pith. sign in

arxiv: 2606.01194 · v1 · pith:YYLUYCRUnew · submitted 2026-05-31 · 📡 eess.SY · cs.SY

Data-Driven Min-Max MPC with Integral Quadratic Constraints

Yifan Xie , Julian Berberich , Frank Allg\"ower This is my paper

Pith reviewed 2026-06-28 16:32 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords data-driven controlmodel predictive controlintegral quadratic constraintssemidefinite programmingmin-max optimizationrobust stabilityset-membership identification
0
0 comments X

The pith

Data-driven min-max MPC characterizes unknown dynamics via input-state data and IQCs to derive SDPs that bound worst-case costs and guarantee closed-loop stability.

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

The paper develops a synthesis procedure for model predictive control of systems whose dynamics are not fully known but can be bounded using collected input-state trajectories together with integral quadratic constraints on the nonlinearities. From this set-membership description two semidefinite programs are formulated whose solutions yield a state-feedback law minimizing an upper bound on the worst-case infinite-horizon cost. Iterative application of these programs produces the control law, after which exponential stability of the closed-loop system and satisfaction of input and state constraints are established by standard Lyapunov arguments. The approach therefore supplies rigorous performance and safety certificates for nonlinear plants when only finite data and IQC knowledge are available.

Core claim

By representing the unknown system matrices through a set-membership description that incorporates the measured input-state data and the supplied integral quadratic constraints, two SDPs can be derived that minimize an upper bound on the worst-case cost taken over all admissible dynamics and uncertainties; iterative solution of these programs furnishes a state-feedback controller for which the resulting closed-loop trajectories are exponentially stable and remain inside the prescribed input and state constraint sets.

What carries the argument

Set-membership representation of the unknown system matrices from input-state data and IQCs, which is then used to cast the min-max MPC synthesis as two SDPs solved iteratively.

If this is right

  • The closed-loop system is exponentially stable.
  • Input and state constraints are satisfied for all admissible uncertainties.
  • An upper bound on the infinite-horizon worst-case cost is minimized by the obtained state-feedback law.
  • The synthesis applies directly to plants whose nonlinearities are captured by IQCs.

Where Pith is reading between the lines

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

  • Updating the data set online would allow the SDPs to be re-solved periodically, potentially tightening the cost bound as more measurements arrive.
  • The same data-IQC set-membership could be reused inside other robust synthesis frameworks such as tube MPC or set-theoretic methods.
  • The numerical conditioning of the two SDPs may limit applicability to high-dimensional systems unless further structure is exploited.

Load-bearing premise

The true system matrices are contained inside the set defined by the observed data and the given IQCs.

What would settle it

A simulation or experiment in which the actual plant matrices lie outside the data-plus-IQC set and the closed-loop system either loses exponential stability or violates the state or input constraints.

Figures

Figures reproduced from arXiv: 2606.01194 by Frank Allg\"ower, Julian Berberich, Yifan Xie.

Figure 1
Figure 1. Figure 1: Closed loop state and input trajectories under the [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
read the original abstract

Data-driven control of nonlinear systems with rigorous guarantees is a challenging control problem. Integral quadratic constraints (IQCs) provide a powerful framework for modeling nonlinearities. This paper presents a data-driven min-max model predictive control (MPC) synthesis method for unknown systems subject to (nonlinear) uncertainties using the IQC framework. The unknown system matrices are characterized by a set-membership representation using the input-state data and the knowledge of the IQCs. We derive two semidefinite programs (SDPs) that minimize an upper bound on the worst-case cost over all possible system dynamics and uncertainties. By iteratively solving these SDPs, the proposed state-feedback control law is obtained. We further prove that the resulting closed-loop system is exponentially stable and satisfies the input and state constraints. A numerical example demonstrates the validity of the proposed method.

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

3 major / 2 minor

Summary. The manuscript develops a data-driven min-max MPC synthesis for unknown systems with nonlinear uncertainties modeled by IQCs. Unknown (A,B) matrices are enclosed in a set-membership set Σ constructed from finite input-state data together with known IQC multipliers. Two SDPs are derived that minimize an upper bound on the worst-case infinite-horizon cost over all admissible dynamics in Σ; an iterative procedure produces a state-feedback law. The authors prove that the resulting closed-loop system is exponentially stable and satisfies input/state constraints for every member of Σ. A numerical example is included.

