I am looking for a reference that explains the asymptotic stability for the mean square consensus of a multi-agent system step by step. I am reading an article: "Sampled-Data Consensus of Linear Multi-agent Systems With Packet Losses", but I cannot understand how they came up with inequalities in equations 12-13.
Any help is much appreciated.
Thank you
Hi Samira,
Eqs. (12)−(13) are the stability conditions that are expressed in the form of Linear Matrix Inequalities (LMIs). Following the Lyapunov stability proof from Eqs. (15)−(18), the LMIs are actually derived from the summarized inequality in Eq. (19) using the Schur complement formula.
Yew-Chung Chak
Thanks a lot Sam. It's more clear to me now. On what basis do you think they have expanded the alpha in 12-13? alpha_bar is the probability and alpha(tk) is a stochastic variable associated with the packet dropout.
Hi Samira Eshghi,
From the multi agent system in Eq. (10), I think the authors expanded α(tk) in Eq. (15) and then regrouped them in {α(tk) − ᾱ} terms. Later, from the implication of Eq. (4) that 𝔼{α(tk) − ᾱ} = 0, the authors eliminated the terms associated with {α(tk) − ᾱ} in the inequality Eq. (19) that looks like a modified discrete time generalized algebraic Riccati equation.
Hi Samira,
I'm not sure what is precisely your doubt. But I consider that a good introductory text is Lectures on Network Systems by Francesco Bullo, you can find the antecedents of mean square consensus in section 12.3.2.
Have a look at "Consensus and Cooperation in Networked Multi-Agent Systems". Its a nice summary paper.
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