertainty in economic nonlinear model predictive control: A comparative case study,”Journal of Process Control, vol. 24, pp. 1247–1259, 2014

    work page 2014

  6. [6]

    Robust model predictive control using tubes,

    W. Langson, S. R. I. Chryssochoos, and D. Q. Mayne, “Robust model predictive control using tubes,”Automatica, vol. 40, no. 1, pp. 125–133, 2004

    work page 2004

  7. [7]

    Tube-based robust nonlinear model predictive control,

    D. Mayne, E. Kerrigan, E. J. van Wyk, and P. Falugi, “Tube-based robust nonlinear model predictive control,”International Journal of Robust and Nonlinear Control, vol. 21, pp. 1341–1353, 2011

    work page 2011

  8. [8]

    S. V . Rakovi´c,Robust Model Predictive Control. London: Springer London, 2019, pp. 1–11

    work page 2019

  9. [9]

    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

    work page 2021

  10. [10]

    Open-loop and closed-loop robust optimal control of batch processes using distributional and worst-case analysis,

    Z. Nagy and R. Braatz, “Open-loop and closed-loop robust optimal control of batch processes using distributional and worst-case analysis,” Journal of Process Control, vol. 14, pp. 411–422, 2004

    work page 2004

  11. [11]

    An efficient algorithm for tube-based robust nonlinear optimal control with optimal linear feedback,

    F. Messerer and M. Diehl, “An efficient algorithm for tube-based robust nonlinear optimal control with optimal linear feedback,” inProceedings of the IEEE Conference on Decision and Control (CDC), 2021

    work page 2021

  12. [12]

    Robust optimal control for nonlinear systems with parametric uncertainties via system level synthesis,

    A. P. Leeman, J. Sieber, S. Bennani, and M. N. Zeilinger, “Robust optimal control for nonlinear systems with parametric uncertainties via system level synthesis,”Proceedings of the IEEE Conference on Decision and Control (CDC), 2023

    work page 2023

  13. [13]

    Robust model predictive control for nonlinear discrete-time systems using iterative time-varying constraint tightening,

    D. D. Leister and J. P. Koeln, “Robust model predictive control for nonlinear discrete-time systems using iterative time-varying constraint tightening,”Proceedings of the American Control Conference (ACC), 2025

    work page 2025

  14. [14]

    Sensitivity- aware model predictive control for robots with parametric uncertainty,

    T. Belvedere, M. Cognetti, G. Oriolo, and P. R. Giordano, “Sensitivity- aware model predictive control for robots with parametric uncertainty,” IEEE Transactions on Robotics, vol. 41, 2025

    work page 2025

  15. [15]

    Robustified time-optimal point-to-point motion planning and control under uncertainty,

    S. Zhang and J. Swevers, “Robustified time-optimal point-to-point motion planning and control under uncertainty,”European Journal of Control, 2025

    work page 2025

  16. [16]

    An approximation technique for robust nonlinear optimization,

    M. Diehl, H. Bock, and E. Kostina, “An approximation technique for robust nonlinear optimization,”Mathematical Programming, vol. 107, pp. 213–230, 2006. [Online]. Available: http://www.springerlink.com/ content/y737861045873673/

    work page 2006

  17. [17]

    Robustness and stability optimization of power generating kite systems in a periodic pumping mode,

    B. Houska and M. Diehl, “Robustness and stability optimization of power generating kite systems in a periodic pumping mode,” in Proceedings of the IEEE Multi-conference on Systems and Control (MSC), Yokohama, Japan, 2010, pp. 2172–2177. [Online]. Available: http://ieeexplore.ieee.org/xpls/abs all.jsp?arnumber=5611288

    work page 2010

  18. [18]

    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,” inProceedings of the IEEE Conference on Decision and Control (CDC), 2018, pp. 6059–6066

    work page 2018

  19. [19]

    Robust optimization of dynamic systems,

    B. Houska, “Robust optimization of dynamic systems,” Ph.D. disserta- tion, KU Leuven, 2011, (ISBN: 978-94-6018-394-2)

    work page 2011

  20. [20]

    Joint synthesis of trajectory and controlled invariant funnel for discrete-time systems with locally lipschitz nonlinearities,

    T. Kim, P. Elango, and B. A c ¸ıkmes ¸e, “Joint synthesis of trajectory and controlled invariant funnel for discrete-time systems with locally lipschitz nonlinearities,”Int J Robust Nonlinear Control, vol. 34, 2024

    work page 2024

  21. [21]

    Cautious model predicitive control using gaussian process regression,

    L. Hewing, J. Kabzan, and M. N. Zeilinger, “Cautious model predicitive control using gaussian process regression,”IEEE Transaction on Control Systems Technology, vol. 28, no. 6, pp. 2736–2743, 2020

    work page 2020

  22. [22]

    An approach to output-feedback MPC of stochastic linear discrete-time systems,

    M. Farina, L. Giulioni, L. Magni, and R. Scattolini, “An approach to output-feedback MPC of stochastic linear discrete-time systems,” Automatica, vol. 55, pp. 140–149, 2015

    work page 2015

  23. [23]

    A dual-control effect preserving formulation for nonlinear output-feedback stochastic model predictive control with constraints,

    F. Messerer, K. Baumg ¨artner, and M. Diehl, “A dual-control effect preserving formulation for nonlinear output-feedback stochastic model predictive control with constraints,”IEEE Control Systems Letters, vol. 7, no. 1171–1176, 2023

    work page 2023

  24. [24]

    Optimization over state feedback policies for robust control with constraints,

    P. J. Goulart, E. C. Kerrigan, and J. M. Maciejowski, “Optimization over state feedback policies for robust control with constraints,”Automatica, vol. 42, pp. 523–533, 2006

    work page 2006

  25. [25]

