Data-driven approximation of regions of attraction via an LP-based selection of PWA Lyapunov functions

arXiv:2605.19961 · math.OC, cs.SY, eess.SY · Submitted 2026-05-19 · 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: "Data-driven approximation of regions of attraction via an LP-based selection of PWA Lyapunov functions".

Dev: This research presents a method to approximate regions of attraction for unknown nonlinear dynamical systems by constructing continuous piecewise affine (PWA) Lyapunov functions through an LP-based selection process.

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

Title and authors: Rosa: So, we've got this paper, "Data-driven approximation of regions of attraction via an LP-based selection of PWA Lyapunov functions." Rosa here. I’m really curious if this stuff is actually practical outside a controlled lab setting for field robots, and what kind of operational time we can expect before it starts degrading.

Dev: That’s a fair question, Rosa; from a control standpoint, the real concern would be the loop rate and latency when you're deploying this on hardware. We need to know if the LP-based selection process for the PWA functions is computationally feasible for real-time updates, especially since we're dealing with complex dynamics.

Taro: I’m more interested in what happens when things go wrong in the field; if the system misbehaves outside of those perfect assumptions, how does this method handle that uncertainty?

Rosa: Well, this paper tackles exactly that by creating a safety region based on what we actually observe, moving away from just theoretical models. It suggests we can use data to build a Lyapunov function candidate without needing the full system equations upfront.

Dev: Exactly; the core idea is constructing a continuous piecewise affine function, or PWA Lyapunov candidate, using an LP selection process based on that observed data and known Lipschitz bounds. It allows us to synthesize a function that respects the uncertainty we've measured.

Taro: So it’s about creating a certificate of stability that only holds true within the boundaries consistent with our training data, which is significant for autonomous systems operating in unpredictable environments?

Rosa: That’s right; it enables data-driven safety certification for nonlinear systems using sparse data, which is a big deal since traditional model-based methods struggle when the exact dynamics aren't known.

Dev: The results show that this approach lets us extract certified Regions of Attraction from relatively sparse data sets through numerical examples. This means we can get a mathematical boundary around our equilibrium point based on what we have collected, rather than just guessing based on a simplified model.

Taro: If this works robustly, it could allow AI agents like autonomous vehicles to operate within mathematically rigorous safety envelopes derived directly from empirical observations instead of relying entirely on idealized theoretical models.

Rosa: It’s really about iterative refinement; the method suggests we can develop an iterative loop where the system collects sparse data, updates the polyhedral uncertainty set and PWA partition via Linear Programming, and generates increasingly tighter and more accurate certified Regions of Attraction.

Dev: From my side, that iterative process sounds promising for online uncertainty quantification; we could potentially update this safety barrier in real-time as new sensor data comes in, allowing for continuous re-certification of safety constraints.

Title and authors: Taro: That capability to continuously update the uncertainty set based on new sensor input is crucial when the physical environment changes during operation.

Rosa: And it helps us design controllers that are guaranteed to maintain stability within that certified region, even if the true dynamics deviate slightly from what we initially modeled, as long as those deviations stay within our known Lipschitz bounds.

Dev: That addresses a major failure mode in traditional control where small model inaccuracies can lead to instability; this method seems designed specifically to mitigate that risk by explicitly incorporating the uncertainty set into the Lyapunov synthesis.

Taro: I’m thinking about how this could apply to systems where the world misbehaves unexpectedly; if we have a reliable data-driven barrier, we might be able to design controllers that actively seek or avoid regions where dynamics become potentially unstable based on these certified bounds.

Rosa: That moves us toward designing robust controllers that use this PWA candidate as a continuous safety barrier around the equilibrium point, giving us a defined area of safe operation derived from our experience.

Dev: The methodology involves defining the uncertainty set FDNd based on data points and then using linear programming to select coefficients for the piecewise affine Lyapunov function over a state-space tessellation. That’s how they enforce the required robust decrease condition across all admissible vector fields.

Taro: I want to press on the geometric structure mentioned; Lemma one shows that each component uncertainty set Qk is a finite union of polyhedral sets, and Lemma two confirms that these projections admit a common refinement defining a finite polyhedral partition independent of the specific component uncertainty.

Rosa: It’s interesting how they manage to establish this common partition across different components, which makes the construction tractable when dealing with multiple state variables simultaneously.

