DO-RRCBF: Disturbance Observer-Based RRCBF

• The paper presents a novel DO-RRCBF framework that integrates disturbance estimation with reciprocal resistance-based control barrier functions to enhance safety and reduce conservatism.
• It utilizes a disturbance observer to compensate for unknown disturbances in control-affine systems, leveraging a reciprocal resistance term that becomes dominant near safety boundaries.
• The approach is validated through simulations, including adaptive cruise control, demonstrating strict safety preservation while achieving closer-to-nominal control performance compared to other robust designs.

Disturbance Observer-Based RRCBF (DO-RRCBF) is a disturbance observer-based reciprocal resistance control barrier function for disturbed affine nonlinear systems. It extends reciprocal resistance-based barrier methods by inserting a disturbance estimate directly into the barrier inequality, while retaining a reciprocal term that grows as the safety boundary is approached. In the formulation introduced in "Enhancing Robustness of Control Barrier Function: A Reciprocal Resistance-based Approach," the objective is twofold: preserve robustness to unknown, bounded disturbances through the reciprocal-resistance mechanism, and reduce the conservatism of pure RRCBF designs by compensating the estimated disturbance in the safety constraint (Wang et al., 25 Jul 2025).

1. Disturbed system model and safety notion

DO-RRCBF is formulated for the disturbed affine nonlinear system

$$\dot{\boldsymbol{x}} = \boldsymbol{f}(\boldsymbol{x}) + \boldsymbol{g}(\boldsymbol{x})\big(\boldsymbol{u} + \boldsymbol{d}(t,\boldsymbol{x})\big),$$

where $$\boldsymbol{x} \in \mathbb{X} \subseteq \mathbb{R}^n$$ is the state, $\boldsymbol{u} \in \mathbb{R}^m$ is the control input, $$\boldsymbol{d}(t,\boldsymbol{x}) \in \mathbb{D} \subseteq \mathbb{R}^m$$ is an unknown but bounded disturbance, and $\boldsymbol{f},\boldsymbol{g}$ are locally Lipschitz. Safety is encoded by a continuously differentiable function $h:\mathbb{R}^n\to\mathbb{R}$, with safe set

$$\mathbb{C} := \{\boldsymbol{x}\in\mathbb{R}^n : h(\boldsymbol{x}) \ge 0\},$$

and interior

$${\rm Int}(\mathbb{C}) := \{\boldsymbol{x}\in\mathbb{R}^n : h(\boldsymbol{x}) > 0\}.$$

Forward invariance means that trajectories starting in the set remain in the set for all future time (Wang et al., 25 Jul 2025).

This setting places DO-RRCBF squarely within the control-barrier-function literature for control-affine systems, but its safety statement is deliberately framed around ${\rm Int}(\mathbb{C})$. That distinction is structurally important. A recurring misconception is to read the method as enforcing invariance of the full closed superlevel set $\mathbb{C}$ in the same manner as a standard zeroing CBF. The underlying reciprocal-resistance construction instead emphasizes strict positivity of the barrier quantity near the boundary, so the forward-invariant object is the interior, together with an invariant inner core generated by the reciprocal term.

2. Reciprocal resistance and the emergence of RRCBF

The precursor to DO-RRCBF is the reciprocal resistance-based barrier construction. Its motivating scalar model is

$$\boldsymbol{x} \in \mathbb{X} \subseteq \mathbb{R}^n$$0

Without disturbances, $$\boldsymbol{x} \in \mathbb{X} \subseteq \mathbb{R}^n$$1 converges to $$\boldsymbol{x} \in \mathbb{X} \subseteq \mathbb{R}^n$$2 and remains strictly positive. With an additive bounded disturbance $$\boldsymbol{x} \in \mathbb{X} \subseteq \mathbb{R}^n$$3,

$$\boldsymbol{x} \in \mathbb{X} \subseteq \mathbb{R}^n$$4

the term $$\boldsymbol{x} \in \mathbb{X} \subseteq \mathbb{R}^n$$5 dominates near $$\boldsymbol{x} \in \mathbb{X} \subseteq \mathbb{R}^n$$6, the dynamics are ISS with respect to $$\boldsymbol{x} \in \mathbb{X} \subseteq \mathbb{R}^n$$7, and $$\boldsymbol{x} \in \mathbb{X} \subseteq \mathbb{R}^n$$8 for all $$\boldsymbol{x} \in \mathbb{X} \subseteq \mathbb{R}^n$$9 if $\boldsymbol{u} \in \mathbb{R}^m$0. The main paper interprets $\boldsymbol{u} \in \mathbb{R}^m$1 as a resistance that increases when $\boldsymbol{u} \in \mathbb{R}^m$2 decreases, thereby creating a positive buffer (Wang et al., 25 Jul 2025).

