High-Order CLF & CBF QP Control Synthesis

• The HOCLF HOCBF QP framework is a unified convex optimization approach that enforces both safety and stabilization in nonlinear systems with high relative degree.
• It extends classical CLF and CBF methodologies by incorporating high-order derivative constraints into a single real-time quadratic program that respects input bounds.
• The framework is applied to autonomous driving, soft robot manipulation, and stochastic system control, with feasibility enhanced via tailored class-K functions and learned constraints.

A High Order Control Lyapunov Function High Order Control Barrier Function Quadratic Programming (HOCLF HOCBF QP) controller is a unified convex optimization framework for synthesizing feedback policies that provably guarantee both stabilization and safety for nonlinear control-affine systems—including those with safety and stabilization objectives of arbitrary relative degree. The methodology extends classical control Lyapunov functions (CLFs) and control barrier functions (CBFs) to the high-order domain, enabling enforcement of safety and stability constraints that only appear in the input after multiple time derivatives. By embedding these constraints into a single real-time quadratic program (QP), the approach achieves online control synthesis that respects input bounds while prioritizing safety.

1. Mathematical Foundations: High-Order Barrier and Lyapunov Functions

High-order barrier and Lyapunov functions formalize constraint and stability enforcement for affine nonlinear systems with high relative degree. Considering a control-affine system

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

a constraint $b(x) \geq 0$ (e.g., for safety or task fulfillment) of relative degree $m$ leads to the following recursive sequences: \begin{align*} &\psi_0(x) := b(x), \ &\psi_i(x) := \dot\psi_{i-1}(x) + \alpha_i(\psi_{i-1}(x)),\quad i=1,...,m, \end{align*} where each $\alpha_i$ is a class-$\mathcal{K}$ function (continuous, strictly increasing, $\alpha_i(0) = 0$) (Xiao et al., 2019).

A scalar function $b$ is a $m$-th order control barrier function (HOCBF) if for all $x$ in the intersection set $C_{\ast} := \bigcap_{i=1}^m C_i$,

$$L_f^m b(x) + L_gL_f^{m-1}b(x)\,u + O_{<m}(b) + \alpha_m(\psi_{m-1}(x)) \geq 0,$$

where $O_{<m}(b)$ collects lower-order Lie derivatives and $\psi_{m-1}$ terms.

Similarly, a high-order control Lyapunov function (HOCLF) $V$ (of relative degree $d$) admits a sequence \begin{align*} &\phi_0(x) := V(x), \ &\phi_i(x) := \dot\phi_{i-1}(x) + \beta_i(\phi_{i-1}(x)),\quad i=1,...,d, \ \end{align*} with class-$\mathcal{K}$ or extended class-$\mathcal{K}_{\infty}$ functions $\beta_i$ (Matias et al., 13 Apr 2025, Chen et al., 14 Dec 2025, Wong et al., 5 May 2025, Sarkar et al., 2020, Xiao et al., 2021). The HOCLF condition enforces

$$L_f^d V(x) + L_gL_f^{d-1}V(x)\,u + \tilde{O}_{<d}(V) + \beta_d(\phi_{d-1}(x)) \leq 0.$$

This construction allows constraint satisfaction and stabilization even when the control input $u$ appears only after several Lie derivatives.

2. Unified Quadratic Programming Formulation

At each sampling instant $t_k$, the HOCLF HOCBF QP paradigm formulates a convex quadratic program: $$\min_{u,\,\delta} \quad \frac{1}{2} \| u - u_{\rm ref} \|^2 + p\,\delta^2$$ subject to high-order safety and stability constraints: \begin{align*} \textrm{(HOCLF constraint)} & :\;\; L_f^d V(x) + L_gL_f^{d-1}V(x)\,u + ... \leq \delta, \ \textrm{(HOCBF constraints)} & :\;\; L_f^{m_j} b_j(x) + L_gL_f^{m_j-1} b_j(x)\,u + ... \geq 0, \quad \forall j, \ \textrm{(input bounds)} & :\;\; u_{\min} \leq u \leq u_{\max}, \ \textrm{(slack nonnegativity)} & :\;\; \delta \geq 0. \end{align*} Here, $u_{\rm ref}$ is a nominal or minimum-effort control input, $\delta$ is a relaxation (slack) for the HOCLF, and $p$ is a large penalty balancing feasibility and convergence (Chen et al., 14 Dec 2025, Xiao et al., 2019, Matias et al., 13 Apr 2025, Wong et al., 5 May 2025).

