arXiv subjects

Andrew Clark

Publications and source records attributed to Andrew Clark.

At least 19 recordsLinked to original sources

arXiv2026

Verification Framework for the Union of Control Barrier Functions

Control Barrier Functions (CBFs) have been proposed to ensure safety of autonomous systems. This paper considers control policies that switch between CBF constraints. Under this approach, we represent a complex non-convex safe region as a union of sets that are computationally tractable to verify. We denote this framework as union-CBFs and make the following contributions. First, considering switching CBF-QP controllers, we propose a sufficient condition that ensures (i) the system undergoes a finite number of switches in any finite time interval and ensures (ii) the forward invariance of the closed-loop system in between switches. Second, we consider two types of switching strategies and propose union-CBFs conditions for each strategy to satisfy (i) and (ii). Third, we formulate Sum-of-Squares (SOS) algorithms to verify the conditions. The experiments show that our union-CBFs framework results in a larger safe region compared to high-degree polynomial CBFs. We also show the efficiency of the verification algorithms using a polynomial system model.

math.OC↗

arXiv2026

Closed-Form Gaussian Estimators for Multi-Source Partial Information Decomposition

Computing multi-source partial information decomposition (PID) for continuous data is hard: existing closed-form Gaussian estimators are restricted to two source variables, while continuous arbitrary-source estimators are typically learning-based and do not provide closed-form expressions. To address this, we develop closed-form Gaussian estimators for multi-source PID. We provide two-source redundancy, multi-source unique information, the K-th order synergistic effect from source subsets of size K, and the total synergistic effect. The estimators are derived from the conditional-independence-based information measures introduced in our earlier work, under which every quantity reduces to a log-determinant expression in covariance blocks of the system. The resulting estimator is plug-in consistent, affine invariant, source-permutation symmetric, and additive over independent systems. We validate it on a controlled Gaussian benchmark, evaluate its computational efficiency against baselines, and confirm its numerical stability in finite-sample regimes. To our knowledge, this is the first covariance-based closed-form estimator that provides multi-source continuous PID measures.

cs.IT↗

arXiv2026

Structural Impossibility of Antichain-Lattice Partial Information Decomposition

Partial Information Decomposition (PID) represents multivariate mutual information via antichain-lattice that aims to specify which source groups can recover which informational components of a target. For three or more sources, widely desired PID axioms become mutually incompatible. This is often treated as an axiomatic tuning issue. This paper argues that the obstruction is representational, rooted in the antichain indexing itself, so that purely axiomatic adjustments within an antichain-lattice structure cannot resolve it in general. We first introduce System Information Decomposition (SID) for the special target-free three-variable setting, obtaining a self-consistent entropy decomposition with an operational redundancy definition. More fundamentally, we then show that for general multivariate PID, there is no universal rule that recovers the decomposed mutual information from the antichain-indexed information atoms. In particular, two systems can share identical atoms regardless of any axioms while having different mutual information. These results reveal the limits of antichain-lattice and motivate relation-based foundations for multivariate information measures.

cs.IT↗

arXiv2026

Distributed Safety-Critical Control of Multi-Agent Systems with Time-Varying Communication Topologies

Coordinating multiple autonomous agents to reach a target region while avoiding collisions and maintaining communication connectivity is a core problem in multi-agent systems. In practice, agents have a limited communication range. Thus, network links can appear and disappear as agents move, making the topology state-dependent and time-varying. Existing distributed solutions to multi-agent reach-avoid problems typically assume a fixed communication topology, and thus are not applicable when encountering discontinuities raised by time-varying topologies. This paper presents a distributed optimization-based control framework that addresses these challenges through two complementary mechanisms. First, we introduce a truncation function that converts the time-varying communication graph into a smoothly state-dependent one, ensuring that constraints remain continuous as communication links are created or removed. Second, we employ auxiliary mismatch variables with two-time-scale dynamics to decouple globally coupled state-dependent constraints, yielding a singular perturbation system that each agent can solve using only local information and neighbor communication. Through singular perturbation analysis, we prove that the distributed controller guarantees collision avoidance, connectivity preservation, and convergence to the target region. We validate the proposed framework through numerical simulations involving multi-agent navigation with obstacles and time-varying communication topologies.

