Discrete-Time High-Order CBFs

• DHCBFs are safety certificates for discrete-time systems that use a hierarchy of auxiliary functions to enforce multi-step forward invariance of safe sets.
• They extend standard control barrier functions to handle arbitrary relative degrees, reducing conservatism in model predictive control applications.
• Their formulation enables convexification and distributed implementation, highlighting a trade-off between computational speed and feasibility.

Searching arXiv for the cited DHCBF literature to ground the article. First, I’ll look up the 2022 DHOCBF-MPC paper and a few adjacent discrete-time CBF papers. Discrete-Time High-Order Control Barrier Functions (DHCBFs), also termed Discrete-Time High Order Control Barrier Functions (DHOCBFs), are barrier-based safety certificates for discrete-time dynamical systems whose safety outputs have arbitrary relative degree. In the formulation developed for the discrete-time nonlinear system $x_{t+1}=l(x_t,u_t)$, a scalar safety function $h:\mathbb{R}^n\to\mathbb{R}$ defines the safe set $\mathcal{C}=\{x\in\mathbb{R}^n\mid h(x)\ge 0\}$, and DHCBFs generalize relative-degree-one discrete-time control barrier functions by constructing a hierarchy of auxiliary functions whose highest-order inequality guarantees forward invariance of the intersection of associated safe sets (Liu et al., 2022). The framework is motivated by safety-critical control and model predictive control (MPC), where many discrete-time systems and constraints exhibit higher relative degree, so enforcing only a one-step barrier condition on $h$ is either inadequate or overly conservative (Liu et al., 2022).

1. Formal setting and core definition

The canonical discrete-time model is

$x_{t+1}=l(x_t,u_t),$

where $x_t\in\mathcal{X}\subset\mathbb{R}^n$, $u_t\in\mathcal{U}\subset\mathbb{R}^q$, and $l:\mathcal{X}\times\mathcal{U}\to\mathbb{R}^n$ is locally Lipschitz (Liu et al., 2022). Safety is encoded by a scalar function $h:\mathbb{R}^n\to\mathbb{R}$, with safe set

$$\mathcal{C}\coloneqq \{x\in\mathbb{R}^n\mid h(x)\ge 0\}.$$

The discrete-time safety requirement is $h:\mathbb{R}^n\to\mathbb{R}$0 for all $h:\mathbb{R}^n\to\mathbb{R}$1, equivalently $h:\mathbb{R}^n\to\mathbb{R}$2 for all $h:\mathbb{R}^n\to\mathbb{R}$3 (Liu et al., 2022).

For relative degree $h:\mathbb{R}^n\to\mathbb{R}$4, DHCBFs introduce the sequence

$h:\mathbb{R}^n\to\mathbb{R}$5

and, for $h:\mathbb{R}^n\to\mathbb{R}$6,

$h:\mathbb{R}^n\to\mathbb{R}$7

with associated sets

$h:\mathbb{R}^n\to\mathbb{R}$8

(Liu et al., 2022). Thus $h:\mathbb{R}^n\to\mathbb{R}$9 is the original safety set, while higher $\mathcal{C}=\{x\in\mathbb{R}^n\mid h(x)\ge 0\}$0 encode multi-step conditions on the evolution of $\mathcal{C}=\{x\in\mathbb{R}^n\mid h(x)\ge 0\}$1.

The formal DHOCBF definition is: $\mathcal{C}=\{x\in\mathbb{R}^n\mid h(x)\ge 0\}$2 is a discrete-time High Order Control Barrier Function with relative degree $\mathcal{C}=\{x\in\mathbb{R}^n\mid h(x)\ge 0\}$3 for $\mathcal{C}=\{x\in\mathbb{R}^n\mid h(x)\ge 0\}$4 if, for all $\mathcal{C}=\{x\in\mathbb{R}^n\mid h(x)\ge 0\}$5,

$\mathcal{C}=\{x\in\mathbb{R}^n\mid h(x)\ge 0\}$6

(Liu et al., 2022). This is an exponential-type condition on the highest-order auxiliary function.

A closely related discrete-time higher-relative-degree construction appears in distributed safety-critical MPC for nonlinear multi-agent systems, where, for each agent, one defines

$\mathcal{C}=\{x\in\mathbb{R}^n\mid h(x)\ge 0\}$7

$\mathcal{C}=\{x\in\mathbb{R}^n\mid h(x)\ge 0\}$8

with

$\mathcal{C}=\{x\in\mathbb{R}^n\mid h(x)\ge 0\}$9

and $h$0, $h$1 (Wang et al., 27 Aug 2025). In that formulation, $h$2 is a DHCBF if $h$3 on $h$4, where $h$5 (Wang et al., 27 Aug 2025).

