# Synchronization of Lienard-Type Oscillators in Uniform Electrical Networks

## Abstract

This paper presents a condition for global asymptotic synchronization of Lienard-type nonlinear oscillators in uniform LTI electrical networks with series R-L circuits modeling interconnections. By uniform electrical networks, we mean that the per-unit-length impedances are identical for the interconnecting lines. We derive conditions for global asymptotic synchronization for a particular feedback architecture where the derivative of the oscillator output current supplements the innate current feedback induced by simply interconnecting the oscillator to the network. Our proof leverages a coordinate transformation to a set of differential coordinates that emphasizes signal differences and the particular form of feedback permits the formulation of a quadratic Lyapunov function for this class of networks. This approach is particularly interesting since synchronization conditions are difficult to obtain by means of quadratic Lyapunov functions when only current feedback is used and for networks composed of series R-L circuits. Our synchronization condition depends on the algebraic connectivity of the underlying network, and reiterates the conventional wisdom from Lyapunov- and passivity-based arguments that strong coupling is required to ensure synchronization.

- Authors:

- Publication Date:

- Research Org.:
- National Renewable Energy Lab. (NREL), Golden, CO (United States)

- Sponsoring Org.:
- NREL Laboratory Directed Research and Development (LDRD)

- OSTI Identifier:
- 1320376

- Report Number(s):
- NREL/CP-5D00-65194

- DOE Contract Number:
- AC36-08GO28308

- Resource Type:
- Conference

- Resource Relation:
- Conference: Presented at the 2016 American Control Conference (ACC), 6-8 July 2016, Boston, Massachusetts

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 24 POWER TRANSMISSION AND DISTRIBUTION; oscillators; synchronization; integrated circuit interconnections; integrated circuit modeling; Lyapunov methods; limit-cycles; mathematical model

### Citation Formats

```
Sinha, Mohit, Dorfler, Florian, Johnson, Brian B., and Dhople, Sairaj V.
```*Synchronization of Lienard-Type Oscillators in Uniform Electrical Networks*. United States: N. p., 2016.
Web. doi:10.1109/ACC.2016.7525600.

```
Sinha, Mohit, Dorfler, Florian, Johnson, Brian B., & Dhople, Sairaj V.
```*Synchronization of Lienard-Type Oscillators in Uniform Electrical Networks*. United States. doi:10.1109/ACC.2016.7525600.

```
Sinha, Mohit, Dorfler, Florian, Johnson, Brian B., and Dhople, Sairaj V. Mon .
"Synchronization of Lienard-Type Oscillators in Uniform Electrical Networks". United States.
doi:10.1109/ACC.2016.7525600.
```

```
@article{osti_1320376,
```

title = {Synchronization of Lienard-Type Oscillators in Uniform Electrical Networks},

author = {Sinha, Mohit and Dorfler, Florian and Johnson, Brian B. and Dhople, Sairaj V.},

abstractNote = {This paper presents a condition for global asymptotic synchronization of Lienard-type nonlinear oscillators in uniform LTI electrical networks with series R-L circuits modeling interconnections. By uniform electrical networks, we mean that the per-unit-length impedances are identical for the interconnecting lines. We derive conditions for global asymptotic synchronization for a particular feedback architecture where the derivative of the oscillator output current supplements the innate current feedback induced by simply interconnecting the oscillator to the network. Our proof leverages a coordinate transformation to a set of differential coordinates that emphasizes signal differences and the particular form of feedback permits the formulation of a quadratic Lyapunov function for this class of networks. This approach is particularly interesting since synchronization conditions are difficult to obtain by means of quadratic Lyapunov functions when only current feedback is used and for networks composed of series R-L circuits. Our synchronization condition depends on the algebraic connectivity of the underlying network, and reiterates the conventional wisdom from Lyapunov- and passivity-based arguments that strong coupling is required to ensure synchronization.},

doi = {10.1109/ACC.2016.7525600},

journal = {},

number = ,

volume = ,

place = {United States},

year = {Mon Aug 01 00:00:00 EDT 2016},

month = {Mon Aug 01 00:00:00 EDT 2016}

}