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arxiv: 2605.23581 · v1 · pith:COWASYSSnew · submitted 2026-05-22 · 🧮 math.OC

Coupling optimization algorithms and monotone control systems: Suboptimal model predictive control as an operator splitting scheme

Till Preuster , Hannes Gernandt , Manuel Schaller This is my paper

Pith reviewed 2026-05-25 04:15 UTC · model grok-4.3

classification 🧮 math.OC
keywords model predictive controlmonotone dynamical systemsoperator splittingsuboptimal MPCport-Hamiltonian systemscontinuous-time optimizationcontrol systems theory
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The pith

Standard suboptimal model predictive control algorithms arise as time discretizations of coupled monotone optimizer-plant dynamics via operator splitting.

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

The paper models both the plant and the optimization routine in model predictive control as dynamical systems rather than treating the optimizer as an instantaneous map. These two systems are interconnected through a structured coupling that preserves monotonicity properties, yielding a well-posed continuous-time closed loop. Within this setting, common iterative optimization algorithms used inside MPC are recovered exactly as discrete-time approximations obtained by applying operator splitting methods to the continuous dynamics. A reader would care because the construction supplies a single dynamical-systems lens that explains why many existing suboptimal MPC schemes work and how they relate to one another.

Core claim

By representing the optimizer as a dynamical system and coupling it with the plant through a structured interconnection of (quasi-)monotone operators, the closed-loop system is well-posed, and standard suboptimal MPC algorithms correspond to time discretizations of this continuous-time dynamics using operator splitting schemes.

What carries the argument

Structured interconnection of monotone dynamical systems (including port-Hamiltonian systems) that produces a closed-loop governed by monotone operators and permits operator-splitting discretizations.

If this is right

  • Suboptimal MPC schemes inherit well-posedness from the underlying monotone operator properties.
  • Iterative optimization algorithms inside MPC become explicit time discretizations of continuous dynamics.
  • Tools from monotone operator theory apply directly to the analysis of the combined plant-optimizer loop.
  • A single continuous-time formulation unifies many existing suboptimal MPC variants.

Where Pith is reading between the lines

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

  • Different splitting methods could generate new families of MPC algorithms whose convergence rates are inherited from the continuous monotone flow.
  • The same interconnection idea might apply to other optimization-based controllers, such as economic MPC or real-time iteration schemes.
  • Stability certificates for the closed loop could be obtained from Lyapunov functions already known for monotone systems rather than from separate optimizer convergence arguments.

Load-bearing premise

Both the plant and the optimizer can be represented as quasi-monotone dynamical systems whose structured interconnection produces a well-posed closed-loop system.

What would settle it

A standard suboptimal MPC algorithm that cannot be recovered, for any choice of splitting step size, as the exact discretization of any well-posed monotone coupled optimizer-plant system.

read the original abstract

We propose a framework for suboptimal model predictive control (MPC) based on the interconnection of monotone dynamical systems, such as port-Hamiltonian systems. In contrast to classical MPC formulations, where the optimizer is treated as an instantaneous mapping, we model both the plant and the optimizer as dynamical systems and couple them through a structured interconnection. This leads to a continuous-time closed-loop formulation governed by (quasi-)monotone operators. Within this setting, we establish well-posedness of the coupled optimizer-plant dynamics and provide a unified interpretation of suboptimal MPC schemes. In particular, we reveal a direct connection between iterative optimization algorithms and dynamical control systems theory by showing that standard suboptimal MPC algorithms can be understood as time discretizations of the underlying continuous-time dynamics via operator splitting methods.

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

2 major / 0 minor

Summary. The manuscript proposes a framework for suboptimal model predictive control (MPC) based on the interconnection of monotone dynamical systems, such as port-Hamiltonian systems. Both the plant and the optimizer are modeled as dynamical systems coupled through a structured interconnection, yielding a continuous-time closed-loop system governed by (quasi-)monotone operators. The authors claim to establish well-posedness of the coupled dynamics and to provide a unified interpretation in which standard suboptimal MPC algorithms arise as time discretizations of the continuous-time dynamics via operator splitting methods.

