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arxiv: 2604.18432 · v1 · submitted 2026-04-20 · 🧮 math.OC

Real-Time Algorithms for Model Predictive Control of Hybrid Dynamical Systems

Armin Nurkanovi\'c , Anton Pozharskiy , Moritz Diehl This is my paper

Pith reviewed 2026-05-10 04:12 UTC · model grok-4.3

classification 🧮 math.OC
keywords model predictive controlhybrid dynamical systemscomplementarity constraintsreal-time algorithmsdiscontinuous feedbackMPCCpiecewise affine approximationrobotic manipulation
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The pith

Real-time MPC for hybrid systems solves quadratic programs with complementarity constraints to approximate discontinuous feedback laws with bounded error.

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

The paper develops real-time algorithms for model predictive control of nonlinear hybrid dynamical systems modeled as dynamical complementarity systems. These systems lead to optimal control problems formulated as mathematical programs with complementarity constraints whose solutions are discontinuous. Standard nonlinear programming approaches can become infeasible at switching times. The new schemes instead solve a quadratic program with complementarity constraints at each sample to produce local piecewise affine approximations of the feedback law. The authors also establish continuity and differentiability properties of parametric MPCCs and conditions that keep the approximation error uniformly bounded despite the discontinuities, as shown in a robotic manipulation example where contact sequences are found online.

Core claim

We introduce three real-time hybrid MPC schemes whose feedback phase solves a quadratic program with complementarity constraints per sample, yielding local discontinuous piecewise affine approximations of the MPC feedback law. We derive continuity and differentiability results for parametric MPCCs, and establish conditions under which the approximation error of our new hybrid MPC algorithms remains uniformly bounded despite solution discontinuities.

What carries the argument

Quadratic program with complementarity constraints solved in the feedback phase, which produces local piecewise affine approximations to the discontinuous solution map of the underlying MPCC.

If this is right

  • The feedback phase remains feasible and computable even when the hybrid system switches between modes.
  • The approximation error of the real-time law stays uniformly bounded under the derived conditions on the MPCC solution map.
  • Contact sequences in manipulation tasks can be discovered automatically during online operation without pre-specification.
  • Standard NLP-based real-time MPC may lose feasibility at switches while the new QPCC-based schemes do not.

Where Pith is reading between the lines

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

  • The continuity results for parametric MPCCs could be reused in other control or optimization settings that encounter complementarity constraints.
  • The piecewise affine local approximations might be combined with explicit MPC techniques to reduce online computation further.
  • If the boundedness conditions fail in practice, one could add regularization or switching logic to restore stability.
  • These schemes open the door to real-time MPC on systems with many discrete modes, such as legged robots or automated vehicles.

Load-bearing premise

The hybrid systems of interest can be represented accurately as dynamical complementarity systems and the stated conditions for uniform boundedness of the approximation error hold for the applications of interest.

What would settle it

A numerical test on a hybrid system with frequent switches where the real-time feedback deviates from the offline optimal MPC solution by an amount that grows without bound as sampling time decreases or near discontinuity points.

Figures

Figures reproduced from arXiv: 2604.18432 by Anton Pozharskiy, Armin Nurkanovi\'c, Moritz Diehl.

Figure 1
Figure 1. Figure 1: Solution map of a parametric MPCC (18), and its QPCC and QP approximations at x¯ = −0.1. The QP predictor matches the QPCC as long as the complementarity active sets agree, and becomes infeasible afterwards. The QPCC predictor captures the solution jump. studied [6], [7], and several computational advances exist [8]– [10]. However, unlike in the smooth MPC case [11], there is currently no established gener… view at source ↗
Figure 3
Figure 3. Figure 3: The left and middle plots show the solution maps of BNLPIa and BNLPIb from Example 3, respectively. The red shaded area indicates where a BNLP solution is not S-stationary. The right plot shows the discontinuous solution mapping of the MPCC. For p ∈ (−1, 1) the MPCC has two isolated local minimizers, and the right plot depicts one possible selection, which tracks Ia, as long as possible. shrinking δ > 0 if… view at source ↗
Figure 4
Figure 4. Figure 4: Tracking error of SQPCC path-following schemes. Left: single￾step SQPCC tracking for different parameter increments ∆x. Right: error reduction with multiple SQPCC iterations per parameter. Accordingly, the QPCC solution map is a piecewise affine, possibly discontinuous, approximation of the MPCC solution [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: Preparation and feedback time for various MPC algorithms, and a scaled Pareto plot comparison. time of 0.1s. The optimal control problem is discretized with an implicit Euler time-stepping scheme for Lagrangian systems, cf. [4], [45]. The first two control intervals have four integrator steps, the third two, and the remaining ones have one per control interval. For closed-loop simulation, in contrast, we s… view at source ↗
read the original abstract

Model predictive control (MPC) of hybrid dynamical systems is challenging because the associated optimization problem is nonsmooth and the resulting feedback law is discontinuous. This paper develops real-time MPC algorithms for nonlinear hybrid systems modeled as dynamical complementarity systems. The resulting optimal control problems are formulated as mathematical programs with complementarity constraints (MPCCs). We show that the solution map of parametric MPCCs is discontinuous, and that standard nonlinear-programming-based approaches may become infeasible when the hybrid system switches. To address this, we introduce three real-time hybrid MPC schemes whose feedback phase solves a quadratic program with complementarity constraints per sample, yielding local discontinuous piecewise affine approximations of the MPC feedback law. Moreover, we derive continuity and differentiability results for parametric MPCCs, and establish conditions under which the approximation error of our new hybrid MPC algorithms remains uniformly bounded despite solution discontinuities. The algorithms are demonstrated on a robotic manipulation example, where contact sequences are discovered online.

