An Exact Method for the Stability Analysis of Linear Consensus Protocols With Time Delay

This technical note presents a methodology for the stability analysis of linear consensus protocols with time-delayed communications. Second order agent dynamics with a fixed and undirected communication topology and uniform delays are considered. This class of group dynamics is very complex and is not fully explored to date. The proposed technique takes advantage of the general structure of the control protocols in performing a state transformation that allows a decomposition of the characteristic equation into a set of factors. These factors distribute the imprint of the delay in the characteristic equation in a much simpler form to achieve the stability analysis in parts. The procedure also prepares the characteristic equation for the deployment of the Cluster Treatment of Characteristic Roots paradigm, a recent method which declares the stability features of the system for various compositions of the time delay and other control parameters. In order to show the effectiveness of this approach, it is applied to different consensus protocols under the assumptions of fixed and undirected communication topologies and uniform communication time delays.

[1]  Bobbi Jo Broxson.  The Kronecker Product , 2006 .

[2]  Junping Du,et al.  Distributed control of multi‐agent systems with second‐order agent dynamics and delay‐dependent communications , 2008 .

[3]  Yingmin Jia,et al.  Further results on decentralised coordination in networks of agents with second-order dynamics , 2009 .

[4]  Richard M. Murray,et al.  Consensus problems in networks of agents with switching topology and time-delays , 2004, IEEE Transactions on Automatic Control.

[5]  Rifat Sipahi,et al.  An exact method for the stability analysis of time-delayed linear time-invariant (LTI) systems , 2002, IEEE Trans. Autom. Control..

[6]  Nejat Olgaç,et al.  Complete stability robustness of third-order LTI multiple time-delay systems , 2005, Autom..

[7]  Nejat Olgac,et al.  Stability Analysis for a Consensus System of a Group of Second Order Dynamics with Time Delays , 2010 .

[8]  Nejat Olgaç,et al.  Exhaustive stability analysis in a consensus system with time delay and irregular topologies , 2011, Proceedings of the 2011 American Control Conference.

[9]  Nejat Olgaç,et al.  Consensus of a group of second order agents with switching irregular communication topologies and time-delay , 2010, 49th IEEE Conference on Decision and Control (CDC).

[10]  Alfred Hanssen,et al.  A theory of polyspectra for nonstationary stochastic processes , 2003, IEEE Trans. Signal Process..

[11]  I. V. Sergienko,et al.  Limiting Representations of Weighted Pseudoinverse Matrices with Positive Definite Weights. Problem Regularization , 2003 .

[12]  Yupu Yang,et al.  Leader-following consensus problem with a varying-velocity leader and time-varying delays , 2009 .

[13]  Yu-Ping Tian,et al.  Consensus of Multi-Agent Systems With Diverse Input and Communication Delays , 2008, IEEE Transactions on Automatic Control.

[14]  Yingmin Jia,et al.  Consensus of second-order discrete-time multi-agent systems with nonuniform time-delays and dynamically changing topologies , 2009, Autom..

[15]  Santa Maria,et al.  Two-Dimensional Fourier Transform Applied to Helicopter Flyover Noise , 1999 .

[16]  Nejat Olgac,et al.  Stability of the Consensus of a Group of Second Order Agents With Time Delayed Communications , 2010 .

[17]  E. Bedrosian,et al.  The output properties of Volterra systems (nonlinear systems with memory) driven by harmonic and Gaussian inputs , 1971 .

[18]  W. Rugh,et al.  On a multidimensional S -transform and the realization problem for homogeneous nonlinear systems , 1977 .

[19]  Wei Ren.  On Consensus Algorithms for Double-Integrator Dynamics , 2008, IEEE Trans. Autom. Control..

[20]  Wilson Rugh,et al.  On transfer function representations for homogeneous nonlinear systems , 1978, 1978 IEEE Conference on Decision and Control including the 17th Symposium on Adaptive Processes.

[21]  Gordon F. Royle,et al.  Algebraic Graph Theory , 2001, Graduate texts in mathematics.

[22]  Jiangping Hu,et al.  Consensus control for multi-agent systems with double-integrator dynamics and time delays , 2010 .

[23]  Nejat Olgac,et al.  A Novel Active Vibration Absorption Technique: Delayed Resonator , 1994 .

[24]  Steven M. Kay,et al.  Rapid estimation of the range-Doppler scattering function , 2003, IEEE Trans. Signal Process..

[25]  Nejat Olgaç,et al.  An improved procedure in detecting the stability robustness of systems with uncertain delay , 2006, IEEE Transactions on Automatic Control.