Control Barrier Functions for Systems with High Relative Degree (1903.04706v2)

Abstract: This paper extends control barrier functions (CBFs) to high order control barrier functions (HOCBFs) that can be used for high relative degree constraints. The proposed HOCBFs are more general than recently proposed (exponential) HOCBFs. We introduce high order barrier functions (HOBF), and show that their satisfaction of Lyapunov-like conditions implies the forward invariance of the intersection of a series of sets. We then introduce HOCBF, and show that any control input that satisfies the HOCBF constraints renders the intersection of a series of sets forward invariant. We formulate optimal control problems with constraints given by HOCBF and control Lyapunov functions (CLF) and analyze the influence of the choice of the class $\mathcal{K}$ functions used in the definition of the HOCBF on the size of the feasible control region. We also provide a promising method to address the conflict between HOCBF constraints and control limitations by penalizing the class $\mathcal{K}$ functions. We illustrate the proposed method on an adaptive cruise control problem.

• The paper introduces HOCBFs that extend traditional control barrier functions to efficiently handle high relative degree constraints and ensure forward invariance via Lyapunov-like conditions.
• It formulates optimal control problems by combining HOCBFs with control Lyapunov functions and demonstrates the effect of class K functions on the feasible control region and system performance.
• The method is validated on an adaptive cruise control example, illustrating how penalized class K functions resolve conflicts between system limitations and control constraints.

Overview of High Order Control Barrier Functions for Systems with High Relative Degree

This paper introduces an advanced framework extending the existing concept of control barrier functions (CBFs) to accommodate systems with constraints of higher relative degrees. The authors propose the high order control barrier functions (HOCBFs), which allow dealing with high relative degree constraints more efficiently than previous methods. By integrating the high order barrier functions (HOBF) into the HOCBF framework, the paper demonstrates that satisfying Lyapunov-like conditions ensures the forward invariance of intersecting sets within the context of complex system dynamics.

Key Contributions

1. Theoretical Foundation of HOCBFs: The authors provide a comprehensive extension of CBFs, transitioning to HOCBFs. This transition facilitates the handling of high order system constraints, enhancing the applicability of control barrier methods across more complex systems. The paper reveals that HOBFs obey Lyapunov-like conditions, which, when satisfied, guarantee the forward invariance of specific sets.
2. Formulation and Optimization: The paper formulates optimal control problems incorporating HOCBF and control Lyapunov functions (CLF). The analysis includes the influence of class $\mathcal{K}$ functions on the feasible control region size. Additionally, it presents a strategy to manage potential conflicts between HOCBF constraints and other system limitations by introducing penalization methods on the class $\mathcal{K}$ functions.
3. Illustrative Application on Adaptive Cruise Control: The authors apply their proposed method to an adaptive cruise control (ACC) problem. The simulations demonstrate that choices of class $\mathcal{K}$ functions considerably affect the results, underscoring their significant impact on system performance and the feasibility of the control input region.

Important Results and Analytical Insights

• Numerical Analysis: The paper introduces various forms of class $\mathcal{K}$ functions (square root, linear, and quadratic) in the HOCBF, demonstrating through simulations how different choices affect system performance and constraint satisfaction.
• System Performance and Feasible Region: The paper discusses the relationship between the definition of HOCBF and the system's feasible control region. It states that higher-order class $\mathcal{K}$ functions expand the feasible region under larger barrier values, potentially enhancing system performance by avoiding over-constrained solutions.
• Conflict Resolution: By adjusting penalties on class $\mathcal{K}$ functions, the proposed method effectively addresses conflicts between system constraints and control limitations, which may otherwise lead to infeasible system operations.

Implications and Future Directions

The theoretical advancements presented in this paper provide a robust framework for addressing complex, high relative degree constraints in control systems, offering practical insights for real-time applications involving strict safety and performance criteria. The methodology has significant potential for extension to a range of advanced control tasks, including those in robotics and autonomous systems, where complex constraints are prevalent.

Future research might explore expanding the HOCBF framework to systems with more dynamic constraints and integrating further aspects of system uncertainty and disturbance rejection. Continued exploration into refined penalty strategies and adaptive tuning of class $\mathcal{K}$ functions could yield broader applicability and enhanced robustness across various dynamic settings.