Dev: And Lemma three is important because it tells us that for any fixed state x within the set X, the uncertainty Qk(x) simplifies to a bounded interval between fmink(x) and fmaxk(x), which keeps things from blowing up during the optimization step.

Taro: So, while they’ve characterized the geometry of the uncertainty set quite well, what are the actual limitations of this data-driven approximation? What does it stop doing?

Rosa: The paper states that their method relies on assuming point-wise evaluations of the vector field and known Lipschitz bounds to construct the uncertainty set FDNd. That means if those initial assumptions about how smoothly the system behaves or how well we can evaluate f(x) are violated, the resulting RoA approximation might not hold.

Dev: They also noted that while this approach is more tractable than some other methods, it still requires a sufficiently dense and representative data set to construct a reliable PWA partition and achieve good certification.

Title and authors: Taro: So the limitation is tied back to the quality of the input; if our operational data is sparse or biased in certain regions, the resulting certified RoA will inherently reflect those limitations.

Rosa: Precisely; it’s not a perfect model-based solution; it’s an approximation whose accuracy is directly tied to the fidelity and coverage of the collected operational data.

Dev: This leads us nicely into how this method fits with other work, like structural sign herdability in temporal networks or learning visual-tactile dexterity, showing that this PWA Lyapunov candidate approach can be combined with other techniques to build more comprehensive safety analyses.

Taro: It seems like the implication is that we can bridge the gap between high-fidelity simulation and real-world deployment by using data to bridge those theoretical gaps in stability certification.

Rosa: That’s the core message: using data not just for learning dynamics, but specifically to generate a mathematically certified safety region, which is a significant step toward deploying complex AI in physical systems responsibly.

Dev: So, we have this method that allows us to synthesize a PWA Lyapunov function through an LP process constrained by observed data and bounds. It’s essentially creating a piecewise approximation of the system's behavior that guarantees stability within the resulting region of attraction.

Taro: I think the biggest impact is shifting stability analysis from being purely model-driven to being data-informed, which opens up avenues for safety guarantees in systems where explicit models are unavailable or too complex to derive fully.

Rosa: It really does give us a new toolset for field robotics, moving us beyond just testing in the lab and allowing us to certify longer operational times outside of ideal conditions.

Dev: We need to keep watching how they handle those online updates; that's where the real engineering challenge will be determining the practical feasibility of running an LP solver fast enough for continuous safety monitoring.

Taro: I’m looking forward to seeing how researchers apply this data-driven approximation in scenarios involving highly dynamic or adversarial environments, pushing these bounds in terms of its applicability.

Rosa: Well, that wraps up our discussion on the "Data-driven approximation of regions of attraction via an LP-based selection of PWA Lyapunov functions." It’s a lot to process, but it shows how powerful combining data characterization with PWA Lyapunov candidates can be for certifying system safety.

Dev: Definitely a paper worth keeping on the radar as we look at how to integrate these types of certified regions into our control loops.

Taro: I think the ability to generate data-backed safety certificates is what really sets this work apart in the context of autonomy research.

The paper's summary: Rosa: So, we're talking about this paper that uses data to build an approximation of where an AI system can safely operate, using these piecewise affine Lyapunov functions constructed through a linear programming process.

Dev: Right, Rosa; basically, they’re taking real-world data and using linear programming to stitch together a continuous function that acts like a safety barrier around the equilibrium point.

Taro: From my angle, this sounds really interesting because it moves stability analysis away from just relying on perfect mathematical models and grounds it in what we actually observe from the system's behavior.

Rosa: Exactly; it lets us get a certified region of attraction even when we don't have the full, explicit equations for the nonlinear system dynamics, which is a big deal for field robotics.

Dev: I’m focused on the practicality here; if this method generates these PWA functions in real-time using an LP solver, we need to know if it keeps up with high-frequency control loops without introducing unacceptable latency.

Taro: If the world misbehaves, what happens when the system deviates outside that certified boundary? Does this data-driven approach give us enough insight to anticipate those misbehaving scenarios?

Rosa: The paper shows that this method allows for online uncertainty quantification; it means we could continuously update the safety barrier as new sensor data comes in, which is crucial for real-world deployment.

Dev: That would mean we’re not just checking stability once at the start, but constantly re-verifying safety constraints based on what the AI is actually experiencing right now.

Taro: I think that continuous re-certification capability addresses a major weakness in traditional methods, where a model that works perfectly in simulation might fail quickly when deployed in a messy physical environment.

