On Spectral Properties of Signed Laplacians With Connections to Eventual Positivity

Chen, Wei; Wang, Dan; Ji, Liu; Chen, Yongxin; Khong, Sei Zhen; Başar, Tamer; Johansson, Karl Henrik; Li, Qiu

doi:10.1109/tac.2020.3008300

Cited by 19 publications

(8 citation statements)

References 46 publications

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“…On the other hand, the Kron-reduction requires loads to be described as positive conductances (see [21]). While research into Kron-reduced networks with negative loads is ongoing (see [22]), the general inclusion of negative loads, e.g., noncontrollable power sources, in Kron-reducible networks remains out of reach at present. Furthermore, consider the case where a DGU can no longer supply or consume the required amount of power, e.g., a fully charged or discharged battery storage.…”

Section: A Literature Reviewmentioning

confidence: 99%

Passivity-Based Power Sharing and Voltage Regulation in DC Microgrids With Unactuated Buses

Malan,

Jané-Soneira,

Strehle

et al. 2024

IEEE Trans. Contr. Syst. Technol.

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In this article, we propose a novel four-stage distributed controller for a dc microgrid that achieves proportional power sharing and average voltage regulation for the voltages at actuated and unactuated buses. The controller is presented for a dc microgrid comprising multiple distributed generation units (DGUs) with time-varying actuation states, dynamic RLC lines, nonlinear constant impedance, current, and power (ZIP) loads, and a time-varying network topology. The controller comprising a nonlinear gain, proportional-integral (PI) controllers, and two dynamic distributed averaging stages is designed for asymptotic stability. This constitutes deriving passivity properties for the dc microgrid, along with each of the controller subsystems. Thereafter, design parameters are found through a passivity-based optimization using the worst-case subsystem properties. The resulting closed loop is robust against DGU actuation changes, network topology changes, and microgrid parameter changes. The stability and robustness of the proposed control are verified via simulations.

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Section: A Literature Reviewmentioning

confidence: 99%

Passivity-Based Power Sharing and Voltage Regulation in DC Microgrids With Unactuated Buses

Malan,

Jané-Soneira,

Strehle

et al. 2024

IEEE Trans. Contr. Syst. Technol.

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“…[9] for its properties). Extensions to signed graphs and negative resistances have been investigated in [32], [26], [33], [34], [31], where positive semidefiniteness of the Laplacian is expressed in terms of effective resistance.…”

Section: Electrical Network and Effective Resistancementioning

confidence: 99%

On the properties of Laplacian pseudoinverses

Fontan¹,

Altafini²

2021

Preprint

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The pseudoinverse of a graph Laplacian is used in many applications and fields, such as for instance in the computation of the effective resistance in electrical networks, in the calculation of the hitting/commuting times for a Markov chain and in continuous-time distributed averaging problems. In this paper we show that the Laplacian pseudoinverse is in general not a Laplacian matrix but rather a signed Laplacian with the property of being an eventually exponentially positive matrix, i.e., of obeying a strong Perron-Frobenius property. We show further that the set of signed Laplacians with this structure (i.e., eventual exponential positivity) is closed with respect to matrix pseudoinversion. This is true even for signed digraphs, and provided that we restrict to Laplacians that are weight balanced also stability is guaranteed.

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“…Other contexts in which signed graphs appear include e.g. small-disturbance angle stability of microgrids [3], [4].…”

Section: Introductionmentioning

confidence: 99%

“…Another complication that arises for "repelling" Laplacians is that strong connectivity of the graph no longer guarantees that the Laplacian has corank 1, meaning that even marginally stable "repelling" Laplacians may fail to lead to consensus when their kernel has dimension larger than 1. In the time-invariant case, conditions for stability are provided in [6], [7], [8], [4] for signed undirected graphs, while for signed digraphs some partial results appear in [9], [10], [11]. In the time-varying case, we are not aware of any systematic study (unlike for the "opposing" signed Laplacian, for which an abundant literature exists [12], [13], [14], [15]).…”

Section: Introductionmentioning

confidence: 99%

Multi-agent consensus over time-invariant and time-varying signed digraphs via eventual positivity

Fontan¹,

Wang²,

Ye³

et al. 2022

Preprint

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Laplacian dynamics on signed digraphs have a richer behavior than those on nonnegative digraphs. In particular, for the so-called "repelling" signed Laplacians, the marginal stability property (needed to achieve consensus) is not guaranteed a priori and, even when it holds, it does not automatically lead to consensus, as these signed Laplacians may loose rank even in strongly connected digraphs. Furthermore, in the time-varying case, instability can occur even when switching in a family of systems each of which corresponds to a marginally stable signed Laplacian with the correct corank. In this paper we present conditions guaranteeing consensus of these signed Laplacians based on the property of eventual positivity, a Perron-Frobenius type of property for signed matrices. The conditions cover both time-invariant and time-varying cases. A particularly simple sufficient condition valid in both cases is that the Laplacians are normal matrices. Such condition can be relaxed in several ways. For instance in the time-invariant case it is enough that the Laplacian has this Perron-Frobenius property on the right but not on the left side (i.e., on the transpose). For the time-varying case, convergence to consensus can be guaranteed by the existence of a common Lyapunov function for all the signed Laplacians. All conditions can be easily extended to bipartite consensus.

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On Spectral Properties of Signed Laplacians With Connections to Eventual Positivity

Cited by 19 publications

References 46 publications

Passivity-Based Power Sharing and Voltage Regulation in DC Microgrids With Unactuated Buses

Passivity-Based Power Sharing and Voltage Regulation in DC Microgrids With Unactuated Buses

On the properties of Laplacian pseudoinverses

Multi-agent consensus over time-invariant and time-varying signed digraphs via eventual positivity

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