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Adaptive Safety with Control Barrier Functions
Andrew J. Taylor, Aaron D. Ames
TL;DR
The paper addresses safety guarantees for nonlinear systems whose model parameters are uncertain, where standard CBF-based guarantees may fail. It develops adaptive Control Barrier Functions alongside adaptive Lyapunov functions in a unified QP framework, and demonstrates safety and tracking in adaptive cruise control. The approach requires bounded initial parameter error and is subject to limitations illustrated by counterexamples.
Problem
Model uncertainty can invalidate safety certification, motivating an online adaptive approach that keeps the system state within a safe set.
Method
The paper formulates adaptive Control Barrier Functions and unifies them with adaptive Control Lyapunov Functions in a QP-based control methodology.
Results
In adaptive cruise control, the aCBF-QP maintains D ≥1.8v for all time under model uncertainty, while the unified aCLF-aCBF controller also achieves zero steady-state tracking error.
Takeaways & Limitations
Adaptive barrier conditions can render systems safe under parametric uncertainty while retaining adaptive stabilization and tracking within the demonstrated framework.
Takeaways & Limitations
The approach requires bounded initial parameter error, and differentiable choices of α and Γ cannot prevent safe-set violation in some R2 examples.
Abstract
from arXiv · showhide
Adaptive Control Lyapunov Functions (aCLFs) were introduced 20 years ago, and provided a Lyapunov-based methodology for stabilizing systems with parameter uncertainty. The goal of this paper is to revisit this classic formulation in the context of safety-critical control. This will motivate a variant of aCLFs in the context of safety: adaptive Control Barrier Functions (aCBFs). Our proposed approach adaptively achieves safety by keeping the systems state within a safe set even in the presence of parametric model uncertainty. We unify aCLFs and aCBFs into a single control methodology for systems with uncertain parameters in the context of a Quadratic Program (QP) based framework. We validate the ability of this unified framework to achieve stability and safety in an adaptive cruise control (ACC) simulation.
I. INTRODUCTION
The paper addresses safety-critical control when model parameters are uncertain, proposing adaptive barrier-function conditions that keep nonlinear systems within safe sets. It builds on adaptive stability methods and targets a unified QP-based methodology.
- Model mismatch can invalidate CBF-based safety guarantees in practical nonlinear control systems.Robotic masses and electrical properties are examples of parameters that may only be approximate.
- The paper seeks online adaptive conditions for maintaining safety despite parametric model uncertainty.Its stated goal is to relate adaptive safety through CBFs to adaptive stability through aCLFs.
- Safety requires keeping the state inside a prescribed safe set at all times, unlike adaptive stability guarantees that permit temporary state growth.The stricter safety objective motivates stronger assumptions on the initial parameter-estimation error.
- Adaptive Control Barrier Functions provide a formal methodology for updating parameter estimates online to ensure safety in nonlinear control-affine systems.The paper presents aCBFs as its main contribution and describes them as an initial step toward unifying online and episodic updates.
- The framework develops adaptive stabilization and safety through CBF-, CLF-, and QP-based controller constructions, followed by an adaptive cruise-control simulation.The paper organization assigns the main adaptive-safety result to Section IV and simulation validation to Section VI.
II. ADAPTIVE CONTROL LYAPUNOV FUNCTIONS
This section reviews adaptive control for systems with unknown parameters, using parameter-estimate updates and adaptive Lyapunov functions to recover stabilization guarantees. An aCLF is equivalent to global adaptive stabilizability and supports QP-based controller synthesis.
- The CLF framework assumes a smooth nonlinear affine system and uses class K∞ functions to formulate Lyapunov decrease conditions.Under these conditions, a CLF implies a state-feedback controller rendering the origin globally asymptotically stable over the state space.
- Adaptive control updates estimates of unknown parameters because explicit controllers robust to unbounded uncertainty cannot generally be designed.The uncertain parameter vector is assumed to enter the dynamics through a smooth matrix-valued function.
- The adaptive controller augments state feedback with a parameter-update law and an adaptive gain matrix.The resulting composite dynamics evolve in the combined state-and-parameter space under regularity and positive-definiteness assumptions.
- Global adaptive stabilizability requires globally bounded state and estimate trajectories with the state converging to the origin.The parameter estimate need not converge to the true parameter for the state to converge.
- An adaptive Control Lyapunov Function makes adaptive controller design equivalent to a non-adaptive CLF design problem.For every parameter value, the adaptive function serves as a CLF for the corresponding estimated-parameter system.
- A system is globally adaptively stabilizable if and only if it possesses an aCLF.The sufficiency argument constructs feedback from the current parameter estimate and uses an update law to establish convergence.
- The composite Lyapunov analysis establishes global boundedness and convergence of the state to the origin via LaSalle’s invariance principle.The controller depends on the current estimate rather than the unknown true parameters.
- A pointwise-optimal QP can synthesize a Lipschitz continuous adaptive controller when the aCLF condition is satisfied.The formulation is guaranteed to have a solution, and a min-norm controller can be obtained from KKT conditions.
III. CONTROL BARRIER FUNCTIONS
Control Barrier Functions formalize safety as forward invariance of a prescribed set and identify controls that maintain the barrier condition. A Lipschitz controller satisfying that condition renders the set safe.
- A safe set is represented as the 0-superlevel set of a continuously differentiable function h.The paper denotes this set by S and calls it the safe set.
