One-Step CBF-CLF QP

• One-Step CBF-CLF QP is a convex optimization framework for immediate control actions, enforcing safety via CBFs and stability via CLFs in control-affine systems.
• It formulates a quadratic program with hard safety constraints and a softened stability constraint, ensuring forward invariance and safe operation.
• The approach supports extensions for robustness, learning, and probabilistic adaptations, making it effective for real-time robotics and aerospace applications.

A One-Step CBF-CLF Quadratic Program is a per-sample or per-event convex optimization problem for control-affine systems that enforces safety through Control Barrier Function constraints and stabilization or tracking through Control Lyapunov Function constraints on a single control action at the current state, rather than over a prediction horizon. In the standard formulation, the system is written as $\dot{x} = f(x) + g(x)u$, the CBF constraint is kept hard to preserve forward invariance of a safe set, and the CLF constraint is often softened by a slack variable so that safety takes priority when the two objectives conflict (Reis et al., 2024, Wong et al., 19 Mar 2026). In the literature, the same “one-step” label also covers event-triggered or sample-and-hold implementations that solve the optimization once at an update instant and then hold the input over a certified interval (Yang et al., 2019).

1. Canonical formulation

For a control-affine system

$$\dot{x} = f(x) + g(x)u,\quad x\in\mathbb{R}^n,\ u\in\mathbb{R}^m,$$

the standard one-step CBF-CLF-QP computes the control by solving, at the current state, a strictly convex quadratic objective subject to affine safety and stability inequalities. A generic formulation used in the literature is

$$\begin{aligned} &\min_{u,\delta}\quad \tfrac{1}{2}u^\top H u + f_q^\top u + \tfrac{w}{2}\delta^2 \ \text{s.t.}\quad &L_f V(x)+L_g V(x)u \le -c\,V(x)+\delta,\ &L_f h_i(x)+L_g h_i(x)u+\alpha_i(h_i(x)) \ge 0,\quad i=1,\dots,p,\ &u\in\mathcal{U}, \end{aligned}$$

where $V$ is a CLF, $h_i$ are CBFs, $H\succ 0$ weights control effort, $c>0$ sets the CLF decay rate, and $w>0$ penalizes relaxation of the CLF constraint (Reis et al., 2024). A closely related baseline minimizes $\|u-u_{\mathrm{nom}}\|_2^2 + p_\delta \delta^2$ subject to the same structure, with $u_{\mathrm{nom}}$ interpreted as a nominal or reference control when such a signal is available (Wong et al., 19 Mar 2026).

The asymmetry between the constraints is central. The CBF inequality is ordinarily hard, because it encodes forward invariance of the safe set

$$\dot{x} = f(x) + g(x)u,\quad x\in\mathbb{R}^n,\ u\in\mathbb{R}^m,$$0

whereas the CLF inequality is commonly softened by $$\dot{x} = f(x) + g(x)u,\quad x\in\mathbb{R}^n,\ u\in\mathbb{R}^m,$$1 so that feasibility can be retained near safety boundaries or under input limits (Wong et al., 19 Mar 2026). This architecture appears in min-norm, reference-tracking, and filter-based forms. Some implementations omit the slack entirely and instead solve

$$\dot{x} = f(x) + g(x)u,\quad x\in\mathbb{R}^n,\ u\in\mathbb{R}^m,$$2

subject to an ECBF constraint and a hard CLF decrease condition, which yields a simpler but less forgiving feasible set (Yang et al., 2019).

The objective is not unique. In some applications the QP is explicitly a minimum-norm projector onto the intersection of CLF and CBF half-spaces; in others it minimizes deviation from a learned policy or a legacy controller. A plausible implication is that the phrase “one-step CBF-CLF-QP” identifies the constraint architecture more than a single cost function: the shared structure is instantaneous optimization over one control move with CBF-enforced safety and CLF-shaped performance.

