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Control Barrier Function Based Quadratic Programs for Safety Critical Systems

Aaron D. Ames, Xiangru Xu, Jessy W. Grizzle, Paulo Tabuada

arXiv:1609.06408v2math.OCeess.SY

TL;DR

Safety-critical control must reconcile safety constraints with potentially conflicting performance objectives. This paper develops reciprocal and zeroing barrier-function conditions, extends them to control barrier functions, and combines them with control Lyapunov functions in quadratic programs demonstrated on automotive control problems.

  • Problem

    The paper addresses how to synthesize controllers that enforce safety and performance or stability objectives that may conflict.

  • Method

    The paper develops reciprocal and zeroing barrier functions, extends them to control barrier functions, and unifies CBFs with CLFs through quadratic programs.

  • Results

    The resulting framework provides barrier-based forward-invariance conditions and is illustrated on adaptive cruise control and lane keeping.

  • Takeaways & Limitations

    The framework offers a real-time optimization-based way to mediate safety and performance specifications while retaining safety guarantees.

Abstract

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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.

I. INTRODUCTION

The paper develops a controller-design methodology that enforces safety through set invariance while addressing potentially conflicting performance or stability objectives. It introduces barrier-function conditions and unifies control barrier functions with control Lyapunov functions in quadratic programs, illustrating the framework on automotive problems.

  • I. INTRODUCTION: The paper targets controller synthesis for cyber-physical systems with coupled safety and performance or stability objectives.The motivating challenge is that separately designed controllers can interact unexpectedly, including in robotic and automotive systems.
  • I. INTRODUCTION: Barrier functions certify forward invariance without requiring computation of the system’s reachable set.The paper uses this certification perspective as the basis for safety-critical controller design.
  • B. Contributions: The paper formulates minimally restrictive derivative conditions for reciprocal and zeroing barrier functions, with implications for input admissibility, robustness, feedback regularity, and convexity.The stated goal is to enlarge the set of inputs compatible with controlled invariance when safety is later combined with performance.
  • B. Contributions: Control barrier functions are unified with control Lyapunov functions through quadratic programs that mediate stabilization and safety specifications.The framework imposes safety constraints while optimizing performance-related control objectives.
  • B. Contributions: The framework is illustrated on adaptive cruise control and lane keeping, which combine performance objectives with safety, actuator, force, or torque constraints.The paper also extends adaptive cruise control to varying lead-vehicle speed with bounded input force and considers lane keeping under the QP framework.
  • I. INTRODUCTION: The paper characterizes relationships between reciprocal and zeroing barrier functions and extends the resulting constructions to control barrier functions.The organization introduces barrier functions, extends them to CBFs, and then develops QPs combining CBFs and CLFs.

2) Reciprocal Barrier Functions and Set Invariance:

This section defines reciprocal barrier functions through boundary growth, Lyapunov-like bounds, and a derivative condition. It proves that the existence of such a function guarantees forward invariance of the interior of the specified set.

  • 2) Reciprocal Barrier Functions and Set Invariance:: A reciprocal barrier function is a continuously differentiable function on Int(C) satisfying class-K bounds and derivative conditions associated with a defining function h.The supplied definition introduces the RBF concept for a set C defined through h.
  • 2) Reciprocal Barrier Functions and Set Invariance:: The condition L_fB ≤ α(1/B) permits B to grow quickly far from ∂C while forcing its growth rate toward zero near the boundary.This condition generalizes a stricter derivative condition and is designed to be less restrictive in the interior.
  • 2) Reciprocal Barrier Functions and Set Invariance:: Theorem 1 establishes that existence of an RBF implies forward invariance of Int(C).The proof bounds B along solutions and concludes that h remains positive throughout the solution interval.
  • 2) Reciprocal Barrier Functions and Set Invariance:: The proof uses a comparison argument and a scalar auxiliary system whose solution exists uniquely for all t ≥ t0.The auxiliary-system construction supplies the bound needed to prevent the barrier from becoming infinite in finite time.

B. Zeroing Barrier Functions

Zeroing barrier functions (ZBFs) provide a boundary-vanishing alternative to reciprocal barriers for certifying set invariance. Under the stated conditions, a ZBF implies forward invariance and can also yield asymptotic stability.

