Convex Co-Design of Control Barrier Functions and Safe Feedback Controllers Under Input Constraints

Core Concepts

Safety in feedback control systems is ensured through the co-design of control barrier functions and feedback controllers, addressing input constraints efficiently.

Abstract

The article discusses the importance of safety in feedback control systems by co-designing control barrier functions (CBF) and linear state feedback controllers. It introduces a method to handle mixed-relative degree problems without explicit safe controllers, incorporating L-norm based input limitations. The study focuses on synthesizing CBFs for safety verification and controller design, especially in high or mixed relative degree scenarios like robotics collision avoidance. The authors propose a convex optimization program to achieve this co-design efficiently. They extend existing literature by considering input constraints like L-1, L-2, and L − ∞ norms as convex constraints in the synthesis program. The paper provides simulation results and concludes with the significance of their proposed method for linear systems.

Stats

For relative degree we mean the number of times we need to differentiate a function whose level set encodes the safe set along the system dynamics until the control explicitly shows [7], [8].
If 1 - a⊤i Ωai = 0, then λi = 1 ≥ 1/2.

Quotes

Key Insights Distilled From

Convex Co-Design of Control Barrier Function and Safe Feedback Controller Under Input Constraints

by Han Wang,Kos... at arxiv.org 03-19-2024

https://arxiv.org/pdf/2403.11763.pdf

Deeper Inquiries

How does the proposed method compare to traditional safety verification techniques

The proposed method of co-designing control barrier functions (CBF) and feedback controllers offers a significant advancement over traditional safety verification techniques. In traditional methods, verifying the safety of a system involves complex computations and often leads to NP-hard problems, especially for high or mixed relative degree cases. The CBF approach provides a more efficient and systematic way to ensure system safety by using continuously differentiable functions that separate safe and unsafe regions in the state space. By formulating the problem as a convex optimization program, the proposed method can handle mixed-relative degree problems without requiring an explicit safe controller. This not only simplifies the design process but also allows for rigorous guarantees of safety while minimizing conservativeness.

What are the implications of handling input constraints in CBF synthesis for real-world applications

Handling input constraints in CBF synthesis has significant implications for real-world applications, particularly in systems where there are limitations on control inputs. By introducing L-norm based input constraints as convex constraints in the optimization program, the proposed method ensures that synthesized controllers adhere to these restrictions while maintaining system safety. This is crucial for practical applications such as robotics collision avoidance problems, where control signals are imposed on accelerations within certain bounds. Addressing input constraints enhances the robustness and reliability of feedback controllers in real-world scenarios, ensuring that they operate within specified limits without compromising safety.

How can this co-design approach be adapted for non-linear systems

The co-design approach presented for linear systems can be adapted for non-linear systems by extending the formulation to accommodate non-linear dynamics. For non-linear systems, additional considerations need to be made regarding stability analysis and Lyapunov function design to ensure system convergence towards desired states while satisfying safety requirements. Non-linearities introduce challenges related to computational complexity and solution feasibility; however, by leveraging tools like sum-of-squares programming and semi-definite programming tailored for non-linear systems, it is possible to extend this co-design methodology effectively. Adapting the approach may involve incorporating non-linear terms into constraint formulations and optimizing over larger parameter spaces while ensuring tractability and efficiency in computation.