### Abstract

In this paper, we introduce an algorithm to decentralize the computation associated with the stability analysis of systems of nonlinear differential equations with a large number of states. The algorithm applies to dynamical systems with polynomial vector fields and checks the local asymptotic stability on hypercubes. We perform the analysis in three steps. First, by applying a multi-simplex version of Polya's theorem to some Lyapunov inequalities, we derive a sequence of stability conditions of increasing accuracy in the form of structured linear matrix inequalities. Then, we design a set-up algorithm to decentralize the computation of the coefficients of the LMIs, among the processing units of a parallel environment. Finally, we use a parallel primal-dual central path algorithm, specifically designed to solve the structured LMIs given by the set-up algorithm. For a sufficiently large number of available processors, the per-core computational complexity of the resulting algorithm is fixed with the accuracy. The algorithm demonstrates a near-linear speed-up in numerical experiments.

Original language | English (US) |
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Title of host publication | Proceedings of the IEEE Conference on Decision and Control |

Publisher | Institute of Electrical and Electronics Engineers Inc. |

Pages | 5858-5863 |

Number of pages | 6 |

ISBN (Print) | 9781467357173 |

DOIs | |

State | Published - 2013 |

Event | 52nd IEEE Conference on Decision and Control, CDC 2013 - Florence, Italy Duration: Dec 10 2013 → Dec 13 2013 |

### Other

Other | 52nd IEEE Conference on Decision and Control, CDC 2013 |
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Country | Italy |

City | Florence |

Period | 12/10/13 → 12/13/13 |

### Fingerprint

### ASJC Scopus subject areas

- Control and Systems Engineering
- Modeling and Simulation
- Control and Optimization

### Cite this

*Proceedings of the IEEE Conference on Decision and Control*(pp. 5858-5863). [6760813] Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/CDC.2013.6760813

**Decentralized Polya's algorithm for stability analysis of large-scale nonlinear systems.** / Kamyar, Reza; Peet, Matthew.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the IEEE Conference on Decision and Control.*, 6760813, Institute of Electrical and Electronics Engineers Inc., pp. 5858-5863, 52nd IEEE Conference on Decision and Control, CDC 2013, Florence, Italy, 12/10/13. https://doi.org/10.1109/CDC.2013.6760813

}

TY - GEN

T1 - Decentralized Polya's algorithm for stability analysis of large-scale nonlinear systems

AU - Kamyar, Reza

AU - Peet, Matthew

PY - 2013

Y1 - 2013

N2 - In this paper, we introduce an algorithm to decentralize the computation associated with the stability analysis of systems of nonlinear differential equations with a large number of states. The algorithm applies to dynamical systems with polynomial vector fields and checks the local asymptotic stability on hypercubes. We perform the analysis in three steps. First, by applying a multi-simplex version of Polya's theorem to some Lyapunov inequalities, we derive a sequence of stability conditions of increasing accuracy in the form of structured linear matrix inequalities. Then, we design a set-up algorithm to decentralize the computation of the coefficients of the LMIs, among the processing units of a parallel environment. Finally, we use a parallel primal-dual central path algorithm, specifically designed to solve the structured LMIs given by the set-up algorithm. For a sufficiently large number of available processors, the per-core computational complexity of the resulting algorithm is fixed with the accuracy. The algorithm demonstrates a near-linear speed-up in numerical experiments.

AB - In this paper, we introduce an algorithm to decentralize the computation associated with the stability analysis of systems of nonlinear differential equations with a large number of states. The algorithm applies to dynamical systems with polynomial vector fields and checks the local asymptotic stability on hypercubes. We perform the analysis in three steps. First, by applying a multi-simplex version of Polya's theorem to some Lyapunov inequalities, we derive a sequence of stability conditions of increasing accuracy in the form of structured linear matrix inequalities. Then, we design a set-up algorithm to decentralize the computation of the coefficients of the LMIs, among the processing units of a parallel environment. Finally, we use a parallel primal-dual central path algorithm, specifically designed to solve the structured LMIs given by the set-up algorithm. For a sufficiently large number of available processors, the per-core computational complexity of the resulting algorithm is fixed with the accuracy. The algorithm demonstrates a near-linear speed-up in numerical experiments.

UR - http://www.scopus.com/inward/record.url?scp=84902338761&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84902338761&partnerID=8YFLogxK

U2 - 10.1109/CDC.2013.6760813

DO - 10.1109/CDC.2013.6760813

M3 - Conference contribution

SN - 9781467357173

SP - 5858

EP - 5863

BT - Proceedings of the IEEE Conference on Decision and Control

PB - Institute of Electrical and Electronics Engineers Inc.

ER -