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Control Barrier Functions for Systems with High Relative Degree

Wei Xiao, Calin Belta

arXiv:1903.04706v2eess.SYcs.RO

TL;DR

Existing barrier functions primarily address relative-degree-one constraints, while higher-relative-degree constraints require more general treatment. This paper develops HOCBFs parameterized by class K functions, establishes forward invariance, and validates their use in adaptive cruise control while analyzing feasible-region and control-limit conflicts.

  • Problem

    Existing barrier functions address relative-degree-one constraints, with prior higher-relative-degree methods shown for only limited cases such as relative degree two.

  • Method

    The paper introduces general HOCBFs and HOBFs based on class K functions, proves forward invariance of intersected sets, and formulates HOCBF-CLF optimal control problems.

  • Results

    The approach was validated on an automatic cruise control problem, and class K function choices were shown to affect feasible control regions and conflicts with control limitations.

  • Takeaways & Limitations

    Penalizing class K functions provides a way to manage conflicts between HOCBF constraints and control limitations while adapting the barrier formulation to different systems and constraints.

  • Takeaways & Limitations

    Class K function tuning imposes control trade-offs: decreasing p increases the minimum control input, while different choices change how closely the HOCBF approaches zero.

Abstract

from arXiv · show

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.

I. INTRODUCTION

Existing barrier and control barrier functions support forward invariance and real-time QP control, but earlier methods primarily address relative-degree-one constraints. Prior extensions reach higher relative degrees, including arbitrarily high degree through exponential barrier functions.

  • Barrier functions are Lyapunov-like tools used for set invariance, verification, control, and multi-objective control.
  • Control barrier functions can combine with control Lyapunov functions as real-time quadratic-program constraints.CLF constraints can be relaxed to reduce conflict with CBF constraints.
  • Earlier barrier-function forms primarily handle constraints with relative degree one, while backstepping and position-based methods address relative degree two.
  • A more general prior method handles arbitrarily high relative degree through input-output linearization and pole placement, yielding an exponential barrier function.
  • The paper proposes HOCBFs that are simpler and more general than exponential HOCBFs, using class K functions and associating barrier conditions with intersections of sets.
  • The paper analyzes how class K function choices affect feasible control regions and system performance, and illustrates the method on adaptive cruise control.

III. HIGH ORDER CONTROL BARRIER FUNCTIONS

The simplified adaptive cruise-control example imposes a constant distance constraint, but the resulting low-order CBF condition does not contain the control input. This prevents direct use of that barrier condition in the paper’s real-time optimization formulation.

  • The simplified adaptive cruise-control problem models each vehicle’s position, velocity, and control input while requiring a constant minimum distance from the preceding vehicle.
  • The preceding vehicle is assumed to travel at constant speed, and the follower’s control input must satisfy the safety constraint for all times.
  • The low-order barrier condition has L_gb(x_i(t)) = 0, so the control input does not appear in the constraint.
  • Because the control input is absent, these barrier functions cannot directly formulate the optimization problem described for real-time CBF-CLF control.

B. High Order Barrier Function (HOBF)

HOBFs construct a hierarchy of auxiliary functions and sets using class K functions. Satisfying the highest-order Lyapunov-like condition propagates nonnegativity backward through the hierarchy, making the intersection of all sets forward invariant.

  • For an mth-order differentiable barrier function, the construction defines auxiliary functions ψ_0 through ψ_m using class K functions α_1 through α_m.
  • The construction also defines a corresponding series of sets C_1(t) through C_m(t).
  • An HOBF requires differentiable class K functions and a highest-order condition over the intersection C_1(t) ∩ ... ∩ C_m(t).
  • If b is an mth-order HOBF, then the intersection C_1(t) ∩ ... ∩ C_m(t) is forward invariant.
  • The proof repeatedly applies a Lyapunov-like lemma from ψ_m−1 backward to ψ_0, establishing membership in every C_i(t).
  • The sets’ intersection must be non-empty at the initial time for the forward-invariance result to apply.

C. High Order Control Barrier Function (HOCBF)

The paper defines HOCBFs for constraints with arbitrary relative degree and proves that satisfying their constraints preserves the forward invariance of an intersection of sets. Unlike exponential CBFs, the construction permits general class K functions.

