Control Barrier Function Based Quadratic Programs for Safety Critical Systems (1609.06408v2)

Abstract: Safety critical systems involve the tight coupling between potentially conflicting control objectives and safety constraints. As a means of creating a formal framework for controlling systems of this form, and with a view toward automotive applications, this paper develops a methodology that allows safety conditions -- expressed as control barrier functions -- to be unified with performance objectives -- expressed as control Lyapunov functions -- in the context of real-time optimization-based controllers. Safety conditions are specified in terms of forward invariance of a set, and are verified via two novel generalizations of barrier functions; in each case, the existence of a barrier function satisfying Lyapunov-like conditions implies forward invariance of the set, and the relationship between these two classes of barrier functions is characterized. In addition, each of these formulations yields a notion of control barrier function (CBF), providing inequality constraints in the control input that, when satisfied, again imply forward invariance of the set. Through these constructions, CBFs can naturally be unified with control Lyapunov functions (CLFs) in the context of a quadratic program (QP); this allows for the achievement of control objectives (represented by CLFs) subject to conditions on the admissible states of the system (represented by CBFs). The mediation of safety and performance through a QP is demonstrated on adaptive cruise control and lane keeping, two automotive control problems that present both safety and performance considerations coupled with actuator bounds.

• The paper presents a methodology that unifies Control Barrier Functions with Control Lyapunov Functions in quadratic programs for balancing safety constraints and performance objectives.
• It introduces novel Reciprocal and Zeroing Barrier Functions that guarantee the forward invariance of safety sets under specific Lyapunov-like conditions.
• High-fidelity simulations in Adaptive Cruise Control and Lane Keeping demonstrate the approach’s ability to effectively manage conflicting objectives in real-time automotive control.

Control Barrier Function Based Quadratic Programs for Safety Critical Systems

This paper authored by Aaron D. Ames, Xiangru Xu, Jessy W. Grizzle, and Paulo Tabuada addresses the challenge of integrating safety and performance objectives in the control of cyber-physical systems, with a specific focus on automotive applications. The authors present a formal methodology for unifying safety constraints through Control Barrier Functions (CBFs) with performance objectives through Control Lyapunov Functions (CLFs), executing this unification within the framework of real-time optimization-based controllers in the form of Quadratic Programs (QPs).

Theoretical Contributions

The authors extend the concept of barrier functions by introducing two novel generalizations: Reciprocal Barrier Functions (RBFs) and Zeroing Barrier Functions (ZBFs). These are utilized to ensure the safety constraints defined by the forward invariance of a set. The paper establishes that if these barrier functions satisfy certain Lyapunov-like conditions, then the forward invariance of the safety set is guaranteed.

Reciprocal Barrier Functions (RBFs)

RBFs provide conditions under which a set remains invariant. The primary condition is that a barrier function $B(x)$ must satisfy: $$\frac{1}{\alpha_1(h(x))} \leq B(x) \leq \frac{1}{\alpha_2(h(x))}$$

$L_f B(x) \leq \alpha_3(h(x))$

where $\alpha_1$, $\alpha_2$, and $\alpha_3$ are class $\mathcal{K}$ functions. This ensures that $B(x)$ tends to infinity as $x$ approaches the boundary of the safety set, preventing state trajectories from exiting the set.

Zeroing Barrier Functions (ZBFs)

ZBFs offer an alternative approach, ensuring the forward invariance of a set by requiring: $L_f h(x) \geq -\alpha(h(x))$ for some extended class $\mathcal{K}$ function $\alpha$. This formulation imposes less restrictive conditions on the interior of the safety set and is particularly useful when dealing with sets defined by more complex constraints.

Control Barrier Functions (CBFs)

In extending barrier functions to control systems, the paper defines Control Barrier Functions (CBFs) for systems modeled by nonlinear affine control dynamics. Both Reciprocal Control Barrier Functions (RCBFs) and Zeroing Control Barrier Functions (ZCBFs) are derived, providing conditions on a control input that ensure the forward invariance of the safety set.

Unification with Control Lyapunov Functions (CLFs)

The integration of CLFs and CBFs in the control design is achieved via quadratic programs (QPs). The QPs are formulated to minimize a cost function subject to the CLF condition (ensuring performance objectives like stabilization) and the CBF condition (ensuring safety constraints), allowing the system to balance multiple objectives.

Practical Applications: Automotive Systems

The utility of the methodology is demonstrated through two automotive applications: Adaptive Cruise Control (ACC) and Lane Keeping (LK).

• Adaptive Cruise Control (ACC): The paper details how CBFs can ensure a safe distance is maintained from a leading vehicle, while CLFs enforce the objective of achieving a desired cruising speed. The CBF-CLF based QP mediates between these objectives, ensuring safety constraints are never violated even if the performance goal must be relaxed.
• Lane Keeping (LK): The LK system demonstrates how lateral displacement constraints and lateral acceleration bounds can be encoded as CBFs, ensuring the vehicle remains centered within the lane. The QP formulation allows for the enforcement of these constraints while also ensuring the vehicle follows a desired path.

Numerical Results and Insights

High-fidelity simulations validate the proposed methods. For the ACC problem, the QP-based controller successfully adjusts vehicle speed in response to changes in lead vehicle behavior while respecting acceleration limits. Similarly, the LK system ensures the vehicle stays within lane boundaries and limits lateral accelerations, demonstrating the practical efficacy of the proposed methodology.

Future Directions

The framework presented opens several avenues for future research. Extending the construction of CBFs to systems with higher relative degrees and integrating multiple safety constraints (e.g., incorporating both ACC and LK constraints) remain challenging but promising directions. The potential for robotic applications, such as collision avoidance and safe maneuvering, presents exciting possibilities for further exploration.

Conclusion

This paper establishes a comprehensive methodology for the real-time optimization-based control of safety-critical systems. By formalizing the interaction between CBFs and CLFs within a QP framework, it offers a robust solution to the often conflicting requirements of safety and performance in cyber-physical systems, with significant implications for autonomous and semi-autonomous vehicle control. The rigorous development and practical validations underscore the methodology's potential to enhance the reliability and safety of complex control systems in real-world applications.