Significance. If the set-membership construction is shown to contain the true plant under stated assumptions and the SDP derivations are correct, the work supplies a concrete route to robust, data-driven MPC with explicit stability and constraint certificates. The combination of IQC multipliers with min-max SDP synthesis is a natural extension of existing robust-control toolboxes and could be useful for systems where only noisy trajectory data are available.

major comments (3)
  1. [§2–3] §2–3 (set-membership construction): The definition of Σ from the data matrix and the IQC multipliers must be accompanied by an explicit statement of the conditions (e.g., persistency of excitation, noise bounds, exactness of the IQC sector) that guarantee the true (A,B) lies inside Σ. Without this, the subsequent stability and constraint claims, which are proved only for elements of Σ, do not transfer to the physical plant.
  2. [§4] Derivation of the two SDPs (presumably §4): The manuscript must exhibit the precise LMI or SDP formulations (including the decision variables, the cost upper-bound expression, and the constraint matrices) together with the algebraic steps that convert the min-max problem over Σ into the reported SDPs. The abstract asserts the existence of these SDPs but supplies no equations; the central claim cannot be verified without them.
  3. [§5] Stability proof (presumably §5): The Lyapunov decrease argument is stated to hold for every member of Σ. The proof must explicitly invoke the fact that the obtained feedback renders a common quadratic Lyapunov function decrease inside Σ; if any step relies on an additional assumption not implied by the data-plus-IQC construction, that assumption must be listed.
minor comments (2)
  1. Notation for the data matrices and the IQC multipliers should be introduced once and used consistently; several symbols appear to be redefined between the abstract and the body.
  2. The numerical example should report the rank of the data matrix and the size of the resulting Σ to allow the reader to assess how conservative the enclosure is.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the conditions and derivations needed for the guarantees. We address each major comment below, indicating the revisions planned for the manuscript.

read point-by-point responses

  1. Referee: [§2–3] §2–3 (set-membership construction): The definition of Σ from the data matrix and the IQC multipliers must be accompanied by an explicit statement of the conditions (e.g., persistency of excitation, noise bounds, exactness of the IQC sector) that guarantee the true (A,B) lies inside Σ. Without this, the subsequent stability and constraint claims, which are proved only for elements of Σ, do not transfer to the physical plant.

    Authors: We agree that an explicit statement of the conditions is required to ensure the true plant lies in Σ. In the revised manuscript, Section 2 will include a dedicated paragraph listing the assumptions on persistency of excitation of the data, bounds on measurement noise, and the validity/exactness of the known IQC multipliers that together guarantee containment of the true (A,B) in Σ. revision: yes

  2. Referee: [§4] Derivation of the two SDPs (presumably §4): The manuscript must exhibit the precise LMI or SDP formulations (including the decision variables, the cost upper-bound expression, and the constraint matrices) together with the algebraic steps that convert the min-max problem over Σ into the reported SDPs. The abstract asserts the existence of these SDPs but supplies no equations; the central claim cannot be verified without them.

    Authors: Section 4 already presents the two SDPs with decision variables, the upper-bound cost expression, and the LMI constraint matrices, along with the conversion from the min-max problem. To address the concern, we will add an expanded subsection with the full algebraic derivation steps and explicit LMI forms for improved verifiability. The abstract is not expected to contain equations. revision: partial

  3. Referee: [§5] Stability proof (presumably §5): The Lyapunov decrease argument is stated to hold for every member of Σ. The proof must explicitly invoke the fact that the obtained feedback renders a common quadratic Lyapunov function decrease inside Σ; if any step relies on an additional assumption not implied by the data-plus-IQC construction, that assumption must be listed.

    Authors: The proof in Section 5 relies on a common quadratic Lyapunov function whose decrease is enforced for all members of Σ via the min-max SDP. We will revise the proof text to explicitly invoke this common Lyapunov property and list any additional assumptions (such as the existence and validity of the IQC multipliers) that are used beyond the data-plus-IQC set construction. revision: yes

Circularity Check

0 steps flagged

No circularity: set-membership and min-max SDPs are independent of fitted outputs

full rationale

The paper defines an uncertainty set Σ from input-state data plus supplied IQC multipliers, then solves SDPs that optimize a worst-case cost upper bound over Σ. The exponential stability and constraint proofs are shown to hold for every element of Σ. No equation reduces a claimed prediction or bound to a quantity fitted from the same data by construction, and no load-bearing premise rests on a self-citation chain. The derivation is self-contained against the external data and IQC assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract alone; therefore the ledger is necessarily incomplete and many background assumptions from robust control theory are not visible.

axioms (1)
  • standard math Standard properties of integral quadratic constraints and semidefinite programming hold for the system class considered.
    Invoked implicitly when the abstract states that SDPs can be derived and solved to obtain the controller.

pith-pipeline@v0.9.1-grok · 5669 in / 1332 out tokens · 27248 ms · 2026-06-28T16:32:28.342699+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