- Safety is defined as forward invariance: every trajectory starting in the safe set remains there throughout its existence interval.The closed-loop system is assumed locally Lipschitz so solutions are uniquely defined on a maximal interval.
- CBFs avoid specifying one particular controller by requiring the existence of admissible controls that satisfy a barrier inequality.The barrier function uses an extended class K∞ function to constrain the system on the safe set.
- The pointwise admissible-control set K_cbf(x) contains inputs satisfying L_fh(x) + L_gh(x)u + α(h(x)) ≥ 0.This condition identifies controls that render the safe set invariant.
- If h is a CBF, any Lipschitz controller selecting controls from K_cbf(x) on S renders the system safe with respect to S.If the condition holds on all of X, S is additionally asymptotically stable in X.
IV. ADAPTIVE CONTROL BARRIER FUNCTIONS
The paper defines adaptive Control Barrier Functions for keeping uncertain nonlinear systems within parameter-dependent safe sets. The framework supports online parameter adaptation, QP-based safe control, and forward-invariance guarantees under bounded initial parameter error and suitable adaptive gains.
- Adaptive safety: The aCBF framework extends barrier-function methods to families of safe sets parameterized by online estimates of unknown system parameters.If the safe set is parameter-independent, the construction reduces to the corresponding nonadaptive form.
- Adaptive safety: Adaptive safety requires keeping the state within a potentially time-varying set even when dynamics parameters are uncertain.The parameter estimates need not converge or remain bounded because safety requires set containment rather than equilibrium convergence.
- Assumptions: The adaptive-gain set G is restricted because the safety proof requires a sufficient lower bound on the smallest eigenvalue of the gain matrix Γ.Greater initial distance from the boundary or smaller initial parameter error can permit smaller adaptive gains.
- Safety condition: Unlike the aCLF constraint, the aCBF condition omits α(ha(x, θ)) because including it does not generally preserve forward invariance of the state safe set.The paper connects this choice to earlier barrier-certificate formulations and gives counterexamples for the alternative constraint.
- Guarantee: An aCBF yields a Lipschitz continuous adaptive controller and implies safety of the estimated safe-set family under parameter uncertainty.The proof establishes forward invariance of superlevel sets of a composite safety function when the initial parameter error is bounded and the gain matrix satisfies the required condition.
- Controller synthesis: A QP-based controller filters a desired, potentially unsafe control action to obtain the nearest safe action using the adaptive barrier constraint.The resulting quadratic program has a closed-form solution, paralleling the adaptive CLF-QP construction.
V. ANALYSIS OF ACBF FORMULATION
The analysis shows that relaxing the aCBF condition can fail to preserve safety: trajectories approach a stable limit cycle that leaves the desired safe set. The construction and counterexample explain why the stronger condition is needed.
- Counterexample: Relaxing the condition from ˙ha ≥ 0 to ˙ha ≥ −α(ha) does not necessarily ensure adaptive safety.The paper analyzes this relaxation using a specific dynamic system.
- Counterexample: The analyzed system has an unstable equilibrium at the origin and is a Liénard system.Its dynamics use F(x) = −1.
- Limit-cycle analysis: Under the stated assumptions, the system has a unique stable limit cycle Φ enclosing the origin.The limit cycle is symmetric and passes through P2 = (x2, ˜θ2), where x2 > a.
- Limit-cycle analysis: Because a = 1 and x2 > a, the limit cycle leaves the state-safe set U while enclosing the origin.The construction also places H0 within Φ ∪ int(Φ).
- Safety consequence: All solutions starting in the 0-superlevel set of h approach Φ and therefore leave the desired state-safe set S.The paper concludes that the relaxation fails to achieve safety, as illustrated in Figure 1.
VI. ADAPTIVE CRUISE CONTROL
The ACC simulation evaluates adaptive and non-adaptive controllers under uncertain friction parameters, combining velocity tracking with distance safety. Adaptive CBF-based controllers preserve safety, while combining aCLF and aCBF achieves both safety and zero steady-state tracking error.
- ACC setup: The ACC objective is to track desired velocity while maintaining a safe distance from a leading vehicle despite uncertain friction parameters.The unknown parameters f0, f1, and f2 are associated with rolling friction and may prevent certifying the safety constraint when inaccurate.
- Adaptive safety construction: The adaptive safety construction makes the composite safety function diminish quadratically near the safety boundary to preserve differentiability.Away from the boundary, the construction can make the update law zero and maintain ḣ = 0.
- Adaptive safety controllers: The aCBF-QP filters a proportional tracking controller to maintain safety under model uncertainty.The filtered controller keeps D ≥1.8v for all time, whereas the proportional controller alone is unsafe.
- Unified adaptive QP: The unified aCLF-aCBF-QP uses separate parameter estimates for tracking and safety adaptation.Separate estimates are maintained because the aCLF and aCBF update laws may not be simultaneously satisfiable with one parameter estimate.
- Controller comparison: The CLF-aCBF controller maintains safety but retains steady-state tracking error.The aCLF-aCBF controller removes that tracking error while remaining safe.
VII. CONCLUSION
The paper presents a method for ensuring safety under parametric uncertainty by extending adaptive Control Lyapunov Function structure to forward-invariant safety sets. It identifies future batched-data work to reduce initial uncertainty and permit less conservative safe sets.
- The paper introduces an approach for ensuring system safety under parametric uncertainty.
- The approach builds on adaptive Control Lyapunov Functions while accounting for the stronger requirements of forward invariance.
- Future work will study batched data to reduce initial parametric uncertainty and permit less conservative safe sets.