Compositional construction of approximate abstractions

In this paper we propose a compositional construction of approximate abstractions of interconnected control systems. Our notion of approximate abstraction is based on so-called simulation functions. The abstraction acts as substitute in the controller design process and is equipped with an interface that is used in lifting a controller, found for the abstraction, to a controller for the concrete system. The error between the abstraction and the concrete system is quantitatively bounded via the simulation function. In the first part of the paper, we provide conditions which facilitate the compositional construction of abstractions together with simulation functions and the associated interfaces of general interconnected control systems, given the abstractions and the simulation functions with the associated interfaces of the subsystems. In the second part of the paper, we characterize a general simulation function with the associated interface for the subclass of linear control systems. This characterization yields algorithmic procedures for the construction of approximate abstractions. Finally, we illustrate our findings with a simple example consisting of three linear subsystems.

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