  • Motivation: ZBFs are motivated by avoiding the unbounded values of reciprocal barrier functions near a set boundary.The paper studies barrier functions that vanish on the boundary of C, which is relevant for real-time and embedded implementations.
  • Definition: An extended class K function is continuous, strictly increasing, and zero at zero.The definition applies on an interval containing zero.
  • Definition: A ZBF is a continuously differentiable function h satisfying an extended class K derivative condition on a domain D containing C.The paper defines h on D with C ⊆ D, allowing analysis beyond the safe set.
  • Invariance: The condition ˙h ≥ −α(h) implies forward invariance of C through Nagumo’s theorem.On the boundary, h(x)=0, so the condition gives ˙h(x) ≥ 0.
  • Stability: A ZBF induces a Lyapunov function outside C, and under forward-completeness or compactness assumptions, C is asymptotically stable.The induced function is zero on C, positive outside C, and satisfies a Lyapunov-like derivative inequality.

C. Relationships of RBFs, ZBFs and Set Invariance

The paper characterizes relationships among reciprocal barrier functions (RBFs), zeroing barrier functions (ZBFs), and forward invariance. Under compactness and boundary-pointing assumptions, the converse relationships hold, while noncompact sets provide counterexamples.

  • Sufficiency: A ZBF is sufficient for forward invariance of C, while an RBF is sufficient for forward invariance of Int(C).These sufficiency results are established by the preceding theorem and proposition statements.
  • ZBF relationship: For nonempty compact C, forward invariance is equivalent to the existence of a ZBF defined on C.The converse is obtained by constructing an extended class K upper bound for the derivative condition.
  • RBF and ZBF relationship: If ˙h is positive on the boundary of compact C, then suitable RBF and ZBF conditions follow for C.Theorem 2 states that 1/h is an RBF on Int(C), while h is a ZBF on C.
  • Set invariance: When C is contractive, C and its interior have the corresponding invariance properties summarized by the barrier-function relationships.Contractivity means the flow points inward on the boundary.
  • Limitations: Without compactness, a forward-invariant set may admit neither a ZBF nor an RBF under the paper’s conditions.The counterexample has forward-invariant C but an infimum of ˙h on level sets equal to −∞, preventing the required class K bounds.
  • Control extension: The proposed barrier conditions are designed to be minimally restrictive in the interior, enlarging the control inputs compatible with controlled invariance.This property supports later integration of safety and performance objectives.

A. Reciprocal Control Barrier Functions

Reciprocal control barrier functions (RCBFs) and zeroing control barrier functions (ZCBFs) convert barrier conditions into control-input inequalities. Controllers satisfying these inequalities guarantee forward invariance, with higher-relative-degree constructions requiring additional assumptions.

  • Motivation: Barrier functions verify invariance but cannot directly design a controller enforcing it, motivating control barrier functions.CBFs extend barrier conditions into constraints on admissible control inputs.
  • Reciprocal CBFs: An RCBF defines an admissible input set K_rcbf(x) through the inequality L_fB(x) + L_gB(x)u − α_3(h(x)) ≤ 0.A locally Lipschitz controller selecting inputs from this set renders Int(C) forward invariant.
  • Zeroing CBFs: A ZCBF defines an admissible input set K_zcbf(x) through L_fh(x) + L_gh(x)u + α(h(x)) ≥ 0.Any Lipschitz controller selecting inputs from this set renders C forward invariant.
  • Caveat: The invariance guarantees hold over the closed-loop system’s maximal existence interval, not necessarily for a forward-complete trajectory.The control-selection condition preserves the relevant set while the solution exists.
  • Higher relative degree: For relative degree greater than one, the basic CBF input constraints can reduce trivially to U or the empty set.The paper constructs a transformed barrier whose relative degree is one under suitable conditions.
  • Higher relative degree: When U ≠ R^m, the higher-relative-degree construction may fail, leaving CBF design under input constraints as an open question.The stated construction assumes unconstrained inputs for this case.

IV. QPS FOR MEDIATING SAFETY AND PERFORMANCE

The paper formulates a quadratic program that selects inputs satisfying safety constraints while pursuing performance objectives. CLFs encode stabilization, CBFs encode safe-set invariance, and relaxation mediates conflicts while preserving safety.

