This paper is concerned with a compositional approach for constructing abstractions of interconnected discrete-time stochastic control systems. The abstraction framework is based on new notions of so-called stochastic simulation functions, using which one can quantify the distance between original interconnected stochastic control systems and their abstractions in the probabilistic setting. Accordingly, one can leverage the proposed results to perform analysis and synthesis over abstract interconnected systems, and then carry the results over concrete ones. In the first part of the paper, we derive sufficient small-gain type conditions for the compositional quantification of the distance in probability between the interconnection of stochastic control subsystems and that of their abstractions. In the second part of the paper, we focus on the class of discrete-time linear stochastic control systems with independent noises in the abstract and concrete subsystems. For this class of systems, we propose a computational scheme to construct abstractions together with their corresponding stochastic simulation functions. We demonstrate the effectiveness of the proposed results by constructing an abstraction (totally 4 dimensions) of the interconnection of four discrete-time linear stochastic control subsystems (together 100 dimensions) in a compositional fashion.
In this paper, we focus on mitigating the computational complexity in abstraction-based controller synthesis for interconnected control systems. To do so, we provide a compositional framework for the construction of abstractions for interconnected systems and a bottom-up controller synthesis scheme. In particular, we propose a notion of approximate composition which makes it possible to compute an abstraction of the global interconnected system from the abstractions (possibly of different types) of its components. Finally, by leveraging our notion of approximate composition, we propose a botto…
In this work, we propose a compositional scheme for the safety controller synthesis of interconnected discrete-time stochastic systems with Markovian switching signals. Our proposed approach is based on a notion of so-called control storage certificates computed for individual subsystems, by leveraging which, one can synthesize state-feedback controllers for interconnected systems to enforce safety specifications over finite time horizons. To do so, we employ a sum-of-squares (SOS) optimization approach to search for multiple storage certificates of each switching subsystem while synthesizing …
In this paper, we introduce a compositional method for the construction of finite abstractions of interconnected discrete-time switched systems. Particularly, we use a notion of so-called alternating simulation function as a relation between each switched subsystem and its finite abstraction. Based on some small-gain type conditions, we use those alternating simulation functions to construct compositionally an overall alternating simulation function as a relation between an interconnection of finite abstractions and that of switched subsystems. This overall alternating simulation function allo…
In this paper, we propose a compositional approach to construct opacity-preserving finite abstractions (a.k.a symbolic models) for networks of discrete-time nonlinear control systems. Particularly, we introduce new notions of simulation functions that characterize the distance between control systems while preserving opacity properties across them. Instead of treating large-scale systems in a monolithic manner, we develop a compositional scheme to construct the interconnected finite abstractions together with the overall opacity-preserving simulation functions. For a network of incrementally i…
This paper formulates and studies the concepts of approximate (alternating) bisimulation relations characterizing equivalence relations between interconnected systems and their abstractions. These equivalence relations guarantee that the symbolic model conserves the original model's dynamics. We develop a compositional approach for abstraction-based controller synthesis by relying on the notions of approximate composition and incremental input-to-state stability. In particular, given a large-scale system consisting of interconnected components, we provide conditions under which the concept of …
We propose a compositional approach for constructing abstractions of general Markov decision processes using approximate probabilistic relations. The abstraction framework is based on the notion of $δ$-lifted relations, using which one can quantify the distance in probability between the interconnected gMDPs and that of their abstractions. This new approximate relation unifies compositionality results in the literature by incorporating the dependencies between state transitions explicitly and by allowing abstract models to have either finite or infinite state spaces. Accordingly, one can lever…
Compositional abstraction and safety synthesis using overlapping symbolic models
KTH Royal Institute of Technology（皇家理工学院）；Laboratoire des signaux et systèmes (L2S), CNRS, CentraleSupélec, Université Paris-Sud, Université Paris-Saclay（信号与系统实验室，法国国家科学研究中心，中央苏佩莱克高等学院，巴黎第十一大学，巴黎萨克雷大学）；Univ. Grenoble Alpes/CNRS, GIPSA-Lab（格勒诺布尔阿尔卑斯大学/法国国家科学研究中心，GIPSA实验室）
In this paper, we propose a compositional approach for the construction of finite abstractions (a.k.a. finite Markov decision processes (MDPs)) for networks of discrete-time stochastic control subsystems that are not necessarily stabilizable. The proposed approach leverages the interconnection topology and a notion of finite-step stochastic storage functions, that describes joint dissipativity-type properties of subsystems and their abstractions, and establishes a finite-step stochastic simulation function as a relation between the network and its abstraction. To this end, we first develop a n…
This paper provides a compositional scheme based on dissipativity approaches for constructing finite abstractions of continuous-time continuous-space stochastic control systems. The proposed framework enjoys the structure of the interconnection topology and employs a notion of stochastic storage functions, that describe joint dissipativity-type properties of subsystems and their abstractions. By utilizing those stochastic storage functions, one can establish a relation between continuous-time continuous-space stochastic systems and their finite counterparts while quantifying probabilistic dist…
Compositional Construction of Infinite Abstractions for Networks of Stochastic Control Systems
Technical University of Munich（慕尼黑工业大学）；Newcastle University（纽卡斯尔大学）；University of Colorado Boulder（科罗拉多大学博尔德分校）；Ludwig Maximilian University of Munich（慕尼黑大学）