    Robust nonlinear optimal control via system level synthesis,

    A. P. Leeman, J. K ¨ohler, A. Zanelli, S. Bennani, and M. N. Zeilinger, “Robust nonlinear optimal control via system level synthesis,”IEEE TAC, vol. 70, no. 7, 2025

    work page 2025

  26. [26]

    Zero-order robust nonlinear model predictive control with ellipsoidal uncertainty sets,

    A. Zanelli, J. Frey, F. Messerer, and M. Diehl, “Zero-order robust nonlinear model predictive control with ellipsoidal uncertainty sets,” Proceedings of the IFAC Conference on Nonlinear Model Predictive Control (NMPC), 2021

    work page 2021

  27. [27]

    Inexact Adjoint-based SQP Algorithm for Real-Time Stochastic nonlinear MPC,

    X. Feng, S. D. Cairano, and R. Quirynen, “Inexact Adjoint-based SQP Algorithm for Real-Time Stochastic nonlinear MPC,” inProceedings of the IFAC World Congress, 2020

    work page 2020

  28. [28]

    acados – a modular open-source framework for fast embedded optimal control,

    R. Verschueren, G. Frison, D. Kouzoupis, J. Frey, N. van Duijkeren, A. Zanelli, B. Novoselnik, T. Albin, R. Quirynen, and M. Diehl, “acados – a modular open-source framework for fast embedded optimal control,” Mathematical Programming Computation, pp. 147–183, Oct 2021

    work page 2021

  29. [29]

    Efficient zero-order robust optimization for real-time model predic- tive control with acados,

    J. Frey, Y . Gao, F. Messerer, A. Lahr, M. N. Zeilinger, and M. Diehl, “Efficient zero-order robust optimization for real-time model predic- tive control with acados,” inProceedings of the European Control Conference (ECC), 2024

    work page 2024

  30. [30]

    Collision-free motion planning for mobile robots by zero-order robust optimization-based mpc,

    Y . Gao, F. Messerer, J. Frey, N. van Duijkeren, and M. Diehl, “Collision-free motion planning for mobile robots by zero-order robust optimization-based mpc,” inProceedings of the European Control Conference (ECC), 2023

    work page 2023

  31. [31]

    Zero-order optimization for Gaussian process-based model predictive control,

    A. Lahr, A. Zanelli, A. Carron, and M. N. Zeilinger, “Zero-order optimization for Gaussian process-based model predictive control,” European Journal of Control, vol. 74, p. 100862, 2023

    work page 2023

  32. [32]

    L4acados: Learning-based models for acados, applied to gaussian process-based predictive control.arXiv preprint arXiv:2411.19258, 2024

    A. Lahr, J. N ¨af, K. P. Wabersich, J. Frey, P. Siehl, A. Carron, M. Diehl, and M. N. Zeilinger, “L4acados: Learning-based models for acados, applied to Gaussian process-based predictive control,” Nov. 2024, arXiv: 2411.19258 tex.pubstate: prepublished

    work page arXiv 2024

  33. [33]

    Ro- bustified time-optimal collision-free motion planning for autonomous mobile robots under disturbance conditions,

    S. Zhang, M. Bos, B. Vandewal, W. Decr´e, J. Gillis, and J. Swevers, “Ro- bustified time-optimal collision-free motion planning for autonomous mobile robots under disturbance conditions,”IEEE International Conference on Robotics and Automation (ICRA), 2024

    work page 2024

  34. [34]

    Fast system level synthesis: Robust model predictive control using riccati recursions,

    A. P. Leeman, J. K ¨ohler, F. Messerer, A. Lahr, M. Diehl, and M. N. Zeilinger, “Fast system level synthesis: Robust model predictive control using riccati recursions,”IFAC-PapersOnLine, vol. 58, no. 18, pp. 173–180, 2024. [Online]. Available: https: //www.sciencedirect.com/science/article/pii/S240589632401406X

    work page 2024

  35. [35]

    CasADi – a software framework for nonlinear optimization and optimal control,

    J. A. E. Andersson, J. Gillis, G. Horn, J. B. Rawlings, and M. Diehl, “CasADi – a software framework for nonlinear optimization and optimal control,”Mathematical Programming Computation, vol. 11, no. 1, pp. 1–36, 2019

    work page 2019

  36. [36]

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

    A. Mesbah, “Stochastic model predictive control: An overview and perspectives for future research,”IEEE Control Systems Magazine, vol. 36, no. 6, pp. 30–44, 2016

    work page 2016

  37. [37]

    R. F. Stengel,Optimal Control and Estimation. Dover, 1986

    work page 1986

  38. [38]

    Nocedal and S

    J. Nocedal and S. J. Wright,Numerical Optimization, 2nd ed., ser. Springer Series in Operations Research and Financial Engineering. Springer, 2006

    work page 2006

  39. [39]

    BLAS- FEO: Basic linear algebra subroutines for embedded optimization,

    G. Frison, D. Kouzoupis, T. Sartor, A. Zanelli, and M. Diehl, “BLAS- FEO: Basic linear algebra subroutines for embedded optimization,” ACM Transactions on Mathematical Software (TOMS), vol. 44, no. 4, pp. 42:1–42:30, 2018

    work page 2018

  40. [40]

    On the implementation of an interior- point filter line-search algorithm for large-scale nonlinear programming,

    A. W ¨achter and L. T. Biegler, “On the implementation of an interior- point filter line-search algorithm for large-scale nonlinear programming,” Mathematical Programming, vol. 106, no. 1, pp. 25–57, 2006

    work page 2006