2. Meaning of “one-step”

The “one-step” qualifier distinguishes these controllers from horizon-based methods such as MPC. The optimization is solved at the current time only, producing one control input that is applied immediately; the procedure is then repeated at the next update (Meinert et al., 19 Mar 2026). In spacecraft close-proximity operations, for example, the runtime filter is described as “one-step” precisely because it optimizes over a single control input $$\dot{x} = f(x) + g(x)u,\quad x\in\mathbb{R}^n,\ u\in\mathbb{R}^m,$$3 at the current time, not over a prediction horizon, and minimally adjusts a learned command to satisfy hard safety constraints while encouraging stability (Meinert et al., 19 Mar 2026).

A stricter interpretation appears in self-triggered implementations, where the QP is solved once at an event time $$\dot{x} = f(x) + g(x)u,\quad x\in\mathbb{R}^n,\ u\in\mathbb{R}^m,$$4, yielding $$\dot{x} = f(x) + g(x)u,\quad x\in\mathbb{R}^n,\ u\in\mathbb{R}^m,$$5, and the input is held under Zeroth-Order Hold: $$\dot{x} = f(x) + g(x)u,\quad x\in\mathbb{R}^n,\ u\in\mathbb{R}^m,$$6 The update time is then determined by certified bounds rather than by a fixed clock. In this setting, the “safe period” $$\dot{x} = f(x) + g(x)u,\quad x\in\mathbb{R}^n,\ u\in\mathbb{R}^m,$$7 is computed so that the CBF inequality remains satisfied continuously over the hold interval, while a separate $$\dot{x} = f(x) + g(x)u,\quad x\in\mathbb{R}^n,\ u\in\mathbb{R}^m,$$8 bounds how long the CLF can continue decreasing under the held control; the next event occurs at

$$\dot{x} = f(x) + g(x)u,\quad x\in\mathbb{R}^n,\ u\in\mathbb{R}^m,$$9

This construction was introduced to address unnecessary controller updates and possible between-sample safety violations under periodic control (Yang et al., 2019).

The self-triggered formulation also changes the interpretation of guarantees. In the double-integrator case study, periodic control with fixed step $$\begin{aligned} &\min_{u,\delta}\quad \tfrac{1}{2}u^\top H u + f_q^\top u + \tfrac{w}{2}\delta^2 \ \text{s.t.}\quad &L_f V(x)+L_g V(x)u \le -c\,V(x)+\delta,\ &L_f h_i(x)+L_g h_i(x)u+\alpha_i(h_i(x)) \ge 0,\quad i=1,\dots,p,\ &u\in\mathcal{U}, \end{aligned}$$0 s violated the position constraint $$\begin{aligned} &\min_{u,\delta}\quad \tfrac{1}{2}u^\top H u + f_q^\top u + \tfrac{w}{2}\delta^2 \ \text{s.t.}\quad &L_f V(x)+L_g V(x)u \le -c\,V(x)+\delta,\ &L_f h_i(x)+L_g h_i(x)u+\alpha_i(h_i(x)) \ge 0,\quad i=1,\dots,p,\ &u\in\mathcal{U}, \end{aligned}$$1, whereas the self-triggered controller updated more frequently near the boundary and the CLF update period converged to approximately $$\begin{aligned} &\min_{u,\delta}\quad \tfrac{1}{2}u^\top H u + f_q^\top u + \tfrac{w}{2}\delta^2 \ \text{s.t.}\quad &L_f V(x)+L_g V(x)u \le -c\,V(x)+\delta,\ &L_f h_i(x)+L_g h_i(x)u+\alpha_i(h_i(x)) \ge 0,\quad i=1,\dots,p,\ &u\in\mathcal{U}, \end{aligned}$$2 s near equilibrium; the reported QP solve time was approximately $$\begin{aligned} &\min_{u,\delta}\quad \tfrac{1}{2}u^\top H u + f_q^\top u + \tfrac{w}{2}\delta^2 \ \text{s.t.}\quad &L_f V(x)+L_g V(x)u \le -c\,V(x)+\delta,\ &L_f h_i(x)+L_g h_i(x)u+\alpha_i(h_i(x)) \ge 0,\quad i=1,\dots,p,\ &u\in\mathcal{U}, \end{aligned}$$3 s per event (Yang et al., 2019). This suggests that “one-step” can refer either to per-sample convex filtering or to a solve-once-and-hold paradigm, provided the optimization remains instantaneous rather than horizon-based.

