Control Barrier Functions: Theory and Applications

Aaron D. Ames; Samuel Coogan; Magnus Egerstedt; Gennaro Notomista; Koushil Sreenath; Paulo Tabuada

Source-linked AI summary

Control Barrier Functions: Theory and Applications

Aaron D. Ames, Samuel Coogan, Magnus Egerstedt, Gennaro Notomista, Koushil Sreenath, Paulo Tabuada

arXiv:1903.11199v1Mar 27, 2019eess.SY

TL;DR

Safety-critical control requires preventing unsafe behavior in dynamical systems, especially autonomous systems operating in unknown and unstructured environments. This paper surveys control barrier functions, their theoretical safety guarantees, and optimization-based applications that minimally modify nominal controllers. The reviewed results show that CBFs can render specified sets safe and support practical safety filters across robotic and automotive systems.

Problem
Safety must be enforced for autonomous systems operating in unknown and unstructured environments, while safety has historically received less attention than liveness in control theory.
Method
The paper introduces and surveys control barrier functions, their theory, and optimization-based safety filters that modify nominal controllers minimally.
Results
Control barrier functions provide safety guarantees by rendering specified sets safe and support applications including automotive and robotic systems.
Takeaways & Limitations
Control barrier functions offer a foundational framework and are presented as a potential essential component of modern control-system design.

Abstract

from arXiv · show

This paper provides an introduction and overview of recent work on control barrier functions and their use to verify and enforce safety properties in the context of (optimization based) safety-critical controllers. We survey the main technical results and discuss applications to several domains including robotic systems.

I. INTRODUCTION

The paper reframes safety as preventing undesirable states through set invariance and introduces control barrier functions as a counterpart to Lyapunov-based stability analysis. It traces barrier-function developments from Nagumo’s theorem to optimization-based controllers that enforce safety with minimal intervention.

Safety means that bad things do not happen, whereas liveness means that good things eventually happen.
Invariance expresses safety because trajectories starting inside a safe set never reach its complement.
A. Brief History of Barrier Functions: Nagumo’s theorem gives necessary and sufficient boundary-derivative conditions for invariance of a smooth safe set.
A. Brief History of Barrier Functions: Barrier certificates extended invariance-based safety reasoning to nonlinear and hybrid systems, including unsafe sets and initial conditions.
A. Brief History of Barrier Functions: Viability theory extended invariance to controlled invariant sets that can be maintained through suitable controller design.
A. Brief History of Barrier Functions: Control barrier functions extend barrier conditions across the safe set and support minimally invasive optimization-based safety controllers.Their conditions are described as necessary and sufficient for compact sets, making the formulation minimally restrictive.
The framework can unify safety through control barrier functions and stability through control Lyapunov functions in optimization-based controllers.

B. Overview of Paper

The paper develops control barrier functions as the safety dual of control Lyapunov functions and uses affine constraints to formulate optimization-based controllers. Its stabilization foundation shows how suitable feedback laws enforce Lyapunov decrease and asymptotic stability.

II. FOUNDATIONS OF CONTROL BARRIER FUNCTIONS: The paper establishes control-barrier-function foundations, formulates optimization-based safety controllers, and describes their unification with control Lyapunov functions.
II. FOUNDATIONS OF CONTROL BARRIER FUNCTIONS: The analysis assumes a nonlinear affine control system with locally Lipschitz dynamics and constrained admissible inputs.
A. Motivation: Control Lyapunov Functions: Stabilization is expressed by finding feedback that drives a positive definite function V toward zero.
A. Motivation: Control Lyapunov Functions: A control Lyapunov function requires the existence of a controller producing a specified negative upper bound on the derivative of V.
A. Motivation: Control Lyapunov Functions: The stabilizing-controller set Kclf(x) is an affine constraint in the input, enabling optimization-based controller synthesis.
A. Motivation: Control Lyapunov Functions: Any Lipschitz continuous feedback controller selected from Kclf(x) asymptotically stabilizes the system to x*=0.

B. Control Barrier Functions

Control barrier functions frame safety as forward invariance of a designated set and provide conditions for enforcing it through feedback control. Under regularity assumptions, controllers satisfying the CBF condition guarantee safety and asymptotic stability of the safe set, with necessity under compactness.

