Barrier Certificates Research Articles

Compositional Synthesis of Controllers via Co-Büchi Barrier Certificates

The success of barrier certificates to ensure the safety of control systems has led to their adoption in designing controllers to guarantee more general ω-regular automata specifications. Recent developments in co-Büchi barrier certificates propose the direct construction of the product of the system and the automaton, ensuring that the accepting states of the automaton are visited finitely often. Despite the benefits, challenges arise when finding these certificates for large-scale systems. To overcome such challenges, we introduce a compositional controller synthesis via co-Büchi barrier certificates. We decompose the large-scale system into an interconnection of smaller subsystems, construct local certificates for these subsystems, and utilize a dissipativity-type condition to guarantee the existence of a certificate for the large-scale system. We design controllers for the subsystems that, when concatenated, act as a formally correct controller for the large-scale system. Finally, we illustrate our approach with a numerical example.

Safe Reach Set Computation via Neural Barrier Certificates

We present a novel technique for online safety verification of autonomous systems, which performs reachability analysis efficiently for both bounded and unbounded horizons by employing neural barrier certificates. Our approach uses barrier certificates given by parameterized neural networks that depend on a given initial set, unsafe sets, and time horizon. Such networks are trained efficiently offline using system simulations sampled from regions of the state space. We then employ a meta-neural network to generalize the barrier certificates to state-space regions that are outside the training set. These certificates are generated and validated online as sound over-approximations of the reachable states, thus either ensuring system safety or activating appropriate alternative actions in unsafe scenarios. We demonstrate our technique on case studies from linear models to nonlinear control-dependent models for online autonomous driving scenarios.

Transfer of Safety Controllers Through Learning Deep Inverse Dynamics Model

Control barrier certificates have proven effective in formally guaranteeing the safety of the control systems. However, designing a control barrier certificate is a time-consuming and computationally expensive endeavor that requires expert input in the form of domain knowledge and mathematical maturity. Additionally, when a system undergoes slight changes, the new controller and its correctness certificate need to be recomputed, incurring similar computational challenges as those faced during the design of the original controller. Prior approaches have utilized transfer learning to transfer safety guarantees in the form of a barrier certificate while maintaining the control invariant. Unfortunately, in practical settings, the source and the target environments often deviate substantially in their control inputs, rendering the aforementioned approach impractical. To address this challenge, we propose integrating inverse dynamics—a neural network that suggests required action given a desired successor state—of the target system with the barrier certificate of the source system to provide formal proof of safety. In addition, we propose a validity condition that, when met, guarantees correctness of the controller. We demonstrate the efficacy of our method with an inverted pendulum case study.

Verification of Hyperproperties for Dynamical Systems via Barrier Certificates

Verification of Hyperproperties for Dynamical Systems via Barrier Certificates

Multiplicative Barrier Certificates for Probabilistic Safety of Markov Jump Systems

This work offers a formal framework for providing safety certificates over discrete-time Markov jump systems (MJSs). Our proposed scheme centers around a new concept of multiplicative barrier certificates (MBCs), enabling us to establish a probabilistic lower bound for the safety property in infinite time horizons within this class of models. In particular, while a summative barrier certificate, as commonly proposed in the relevant literature, may not be available to ensure the safety of MJS models, we introduce an alternative approach in a multiplicative form, as opposed to the conventional “decreasing in mean” formulation. We demonstrate that in scenarios where a summative barrier does not exist, there is potential for the existence of a multiplicative counterpart. We also provide a systematic methodology grounded on the counterexample-guided inductive synthesis (CEGIS) scheme to systematically construct the proposed multiplicative certificates. We showcase the efficacy of our approach through its application in an air traffic management system.

Learning safe control for multi-robot systems: Methods, verification, and open challenges

In this survey, we review the recent advances in control design methods for robotic multi-agent systems (MAS), focusing on learning-based methods with safety considerations. We start by reviewing various notions of safety and liveness properties, and modeling frameworks used for problem formulation of MAS. Then we provide a comprehensive review of learning-based methods for safe control design for multi-robot systems. We start with various shielding-based methods, such as safety certificates, predictive filters, and reachability tools. Then, we review the current state of control barrier certificate learning in both a centralized and distributed manner, followed by a comprehensive review of multi-agent reinforcement learning with a particular focus on safety. Next, we discuss the state-of-the-art verification tools for the correctness of learning-based methods. Based on the capabilities and the limitations of the state-of-the-art methods in learning and verification for MAS, we identify various broad themes for open challenges: how to design methods that can achieve good performance along with safety guarantees; how to decompose single-agent-based centralized methods for MAS; how to account for communication-related practical issues; and how to assess transfer of theoretical guarantees to practice.

