Gradient and passive circuit structure in a class of non-linear dynamics on a graph
Section snippets
Motivation
Gradient methods provide an elegant way to physics motivated modeling [1], [2] and are closely linked to passivity theory and the circuit concept [3], [4]. They are a basic tool in studying and designing non-linear systems on a graph, e.g., in distributed optimization [5] or in multi-robot problems such as coverage or formation control, cf., e.g., [6], [7], and references therein.
Another pillar in network system studies is the classical consensus problem [8]. An equivalence between the
Problem description and related literature
Let be a weighted directed graph, where is the set of nodes, denotes the set of branches whose elements are ordered pairs denoting an edge from node to , and is a weighting function, such that , if , else . Associated to a graph is the Laplace matrix , defined component-wise as , . For strongly connected graphs, denote the positive left-eigenvector associated to the unique zero eigenvalue of the
Gradient representation
With the following result we provide a procedure to bring a dynamics (2) into the form (3). By that we characterize a family of sum-separable Lyapunov functions characterizing asymptotic stability of agreement states, i.e., states where all components are equal. Theorem 1 Consider a network system dynamics governed by the protocol (2) on a strongly connected graph such that detailed balance (1) holds for some . Define the new state , and consider the sum-separable function
Circuit formulation
In circuit theory the dynamical behavior of a system is seen as the result of the interaction of a finite number of interconnected circuit elements. Circuit elements are single-input-single-output systems among which lossless, dynamical ones that can store energy and possess memory, e.g. capacitors in electric circuits, and memoryless, non-dynamic ones that dissipate energy, e.g. resistors, play important roles. Sum-separability of stored energy in this context thus arises naturally, as it is
Coupled oscillator models and electric power grids
A generic model in the study of phase-coupled oscillator networks is given by the ODE system on a graph , where is the vector of natural (driving) frequencies, and the state is an -vector of angles as elements of the -Torus.
If we set , for all , then (7) represents Kuramoto’s oscillator model [37]. If , a voltage magnitude, the complex admittance of a line , and a damping
Conclusion
In this paper we established for a general class of non-linear dynamics on a graph with detailed balance weighting a family of gradient structures associated to sum-separable strictly convex energy functions. This structure extends known gradient results in dynamical systems to pairwise couplings involving non-separable non-linearity. Based on our gradient formalism we made several connections between previously separated results in dynamical network systems, Markov chains, information and
Acknowledgments
The authors would like to thank the anonymous reviewers for their useful comments and in particular for pointing out Ref. [12]. The second author was supported by the Interuniversity Attraction Pole “Dynamical Systems, Control and Optimization (DYSCO)”, initiated by the Belgian State, Prime Minister’s Office, and the Action de Recherche Concertée funded by the Federation Wallonia-Brussels. The third author was supported by the National Science Foundation under the grant EECS-1135843, “CPS:
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