Rosa: That's the core implication; it enables us to design controllers that are guaranteed to stay within the safe region derived from our collected data, even if the true dynamics are slightly different from what we expected.

Dev: It’s about building robustness into the control law itself by explicitly incorporating the uncertainty set derived from empirical observations into how we define stability.

Taro: This really pushes us toward designing autonomous agents that aren't just robust against known disturbances, but are certified safe within a region defined by their own operational experience.

Rosa: So, it’s about creating a way to generate data-backed safety certificates for complex nonlinear systems using sparse observations and linear programming techniques.

Dev: And the results suggest that this approach can provide a mathematically rigorous boundary around an equilibrium point based on what we have collected, rather than just relying on simplified theoretical models.

Taro: This could open up whole new avenues for autonomy research by providing a practical tool to bridge the gap between idealized simulations and unpredictable real-world operation.

Rosa: It really gives us a new toolset for field robotics, allowing us to certify longer operational times outside of ideal conditions where explicit system models are hard to get.

The paper's improvements: Rosa: So, we're looking at how the authors suggest they can make this data-driven approximation even better by refining their methodology.

Dev: They propose an iterative refinement loop, suggesting that instead of just running one LP optimization, the system should collect data, update the polyhedral partition based on that new info, and then re-run the selection process.

Taro: That sounds like a way to improve accuracy over time; if we can continuously refine our safety certificate as we gather more operational data, it makes more sense for handling evolving environments.

Rosa: Exactly; it shifts the method from a one-time calculation to an iterative process, which means the certified region of attraction gets tighter and more accurate with every piece of data we feed in.

Dev: From a control standpoint, that iterative update is something we could potentially implement online; it would allow us to continuously monitor and adjust our safety barriers as the system operates.

Taro: If the uncertainty set evolves alongside the system's behavior, then this refinement process could give us a way to anticipate and adapt to unpredictable changes in the environment.

Rosa: It means we can build a controller that is constantly learning its own safety boundaries based on what it sees in real-time, rather than relying on a static model.

Dev: That addresses the failure modes where our initial assumptions about the system's behavior might become outdated over time; this method seems designed to handle that drift in uncertainty.

Taro: I wonder if this iterative refinement could also be used to explore different control strategies; maybe we could use it not just for stability, but for finding control laws that keep the system safe under varying data conditions.

Rosa: That’s a possibility; combining this with other techniques like those from the Dex-X paper might allow us to explore how different manipulation behaviors affect the resulting certified regions.

Dev: We have to consider the computational cost here; if each iteration requires a full LP re-solution, we need to ensure that it doesn't take so long that it compromises our real-time performance requirements.

Taro: The authors mention that the method can handle multiple state variables simultaneously, which is impressive because in complex autonomy tasks, we’re often dealing with coupled dynamics.

Rosa: It really does; the geometric structure they established—that common refinement partition—is what makes this approach work well even when you have many interconnected variables.

Dev: So, the implication here is that we can achieve a higher degree of certification fidelity by making the method adaptive to real-world data rather than relying on a fixed set of initial assumptions.

Taro: This moves us closer to having safety guarantees for more complex, coupled systems in autonomous applications, which is where I see the biggest potential impact.

Rosa: It really suggests that the future of safety certification in robotics isn't just about building better models, but about building better processes for learning and certifying those models directly from experience.

Dev: We need to focus on how they handle the complexity of updating that polyhedral set; if they can manage that efficiently, this has serious implications for deploying AI in physically demanding tasks.

Conclusion: Rosa: So, to wrap up our discussion on "Data-driven approximation of regions of attraction via an LP-based selection of PWA Lyapunov functions," we've seen how this method uses data and linear programming to synthesize a safety barrier around an equilibrium point for unknown nonlinear systems.

Dev: It’s clear that the core strength lies in generating a mathematically sound piecewise affine candidate function that respects the observed uncertainty, which is something we need for robust control design.

Taro: I think this work really pushes us toward autonomy because it gives us a way to get certified safety regions even when we lack perfect system models for complex environments.

Rosa: Exactly; it moves stability analysis from purely theoretical modeling to something grounded in empirical observations, which is huge for field robotics applications.

Dev: We just need to keep pushing on the computational feasibility of that LP selection process so that we can actually integrate this into our real-time control loops without introducing unacceptable lag.

