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Input-to-State Safety With Control Barrier Functions
Shishir Kolathaya, Aaron D. Ames
TL;DR
The paper addresses safety of nonlinear systems under input disturbances, where ordinary safety requires exact forward invariance. It constructs ISSf-CBFs from CBFs and combines them with CLFs in unified QPs; the resulting formulation provides ISSf in an example where a standard QP does not.
Problem
Input disturbances can undermine safety guarantees, motivating barrier-function methods that keep states inside or close to safe sets under bounded disturbances.
Method
The paper defines ISSf and ISSf-CBFs, constructs ISSf-CBFs from existing CBFs, and combines ISSf-CBF and CLF constraints in unified quadratic programs.
Results
A standard QP does not guarantee ISSf in the reported example, whereas the ISSf-QP yields ISSf of the set under tested disturbance values.
Takeaways & Limitations
The formulations provide a basis for safety-critical control intended to be robust to disturbances while combining safeguarding and stabilization.
Abstract
from arXiv · showhide
This letter presents a new notion of input-to-state safe control barrier functions (ISSf-CBFs), which ensure safety of nonlinear dynamical systems under input disturbances. Similar to how safety conditions are specified in terms of forward invariance of a set, input-to-state safety (ISSf) conditions are specified in terms of forward invariance of a slightly larger set. In this context, invariance of the larger set implies that the states stay either inside or very close to the smaller safe set; and this closeness is bounded by the magnitude of the disturbances. The main contribution of the letter is the methodology used for obtaining a valid ISSf-CBF, given a control barrier function (CBF). The associated universal control law will also be provided. Towards the end, we will study unified quadratic programs (QPs) that combine control Lyapunov functions (CLFs) and ISSf-CBFs in order to obtain a single control law that ensures both safety and stability in systems with input disturbances.
I. INTRODUCTION
The letter addresses uncertainty in real-time safety-critical control by developing input-to-state safe control barrier functions and unified QPs for safety and stability under input disturbances.
- Existing CBF-based controllers did not support imposing safety atop an existing controller or alongside stability conditions.
- Uncertainties such as sensing errors and actuation or sensing delays can cause lane-boundary violations and collisions.
- Input-to-state safety extends robustness analysis to characterize safety under input disturbances.
- The unified formulation is intended to provide a single controller ensuring safety and stability in nonlinear systems with input disturbances.
- The paper constructs ISSf-CBFs from existing CBFs and develops a unified QP combining CLF and ISSf-CBF constraints.
II. PRELIMINARY ON CONTROL BARRIER FUNCTIONS
This section introduces barrier-function preliminaries by defining forward invariance as the basis for safety and specifying the system, safe set, and comparison-function notation.
- A set is safe when it is forward invariant: every trajectory starting in the set remains there throughout its solution interval.
- The system assumes x ∈ R^n and a locally Lipschitz vector field f, with unique solutions over a maximal time interval.
- The safe set C is a closed subset of R^n represented through a continuously differentiable function h, with nonempty interior and no isolated points.
- Class K, class K∞, and extended class K functions provide the comparison-function notation used later in the barrier conditions.
- The disturbance signal d is characterized using its essential-supremum norm ∥d∥∞.
A. Barrier functions
Barrier functions certify forward invariance on an open domain containing the safe set, with the domain restricted so the comparison condition remains valid.
- The paper studies safety of a smaller set C by establishing invariance of a larger open set D containing C.
- A barrier function h is continuously differentiable and satisfies L_fh(x) ≥ −α(h(x)) throughout D for an appropriately chosen extended class K function α.
- The boundaries −b and c must ensure that h(x) lies within the domain of α, keeping α(h(x)) well defined.
- The domain D can exclude regions where the derivative of h vanishes and h therefore cannot remain a valid barrier function.
B. Control barrier functions
Control barrier functions extend barrier conditions to affine control systems, enabling safeguarding controllers that enforce forward invariance when natural dynamics do not.
- For affine systems, a safeguarding controller can be chosen to make a set C forward invariant when the natural dynamics fail to preserve it.
- The affine model uses locally Lipschitz drift and input fields, with state x ∈ R^n and control u ∈ U ⊂ R^m.
- A CBF is defined on an open set D when there exists an admissible control set and an extended class K function satisfying the control-dependent barrier condition.
- When C is compact, a barrier function ensures both forward invariance and asymptotic stability of C.
III. PRELIMINARY ON INPUT-TO-STATE SAFETY
Input-to-state safety enlarges the safe set according to disturbance magnitude, requiring the enlarged set to remain forward invariant. The motivating example shows why ordinary safeguarding can fail under disturbances and motivates disturbance-aware control.
- Problem setup: The safeguarding controller is applied with an added disturbance through the dynamics ẋ = f̄(x) + g(x)d(t).Here f̄(x) incorporates the nominal safeguarding controller.
