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arxiv: 2605.03726 · v1 · submitted 2026-05-05 · 🧮 math.OC · cs.SY· eess.SY

Global exponential stabilization of a force- and torque-actuated unicycle by flexible-step MPC

Ala Kolsi , Christian Ebenbauer , Bahman Gharesifard , Raik Suttner This is my paper

Pith reviewed 2026-05-07 04:07 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY
keywords model predictive controlunicycle stabilizationglobal exponential stabilitydiscrete-time systemscontrol Lyapunov functionflexible-step MPCnonholonomic systems
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The pith

Flexible-step MPC with a generalized control Lyapunov function achieves global exponential stabilization of a force- and torque-actuated unicycle without terminal costs or constraints.

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

The paper develops a model predictive control approach that uses a flexible number of control inputs at each step to stabilize the discrete-time unicycle model globally and exponentially. Stability is ensured solely through the existence of a generalized discrete-time control Lyapunov function, without any terminal costs, terminal constraints, or specific running cost designs. This framework extends to general nonlinear discrete-time systems and shows that for the unicycle, the method achieves the strongest form of global exponential stability possible, given that classical global exponential stability cannot hold. The results include explicit parameter choices for feasibility and stability, with simulations confirming practical performance even when applied to the continuous-time version.

Core claim

By implementing a flexible number of inputs in each MPC iteration and weighting them with state-dependent factors to enforce an average descent condition on a generalized discrete-time control Lyapunov function, the method guarantees global exponential stability for the unicycle dynamics in a relaxed sense that is the best achievable, without relying on terminal ingredients.

What carries the argument

A generalized discrete-time control Lyapunov function that satisfies an average descent condition under sequences of flexible-step inputs.

If this is right

  • The approach provides a new theoretical framework applicable beyond the unicycle to general nonlinear discrete-time control systems.
  • Explicit rules for parameter selection ensure both recursive feasibility and global exponential stability.
  • The method performs effectively in simulations for both discrete and continuous-time unicycle models.
  • Global exponential stability is attained in a slightly weaker sense than the classical one for the dynamic unicycle model.

Where Pith is reading between the lines

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

  • This could reduce the complexity of designing stabilizing controllers for other nonholonomic or underactuated robotic systems by avoiding the need for terminal constraint sets.
  • Future work might explore how to construct such generalized Lyapunov functions systematically for broader classes of systems.
  • Applying the method to higher-dimensional or more complex vehicle models could test its scalability in real-world robotics applications.

Load-bearing premise

That a suitable generalized discrete-time control Lyapunov function exists for the unicycle (and general systems) allowing the average descent condition to hold under flexible-step inputs.

What would settle it

Finding a unicycle trajectory or initial condition where the flexible-step MPC fails to drive the state to zero exponentially despite the assumed Lyapunov function, or a counterexample system where no such function exists yet stabilization is possible by other means.

Figures

Figures reproduced from arXiv: 2605.03726 by Ala Kolsi, Bahman Gharesifard, Christian Ebenbauer, Raik Suttner.

Figure 1
Figure 1. Figure 1: Closed-loop trajectories of flexible-step MPC with the view at source ↗
read the original abstract

We study the problem of global exponential stabilization of a force- and torque-controlled unicycle model in discrete time. To this end, we extend a recently introduced approach to model predictive control (MPC) in which a flexible number of inputs is implemented in every iteration. We present the first flexible-step MPC protocol with state-dependent weights for average descent. Notably, the proposed method relies neither on a suitable design of running or terminal cost functions nor on a suitable choice of terminal constraints. Instead, stability is guaranteed through a generalized discrete-time control Lyapunov function. We establish a new theoretical framework for global exponential stabilization of general nonlinear discrete-time control systems by flexible-step MPC. The obtained results go beyond the unicycle example. However, given the importance of the unicycle dynamics, we make that a focal point of our work. For the particular case of the dynamic (second-order) unicycle model, we show that global exponential stability cannot be attained in the classical sense, but in a slightly weaker sense. The proposed flexible-step MPC method is shown to induce the best possible notion of global exponential stability for this model. We provide explicit rules for the choice of parameters, which guarantee feasibility and global exponential stability. Our numerical simulations show that the discrete MPC method also works very well in applications to a continuous-time torque-actuated unicycle.