Taro: If we can get this operational, it opens up possibilities for designing agents that are certified safe within empirical envelopes, which is a major step for autonomous vehicles navigating unpredictable situations.

Rosa: It really does suggest that the future of safety certification in robotics isn't just about building better simulations, but about developing processes to learn and certify safety regions directly from real-world data.

Dev: I agree; the iterative refinement loop they proposed is what makes this method potentially useful for online monitoring, allowing us to continuously verify those safety constraints as the system operates.

Taro: And that continuous verification capability is essential when dealing with dynamic environments where the system's operational envelope might change during a mission.

Rosa: So, while this paper provides a strong foundation for data-backed safety certification using PWA Lyapunov functions, we still need to figure out the practical deployment of that LP solver in high-speed hardware.

Dev: That’s the next big engineering hurdle; we have to make sure the solution is fast enough and reliable enough for actual deployment on our systems.

Taro: I'm looking forward to seeing how researchers apply this method in more complex scenarios involving adversarial environments, testing its limits where things get really messy.

Rosa: Well, that wraps up our conversation on "Data-driven approximation of regions of attraction via an LP-based selection of PWA Lyapunov functions," and it’s a lot to consider about what we can achieve with data.

math.OC, cs.SY, eess.SY

Submitted: 2026-05-19

Updated: 2026-10-01

Comments: Submitted to CDC 2026

Code: https://github.com/OumaymaK/LP

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 73/100

The gist: This research presents a method to approximate regions of attraction for unknown nonlinear dynamical systems by constructing continuous piecewise affine (PWA) Lyapunov functions through an LP-based

Key concepts

Region of Attraction (RoA)
The set of all initial states from which a nonlinear dynamical system will eventually converge to a specific stable equilibrium point. The goal is to approximate this set using data, as the exact mathematical model might be missing.
Piecewise Affine (PWA) Lyapunov Function
A function defined by different linear equations over different regions of space. This function is used to prove stability because it ensures that the system's energy decreases when moving within certain boundaries defined by these linear pieces.
LP-Based Selection Process
Using Linear Programming (LP) as a tool to select the optimal vertices for partitioning the state space. The LP minimizes a cost function while ensuring that the resulting piecewise function satisfies necessary stability conditions, leading to a certified approximation.

Terminology

Summary

This research presents a method to approximate regions of attraction for unknown nonlinear dynamical systems by constructing continuous piecewise affine (PWA) Lyapunov functions through an LP-based selection process. This approach is significant because it enables the certification of a region of attraction consistent with available data, overcoming the limitations of traditional model-based stability analysis when explicit system models are unavailable.

Problem Formulation and Assumptions

The primary goal is to approximate the region of attraction (RoA) for a stable equilibrium within a convex, compact polyhedral set X. The method proceeds in three main steps: identifying the uncertainty set consistent with data and Lipschitz bounds, partitioning the state space to construct a PWA Lyapunov candidate, and extracting a certified RoA from its level set. Key assumptions include:

  1. The system admits a unique locally asymptotically stable equilibrium within X (taken as the origin).

  2. The vector field f is Lipschitz continuous on X with an upper bound M known (interpreted with respect to the L∞-norm: f(x) − f(y) ≤ Mx − y∞, ∀x, y ∈ X).

  3. The function f is accessible for point-wise evaluation via a data set DNd = ∪ i∈[Nd] (xi, fi), where xi ∈ X and fi = f(xi).

  4. An arbitrarily small polyhedral set A ⊂ X containing the equilibrium is available, ensuring safe behavior in that region.

Characterizing Uncertainty from Data

The uncertainty set consistent with data is defined as FDNd = ∪ (x, z) ∈ X × Rn such that z − fi ≤ Mx − xi∞ for all i ∈ [Nd] (Equation 2). This basic characterization is refined by considering the component-wise Lipschitz inequality in vector form:

fκ(x) − fκ,i ≤ Mκmax k∈[n] (xk − xk,i) (Equation 4). This leads to two alternatives for the admissible uncertainty in each component of f(x), defined by cones P(+) and P(-), which are characterized by polyhedral representations (Equations 7 and 8). The full uncertainty set of a component fκ(x) over X given DNd is obtained as Qκ = ∪ i∈[Nd] Pi,κ (Equation 10).

