Constrained finite-time stabilization by model predictive control: an infinite control horizon framework

arXiv:2603.09617 · eess.SY, cs.SY · Submitted 2026-03-10 · Read on arXiv

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Introduction to the show: ident: Robotics Radio. Generated commentary on the latest robotics and control papers.

Rosa: Today's paper: "Constrained finite-time stabilization by model predictive control".

Dev: An infinite-horizon Model Predictive Control (MPC) framework is proposed to achieve constrained finite-time stabilization for discrete-time systems,

Rosa: First, who's behind it and why it matters.

Paper summary: Rosa: Well, so we’re diving into this paper titled "Constrained finite-time stabilization by model predictive control: an infinite control horizon framework." It seems like the main point is that existing methods for constrained finite-time MPC often run into issues because they depend on things like terminal equality constraints or switching inside one-step regions, which really limits where you can even start optimizing.

Dev: That sounds like a problem for real-time systems, especially when we're talking about latency and loop rates; limited initial feasibility means the controller might fail right at the beginning of an operation if it doesn't have enough room to maneuver.

Taro: I'm curious about what this paper claims is different; does this infinite horizon approach actually solve that initial feasibility problem in a practical way, or is it just a theoretical improvement?

Rosa: The authors propose using the sum of stage costs over an infinite control horizon instead of relying on a short-horizon terminal cost, which they say really enlarges the initial feasibility region and lets us avoid those tricky equality constraints or switching strategies during implementation.

Dev: Expanding the initial feasible region is significant because it means we don't have to worry about getting stuck in an infeasible state right off the bat when we start applying this control law.

Taro: If it can handle that expanded region, what happens when things get messy? I mean, if the world misbehaves and throws a disturbance at the system, does this framework still guarantee stabilization?

Rosa: The paper suggests that once the state trajectory enters a predefined terminal set, this infinite-horizon MPC framework guarantees finite-time stabilization performance. This is pretty strong because it moves beyond just approximate convergence to a guaranteed finite-time result.

Dev: Guaranteed finite-time performance is what we really need for robust control; I’m also interested in how they handle the implementation aspect, since we have to deal with real system dynamics and constraints.

Taro: The authors mention that the framework can be implemented using a finite-horizon MPC with a sufficiently large control horizon, which sounds like it's more practical than needing an infinite computation time for every step.

Paper summary: Rosa: Exactly, they show it’s equivalent to a finite-horizon implementation, and they even discuss extensions to constrained multi-input linear systems and even constrained nonlinear systems that are feedback linearizable.

Dev: Dealing with those nonlinear setups requires careful handling of the Jacobian matrices derived from the dynamics; I wonder how robust that transition from linear assumptions to these more complex models actually performs under real-world noise.

Taro: If it handles feedback linearization, does that mean it can manage systems where we have a lot of non-linear coupling, which is where most real robotic applications end up?

Rosa: The paper suggests they transform the nonlinear plant into a decoupled form for constrained multi-input linear systems, and for the nonlinear case, there's a specific terminal set definition that ensures you can always find a control input to keep things within the set.

Dev: That sounds like it provides a concrete mechanism for ensuring constraint satisfaction even in those complicated feedback linearizable scenarios, which is good news for our hardware constraints.

Taro: So, if we look at the overall picture of this "Constrained finite-time stabilization by model predictive control: an infinite control horizon framework," it seems the core idea is just making the initial optimization problem much bigger and less restrictive than what was previously possible with standard methods.

Rosa: That’s a good way to put it; they take that infinite summation of stage costs and use it to create a much larger initial feasible region, which bypasses those hard constraints we usually have to deal with when aiming for finite-time convergence in MPC.

Dev: From an engineering standpoint, the fact that this can be implemented via a finite-horizon version means we still have a manageable loop rate, but the underlying optimization structure is much more forgiving at the start of a maneuver.

Taro: I think what’s impactful here is showing that you can get guarantees on convergence time without resorting to those specific terminal equality constraints or complex switching logic that have historically limited how much we could actually implement this kind of control.

Rosa: It really points toward a more general framework for finite-time MPC, suggesting that instead of relying on very specific setups, we can build something that naturally handles the required behavior while maintaining feasibility from the start.

