Synchronization of linear oscillators coupled through a dynamic network with interior nodes

Abstract

Synchronization is studied in an array of identical linear oscillators of arbitrary order, coupled through a dynamic network comprising dissipative connectors (e.g., dampers) and restorative connectors (e.g., springs). The coupling network is allowed to contain interior nodes, i.e., those that are not directly connected to an oscillator. It is shown that the oscillators asymptotically synchronize if and only if the Schur complement (with respect to the boundary nodes) of the complex-valued Laplacian matrix representing the coupling has a single eigenvalue on the imaginary axis.

Introduction

Consider the dynamic coupling network with four nodes, shown in Fig. 1; dynamic because it contains energy storage components (inductors). This network can be represented by a pair of Laplacian matrices $(D,R)$ where $D$ locates the dissipative connectors (i.e., the resistor with conductance $g_{13}$) and $R$ the restorative connectors (i.e., the inductors with inductances ${\ell }_{12},{\ell }_{23},{\ell }_{34}$) as

$$D=\left[\begin{array}{cccc}\hfill g_{13}\hfill & \hfill 0\hfill & \hfill -g_{13}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -g_{13}\hfill & \hfill 0\hfill & \hfill g_{13}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right],$$$$R=\left[\begin{array}{cccc}\hfill {\ell }_{12}^{-1}\hfill & \hfill -{\ell }_{12}^{-1}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -{\ell }_{12}^{-1}\hfill & \hfill {\ell }_{12}^{-1}+{\ell }_{23}^{-1}\hfill & \hfill -{\ell }_{23}^{-1}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -{\ell }_{23}^{-1}\hfill & \hfill {\ell }_{23}^{-1}+{\ell }_{34}^{-1}\hfill & \hfill -{\ell }_{34}^{-1}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -{\ell }_{34}^{-1}\hfill & \hfill {\ell }_{34}^{-1}\hfill \end{array}\right].$$

Clearly, the pair $(D,R)$ can be fused into a single entity $[D+jR]$, a complex-valued Laplacian, without any loss of information. It turns out that this fusion has merits beyond mere notational convenience: The collective behavior of an array of coupled oscillators is closely related to the spectral properties of the complex-valued Laplacian representing the network through which the oscillators are coupled. Let us elaborate on this point.

Suppose now to the network in Fig. 1 we connect four identical harmonic oscillators (LC-tanks) as shown in Fig. 2. Will these second-order circuits eventually oscillate in unison? The answer to this question was given in a recent work (Tuna, 2019b), where it was shown that a coupled array of harmonic oscillators asymptotically synchronize when (and only when) the coupling matrix $[D+jR]$ has a single eigenvalue on the imaginary axis. A somewhat interesting feature of this result is that synchronization (or its absence) is independent of the characteristic frequency of the harmonic oscillators. This immediately brings up the question: Does this independence exist as well for higher-order oscillators, having two or more characteristic frequencies? which motivates us to conduct the analysis presented in this paper. To illustrate the setup associated to this question, let us visit once again the network in Fig. 1. Consider this time the case where we employ our network to couple fourth-order linear oscillator circuits, each having two characteristic frequencies. Furthermore, suppose that the oscillator connected to the third node becomes dysfunctional during operation and therefore is disconnected from the network (or has never been connected to the network in the first place). This scenario is depicted in Fig. 3, where the node ③ is said to be an interior node¹ of the network, for it is not directly connected to an oscillator. The simple circuitry shown in Fig. 3 exemplifies the general setup we explore in this paper: an array of identical linear time-invariant (LTI) oscillators coupled through an LTI network with interior nodes, containing not only dissipative components (e.g., resistors, dampers) but restorative ones (e.g., inductors, springs) as well. And what we study in this setup is the problem of synchronization. We state our findings next.

For the setup described above (see the next section for the formal description) we show that what has been established in Tuna (2019b) for harmonic oscillators is true in general for higher-order systems. Namely, in the absence of interior nodes, an array of coupled oscillators asymptotically synchronize if and only if the complex-valued Laplacian matrix representing the coupling network has a single eigenvalue on the imaginary axis. For the more general case, where the network does have interior nodes, this eigenvalue test remains valid, except for the difference that instead of the Laplacian itself it has to be applied to the Schur complement of the Laplacian with respect to the set of non-interior nodes.

