Stochastic Simulation Functions Research Articles

Chloride ion transport and service life prediction of aeolian sand concrete under dry–wet cycles

Abstract In this study, the dry–wet cycle test of chloride salt was carried out on aeolian sand concrete with different contents, and the chloride ion content in aeolian sand concrete was determined by taking powder from different depths on one side of the test block. And then combined with Monte Carlo stochastic statistical simulation and Weibull probability distribution function, the service life prediction model of aeolian sand concrete against chloride ion erosion is established. The results show that the free chloride ion content in aeolian sand concrete decreases with the increase in the depth from the surface of the specimen. At the same depth from the surface of the specimen, the free chloride ion content gradually increases with the increase in the number of dry–wet cycles. Through the analysis of life prediction, it is concluded that with the increase in aeolian sand content, the service life of aeolian sand concrete increases first and then decreases. The service life value of concrete with 75% aeolian sand content is the largest, and the greater the thickness of the protective layer, the more favorable the service life value.

Compositional (In)Finite Abstractions for Large-Scale Interconnected Stochastic Systems

This article is concerned with a compositional approach for constructing both infinite (reduced-order models) and finite abstractions [a.k.a. finite Markov decision processes (MDPs)] of large-scale interconnected discrete-time stochastic systems. The proposed framework is based on the notion of stochastic simulation functions enabling us to employ an abstract system as a substitution of the original one in the controller design process with guaranteed error bounds. In the first part of this article, we derive sufficient small-gain-type conditions for the compositional quantification of the probabilistic distance between the interconnection of stochastic control subsystems and that of their infinite abstractions. We then construct infinite abstractions together with their corresponding stochastic simulation functions for a particular class of discrete-time nonlinear stochastic control systems. In the second part of this article, we leverage small-gain-type conditions for the compositional construction of finite abstractions. We propose an approach to construct finite MDPs as finite abstractions of concrete models or their reduced-order versions satisfying an incremental input-to-state stability property. We also show that for the particular class of nonlinear stochastic control systems, the aforementioned property can be readily checked by matrix inequalities. We demonstrate the effectiveness of the proposed results by applying our approaches to a fully interconnected network of 20 nonlinear subsystems (totally 100 dimensions). We construct finite MDPs from their reduced-order versions (together 20 dimensions) with guaranteed error bounds on their output trajectories. We also apply the proposed results to a temperature regulation in a circular building and construct compositionally a finite abstraction of a network containing 1000 rooms. We employ the constructed finite abstractions as substitutes to compositionally synthesize policies regulating the temperature in each room for a bounded time horizon.

Compositional abstraction-based synthesis for continuous-time stochastic hybrid systems

In this paper, we propose a compositional framework for the construction of discrete-time finite abstractions, also known as finite Markov decision processes, from continuous-time stochastic hybrid systems by quantifying the distance between their outputs in a probabilistic setting. The proposed scheme is based on the notion of stochastic simulation functions, which is used to relate continuous-time stochastic systems with their discrete-time counterparts. Accordingly, one can employ discrete-time abstract systems as substitutions of the continuous-time ones in the controller design process with guaranteed error bounds on their output trajectories. To this end, we first derive sufficient small-gain type conditions for the compositional quantification of the probabilistic distance between the interconnection of original continuous-time stochastic hybrid systems and their discrete-time (finite or infinite) abstractions. We then construct finite abstractions together with their corresponding stochastic simulation functions for a particular class of nonlinear stochastic hybrid systems having some stability property. We illustrate the effectiveness of the proposed results by applying our approaches to the temperature regulation in a circular building and constructing compositionally a discrete-time abstraction from its original continuous-time dynamics in a network containing 1000 rooms. We employ the constructed discrete-time abstractions as substitutes to compositionally synthesize policies regulating the temperature of each room for a bounded time horizon.

Compositional abstraction of large-scale stochastic systems: A relaxed dissipativity approach

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 new type of compositionality conditions which is less conservative than the existing ones. In particular, using a relaxation via a finite-step stochastic simulation function, it is possible to construct finite abstractions such that stabilizability of each subsystem is not necessarily required. We then propose an approach to construct finite MDPs together with their corresponding finite-step storage functions for general discrete-time stochastic control systems satisfying an incremental passivability property. We also construct finite MDPs for a particular class of nonlinear stochastic control systems. To demonstrate the effectiveness of the proposed results, we first apply our approach to an interconnected system composed of 4 subsystems such that 2 of them are not stabilizable. We then consider a road traffic network in a circular cascade ring composed of 50 cells, and construct compositionally a finite MDP of the network. We employ the constructed finite abstractions as substitutes to compositionally synthesize policies keeping the density of the traffic lower than 20 vehicles per cell. Finally, we apply our proposed technique to a fully interconnected network of 500 nonlinear subsystems and construct their finite MDPs with guaranteed error bounds on the probabilistic distance between their output trajectories.

