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arxiv: 2604.21989 · v1 · submitted 2026-04-23 · 🧮 math.OC · cs.AI· cs.RO· cs.SY· eess.SY· math.DS

Model Predictive Control of Hybrid Dynamical Systems

Ricardo G. Sanfelice , Berk Altin This is my paper

Pith reviewed 2026-05-09 20:41 UTC · model grok-4.3

classification 🧮 math.OC cs.AIcs.ROcs.SYeess.SYmath.DS
keywords model predictive controlhybrid dynamical systemsasymptotic stabilitycontrol Lyapunov functionhybrid time domainsstage costterminal cost
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The pith

Sufficient conditions on costs and state-feedback laws guarantee asymptotic stability of a target set under hybrid model predictive control.

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

The paper formulates a model predictive control strategy tailored to hybrid dynamical systems that evolve according to both differential equations and difference equations. It derives structural properties of the resulting optimization problem and its value function when horizons are constructed over hybrid time domains. Checkable conditions are given in terms of the stage cost, terminal cost, and the existence of static state-feedback laws that together satisfy a control Lyapunov function inequality. A reader would care because many engineered systems combine continuous flows with discrete jumps, and this supplies a direct way to obtain provable stability from an optimization-based controller. If the conditions hold, the closed-loop hybrid system converges to the target set while respecting input and state constraints.

Core claim

We formulate the hybrid MPC problem using prediction and control horizons adapted to hybrid time domains. We establish that the optimization problem possesses structural properties that make its feasible set and value function well-behaved. Under the assumption that static state-feedback laws exist satisfying a control Lyapunov function condition, together with suitable properties on the stage and terminal costs, the value function decreases along closed-loop solutions and the target set is asymptotically stable.

What carries the argument

The hybrid MPC optimization problem whose feasible set and value function are shown to inherit invariance and decrease properties from the control Lyapunov function condition on the stage and terminal costs.

If this is right

  • The closed-loop system is asymptotically stable to the target set.
  • The value function of the optimization problem serves as a Lyapunov function for the hybrid closed-loop dynamics.
  • Feasibility of the MPC problem is preserved along solutions when the initial condition lies in the feasible set.
  • The approach applies to any hybrid system that can be written in hybrid equation form with the required feedback laws.
  • Examples illustrate that the conditions can be verified on concrete systems such as those with impacts or switches.

Where Pith is reading between the lines

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

  • The same cost and feedback conditions might be used to certify stability for other optimization-based controllers on hybrid systems.
  • Numerical checks of the control Lyapunov function inequality could be automated to certify candidate costs before deployment.
  • The framework suggests a route to robust versions by relaxing the exact CLF inequality to an inequality with a margin.
  • Real-time implementation would require efficient solvers for the hybrid optimization problem at each step.

Load-bearing premise

There exist static state-feedback laws that satisfy the control Lyapunov function condition together with the stated structural properties of the hybrid optimization problem.

What would settle it

A concrete hybrid system together with stage and terminal costs for which the control Lyapunov function condition holds, yet some closed-loop trajectory fails to converge to the target set.

Figures

Figures reproduced from arXiv: 2604.21989 by Berk Altin, Ricardo G. Sanfelice.

Figure 1
Figure 1. Figure 1: Hybrid prediction horizon for the case of J = 1, t0 = 2, t1 = 1, and t2 = 0. Remark 3.3 (On Particular Constructions of T ): A natural construction of T is one that independently limits the amount of flow and the number of jumps. Similar to discrete time MPC, the parameter N ∈ {1, 2, . . .} denotes the maximum number of jumps allowed in the prediction. To limit the amount of flow allowed, let δ > 0 and def… view at source ↗
Figure 2
Figure 2. Figure 2: Hybrid time domains associated with the hybrid MPC algorithm for the case of N = 4 and Nc = 2. The hybrid time instances (T1, J1), (T2, J2) and (T3, J3) at which Problem 3.5 is solved are circled. 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Hybrid time domain of the optimal trajectory associated to the hybrid time domains in [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Position and velocity trajectories of the bouncing ball controlled by hybrid MPC projected onto ordinary time t. 0 0.5 1 1.5 2 2.5 3 3.5 4 t [sec] 0 5 10 15 20 25 30 35 40 W [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Evolution of the total energy of the bouncing ball controlled by hybrid MPC projected onto ordinary time t. function. The terminal cost V is chosen as V (x) = exp(−σx2)  x ⊤ 1 exp(A⊤ f (Ts − x2))P exp(Af (Ts − x2))x1  , where σ > 0, Af =  A B 0 0  , and P is a symmetric positive definite matrix, chosen next. The matrices QC and P are chosen as follows. Given matrices A ∈ R nz × R nz and B ∈ R nz × R mz… view at source ↗
read the original abstract

The problem of controlling hybrid dynamical systems using model predictive control (MPC) is formulated and sufficient conditions for asymptotic stability of a set are provided. Hybrid dynamical systems are modeled in terms of hybrid equations, involving a differential equation and a difference equation with inputs and constraints. The proposed hybrid MPC algorithm uses a suitable prediction and control horizon construction inspired by hybrid time domains. Structural properties of the hybrid optimization problem, its feasible set, and its value function are provided. Checkable conditions to guarantee asymptotic stability of a set are provided. These conditions are given in terms of properties on the stage cost, terminal cost, and the existence of static state-feedback laws, related through a control Lyapunov function condition. Examples illustrate the results throughout the paper.