- Definition of ISSf: ISSf requires a disturbance-dependent enlarged set Cd containing C to be forward invariant, so states remain in or close to the safe set.The allowed closeness is directly related to the disturbance magnitude.
- Definition of ISSf: The ISSf set depends on a gain γ and disturbance bound d̄, with conditions ensuring Cd remains inside the open set D.The construction is local when the disturbance bound is restricted below a.
- Scope and limitation: The global ISSf notion is less useful because large disturbances can make Cd large and place states far into the unsafe zone.For finite a, disturbances with ∥d∥∞≥a can also invalidate the definition because Cd may not be contained in D.
- Motivating example: A small constant disturbance can drive the state unboundedly into the unsafe zone despite a safeguarding controller.In the example, x(0)=2 and d(t)=1 produce this failure.
- Motivating example: Replacing the disturbance in the controller with its norm keeps states close to or inside the safe zone for small disturbances, with eventual entry as ∥d∥∞→0.The resulting controller is an input-to-state safeguarding controller.
A. Input-to-state safe barrier function
An ISSf barrier function adds a disturbance-dependent penalty to the barrier derivative condition. Theorem 1 shows that such a function guarantees ISSf by making the corresponding enlarged set forward invariant.
- Definition: An ISSf-BF requires L_f̄h(x) + L_gh(x)μ ≥ −α(h(x)) − ι(|μ|) for bounded auxiliary inputs μ.The condition is imposed on an open set D containing C.
- Guarantee: Theorem 1 states that an ISSf-BF for C guarantees that C is ISSf.The proof establishes forward invariance of the disturbance-dependent set Cd.
- Proof mechanism: The proof checks the barrier derivative on the boundary of Cd, where the transformed function η equals zero, and applies the standard invariance argument.At the boundary, the derivative condition reduces to the form needed for invariance.
- Extension: For exponential barrier functions with α(h(x))=λh(x), λ>0, the same result yields a corresponding invariant set.The exponential choice can be substituted into the general construction.
- Example: In the motivating example, the resulting set is Cd = {x : 2−x + ∥d∥∞^2/4 ≥ 0}.This expression is the disturbance-dependent safe enlargement for that example.
IV. INPUT-TO-STATE SAFE CONTROL BARRIER
The section defines ISSf-CBFs and constructs them from existing safeguarding controllers, providing a universal formula for safety under bounded input disturbances.
- ISSf-CBFs are introduced to ensure input-to-state safety for a set under bounded disturbance inputs.The definition uses a larger forward-invariant set and disturbance-dependent conditions on the controller.
- Given a safeguarding controller k(x), the paper constructs a controller claimed to render the safe set C input-to-state safe.The construction is motivated by Sontag-style universal stabilization formulas.
- Theorem 2 states that satisfying the proposed condition for h on an open set D makes h an ISSf-CBF on D.The condition is established by substituting the constructed controller into the derivative of h.
- The universal input-to-state safeguarding formula is obtained from the constructed controller and Theorem 2.The paper presents this formula as analogous to Sontag’s universal stabilization formula.
V. CONTROL LYAPUNOV FUNCTIONS AND CONTROL BARRIER FUNCTIONS
This section combines CLFs and ISSf-CBFs in quadratic programs to address stability and safety under input disturbances, then illustrates the formulation in scalar and robotic examples.
- CLF-CBF formulation: The paper extends CLF-CBF quadratic-program formulations to systems with input disturbances, combining stabilization and safety constraints.The standard QP uses a CLF constraint with relaxation and a CBF constraint for forward invariance.
- CLF-CBF formulation: The standard QP may fail to preserve forward invariance of C when input disturbances are present.This limitation motivates the ISSf-QP formulation.
- Example 2: In Example 2, the CBF keeps x in [-2, 2], but the standard QP does not guarantee ISSf under x(0) = 0.1, u = 0, and d(t) = 10.The ISSf-QP does yield ISSf of C, with Fig. 3 comparing different disturbance and ε values.
- Example 3: The robotic example drives q = (θ, r) toward desired values qd using a CLF based on configuration error and velocity energy.The inertia matrix D and identity proportional and derivative gains define the CLF-based controller ingredients.
- Example 3: The robotic ISSf-CBF constrains the displacement safe set under bounded disturbances using an exponential barrier because h has relative degree two.For disturbance d = 5, Fig. 4 reports r responses for different ε values; Cd shrinks as ε increases.
VI. CONCLUSIONS
The letter defines input-to-state safety and ISSf-CBFs for maintaining forward invariance under input disturbances. It also presents a construction from existing CBFs, establishing groundwork for disturbance-robust safety-critical control.
- The letter formally defines input-to-state safety with respect to sets and associated ISSf-CBFs for forward invariance under input disturbances.
- It presents methods for constructing ISSf-CBFs from existing CBF formulations.
- The proposed formulations are intended to lay groundwork for safety-critical control that is robust to disturbances.