3. Constraint architecture: CBFs, CLFs, and relative degree

The simplest one-step CBF-CLF-QP uses a relative-degree-one CBF,

$$\begin{aligned} &\min_{u,\delta}\quad \tfrac{1}{2}u^\top H u + f_q^\top u + \tfrac{w}{2}\delta^2 \ \text{s.t.}\quad &L_f V(x)+L_g V(x)u \le -c\,V(x)+\delta,\ &L_f h_i(x)+L_g h_i(x)u+\alpha_i(h_i(x)) \ge 0,\quad i=1,\dots,p,\ &u\in\mathcal{U}, \end{aligned}$$4

and an exponential CLF inequality,

$$\begin{aligned} &\min_{u,\delta}\quad \tfrac{1}{2}u^\top H u + f_q^\top u + \tfrac{w}{2}\delta^2 \ \text{s.t.}\quad &L_f V(x)+L_g V(x)u \le -c\,V(x)+\delta,\ &L_f h_i(x)+L_g h_i(x)u+\alpha_i(h_i(x)) \ge 0,\quad i=1,\dots,p,\ &u\in\mathcal{U}, \end{aligned}$$5

both affine in $$\begin{aligned} &\min_{u,\delta}\quad \tfrac{1}{2}u^\top H u + f_q^\top u + \tfrac{w}{2}\delta^2 \ \text{s.t.}\quad &L_f V(x)+L_g V(x)u \le -c\,V(x)+\delta,\ &L_f h_i(x)+L_g h_i(x)u+\alpha_i(h_i(x)) \ge 0,\quad i=1,\dots,p,\ &u\in\mathcal{U}, \end{aligned}$$6. This is the setting of zeroing CBFs and exponentially stabilizing CLFs in the standard control-affine model (Yang et al., 2019). When the barrier has higher relative degree, the CBF constraint must be lifted. A common construction is the Exponential CBF, which replaces the first-order inequality by

$$\begin{aligned} &\min_{u,\delta}\quad \tfrac{1}{2}u^\top H u + f_q^\top u + \tfrac{w}{2}\delta^2 \ \text{s.t.}\quad &L_f V(x)+L_g V(x)u \le -c\,V(x)+\delta,\ &L_f h_i(x)+L_g h_i(x)u+\alpha_i(h_i(x)) \ge 0,\quad i=1,\dots,p,\ &u\in\mathcal{U}, \end{aligned}$$7

with $$\begin{aligned} &\min_{u,\delta}\quad \tfrac{1}{2}u^\top H u + f_q^\top u + \tfrac{w}{2}\delta^2 \ \text{s.t.}\quad &L_f V(x)+L_g V(x)u \le -c\,V(x)+\delta,\ &L_f h_i(x)+L_g h_i(x)u+\alpha_i(h_i(x)) \ge 0,\quad i=1,\dots,p,\ &u\in\mathcal{U}, \end{aligned}$$8 chosen so that the associated companion dynamics are Hurwitz (Yang et al., 2019). High-order and input-constrained variants play the same role in other domains, including marine vehicles and spacecraft safety filters (Garg et al., 2020, Meinert et al., 19 Mar 2026).

Multiple safety constraints are typically handled by stacking linear inequalities. For soft continuum manipulators, one baseline one-step QP uses one CLF for tip regulation and a separate CBF for every robot–obstacle sphere pair,

$$\begin{aligned} &\min_{u,\delta}\quad \tfrac{1}{2}u^\top H u + f_q^\top u + \tfrac{w}{2}\delta^2 \ \text{s.t.}\quad &L_f V(x)+L_g V(x)u \le -c\,V(x)+\delta,\ &L_f h_i(x)+L_g h_i(x)u+\alpha_i(h_i(x)) \ge 0,\quad i=1,\dots,p,\ &u\in\mathcal{U}, \end{aligned}$$9

together with a soft CLF inequality and optional input bounds (Wong et al., 19 Mar 2026). Because this can create a large number of constraints, an alternative replaces all pairwise barriers by a single log-sum-exp aggregate

$V$0

for which the paper states the implication $V$1 (Wong et al., 19 Mar 2026).

