Compositional abstraction of large-scale stochastic systems: A relaxed dissipativity approach. (May 2020) - Lavaei, Abolfazl; Soudjani, Sadegh; Zamani, Majid
Abstract: 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.