Duality and network theory in passivity-based cooperative control

Abstract

This paper presents a class of passivity-based cooperative control problems that have an explicit connection to convex network optimization problems. The new notion of maximal equilibrium independent passivity is introduced and it is shown that networks of systems possessing this property asymptotically approach the solutions of a dual pair of network optimization problems, namely an optimal potential and an optimal flow problem. This connection leads to an interpretation of the dynamic variables, such as system inputs and outputs, to variables in a network optimization framework, such as divergences and potentials, and reveals that several duality relations known in convex network optimization theory translate directly to passivity-based cooperative control problems. The presented results establish a strong and explicit connection between passivity-based cooperative control theory on the one side and network optimization theory on the other, and they provide a unifying framework for network analysis and optimal design. The results are illustrated on a nonlinear traffic dynamics model that is shown to be asymptotically clustering. IntroductionOne of the most profound concepts in mathematics is the notion of duality. In many ways, duality theory is the mathematical answer to the idiom "there are two sides to every coin." This powerful concept manifests itself across many mathematical disciplines, but perhaps the most elegant and complete notion of duality is the celebrated Lagrange duality in convex optimization theory [1]. One of the most complete expositions of this duality theory relates to a class of optimization problems over networks, generally known as network optimization [2], [3]. In [2], a unifying framework for network optimization was established, with the key elements being a pair of dual optimization problems: the optimal network flow problem and the optimal potential problem. This dual pair of optimization problems characterizes the majority of network decision problems. The notion of duality also has a long history within the theory of control systems, as control problems are often intimately related to optimization problems and their respective duals have again a controls interpretation, see e.g. [4].A recent trend in modern control theory is the study of cooperative control problems amongst groups of dynamical systems that interact over an information exchange network. A fundamental goal for the analysis of these systems is to reveal the interplay between properties of the individual dynamic agents, the underlying network topology, and the interaction protocols that influence the functionality of the overall system [5]. Amongst the numerous control theoretic approaches being pursued to define a general theory for networks of dynamical systems, passivity [6] takes an outstanding role; see e.g., [7]. The conceptual idea underlying passivity-based cooperative control is to separate the network analysis and synthesis into two layers. On the systems layer, each dynamical system comprising the network ...

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