eess.SY↗

arXiv2025

Validity Is What You Need

While AI agents have long been discussed and studied in computer science, today's Agentic AI systems are something new. We consider other definitions of Agentic AI and propose a new realist definition. Agentic AI is a software delivery mechanism, comparable to software as a service (SaaS), which puts an application to work autonomously in a complex enterprise setting. Recent advances in large language models (LLMs) as foundation models have driven excitement in Agentic AI. We note, however, that Agentic AI systems are primarily applications, not foundations, and so their success depends on validation by end users and principal stakeholders. The tools and techniques needed by the principal users to validate their applications are quite different from the tools and techniques used to evaluate foundation models. Ironically, with good validation measures in place, in many cases the foundation models can be replaced with much simpler, faster, and more interpretable models that handle core logic. When it comes to Agentic AI, validity is what you need. LLMs are one option that might achieve it.

cs.AI↗

arXiv2025

BOND: License to Train with Black-Box Functions

We introduce Bounded Numerical Differentiation (BOND), a perturbative method for estimating the gradients of black-box functions. BOND is distinguished by its formulation, which adaptively bounds perturbations to ensure accurate sign estimation, and by its implementation, which operates at black-box interfaces. This enables BOND to be more accurate and scalable compared to existing methods, facilitating end-to-end training of architectures that incorporate non-autodifferentiable modules. We observe that these modules, implemented in our experiments as frozen networks, can enhance model performance without increasing the number of trainable parameters. Our findings highlight the potential of leveraging fixed transformations to expand model capacity, pointing to hybrid analogue - digital devices as a path to scaling networks, and provides insights into the dynamics of adaptive optimizers.

cs.LG↗

arXiv2025

The Whole Is Less than the Sum of Parts: Subsystem Inconsistency in Partial Information Decomposition

Partial Information Decomposition (PID) was proposed by Williams and Beer in 2010 as a tool for analyzing fine-grained interactions between multiple random variables, and has since found numerous applications ranging from neuroscience to privacy. However, a unified theoretical framework remains elusive due to key conceptual and technical challenges. We identify and illustrate a crucial problem: PID violates the set-theoretic principle that the whole equals the sum of its parts (WESP). Through a counterexample in a three-variable system, we demonstrate how such violations naturally arise, revealing a fundamental limitation of current lattice-based PID frameworks. To address this issue, we introduce a new axiomatic framework, termed System Information Decomposition (SID), specifically tailored for three-variable systems. SID resolves the WESP violation by redefining the summation rules of decomposed information atoms based on synergistic relationships. However, we further show that for systems with four or more variables, no partial summation approach within the existing lattice-based structures can fully eliminate WESP inconsistencies. Our results thus highlight the inherent inadequacy of (antichain) lattice-based decompositions for general multivariate systems.

cs.IT↗

arXiv2025

Multivariate Partial Information Decomposition: Constructions, Inconsistencies, and Alternative Measures

While mutual information effectively quantifies dependence between two variables, it does not by itself reveal the complex, fine-grained interactions among variables, i.e., how multiple sources contribute redundantly, uniquely, or synergistically to a target in multivariate settings. The Partial Information Decomposition (PID) framework was introduced to address this by decomposing the mutual information between a set of source variables and a target variable into fine-grained information atoms such as redundant, unique, and synergistic components. In this work, we review the axiomatic system and desired properties of the PID framework and make three main contributions. First, we resolve the two-source PID case by providing explicit closed-form formulas for all information atoms that satisfy the full set of axioms and desirable properties. Second, we prove that for three or more sources, PID suffers from fundamental inconsistencies: we review the known three-variable counterexample where the sum of atoms exceeds the total information, and extend it to a comprehensive impossibility theorem showing that no lattice-based decomposition can be consistent for all subsets when the number of sources exceeds three. Finally, we deviate from the PID lattice approach to avoid its inconsistencies, and present explicit measures of multivariate unique and synergistic information. Our proposed measures, which rely on new systems of random variables that eliminate higher-order dependencies, satisfy key axioms such as additivity and continuity, provide a robust theoretical explanation of high-order relations, and show strong numerical performance in comprehensive experiments on the Ising model. Our findings highlight the need for a new framework for studying multivariate information decomposition.