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 / 2 minor

Summary. The paper formulates a model predictive control (MPC) scheme for hybrid dynamical systems modeled via hybrid equations (flows and jumps with inputs and constraints). It introduces a prediction horizon construction based on hybrid time domains, establishes structural properties of the resulting hybrid optimization problem (including feasibility set and value function), and states sufficient conditions for asymptotic stability of a set. These conditions are expressed in terms of properties of the stage cost, terminal cost, and the existence of static state-feedback laws satisfying a control Lyapunov function (CLF) inequality along flows and jumps.

Significance. If the derivations hold, the work provides a stability framework for MPC on hybrid systems that leverages standard Lyapunov and cost-function ideas while respecting the hybrid time-domain structure. The structural results on the optimization problem and feasible set are potentially useful for future analysis. However, the practical impact is limited by the non-constructive nature of the key hypothesis.

major comments (2)
  1. [Abstract] Abstract and main stability theorem: the claim of 'checkable conditions' is undermined by the load-bearing hypothesis that there exist static state-feedback laws satisfying the CLF condition. No general construction, algorithmic test, or sufficient condition for the existence of such laws (that are consistent across flows and jumps without inducing Zeno behavior) is supplied; verification for arbitrary hybrid systems may be as difficult as the original stability question.
  2. [Main stability result] The proofs establish that if the feedback laws exist and satisfy the CLF inequality, then the MPC value function decreases and the set is asymptotically stable. This is a valid implication, but the manuscript does not address how to obtain or certify the required feedbacks for systems beyond the provided examples.
minor comments (2)
  1. [Section on algorithm formulation] Clarify the precise definition of the hybrid time-domain prediction horizon and how it differs from standard continuous or discrete MPC horizons.
  2. [Examples] Ensure all examples explicitly verify or assume the existence of the required state-feedback laws so readers can see the conditions in action.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed review and valuable feedback on our manuscript. We provide point-by-point responses to the major comments below.

read point-by-point responses

  1. Referee: [Abstract] Abstract and main stability theorem: the claim of 'checkable conditions' is undermined by the load-bearing hypothesis that there exist static state-feedback laws satisfying the CLF condition. No general construction, algorithmic test, or sufficient condition for the existence of such laws (that are consistent across flows and jumps without inducing Zeno behavior) is supplied; verification for arbitrary hybrid systems may be as difficult as the original stability question.

    Authors: The term 'checkable' in the abstract refers to the fact that, given candidate static state-feedback laws, the CLF inequalities along flows and jumps, as well as the properties of the stage and terminal costs, can be verified directly. This is analogous to standard assumptions in MPC literature for nonlinear systems, where a control Lyapunov function is assumed to exist. While we do not provide a general algorithmic procedure for constructing such laws for arbitrary hybrid systems (as this would essentially solve the stabilization problem), the framework is useful when such laws are available or can be found via system-specific analysis, as shown in the examples. We will revise the abstract and introduction to better clarify the scope of the checkable conditions and acknowledge the design effort required for the feedback laws. revision: partial

  2. Referee: [Main stability result] The proofs establish that if the feedback laws exist and satisfy the CLF inequality, then the MPC value function decreases and the set is asymptotically stable. This is a valid implication, but the manuscript does not address how to obtain or certify the required feedbacks for systems beyond the provided examples.

    Authors: The primary focus of the paper is to establish the MPC scheme and prove that the stated conditions suffice for asymptotic stability. The proofs are valid under the given hypotheses. The examples demonstrate concrete cases where the required feedbacks can be obtained and certified. For broader classes of systems, additional tools from hybrid systems theory (such as those based on Lyapunov functions or hybrid invariance principles) may be employed to find the feedbacks. We agree that expanding on methods to certify the feedbacks could enhance the manuscript and will consider adding a remark or section discussing this aspect if space permits. revision: partial

Circularity Check

0 steps flagged

No circularity: stability conditions are independent sufficient criteria

full rationale

The derivation establishes structural properties of the hybrid MPC problem (feasibility, value function) and then shows that if stage/terminal costs and static state-feedback laws satisfy a CLF inequality along flows and jumps, the MPC value function is decreasing and the target set is asymptotically stable. These are standard Lyapunov-style sufficient conditions; the existence of the feedbacks is an external hypothesis, not defined in terms of the stability conclusion or fitted from the result. No self-citation chain, self-definition, or renaming of known results is used to close the argument. The paper is therefore self-contained against external benchmarks for the claimed implications.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, invented entities, or non-standard axioms are stated. The modeling framework relies on the standard hybrid equations formalism.

axioms (1)
  • domain assumption Hybrid dynamical systems are modeled by hybrid equations involving differential and difference inclusions with inputs and constraints.
    This is the foundational modeling choice stated in the abstract and is standard in the hybrid systems literature.

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