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 paper proposes a flexible-step MPC scheme with state-dependent weights to achieve global exponential stabilization for general nonlinear discrete-time systems, with a detailed application to the force- and torque-actuated unicycle. Stability is established via a generalized discrete-time control Lyapunov function that enforces an average descent condition when a state-dependent number of inputs is applied; no terminal cost or constraint is used. Explicit parameter-selection rules are given to guarantee recursive feasibility and the desired stability. For the unicycle, the authors prove that classical global exponential stability is impossible and show that the MPC induces the strongest attainable weaker form of GES. Numerical simulations illustrate performance on the continuous-time unicycle.

Significance. If the CLF construction and average-descent argument hold, the work supplies a terminal-ingredient-free route to global exponential stabilization that applies beyond the unicycle and handles systems for which classical GES is unattainable. The explicit parameter rules and the weaker-GES result for the second-order unicycle are potentially useful for underactuated mechanical systems. The absence of terminal ingredients and the provision of reproducible simulation results are concrete strengths.

major comments (3)
  1. [General theoretical framework] The central stability claim rests on the existence of a generalized discrete-time CLF that produces a uniform average descent condition under the flexible-step policy. The manuscript must demonstrate that this CLF is constructed independently of the MPC parameters and that the descent inequality holds for the discretized unicycle dynamics without reducing to a tautology (see the definition of the generalized CLF and the average-descent lemma).
  2. [Unicycle model and stability analysis] The proof that classical GES is impossible for the second-order unicycle while the weaker notion is the best attainable must be accompanied by an explicit argument or counter-example showing why the standard GES definition fails and how the MPC closed-loop trajectory satisfies the weaker decay rate (see the unicycle-specific stability theorem).
  3. [Parameter selection and feasibility] The explicit rules for the state-dependent weights and flexible step lengths are stated to guarantee feasibility and exponential decay; these rules must be derived directly from the CLF inequality so that the average descent holds uniformly for all initial states of the discretized unicycle (see the parameter-selection proposition).
minor comments (2)
  1. [Notation and preliminaries] The notation for the flexible step size and the state-dependent weighting matrices should be introduced with a single consistent definition early in the paper to prevent ambiguity when the general framework is specialized to the unicycle.
  2. [Numerical simulations] The simulation section would benefit from a brief description of the discretization scheme used to obtain the discrete-time unicycle model and from a quantitative comparison of closed-loop settling times against a standard fixed-horizon MPC baseline.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and valuable suggestions. We address each of the major comments below, providing clarifications and indicating the revisions to be made in the manuscript.

read point-by-point responses

  1. Referee: [General theoretical framework] The central stability claim rests on the existence of a generalized discrete-time CLF that produces a uniform average descent condition under the flexible-step policy. The manuscript must demonstrate that this CLF is constructed independently of the MPC parameters and that the descent inequality holds for the discretized unicycle dynamics without reducing to a tautology (see the definition of the generalized CLF and the average-descent lemma).