Verification of Diagnosability for Cyber-Physical Systems via Hybrid Barrier Certificates

In this paper, we provide an automata-based framework for verifying diagnosability property of Cyber-Physical Systems leveraging a notion of so-called hybrid barrier certificates. Concretely, we first construct a so-called (δ,K)-deterministic finite automata ((δ,K)-DFA) associated with the desired diagnosability property, which captures the occurrence of the fault to be diagnosed. Having a (δ,K)-DFA, we show that the verification of diagnosability properties is equivalent to a safety verification problem over a product system between this DFA and the dynamical system of interest. We further show that such a verification problem can be solved via computing hybrid barrier certificates for the product system. To compute the hybrid barrier certificates, we provide a systematic technique leveraging a counter-example guided inductive synthesis framework. Finally, we showcase the effectiveness of our results through a case study.

Distributed Safety Controller Synthesis for Unknown Interconnected Systems via Graph Neural Networks

This paper focuses on distributed controller synthesis for ensuring the safety of large-scale systems with unknown dynamics and known interconnection structures, employing control barrier certificates. Our approach centers on a distributed framework tailored to create control barrier certificates for safety, utilizing the interconnections within large-scale systems. We introduce a novel application of graph neural networks to synthesize these certificates and distributed controllers in a data-driven fashion. The formal correctness of trained networks is subsequently validated through data-driven techniques involving the solution of a sampling-based optimization problem. To illustrate the effectiveness of our methodology, we conduct experiments on a complex interconnected system comprising as many as 1000 components.

Formal Synthesis of Safety Controllers via $k$-Inductive Control Barrier Certificates

Formal Synthesis of Safety Controllers via $k$-Inductive Control Barrier Certificates

A Compositional Dissipativity Approach for Data-Driven Safety Verification of Large-Scale Dynamical Systems

This work is concerned with a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">compositional data-driven approach</i> for formal safety verification of large-scale continuous-time dynamical systems with unknown models. The proposed framework enjoys the interconnection matrix and joint dissipativity-type properties of subsystems, described by the notion of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">stochastic storage certificates</i> . In the first part of the paper, we cast the required conditions for constructing storage certificates as a robust optimization program (ROP). Since the proposed ROP is not tractable due to the unknown model appearing in one of its constraints, we propose a scenario optimization program (SOP) corresponding to the original ROP by collecting finite numbers of data from trajectories of each subsystem. By establishing a probabilistic relation between the optimal value of SOP and that of ROP, we construct a storage certificate for each unknown subsystem based on the number of data and a required level of confidence. We accordingly propose a compositional technique based on <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dissipativity reasoning</i> to construct <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">stochastic barrier certificates</i> of interconnected systems based on storage certificates of individual subsystems. By leveraging the acquired barrier certificate, we quantify a lower bound on the probability that an interconnected system never reaches a certain unsafe region in finite time horizons with an a-priori guaranteed confidence. We also propose an auxiliary compositional approach without requiring any compositionality condition but at the cost of providing a potentially conservative safety guarantee. In the second part of the paper, we propose our approaches for <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">deterministic</i> continuous-time systems with unknown dynamics. We verify our results over an unknown room temperature network containing 100 rooms.

Data-driven verification and synthesis of stochastic systems via barrier certificates

In this work, we study verification and synthesis problems for safety specifications over unknown discrete-time stochastic systems. When a model of the system is available, barrier certificates have been successfully applied for ensuring the satisfaction of safety specifications. In this work, we formulate the computation of barrier certificates as a robust convex program (RCP). Solving the acquired RCP is hard in general because the model of the system that appears in one of the constraints of the RCP is unknown. We propose a data-driven approach that replaces the uncountable number of constraints in the RCP with a finite number of constraints by taking finitely many random samples from the trajectories of the system. We thus replace the original RCP with a scenario convex program (SCP) and show how to relate their optimizers. We guarantee that the solution of the SCP is a solution of the RCP with a priori guaranteed confidence when the number of samples is larger than a specific value. This provides a lower bound on the safety probability of the original unknown system together with a controller in the case of synthesis. We also discuss an extension of our verification approach to a case where the associated robust program is non-convex and show how a similar methodology can be applied. Finally, the applicability of our proposed approach is illustrated through three case studies.

