Structural Analysis and Control of Dynamical Networks

A dynamical network is comprised of a finite number of subsystems, each having its own dynamics, which interact according to a given interconnection topology. Dynamical networks are a powerful modelling tool to represent a large number of systems in different contexts, ranging from natural to man-made systems, and have a peculiar feature: the global behaviour is the outcome of an ensemble of local interactions. Hence, dynamical networks can be analysed so as to understand how local events can lead to global consequences and can be controlled by acting locally so as to achieve the desired global behaviour. The analysis and the control of dynamical networks are structural when they are grounded on the topology of the interconnection graph, along with qualitative, parameter-free specifications. Structural analysis aims at assessing properties for a whole family of systems having the same structure and is particularly suited for natural systems, which can exhibit an extraordinary robustness in spite of large uncertainties and intrinsic variability. In this thesis, results and procedures are presented to structurally assess relevant properties, such as stability, boundedness and the sign of steady-state input-output influences, for a wide class of systems whose Jacobian admits the so-called BDC-decomposition, which embodies the sum of the effects of single local interactions. A structural classification is also proposed, to discriminate between systems that can possibly or exclusively admit instability related to oscillations or to multistationarity, for systems with a sign-definite Jacobian and for systems composed of the interconnection of stable monotone subsystems; a graph-based classification is given and applied to examples of artificial biomolecular networks. In a dynamical network described by a graph, subsystems are associated with nodes and interactions with arcs. When the interactions are not given, they can be decided by a control system. In particular, network-decentralised control aims at governing the global behaviour of a dynamical network through controllers that are associated with the arcs of the interconnection graph, hence act locally and have access to local information only. Despite the restricted information constraint, a large class of systems can be always stabilised resorting to a network-decentralised controller. Both linear systems composed of independent subsystems, connected by the control action, and nonlinear compartmental systems are considered; the robustness and optimality properties of the devised network-decentralised control are investigated and several application examples are proposed, spanning from traffic control and data transmission to synchronisation and vehicle platooning. Network-decentralised estimation is also considered, for systems composed of identical agents; a robustness result is provided, exploiting the smallest eigenvalue of the generalised Laplacian matrix associated with the interaction graph. Structural analysis and network-decentralised control synthesis are presented in this work as complementary facets of the same approach, which can streamline each other. Structural analysis can help explain the robustness of natural systems, so that the clever resources of nature can be mimicked to improve the control strategies designed for man-made systems; at the same time, local interactions can be engineered in biomolecular systems, as is done for artificial systems, to obtain the desired global behaviour. This virtuous circle will hopefully result in innovative approaches for biotechnologies and large-scale network engineering, aimed at improving the quality of our daily life.