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 / 3 minor

Summary. The paper develops real-time MPC algorithms for nonlinear hybrid dynamical systems modeled as dynamical complementarity systems. Optimal control problems are cast as MPCCs, which are shown to have discontinuous solution maps. Three new schemes are introduced whose feedback phase solves a QP with complementarity constraints at each sample, producing local discontinuous piecewise-affine approximations to the MPC feedback law. Continuity and differentiability results are derived for parametric MPCCs, together with conditions guaranteeing that the approximation error remains uniformly bounded despite solution discontinuities. The algorithms are illustrated on a robotic manipulation example in which contact sequences are discovered online.

Significance. If the regularity conditions hold, the work supplies a principled route to real-time feedback for hybrid systems whose mode switches produce discontinuities, with explicit error bounds that standard NLP-based MPC lacks. The derivation of continuity/differentiability properties for parametric MPCCs and the construction of three distinct real-time schemes constitute concrete technical contributions that could be useful in contact-rich robotics and other hybrid control settings.

major comments (2)
  1. [the section establishing bounded-error conditions for the hybrid MPC schemes] The uniform-boundedness claim for the approximation error (established after the continuity/differentiability results for parametric MPCCs) rests on strong regularity, non-degeneracy of the complementarity constraints, and Lipschitz continuity of the solution map away from jumps. In the robotic manipulation demonstration the contact sequence is discovered online, yet no verification is provided that these regularity conditions are satisfied at every discovered switch; only performance on the shown trajectories is reported. If any switch violates the conditions, the uniform error bound fails and the theoretical guarantee for the real-time schemes does not apply.
  2. [the algorithmic description of the three schemes] The three real-time schemes are stated to yield local discontinuous PWA approximations, but the precise relationship between the QP-with-complementarity-constraints subproblem solved at each sample and the original MPCC is not shown to preserve the derived continuity/differentiability properties when the hybrid system switches. A concrete counter-example or additional assumption would be needed to confirm that the feedback-phase QP remains well-posed and the error bound carries over.
minor comments (3)
  1. [Introduction and algorithmic section] The abstract states that three schemes are introduced, but the body does not clearly delineate their differences (e.g., warm-starting strategy, constraint linearization, or handling of active-set changes) until late in the algorithmic section; an early comparison table would improve readability.
  2. [preliminaries on MPCCs] Notation for the complementarity constraints and the parametric MPCC is introduced without an explicit list of standing assumptions (e.g., LICQ, MFCQ, or strong regularity) that are later invoked; collecting these in a single preliminary subsection would reduce cross-referencing.
  3. [numerical demonstration] The robotic example would benefit from an additional plot or table quantifying the realized approximation error across multiple random initial conditions and contact sequences, rather than only trajectory snapshots, to give visual support for the uniform-boundedness claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below, indicating planned revisions where appropriate.

read point-by-point responses

  1. Referee: [the section establishing bounded-error conditions for the hybrid MPC schemes] The uniform-boundedness claim for the approximation error (established after the continuity/differentiability results for parametric MPCCs) rests on strong regularity, non-degeneracy of the complementarity constraints, and Lipschitz continuity of the solution map away from jumps. In the robotic manipulation demonstration the contact sequence is discovered online, yet no verification is provided that these regularity conditions are satisfied at every discovered switch; only performance on the shown trajectories is reported. If any switch violates the conditions, the uniform error bound fails and the theoretical guarantee for the real-time schemes does not apply.

    Authors: We agree that explicit verification of the regularity conditions would strengthen the link between the theoretical guarantees and the numerical demonstration. The bounded-error result relies on these conditions holding at the relevant points. In the revised version, we will add a short subsection to the robotic manipulation example that examines the discovered contact sequences and confirms (or discusses the satisfaction of) strong regularity, non-degeneracy, and local Lipschitz continuity of the solution map for the reported trajectories. revision: yes

  2. Referee: [the algorithmic description of the three schemes] The three real-time schemes are stated to yield local discontinuous PWA approximations, but the precise relationship between the QP-with-complementarity-constraints subproblem solved at each sample and the original MPCC is not shown to preserve the derived continuity/differentiability properties when the hybrid system switches. A concrete counter-example or additional assumption would be needed to confirm that the feedback-phase QP remains well-posed and the error bound carries over.

    Authors: Each feedback-phase subproblem is a QPCC obtained by instantiating the parametric MPCC at the current measured state (with the complementarity constraints retained). Consequently, the continuity and differentiability results derived for parametric MPCCs apply directly to these subproblems. When a mode switch occurs, the parameter vector jumps, but a fresh QPCC is solved at the new point; the local PWA approximation is therefore rebuilt around the current operating condition. We will insert a clarifying remark (or short lemma) in the algorithmic section that makes this inheritance explicit and confirms that the uniform error bound carries over under the same regularity conditions already stated for the MPCCs. revision: yes

Circularity Check

0 steps flagged

No circularity in derivation chain; results derived from standard parametric optimization

full rationale

The paper formulates hybrid MPC as MPCCs, introduces three new real-time schemes that solve per-sample QPs with complementarity constraints to produce local discontinuous PWA approximations, and derives continuity/differentiability properties plus uniform boundedness conditions for the approximation error under explicit regularity assumptions on the solution map. These steps rely on standard results from nonlinear programming and parametric optimization rather than self-definitions, fitted inputs renamed as predictions, or load-bearing self-citations. The robotic manipulation example serves only as illustration; the theoretical claims remain independent of it and do not reduce to their inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No specific free parameters, axioms, or invented entities are identifiable from the abstract alone; the work relies on standard modeling of hybrid systems as dynamical complementarity systems.

pith-pipeline@v0.9.0 · 5462 in / 1270 out tokens · 61937 ms · 2026-05-10T04:12:28.121235+00:00 · methodology

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