A Barrier-Certificated Reinforcement Learning Approach for Enhancing Power System Transient Stability

Increasing integration of renewable resources brings more flexibility and poses new challenges to modern power systems, leading to highly nonlinear and complex dynamics. This paper aims to provide a general solution framework to traditional control problems, such as frequency control and voltage control, which attempt to maintain the stability of either synchronous generators-governed or inverter-governed systems when subjected to a disturbance and simultaneously guarantee operational constraints, providing a complete complement to existing works on control design. Building on reinforcement learning (RL) and control barrier functions, the framework includes two subsystems, i.e., a model-free controller and a barrier-certification system, which discover RL-based control actions and sequentially filter them using a barrier certificate to satisfy operational constraints. Calculating a barrier function is generally challenging for a complex power system. This is addressed by representing the barrier function using neural networks (NNs) and data-based approaches. An adaptive method is introduced to certify the neural barrier function that perseveres barrier conditions, which is more compatible with online implementation. The proposed framework synthesizes a stabilizing controller that satisfies predefined safety regions. The effectiveness of the proposed framework is demonstrated via several comparative case studies.

Compositional synthesis of control barrier certificates for networks of stochastic systems against [formula omitted]-regular specifications

This paper is concerned with a compositional scheme for the construction of control barrier certificates for interconnected discrete-time stochastic systems. The main objective is to synthesize switching controllers against ω-regular properties that can be described by accepting languages of deterministic Streett automata (DSA) along with providing probabilistic guarantees for the satisfaction of such specifications. The proposed framework leverages the interconnection topology and a notion of so-called control sub-barrier certificates of subsystems, which are used to compositionally construct control barrier certificates of interconnected systems by imposing some dissipativity-type compositionality conditions. We propose a systematic approach to decompose high-level ω-regular specifications into simpler tasks by utilizing the automata corresponding to the specifications. In addition, we formulate an alternating direction method of multipliers (ADMM) optimization problem in order to obtain suitable control sub-barrier certificates of subsystems while satisfying compositionality conditions. For systems with polynomial dynamics, we provide a sum-of-squares (SOS) optimization problem for the computation of control sub-barrier certificates and local controllers of subsystems. Finally, we demonstrate the effectiveness of our proposed approaches by applying them to a physical case study.

Formal Synthesis of Neural Barrier Certificates for Continuous Systems via Counterexample Guided Learning

This paper presents a novel approach to safety verification based on neural barrier certificates synthesis for continuous dynamical systems. We construct the synthesis framework as an inductive loop between a Learner and a Verifier based on barrier certificate learning and counterexample guidance. Compared with the counterexample-guided verification method based on the SMT solver, we design and learn neural barrier functions with special structure, and use the special form to convert the counterexample generation into a polynomial optimization problem for obtaining the optimal counterexample. In the verification phase, the task of identifying the real barrier certificate can be tackled by solving the Linear Matrix Inequalities (LMI) feasibility problem, which is efficient and makes the proposed method formally sound. The experimental results demonstrate that our approach is more effective and practical than the traditional SOS-based barrier certificates synthesis and the state-of-the-art neural barrier certificates learning approach.

Safety verification for Regime-Switching Jump Diffusions via barrier certificates

It is well known that for a stochastic hybrid system, if its failure probability does not exceed a given safety threshold, the safety of the system can be guaranteed. Thus, in this paper, we concern with the upper bound of failure probability for failure analysis of a class of nonautonomous stochastic hybrid systems, denoted as Regime-Switching Jump Diffusions (RSJDs). We start with the definition of RSJDs, which contain not only continuous flows described by stochastic differential equations, but also instantaneous behavior described by Markovian switching and instantaneous behavior described by Lévy jump. Then we decompose the failure probability of RSJDs in [t0,+∞) into two segments: one is defined over [T,+∞), and the other is defined over [t0,T]. For failure probability over [T,+∞), by utilizing multiple vectorial barrier certificates, an asymptotically decreasing bound of failure probability with respect to T is established, where a general nonnegative matrix is used instead of a special nonnegative matrix defined by the exponent of essentially nonnegative matrix for a broader applicability. For failure probability over [t0,T], a generalized c-martingale condition is adopted to obtain a T-dependent failure probability bound, in which two non-negative scalar functions are utilized to relax the conservativeness of infinitesimal generators. Finally, for rational RSJDs, we transform the decomposed failure analysis problems into two semi-definite programming (SDP) problems, and then solve them via sum of squares programming. The applicabilities and effectiveness of our computable decomposition methodology are illustrated through three examples.

