From Dissipativity Theory to Compositional Synthesis of Large-Scale Stochastic Switched Systems

Abstract

This work is concerned with a compositional technique for the construction of finite abstractions ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a.k.a.,</i> finite Markov decision processes (MDPs)) for networks of discrete-time stochastic switched systems. We propose a framework based on a notion of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">stochastic simulation functions</i> , using which one can quantify the probabilistic distance between original interconnected stochastic switched systems and their finite MDPs by leveraging dissipativity-type compositional conditions. We show that the proposed compositionality conditions can enjoy the structure of the interconnection topology and be potentially fulfilled <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">independently</i> of the number or gains of subsystems. We also propose an approach to construct finite MDPs together with their corresponding stochastic simulation functions for nonlinear stochastic switched systems satisfying some <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">incremental passivity</i> property. We show that for a particular class of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">nonlinear</i> stochastic switched systems whose nonlinearities satisfy an incremental quadratic inequality, the aforementioned property can be readily checked by some linear matrix inequalities. To demonstrate the effectiveness of the proposed results, we apply our approaches to the following two different case studies: a road traffic network, and a fully interconnected network of nonlinear switched systems accepting different dissipativity properties.