  • The QP selects control inputs that satisfy safety requirements while also enforcing liveness or stability objectives.This directly addresses the problem of choosing among safety-compatible inputs.
  • An ES-CLF defines a constraint set of inputs whose satisfaction yields exponential stabilization toward the zero dynamics.The admissible CLF inputs satisfy L_fV(x) + L_gV(x)u + c_3V(x) ≤ 0.
  • The QP formulation incorporates actuator bounds and has been executed in real time at sample rates from 200 Hz to 1 kHz.Reported implementations include bipedal walking and scale-car systems.
  • Combining CLFs and CBFs lets the QP pursue performance subject to trajectories remaining in safe sets.The CLF constraint can be relaxed and weighted so the QP mediates performance against safety while guaranteeing safety.
  • Under local Lipschitz assumptions and the relative-degree-one condition L_gB(x) ≠ 0, the CLF-CBF QP solution is locally Lipschitz in the interior of the safe set.The theorem also states that a closed-form expression for the optimizer can be given.
  • If the performance and safety objectives do not conflict, suitable weights can produce a solution with relaxation δ approximately zero; added input constraints ensure QP feasibility but do not currently guarantee Lipschitz continuity.The latter qualification applies when the additional force-related constraint is included.

V. TWO AUTOMOTIVE SAFETY PROBLEMS VIA QPS

The paper demonstrates the CLF-CBF QP framework on adaptive cruise control and lane keeping. These automotive problems combine performance goals with safety requirements and physical constraints.

  • Adaptive cruise control and lane keeping illustrate how CLF-CBF QPs meet performance objectives subject to safety requirements.
  • ACC seeks convergence to a fixed cruising speed while maintaining a guaranteed lower bound on time headway or following distance when encountering a slower vehicle.
  • The ACC model treats the following vehicle as the controlled system and the lead vehicle as a disturbance to its cruising-speed objective.

1) ACC problem setup:

The ACC setup models vehicle velocities and separation, then encodes safe following distance as a hard constraint and desired speed as a soft CLF objective. The resulting QP balances speed tracking, safety, and solvability.

  • The ACC state comprises following-car velocity, lead-car velocity, and intervehicle distance, with wheel force as the control input.The initial model assumes an unbounded input set U = R.
  • The hard ACC constraint requires maintaining a safe distance from the vehicle ahead, while the soft constraint targets desired speed when adequate headway is assured.
  • The function h(x) = D − τ_dv_f defines the admissible safe set, from which a candidate reciprocal control barrier function is constructed.
  • The constructed barrier function is a valid reciprocal control barrier function for U = R.
  • The ACC QP combines CLF and CBF inequalities, with relaxation δ preventing infeasibility when exact CLF convergence conflicts with the safety constraint.Setting δ = 0 would make the CLF constraint hard and could make the QP infeasible.
  • In simulation, the controlled vehicle is evaluated against a desired cruising speed of 79.2 km/h, with vehicle acceleration and hard-constraint satisfaction also reported.
  • Physical wheel-force limits require a bounded admissible input set and motivate additional force constraints for realistic ACC control.

3) Force Constraints and CBFs:

The force-constrained ACC formulation adds acceleration and deceleration bounds to the safety problem. A force-based barrier function is used to characterize states where safety and braking limits can be satisfied together.

  • Force constraints limit acceleration and deceleration to specified fractions of gravitational acceleration.
  • The force-based safe set consists of states from which the ACC vehicle can satisfy the following-distance constraint while respecting maximum braking.
  • The simpler force-based CBF gives a more conservative safe-set approximation than the optimal force-based CBF.
  • The optimal force-based construction provides the maximal safe set compatible with the force-based barrier constraint and force bounds.
  • The modified force-based CLF-CBF QP incorporates the CLF, comfort constraints, force-based CBF, and admissible input bounds.
  • Simulation results for the force-constrained controller are compared with those of the unconstrained-force ACC QP.

B. Lane Keeping Via QPs

The lane-keeping formulation models lateral vehicle motion and encodes lane-position and acceleration requirements through a CBF-based quadratic program. Forward invariance of the constructed safe set preserves the hard lane constraint, while bounded acceleration yields bounded yaw behavior.

