Improved Weighted Average Prediction for Multi-Agent Networks

Abstract

In sense of communication delays, an improved robust consensus algorithm for multi-agent networks and its the convergence rate have been investigated in this paper. Precisely, an improved weighted average prediction has been introduced to reformulate the network model into a neutral network fashion. By virtue of analyzing the Hopf bifurcation, an upper bound of the communication delay is derived for the multi-agent network, which could guarantee the network to achieve weighted average consensus. In addition, the main results show that not only can the proposed method promote the robustness but also improve its convergence rate. Finally, two numerical simulations are provided, which demonstrates the effectiveness of the method.

Appendix: A Simple Derivation of (4)

In this appendix, it provides a simple derivation of the protocol (4). Since

$$\hat{x}_{j}(t-\tau+\sigma) = x_{j}(t-\tau) + \delta _1(\sigma) \dot{x}_{j}(t-\tau),$$

where δ ₁(σ)=(e ^σ−1)(1−[tanh(σ)]²)+tanh(σ), it follows that $\hat{x}_{j}(\theta) = x_{j}(t-\tau) + \delta_{1}(\theta-t+\tau) \dot {x}_{j}(t-\tau)$, then substituting this into (3), we have

$$\begin{aligned} &\bar{\hat{x}}_{j}(t;\tau,\sigma) \\ &\quad= \frac{\int_{t-\tau}^{t-\tau+\sigma} e^{t-\tau-\theta} \hat{x}_j(\theta) \,\mathrm{d}\theta}{\int_{t-\tau }^{t-\tau+\sigma} e^{t-\tau-\theta} \,\mathrm{d}\theta} \\ &\quad= \frac{x_{j}(t-\tau) \int_{t-\tau}^{t-\tau+\sigma} e^{t-\tau-\theta} \,\mathrm{d}\theta+ \dot{x}_{j}(t-\tau) \int_{t-\tau}^{t-\tau+\sigma} e^{t-\tau-\theta} \delta_1(\theta-t+\tau) \,\mathrm{d}\theta}{\int_{t-\tau }^{t-\tau+\sigma} e^{t-\tau-\theta} \,\mathrm{d}\theta} \\ &\quad= x_{j}(t-\tau) + \frac{\dot{x}_{j}(t-\tau)}{1 - e^{-\sigma}} \int_{t-\tau}^{t-\tau+\sigma} e^{t-\tau-\theta} \delta_1(\theta-t+\tau) \,\mathrm{d}\theta. \end{aligned}$$

(24)

Let s=θ−t+τ, after substituting δ ₁(⋅) into (24), one obtains

$$\begin{aligned} \bar{\hat{x}}_{j}(t;\tau,\sigma) &= x_{j}(t- \tau) + \frac{\dot {x}_{j}(t-\tau)}{1 - e^{-\sigma}} \int_{0}^{\sigma} e^{-s} {\bigl[} \bigl(e^{s} - 1\bigr) \bigl(1 - \bigl[ \tanh(s)\bigr]^{2}\bigr) + \tanh(s) {\bigr]} \,\mathrm{d}s \\ &= x_{j}(t-\tau) + \frac{\dot{x}_{j}(t-\tau)}{1 - e^{-\sigma}} {\biggl[} \int _{0}^{\sigma} \bigl(1 - \bigl[\tanh(s) \bigr]^{2}\bigr) \,\mathrm{d}s \\&\quad{} - \int_{0}^{\sigma} e^{-s} \bigl(1 - \bigl[\tanh(s)\bigr]^{2}\bigr) \,\mathrm{d}s + \int_{0}^{\sigma} e^{-s} \tanh (s) \,\mathrm{d}s {\biggr]} \\ &= x_{j}(t-\tau) + \frac{\dot{x}_{j}(t-\tau)}{1 - e^{-\sigma}} {\biggl[} \tanh(\sigma) - \int _{0}^{\sigma} e^{-s} \bigl(1 - \bigl[\tanh(s) \bigr]^{2}\bigr) \mathrm {d}s \\&\quad{} + \int_{0}^{\sigma} e^{-s} \tanh(s) \,\mathrm{d}s {\biggr]} \\ &= x_{j}(t-\tau) + \frac{\dot{x}_{j}(t-\tau)}{1 - e^{-\sigma}} {\biggl[} \tanh(\sigma) \\&\quad{} - \int _{0}^{\sigma} e^{-s} \bigl(1 - \bigl[\tanh(s) \bigr]^{2}\bigr) \mathrm {d}s - e^{-\sigma} \tanh(\sigma) + \int _{0}^{\sigma} e^{-s} \mathrm {d}\tanh(s) { \biggr]} \\ &= x_{j}(t-\tau) + \frac{\dot{x}_{j}(t-\tau)}{1 - e^{-\sigma}} {\bigl[} \tanh(\sigma) - e^{-\sigma} \tanh(\sigma) {\bigr]} \\ &= x_{j}(t-\tau) + \tanh(\sigma) \dot{x}_{j}(t-\tau). \end{aligned}$$