cs.IT↗

arXiv2025

Stochastic Neural Control Barrier Functions

Control Barrier Functions (CBFs) are utilized to ensure the safety of control systems. CBFs act as safety filters in order to provide safety guarantees without compromising system performance. These safety guarantees rely on the construction of valid CBFs. Due to their complexity, CBFs can be represented by neural networks, known as neural CBFs (NCBFs). Existing works on the verification of the NCBF focus on the synthesis and verification of NCBFs in deterministic settings, leaving the stochastic NCBFs (SNCBFs) less studied. In this work, we propose a verifiably safe synthesis for SNCBFs. We consider the cases of smooth SNCBFs with twice-differentiable activation functions and SNCBFs that utilize the Rectified Linear Unit or ReLU activation function. We propose a verification-free synthesis framework for smooth SNCBFs and a verification-in-the-loop synthesis framework for both smooth and ReLU SNCBFs. and we validate our frameworks in three cases, namely, the inverted pendulum, Darboux, and the unicycle model.

eess.SY↗

arXiv2025

Swarm-STL: A Framework for Motion Planning in Large-Scale, Multi-Swarm Systems

In multi-agent systems, signal temporal logic (STL) is widely used for path planning to accomplish complex objectives with formal safety guarantees. However, as the number of agents increases, existing approaches encounter significant computational challenges. Recognizing that many complex tasks require cooperation among multiple agents, we propose swarm STL specifications to describe the collective tasks that need to be achieved by a team of agents. Next, we address the motion planning problem for all the agents in two stages. First, we abstract a group of cooperating agents as a swarm and construct a reduced-dimension state space whose dimension does not increase with the number of agents. The path planning is performed at the swarm level, ensuring the safety and swarm STL specifications are satisfied. Then, we design low-level control strategies for agents within each swarm based on the path synthesized in the first step. The trajectories of agents generated by the two-step policy ensure satisfaction of the STL specifications. We evaluate our two-stage approach in both single-swarm and multi-swarm scenarios. The results demonstrate that all tasks are completed with safety guarantees. Compared to the baseline multi-agent planning approach, our method maintains computational efficiency as the number of agents increases, since the computational time scales with the number of swarms rather than the number of agents.

eess.SY↗

arXiv2024

Modeling and Designing Non-Pharmaceutical Interventions in Epidemics: A Submodular Approach

This paper considers the problem of designing non-pharmaceutical intervention (NPI) strategies, such as masking and social distancing, to slow the spread of a viral epidemic. We formulate the problem of jointly minimizing the infection probabilities of a population and the cost of NPIs based on a Susceptible-Infected-Susceptible (SIS) propagation model. To mitigate the complexity of the problem, we consider a steady-state approximation based on the quasi-stationary (endemic) distribution of the epidemic, and prove that the problem of selecting a minimum-cost strategy to satisfy a given bound on the quasi-stationary infection probabilities can be cast as a submodular optimization problem, which can be solved in polynomial time using the greedy algorithm. We carry out experiments to examine effects of implementing our NPI strategy on propagation and control of epidemics on a Watts-Strogatz small-world graph network. We find the NPI strategy reduces the steady state of infection probabilities of members of the population below a desired threshold value.

eess.SY↗

arXiv2024

SEEV: Synthesis with Efficient Exact Verification for ReLU Neural Barrier Functions