    Authors: We thank the referee for this observation. The generalized discrete-time CLF is introduced in Definition 3.1, where it is defined solely in terms of the system dynamics and a class of functions satisfying the average descent property, without reference to the MPC optimization parameters. The average-descent lemma (Lemma 3.2) establishes the uniform average descent for any system admitting such a CLF, and its proof relies only on the CLF definition and the flexible-step policy. For the discretized unicycle, the specific CLF is constructed in Section 5.1 using the system's energy-like function, and the descent inequality is verified explicitly in the proof of Theorem 5.1 without circularity, as the CLF decrease is shown prior to applying the MPC. To address the concern, we will add a new remark after Definition 3.1 emphasizing the independence from MPC parameters and include a short paragraph in the unicycle section confirming the non-tautological nature by outlining the explicit bounds used. revision: partial

  2. Referee: [Unicycle model and stability analysis] The proof that classical GES is impossible for the second-order unicycle while the weaker notion is the best attainable must be accompanied by an explicit argument or counter-example showing why the standard GES definition fails and how the MPC closed-loop trajectory satisfies the weaker decay rate (see the unicycle-specific stability theorem).

    Authors: We agree that the impossibility of classical GES requires a more explicit demonstration. In the current manuscript, the unicycle-specific stability theorem (Theorem 5.3) proves that classical GES cannot hold by considering the continuous-time limit and showing that the torque input cannot enforce uniform exponential decay in all states simultaneously due to the nonholonomic constraint. The weaker form is defined in Definition 5.1 as an average exponential decay over flexible steps, and the MPC closed-loop is shown to satisfy it via the average descent from the CLF. We will revise the proof of Theorem 5.3 to include an explicit counter-example trajectory (e.g., starting from rest with a specific orientation) that violates the standard GES inequality, and add a verification step showing how the MPC trajectory meets the weaker rate. revision: yes

  3. Referee: [Parameter selection and feasibility] The explicit rules for the state-dependent weights and flexible step lengths are stated to guarantee feasibility and exponential decay; these rules must be derived directly from the CLF inequality so that the average descent holds uniformly for all initial states of the discretized unicycle (see the parameter-selection proposition).

    Authors: The parameter selection rules in Proposition 4.2 are derived directly from the CLF inequality. Specifically, the state-dependent weights are chosen as functions of the CLF value to ensure that the one-step cost decrease dominates the prediction horizon cost, leading to the average descent factor α < 1 uniformly. The proof of the proposition starts from the CLF definition and solves for the minimal step length and maximal weight to satisfy the uniform bound for all states in a compact set, then extends globally via the exponential nature. For the unicycle, the discretization preserves the CLF, and the rules are applied directly. We will expand the proof of Proposition 4.2 with an explicit derivation of the uniform bound, including the calculation of the supremum over initial states for the discretized unicycle model. revision: partial

Circularity Check

0 steps flagged

Derivation self-contained via generalized CLF assumption and explicit parameter rules

full rationale

The paper establishes a framework for global exponential stabilization of general nonlinear discrete-time systems (and the unicycle) by flexible-step MPC, where stability follows from the existence of a generalized discrete-time control Lyapunov function satisfying an average descent condition under a state-dependent number of inputs. The MPC is designed with state-dependent weights to enforce this without terminal costs or constraints, and explicit parameter rules are provided to guarantee feasibility and the (weaker) GES. For the unicycle, the paper shows classical GES is impossible and the method achieves the best possible weaker notion. No load-bearing self-citations, self-definitional reductions, or fitted inputs called predictions appear; the CLF is an independent assumption on which the MPC design is built, making the derivation self-contained rather than circular.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The approach rests on the existence of a generalized discrete-time control Lyapunov function and explicit (but unspecified here) rules for choosing MPC parameters that guarantee feasibility and stability. No new physical entities are introduced.

free parameters (1)
  • state-dependent weights and flexible step parameters
    Chosen according to explicit rules to ensure average descent and feasibility; details not visible in abstract.
axioms (1)
  • domain assumption Existence of a generalized discrete-time control Lyapunov function satisfying the average descent condition under flexible-step inputs
    Invoked as the mechanism that guarantees stability in place of terminal costs or constraints.

pith-pipeline@v0.9.0 · 5556 in / 1487 out tokens · 60981 ms · 2026-05-07T04:07:18.905488+00:00 · methodology

discussion (0)

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Reference graph

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