This QP structure generalizes to stochastic systems by including diffusion terms in the Lie derivatives (Sarkar et al., 2020). For multiple control inputs with possibly heterogeneous relative degrees, the HOCLF HOCBF QP accommodates either by integral control augmentation or constraint transformation to a uniform relative degree (Xiao et al., 2022).

3. Theoretical Properties: Invariance, Stabilization, and Feasibility

The HOCBF condition guarantees forward invariance of the intersection set $C_{\ast}$, i.e., any (Lipschitz) controller $u(\cdot)$ satisfying the HOCBF constraints ensures $x(t_0)\in C_{\ast} \implies x(t)\in C_{\ast}$ for all $t\geq t_0$ (Xiao et al., 2019, Chen et al., 14 Dec 2025, Matias et al., 13 Apr 2025). HOCLF constraints guarantee convergence (asymptotic or finite-time depending on the nonlinearity parameterization) to the designated set or goal.

A crucial aspect is the selection of class-$\mathcal{K}$ functions $\alpha_i, \beta_i$. Larger growth (e.g., quadratic) increases the right-hand-side of the barrier inequality, thereby enlarging the feasible region for $u$. Sublinear choice can overconstrain the QP and reduce feasibility, notably close to the constraint boundary (Xiao et al., 2019, Chen et al., 14 Dec 2025).

To resolve infeasibility due to tight input bounds or conflicting constraints, HOCBFs may introduce penalties $p_i$ in the recursive definition or incorporate learned feasibility constraints as additional HOCBFs learned via machine learning classification (Xiao et al., 2023). Theoretically, provided HOCBF constraints are satisfied at all times (with sufficient regularity of dynamics and class-$\mathcal{K}$ functions), forward invariance and global asymptotic stabilization are ensured (Xiao et al., 2019, Matias et al., 13 Apr 2025, Xiao et al., 2021).

4. Extensions and Adaptations

Stochastic and Uncertain Systems

For systems with stochastic dynamics,

$$\mathrm{d}x = f(x)\,\mathrm{d}t + g(x)\,u\,\mathrm{d}t + \sigma(x)\,\mathrm{d}w_t,$$

the QP constraints extend to include diffusion (Itô) terms in the high-order Lyapunov and barrier inequalities. Safety and stabilization are enforced in a sample-path sense, with possible chance-constraint reformulations (Sarkar et al., 2020).

Multi-Input and High-Order Systems

For multi-input systems with nonuniform input appearance (varying relative degree), two strategies exist: (i) augment the original system via auxiliary integrator chains (integral HOCBFs, iHOCBF), thus ensuring all inputs affect the barrier constraint; (ii) transform the original state constraint to a geometric one of uniform relative degree, resulting in a barrier inequality linear in all control components (Xiao et al., 2022).

Temporal Logic and Hybrid Strategies

HOCLF and HOCBF constraints can encode high-level temporal logic specifications (e.g., STL), enabling Boolean satisfaction of “always,” “eventually,” and disjunctive-conjunctive formulae. The approach unites high-order constraint satisfaction and set stabilization in a single QP-based policy (Xiao et al., 2021). Hybrid backstepping strategies further extend the methodology to strict-feedback forms and guarantee finite-time progress and safety through mode switching (Matias et al., 13 Apr 2025).