A safe set C is represented as the superlevel set of a continuously differentiable function h, with interior Int(C) = {x ∈ D : h(x) > 0}.
Safety is defined as forward invariance: every trajectory starting in the safe set remains there throughout its interval of existence.The closed-loop vector field is assumed locally Lipschitz so solutions are unique on a maximum interval.
A control barrier function extends boundary invariance conditions across the safe set using an extended class K∞ function α.The resulting condition generalizes the boundary requirement ˙h ≥ 0 and includes the special case ˙h ≥ −h when α(r) = r.
The admissible safe controls are Kcbf(x) = {u ∈ U : Lfh(x) + Lgh(x)u + α(h(x)) ≥ 0}.This set collects all control inputs that keep the system safe at a given state.
Any Lipschitz controller selecting u(x) ∈ Kcbf(x) renders C safe and makes C asymptotically stable, provided the gradient of h is nonzero on ∂C.The asymptotic-stability result means controllers drive the system back toward C after perturbations such as noise or modeling errors.
For compact C with the same boundary regularity, the CBF condition is necessary as well as sufficient whenever some feedback controller renders C safe.

C. Optimization Based Control

Optimization-based safety-critical control modifies an existing feedback law minimally when it violates the CBF admissibility condition. Without input constraints, the resulting single-constraint CBF-QP has a closed-form min-norm solution, while relaxation variables can preserve solvability when safety and stability constraints conflict.

Optimization-based controllers modify a nominal feedback law minimally so that the resulting input satisfies the CBF constraint.This formulation is motivated by preserving an existing controller while guaranteeing safety.
With U = R^m and one inequality constraint, the CBF-QP has a closed-form solution given by the min-norm controller.The solution follows from the KKT conditions.
A relaxation variable δ penalized by p > 0 ensures QP solvability by relaxing the stability condition when necessary to guarantee safety.The formulation uses a positive-definite matrix H(x), and the resulting controller was established to be Lipschitz continuous.

III. CBFS FOR SYSTEMS WITH ACTUATION

The paper constructs control barrier functions as safe subsets of allowable states when the full allowable set cannot be made invariant, using nominal-controller trajectories and computational synthesis methods.

Constructing safe subsets: When the allowable set A cannot be rendered forward invariant, the objective is to construct a safe set C that is a strict subset of A.This limitation can arise from restricted control actions or higher-relative-degree dynamics.
Constructing safe subsets: A nominal controller β is used to define trajectories and need not itself guarantee safety a priori.For an autonomous mobile agent, β may represent a swerving or rapid-deceleration maneuver.
Trajectory-based barrier construction: The barrier h assigns each state the infimum performance value attained along its β-generated trajectory, yielding a CBF under mild conditions.The corresponding superlevel set C is constructed from this trajectory-based barrier.
Trajectory-based barrier construction: Theorem 4 guarantees that h is a CBF, C ⊆ A, and β(x) ∈ K_cbf(x) for all x ∈ C under its stated regularity and uniqueness assumptions.The result links the trajectory construction to both allowable-state containment and admissibility of the nominal controller.
Computational synthesis: Numerically approximating h and its gradient through finite-horizon simulation becomes computationally challenging as system dimension grows.Using h in a quadratic program requires evaluating its gradient.
Computational synthesis: Polynomial CBF synthesis uses sums-of-squares constraints to enforce containment in A and the barrier condition, but bilinear decision-variable products prevent direct semidefinite-program conversion.Iterative schemes fix one factor in each problematic product to obtain an efficient numerical procedure.

IV. EXPONENTIAL CONTROL BARRIER FUNCTIONS

Exponential control barrier functions extend CBF methods to safety constraints with arbitrary relative degree. They transform higher-order barrier dynamics into a controllable linear form and enforce safety through a QP-compatible condition.