  • Problem setup: The lane-keeping problem seeks steering inputs that keep a vehicle centered in a possibly curved lane under constant longitudinal speed.The model uses lateral displacement, lateral velocity, yaw-angle error, and yaw rate as states, with steering angle as input.
  • Constraint encoding: The hard lane constraint requires the vehicle’s displacement from the lane center to satisfy |y| ≤0.9.The value ymax is motivated by a 12-foot lane and a typical 6-foot car width.
  • Constraint encoding: The acceleration constraint bounds lateral acceleration because of vehicle force limits and driver comfort requirements.The hard and acceleration constraints are encoded formally before incorporation into the lane-keeping QP.
  • CBF construction: The barrier function hF defines a safe set CF whose interior is controlled invariant while acceleration remains within its allowable bound.The construction uses the maximal allowable acceleration and establishes a CBF condition for Int(CF).
  • Forward invariance: Any feedback controller rendering Int(CF) forward invariant also renders the lane-constraint set CLK forward invariant.At the lane boundaries, the lateral velocity points back toward the admissible region.
  • Performance and stability: A controller rendering CLK forward invariant with bounded lateral acceleration produces ultimately bounded yaw angle and yaw rate.The resulting companion-form linear system is exponentially stable and input-to-state stable, with bounded lateral velocity, acceleration, and desired yaw-rate inputs.
  • QP formulation: The QP combines the lane-keeping CBF constraints with a linear performance controller, u = −K(x −xff), to mediate safety and control performance.The paper notes that a quadratic Lyapunov function could alternatively serve as a CLF for the open-loop system.

3) CBF-based QP for LK:

The QP-based controller integrates safety constraints and performance objectives for adaptive cruise control and lane keeping, including actuator bounds. Simulations show that the resulting controllers satisfy the stated automotive safety constraints.

  • ACC simulation: ACC-QP2 incorporates force constraints, causing slower convergence to the desired speed and earlier braking than ACC QP.The comparison uses optimal and conservative reciprocal control barrier functions.
  • ACC simulation: The optimal RCBF permits a smaller following distance while the specified time-headway constraint remains satisfied.The hard constraint remains positive, indicating satisfaction.
  • ACC simulation: The ACC-QP2 controller satisfies the force constraints for all time and unifies the ACC control objectives and constraints.This result is obtained using a force-based RCBF within the QP controller.
  • CBF comparison: ZCBFs produce smoother input trajectories than RCBFs while satisfying the force constraints.The comparison covers both optimal and conservative CBFs; the proposed explanation is that an RCBF’s local Lipschitz constant may become arbitrarily large near the safe-set boundary.
  • LK simulation: In lane keeping, lateral displacement remains bounded by 0.9m and lateral acceleration by 0.3g.Both displacement and acceleration constraints are reported as satisfied in the curved-road simulation.
  • Framework: The framework unifies CBF-based safety conditions with CLF-based control objectives through affine input inequalities in a quadratic program.The paper presents this as a framework for enforcing forward invariance while pursuing performance objectives.

APPENDIX

The appendix develops optimal and conservative control barrier functions for adaptive cruise control under input constraints. The constructions use braking-based stopping-distance reasoning and trade conservatism against closed-form simplicity.

  • Optimal CBF for ACC: The optimal ACC RCBF is constructed from the maximum decrease in distance headway during braking.The decrease depends on whether the lead or following car stops first.
  • Optimal CBF for ACC: The stopping-time cases distinguish whether the lead car or the following car can stop first.The cases are characterized by Tf > Tl and Tf ≤ Tl, respectively.
  • Conservative CBF for ACC: A conservative safe-set estimate replaces τd(vf − afgt) with τd vf, yielding a simpler closed-form expression.The associated conservative RCBF represents a more conservative safety set than the optimal construction.
  • Validity under the ACC model: The conservative RCBF is valid for the ACC safe set under the simplified model that drops aerodynamic drag.Because drag augments braking, a barrier function satisfying the comfort constraint for the simplified dynamics also applies to the original model.
  • Zeroing CBFs: Optimal and conservative zeroing CBFs correspond to the optimal and conservative constructions developed above.The appendix identifies the two ZCBFs with the corresponding functions for the ACC problem.
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