Pol Mestres

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.

Kazuya Echigo, David E. J. van Wijk, Pol Mestres et al.

Safety-critical control systems, such as spacecraft performing proximity operations, must provide formal safety guarantees despite stochastic uncertainties from state estimation and unmodeled dynamics. Although Control Barrier Functions (CBFs) have been extended to stochastic systems, existing approaches typically face a trade-off between the tightness of probabilistic guarantees and computational tractability. This paper presents a particle-based probabilistic CBF framework that overcomes this limitation by exploiting the sub-Gaussian structure of the barrier function increment under Gaussian uncertainties. We establish that Gaussian uncertainties propagating through Lipschitz-continuous control-affine dynamics preserve sub-Gaussianity of the barrier function increment, with explicit tail bounds. Leveraging this structure, we derive finite-sample bounds on the approximation error between particle-based Conditional Value at Risk (CVaR) estimates and ground-truth probabilistic constraints; applying this yields a tractable optimization problem formulation with finite-sample safety certificates. We show through numerical experiments how the proposed approach provides tight yet provably valid probabilistic safety guarantees.

Yiting Chen, Pol Mestres, Emiliano Dall'Anese et al.

Control barrier function (CBF)-based safety filters provide a systematic way to enforce state constraints, but they can significantly alter the closed-loop dynamics induced by a nominal, stabilizing controller. In particular, the resulting safety-filtered system may exhibit undesirable behaviors including limit cycles, unbounded trajectories, and undesired equilibria. This paper develops a policy optimization framework to maximally enhance the stability properties of safety-filtered controllers. Focusing on linear systems with linear nominal controllers, we jointly parameterize the nominal feedback gain and safety-filter components, and optimize them using trajectory-based objectives computed from closed-loop rollouts. To ensure that the nominal controller remains stabilizing throughout training, we encode Lyapunov-based stability conditions as smooth scalar constraints and enforce them using robust safe gradient flow. This guarantees feasibility of the stability constraints along the optimization iterates and therefore avoids instability during training. Numerical experiments on obstacle-avoidance problems show that the proposed approach can remove asymptotically stable undesired equilibria and improve convergence behavior while maintaining forward invariance of the safe set.

Shima Sadat Mousavi, Pol Mestres, Aaron D. Ames

Control barrier function (CBF)-QP safety filters enforce safety by minimally modifying a nominal controller. While prior work has mainly addressed robustness of safety under uncertainty, robustness of the resulting closed-loop \emph{stability} is much less understood. This issue is important because once the safety filter becomes active, it modifies the nominal dynamics and can reduce stability margins or even destabilize the system, despite preserving safety. For linear systems with a single affine safety constraint, we show that the active-mode dynamics admit an exact scalar loop representation, leading to a classical robust-control interpretation in terms of gain, phase, and delay margins. This viewpoint yields exact stability-margin characterizations and tractable linear matrix inequality (LMI)-based certificates and synthesis conditions for controllers with certified robustness guarantees. Numerical examples illustrate the proposed analysis and the enlargement of certified stability margins for safety-filtered systems.

Shima Sadat Mousavi, Max H. Cohen, Pol Mestres et al.

Safety filters based on control barrier functions (CBFs) and high-order control barrier functions (HOCBFs) are often implemented through quadratic programs (QPs). In general, especially in the presence of multiple constraints, feasibility is difficult to certify before solving the QP and may be lost as the state evolves. This paper addresses this issue for linear time-invariant (LTI) systems with affine safety constraints. Exploiting the resulting geometry of the constraint normals, and considering both unbounded and bounded inputs, we characterize feasibility for several structured classes of constraints. For certain such cases, we also derive closed-form safety filters. These explicit filters avoid online optimization and provide a simple alternative to QP-based implementations. Numerical examples illustrate the results.