2. Forward invariance and relative degree

The defining property of DHCBFs is forward invariance. In the 2022 DHOCBF framework, if $h$6 and the control satisfies

$h$7

for all $h$8, then $h$9 is forward invariant for $x_{t+1}=l(x_t,u_t),$0 (Liu et al., 2022). The recursive construction propagates nonnegativity backward through the hierarchy of $x_{t+1}=l(x_t,u_t),$1, so nonnegativity of the highest-order term yields nonnegativity of all lower-order terms (Liu et al., 2022).

The relative-degree interpretation is central. In discrete time, the relative degree of $x_{t+1}=l(x_t,u_t),$2 with respect to $x_{t+1}=l(x_t,u_t),$3 is informally the smallest number of steps after which $x_{t+1}=l(x_t,u_t),$4 directly affects the evolution of a function of $x_{t+1}=l(x_t,u_t),$5 (Liu et al., 2022). The multi-agent DSMPC formulation states this through derivatives of composed maps: the relative degree of the output $x_{t+1}=l(x_t,u_t),$6 with respect to $x_{t+1}=l(x_t,u_t),$7 is $x_{t+1}=l(x_t,u_t),$8 if

$x_{t+1}=l(x_t,u_t),$9

and

$x_t\in\mathcal{X}\subset\mathbb{R}^n$0

(Wang et al., 27 Aug 2025). This is the discrete-time analogue of high relative degree in continuous-time HOCBF theory.

The framework also supports a weaker enforcement principle. The 2022 paper remarks that satisfying the $x_t\in\mathcal{X}\subset\mathbb{R}^n$1-th order CBF constraint is sufficient to render $x_t\in\mathcal{X}\subset\mathbb{R}^n$2 forward invariant, so it is not strictly necessary to go all the way to order $x_t\in\mathcal{X}\subset\mathbb{R}^n$3 in an optimal control problem; one can choose a suitable $x_t\in\mathcal{X}\subset\mathbb{R}^n$4 to reduce computational complexity (Liu et al., 2022). This suggests a design trade-off between barrier order and solver burden.

3. Relation to standard discrete-time CBFs, sampled-data CBFs, and continuous-time HOCBFs

The relative-degree-one discrete-time CBF condition used in prior MPC-DCBF and NMPC-DCBF formulations is

$x_t\in\mathcal{X}\subset\mathbb{R}^n$5

with slack variable $x_t\in\mathcal{X}\subset\mathbb{R}^n$6 (Liu et al., 2022). When $x_t\in\mathcal{X}\subset\mathbb{R}^n$7, the DHOCBF recursion collapses to $x_t\in\mathcal{X}\subset\mathbb{R}^n$8, and the highest-order condition applies directly to $x_t\in\mathcal{X}\subset\mathbb{R}^n$9, so the relative-degree-one case is recovered (Liu et al., 2022). The distinction is therefore not categorical but structural: DHCBFs lift the safety condition into auxiliary functions so that systems with multi-step input influence can still be constrained through an exponential-type inequality.

This differs from the discrete-time CBF notion that is necessary and sufficient for controlled invariance in a one-step sense,

$u_t\in\mathcal{U}\subset\mathbb{R}^q$0

with safe input set

$u_t\in\mathcal{U}\subset\mathbb{R}^q$1

(Cavorsi et al., 2020). That formulation is explicitly least restrictive among several discrete-time barrier inequalities discussed there, because prior relaxed forms imply it but not conversely (Cavorsi et al., 2020). By contrast, DHCBFs deliberately introduce higher-order auxiliary conditions to account for relative degree rather than to relax one-step invariance.

Sampled-data CBF work provides a parallel discrete-time viewpoint even when the plant evolves continuously between samples. In that setting, a sampled-data barrier condition constrains the increment of $u_t\in\mathcal{U}\subset\mathbb{R}^q$2 over one sample,

$u_t\in\mathcal{U}\subset\mathbb{R}^q$3

and practical safety is established for the exact sampled-data system when the approximate discrete-time model is one-step consistent with the exact flow (Taylor et al., 2022). Although that framework does not explicitly define DHCBFs, it provides a direct bridge from sampled implementation to discrete-time barrier design and shows how approximate discrete-time models and Runge–Kutta maps can support convex synthesis for higher-relative-degree structures (Taylor et al., 2022).

