Lyapunov-Net: A Deep Neural Network Architecture for Lyapunov Function Approximation

Gaby, Nathan; Zhang, Fumin; Ye, Xiaojing

doi:10.48550/arxiv.2109.13359

Cited by 4 publications

(6 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…By the universal approximation theorem, there exists a neural network φ approximating f such that f (x, κ(x) − φ(x, κ(x) ∞ < β M on the D \ { x < ε}. As in (10), the following holds…”

Section: Discussionmentioning

confidence: 96%

“…As both the right-hand side of a differential equation and the Lyapunov function can be regarded as nonlinear (or linear) functions, mapping from the state spaces to some vector spaces, a natural question is: can we use neural networks to approximate the dynamics and find a Lyapunov function to attain a larger estimate of the ROA, compared with widely used linear-quadratic regulator (LQR) and sum of squares (SOS) methods [8,9]? Various applications of neural networks for system identification and Lyapunov stability have been proposed recently [5,10]. For instance, Wang et al [11] use a recurrent neural network to generate the state-space representation of an unknown dynamical system.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Neural Lyapunov Control of Unknown Nonlinear Systems with Stability Guarantees

Zhou¹,

Quartz²,

Sterck³

et al. 2022

Preprint

View full text Add to dashboard Cite

Learning for control of dynamical systems with formal guarantees remains a challenging task. This paper proposes a learning framework to simultaneously stabilize an unknown nonlinear system with a neural controller and learn a neural Lyapunov function to certify a region of attraction (ROA) for the closed-loop system. The algorithmic structure consists of two neural networks and a satisfiability modulo theories (SMT) solver. The first neural network is responsible for learning the unknown dynamics. The second neural network aims to identify a valid Lyapunov function and a provably stabilizing nonlinear controller. The SMT solver then verifies that the candidate Lyapunov function indeed satisfies the Lyapunov conditions. We provide theoretical guarantees of the proposed learning framework in terms of the closed-loop stability for the unknown nonlinear system. We illustrate the effectiveness of the approach with a set of numerical experiments.Preprint. Under review.

show abstract

Section: Discussionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Neural Lyapunov Control of Unknown Nonlinear Systems with Stability Guarantees

Zhou¹,

Quartz²,

Sterck³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Methodology Certificate Generic Affine Hybrid Markovian Model-Free RL Supervised L&NP LF/CLF BF/CBF [29] ✓ ✓ ✓…”

Section: Model Typementioning

confidence: 99%

Stability and Safety Learning Methods for Legged Robots

Arena,

Li Noce,

Patanè

2024

Robotics

View full text Add to dashboard Cite

Learning-based control systems have shown impressive empirical performance on challenging problems in all aspects of robot control and, in particular, in walking robots such as bipeds and quadrupeds. Unfortunately, these methods have a major critical drawback: a reduced lack of guarantees for safety and stability. In recent years, new techniques have emerged to obtain these guarantees thanks to data-driven methods that allow learning certificates together with control strategies. These techniques allow the user to verify the safety of a trained controller while providing supervision during training so that safety and stability requirements can directly influence the training process. This survey presents a comprehensive and up-to-date study of the evolving field of stability certification of neural controllers taking into account such certificates as Lyapunov functions and barrier functions. Although specific attention is paid to legged robots, several promising strategies for learning certificates, not yet applied to walking machines, are also reviewed.

show abstract

“…While both the papers [17], [18] empirically verify their findings, neither provide hardware experimental validation of their methods. Neural certificates, which use neural networks to construct safety barrier certificates, have been demonstrated in [19]- [22]. These neural certificates provide a datadriven approach to learning-based controllers and provide formal proofs of correctness.…”

Section: Introductionmentioning

confidence: 99%

Gaussian Control Barrier Functions : A Non-Parametric Paradigm to Safety

Khan¹,

Ibuki²,

Chatterjee³

2022

Preprint

View full text Add to dashboard Cite

Inspired by the success of control barrier functions (CBFs) in addressing safety, and the rise of data-driven techniques for modeling functions, we propose a non-parametric approach for online synthesis of CBFs using Gaussian Processes (GPs). Mathematical constructs such as CBFs have achieved safety by designing a candidate function a priori. However, designing such a candidate function can be challenging. A practical example of such a setting would be to design a CBF in a disaster recovery scenario where safe and navigable regions need to be determined. The decision boundary for safety in such an example is unknown and cannot be designed a priori. In our approach, we work with safety samples or observations to construct the CBF online by assuming a flexible GP prior on these samples, and term our formulation as a Gaussian CBF. GPs have favorable properties, in addition to being non-parametric, such as analytical tractability and robust uncertainty estimation. This allows realizing the posterior components with high safety guarantees by incorporating variance estimation, while also computing associated partial derivatives in closed-form to achieve safe control. Moreover, the synthesized safety function from our approach allows changing the corresponding safe set arbitrarily based on the data, thus allowing non-convex safe sets. We validate our approach experimentally on a quadrotor by demonstrating safe control for fixed but arbitrary safe sets and collision avoidance where the safe set is constructed online. Finally, we juxtapose Gaussian CBFs with regular CBFs in the presence of noisy states to highlight its flexibility and robustness to noise. The experiment video can be seen at: https://youtu.be/HX6uokvCiGk.

show abstract

Lyapunov-Net: A Deep Neural Network Architecture for Lyapunov Function Approximation

Cited by 4 publications

References 28 publications

Neural Lyapunov Control of Unknown Nonlinear Systems with Stability Guarantees

Neural Lyapunov Control of Unknown Nonlinear Systems with Stability Guarantees

Stability and Safety Learning Methods for Legged Robots

Gaussian Control Barrier Functions : A Non-Parametric Paradigm to Safety

Contact Info

Product

Resources

About