1,571 research outputs found
Compositional (In)Finite Abstractions for Large-Scale Interconnected Stochastic Systems
This paper 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 the paper, 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 the
paper, 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 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.Comment: This work is accepted as a full paper at IEEE Transactions on
Automatic Contro
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.Comment: 26 pages, 7 figure
From Dissipativity Theory to Compositional Abstractions of Interconnected Stochastic Hybrid Systems
In this work, we derive conditions under which compositional abstractions of
networks of stochastic hybrid systems can be constructed using the
interconnection topology and joint dissipativity-type properties of subsystems
and their abstractions. In the proposed framework, the abstraction, itself a
stochastic hybrid system (possibly with a lower dimension), can be used as a
substitute of the original system in the controller design process. Moreover,
we derive conditions for the construction of abstractions for a class of
stochastic hybrid systems involving nonlinearities satisfying an incremental
quadratic inequality. In this work, unlike existing results, the stochastic
noises and jumps in the concrete subsystem and its abstraction need not to be
the same. We provide examples with numerical simulations to illustrate the
effectiveness of the proposed dissipativity-type compositional reasoning for
interconnected stochastic hybrid systems
Approximate abstractions of control systems with an application to aggregation
Previous approaches to constructing abstractions for control systems rely on
geometric conditions or, in the case of an interconnected control system, a
condition on the interconnection topology. Since these conditions are not
always satisfiable, we relax the restrictions on the choice of abstractions,
instead opting to select ones which nearly satisfy such conditions via
optimization-based approaches. To quantify the resulting effect on the error
between the abstraction and concrete control system, we introduce the notions
of practical simulation functions and practical storage functions. We show that
our approach facilitates the procedure of aggregation, where one creates an
abstraction by partitioning agents into aggregate areas. We demonstrate the
results on an application where we regulate the temperature in three separate
zones of a building.Comment: 24 pages, 3 figures, 1 tabl
Compositional Synthesis of Symbolic Models for Networks of Switched Systems
In this paper, we provide a compositional methodology for constructing
symbolic models for networks of discrete-time switched systems. We first define
a notion of so-called augmented-storage functions to relate switched subsystems
and their symbolic models. Then we show that if some dissipativity type
conditions are satisfied, one can establish a notion of so-called alternating
simulation function as a relation between a network of symbolic models and that
of switched subsystems. The alternating simulation function provides an upper
bound for the mismatch between the output behavior of the interconnection of
switched subsystems and that of their symbolic models. Moreover, we provide an
approach to construct symbolic models for discrete-time switched subsystems
under some assumptions ensuring incremental passivity of each mode of switched
subsystems. Finally, we illustrate the effectiveness of our results through two
examples
Automated Verification and Synthesis of Stochastic Hybrid Systems: A Survey
Stochastic hybrid systems have received significant attentions as a relevant
modelling framework describing many systems, from engineering to the life
sciences: they enable the study of numerous applications, including
transportation networks, biological systems and chemical reaction networks,
smart energy and power grids, and beyond. Automated verification and policy
synthesis for stochastic hybrid systems can be inherently challenging: this is
due to the heterogeneity of their dynamics (presence of continuous and discrete
components), the presence of uncertainty, and in some applications the large
dimension of state and input sets. Over the past few years, a few hundred
articles have investigated these models, and developed diverse and powerful
approaches to mitigate difficulties encountered in the analysis and synthesis
of such complex stochastic systems. In this survey, we overview the most recent
results in the literature and discuss different approaches, including
(in)finite abstractions, verification and synthesis for temporal logic
specifications, stochastic similarity relations, (control) barrier
certificates, compositional techniques, and a selection of results on
continuous-time stochastic systems; we finally survey recently developed
software tools that implement the discussed approaches. Throughout the
manuscript we discuss a few open topics to be considered as potential future
research directions: we hope that this survey will guide younger researchers
through a comprehensive understanding of the various challenges, tools, and
solutions in this enticing and rich scientific area
Compositional Abstraction-based Synthesis of General MDPs via Approximate Probabilistic Relations
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 -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
leverage the proposed results to perform analysis and synthesis over abstract
models, and then carry the results over concrete ones. To this end, we first
propose our compositionality results using the new approximate probabilistic
relation which is based on lifting. We then focus on a class of stochastic
nonlinear dynamical systems and construct their abstractions using both model
order reduction and space discretization in a unified framework. We provide
conditions for simultaneous existence of relations incorporating the structure
of the network. Finally, we demonstrate the effectiveness of the proposed
results by considering a network of four nonlinear dynamical subsystems
(together 12 dimensions) and constructing finite abstractions from their
reduced-order versions (together 4 dimensions) in a unified compositional
framework. We benchmark our results against the compositional abstraction
techniques that construct both infinite abstractions (reduced-order models) and
finite MDPs in two consecutive steps. We show that our approach is much less
conservative than the ones available in the literature
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 passivablity property. We also construct
finite MDPs for a particular class of nonlinear stochastic control systems. To
demonstrate the effectiveness of the proposed results, we apply our results on
three different case studies.Comment: This work is accepted at Nonlinear Analysis: Hybrid Systems. arXiv
admin note: text overlap with arXiv:1712.0779
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 for a particular class of
nonlinear stochastic switched systems. 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.Comment: This work is accepted as a regular paper at Automatica. arXiv admin
note: text overlap with arXiv:1902.01223, arXiv:1808.0089
Verification of Initial-State Opacity for Switched Systems: A Compositional Approach
The security in information-flow has become a major concern for
cyber-physical systems (CPSs). In this work, we focus on the analysis of an
information-flow security property, called opacity. Opacity characterizes the
plausible deniability of a system's secret in the presence of a malicious
outside intruder. We propose a methodology of checking a notion of opacity,
called approximate initial-state opacity, for networks of discrete-time
switched systems. Our framework relies on compositional constructions of finite
abstractions for networks of switched systems and their so-called approximate
initial-state opacity-preserving simulation functions (InitSOPSFs). Those
functions 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 subsystem separately. Additionally, assuming
certain stability property of switched systems, we also provide a technique on
constructing their finite abstractions together with the corresponding local
InitSOPSFs. Finally, we illustrate the effectiveness of our results through an
example.Comment: 19 pages, 7 figure
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