Continuous-time HOCBFs instead use derivatives and Lie derivatives,

$u_t\in\mathcal{U}\subset\mathbb{R}^q$4

and enforce a highest-order inequality involving $u_t\in\mathcal{U}\subset\mathbb{R}^q$5 and $u_t\in\mathcal{U}\subset\mathbb{R}^q$6 (Aali et al., 2024). Discrete-time DHCBFs replace derivatives with finite differences and use inequalities like

$u_t\in\mathcal{U}\subset\mathbb{R}^q$7

or the equivalent multi-step Taylor-based conditions discussed below (Liu et al., 2022, Xu et al., 19 Mar 2025).

4. Embedding in model predictive control

A principal application of DHCBFs is safety-critical MPC. The baseline nonlinear MPC with a relative-degree-one discrete-time CBF minimizes

$u_t\in\mathcal{U}\subset\mathbb{R}^q$8

subject to dynamics, input/state constraints, and

$u_t\in\mathcal{U}\subset\mathbb{R}^q$9

over the horizon (Liu et al., 2022). The DHOCBF extension preserves the MPC structure but replaces the one-step barrier with high-order recursion on a candidate barrier $l:\mathcal{X}\times\mathcal{U}\to\mathbb{R}^n$0 and its auxiliary functions $l:\mathcal{X}\times\mathcal{U}\to\mathbb{R}^n$1 (Liu et al., 2022).

In the iterative MPC-DCBF formulation, the dynamics and barrier functions are linearized at each time step. The nonlinear dynamics are approximated by

$l:\mathcal{X}\times\mathcal{U}\to\mathbb{R}^n$2

with

$l:\mathcal{X}\times\mathcal{U}\to\mathbb{R}^n$3

(Liu et al., 2022). The nonlinear safety function is replaced by a linearized obstacle boundary $l:\mathcal{X}\times\mathcal{U}\to\mathbb{R}^n$4, obtained from the tangent of the obstacle boundary at an intersection point $l:\mathcal{X}\times\mathcal{U}\to\mathbb{R}^n$5, and this linearized function becomes $l:\mathcal{X}\times\mathcal{U}\to\mathbb{R}^n$6 in the high-order recursion (Liu et al., 2022).

To improve feasibility, per-order, per-time slack variables $l:\mathcal{X}\times\mathcal{U}\to\mathbb{R}^n$7 are introduced: $l:\mathcal{X}\times\mathcal{U}\to\mathbb{R}^n$8 (Liu et al., 2022). Because this form can remain nonconvex, the paper derives a linear convex reparameterization,

$l:\mathcal{X}\times\mathcal{U}\to\mathbb{R}^n$9

where $h:\mathbb{R}^n\to\mathbb{R}$0 depend only on $h:\mathbb{R}^n\to\mathbb{R}$1 and $h:\mathbb{R}^n\to\mathbb{R}$2 (Liu et al., 2022). This yields linear and convex DHOCBF constraints over the MPC horizon.

A distributed variant for nonlinear multi-agent systems uses DHCBFs inside distributed safety-critical MPC alongside discrete-time control Lyapunov functions (DCLFs). Each agent enforces

$h:\mathbb{R}^n\to\mathbb{R}$3

as a hard MPC constraint, with neighbor states replaced by estimated neighbor trajectories to enable distributed implementation (Wang et al., 27 Aug 2025). The resulting DSMPC formulation includes dynamics, box constraints, DHCBF constraints, compatibility bounds on estimation errors, and DCLF-based terminal constraints (Wang et al., 27 Aug 2025).

5. Computational structure, convexity, and performance

The computational challenge of DHCBFs is explicit in the literature: nonlinear dynamics and nonlinear barrier functions render MPC nonconvex, while higher-order barriers introduce more auxiliary functions and more constraints across larger horizons (Liu et al., 2022). The iterative convexification strategy addresses this by solving a sequence of convex subproblems based on local linearization (Liu et al., 2022).

The reported numerical benchmark compares NMPC-DCBF and iMPC-DCBF for relative degree $h:\mathbb{R}^n\to\mathbb{R}$4 and $h:\mathbb{R}^n\to\mathbb{R}$5, with horizons $h:\mathbb{R}^n\to\mathbb{R}$6 (Liu et al., 2022). The key observations are summarized below.