  • HOCBFs are defined using differentiable class K functions α1, α2, …, αm and associated sets C1(t), C2(t), …, Cm(t).
  • For a relative-degree-m constraint, the HOCBF condition defines admissible controls through the highest-order control-dependent Lie derivative and remaining lower-order terms.The remaining terms include Lie derivatives along f and time derivatives of degree at most m − 1.
  • Any Lipschitz continuous controller in the HOCBF admissible-control set renders C1(t) ∩ C2(t) ∩ … ∩ Cm(t) forward invariant from an initially feasible state.
  • The number of associated sets equals the relative degree m of the constraint.
  • Time-varying HOCBFs apply to general time-varying constraints and systems, while the paper focuses on time-invariant versions for the ACC problem.
  • Choosing linear class K functions with positive coefficients recovers the exponential CBF formulation, whereas general class K functions extend beyond it.
  • For the SACC example, the safety constraint has relative degree 2, so a second-order HOCBF is constructed using quadratic class K functions.

D. Optimal Control for Time-Invariant Constraints

The paper embeds time-invariant HOCBF safety constraints in optimal control problems, optionally alongside CLF constraints for convergence. The resulting receding-horizon procedure preserves the safety constraint when each step satisfies the required conditions.

  • The optimal-control formulation minimizes a cost over system trajectories subject to a time-invariant safety constraint b(x) ≥ 0 of relative degree m.
  • The initial state must lie in the intersection C1(t0) ∩ C2(t0) ∩ … ∩ Cm(t0).
  • When convergence is also required, HOCBF and CLF constraints are imposed at each discretized time step while the dynamics are updated iteratively.
  • Under this procedure, the intersection of the associated sets remains forward invariant, so b(x) ≥ 0 is satisfied throughout [t0, tf].

E. Time-invariant HOCBF Properties

The choice of class K functions changes both the feasible control region and system performance. Higher-order polynomial choices can enlarge feasibility away from the boundary, while penalties can help reconcile HOCBF constraints with control limits.

  • The analysis assumes LgLm−1f b(x(t)) does not change sign over the time horizon.
  • 1) Feasible Region of Control Input:: Higher-order polynomial class K functions produce a larger feasible control region when b(x), ψ1(x), …, ψm−1(x) are large.
  • 1) Feasible Region of Control Input:: When b(x), ψ1(x), …, ψm−1(x) become small, higher-order polynomial choices can make the HOCBF right-hand side smaller, usually negative, than low-order choices.
  • 1) Feasible Region of Control Input:: A larger feasible control region can prevent the HOCBF constraint from over-constraining the optimal-control problem and reducing system performance.
  • 1) Feasible Region of Control Input:: If the HOCBF constraint conflicts with a control bound, the optimal-control problem can become infeasible.
  • 1) Feasible Region of Control Input:: Minimum-braking-distance treatments can require approximation for nonlinear dynamics and cooperative optimization, making the conflict difficult to address in high-dimensional systems.
  • 1) Feasible Region of Control Input:: Penalties p1, p2, …, pm on the class K functions can help make HOCBF constraints comply with control limitations.
  • 1) Feasible Region of Control Input:: These penalties also limit the feasible control region, but that limitation is weak when the barrier variables are much larger than 1; initial set-membership conditions remain necessary.

IV. ACC PROBLEM FORMULATION

The ACC formulation uses more realistic vehicle dynamics and imposes speed, acceleration, and inter-vehicle-distance constraints. HOCBFs enforce safety, CLFs target desired speed, and the optimization cost represents energy consumption.

  • A. Vehicle Dynamics: The realistic ACC problem replaces simplified dynamics with more accurate vehicle dynamics for each vehicle in the urban-area index set S(t).
  • A. Vehicle Dynamics: Vehicle dynamics include control input, vehicle mass, velocity, and a resistance force modeled with empirically determined coefficients.
  • A. Vehicle Dynamics: The resistance model separates Coulomb friction, viscous friction, and aerodynamic drag.
  • A. Vehicle Dynamics: The state combines lane position and velocity, with the position component denoting the vehicle’s position in the lane.
  • A. Vehicle Dynamics: Vehicle limitations constrain each vehicle’s speed and acceleration, including maximum and minimum allowed speeds and mass-dependent control bounds.
  • A. Vehicle Dynamics: The safety constraint requires the distance between a vehicle and its immediately preceding vehicle to satisfy a specified lower bound.
  • A. Vehicle Dynamics: The objectives are to reach desired speed and minimize energy consumption for each vehicle.
  • A. Vehicle Dynamics: HOCBF constraints enforce safety, a CLF pursues desired speed, and energy consumption is included in the optimization cost.