This mechanism is transferred to the barrier function $\boldsymbol{u} \in \mathbb{R}^m$3. For the autonomous system $\boldsymbol{u} \in \mathbb{R}^m$4, a continuously differentiable $\boldsymbol{u} \in \mathbb{R}^m$5 is an RRBF if there exist extended class $\boldsymbol{u} \in \mathbb{R}^m$6 functions $\boldsymbol{u} \in \mathbb{R}^m$7 such that

$\boldsymbol{u} \in \mathbb{R}^m$8

for all $\boldsymbol{u} \in \mathbb{R}^m$9. Relative to a standard zeroing-barrier inequality, the additional term $$\boldsymbol{d}(t,\boldsymbol{x}) \in \mathbb{D} \subseteq \mathbb{R}^m$$0 acts as a positive reciprocal resistance in the induced dynamics of $$\boldsymbol{d}(t,\boldsymbol{x}) \in \mathbb{D} \subseteq \mathbb{R}^m$$1 (Wang et al., 25 Jul 2025).

A central object is the buffer set

$$\boldsymbol{d}(t,\boldsymbol{x}) \in \mathbb{D} \subseteq \mathbb{R}^m$$2

where $$\boldsymbol{d}(t,\boldsymbol{x}) \in \mathbb{D} \subseteq \mathbb{R}^m$$3 is the unique positive solution of

$$\boldsymbol{d}(t,\boldsymbol{x}) \in \mathbb{D} \subseteq \mathbb{R}^m$$4

$$\boldsymbol{d}(t,\boldsymbol{x}) \in \mathbb{D} \subseteq \mathbb{R}^m$$5 is a strict subset of $$\boldsymbol{d}(t,\boldsymbol{x}) \in \mathbb{D} \subseteq \mathbb{R}^m$$6. For the undisturbed autonomous system, trajectories starting in $$\boldsymbol{d}(t,\boldsymbol{x}) \in \mathbb{D} \subseteq \mathbb{R}^m$$7 remain in $$\boldsymbol{d}(t,\boldsymbol{x}) \in \mathbb{D} \subseteq \mathbb{R}^m$$8, while trajectories starting in $$\boldsymbol{d}(t,\boldsymbol{x}) \in \mathbb{D} \subseteq \mathbb{R}^m$$9 converge to $\boldsymbol{f},\boldsymbol{g}$0. For the disturbed autonomous system $\boldsymbol{f},\boldsymbol{g}$1, the reciprocal term yields robust invariance of $\boldsymbol{f},\boldsymbol{g}$2 under arbitrary bounded disturbances, without requiring explicit knowledge of disturbance magnitude. This is the basis for the reciprocal resistance-based control barrier function,

$\boldsymbol{f},\boldsymbol{g}$3

whose admissible-control set renders $\boldsymbol{f},\boldsymbol{g}$4 forward invariant under bounded disturbances (Wang et al., 25 Jul 2025).

The significance of this step is not merely algebraic. RRCBF replaces explicit disturbance-margin design by a state-dependent reciprocal barrier pressure that becomes strongest exactly where safety is most fragile. The method therefore shifts robustness from a priori worst-case bound specification toward boundary-dominating barrier geometry.

3. Disturbance observer integration and the DO-RRCBF definition

DO-RRCBF augments RRCBF with a disturbance observer. The paper recalls a Chen-style nonlinear disturbance observer for each component $\boldsymbol{f},\boldsymbol{g}$5 of the disturbance: $\boldsymbol{f},\boldsymbol{g}$6 where $\boldsymbol{f},\boldsymbol{g}$7 is an auxiliary observer state, $\boldsymbol{f},\boldsymbol{g}$8 is a designed nonlinear function, and $\boldsymbol{f},\boldsymbol{g}$9 is the nonlinear observer gain. Under the assumption that $h:\mathbb{R}^n\to\mathbb{R}$0 exists and $h:\mathbb{R}^n\to\mathbb{R}$1 with known $h:\mathbb{R}^n\to\mathbb{R}$2, the estimation error $h:\mathbb{R}^n\to\mathbb{R}$3 satisfies

$h:\mathbb{R}^n\to\mathbb{R}$4

With suitable $h:\mathbb{R}^n\to\mathbb{R}$5, this error system is ISS in $h:\mathbb{R}^n\to\mathbb{R}$6, so the estimation error can be made small and is ultimately bounded (Wang et al., 25 Jul 2025).