Neural Control Barrier Functions (NCBFs) have shown significant promise in enforcing safety constraints on nonlinear autonomous systems. State-of-the-art exact approaches to verifying safety of NCBF-based controllers exploit the piecewise-linear structure of ReLU neural networks, however, such approaches still rely on enumerating all of the activation regions of the network near the safety boundary, thus incurring high computation cost. In this paper, we propose a framework for Synthesis with Efficient Exact Verification (SEEV). Our framework consists of two components, namely (i) an NCBF synthesis algorithm that introduces a novel regularizer to reduce the number of activation regions at the safety boundary, and (ii) a verification algorithm that exploits tight over-approximations of the safety conditions to reduce the cost of verifying each piecewise-linear segment. Our simulations show that SEEV significantly improves verification efficiency while maintaining the CBF quality across various benchmark systems and neural network structures. Our code is available at https://github.com/HongchaoZhang-HZ/SEEV.

eess.SY↗

arXiv2024

Who is Responsible? Explaining Safety Violations in Multi-Agent Cyber-Physical Systems

Multi-agent cyber-physical systems are present in a variety of applications. Agent decision-making can be affected due to errors induced by uncertain, dynamic operating environments or due to incorrect actions taken by an agent. When an erroneous decision that leads to a violation of safety is identified, assigning responsibility to individual agents is a key step toward preventing future accidents. Current approaches to carrying out such investigations require human labor or high degree of familiarity with operating environments. Automated strategies to assign responsibility can achieve a significant reduction in human effort and associated cognitive burden. In this paper, we develop an automated procedure to assign responsibility for safety violations to actions of any single agent in a principled manner. We base our approach on reasoning about safety violations in road safety. Given a safety violation, we use counterfactual reasoning to create alternative scenarios, showing how different outcomes could have occurred if certain actions had been replaced by others. We introduce the degree of responsibility (DoR) metric for each agent. The DoR, using the Shapley value, quantifies each agent's contribution to the safety violation, providing a basis to explain and justify decisions. We also develop heuristic techniques and methods based on agent interaction structures to improve scalability as agent numbers grow. We examine three safety violation cases from the National Highway Traffic Safety Administration (NHTSA). We run experiments using CARLA urban driving simulator. Results show the DoR improves the explainability of decisions and accountability for agent actions and their consequences.

eess.SY↗

arXiv2024

Verification and Synthesis of Compatible Control Lyapunov and Control Barrier Functions

Safety and stability are essential properties of control systems. Control Barrier Functions (CBFs) and Control Lyapunov Functions (CLFs) are powerful tools to ensure safety and stability respectively. However, previous approaches typically verify and synthesize the CBFs and CLFs separately, satisfying their respective constraints, without proving that the CBFs and CLFs are compatible with each other, namely at every state, there exists control actions within the input limits that satisfy both the CBF and CLF constraints simultaneously. Ignoring the compatibility criteria might cause the CLF-CBF-QP controller to fail at runtime. There exists some recent works that synthesized compatible CLF and CBF, but relying on nominal polynomial or rational controllers, which is just a sufficient but not necessary condition for compatibility. In this work, we investigate verification and synthesis of compatible CBF and CLF independent from any nominal controllers. We derive exact necessary and sufficient conditions for compatibility, and further formulate Sum-Of-Squares programs for the compatibility verification. Based on our verification framework, we also design a nominal-controller-free synthesis method, which can effectively expands the compatible region, in which the system is guaranteed to be both safe and stable. We evaluate our method on a non-linear toy problem, and also a 3D quadrotor to demonstrate its scalability. The code is open-sourced at \url{https://github.com/hongkai-dai/compatible_clf_cbf}.