V. ACC PROBLEM REFORMULATION

The ACC reformulation combines HOCBF constraints for speed limits with a relaxed CLF constraint for tracking desired speed, while direct control limits remain explicit constraints.

  • HOCBF and CLF formulation: Three class K function types define the HOCBF for the safety constraint in the QP-based ACC formulation.The considered forms are square root, linear, and quadratic functions.
  • HOCBF and CLF formulation: The CLF V_acc(x_i(t)) := (v_i(t) − v_d)^2 stabilizes vehicle speed toward the desired speed and is softened with a relaxation variable.The relaxation variable δ_acc(t) makes the corresponding CLF constraint soft.
  • Speed limitations: Because speed limitations have relative degree one, HOCBFs with m = 1 map upper and lower speed bounds to control-input constraints.The bounds are b_i,1(x_i(t)) := v_max − v_i(t) and b_i,2(x_i(t)) := v_i(t) − v_min, with linear α_1 functions.
  • Control limitations: Control limitations are imposed directly because they already constrain the control input.

C. Safety Constraint (Constraint 2)

The safety constraint has relative degree two, so the reformulation uses a second-order HOCBF with selectable class K functions and penalization, solved through piecewise-constant QPs.

  • Safety HOCBF construction: The relative-degree-two safety constraint is represented with b(x_i(t)) := z_i,ip(t) − δ and a HOCBF with m = 2.Three class K function forms are considered, with a positive penalty p applied to α_1 and α_2.
  • Piecewise-constant QP implementation: The optimization is discretized into equal time intervals, with constant control within each interval and a QP solved at every interval start.The optimal control is then applied over the interval before the dynamics are updated.
  • Piecewise-constant QP implementation: The QP uses a quadratic objective and constraint parameters derived from the system dynamics and the selected HOCBF form.The listed expressions include Lie-derivative terms and form-dependent penalty terms.
  • Piecewise-constant QP implementation: After each QP solution, the dynamics are updated using u*(t) throughout the corresponding interval.
  • Control-limit handling: The minimum control constraint is omitted from the QP because a sufficiently small penalty p may ensure that it is satisfied.

VI. IMPLEMENTATION AND RESULTS

The implementation compares class K function choices through QP-based simulations, examining when HOCBF constraints activate and how they shape the feasible control region.

  • Simulation implementation: The implementation uses MATLAB, quadprog for quadratic programs, and ode45 for integrating the dynamics.
  • Feasible control region: The case study is organized as a comparison of feasible control regions for square-root, linear, and quadratic functions.
  • Feasible control region: The simulations compare square-root, linear, and quadratic class K functions by tracking optimal and HOCBF-boundary controls as b(x_i(t)) approaches zero.Dashed curves represent HOCBF constraint boundaries, while solid curves show optimal controls; coincidence indicates an active constraint.
  • Feasible control region: Form 1 becomes over-constrained because its HOCBF constraint is active from b(x_i(t)) = 90, reducing vehicle performance.The quadratic form provides a larger feasible region than the linear form for large b(x_i), but can require larger control after activation.

B. Case 2: conflict between braking limitation and HOCBF constraint

The braking-conflict study varies the penalty p for linear and quadratic class K functions, then evaluates speed tracking and forward invariance under the selected forms.

  • Braking-limit conflict: The HOCBF constraint avoids conflict with braking limits at p = 1 for linear and p = 0.02 for quadratic class K functions.The minimum control input increases as p decreases.
  • ACC profiles and invariance: The profiles use p = 1, 1, and 0.02 for square-root, linear, and quadratic forms, respectively, while evaluating speed, control, and set invariance.The evaluated invariant set is C_1(t) ∩ C_2(t), with C_1 defined by b(x_i(t)) ≥ 0 and C_2 by ψ_1(x_i(t)) ≥ 0.
  • Effect of penalty and function choice: Increasing Form 1’s p to 2 prevents its HOCBF constraint from being over-constrained, while b(x_i(t)) decreases fastest for square-root functions and stays farthest from zero for quadratic functions.
  • Effect of penalty and function choice: At t = 15s, b(x_i(t)) equals 0.0193, 0.0413, and 15.6669 for square-root, linear, and quadratic functions, respectively.
  • Study scope: The paper validates HOCBFs on an automatic cruise-control problem with a constant safety constraint and identifies more complex future applications.Future applications include differential flatness in high relative degree systems and bipedal walking.
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