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.

在本文中，我们通过量化概率设置中输出之间的距离，为连续时间随机混合系统提出了离散时间有限抽象（也称为有限马尔可夫决策过程）的构造框架。所提出的方案基于随机仿真功能的概念，用于将连续时间随机系统与其离散时间对应系统相关联。因此，可以在控制器设计过程中采用离散时间抽象系统代替连续时间抽象系统，并在其输出轨迹上保证误差范围。为此，我们首先导出足够的小增益类型条件，以对原始连续时间随机混合系统的互连与其离散时间（有限或无限）抽象之间的概率距离进行组成量化。然后，我们针对具有某些稳定性的特定类别的非线性随机混合系统构造有限抽象及其相应的随机模拟函数。我们通过将我们的方法应用于圆形建筑的温度调节并在包含1000个房间的网络中从其原始连续时间动态构造离散时间抽象，来说明所提出结果的有效性。我们使用构造的离散时间抽象作为替代品，以合成合成的策略来调节有限时间范围内每个房间的温度。