Pio Ong

Pol Mestres, Shima Sadat Mousavi, Pio Ong et al.

This paper studies the efficient implementation of safety filters that are designed using control barrier functions (CBFs), which minimally modify a nominal controller to render it safe with respect to a prescribed set of states. Although CBF-based safety filters are often implemented by solving a quadratic program (QP) in real time, the use of off-the-shelf solvers for such optimization problems poses a challenge in applications where control actions need to be computed efficiently at very high frequencies. In this paper, we introduce a closed-form expression for controllers obtained through CBF-based safety filters. This expression is obtained by partitioning the state-space into different regions, with a different closed-form solution in each region. We leverage this formula to introduce a resource-aware implementation of CBF-based safety filters that detects changes in the partition region and uses the closed-form expression between changes. We showcase the applicability of our approach in examples ranging from aerospace control to safe reinforcement learning.

Max H. Cohen, Pio Ong, Pol Mestres et al.

Control barrier functions (CBFs) provide a rigorous framework for designing controllers enforcing safety constraints. While CBF theory is well-developed for a finite number of safety constraints, certain applications, e.g., backup CBFs, require an infinite number of constraints. Despite the practical success of CBFs, several fundamental questions remain unanswered when safe sets are defined with an infinite numbers of constraints, including: necessary and sufficient conditions for forward set invariance, the actual definition of CBFs associated with these sets, the regularity properties of the resulting controllers, and the ability to reduce a collection of infinite constraints to a finite number. This paper addresses these questions by extending CBF theory to the infinite constraint setting. We identify regularity conditions under which Nagumo's Theorem reduces to barrier-like inequalities and when the associated CBF controllers are at least continuous. We further connect these results to optimal-decay CBFs, bridging theoretical conditions for invariance and practical instantiations of the resulting controller. Finally, we illustrate how the developed theory addresses limitations of backup CBFs.

Pio Ong, David E. J. van Wijk, Massimiliano de Sa et al.

Control barrier functions (CBFs) provide a principled framework for enforcing safety in control systems -- yet the certified safe operating region in practice is often conservative, especially under input bounds. In many applications, multiple smaller safe sets can be certified independently, e.g., around distinct equilibria with different stabilizing controllers. This paper proposes a framework for uniting such regions into a single certified safe set using \emph{combinatorial CBFs}. We refine the combinatorial CBF framework by introducing an auxiliary variable that enables logical compositions of individual CBFs. In the proposed framework, we show that such compositions yield a \emph{generalized combinatorial CBF} under a condition termed \emph{conjunctive compatibility}. Building on this result, we extend the framework to enable the aggregation of multiple implicit safe sets generated by the backup CBF framework. We show that the resulting CBF-based quadratic program yields a continuous safety filter over the aggregated safe region. The approach is demonstrated on two spacecraft safety problems, safe attitude control and safe station keeping, where multiple certified safe regions are combined to expand the operational envelope.

Samuel G. Gessow, Pio Ong, Aaron D. Ames et al.

Control barrier functions (CBFs) provide an effective framework for enforcing safety in dynamical systems with scalar constraints. However, many safety constraints are more naturally expressed as matrix-valued conditions, such as positive definiteness or eigenvalue bounds - scalar formulations introduce potential nonsmoothness that complicates analysis. Matrix control barrier functions (MCBFs) address this limitation by directly enforcing matrix-valued safety constraints. Yet for constraints where the control input does not appear in the first derivative, high-order formulations are required. While such extensions are well understood in the scalar case, they remain largely unexplored in the matrix case. This paper develops high-order matrix control barrier functions (HOMCBFs) and establishes conditions ensuring well-posedness and feasibility of the associated constraints, enabling enforcement of matrix-valued safety constraints for systems with high-order dynamics. We further show that, using an optimal-decay HOMCBF formulation, forward invariance can be ensured while requiring control only over the minimum eigenspace. The framework is demonstrated on a localization safety problem by enforcing positive definiteness of the information matrix for a double integrator system with a nonlinear measurement model.

Yana Lishkova, Pio Ong, Sander Tonkens et al.

In applications such as autonomous landing and navigation, it is often desirable to steer toward a target while retaining the ability to divert to at least $r$ (out of $p$) alternative sites if conditions change. In this work, we formalize this combinatorial contingency requirement and develop tractable control filters for enforcement. Combinatorial stabilization requires asymptotic stability of a selected equilibrium while ensuring the trajectory remains within the safe region of attraction of at least $r$-out-of-$p$ candidates. To enforce this requirement, we use control Lyapunov functions (CLFs) to construct regions of attraction, which are combined combinatorially within an optimization-based filter. Combinatorial targeting extends this framework to finite-horizon problems using Hamilton-Jacobi backward reach-avoid sets, accommodating shrinking reachable regions due to finite horizons or resource depletion. In both formulations, the resulting combinatorial stability filter and combinatorial reach-avoid filter require only $p+1$ constraints, preventing combinatorial blow-up and enabling safe real-time switching between targets. The framework is demonstrated on two examples where the filters ensure steering with contingency and enable safe diversion.