Other constructions alter the barrier rather than the QP. For relative-degree-two spacecraft keep-out and approach-corridor constraints, an input-constrained barrier

$V$2

is used so that the admissibility condition remains affine in the commanded acceleration and explicitly accounts for actuator limits (Meinert et al., 19 Mar 2026). For Cassie, multiple non-overlapping obstacles are encoded by a single continuously differentiable composite barrier

$V$3

where each $V$4 is an individual obstacle barrier and $V$5 is a $V$6 saturation; the resulting QP size remains constant even as the number of obstacles changes (Liu et al., 2023).

These variants preserve the one-step character of the controller while changing its geometry. The underlying pattern is stable: the optimization remains low-dimensional in the control input, but the barrier layer can range from a single first-order inequality to stacked ECBFs, HOCBFs, or smooth aggregated barriers.

4. Feasibility, compatibility, and equilibrium pathologies

A central difficulty of one-step CBF-CLF-QPs is that safety and stability constraints can be simultaneously infeasible. This is especially pronounced when fixed CBF decay rates are combined with actuator saturation. One proposed remedy introduces an online CBF decay-rate variable $V$7 and solves

$V$8

and the corresponding theorem states that, for any $V$9 with $h_i$0, the one-step optimal-decay CBF-QP and CLF-CBF-QP are feasible and yield a unique optimizer when $h_i$1 is convex and nonempty (Zeng et al., 2021).

Feasibility, however, is not the only structural issue. In the driftless, full-rank setting with quadratic CLFs and multiple quadratic CBFs, it is proved that undesirable equilibrium points occur for most systems, that all nontrivial equilibria lie on safety boundaries, and that their stability depends on the CLF and CBF geometrical properties (Reis et al., 2024). In that analysis, stable boundary equilibria arise when the CLF descent direction is constrained by active barriers in a way that halts progress, and the paper introduces CLF-CBF compatibility as the property that the only stable equilibrium is the CLF global minimum (Reis et al., 2024).

A different response is to alter the filter itself. A relaxed-compatibility CBF-CLF-QP replaces the usual stabilizing nominal-controller assumption with a milder boundary condition and solves

$h_i$2

subject to

$h_i$3

Under relaxed compatibility, this program is guaranteed feasible, the optimal control law is locally Lipschitz continuous, safety is guaranteed, local asymptotic stability follows, and there are no equilibrium points in the interior of the control invariant set except at the origin (Wang et al., 2024).

For scalar-input systems, recent work goes further and removes online optimization altogether. A necessary and sufficient compatibility condition is derived for simultaneous satisfaction of the CLF and CBF inequalities, and when it holds, two explicit continuous feedback laws built from the Lie-derivative data yield asymptotic stabilization and forward invariance without online quadratic programming (Wang et al., 24 Mar 2026). This suggests that, at least in the single-input case, one-step QP feasibility can be recast as a purely geometric compatibility problem rather than a numerical one.

5. Robust, probabilistic, and learning-augmented extensions

The one-step CBF-CLF-QP has been extended to uncertainty, estimation error, and learned model mismatch by modifying either the constraints or the optimization class. Under additive disturbances and state-estimation errors, a robust CBF-FxT-CLF-QP uses tightened barrier inequalities and a fixed-time CLF condition. At each sampling instant, it solves for $h_i$4 and a CLF slack $h_i$5 so as to enforce robust static and dynamic CBFs together with

$h_i$6

which the paper uses to guarantee robust forward invariance of the safe sets and fixed-time convergence to the goal set under its assumptions (Garg et al., 2020).

