Contribution
A robust version of control barrier functions (CBFs) for affine systems with matched input disturbances, by analogy with Sontag’s input-to-state stability. Before it there was “no existing literature that allows to make an estimate of this buffer while providing a means to study safety under uncertainties” (p. 1). The letter defines input-to-state safe (ISSf) sets and ISSf barrier functions, gives a sufficient condition (Theorem 1), a construction from an existing CBF (Theorem 2), a universal formula (Remark 5), and an ISSf CLF–CBF quadratic program, demonstrated on two one-dimensional systems and a two-link robot.
Key insight
Do not ask that the safe set stay invariant under disturbance, which a nominal CBF controller cannot guarantee. Ask that a slightly larger set, enlarged by an amount that vanishes with the disturbance, stay invariant. A Sontag-style damping term, pushing harder along L_g h, buys that guarantee, and its gain ε trades control effort for the size of the enlargement.
Assumptions
- Affine control system ẋ = f(x) + g(x)u, f and g locally Lipschitz (eq. 9); the disturbance enters through the input, u = k(x) + d(t), d ∈ L∞ (eq. 11). Disturbances outside the input channel are not treated.
- Safe set C = {h ≥ 0} with h continuously differentiable, non-empty interior and no isolated points (eqs. 2–4).
- A domain D = {h + b > 0} on which the barrier inequality holds, and extended class-K functions α on (−b, c) (Definition 1, eq. 8).
- A disturbance bound d̄ < a, small enough that C_d stays inside D. The definition is local; the global version is “less interesting” because large disturbances make C_d large (p. 2).
Key results
- Definition 3 (ISSf set). C is ISSf if there are γ ∈ K and d̄ such that for every ‖d‖∞ ≤ d̄ the set C_d = {h(x) + γ(‖d‖∞) ≥ 0} is forward invariant (eqs. 12–14).
- Theorem 1. If L_f̄ h(x) + L_g h(x)µ ≥ −α(h(x)) − ι(|µ|) on D for |µ| ≤ d̄ (an ISSf barrier function, eq. 20), then C is ISSf, with γ = β⁻¹∘ι, β(r) = −α(−r). Proof: on ∂C_d, η = h + γ = 0 and η̇ ≥ −α(−γ) − ι = β(γ) − ι = 0 (eqs. 21–23); then Nagumo-type invariance. With α(h) = λh, C_d = {h + ι(‖d‖∞)/λ ≥ 0} (eq. 25).
- Theorem 2. If sup_u [L_f h + L_g h u − L_g h L_g hᵀ] ≥ −α(h) on D, then h is an ISSf-CBF, since |L_g h|² − |L_g h|‖d‖ ≥ −‖d‖²/4; with the gain ε, the bound is −‖d‖²/(4ε).
- Remark 5. A universal ISSf safeguarding formula, Sontag’s universal formula plus the damping term.
- Example 1. ẋ = −x + x²u, C = {x ≤ 2}: k ≡ 0 is safe without disturbance, but d = 1 from x = 2 sends the state off to infinity; k(x) = −x² gives C_d = {2 − x + ‖d‖²/4 ≥ 0} (checked; the bound is conservative).
- Example 2. With the ordinary CBF–QP, ISSf fails (x(0) = 0.1, u = 0, d = 10, checked); the ISSf-QP keeps the state bounded, and “C_d shrinks for larger values of ε” (Fig. 3–4 captions).
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | An ISSf barrier inequality implies forward invariance of the enlarged set C_d | strong | Theorem 1, short proof checked |
| C2 | Adding the damping term L_g hᵀ (gain ε) to any CBF controller yields ISSf with enlargement ‖d‖²/(4ε) in h | strong | Theorem 2, completing the square, checked |
| C3 | Ordinary CBF–QP controllers need not be ISSf | strong | Example 2, a counterexample, checked |
| C4 | The ISSf-QP unifies safety and stability under input disturbance on a robot | weak | one simulated two-link example, figures only |
Method
The ISSf-QP (p. 4): minimize ½uᵀHu + Fᵀu over (u, δ) subject to the CLF constraint with relaxation δ and the barrier constraint L_f h + L_g h u − εL_g h L_g hᵀ ≥ −α(h). For relative degree two, an exponential-CBF version with gains k_p, k_d (eq. 37).
Concepts
- Safeguarding controller — one that renders C forward invariant (p. 2).
- Input-to-state safety (ISSf) — invariance of C_d ⊇ C whose enlargement is a class-K function of ‖d‖∞ (Def. 3). Defined here with respect to sets, unlike Romdlony and Jayawardhana’s definition with respect to systems (Remark 2).
- ISSf barrier / control barrier function — eq. (20) / eq. (26).
Connections
- Nominal CBFs (Ames et al. 2014, 2017) and barrier certificates (Prajna and Rantzer 2005) are the base. ISSf is due to Romdlony and Jayawardhana (2016, 2017); this letter redefines it for superlevel sets.
- Sontag’s ISS and universal formula are the templates (Theorem 2 follows Sontag 1989’s k(x) − L_g Vᵀ).
- In the record, forward invariance of a constraint set is viability’s invariance (THEORY-134): a barrier inequality is a Nagumo tangency condition made quantitative. ISSf adds a statement about what happens when the tangency condition is violated by a bounded disturbance: the state leaves C but by a bounded amount. Contraction under singular perturbation (THEORY-771) is the stability-side analogue of the same ISS template.
Bearing on the record
- The registration’s summary needs one correction. It says the state stays “within a set enlarged in proportion to the disturbance bound”. The enlargement is a class-K function γ of ‖d‖∞, measured in units of h rather than distance, and in the paper’s own construction it is quadratic, ‖d‖²/(4ε) (or ι/λ for an exponential CBF).
- CASE-190 says “Barrier certificates of this kind are Kolathaya and Ames’s”. That is loose. The case’s box certificate checks that every face’s inward margin exceeds the disturbance bound (margins +0.16 to +0.568 against disturbances of 0.02 for u and v). That is robust forward invariance of the box itself, a Nagumo or nominal barrier condition with a disturbance margin, and it is THEORY-134’s notion. ISSf is the different, weaker guarantee for when the disturbance can exceed the margin: the state may leave the box, but only into a set enlarged by γ(‖d‖). The case would gain from ISSf exactly where its certificate stops, for degradations beyond the stated bounds. The citation should read “robust invariance of the box (THEORY-134); Kolathaya and Ames’s ISSf is the graded version that would apply beyond the margin”.
- Topic. The letter is control theory. Under the current vocabulary mathematics is the right primary tag; complex-systems is weak. The vocabulary still has no control-theory topic, which is left to the owner.
- No instruction for ML practice.
Limitations
- Disturbances only through the input channel (matched), bounded in L∞.
- Local: requires d̄ small enough that C_d stays in D.
- The enlargement is in barrier-function units, so how far in state space C_d reaches depends on h.
- The examples are illustrations; no comparison with robust or tube MPC.
- Theorem 1’s proof is a sketch that defers to Ames et al. 2017’s Proposition 1, and the statement has a typo (lim_{r→a} γ(a) = b).
Open questions
- Is there a converse: does every ISSf set admit an ISSf barrier function?
- How does ISSf compose for interconnected systems, the small-gain question the exchange gestured at without a reference?