Learning-Integrated Feasibility Enhancement

Feasibility of HOCLF HOCBF QPs, particularly under high relative degree and tight bounds, can be improved by learning a feasibility classifier—which defines an additional barrier and restricts the domain to statically or dynamically feasible regions, thus maintaining recursive feasibility (Xiao et al., 2023).

5. Applications and Practical Implementations

The HOCLF HOCBF QP paradigm is broadly applicable to safety-critical autonomous systems:

• Autonomous Driving for VRU Safety: Ensures robust collision avoidance and trajectory tracking in the presence of bicyclists and complex, dynamic traffic, enabling real-time maneuvers that avoid collision while minimizing deviation from the planned trajectory. Simulation studies report maximum lateral error $<0.2\,\text{m}$ and minimum inter-agent clearance $>2\,\text{m}$ in high-risk bicyclist crash scenarios (Chen et al., 14 Dec 2025).
• Soft Robot Manipulation: Embeds whole-body collision detection and strict contact force constraints for continuum manipulators, leveraging a high-order geometric barrier derived from conservative, differentiable separating axis metrics. Controllers simultaneously achieve shape/task regulation and safety at real-time execution rates (400–800 Hz) (Wong et al., 5 May 2025).
• Robotic Navigation and Overtaking: Multi-input systems with tight bounds (e.g., unicycle models) benefit from improved feasibility and performance by deploying iHOCBF or transformed HOCBF approaches (Xiao et al., 2022, Xiao et al., 2023).
• Nonlinear and Stochastic Systems: HOCLF and HOCBF constructions enable provable safety and stabilization of underactuated (e.g., relative degree 4) systems subject to noise, as seen in high-DOF manipulator and vehicle dynamics (Sarkar et al., 2020).

In all applications, the online QP solution computes the minimum-effort input satisfying all constraints, with most reported computational times on the order required for real-time embedded deployment (controller frequencies $10$–$800$ Hz, depending on model complexity and solver choice; e.g., OSQP, CBFpy) (Wong et al., 5 May 2025, Chen et al., 14 Dec 2025).

6. Limitations and Tuning Considerations

Feasibility and real-time performance may be degraded by poor tuning of class-$\mathcal{K}$ parameters, high constraint density (leading to conservative QPs), or insufficient actuation authority. The literature highlights the dependence of feasible QP regions on $\alpha_i,\beta_i$ choices and the need for empirical or automated tuning (Chen et al., 14 Dec 2025, Xiao et al., 2019). For scenarios with rapidly time-varying or highly uncertain obstacles, future directions include adaptive or learning-based gain adjustment, and integration with sensor fusion or prediction frameworks.

7. Summary Table: Key Components in HOCLF HOCBF QP

Component	Notation/Property	Reference Papers
High-Order Safety	$b(x)$, $\psi_i$, relative degree $m$, class-$\mathcal{K}$ $\alpha_i$	(Xiao et al., 2019, Chen et al., 14 Dec 2025)
High-Order Stability	$V(x)$, $\phi_i$, relative degree $d$, class-$\mathcal{K}$ $\beta_i$	(Matias et al., 13 Apr 2025, Wong et al., 5 May 2025)
QP Structure	Minimize input and slack, enforce HOCBF/HOCLF inequalities	(Chen et al., 14 Dec 2025, Xiao et al., 2022)
Feasibility Enhancements	Penalty $p_i$, learned constraints, integrator augmentation	(Xiao et al., 2023, Xiao et al., 2019)
Application Domains	Autonomous driving, soft robotics, stochastic/nonlinear systems	(Chen et al., 14 Dec 2025, Wong et al., 5 May 2025, Sarkar et al., 2020)

This table summarizes the canonical features underpinning the HOCLF HOCBF QP approach across recent research. For new deployments, exact implementation details (e.g., penalty weights, class-$\mathcal{K}$ gain tuning, QP solver) must be adapted to system dynamics and application constraints. Empirical validation is essential for confirming expected safety and stability guarantees in closed-loop operation.