Compositional abstraction-based synthesis for networks of stochastic switched systems

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 finite abstractions via stochastic simulation functions. We then leverage sufficient small-gain type conditions to show compositionality results of this work. Afterwards, we show that under standard assumptions ensuring incremental input-to-state stability of switched systems (i.e., existence of common incremental Lyapunov functions, or multiple incremental Lyapunov functions with dwell-time), one can construct finite MDPs for the general setting of nonlinear stochastic switched systems. We also propose an approach to construct finite MDPs together with their corresponding stochastic simulation functions for a particular class of nonlinear stochastic switched systems. We show that for this class of systems, the aforementioned incremental stability property can be readily checked by matrix inequalities. To demonstrate the effectiveness of our proposed results, we first apply our approaches to a road traffic network in a circular cascade ring composed of 200 cells, and construct compositionally a finite MDP of the network. We employ the constructed finite abstractions as substitutes to compositionally synthesize policies keeping the density of the traffic lower than 20 vehicles per cell. We then apply our proposed techniques to a fully interconnected network of 500 nonlinear subsystems (totally 1000 dimensions), and construct their finite MDPs with guaranteed error bounds. We compare our proposed results with those available in the literature.

Synthesis of Stochastic Systems with Partial Information via Control Barrier Functions

Synthesis of controllers for stochastic control systems ensuring safety constraints has gained considerable attention in the last few years. In this paper, we consider the problem of synthesizing controllers for partially observed stochastic control systems to ensure finite-time safety. Given an estimator with a probabilistic guarantee on the accuracy of the estimations, we provide an approach to compute a controller providing a lower bound on the probability that the trajectories of the stochastic control system remain safe over a finite time-horizon. To obtain such controllers, we utilize a notion of control barrier functions. We also provide an approach to compute a probability bound on estimator accuracy by using a notion of so-called stochastic simulation function. The proposed result is illustrated on a case study.

Compositional construction of infinite abstractions for networks of stochastic control systems

This paper is concerned with a compositional approach for constructing infinite abstractions of interconnected discrete-time stochastic control systems. The proposed approach uses the interconnection matrix and joint dissipativity-type properties of subsystems and their abstractions described by new notions of so-called stochastic storage functions. The interconnected abstraction framework is based on new notions of so-called stochastic simulation functions, constructed compositionally using stochastic storage functions of components. Using stochastic simulation functions, one can quantify the distance between original interconnected stochastic systems and interconnected 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 back over concrete ones. In the first part of the paper, we derive dissipativity-type compositional reasoning for the quantification of the distance in probability between the interconnection of stochastic control subsystems and that of their abstractions. Moreover, we focus on a class of discrete-time nonlinear stochastic control systems with independent noises in the abstract and concrete subsystems, and propose a computational scheme to construct abstractions together with their corresponding stochastic storage functions. In the second part of the paper, we consider specifications expressed as syntactically co-safe linear temporal logic formulae and show how a synthesized policy for the abstract system can be refined to a policy for the original system while providing a guarantee on the probability of satisfaction. We demonstrate the effectiveness of the proposed results by constructing an abstraction (totally 3 dimensions) of the interconnection of three discrete-time nonlinear stochastic control subsystems (together 222 dimensions) in a compositional fashion such that the compositionality condition does not require any constraint on the number or gains of the subsystems. We also employ the constructed abstraction as a substitute to synthesize a controller enforcing a syntactically co-safe linear temporal logic specification.

The effect of yeast supplementation on equine fecal microbial population dynamics following abrupt dietary changes

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 new type of compositionality conditions which is less conservative than the existing ones. In particular, using a relaxation via a finite-step stochastic simulation function, it is possible to construct finite abstractions such that stabilizability of each subsystem is not necessarily required. We then propose an approach to construct finite MDPs together with their corresponding finite-step storage functions for general discrete-time stochastic control systems satisfying an incremental passivability property. We also construct finite MDPs for a particular class of nonlinear stochastic control systems. To demonstrate the effectiveness of the proposed results, we first apply our approach to an interconnected system composed of 4 subsystems such that 2 of them are not stabilizable. We then consider a road traffic network in a circular cascade ring composed of 50 cells, and construct compositionally a finite MDP of the network. We employ the constructed finite abstractions as substitutes to compositionally synthesize policies keeping the density of the traffic lower than 20 vehicles per cell. Finally, we apply our proposed technique to a fully interconnected network of 500 nonlinear subsystems and construct their finite MDPs with guaranteed error bounds on the probabilistic distance between their output trajectories.

