Calin A. Belta

Wei Xiao, Christos G. Cassandras, Calin A. Belta

It has been shown that optimizing quadratic costs while stabilizing affine control systems to desired (sets of) states subject to state and control constraints can be reduced to a sequence of Quadratic Programs (QPs) by using Control Barrier Functions (CBFs) and Control Lyapunov Functions (CLFs). In this paper, we employ machine learning techniques to ensure the feasibility of these QPs, which is a challenging problem, especially for high relative degree constraints where High Order CBFs (HOCBFs) are required. To this end, we propose a sampling-based learning approach to learn a new feasibility constraint for CBFs; this constraint is then enforced by another HOCBF added to the QPs. The accuracy of the learned feasibility constraint is recursively improved by a recurrent training algorithm. We demonstrate the advantages of the proposed learning approach to constrained optimal control problems with specific focus on a robot control problem and on autonomous driving in an unknown environment.

Reza Moazzez Estanjini, Xu Chu Ding, Morteza Lahijanian et al.

We consider the problem of finding a control policy for a Markov Decision Process (MDP) to maximize the probability of reaching some states while avoiding some other states. This problem is motivated by applications in robotics, where such problems naturally arise when probabilistic models of robot motion are required to satisfy temporal logic task specifications. We transform this problem into a Stochastic Shortest Path (SSP) problem and develop a new approximate dynamic programming algorithm to solve it. This algorithm is of the actor-critic type and uses a least-square temporal difference learning method. It operates on sample paths of the system and optimizes the policy within a pre-specified class parameterized by a parsimonious set of parameters. We show its convergence to a policy corresponding to a stationary point in the parameters' space. Simulation results confirm the effectiveness of the proposed solution.

Shuo Liu, Wei Xiao, Calin A. Belta

In safety-critical control systems, ensuring both safety and feasibility under sampled-data implementations is crucial for practical deployment. Existing Control Barrier Function (CBF) frameworks, such as High-Order CBFs (HOCBFs), effectively guarantee safety in continuous time but may become unsafe when executed under zero-order-hold (ZOH) controllers due to inter-sampling effects. Moreover, they do not explicitly handle finite-time reach-and-remain requirements or multiple simultaneous constraints, which often lead to conflicts between safety and reach-and-remain objectives, resulting in feasibility issues during control synthesis. This paper introduces Sampling-Aware Control Barrier Functions (SACBFs), a unified framework that accounts for sampling effects and high relative-degree constraints by estimating and incorporating Taylor-based upper bounds on barrier evolution between sampling instants. The proposed method guarantees continuous-time forward invariance of safety and finite-time reach-and-remain sets under ZOH control. To further improve feasibility, a relaxed variant (r-SACBF) introduces slack variables for handling multiple constraints realized through time-varying CBFs. Simulation studies on a unicycle robot demonstrate that SACBFs achieve safe and feasible performance in scenarios where traditional HOCBF methods fail.

Shuo Liu, Zhe Huang, Calin A. Belta

Obstacle avoidance of polytopic obstacles by polytopic robots is a challenging problem in optimization-based control and trajectory planning. Many existing methods rely on smooth geometric approximations, such as hyperspheres or ellipsoids, which allow differentiable distance expressions but distort the true geometry and restrict the feasible set. Other approaches integrate exact polytope distances into nonlinear model predictive control (MPC), resulting in nonconvex programs that limit real-time performance. In this paper, we construct linear discrete-time control barrier function (DCBF) constraints by deriving supporting hyperplanes from exact closest-point computations between convex polytopes. We then propose a novel iterative convex MPC-DCBF framework, where local linearization of system dynamics and robot geometry ensures convexity of the finite-horizon optimization at each iteration. The resulting formulation reduces computational complexity and enables fast online implementation for safety-critical control and trajectory planning of general nonlinear dynamics. The framework extends to multi-robot and three-dimensional environments. Numerical experiments demonstrate collision-free navigation in cluttered maze scenarios with millisecond-level solve times.

Shuo Liu, Wei Xiao, Christos G. Cassandras et al.

This paper studies safety-critical control for nonlinear systems under sampled-data implementations of the controller. The recently proposed Taylor--Lagrange Control (TLC) method provides rigorous safety guarantees but relies on a fixed discretization-related parameter, which can lead to infeasibility or unsafety in the presence of input constraints and inter-sampling effects. To address these limitations, we propose an adaptive Taylor--Lagrange Control (aTLC) framework with an event-triggered implementation, where the discretization-related parameter defines the discretization time scale and is selected online as state-dependent rather than fixed. This enables the controller to dynamically balance feasibility and safety by adjusting the effective time scale of the Taylor expansion. The resulting controller is implemented as a sequence of Quadratic Programs (QPs) with input constraints. We further introduce a selection rule to choose the discretization-related parameter from a finite candidate set, favoring feasible inputs and improved safety. Simulation results on an adaptive cruise control (ACC) problem demonstrate that the proposed approach improves feasibility, guarantees safety, and achieves smoother control actions compared to TLC while requiring a single automatically tuned parameter.

Shuo Liu, Wenliang Liu, Wei Xiao et al.

