### Abstract

In a network of n agents, consensus means that all n agents reach an agreement on a specific value of some quantity via local interactions. A linear consensus process can typically be modeled by a discrete-time linear recursion equation or a continuous-time linear differential equation, whose equilibria include nonzero states of the form a1 where a is a constant and 1 is a column vector in ^{n} whose entries all equal 1. Using a suitably defined semi-norm, this paper extends the standard notions of uniform asymptotic stability and exponential stability from linear systems to linear recursions and differential equations of this type. It is shown that these notions are equivalent just as they are for conventional linear systems. The main contributions of this paper are first to provide a simple, direct proof of the necessary graph-theoretic condition given in [1] for a discrete-time linear consensus process to be exponentially stable, and second to derive a necessary graph-theoretic condition for a piecewise time-invariant continuous-time linear consensus process to be exponentially stable.

Original language | English (US) |
---|---|

Article number | 7039499 |

Pages (from-to) | 922-927 |

Number of pages | 6 |

Journal | Proceedings of the IEEE Conference on Decision and Control |

Volume | 2015-February |

Issue number | February |

DOIs | |

State | Published - Jan 1 2014 |

Event | 2014 53rd IEEE Annual Conference on Decision and Control, CDC 2014 - Los Angeles, United States Duration: Dec 15 2014 → Dec 17 2014 |

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### 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*,

*2015-February*(February), 922-927. [7039499]. https://doi.org/10.1109/CDC.2014.7039499

**Internal stability of linear consensus processes.** / Liu, Ji; Morse, A. Stephen; Nedic, Angelia; Basar, Tamer.

Research output: Contribution to journal › Conference article

*Proceedings of the IEEE Conference on Decision and Control*, vol. 2015-February, no. February, 7039499, pp. 922-927. https://doi.org/10.1109/CDC.2014.7039499

}

TY - JOUR

T1 - Internal stability of linear consensus processes

AU - Liu, Ji

AU - Morse, A. Stephen

AU - Nedic, Angelia

AU - Basar, Tamer

PY - 2014/1/1

Y1 - 2014/1/1

N2 - In a network of n agents, consensus means that all n agents reach an agreement on a specific value of some quantity via local interactions. A linear consensus process can typically be modeled by a discrete-time linear recursion equation or a continuous-time linear differential equation, whose equilibria include nonzero states of the form a1 where a is a constant and 1 is a column vector in n whose entries all equal 1. Using a suitably defined semi-norm, this paper extends the standard notions of uniform asymptotic stability and exponential stability from linear systems to linear recursions and differential equations of this type. It is shown that these notions are equivalent just as they are for conventional linear systems. The main contributions of this paper are first to provide a simple, direct proof of the necessary graph-theoretic condition given in [1] for a discrete-time linear consensus process to be exponentially stable, and second to derive a necessary graph-theoretic condition for a piecewise time-invariant continuous-time linear consensus process to be exponentially stable.

AB - In a network of n agents, consensus means that all n agents reach an agreement on a specific value of some quantity via local interactions. A linear consensus process can typically be modeled by a discrete-time linear recursion equation or a continuous-time linear differential equation, whose equilibria include nonzero states of the form a1 where a is a constant and 1 is a column vector in n whose entries all equal 1. Using a suitably defined semi-norm, this paper extends the standard notions of uniform asymptotic stability and exponential stability from linear systems to linear recursions and differential equations of this type. It is shown that these notions are equivalent just as they are for conventional linear systems. The main contributions of this paper are first to provide a simple, direct proof of the necessary graph-theoretic condition given in [1] for a discrete-time linear consensus process to be exponentially stable, and second to derive a necessary graph-theoretic condition for a piecewise time-invariant continuous-time linear consensus process to be exponentially stable.

UR - http://www.scopus.com/inward/record.url?scp=84988241322&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84988241322&partnerID=8YFLogxK

U2 - 10.1109/CDC.2014.7039499

DO - 10.1109/CDC.2014.7039499

M3 - Conference article

AN - SCOPUS:84988241322

VL - 2015-February

SP - 922

EP - 927

JO - Proceedings of the IEEE Conference on Decision and Control

JF - Proceedings of the IEEE Conference on Decision and Control

SN - 0191-2216

IS - February

M1 - 7039499

ER -