An official website of the United States government
Vector Lyapunov functions are a multi-dimensional extension of the more familiar (scalar) Lyapunov functions, commonly used to prove stability properties in systems described by non-linear ordinary differential equations (ODEs). This paper explores an analogous vector extension for so-called barrier certificates used in safety verification. As with vector Lyapunov functions, the approach hinges on constructing appropriate comparison systems, i.e., related differential equation systems from which properties of the original system may be inferred. The paper presents an accessible development of the approach, demonstrates that most previous notions of barrier certificate are special cases of comparison systems, and discusses the potential applications of vector barrier certificates in safety verification and invariant synthesis.
more »
« less
Guaranteed Safe Path and Trajectory Tracking via Reachability Analysis Using Differential Inequalities
In many automated motion planning systems, vehicles are tasked with tracking a reference path or trajectory that is safe by design. However, due to various uncertainties, real vehicles may deviate from such references, potentially leading to collisions. This paper presents rigorous reachable set bounding methods for rapidly enclosing the set of possible deviations under uncertainty, which is critical information for online safety verification. The proposed approach applies recent advances in the theory of differential inequalities that exploit redundant model equations to achieve sharp bounds using only simple interval calculations. These methods have been shown to produce very sharp bounds at low cost for nonlinear systems in other application domains, but they rely on problem-specific insights to identify appropriate redundant equations, which makes them difficult to generalize and automate. Here, we demonstrate the application of these methods to tracking problems for the first time using three representative case studies. We find that defining redundant equations in terms of Lyapunov-like functions is particularly effective. The results show that this technique can produce effective bounds with computational times that are orders of magnitude less than the planned time horizon, making this a promising approach for online safety verification. This performance, however, comes at the cost of low generalizability, specifically due to the need for problem-specific insights and advantageous problem structure, such as the existence of appropriate Lyapunov-like functions.
more »
« less
- Award ID(s):
- 1949748
- PAR ID:
- 10431461
- Date Published:
- Journal Name:
- Journal of intelligent robotic systems
- ISSN:
- 1573-0409
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
This tutorial paper focuses on safe physics- informed machine learning in the context of dynamics and control, providing a comprehensive overview of how to integrate physical models and safety guarantees. As machine learning techniques enhance the modeling and control of complex dynamical systems, ensuring safety and stability remains a critical challenge, especially in safety-critical applications like autonomous vehicles, robotics, medical decision-making, and energy systems. We explore various approaches for embedding and ensuring safety constraints, including structural priors, Lyapunov and Control Barrier Functions, predictive control, projections, and robust optimization techniques. Additionally, we delve into methods for uncertainty quantification and safety verification, including reachability analysis and neural network verification tools, which help validate that control policies remain within safe operating bounds even in uncertain environments. The paper includes illustrative examples demonstrating the implementation aspects of safe learning frameworks that combine the strengths of data-driven approaches with the rigor of physical principles, offering a path toward the safe control of complex dynamical systems.more » « less
-
Abstract Advances in deep learning have revolutionized cyber‐physical applications, including the development of autonomous vehicles. However, real‐world collisions involving autonomous control of vehicles have raised significant safety concerns regarding the use of deep neural networks (DNNs) in safety‐critical tasks, particularly perception. The inherent unverifiability of DNNs poses a key challenge in ensuring their safe and reliable operation. In this work, we propose perception simplex ( ), a fault‐tolerant application architecture designed for obstacle detection and collision avoidance. We analyse an existing LiDAR‐based classical obstacle detection algorithm to establish strict bounds on its capabilities and limitations. Such analysis and verification have not been possible for deep learning‐based perception systems yet. By employing verifiable obstacle detection algorithms, identifies obstacle existence detection faults in the output of unverifiable DNN‐based object detectors. When faults with potential collision risks are detected, appropriate corrective actions are initiated. Through extensive analysis and software‐in‐the‐loop simulations, we demonstrate that provides deterministic fault tolerance against obstacle existence detection faults, establishing a robust safety guarantee.more » « less
-
Recent studies suggest that deep neural networks (DNNs) have the potential to outperform neural networks (NNs) in approximating complex dynamics, which may enhance the tracking performance of a control system. However, unlike NN-based nonlinear control systems, designing update policies for the inner-layer weights of a DNN using Lyapunov-based stability methods is problematic since the inner-layer weights are nested within activation functions. Traditional DNN training methods (e.g., gradient descent) may improve the approximation capability of a DNN and thus could enhance a DNN-based controller’s tracking performance; however, traditional DNN-based control approaches lack stability guarantees and may be ineffective in training the DNN without large data sets, which could hinder the tracking performance. In this work, a DNN-based control structure is developed for a hybrid exoskeleton, which combines a rehabilitative robot with functional electrical stimulation (FES). The proposed control system updates the DNN weights in real-time and a rigorous Lyapunov-based stability analysis is performed to ensure semi-global asymptotic trajectory tracking, even without the presence of a data set. Specifically, a Lyapunov-based update law is developed for the output-layer DNN weights and Lyapunov-based constraints are established for the adaptation laws of the inner-layer DNN weights. Additionally, the DNN-based FES controller was designed to be saturated to increase the comfort and safety of the participant.more » « less
-
We propose new methods for learning control policies and neural network Lyapunov functions for nonlinear control problems, with provable guarantee of stability. The framework consists of a learner that attempts to find the control and Lyapunov functions, and a falsifier that finds counterexamples to quickly guide the learner towards solutions. The procedure terminates when no counterexample is found by the falsifier, in which case the controlled nonlinear system is provably stable. The approach significantly simplifies the process of Lyapunov control design, provides end-to-end correctness guarantee, and can obtain much larger regions of attraction than existing methods such as LQR and SOS/SDP. We show experiments on how the new methods obtain high-quality solutions for challenging robot control problems such as path tracking for wheeled vehicles and humanoid robot balancing.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- The DOI is not currently available.