Geometric Structure of the Uncertainty Set

The analysis reveals crucial geometric properties for tractability. Lemma 1 proves that for any κ ∈ [n], the set Qκ is a finite union of polyhedral sets, and if X is bounded, Qκ is also bounded. Lemma 2 demonstrates that the projections onto the state space of these regions admit a common refinement that defines a finite polyhedral partition of X, which is independent of κ. Lemma 3 further establishes that for any fixed x ∈ X, the uncertainty Qκ(x) is a bounded interval [fminκ(x), fmaxκ(x)]. Consequently, the global uncertainty set FDNd can be equivalently written as the graph of a set-valued map F: X ⇒ Rn, where F(x) = Q1(x) × · · · × Qn(x), and its graph is represented by a finite union of polyhedral sets (Proposition 1). The vertices of this hyperbox uncertainty set are fully characterized by Fv(x) = ∪ z ∈ Rn zk ∈ [fmink(x), fmaxk(x)], ∀k ∈ [n] (Equation 12).

LP-Based PWA Candidate Selection

The construction of the PWA Lyapunov function V is achieved by treating the state-space tessellation as a design parameter. The tessellation vertices are chosen to capture the extreme points of the uncertainty set, ensuring consistency between the partition and the uncertainty set. The candidate function V is defined piecewise over polyhedral cells Yj: V(x) = aTj x + bj, ∀x ∈ Yj (Equation 13). To enforce Lyapunov properties, coefficients are optimized via a Linear Program (LP):

min X u su (17a) s.t. su ≥ −µ ∀u, (17b); aTj vu + bj ≥ 0 ∀u, j: vu ∈ Yj, and the continuity constraint: aTj vu + bj = aTj'vu + bj' ∀u, j, j′: vu ∈ Yj ∩ Yj′. The critical decrease condition (16) is relaxed using slack variables su to enforce negativity of the Lyapunov decrease condition at the vertices of the uncertainty set Fv(vu), leading to constraint (17e): aTj z ≤ su ∀u, ∀j: vu ∈ Yj, ∀z ∈ Fv(vu).

Improvements for AI systems

Based on the provided scientific paper, here are the specific improvements that can be made to AI systems, and what those improved systems can achieve:

The core contribution of this research is a data-driven method for certifying stability (Regions of Attraction) for unknown nonlinear dynamical systems by constructing Piecewise Affine (PWA) Lyapunov functions. The primary improvements lie in moving from purely learned models to certified, robust safety guarantees derived directly from operational data.

Here are the specific improvements and capabilities:

  1. Improve the ability of AI/Control systems to operate reliably in environments where the true dynamics are unknown or only partially observed (e.g., complex physical processes, real-world robotics).

  2. Enable Data-Driven Safety Certification for nonlinear systems using sparse data, which is a major limitation of traditional model-based control methods.

  3. Provide explicit, provable bounds on the stability region (Region of Attraction) consistent with the collected training or operational data.

The improved AI/Control system can achieve the following specific capabilities:

  1. Implement a control law that is guaranteed to maintain stability within a certified region, even if the true dynamics deviate slightly from the learned model (within specified Lipschitz bounds).

  2. Perform online uncertainty quantification for unknown nonlinear systems by continuously updating an uncertainty set based on new sensor data, allowing for real-time re-certification of safety constraints.

  3. Design robust controllers that actively seek or avoid regions where the system dynamics might become unstable, using the PWA Lyapunov candidate as a continuous safety barrier.

  4. Certify safe operating envelopes for AI agents (e.g., autonomous vehicles, industrial robots) by defining a mathematically rigorous boundary around an equilibrium point based on empirical observations, rather than relying solely on idealized theoretical models.

  5. Develop an iterative refinement loop where the system collects sparse data, updates the polyhedral uncertainty set and PWA partition via Linear Programming (LP), and generates increasingly tighter and more accurate certified Regions of Attraction (RoA).

In summary, this research transforms AI/Control from a purely model-based discipline into a framework capable of generating data-backed safety certificates for complex, non-linear systems.

Abstract

This paper presents a method to approximate regions of attraction of unknown nonlinear dynamical systems from data. Assuming point-wise evaluations of the vector field and known Lipschitz bounds, a polyhedral uncertainty set of admissible dynamics is constructed. This uncertainty description enables the synthesis of a continuous piece-wise affine Lyapunov candidate via a linear program, enforcing a robust decrease condition for all admissible vector fields. The approach allows certification of a region of attraction consistent with the available data. Numerical examples illustrate the effectiveness of the proposed method in extracting certified regions of attraction from sparse data.

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