High Relative-Degree Safety Constraints: The section addresses the restrictive relative-degree-one assumption by considering safety functions with arbitrarily high relative degree r ≥ 1.Earlier approaches included only relative-degree-two position constraints or challenging backstepping designs for higher relative degree.
High Relative-Degree Safety Constraints: For relative degree r, the vector η_b(x) collects h(x) and its first r−1 derivatives, while the control affects the rth derivative through a scalar μ.The resulting barrier dynamics can be written as a linear system in controllable canonical form.
High Relative-Degree Safety Constraints: Choosing μ = −K_αη_b(x), or enforcing μ ≥ −K_αη_b(x), yields the exponential bound h(x(t)) ≥ C e^(F−GK_α)t η_b(x0).This comparison-based condition is the basis for the higher-relative-degree safety guarantee.
High Relative-Degree Safety Constraints: An exponential control barrier function exists when a row vector K_α makes the closed-loop canonical dynamics satisfy the required barrier condition.The construction extends the relative-degree-one exponential CBF definition to higher-relative-degree functions.
High Relative-Degree Safety Constraints: The resulting condition can be implemented using an optimization-based quadratic-program controller.The QP extends the earlier optimization-based safety-critical control methodology.

B. Designing Exponential Control Barrier Functions

The paper designs exponential CBFs through pole placement and recursively defined constraint sets, selecting real negative poles that also account for initial higher-order barrier derivatives.

B. Designing Exponential Control Barrier Functions: The closed-loop matrix F − GK_α is placed in controllable canonical form, with its characteristic polynomial roots determining poles p_1, …, p_r.The polynomial coefficients α_i and roots p_i are related through standard polynomial-root relations.
B. Designing Exponential Control Barrier Functions: The recursively defined functions ν_i(x) = ν̇_i−1(x) + p_iν_i−1(x) generate superlevel sets C_i, with C_0 identical to the original safe set C.The final set C_r is defined by ν_r(x) ≥ 0.
B. Designing Exponential Control Barrier Functions: If C_i is forward invariant, then C_i−1 is forward invariant when p_i > 0 and the initial state lies in C_i ∩ C_i−1.Recursive application propagates invariance backward from the highest-order set to C.
B. Designing Exponential Control Barrier Functions: The design requires F − GK_α to be Hurwitz and total negative, producing real negative poles whose placement depends on initial higher-order barrier derivatives.The initial-state condition imposes bounds involving −ν̇_i−1(x0)/ν_i−1(x0).
B. Designing Exponential Control Barrier Functions: Classical pole-placement strategies can therefore design exponential CBFs while incorporating initial-condition-dependent derivative information.The selected poles must satisfy both stability and initial-state conditions.

V. APPLICATIONS: CBFS FOR ROBOTIC SYSTEMS

The paper applies CBFs to robotic systems, including hybrid walking robots, where safety constraints require precise foot placement on discrete stepping-stone terrains.

V. APPLICATIONS: CBFS FOR ROBOTIC SYSTEMS: The applications cover single-agent walking robots, cars, and Segways, followed by multi-agent robotic systems.These domains illustrate practical uses of CBFs after the theoretical development.
Walking Robots: Stepping-stone locomotion is safety-critical because missing a foothold by a few centimeters can cause a dramatic fall.The terrain contains discrete gaps and requires precise foot placement.
Walking Robots: Legged robots are modeled as hybrid systems with continuous single-support dynamics and instantaneous double-support impacts across a switching surface.The impact map causes an instantaneous jump in the system state.
Walking Robots: An HZD-based offline constrained optimization designs a stable periodic walking orbit, while a RES-CLF regulates outputs and contracts faster than impact-induced expansion.The RES-CLF condition is used to obtain rapid exponential stability for the hybrid system.
Walking Robots: The end-time foot-placement constraint is converted into a pointwise barrier constraint using swing-foot distances from the inner and outer circles defining the foothold.Because the position constraints have relative degree 2, exponential CBF tools design the associated gains and control input.
Walking Robots: Enforcing the resulting inequalities keeps h_i(x) ≥ 0 during dynamic walking and supports terrains with changing step length, width, or height.The paper illustrates nonnegative h_1 and h_2 over time and simulated walking across stepping-stone terrains.

B. Automotive Systems: Automatic Cruise Control and Lane Keeping

The automotive examples use CBFs to enforce lane keeping and adaptive following-distance constraints while CLFs regulate performance objectives. Experiments demonstrate simultaneous enforcement on a Khepera robot.

