Elsevier

Systems & Control Letters

Volume 96, October 2016, Pages 30-36
Systems & Control Letters

Gradient and passive circuit structure in a class of non-linear dynamics on a graph

https://doi.org/10.1016/j.sysconle.2016.06.019Get rights and content

Abstract

We consider a class of non-linear dynamics on a graph that contains and generalizes various models from network systems and control and study convergence to uniform agreement states using gradient methods. In particular, under the assumption of detailed balance, we provide a method to formulate the governing ODE system in gradient descent form of sum-separable energy functions, which thus represent a class of Lyapunov functions; this class coincides with Csiszár’s information divergences. Our approach bases on a transformation of the original problem to a mass-preserving transport problem and it reflects a little-noticed general structure result for passive network synthesis obtained by B.D.O. Anderson and P.J. Moylan in 1975. The proposed gradient formulation extends known gradient results in dynamical systems obtained recently by M. Erbar and J. Maas in the context of porous medium equations. Furthermore, we exhibit a novel relationship between inhomogeneous Markov chains and passive non-linear circuits through gradient systems, and show that passivity of resistor elements is equivalent to strict convexity of sum-separable stored energy. Eventually, we discuss our results at the intersection of Markov chains and network systems under sinusoidal coupling.

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 G=(N,B,w) be a weighted directed graph, where N={1,2,,n} is the set of nodes, B={1,2,,b}N×N denotes the set of branches whose elements are ordered pairs (j,i) denoting an edge from node j to i, and w:BR>0 is a weighting function, such that w((j,i))wij, if (j,i)B, else wij=0. Associated to a graph is the Laplace matrix L, defined component-wise as [L]ij=wij, [L]ii=jwij. 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 G such that detailed balance   (1)   holds for some C. Define the new state qCx, and consider the sum-separable functionE(q)iNciH(ci1q

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 G, θ̇i=ωi+j:(j,i)Bwijsin(θjθi),iN, where ω=(ω1,ω2,,ωn)Rn is the vector of natural (driving) frequencies, and the state θTn is an n-vector of angles as elements of the n-Torus.

If we set wij=Kn, K>0 for all (j,i)B, then (7) represents Kuramoto’s oscillator model  [37]. If wij=|vi||vj|(yij)Di, |vi| a voltage magnitude, yij the complex admittance of a line (j,i), and Di>0 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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