Symbolic Abstractions with Guarantees: A Data-Driven Divide-and-Conquer Strategy

This article is concerned with a data-driven divide-and-conquer strategy to construct symbolic abstractions for interconnected control networks with unknown mathematical models. We employ a notion of alternating bisimulation functions (ABF) to quantify the closeness between state trajectories of an interconnected network and its symbolic abstraction. Consequently, the constructed symbolic abstraction can be leveraged as a beneficial substitute for the formal verification and controller synthesis over the interconnected network. In our data-driven framework, we first establish a relation between each unknown subsystem and its data-driven symbolic abstraction, so-called alternating pseudo-bisimulation function (APBF), with a guaranteed probabilistic confidence. We then provide compositional conditions based on max-type small-gain techniques to construct an ABF for an unknown interconnected network using APBF of its individual subsystems, constructed from data. We demonstrate the efficacy of our data-driven approach over a room temperature network composing 100 rooms with unknown models. We construct a symbolic abstraction from data for each room as an appropriate substitute of original system and compositionally synthesize controllers regulating the temperature of each room within a safe zone with some guaranteed probabilistic confidence

Safety Barrier Certificates for Stochastic Control Systems with Wireless Communication Networks

This work is concerned with a formal approach for safety controller synthesis of stochastic control systems with both process and measurement noises while considering wireless communication networks between sensors, controllers, and actuators. The proposed scheme is based on control barrier certificates (CBC), which allows us to provide safety certifications for wirelessly-connected stochastic control systems. Despite the available literature on designing control barrier certificates, there has been unfortunately no consideration of wireless communication networks to capture potential packet losses and end-to-end delays, which is absolutely crucial in safety-critical real-world applications. In our proposed setting, the key objective is to construct a control barrier certificate together with a safety controller while providing a lower bound on the satisfaction probability of the safety property over a finite time horizon. We propose a systematic approach in the form of sum-of-squares optimization and matrix inequalities for the synthesis of CBC and its associated controller. We demonstrate the efficacy of our approach on a permanent magnet synchronous motor. For the application of automotive electric steering under a wireless communication network, we design a CBC together with a safety controller to maintain the electrical current of the motor in a safe set within a finite time horizon while providing a formal probabilistic guarantee

From Small-Gain Theory to Compositional Construction of Barrier Certificates for Large-Scale Stochastic Systems

This paper is concerned with a compositional approach for the construction of control barrier certificates for large-scale interconnected stochastic systems while synthesizing hybrid controllers against high-level logic properties. Our proposed methodology involves decomposition of interconnected systems into smaller subsystems and leverages the notion of control sub-barrier certificates of subsystems, enabling one to construct control barrier certificates of interconnected systems by employing some $\max$-type small-gain conditions. The main goal is to synthesize hybrid controllers enforcing complex logic properties including the ones represented by the accepting language of deterministic finite automata (DFA), while providing probabilistic guarantees on the satisfaction of given specifications in bounded-time horizons. To do so, we propose a systematic approach to first decompose high-level specifications into simple reachability tasks by utilizing automata corresponding to the complement of specifications. We then construct control sub-barrier certificates and synthesize local controllers for those simpler tasks and combine them to obtain a hybrid controller that ensures satisfaction of the complex specification with some lower-bound on the probability of satisfaction. To compute control sub-barrier certificates and corresponding local controllers, we provide two systematic approaches based on sum-of-squares (SOS) optimization program and counter-example guided inductive synthesis (CEGIS) framework. We finally apply our proposed techniques to two physical case studies

Compositional Synthesis of Control Barrier Certificates for Networks of Stochastic Systems against ω\omega-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 control policies against $\omega$-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 $\omega$-regular specifications into simpler tasks by utilizing the automata corresponding to the complement of 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. We also provide a sum-of-squares (SOS) optimization problem for the computation of control sub-barrier certificates and local control policies of subsystems. Finally, we demonstrate the effectiveness of our proposed approaches by applying them to a physical case study

Data-driven verification and synthesis of stochastic systems through 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 pre-computed 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