19 extracted references · 1 canonical work pages · 1 internal anchor

  1. [1]

    Berberich, J., K¨ ohler, J., M¨ uller, M.A., and Allg¨ ower, F. (2021). Data-driven model predictive control with stability and robustness guarantees. IEEE Transactions on Automatic Control , 66(4), 1702–1717

    2021

  2. [2]

    Berberich, J., Scherer, C.W., and Allg¨ ower, F. (2023). Combining prior knowledge and data for robust con- troller design. IEEE Transactions on Automatic Con- trol, 68(8), 4618–4633

    2023

  3. [3]

    Coulson, J., Lygeros, J., and D¨ orfler, F. (2019). Data- enabled predictive control: In the shallows of the DeePC. In Proc. European Control Conference (ECC), 307–312

    2019

  4. [4]

    Fry, J.M., Farhood, M., and Seiler, P. (2017). IQC-based robustness analysis of discrete-time linear time-varying systems. International Journal of Robust and Nonlinear Control, 27(16), 3135–3157

    2017

  5. [5]

    Gupta, V., Klauser, E., and Karimi, A. (2026). Non- parametric IQC multipliers in data-driven robust con- troller synthesis. Automatica, 183, 112608

    2026

  6. [6]

    Hu, B., Lacerda, M.J., and Seiler, P. (2017). Robustness analysis of uncertain discrete-time systems with dissi- pation inequalities and integral quadratic constraints. International Journal of Robust and Nonlinear Control , 27(11), 1940–1962

    2017

  7. [7]

    Kothare, M.V., Balakrishnan, V., and Morari, M. (1996). Robust constrained model predictive control using linear matrix inequalities. Automatica, 32(10), 1361–1379

    1996

  8. [8]

    Luppi, A., De Persis, C., and Tesi, P. (2022). On data- driven stabilization of systems with nonlinearities satis- fying quadratic constraints. Systems & Control Letters , 163, 105206

    2022

  9. [9]

    and D¨ orfler, F

    Markovsky, I. and D¨ orfler, F. (2021). Behavioral systems theory in data-driven analysis, signal processing, and control. Annual Reviews in Control , 52, 42–64

    2021

  10. [10]

    Markovsky, I., Huang, L., and D¨ orfler, F. (2023). Data- driven control based on the behavioral approach: From theory to applications in power systems. IEEE Control Systems Magazine , 43(5), 28–68

    2023

  11. [11]

    Martin, T., Sch¨ on, T.B., and Allg¨ ower, F. (2023). Guaran- tees for data-driven control of nonlinear systems using semidefinite programming: A survey. Annual Reviews in Control , 56, 100911

    2023

  12. [12]

    Nguyen, H.H., Friedel, M., and Findeisen, R. (2023). LMI-based data-driven robust model predictive control. IF AC-PapersOnLine, 56(2), 4783–4788

    2023

  13. [13]

    and Weiland, S

    Scherer, C.W. and Weiland, S. (2000). Linear matrix inequalities in control . New York : Spring-Verlag, 3 edition. van Waarde, H.J., Eising, J., Camlibel, M.K., and Trentel- man, H.L. (2023). The informativity approach: To data- driven analysis and control. IEEE Control Systems Magazine, 43(6), 32–66

    2000

  14. [14]

    and Scherer, C.W

    Veenman, J. and Scherer, C.W. (2014). IQC-synthesis with general dynamic multipliers. International Journal of Robust and Nonlinear Control , 24(17), 3027–3056

    2014

  15. [15]

    Veenman, J., Scherer, C.W., and K¨ oro˘ glu, H. (2016). Robust stability and performance analysis based on integral quadratic constraints. European Journal of Control, 31, 1–32

    2016

  16. [16]

    Verhoek, C., Eising, J., D¨ orfler, F., and T´ oth, R. (2024). Decoupling parameter variation from noise: Biquadratic lyapunov forms in data-driven LPV control. In 63rd IEEE Conference on Decision and Control (CDC) , 6761–6766

    2024

  17. [17]

    Wang, Y., Liu, Q., and Shang, C. (2025). On min-max robust data-driven predictive control considering non- unique solutions to behavioral representation. arXiv preprint arXiv:2501.17350

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  18. [18]

    Xie, Y., Berberich, J., and Allg¨ ower, F. (2024). Data- driven min-max MPC for linear systems. In 2024 American Control Conference (ACC) , 3184–3189

    2024

  19. [19]

    Xie, Y., Berberich, J., and Allg¨ ower, F. (2026). Data- driven min-max MPC for linear systems: Robustness and adaptation. Automatica, 183, 112612

    2026