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

Abstract

This paper is concerned with a compositional approach for constructing both infinite (reducedorder models) and finite abstractions (a.k.a. finite Markov decision processes) of large-scale interconnected discrete-time stochastic control systems. The proposed framework is based on the notion of stochastic simulation functions 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 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 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 Markov decision processes (MDPs) of the concrete models (or their reduced-order versions) satisfying an incremental input-tostate stability property. We also show that for a 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 the temperature regulation in a circular building and constructing 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. We also apply our proposed techniques to a fully connected network of 20 nonlinear subsystems (totally 100 dimensions) and construct finite MDPs from their reduced-order versions (together 20 dimensions) with guaranteed error bounds on their output trajectories.

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

“…However, the major bottleneck of finite-abstraction techniques is their dependency in the state and input set discretization parameters, and consequently, they suffer from the curse of dimensionality: the computational complexity grows exponentially as the dimension of the system increases. To alleviate this issue, compositional techniques have been introduced in the past few years to construct finite abstractions of interconnected systems based on abstractions of smaller subsystems [HHHK13,SAM17,LSZ20a,LSZ20c,LSZ20d,LSZ19a,LSZ18,LSMZ17,LZ19,LSZ19b,LSZ20b,Lav19,NSZ20a,NZ20].…”

Compositional abstraction-based synthesis for networks of stochastic switched systems

“…Compositional construction of infinite abstractions (reduced-order models) for networks of stochastic control systems is proposed in [LSMZ17] and [LSZ19c] using small-gain type conditions and dissipativity-type properties of subsystems and their abstractions, respectively. Compositional construction of finite abstractions is presented in [LSZ18b] and [LSZ19d] using respectively dissipativity-type reasoning and small-gain conditions, both for discrete-time stochastic control systems. In comparison with the current work, the proposed results in [LSZ18b], [LSMZ17], [LSZ19c], [LSZ19d] are about the compositional construction of (in)finite abstractions for stochastic control systems, while here for the first time we enlarge the class of systems to switched ones.…”

“…Compositional construction of finite abstractions is presented in [LSZ18b] and [LSZ19d] using respectively dissipativity-type reasoning and small-gain conditions, both for discrete-time stochastic control systems. In comparison with the current work, the proposed results in [LSZ18b], [LSMZ17], [LSZ19c], [LSZ19d] are about the compositional construction of (in)finite abstractions for stochastic control systems, while here for the first time we enlarge the class of systems to switched ones. If switched systems accept common Lyapunov functions, our proposed results here recover the ones presented in the previous works by considering switching signals as discrete inputs.…”

Compositional construction of infinite abstractions for networks of stochastic control systems

“…Recently, compositional synthesis of large-scale stochastic systems using a relaxed dissipativity approach is proposed in [LSZ19]. Compositional (in)finite abstractions for large-scale interconnected stochastic systems using small-gain type conditions are proposed in [LSZ18a].…”