When uncertainty is modeled nonparametrically, the one-step QP often becomes a second-order cone program. In GP-CBF-CLF-SOCP, Gaussian-process posterior mean and variance terms are inserted into the nominal CLF and CBF Lie derivatives, producing chance constraints of the form

$h_i$7

and analogous CLF inequalities. The resulting deterministic reformulation is convex, and the paper derives necessary and sufficient pointwise feasibility conditions for the SOC CBF constraint (Castañeda et al., 2021).

Learning-based formulations preserve the one-step filter but learn the model errors inside it. In RL-CBF-CLF-QP, reinforcement learning estimates affine corrections to the CLF, CBF, and other control-affine constraints, and the online controller solves a one-step QP over the transverse input $h_i$8 using these learned corrections (Choi et al., 2020). In a separate output-feedback line, confidence-aware safe and stable control uses an observer state $h_i$9, an EKF-style confidence matrix $H\succ 0$0, and a convex one-step program with objective

$H\succ 0$1

so that the controller simultaneously enforces a CLF-CBF filter and increases the observer’s slowest-mode confidence (Wei et al., 2024).

Attack models lead to yet another modification. For control-input false data injection attacks satisfying an at-most-exponential envelope, an attack-resilient one-step CLF-CBF-QP adds adaptive compensation terms to both constraints and proves uniformly ultimately bounded stability and uniform ultimate safety under the stated growth assumptions (Rajabinezhad et al., 19 May 2026). Across these extensions, the basic one-step idea remains intact: the controller still computes a single control move from current information, but the admissible half-spaces are replaced by robust, probabilistic, learned, or adaptive surrogates.

6. Applications, computation, and departures from QP

One-step CBF-CLF-QPs are used as real-time filters in settings where the optimization dimension is small enough for online deployment. In spacecraft close-proximity operations, a runtime filter based on CWH dynamics solves

$H\succ 0$2

at $H\succ 0$3 s while enforcing hard barrier constraints for a spherical keep-out zone or conical approach corridor and a soft CLF centered at decision points. Representative compute times are reported as an average of $H\succ 0$4 ms on a PC and $H\succ 0$5 ms on an ESP32-S3-N16R8, both within the $H\succ 0$6 Hz update budget (Meinert et al., 19 Mar 2026).

In tendon-driven soft continuum robots, the baseline one-step CLF-CBF-QP is used for whole-body obstacle avoidance with pairwise sphere barriers. The paper reports a QP solver runtime of $H\succ 0$7s/call and a closed-form replacement runtime of $H\succ 0$8s/call on CPU, corresponding to a speedup of approximately $H\succ 0$9; it also reports that sampling-based planning took approximately $c>0$0 s in one setpoint experiment (Wong et al., 19 Mar 2026). That comparison is significant because it shows why alternatives to one-step QPs are being pursued: the QP is already real-time, but analytic reductions can matter when the number of barriers grows with spatial discretization.

Locomotion and navigation supply a different class of demonstrations. Cassie uses a one-step CLF-CBF-QP together with a single continuously differentiable multi-obstacle barrier built from LiDAR-derived obstacle approximations, enabling safe planning around multiple non-overlapping obstacles in simulation and experiment (Liu et al., 2023). In multi-goal reach-and-avoid navigation, the QP itself can remain unchanged while the engaged CLF is switched according to a conflict metric based on cosine similarity between $c>0$1 and $c>0$2; in multi-agent multi-goal experiments, this conflict-aware switching reduced both completion time and timeout rates relative to a baseline sequential goal policy (Walia et al., 19 Jun 2026).

The computational record therefore supports two complementary conclusions. First, one-step CBF-CLF-QPs are sufficiently light for many embedded and robotic applications. Second, their known limitations—constraint conflict, conservatism, scaling with many barriers, and equilibrium pathologies—have motivated closed-form controllers, compatibility-based redesigns, and switching or aggregation mechanisms rather than abandonment of the one-step paradigm itself.