Aspect	Observation	Context
Computation time	iMPC-DCBF achieves about 10x faster computation times	Compared with NMPC-DCBF
Infeasibility rate	iMPC-DCBF has about 30% more infeasibility	Trade-off with speed
Relative degree handling	Both $h:\mathbb{R}^n\to\mathbb{R}$7 and $h:\mathbb{R}^n\to\mathbb{R}$8 are evaluated	High-order handled in iMPC

These results indicate a speed–feasibility trade-off rather than a universal dominance relation (Liu et al., 2022). In distributed MPC, simulation results similarly report improved performance and reduced computation time relative to MPC with distance constraints, while retaining obstacle avoidance and formation control feasibility under a mild assumption (Wang et al., 27 Aug 2025). This suggests that DHCBFs can reduce the need for longer horizons by imposing more informative safety structure at each step.

A different computational route appears in sampled-data discrete-time CBF synthesis using approximate discrete-time models. For systems with block-integrator structure, if the barrier depends on the first $h:\mathbb{R}^n\to\mathbb{R}$9 blocks and is concave in its last argument, then choosing Runge–Kutta order $$\mathcal{C}\coloneqq \{x\in\mathbb{R}^n\mid h(x)\ge 0\}.$$0 makes the function

$$\mathcal{C}\coloneqq \{x\in\mathbb{R}^n\mid h(x)\ge 0\}.$$1

convex in $$\mathcal{C}\coloneqq \{x\in\mathbb{R}^n\mid h(x)\ge 0\}.$$2 (Taylor et al., 2022). This result is not stated as a DHCBF theorem, but it supplies a convexity template for discrete-time higher-relative-degree barriers.

6. Alternative formulations, robustness, and verification

Several adjacent directions broaden the DHCBF landscape.

A truncated Taylor formulation proposes a discrete-time high-order condition by approximating the one-step barrier increment with derivatives up to the relative degree $$\mathcal{C}\coloneqq \{x\in\mathbb{R}^n\mid h(x)\ge 0\}.$$3. For high-order safety constraints, the Truncated Taylor CBF (TTCBF) enforces

$$\mathcal{C}\coloneqq \{x\in\mathbb{R}^n\mid h(x)\ge 0\}.$$4

and requires only one class-$$\mathcal{C}\coloneqq \{x\in\mathbb{R}^n\mid h(x)\ge 0\}.$$5 function, independent of the relative degree (Xu et al., 21 Jan 2026). The earlier continuous-time-to-discrete-time exposition of TTCBF presents the same design motivation: standard HOCBFs require multiple class-$$\mathcal{C}\coloneqq \{x\in\mathbb{R}^n\mid h(x)\ge 0\}.$$6 functions whose tuning scales with relative degree, whereas the truncated Taylor approach reduces design complexity while ensuring safety in sampled-data settings (Xu et al., 19 Mar 2025). This suggests a distinct DHCBF interpretation: rather than constructing auxiliary forward-difference sequences, one can enforce an approximate $$\mathcal{C}\coloneqq \{x\in\mathbb{R}^n\mid h(x)\ge 0\}.$$7-step discrete-time CBF condition using truncated Taylor expansions.

Robust sampled-data CBF methods with zero-order hold and input delay offer complementary tools. They derive CBF constraints affine in the input by evaluating barrier inequalities at predicted delayed states and robustifying them over reachable sets or uncertainty sets (Singletary et al., 2020). The paper explicitly notes that it does not define DHCBFs, but it provides a robust sampled-data barrier framework and delay-handling methodology directly useful for discrete-time high-order barrier design under ZOH, delays, and uncertainty (Singletary et al., 2020).

Verification has also become a distinct topic. An optimization-based branch-and-bound method verifies or falsifies candidate discrete-time CBFs by solving global nonconvex optimization problems over the state set and admissible inputs (Shakhesi et al., 2024). Although it targets first-order DTCBFs, the paper states that the same machinery extends conceptually to DHCBFs by replacing the single-step expression $$\mathcal{C}\coloneqq \{x\in\mathbb{R}^n\mid h(x)\ge 0\}.$$8 with multi-step barrier expressions involving iterated dynamics and control sequences (Shakhesi et al., 2024). This suggests that high-order barrier verification is an optimization problem over more complicated but structurally similar inequalities.

Robust discrete-time CBF synthesis under bounded disturbances uses a sufficient condition

$$\mathcal{C}\coloneqq \{x\in\mathbb{R}^n\mid h(x)\ge 0\}.$$9

to replace infinitely many disturbance constraints by a single inequality involving a Lipschitz constant and a disturbance bound (Shakhesi et al., 16 Jun 2025). That work does not explicitly define DHCBFs, but it provides a counterexample-guided synthesis and verification loop for robust discrete-time barrier functions, which a plausible extension would carry over to multi-step or higher-order barrier conditions.