Adaptive Cruise Control: Adaptive cruise control regulates user-set speed when no vehicle is ahead and maintains a safe following distance when one is detected.The following-distance requirement is formulated as a safety-critical CBF constraint.
System Model: The Khepera model represents position, orientation, longitudinal and angular velocities, with longitudinal force and angular torque as control inputs.It is written as a nonlinear control-affine system suitable for CBF and CLF conditions.
Lane Keeping: Lane keeping enforces the safety constraint ylat ≤ dmax, where lateral distance must remain within the lane boundaries.The lane-width limit is enforced through a lane-keeping CBF.
Unified Controller: CLFs regulate longitudinal velocity, path-following smoothness, and path tracking, while a unified CLF-CBF QP enforces performance and safety.The objectives are v → vd, ω → 0, and (x, y) → Rd.
Experimental Demonstration: Experiments simultaneously enforce lane keeping and adaptive speed regulation on the Khepera robot.The corresponding CBF values are non-negative in simulation and experiment, demonstrating constraint enforcement.

C. Dynamic Balancing on Segways

The Segway example uses an Active Set Invariance Filter to minimally modify a nominal controller while enforcing uprightness and physical constraints. Hardware experiments show safety under an excessive command and an external kick only when the filter is active.

Safety Filter: The ASIF filters a nominal control input through a CBF-QP to ensure safety while modifying the controller minimally.The nominal input udes is transformed into a safe applied input.
Safety Constraints: Segway safety is defined by keeping the pendulum angle within φ ∈ [−π/12, π/12] rad while satisfying additional physical constraints.The implementation also considers bounds on motor voltage, pendulum angular rate, and forward velocity.
Safe Set Construction: Simple angle barrier functions alone do not enforce all constraints required for experimental implementation.Hamilton-Jacobi reachability over a 75x75x75 state-space grid is used to determine the safe set.
Experimental Demonstration: The safety filter remains safe under an excessive sinusoidal angle command, whereas the unfiltered system does not.The command amplitude exceeds the π/12 angle constraint.
Experimental Demonstration: With an external kick, the Segway stays upright with CBFs but fails without them.This comparison demonstrates the filter’s experimental safety effect under disturbance.

D. Long Duration Autonomy

The long-duration autonomy example combines CBF constraints for energy survivability, collision avoidance, and environmental coverage in a multi-robot QP. A Robotarium experiment demonstrates monitoring beyond simulated battery life while robots avoid obstacles and recharge.

Motivation: Long-duration autonomy requires robots to execute monitoring and related tasks over time horizons exceeding a single battery charge.Survivability is encoded as a control constraint rather than treated separately from task execution.
Robot and Energy Model: Each robot’s augmented state includes battery energy, with charging stations represented as regions of the workspace.The model combines robot dynamics with energy dynamics and position-dependent energy requirements.
Survivability Constraints: The survivability constraint hc,i(χi) = Ei − Emin − ρi(p(xi)) ≥ 0 ensures enough energy to reach a charging station with minimum reserve Emin.An additional constraint prevents overcharging, and the two are combined through a logical-and condition.
Coverage and Collision Avoidance: CBFs encode environmental coverage and collision avoidance, including a minimum separation distance Δ between robots.Coverage objectives are combined with energy and safety constraints in differential constraints.
Controller: A relaxation variable δ trades coverage-task execution against safety and energy requirements in the controller QP.The QP minimizes control effort and relaxation cost subject to CBF-derived constraints.
Experimental Demonstration: Six Robotarium robots monitored a domain beyond simulated battery life while avoiding two obstacles, recharging, and maintaining coverage.The experiment used six charging stations and the controller solution of (25).

VI. CONCLUSIONS

The paper summarizes theoretical foundations and applications of control barrier functions for safety-critical control. It concludes that CBFs provide a foundational optimization-based framework and may become essential in modern control design.

Conclusion: The paper presents recent results in safety-critical control based on a novel form of control barrier functions.Its scope includes theoretical foundations and selected application domains.
Conclusion: Optimization-based controllers use this formulation to modify desired controllers in a minimally invasive fashion while providing a foundational framework for safety-critical control.The conclusion connects CBF-based safety enforcement with controller synthesis.