Abstract
To our minds, the real world appears as a composition of different interacting entitites, which demonstrate complex behavior. In the current paper, we primarly aim to study such networked systems by developing corresponding approaches to modeling them, given a class of tasks. We derive it from the primary concept of information and a system, with corresponding dynamics emerging from interactions between system components. As we progress through the study, we discover three possible levels of certain synchronous pattern composition in complex systems: microscopic (the level of elementary components), mesoscopic (the level of clusters), and macroscopic (the level of the whole system). Above all, we focus on the clusterization phenomenon, which allows to reduce system complexity by regarding only a small number of stable manifolds, corresponding to cluster synchronization of system component states—as opposed to regarding the system as a whole or each elementary component separately. Eventually, we demonstrate how an optimization problem for cluster control synthesis can be formulated for a simple discrete linear system with clusterization.
Highlights
Since the ancient times, humanity was always deeply curious about reasons and causes of things
2.1 Understanding of Complex Systems As we study patterns and their evolution in time, we may find that some generalization and corresponding dynamic analysis problems may appear exceptionally challenging
We proposed an approach to optimal cluster control synthesis for complex systems
Summary
Humanity was always deeply curious about reasons and causes of things. At the most basic level, some objects appear to our minds as identical, while other ones can be highlighted from the rest by some distinctive features This ability to generalize the perceived environment—but not in all its manifestations, just in certain patterns—originates mathematics and cybernetics. To further motivate the need to highlight time and dynamical nature of things, consider irreversible chaotic systems, behavior of which is notably hard to predict in the future, if their initial state is provided inaccurately in the past or in the present Such inaccuracy may arise in presence of unknown (stochastic) disturbances or noise, which we are unable to predict due to bio- or technological limitations. This definition is very similar to the static one, except locality can be regarded in space and time separately, which may become convenient in control of systems with complex unpredictable dynamics
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