In this paper, we provide a compositional framework for synthesizing hybrid controllers for interconnected discrete-time control systems enforcing specifications expressed by co-Buchi automata. In particular, we first decompose the given specification to simpler reachability tasks based on automata representing the complements of original co-Buchi automata. Then, we provide a systematic approach to solve those simpler reachability tasks by computing corresponding control barrier functions. We show that such control barrier functions can be constructed compositionally by assuming some small-gai…
In this paper, we propose a compositional framework for the construction of control barrier certificates for large-scale stochastic switched systems accepting multiple control barrier certificates with some dwell-time conditions. The proposed scheme is based on a notion of so-called augmented pseudo-barrier certificates computed for each switched subsystem, using which one can compositionally synthesize state-feedback controllers for interconnected systems enforcing safety specifications over a finite-time horizon. In particular, we first leverage sufficient max-type small-gain conditions to c…
In this work, we present a compositional safety controller synthesis approach for the class of discrete-time linear control systems. Here, we leverage a state-of-the-art result on the computation of robust controlled invariant sets. To tackle the complexity of controller synthesis over complex interconnected systems, this paper introduces a decentralized controller synthesis scheme. Rather than treating the interconnected system as a whole, we first design local safety controllers for each subsystem separately to enforce local safety properties, with polytopic state and input constraints as we…
In this paper, we provide a compositional approach for constructing finite abstractions (a.k.a. finite Markov decision processes (MDPs)) of interconnected discrete-time stochastic switched systems. The proposed framework is based on a notion of stochastic simulation functions, using which one can employ an abstract system as a substitution of the original one in the controller design process with guaranteed error bounds on their output trajectories. To this end, we first provide probabilistic closeness guarantees between the interconnection of stochastic switched subsystems and that of their f…
In this paper, we develop a compositional scheme for the construction of continuous approximations for interconnections of infinitely many discrete-time switched systems. An approximation (also known as abstraction) is itself a continuous-space system, which can be used as a replacement of the original (also known as concrete) system in a controller design process. Having designed a controller for the abstract system, it is refined to a more detailed one for the concrete system. We use the notion of so-called simulation functions to quantify the mismatch between the original system and its app…
We construct compositional continuous approximations for an interconnection of infinitely many discrete-time switched systems. An approximation (known as abstraction) is itself a continuous-space system, which can be used as a replacement of the original (known as concrete) system in a controller design process. Having synthesized a controller for the abstract system, the controller is refined to a more detailed controller for the concrete system. To quantify the mismatch between the output trajectory of the approximation and of that the original system, we use the notion of so-called simulati…
In this work, we propose a compositional framework for the construction of control barrier functions for networks of continuous-time stochastic hybrid systems enforcing complex logic specifications expressed by finite-state automata. The proposed scheme is based on a notion of so-called pseudo-barrier functions computed for subsystems, by employing which one can synthesize hybrid controllers for interconnected systems enforcing complex specifications over a finite-time horizon. Particularly, we first leverage sufficient small-gain type conditions to compositionally construct control barrier fu…
In this paper, we propose a compositional framework for the synthesis of safety controllers for networks of partially-observed discrete-time stochastic control systems (a.k.a. continuous-space POMDPs). Given an estimator, we utilize a discretization-free approach to synthesize controllers ensuring safety specifications over finite-time horizons. The proposed framework is based on a notion of so-called local control barrier functions computed for subsystems in two different ways. In the first scheme, no prior knowledge of estimation accuracy is needed. The second framework utilizes a probabilit…
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 $ω$-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 \emph{control sub-barrier certificates} of subsystems, which are used to composition…
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 com…
In this work, we propose a compositional framework for the verification of approximate initial-state opacity for networks of discrete-time switched systems. The proposed approach is based on a notion of approximate initial-state opacity-preserving simulation functions (InitSOPSFs), which characterize how close concrete networks and their finite abstractions are in terms of the satisfaction of approximate initial state opacity. We show that such InitSOPSFs can be obtained compositionally by assuming some small-gain type conditions and composing so-called local InitSOPSFs constructed for each su…
This paper offers a finite-state abstraction of traffic coordination and congestion in a network of interconnected roads (NOIR). By applying mass conservation, we model traffic coordination as a Markov process. Model Predictive Control (MPC) is applied to control traffic congestion through the boundary of the traffic network. The optimal boundary inflow is assigned as the solution of a constrained quadratic programming problem. Additionally, the movement phases commanded by traffic signals are determined using receding horizon optimization. In simulation, we show how traffic congestion can be …
Complex, interconnected Cyber-physical Systems (CPS) are increasingly common in applications including smart grids and transportation. Ensuring safety of interconnected systems whose dynamics are coupled is challenging because the effects of faults and attacks in one sub-system can propagate to other sub-systems and lead to safety violations. In this paper, we study the problem of safety-critical control for CPS with coupled dynamics when some sub-systems are subject to failure or attack. We first propose resilient-safety indices (RSIs) for the faulty or compromised sub-systems that bound the …