In order to lay bare the contours of our contribution, we now attempt to place the above-mentioned main theorem of this paper with respect to some of the related work in the literature. To this end, let us impart some details first. The isolated (uncoupled) oscillator dynamics we study here read (Lax, 1996 Ch. 11)

$$M{\ddot{x}}_i+K{x}_i=0$$

where the constant matrices $M,K$ are symmetric positive definite and $x_i$ is a vector. Note that this LTI differential equation is a natural generalization of the dynamics of an harmonic oscillator, for which $M$, $K$, and $x_i$ are scalar. Even though we have so far motivated our problem through electrical circuits, Eq. (1) plays an even more important role in mechanics (Landau & Lifshitz, 1976 Ch. V). The linearization of a Lagrangian system about a stable equilibrium enjoys the form (1), which successfully represents the behavior of the actual system undergoing small vibrations, e.g., an $n$-link pendulum. This allows, for instance, the main theorem of this paper to determine whether an array of mechanical oscillators (undergoing small vibrations) coupled through a network consisting of springs and dampers asymptotically synchronize or not; see Fig. 4. To the best of our knowledge, works that investigate conditions on the coupling network that ensure synchronization in an array of oscillators with individual units obeying (1) are relatively few, save the special case of harmonic oscillators; see, for instance, Ren, 2008, Su et al., 2009, Sun et al., 2015, Tuna, 2017 and Zhou, Zhang, Xiang, and Wu (2012). There are, it is true, important works on synchronization (Li et al., 2010, Li et al., 2015) that study LTI multi-agent systems in a very general setting which allows the linear oscillators (2) we consider here, as the individual building blocks of the assembly. Still, the problem we consider in this paper does not fit into the framework therein due to the two features it has:

The first feature has to do with the type of coupling. The type of interconnection we study here has to be represented by two separate graphs (as opposed to one single graph) thereby making it an instance of the so called multi-layer networks (Boccaletti et al., 2014). Relevant to the problem we study are the works He et al. (2017) and Lee and Shim (2017), where, in the former synchronization is established under the assumption that all the Laplacian matrices (each representing a layer/graph) are simultaneously diagonalizable, whereas the latter assumes the overall interconnection matrix (sometimes called the supra-Laplacian) has block diagonal structure. We here make neither of those assumptions.

The second feature that locates our work at some relative distance from the main stream multi-agent systems literature is that we allow in our network the interior nodes, i.e., the nodes that are not directly connected to an external unit (an oscillator in our case). This, of course, is anything but novelty within the realm of electrical networks. Whether within a microchip no larger than a fingertip or through a power distribution network spread across half a continent, interior nodes are omnipresent (though not always desirable) constituents of electric circuitry (Rommes and Schilders, 2010, Ward, 1949). Although it is in general not easy (or possible) to dispense with interior nodes physically they can be efficiently removed in analysis through the powerful method known as Kron reduction (Caliskan and Tabuada, 2014, Dobson, 2012, Dörfler and Bullo, 2013, Dörfler et al., 2018), which basically is the representation of the network admittance matrix by its (smaller size) Schur complement (Zhang, 2005). This, indeed is the very method we employ in our synchronization analysis. To be more precise, we work with the Schur complement of the complex-valued Laplacian matrix representing the coupling. Among notable works that study synchronization through networks with interior nodes is Dhople, Johnson, Dörfler, and Hamadeh (2014), where it is assumed that the (nonlinear) oscillators are coupled through an LTI electrical network that is either homogeneous or uniform. We here assume neither homogeneity nor uniformity (as defined therein) of the coupling network.

Lastly, we note that a problem setup similar to ours has been studied in Tuna (2019b). There the oscillators are coupled through a pair of matrix-weighted Laplacian matrices (recall that the Laplacians we work with here are scalar-weighted), but the network is free of interior nodes. Besides the absence of interior nodes, it is also assumed in Tuna (2019b) that the imaginary part of the complex (matrix-weighted) Laplacian is sufficiently small (e.g., the springs in Fig. 4 are weak) compared to that of the real part; and within that framework certain eigenvalue tests (similar to the one we establish here) are presented for synchronization of oscillators in various situations. The closest to our main theorem and founded on the two above-mentioned assumptions (which we do not make in this paper) is the result (Tuna, 2019b Cor. 14),² where the coupling is represented by scalar-weighted Laplacian matrices.