Distributed control of complex interconnected systems: a convex approach
Raffaello D’Andrea, Cédric Langbort · 2005
Many modern engineering problems involve a large number of interacting units. When controlling such systems, the classical approach that consists in lumping all subsystems together and controlling them as a single large-scale system is not always appropriate. For example, in the case of an automated highway or formation flight; it seems more reasonable to regulate the system by equipping each vehicle with its own controller than to rely on a centralized control authority that observes and acts on all units at once. In the recent literature, it has often been taken for granted that the natural alternative to such a centralized control architecture was to adopt a fully decentralized one, where a controller is assigned to each plant's subsystem, providing it with corrective action based solely on local measurements. The position taken in this thesis is that distributed control is a better alternative for regulating complex systems in general, because it gives rise to convex synthesis conditions. The essential difference between decentralized and distributed control is that, in the latter case, one allows the controller's subsystems to communicate with each other whenever the plant's units are interconnected. It is the extra degrees of freedom provided by this architecture that allow to convexify the control design problem, even when decentralized control design in inherently intractable. We present a theory of distributed control for systems composed of different linear-time invariant subsystems coupled over an arbitrary graph. The tools used are inspired by dissipativity and robust control theory and allow us to derive sufficient conditions for the existence of a controller with the same structure as the plant. These conditions take the form of linear matrix inequalities (LMIs). While typically of large size, these LMIs are coupled in a way that reflects the spatial structure of the problem and can be exploited to design parallel, distributed synthesis algorithms. We also show how using interconnection symmetries (spatial invariance and spatial reversibility) can drastically reduce the number of variables involved and collapse our conditions to a single, tractable LMI.