On this basis, a continuously differentiable $h:\mathbb{R}^n\to\mathbb{R}$7 is a DO-RRCBF if there exist extended class $h:\mathbb{R}^n\to\mathbb{R}$8 functions $h:\mathbb{R}^n\to\mathbb{R}$9 such that

$$\mathbb{C} := \{\boldsymbol{x}\in\mathbb{R}^n : h(\boldsymbol{x}) \ge 0\},$$0

for all $$\mathbb{C} := \{\boldsymbol{x}\in\mathbb{R}^n : h(\boldsymbol{x}) \ge 0\},$$1. The associated admissible input set is

$$\mathbb{C} := \{\boldsymbol{x}\in\mathbb{R}^n : h(\boldsymbol{x}) \ge 0\},$$2

Any Lipschitz controller $$\mathbb{C} := \{\boldsymbol{x}\in\mathbb{R}^n : h(\boldsymbol{x}) \ge 0\},$$3 renders $$\mathbb{C} := \{\boldsymbol{x}\in\mathbb{R}^n : h(\boldsymbol{x}) \ge 0\},$$4 forward invariant when $$\mathbb{C} := \{\boldsymbol{x}\in\mathbb{R}^n : h(\boldsymbol{x}) \ge 0\},$$5 (Wang et al., 25 Jul 2025).

The proof pattern mirrors the RRCBF case. Along system trajectories,

$$\mathbb{C} := \{\boldsymbol{x}\in\mathbb{R}^n : h(\boldsymbol{x}) \ge 0\},$$6

and the DO-RRCBF constraint yields

$$\mathbb{C} := \{\boldsymbol{x}\in\mathbb{R}^n : h(\boldsymbol{x}) \ge 0\},$$7

The residual disturbance is therefore not the full unknown disturbance but the estimation error. Because that error is bounded through the disturbance observer design, the reciprocal term remains able to dominate near the boundary and enforce $$\mathbb{C} := \{\boldsymbol{x}\in\mathbb{R}^n : h(\boldsymbol{x}) \ge 0\},$$8 (Wang et al., 25 Jul 2025).

A common misunderstanding is to treat the observer estimate as if it were itself the safety guarantee. In the DO-RRCBF construction, safety is not delegated to observer convergence alone. The observer attenuates the effective disturbance, but strict safety still relies on the reciprocal term dominating the bounded residual $$\mathbb{C} := \{\boldsymbol{x}\in\mathbb{R}^n : h(\boldsymbol{x}) \ge 0\},$$9.

4. Conservatism reduction and relation to adjacent robust CBF formulations

The main comparison in the DO-RRCBF paper is among four barrier styles. A robust CBF in the sense of Janković 2018 uses a fixed worst-case disturbance term $${\rm Int}(\mathbb{C}) := \{\boldsymbol{x}\in\mathbb{R}^n : h(\boldsymbol{x}) > 0\}.$$0, which is conservative when the disturbance bound is large or poorly known. A DO-CBF of the Das type uses $${\rm Int}(\mathbb{C}) := \{\boldsymbol{x}\in\mathbb{R}^n : h(\boldsymbol{x}) > 0\}.$$1 together with an explicit error-envelope term $${\rm Int}(\mathbb{C}) := \{\boldsymbol{x}\in\mathbb{R}^n : h(\boldsymbol{x}) > 0\}.$$2, reducing conservatism as the observer converges but still requiring prior disturbance information and a designed estimation-error bound. Pure RRCBF removes explicit disturbance bounds altogether, but the reciprocal term is always active and may remain conservative, especially far from the boundary. DO-RRCBF combines direct disturbance compensation with the reciprocal term and does so without an explicit $${\rm Int}(\mathbb{C}) := \{\boldsymbol{x}\in\mathbb{R}^n : h(\boldsymbol{x}) > 0\}.$$3 term in the barrier condition (Wang et al., 25 Jul 2025).

The intended reduction in conservatism is therefore specific. When $${\rm Int}(\mathbb{C}) := \{\boldsymbol{x}\in\mathbb{R}^n : h(\boldsymbol{x}) > 0\}.$$4, the effective disturbance in the barrier dynamics is small, so control action can stay closer to nominal. The reciprocal term then mainly guards against residual estimation error rather than against the full disturbance. The paper states three key points: DO-RRCBF does not require a priori disturbance bounds $${\rm Int}(\mathbb{C}) := \{\boldsymbol{x}\in\mathbb{R}^n : h(\boldsymbol{x}) > 0\}.$$5 or a designed function $${\rm Int}(\mathbb{C}) := \{\boldsymbol{x}\in\mathbb{R}^n : h(\boldsymbol{x}) > 0\}.$$6; it still guarantees strict safety because $${\rm Int}(\mathbb{C}) := \{\boldsymbol{x}\in\mathbb{R}^n : h(\boldsymbol{x}) > 0\}.$$7 is chosen to dominate bounded estimation error; and it is less conservative than RRCBF because disturbance compensation is explicit (Wang et al., 25 Jul 2025).

