- The paper introduces Control Barrier-Value Functions (CBVF) that merge HJ reachability with Control Barrier Functions to ensure robust safety guarantees.
- It develops a rigorous mathematical formulation using a Hamilton-Jacobi-Isaacs variational inequality and a QP-based synthesis method to achieve feasible, optimal control strategies.
- The research demonstrates the CBVF approach on double-integrator and Dubins car models, showing adjustable conservativeness through discount parameters for scalable, real-time safety control.
Robust Control Barrier--Value Functions for Safety-Critical Control
This paper addresses a critical issue within the field of safety-critical control systems by introducing a novel framework that unifies Hamilton-Jacobi (HJ) reachability and Control Barrier Functions (CBFs). These two methodologies have each presented unique benefits and challenges when it comes to ensuring the safety of dynamical systems. HJ reachability provides strong safety guarantees and can be applied to calculate value functions directly. However, it often results in conservative control strategies that can lead to chattering and suboptimal performance. On the other hand, the CBF framework applies point-wise optimization using quadratic programs (CBF-QP) to enforce safety constraints, but constructing a valid CBF for general systems is nontrivial and usually requires application-specific solutions.
To overcome the existing limitations and harness the strengths of both methods, the authors propose the concept of Control Barrier-Value Functions (CBVF). The paper presents a detailed mathematical formulation, showing that CBVFs act as value functions adaptable to the structure of CBFs, thereby providing robust safety assurances akin to HJ reachability, while also facilitating computational solutions akin to CBFs. The CBVF is shown to be a viscosity solution to a newly defined Hamilton-Jacobi-Isaacs Variational Inequality, effectively merging the conceptual frameworks of the two approaches.
The robustness of the CBVF is demonstrated through its application to systems affine in control and disturbance. The CBVF's optimal control strategy, derived from a QP-based synthesis method, is always feasible within the boundaries defined by the safe set—a fundamental advantage in practical applications. The paper meticulously illustrates this concept using double-integrator and Dubins car model simulations, highlighting the CBVF’s capability to mitigate conservativeness while remaining resilient against disturbances.
Numerically, the paper illustrates the potential real-world impacts of a CBVF-based approach. For instance, by adjusting the discount rate parameter, the resultant control policy’s aggressiveness can be modulated—allowing controllers to achieve objectives with less conservativeness compared to traditional HJ reachability-based methods.
The theoretical implications of this research extend beyond its immediate practical applications. It opens avenues for further exploration into combining different mathematical frameworks for reachability analysis and control synthesis, specifically within high-dimensional state spaces where the curse of dimensionality impedes conventional numerical methods. This synthesis approach could be pivotal in developing scalable, real-time solutions for enforcing safety in complex autonomous systems.
In summary, this paper presents a significant stride towards synthesizing robust, efficient safety controls by unifying HJ reachability techniques with CBF methodologies. The implications for both theoretical research and applied systems control are substantial, providing a foundation for developing innovative solutions in safety-critical control systems. Future research could expand this unification to include learning-based adaptability, enabling control systems to evolve and respond optimally to dynamic real-world environments.