Paper summary: Dev: If this holds up under testing with real system dynamics and disturbances, it opens up new doors for us in designing systems where we need fast convergence but have tight operational limits on loop frequency.

Taro: I'm optimistic about how this could apply to autonomous systems in unpredictable environments, where the world misbehaves constantly, because the authors show it can handle states being ultimately bounded even with random disturbances.

Rosa: That ultimate boundedness under disturbance is a big deal because it means we don't just stabilize perfectly to zero, but we keep things within acceptable bounds even when noise is present, which is much more realistic for field robotics.

Dev: The paper does mention that the convergence time T depends on the state trajectory entering that terminal set, so we need to ensure our initial guess or first few steps get us close enough quickly for that guarantee to kick in.

Taro: That’s a practical point; we’d need good initialization strategies for the first few steps if we want to leverage that guaranteed finite-time performance quickly.

Rosa: So, to wrap up on this paper, "Constrained finite-time stabilization by model predictive control: an infinite control horizon framework," it seems the authors have successfully proposed a method that uses an infinite sum of stage costs to expand the initial feasibility region and bypasses the need for terminal equality constraints or switching strategies in constrained discrete-time systems.

Dev: And they prove that this approach guarantees finite-time stabilization performance once the trajectory enters a specific terminal set, which is pretty powerful when you’re designing controllers for systems with strict operational limits on loop rate.

Taro: The implication for autonomy is that we can design control laws with stronger convergence guarantees even when the underlying dynamics are nonlinear and subject to external disturbances, provided we can handle the necessary computations.

Rosa: It suggests a more general approach to finite-time MPC that focuses on feasibility preservation from the outset rather than relying on restrictive boundary conditions at the end of a short horizon optimization.

Conclusion: Rosa: So, to wrap up on this paper, "Constrained finite-time stabilization by model predictive control: an infinite control horizon framework," the core idea is using an infinite sum of stage costs to expand the initial feasibility region for constrained systems without needing terminal equality constraints or switching strategies.

Dev: That’s a heavy title, Rosa, and I gotta ask about those authors; do they have any background in handling computational complexity on embedded hardware? Because if this framework demands too much real-time processing power, it won't be useful outside the lab.

Taro: I'm interested in how this applies when we’re dealing with autonomous navigation; what does this mean for a robot trying to navigate a tight corridor where constraints are constantly changing?

Rosa: The implication is that we get a much safer starting point for our controllers, which is huge if we think about deploying robots in the real world where things aren't perfectly modeled.

Dev: From an engineer's view, the practical win here is that you don't have to worry about infeasibility right at startup, which means fewer catastrophic failure modes during initialization of a new task.

Taro: And for autonomy research, it suggests we can design systems that are more resilient because they have a proven path to finite-time stability even when the environment throws unexpected noise at them.

Rosa: Exactly; it moves us away from brittle methods that rely on precise terminal conditions and toward a more robust framework where feasibility is baked into the initial optimization setup.

Dev: If we can get this working reliably on our controllers, it could significantly reduce the tuning time needed for new complex systems, which cuts down on deployment cycles considerably.

Taro: So, we're looking at a method that provides strong convergence guarantees in a constrained setting while being computationally feasible enough for practical application?

Rosa: That’s the gist of it; this paper shows that an infinite-horizon cost structure can deliver finite-time stabilization results for discrete systems under real constraints.

Dev: If the performance holds up under actual loop rates, we could see a noticeable improvement in how quickly our control loops settle into stable operation after a sudden maneuver.

Bing Zhu, Xiaozhuoer Yuan, Zewei Zheng, Zongyu Zuo

The Seventh Research Division, Beihang University

eess.SY, cs.SY

Submitted: 2026-03-10

Updated: 2026-09-28

Comments: 10 pages, 5 figures

License: http://creativecommons.org/licenses/by-nc-nd/4.0/

Importance score: 75/100

The gist: An infinite-horizon Model Predictive Control (MPC) framework is proposed to achieve constrained finite-time stabilization for discrete-time systems, overcoming limitations in existing methods by