Related DOB-CBF literature clarifies how unusual this combination is. Earlier disturbance-observer-based robust CBF designs tightened the barrier inequality using quantified transient and steady-state observer-error bounds (Alan et al., 2022), or incorporated disturbance estimates together with robust margins derived from DOB error bounds for matched and mismatched disturbances (Wang et al., 2022). A different line, disturbance observer-parameterized CBFs, parameterizes the barrier itself by the disturbance estimate and proves forward invariance through a composite barrier $${\rm Int}(\mathbb{C}) := \{\boldsymbol{x}\in\mathbb{R}^n : h(\boldsymbol{x}) > 0\}.$$8 and a robust term $${\rm Int}(\mathbb{C}) := \{\boldsymbol{x}\in\mathbb{R}^n : h(\boldsymbol{x}) > 0\}.$$9 (Yang et al., 2024). DO-RRCBF differs in that its primary robustness mechanism remains the reciprocal resistance term, while the observer is used to reduce the effective disturbance seen by that term (Wang et al., 25 Jul 2025).

The main controversy in practice is not whether observer-based compensation helps, but where the remaining safety margin should reside. DO-RRCBF places it in the reciprocal barrier geometry. DOB-CBF formulations of the earlier type place it in an explicit worst-case residual bound. DOp-CBF places part of it in an adaptive barrier parameterization. These are distinct robustification strategies, even when all three use disturbance observers.

5. High-order extension and quadratic-program realization

For constraints of relative degree ${\rm Int}(\mathbb{C})$0, the paper defines the chain

${\rm Int}(\mathbb{C})$1

with extended class ${\rm Int}(\mathbb{C})$2 functions ${\rm Int}(\mathbb{C})$3. Let ${\rm Int}(\mathbb{C})$4 denote the associated superlevel sets and

${\rm Int}(\mathbb{C})$5

Assuming input relative degree ${\rm Int}(\mathbb{C})$6, a high-order RRCBF satisfies

${\rm Int}(\mathbb{C})$7

for all ${\rm Int}(\mathbb{C})$8, where

${\rm Int}(\mathbb{C})$9

Any Lipschitz controller in the corresponding admissible set renders $\mathbb{C}$0 forward invariant (Wang et al., 25 Jul 2025).

Controller synthesis is implemented by a standard CBF-QP safety filter. For RRCBF with nominal controller $\mathbb{C}$1, $$\boldsymbol{x} \in \mathbb{X} \subseteq \mathbb{R}^n$$27 For DO-RRCBF, the constraint is modified by replacing $\mathbb{C}$2 with $\mathbb{C}$3: $\mathbb{C}$4 The paper emphasizes that this is a convex quadratic program with a single linear constraint and can be solved efficiently in real time. For high-relative-degree constraints, the HO-RRCBF expression supplies the linear constraint in the QP (Wang et al., 25 Jul 2025).

This implementation places DO-RRCBF within a broader family of disturbance-observer barrier filters for high-relative-degree systems. Related work on disturbance observer-based integral control barrier functions similarly augments the dynamics with observer states, derives a time-varying bound on estimation error, and inserts that bound into a QP-based safety filter for nonlinear systems with high relative degree (Zinage et al., 2023). The direct formulations differ, but the architectural pattern is the same: nominal control, observer-based disturbance attenuation, and a convex safety projection.

6. Numerical demonstrations and broader research setting

The paper validates the reciprocal-resistance framework with two simulation studies. The first is the disturbed second-order linear system