Control Barrier Functions (CBFs) have emerged as a powerful tool for enforcing safety in optimization-based controllers, and their integration with Signal Temporal Logic (STL) has enabled the specification-driven synthesis of complex robotic behaviors. However, existing CBF-STL approaches typically rely on fixed hyperparameters and myopic, per-time step optimization, which can lead to overly conservative behavior, infeasibility near tight input limits, and difficulty satisfying long-horizon STL tasks. To address these limitations, we propose a feasibility-aware learning framework that embeds trainable, time-varying High Order Control Barrier Functions (HOCBFs) into a differentiable Quadratic Program (dQP). Our approach provides a systematic procedure for constructing time-varying HOCBF constraints for a broad fragment of STL and introduces a unified robustness measure that jointly captures STL satisfaction, QP feasibility, and control-bound compliance. Three neural networks-InitNet, RefNet, and an extended BarrierNet-collaborate to generate reference inputs and adapt constraint-related hyperparameters automatically over time and across initial conditions, reducing conservativeness while maximizing robustness. The resulting controller achieves STL satisfaction with strictly feasible dQPs and requires no manual tuning. Simulation results demonstrate that the proposed framework maintains high STL robustness under tight input bounds and significantly outperforms fixed-parameter and non-adaptive baselines in complex environments.

Wei Xiao, Calin A. Belta, Christos G. Cassandras

Recent work has shown that stabilizing an affine control system to a desired state while optimizing a quadratic cost subject to state and control constraints can be reduced to a sequence of Quadratic Programs (QPs) by using Control Barrier Functions (CBFs) and Control Lyapunov Functions (CLFs). In our own recent work, we defined High Order CBFs (HOCBFs) for systems and constraints with arbitrary relative degrees. In this paper, in order to accommodate initial states that do not satisfy the state constraints and constraints with arbitrary relative degree, we generalize HOCBFs to High Order Control Lyapunov-Barrier Functions (HOCLBFs). We also show that the proposed HOCLBFs can be used to guarantee the Boolean satisfaction of Signal Temporal Logic (STL) formulae over the state of the system. We illustrate our approach on a safety-critical optimal control problem (OCP) for a unicycle.

Wei Xiao, Christos G. Cassandras, Calin A. Belta

We address the problem of optimizing the performance of a dynamic system while satisfying hard safety constraints at all times. Implementing an optimal control solution is limited by the computational cost required to derive it in real time, especially when constraints become active, as well as the need to rely on simple linear dynamics, simple objective functions, and ignoring noise. The recently proposed Control Barrier Function (CBF) method may be used for safety-critical control at the expense of sub-optimal performance. In this paper, we develop a real-time control framework that combines optimal trajectories generated through optimal control with the computationally efficient CBF method providing safety guarantees. We use Hamiltonian analysis to obtain a tractable optimal solution for a linear or linearized system, then employ High Order CBFs (HOCBFs) and Control Lyapunov Functions (CLFs) to account for constraints with arbitrary relative degrees and to track the optimal state, respectively. We further show how to deal with noise in arbitrary relative degree systems. The proposed framework is then applied to the optimal traffic merging problem for Connected and Automated Vehicles (CAVs) where the objective is to jointly minimize the travel time and energy consumption of each CAV subject to speed, acceleration, and speed-dependent safety constraints. In addition, when considering more complex objective functions, nonlinear dynamics and passenger comfort requirements for which analytical optimal control solutions are unavailable, we adapt the HOCBF method to such problems. Simulation examples are included to compare the performance of the proposed framework to optimal solutions (when available) and to a baseline provided by human-driven vehicles with results showing significant improvements in all metrics.

Wei Xiao, Calin A. Belta, Christos G. Cassandras

Optimal control problems with constraints ensuring safety and convergence to desired states can be mapped onto a sequence of real time optimization problems through the use of Control Barrier Functions (CBFs) and Control Lyapunov Functions (CLFs). One of the main challenges in these approaches is ensuring the feasibility of the resulting quadratic programs (QPs) if the system is affine in controls. The recently proposed penalty method has the potential to improve the existence of feasible solutions to such problems. In this paper, we further improve the feasibility robustness (i.e., feasibility maintenance in the presence of time-varying and unknown unsafe sets) through the definition of a High Order CBF (HOCBF) that works for arbitrary relative degree constraints; this is achieved by a proposed feasibility-guided learning approach. Specifically, we apply machine learning techniques to classify the parameter space of a HOCBF into feasible and infeasible sets, and get a differentiable classifier that is then added to the learning process. The proposed feasibility-guided learning approach is compared with the gradient-descent method on a robot control problem. The simulation results show an improved ability of the feasibility-guided learning approach over the gradient-decent method to determine the optimal parameters in the definition of a HOCBF for the feasibility robustness, as well as show the potential of the CBF method for robot safe navigation in an unknown environment.

Xu Chu Ding, Jing Wang, Morteza Lahijanian et al.

In this paper, we consider the problem of deploying a robot from a specification given as a temporal logic statement about some properties satisfied by the regions of a large, partitioned environment. We assume that the robot has noisy sensors and actuators and model its motion through the regions of the environment as a Markov Decision Process (MDP). The robot control problem becomes finding the control policy maximizing the probability of satisfying the temporal logic task on the MDP. For a large environment, obtaining transition probabilities for each state-action pair, as well as solving the necessary optimization problem for the optimal policy are usually not computationally feasible. To address these issues, we propose an approximate dynamic programming framework based on a least-square temporal difference learning method of the actor-critic type. This framework operates on sample paths of the robot and optimizes a randomized control policy with respect to a small set of parameters. The transition probabilities are obtained only when needed. Hardware-in-the-loop simulations confirm that convergence of the parameters translates to an approximately optimal policy.