Sample-efficient certification of discrete-time CBFs via Lipschitz arguments similarly focuses on first-order conditions,

$h:\mathbb{R}^n\to\mathbb{R}$00

and certifies them on the whole zero-sublevel set using $h:\mathbb{R}^n\to\mathbb{R}$01-nets and level-wise segmentation (Mulagaleti et al., 4 Sep 2025). The paper explicitly notes that the method can be generalized to multi-step maps $h:\mathbb{R}^n\to\mathbb{R}$02 and higher-order discrete-time barrier conditions involving forward differences and multi-step compositions (Mulagaleti et al., 4 Sep 2025). This suggests a verification pipeline for DHCBFs based on multi-step Lipschitz constants.

7. Related constructions and open directions

A distinct branch concerns synthesis rather than online enforcement. For control-affine polynomial discrete-time systems with input constraints and semi-algebraic safe sets, an alternating-descent sum-of-squares framework synthesizes quadratic DTCBFs and polynomial control policies, then extends to higher-degree polynomial DTCBFs (Shakhesi et al., 27 Apr 2025). That work does not use “high-order” in the relative-degree sense, but it directly addresses higher-degree polynomial barrier construction in discrete time and is relevant whenever DHCBF implementations need expressive polynomial certificates (Shakhesi et al., 27 Apr 2025).

Discrete-time CBF compositions also matter. A tractable formulation for partially control-affine discrete-time systems introduces nonlinear DT-CBFs that remain affine in control and develops mixed-integer constraints for conjunction, disjunction, implication, exclusive OR, equivalence, and piecewise barrier functions (Cavorsi et al., 2020). This provides a route to expressing mode-dependent safety logic, which a plausible implication is especially useful when DHCBFs must switch between obstacle-avoidance modes or region-specific higher-order constraints.

Beyond model-based constructions, a continuous-time model-free approach to higher-order CBFs for systems of arbitrary relative degree uses funnel control to construct vector-valued auxiliary variables $h:\mathbb{R}^n\to\mathbb{R}$03 and a first-order zeroing CBF

$h:\mathbb{R}^n\to\mathbb{R}$04

for the highest-order auxiliary variable (Lanza et al., 3 Jun 2026). The paper explicitly demonstrates that a straightforward extension of relative-degree-one model-free CBF design to higher-order systems fails, and replaces scalar HOCBF recursion with a funnel-based recursive structure (Lanza et al., 3 Jun 2026). While the treatment is continuous-time, this suggests that discrete-time higher-order barrier design need not always mirror the standard HOCBF recursion; alternative recursive transforms may be viable.

Across the literature, the recurring issues are clear. DHCBFs are introduced to handle arbitrary relative degree in discrete time (Liu et al., 2022, Wang et al., 27 Aug 2025), but this raises computational complexity, tuning complexity, and feasibility concerns (Liu et al., 2022, Xu et al., 19 Mar 2025). Sampled-data formulations emphasize the gap between continuous-time theory and digital implementation (Singletary et al., 2020, Taylor et al., 2022). Verification and synthesis remain nontrivial global optimization problems (Shakhesi et al., 2024, Shakhesi et al., 27 Apr 2025, Shakhesi et al., 16 Jun 2025, Mulagaleti et al., 4 Sep 2025).

This suggests three broad research directions already visible in the cited works. One is simplification of high-order design, exemplified by TTCBFs requiring only one class-$h:\mathbb{R}^n\to\mathbb{R}$05 function (Xu et al., 19 Mar 2025, Xu et al., 21 Jan 2026). A second is tractable synthesis and verification, through convexification, branch-and-bound, or sum-of-squares methods (Liu et al., 2022, Shakhesi et al., 2024, Shakhesi et al., 27 Apr 2025). A third is robustness to sampled implementation artifacts, delays, uncertainty, and distributed coupling (Singletary et al., 2020, Taylor et al., 2022, Wang et al., 27 Aug 2025).

In that sense, DHCBFs are best understood not as a single formula but as a family of discrete-time safety constructions for higher-relative-degree constraints. Their common objective is the same: to make the safe set, or an auxiliary hierarchy of safe sets, forward invariant under discrete-time dynamics when the control acts on the safety output only through multi-step evolution.