$\mathbb{C}$5

with safety function $\mathbb{C}$6, safe set $\mathbb{C}$7, nominal controller $\mathbb{C}$8, and QP parameters $\mathbb{C}$9, $$\boldsymbol{x} \in \mathbb{X} \subseteq \mathbb{R}^n$$00. Without disturbance, ZCBF, RCBF with $$\boldsymbol{x} \in \mathbb{X} \subseteq \mathbb{R}^n$$01, and RRCBF all enforce safety, but RRCBF creates the invariant inner set $$\boldsymbol{x} \in \mathbb{X} \subseteq \mathbb{R}^n$$02. With disturbance $$\boldsymbol{x} \in \mathbb{X} \subseteq \mathbb{R}^n$$03, ZCBF and RCBF fail to maintain invariance of $$\boldsymbol{x} \in \mathbb{X} \subseteq \mathbb{R}^n$$04, while RRCBF maintains invariance of $$\boldsymbol{x} \in \mathbb{X} \subseteq \mathbb{R}^n$$05 and prevents $$\boldsymbol{x} \in \mathbb{X} \subseteq \mathbb{R}^n$$06 from approaching zero. The parameter $$\boldsymbol{x} \in \mathbb{X} \subseteq \mathbb{R}^n$$07 tunes the conservatism of the buffer set $$\boldsymbol{x} \in \mathbb{X} \subseteq \mathbb{R}^n$$08; smaller $$\boldsymbol{x} \in \mathbb{X} \subseteq \mathbb{R}^n$$09 enlarges $$\boldsymbol{x} \in \mathbb{X} \subseteq \mathbb{R}^n$$10 (Wang et al., 25 Jul 2025).

The second example is adaptive cruise control with dynamics

$$\boldsymbol{x} \in \mathbb{X} \subseteq \mathbb{R}^n$$11

where the safety objective is $$\boldsymbol{x} \in \mathbb{X} \subseteq \mathbb{R}^n$$12 with barrier $$\boldsymbol{x} \in \mathbb{X} \subseteq \mathbb{R}^n$$13, which has relative degree $$\boldsymbol{x} \in \mathbb{X} \subseteq \mathbb{R}^n$$14 with respect to $$\boldsymbol{x} \in \mathbb{X} \subseteq \mathbb{R}^n$$15. The disturbance is

$$\boldsymbol{x} \in \mathbb{X} \subseteq \mathbb{R}^n$$16

the control bounds are $$\boldsymbol{x} \in \mathbb{X} \subseteq \mathbb{R}^n$$17, and the parameters are $$\boldsymbol{x} \in \mathbb{X} \subseteq \mathbb{R}^n$$18, $$\boldsymbol{x} \in \mathbb{X} \subseteq \mathbb{R}^n$$19, $$\boldsymbol{x} \in \mathbb{X} \subseteq \mathbb{R}^n$$20, $$\boldsymbol{x} \in \mathbb{X} \subseteq \mathbb{R}^n$$21, $$\boldsymbol{x} \in \mathbb{X} \subseteq \mathbb{R}^n$$22, and observer gain $$\boldsymbol{x} \in \mathbb{X} \subseteq \mathbb{R}^n$$23. Five controllers are compared: standard CBF, robust CBF, DO-CBF without a conservative estimation-error bound in the constraint, RRCBF, and DO-RRCBF. The reported observations are that naive CBF and the improperly designed DO-CBF can violate the safety distance, RoCBF and RRCBF ensure strict safety but are more conservative, and DO-RRCBF maintains strict safety while recovering nominal control performance better than RoCBF and RRCBF. In particular, $$\boldsymbol{x} \in \mathbb{X} \subseteq \mathbb{R}^n$$24 tracks $$\boldsymbol{x} \in \mathbb{X} \subseteq \mathbb{R}^n$$25 closely, $$\boldsymbol{x} \in \mathbb{X} \subseteq \mathbb{R}^n$$26 stays nonnegative with a smaller safety margin than RoCBF and RRCBF, and the control input stays within limits and is less aggressive than purely robust designs (Wang et al., 25 Jul 2025).

Within the broader literature, DO-RRCBF belongs to an expanding disturbance-observer safety landscape. Disturbance-observer-based CBF filters have been used for safe and efficient reinforcement learning, where the RL action is minimally modified by a QP whose robust CBF constraint uses the observer estimate and its error bound (Cheng et al., 2022). Related work has combined residual model learning and DOBs inside robust CBF safety filters for safe RL under internal and external disturbances (Kalaria et al., 2024). In discrete-time linear settings, the disturbance-estimation layer itself can be analyzed through a set-membership lens: bounded disturbance observers exist if and only if a rank condition holds, and an SMF-based disturbance observer is bounded if and only if bounded DOs exist (Li et al., 2023). A plausible implication is that future discrete-time or sampled-data DO-RRCBF designs may benefit from importing such observer-existence checks before barrier synthesis.

DO-RRCBF is therefore best understood not as an isolated barrier variant, but as a specific synthesis of three mechanisms: barrier-based forward invariance, reciprocal resistance-induced buffering near the safety boundary, and disturbance observer compensation that shifts the robust burden from the full disturbance to the residual estimation error. Its defining claim is precisely that this combination can maintain strict safety under bounded disturbances while bringing the closed-loop input closer to nominal behavior than either worst-case robust CBFs or observer-free RRCBFs (Wang et al., 25 Jul 2025).