Significance. If the modeling framework and the operator-splitting equivalence can be rigorously established, the work would offer a novel bridge between iterative optimization algorithms and monotone dynamical systems theory, potentially allowing new stability or convergence analyses for suboptimal MPC. The explicit use of port-Hamiltonian structure and monotone-operator theory is a conceptual strength when the interconnection preserves the required monotonicity properties.

major comments (2)
  1. [Abstract] Abstract (paragraph 2): the well-posedness of the coupled optimizer-plant dynamics is asserted without any theorem statement, proof sketch, or reference to an existence result for the monotone-operator differential inclusion; this is load-bearing for the entire framework.
  2. [Abstract] Abstract (final sentence): the claim that 'standard suboptimal MPC algorithms can be understood as time discretizations ... via operator splitting methods' is stated without exhibiting a concrete algorithm (e.g., projected gradient or ADMM) and the corresponding splitting operator, or showing that the discrete iteration matches the continuous flow at the claimed time step.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful comments on our manuscript. We address the major comments point by point below, clarifying where the supporting results appear in the body of the paper while keeping the abstract at an appropriate level of detail.

read point-by-point responses

  1. Referee: [Abstract] Abstract (paragraph 2): the well-posedness of the coupled optimizer-plant dynamics is asserted without any theorem statement, proof sketch, or reference to an existence result for the monotone-operator differential inclusion; this is load-bearing for the entire framework.

    Authors: The abstract is intended as a concise summary. The well-posedness result is stated and proved as Theorem 3.1 in Section 3, which establishes existence and uniqueness for the coupled differential inclusion by appealing to standard theory of maximal monotone operators (citing Brezis, 1973, and related results on quasi-monotone inclusions). A self-contained proof sketch follows the theorem statement, using the monotonicity of the interconnection and the port-Hamiltonian structure to verify the required conditions. We are happy to insert a parenthetical reference to Theorem 3.1 in the abstract if the editor and referee consider it helpful for readability. revision: partial

  2. Referee: [Abstract] Abstract (final sentence): the claim that 'standard suboptimal MPC algorithms can be understood as time discretizations ... via operator splitting methods' is stated without exhibiting a concrete algorithm (e.g., projected gradient or ADMM) and the corresponding splitting operator, or showing that the discrete iteration matches the continuous flow at the claimed time step.

    Authors: Again, the abstract summarizes the contribution at a high level. Concrete realizations are developed in Section 4. Proposition 4.1 shows that the projected-gradient iteration is exactly the forward-backward splitting discretization of the continuous-time monotone flow, with the discrete update matching the continuous trajectory at each step size h. Proposition 4.2 likewise identifies ADMM with the Douglas-Rachford splitting and verifies the exact correspondence. These derivations are fully explicit and include the splitting operators. We do not believe the abstract needs to reproduce these examples, but we can add a short illustrative sentence if space allows in a revision. revision: no

Circularity Check

0 steps flagged

No significant circularity; framework derived from external monotone-operator theory

full rationale

The paper's central contribution is a modeling framework that represents both plant and optimizer as (quasi-)monotone dynamical systems and interprets standard suboptimal MPC schemes as operator-splitting discretizations of the resulting continuous-time interconnection. This is presented as an application of existing port-Hamiltonian and monotone-operator theory rather than a derivation that reduces any claimed result to a fitted parameter or self-citation by construction. No load-bearing step equates a prediction to its own inputs; well-posedness follows from standard properties of monotone operators. The modeling choice itself is the premise being explored, not a hidden tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on domain assumptions about monotonicity rather than new free parameters or invented entities.

axioms (1)
  • domain assumption Plant and optimizer admit representations as (quasi-)monotone dynamical systems that admit a structured interconnection preserving monotonicity
    Invoked to obtain the closed-loop governed by (quasi-)monotone operators (abstract).

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