Safety Verification of Nonlinear Systems with Bayesian Neural Network Controllers

Bayesian neural networks (BNNs) retain NN structures with a probability distribution placed over their weights. With the introduced uncertainties and redundancies, BNNs are proper choices of robust controllers for safety-critical control systems. This paper considers the problem of verifying the safety of nonlinear closed-loop systems with BNN controllers over unbounded-time horizon. In essence, we compute a safe weight set such that as long as the BNN controller is always applied with weights sampled from the safe weight set, the controlled system is guaranteed to be safe. We propose a novel two-phase method for the safe weight set computation. First, we construct a reference safe control set that constraints the control inputs, through polynomial approximation to the BNN controller followed by polynomial-optimization-based barrier certificate generation. Then, the computation of safe weight set is reduced to a range inclusion problem of the BNN on the system domain w.r.t. the safe control set, which can be solved incrementally and the set of safe weights can be extracted. Compared with the existing method based on invariant learning and mixed-integer linear programming, we could compute safe weight sets with larger radii on a series of linear benchmarks. Moreover, experiments on a series of widely used nonlinear control tasks show that our method can synthesize large safe weight sets with probability measure as high as 95% even for a large-scale system of dimension 7.

Higher Order Barrier Certificates for Leader–Follower Multiagent Systems

This paper presents control strategies based on time-varying convergent higher order control barrier functions for a class of leader-follower multi-agent systems under signal temporal logic (STL) tasks. Each agent is assigned a local STL task which may be dependent on the behavior of agents involved in other tasks. We consider one or more than one leader in the multi-agent system. The leader has knowledge on the associated tasks and controls the performance of the subgroup involved agents. The followers are not aware of the tasks, and do not have any control authority to reach them. They follow the leader commands indirectly, according to their dynamics interconnections, for the task satisfaction. We further assume that the input-to-state stability (ISS) property for the multi-agent system is fulfilled. First, robust solutions for the task satisfaction, based on the leader's accessibility to the follower agents' states are suggested. In addition, using the notion of higher order barrier functions, individual barrier certificates for each agent evolving in a formation dynamic structure are proposed. For the case of presence of more leaders in the subgroups, we provide decentralized barrier certificates. Our approach finds solutions to guarantee the satisfaction of STL tasks independent of the agents' initial conditions.

Model-Free Safe Reinforcement Learning Through Neural Barrier Certificate

Safety is a critical concern when applying reinforcement learning (RL) to real-world control tasks. However, existing safe RL works either only consider expected safety constraint violations and fail to maintain safety guarantees, or use overly conservative safety certificate tools borrowed from safe control theory, which sacrifices reward optimization and relies on analytic system models. This letter proposes a model-free safe RL algorithm that achieves near-zero constraint violations with high rewards. Our key idea is to jointly learn a policy and a neural barrier certificate under stepwise state constraint setting. The barrier certificate is learned in a model-free manner by minimizing the violations of appropriate barrier properties on transition data collected by the policy. We extend the single-step invariant property of the barrier certificate to a multi-step version and construct the corresponding multi-step invariant loss. This loss balances the bias and variance of the barrier certificate and enhances both the safety and performance of the policy. The policy is optimized under the constraint of the multi-step invariant property using the Lagrangian method. We optimize the policy in a model-free manner by introducing an importance sampling weight in the constraint. We test our algorithm on multiple problems, including classic control tasks, robot collision avoidance, and autonomous driving. Results show that our algorithm achieves near-zero constraint violations and high performance compared to the baselines. Moreover, the learned barrier certificates successfully identify the feasible regions on multiple tasks.

LQG Reference Tracking With Safety and Reachability Guarantees Under Unknown False Data Injection Attacks

We investigate a linear quadratic Gaussian (LQG) tracking problem with safety and reachability constraints in the presence of an adversary who mounts a false data injection attack on an unknown set of sensors. For each possible set of compromised sensors, we maintain a state estimator disregarding the sensors in that set, and calculate the optimal LQG control input at each time based on this estimate. We propose a control policy which constrains the control input to lie within a fixed distance of the optimal control input corresponding to each state estimate. The control input is obtained at each time step by solving a quadratically constrained quadratic program. We prove that our policy can achieve a desired probability of safety and reachability using the barrier certificate method. Our control policy is evaluated via a numerical case study.

Safety Verification and Controller Synthesis for Systems with Input Constraints

In this paper we consider the safety verification and safe controller synthesis for nonlinear control systems. The Control Barrier Certificates (CBC) approach is proposed as an extension to the barrier certificates approach. Our approach can be used to characterize control invariance of a given set in terms of safety of a general nonlinear control system subject to input constraints. From the point of view of controller design, the proposed method provides an approach to synthesize a safe control law that guarantees that the trajectories of the system starting from a given initial set do not enter an unsafe set. Unlike the related control barrier functions formulations, our formulation only considers the vector field within the tangent cone of the zero level set defined by the certificates, and is shown to be less conservative by means of numerical evidence. For polynomial systems with semi-algebraic initial and safe sets, CBCs and safe control laws can be synthesized using sum-of-squares decomposition and semi-definite programming. Numerical examples demonstrate the efficacy of our approach.