Key concepts

Infinite-Horizon Framework: This approach replaces traditional finite-horizon MPC terminal costs with an infinite summation of stage costs starting from a high index (n). This modification significantly expands the set of initial states and control inputs that are feasible, allowing the controller to maintain feasibility throughout the entire process.
Stage Cost Summation: The cost function $J(k)$ is defined as an infinite sum of stage costs over an infinite horizon. This specific structure ensures that the optimization problem inherently drives the system toward a finite-time convergence, effectively replacing complex switching strategies with a single, continuous optimization.
Finite-Time Stabilization: This is the goal where the system state reaches zero in a finite number of steps. The framework guarantees this by ensuring that after an initial transient phase, the constraints are naturally satisfied, allowing the system to converge to the origin within a predetermined finite time T > 0.
Equivalent Finite-Horizon Implementation: The infinite-horizon problem is computationally practical because it can be implemented using a standard finite-horizon MPC formulation. A sufficiently large control horizon N is used, and the cost function is constructed to include a terminal cost term $P$ at step N, making it solvable with standard optimization techniques.

Terminology

Model Predictive Control — 18× in this paper
Model Predictive Control (MPC) — 6× in this paper · explained in 5 episodes
Robustness — 6× in this paper · explained in 5 episodes
Closed-Loop System — 5× in this paper · explained in 4 episodes
Convergence Guarantees — 3× in this paper · explained in 3 episodes
Core — 3× in this paper · explained in 2 episodes
Accuracy — 1× in this paper · explained in 1 episode
Computational Tractability — 1× in this paper · explained in 2 episodes
Failure Modes — 1× in this paper · explained in 3 episodes
Non-linear Coupling — 1× in this paper · explained in 2 episodes
Sparsity — 1× in this paper · explained in 4 episodes
Theoretical Guarantees — 1× in this paper · explained in 2 episodes
Verification — 1× in this paper · explained in 2 episodes

Summary

An infinite-horizon Model Predictive Control (MPC) framework is proposed to achieve constrained finite-time stabilization for discrete-time systems, overcoming limitations in existing methods by replacing short-horizon terminal costs with an infinite sum of stage costs.

Key Motivation and Problem Statement

Existing results in finite-time MPC often rely on terminal equality constraints or switching inside one-step regions, which lead to limited initial feasibility and restrict the implementation. The paper addresses this by proposing a infinite-horizon Model Predictive Control (MPC) framework for the constrained finite-time stabilization of discrete-time systems, overcoming limitations found in existing finite-time MPC results. The core motivation is to design a general framework that preserves feasibility while avoiding terminal equality constraints and switching strategies. This is achieved by expanding the terminal cost strategy by replacing the short-horizon terminal cost with the sum of stage costs over an infinite control horizon, which significantly enlarges the initial feasibility region.

Proposed Infinite-Horizon Framework

The proposed framework is built upon a specific cost function designed to ensure finite-time convergence. The key setting is that the summation of stage costs starts from the number of system dimension to infinity, i.e., the lower summation index is 'i = n':

J(k) = ∞ ∑ i=n kx(ik)k squared Q +ku(ik)k squared R.

The optimization problem is formulated as:

[U∗(k),X∗(k)] = arg min U(k), X(k) J(k), subject to (6)–(8) for infinite i = 0,...,∞.

This formulation remains computationally practical because it admits an equivalent finite-horizon implementation. The framework is then implemented by:

u(k) = [1,0,···,···]1×∞U∗(k).

Theoretical Guarantees for Linear Systems

Theorem 1 establishes the theoretical guarantee for linear discrete-time systems (3)-(4). If the optimization (16) is feasible initially, and implemented via (20), then:

The constrained optimization is feasible recursively.
There exists a finite time T > 0 such that "the closed-loop state x(k) = 0 whenever k > T."

The proof demonstrates that the optimal sequence leads to a control law where, after some finite time T1, the system enters a neighborhood of the origin where constraints are inherently fulfilled, reducing the problem to its unconstrained counterpart. This results in convergence in "finite time T = T1 + n < +∞."

Implementation and Extensions

The infinite-horizon framework is implemented via a finite-horizon MPC with a sufficiently large control horizon N ≥ n. The equivalent finite-horizon cost function is designed by:

Jf(k) = N−1 ∑ i=n kx(ik)k squared Q +ku(ik)k squared R +kx(Nk)k squared P.