eess.SY↗

arXiv2024

Learning a Formally Verified Control Barrier Function in Stochastic Environment

Safety is a fundamental requirement of control systems. Control Barrier Functions (CBFs) are proposed to ensure the safety of the control system by constructing safety filters or synthesizing control inputs. However, the safety guarantee and performance of safe controllers rely on the construction of valid CBFs. Inspired by universal approximatability, CBFs are represented by neural networks, known as neural CBFs (NCBFs). This paper presents an algorithm for synthesizing formally verified continuous-time neural Control Barrier Functions in stochastic environments in a single step. The proposed training process ensures efficacy across the entire state space with only a finite number of data points by constructing a sample-based learning framework for Stochastic Neural CBFs (SNCBFs). Our methodology eliminates the need for post hoc verification by enforcing Lipschitz bounds on the neural network, its Jacobian, and Hessian terms. We demonstrate the effectiveness of our approach through case studies on the inverted pendulum system and obstacle avoidance in autonomous driving, showcasing larger safe regions compared to baseline methods.

cs.RO↗

arXiv2024

Fault Tolerant Neural Control Barrier Functions for Robotic Systems under Sensor Faults and Attacks

Safety is a fundamental requirement of many robotic systems. Control barrier function (CBF)-based approaches have been proposed to guarantee the safety of robotic systems. However, the effectiveness of these approaches highly relies on the choice of CBFs. Inspired by the universal approximation power of neural networks, there is a growing trend toward representing CBFs using neural networks, leading to the notion of neural CBFs (NCBFs). Current NCBFs, however, are trained and deployed in benign environments, making them ineffective for scenarios where robotic systems experience sensor faults and attacks. In this paper, we study safety-critical control synthesis for robotic systems under sensor faults and attacks. Our main contribution is the development and synthesis of a new class of CBFs that we term fault tolerant neural control barrier function (FT-NCBF). We derive the necessary and sufficient conditions for FT-NCBFs to guarantee safety, and develop a data-driven method to learn FT-NCBFs by minimizing a loss function constructed using the derived conditions. Using the learned FT-NCBF, we synthesize a control input and formally prove the safety guarantee provided by our approach. We demonstrate our proposed approach using two case studies: obstacle avoidance problem for an autonomous mobile robot and spacecraft rendezvous problem, with code available via https://github.com/HongchaoZhang-HZ/FTNCBF.

cs.RO↗

arXiv2024

Explicit Formula for Partial Information Decomposition

Mutual information between two random variables is a well-studied notion, whose understanding is fairly complete. Mutual information between one random variable and a pair of other random variables, however, is a far more involved notion. Specifically, Shannon's mutual information does not capture fine-grained interactions between those three variables, resulting in limited insights in complex systems. To capture these fine-grained interactions, in 2010 Williams and Beer proposed to decompose this mutual information to information atoms, called unique, redundant, and synergistic, and proposed several operational axioms that these atoms must satisfy. In spite of numerous efforts, a general formula which satisfies these axioms has yet to be found. Inspired by Judea Pearl's do-calculus, we resolve this open problem by introducing the do-operation, an operation over the variable system which sets a certain marginal to a desired value, which is distinct from any existing approaches. Using this operation, we provide the first explicit formula for calculating the information atoms so that Williams and Beer's axioms are satisfied, as well as additional properties from subsequent studies in the field.

cs.IT↗

arXiv2023

Almost-Sure Safety Guarantees of Stochastic Zero-Control Barrier Functions Do Not Hold

The 2021 paper "Control barrier functions for stochastic systems" provides theorems that give almost sure safety guarantees given stochastic zero control barrier function (ZCBF). Unfortunately, both the theorem and its proof is invalid. In this letter, we illustrate on a toy example that the almost sure safety guarantees for stochastic ZCBF do not hold and explain why the proof is flawed. Although stochastic reciprocal barrier functions (RCBF) also uses the same proof technique, we provide a different proof technique that verifies that stochastic RCBFs are indeed safe with probability one. Using the RCBF, we derive a modified ZCBF condition that guarantees safety with probability one. Finally, we provide some discussion on the role of unbounded controls in the almost-sure safety guarantees of RCBFs, and show that the rate of divergence of the ratio of the drift and diffusion is the key for whether a system has almost sure safety guarantees.

math.OC↗