Compositional Synthesis of Finite Abstractions for Networks of Systems: A Dissipativity Approach

Abstract

IONS In the previous sections, concrete systems and their abstractions were considered as general discrete-time control systems, deterministic or nondeterministic, finite or infinite, that can be related to each other through a storage function (in the case of subsystems) or an alternating simulation function (in the case of interconnected systems). In this section, we consider infinite, deterministic, control subsystems and provide a way of constructing their finite abstractions together with their corresponding storage functions. The construction of the finite abstraction is performed in a straightforward way. Simply, the finite state and input sets of the finite abstraction are constructed by gridding the state and input sets of the concrete subsystem with suitable grid sizes. Moreover, the transitions between those finite states are established as follows: given an initial cell and a discrete input, the concrete system is simulated for one iteration starting from the center of the cell and under the discrete input. The simulated point is contained in a cell of the grid. This implies existence of a transition between the center of the initial cell and the one containing the simulated point under the given discrete input. This is performed for all grid cells and all possible discrete inputs as defined formally in [9, Definition 7]. 5 CONSTRUCTION OF STORAGE FUNCTIONS The storage function from the finite abstraction to the concrete subsystem and vice versa is established under the assumption that the original discrete-time control subsystem is so-called incrementally passivable [9, Definition 5]. Such an incremental passivity property is described based on the existence of a function satisfying some conditions. Then, under some mild assumptions, it can be shown that this function is actually a storage function from the concrete subsystem to its finite abstraction and vice versa. Note that any stabilizable linear control system and some incrementally stabilizable control systems satisfy this property [9, Remark 3].