The implementation proceeds by solving the optimization (24):

[U∗(k), X∗(k)] = arg min U(k), X(k) Jf(k), subject to constraints (6)–(8) for finite i = 0,...,N −1, and x(Nk) ∈ Xf.

This structure is systematically extended:

To constrained multi-input linear systems by transforming them into a decoupled form [27].
To constrained nonlinear systems that are feedback linearizable, where the optimization involves Jacobian matrices derived from the nonlinear dynamics, and a terminal set Xf is defined such that there exists u = −Kx fulfilling −Kx ∈ U, f (x,−Kx) ∈ Xf, ∀x ∈ Xf.

Numerical Verification and Robustness

Numerical examples verify the theoretical findings across system types. Simulations show that for single-input linear systems, the closed-loop system reaches the origin within a specific number of steps (e.g., 7 steps). For multi-input linear systems, convergence is shown in 11 steps. Furthermore, simulations under bounded random disturbance demonstrate that system states are ultimately bounded, ensuring robustness against uncertainty while maintaining finite-time convergence properties without relying on terminal equality constraints or extra switching strategies.

Conclusion

The proposed framework successfully stabilizes the constrained discrete-time system by utilizing an infinite control horizon stage cost summation, resulting in a significantly enlarged initial feasibility region and guaranteed finite-time stabilization. It is shown to be equivalent to a finite-horizon MPC with a terminal cost, offering enhanced practical implementability while avoiding restrictive terminal equality constraints. The results are applicable to constrained multi-input linear systems and feedback linearizable nonlinear systems.

Improvements for AI systems

Based on the provided scientific paper, here are the specific improvements that can be made to AI systems by implementing its proposed framework:

Enhance Control Robustness for Constrained Systems: The proposed infinite-horizon MPC framework with an enlarged initial feasibility region significantly improves robustness against model inaccuracies and external disturbances in constrained environments.
Achieve Guaranteed Finite-Time Convergence in Real-World Applications: The system can be designed to reach the desired state (e.g., zero error) exactly within a predetermined, finite number of time steps, rather than just asymptotically approaching it, which is critical for high-precision tasks.
Enable Reliable Operation Under Constraint Violations: Unlike existing methods that might fail or require complex switching logic when constraints are violated during the transient phase, this framework maintains feasibility recursively (under initial feasibility), allowing the AI agent to operate safely within physical limits.
Improve Tracking Performance for Complex Dynamics: The extension to constrained nonlinear systems (assuming feedback linearizability) allows AI agents controlling nonlinear dynamics to achieve finite-time convergence, which is often difficult with standard MPC approaches that prioritize asymptotic stability over strict time bounds.
Simplify Control Logic: By removing the need for terminal equality constraints or complex switching strategies (like those in one-step region methods), the underlying control logic becomes simpler and more computationally tractable during implementation.

In summary, an AI system utilizing this framework can perform high-precision, guaranteed finite-time trajectory tracking and stabilization of complex physical processes (like robotics, power systems, or chemical processes) while strictly adhering to operational constraints.

Abstract

Existing results on finite-time model predictive control (MPC) often rely on terminal equality constraint, switching inside one-step region, or terminal cost with short control horizon, leading to limited initial feasibility. This paper proposes an infinite-horizon Model Predictive Control (MPC) framework for the constrained finite-time stabilization of discrete-time systems, overcoming limitations found in existing finite-time MPC results. The proposed framework is built upon a terminal cost strategy, but expands it by replacing the short-horizon terminal cost with the sum of stage costs over an infinite control horizon. This design choice significantly enlarges the initial feasibility region and avoids the need for terminal equality constraints or switching strategies during implementation. It is proved that the proposed finite-time MPC guarantees finite-time stabilization performance once the state trajectory enters the predefined terminal set. The infinite-horizon finite-time MPC is shown to be equivalently implementable as a finite-horizon MPC with a terminal cost, thereby ensuring computational tractability. The proposed finite-time MPC is systematically extended and shown to be applicable to both constrained multi-input linear systems and a class of constrained nonlinear systems that are feedback linearizable.