Compositional Construction of Finite MDPs for Large-Scale Stochastic Switched Systems: A Dissipativity Approach

Abstract In this paper, we provide a compositional technique for constructing finite abstractions (a.k.a. finite Markov decision processes) for networks of discrete-time stochastic switched systems. The proposed framework is based on the notion of stochastic simulation functions, using which one can employ a finite MDP as a substitution of the original one in the controller design process with guaranteed error bounds on their output trajectories. In this respect, we first leverage dissipativity-type compositional conditions for quantifying the error between the interconnection of stochastic switched subsystems and that of their finite abstractions. We then propose an approach to construct finite MDPs together with their corresponding stochastic simulation functions for a particular class of nonlinear stochastic switched systems. To demonstrate the effectiveness of our proposed results, we apply our approaches to two different case studies.

Compositional Synthesis of Finite Abstractions for Continuous-Space Stochastic Control Systems: A Small-Gain Approach

This paper is concerned with a compositional approach for constructing finite abstractions (a.k.a. finite Markov decision processes) of interconnected discrete-time stochastic control systems. The proposed framework is based on a notion of so-called stochastic simulation function enabling us to use an abstract system as a substitution of the original one in the controller design process with guaranteed error bounds. 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 (finite or infinite) abstractions. In the second part of the paper, we construct finite abstractions together with their corresponding stochastic simulation functions for the class of linear stochastic control systems. We apply our results to the temperature regulation in a circular building by constructing compositionally a finite abstraction of a network containing 1000 rooms. We use the constructed finite abstractions as substitutes to synthesize policies compositionally regulating the temperature in each room for a bounded time horizon.

Approximations of Stochastic Hybrid Systems: A Compositional Approach

In this paper we propose a compositional framework for the construction of approximations of the interconnection of a class of stochastic hybrid systems. As special cases, this class of systems includes both jump linear stochastic systems and linear stochastic hybrid automata. In the proposed framework, an approximation is itself a stochastic hybrid system, which can be used as a replacement of the original stochastic hybrid system in a controller design process. We employ a notion of so-called stochastic simulation function to quantify the error between the approximation and the original system. In the first part of the paper, we derive sufficient conditions which facilitate the compositional quantification of the error between the interconnection of stochastic hybrid subsystems and that of their approximations using the quantified error between the stochastic hybrid subsystems and their corresponding approximations. In particular, we show how to construct stochastic simulation functions for approximations of interconnected stochastic hybrid systems using the stochastic simulation function for the approximation of each component. In the second part of the paper, we focus on a specific class of stochastic hybrid systems, namely, jump linear stochastic systems, and propose a constructive scheme to determine approximations together with their stochastic simulation functions for this class of systems. Finally, we illustrate the effectiveness of the proposed results by constructing an approximation of the interconnection of four jump linear stochastic subsystems in a compositional way.

Approximations of Stochastic Hybrid Systems

This paper develops a notion of approximation for a class of stochastic hybrid systems that includes, as special cases, both jump linear stochastic systems and linear stochastic hybrid automata. Our approximation framework is based on the recently developed notion of the so-called stochastic simulation functions. These Lyapunov-like functions can be used to rigorously quantify the distance or error between a system and its approximate abstraction. For the class of jump linear stochastic systems and linear stochastic hybrid automata, we show that the computation of stochastic simulation functions can be cast as a tractable linear matrix inequality problem. This enables us to compute the modeling error incurred by abstracting some of the continuous dynamics, or by neglecting the influence of stochastic noise, or even the influence of stochastic discrete jumps.

Geometric and Topological Analysis of Three-Dimensional Porous Media: Pore Space Partitioning Based on Morphological Skeletonization

This article presents a versatile, rigorous, and efficient methodology for extracting various geometric and topological parameters of 3D discrete porous media. The new approach takes advantage of the morphological skeleton of the pore structure—a lower dimensional representation of the pore space akin to the topological “deformation retract”. The skeleton is derived by a fully parallel thinning algorithm that fulfils two essential requirements: it generates a medial axis and preserves the connectivity of the pore space. Topological analysis is accomplished by classifying all skeleton points as node or link (branch) points according to the concept of λ-adjacency in 3D discrete space. In this manner, node coordination number and link length distributions are directly obtained from the skeleton. Pore necks (throats) are identified through a search for minima in the hydraulic radius of individual pore space channels outlined by skeleton links. In addition to the determination of the size distribution of the constrictions (pore necks) that control nonwetting phase invasion, improved estimates of the distributions of effective hydraulic and electric conductivity of individual pore space channels are obtained. Furthermore, erection of planes at the location of pore necks results in partitioning of the pore space into its constituent pores. This enables the characterization of the pore space in terms of a pore volume distribution. The new methodology is illustrated by application to a regular cubic pore network and irregularly shaped 2D and 3D pore networks generated by stochastic simulation. In the latter case, important new results are obtained concerning the sensitivity of geometric and topological properties of the microstructure to the parameters of stochastic simulation, namely, the porosity and correlation function. It is found that model porous media reconstructed from the same porosity and correlation function can exhibit marked differences in geometry and connectivity, which correlate with differences in specific surface area.