[Y. Li, C. Tang, S. Peeta and Y. Wang, “Nonlinear Consensus-Based Connected Vehicle Platoon Control Incorporating Car-Following Interactions and Heterogeneous Time Delays,” in IEEE Transactions on Intelligent Transportation Systems, vol. 20, no. 6, pp. 2209-2219, June 2019, doi: 10.1109/TITS.2018.2865546.]

文章目录

  • 2 Preliminaries and Problem Statement
    • 2.1 Graph Theory
    • 2.2 Mathematical Preliminaries
    • 2.3 Consensus Problem Statement
  • 3 Nonlinear Consensus Algorithm
    • Remark 1
  • 4 Convergence Analysis
    • 4.1 Convergence Analysis
    • 4.2 Convergence Speed Analysis
  • 5 Numerical Experiments
    • 5.1 Simulation Setting
    • 5.2 Discussion of Results
    • 5.3 Comparison to Existing Approaches
  • ZGC Codes
    • zgc.1 Leader's states
    • zgc.2 Leader and some followers' states
    • zgc.3 Leader and all followers' states (no time delays)
    • zgc.4 Leader and all followers' states (heterogeneous time delays)
    • zgc.5 Leader and all followers' states (homogeneous time delays)

2 Preliminaries and Problem Statement

2.1 Graph Theory

为了有效地利用通信容量，我们提出了在连通环境下车辆之间的连通性特征的上届领导拓扑(PLF)。也就是说，每一辆后面的车辆都可以通过V2V通信链路，从前面一辆车和领导者获取实时信息(如位置和速度)。

2.2 Mathematical Preliminaries

观察图可知，每个车子仅由 Leader 和前车 follower 影响。

2.3 Consensus Problem Statement

系统采用的是二阶积分器模型：
{x˙i(t)=vi(t)v˙i(t)=ui(t),i=L,1,2,⋯,n,(9)\left\{\begin{aligned} &\dot{x}_i(t) = v_i(t) \\ &\dot{v}_i(t) = u_i(t), ~~~ i=L,1,2,\cdots, n, \\ \end{aligned}\right.\tag{9}{​x˙i​(t)=vi​(t)v˙i​(t)=ui​(t),   i=L,1,2,⋯,n,​(9)

控制目标是保持安全距离 hch_chc​ 的一字型 Leader 编队控制

3 Nonlinear Consensus Algorithm

分布式非线性时滞控制算法为
ui(t)=∑j=1nai,j[α(Vi(hi,j(t))−vi(t))+β(vj(t−τij(t))−vi(t))+γ(xj(t−τij(t))−xi(t))+vL(t−τiL(t))τij(t)−ri,j)]+ki,L[β(vL(t−τiL(t))−vi(t))+γ(xL(t−τiL(t))+vL(t−τiL)τiL(t)−xi(t)−ri,L)](13)\begin{aligned} u_i(t) =& \sum_{j=1}^{n} \red{a_{i,j}} [ \green{\alpha} (~V_i(h_{i,j}(t)) - v_i(t)~) \\ & ~~~~~~~~~+ \green{\beta} (~\blue{v_j(t-\tau_{ij}(t))} - v_i(t)~) \\ & ~~~~~~~~~+ \green{\gamma} (~\blue{x_j(t-\tau_{ij}(t))} - x_i(t)~) \\ & ~~~~~~~~~+ \textcolor{#9932CC}{v_L (t-\tau_{iL}(t)) \tau_{ij}(t)} - \textcolor{#FF8C00}{r_{i,j}}) ~] \\ &+ \red{k_{i,L}} [ ~ \green{\beta} ( ~ \blue{v_L(t-\tau_{iL}(t))} - v_i(t) ~ ) \\ & ~~~~~~~~~ +\green{\gamma} ( ~ \blue{x_L(t-\tau_{iL}(t))} + \textcolor{#9932CC}{v_L(t-\tau_{iL}) \tau_{iL}(t)} - x_i(t) - \textcolor{#FF8C00}{r_{i,L}} ~ ) ~ ] \end{aligned}\tag{13}ui​(t)=​j=1∑n​ai,j​[α( Vi​(hi,j​(t))−vi​(t) )         +β( vj​(t−τij​(t))−vi​(t) )         +γ( xj​(t−τij​(t))−xi​(t) )         +vL​(t−τiL​(t))τij​(t)−ri,j​) ]+ki,L​[ β( vL​(t−τiL​(t))−vi​(t) )         +γ( xL​(t−τiL​(t))+vL​(t−τiL​)τiL​(t)−xi​(t)−ri,L​ ) ]​(13)

先分解这个控制协议。

ai,j\red{a_{i,j}}ai,j​ ： 是 follower 之间通信链接。
ki,L\red{k_{i,L}}ki,L​ ： 是follower 与 leader 的通信链接。
α,β,γ\green{\alpha, \beta, \gamma}α,β,γ : 控制增益常数。
ri,L=i⋅(lc+hc)\textcolor{#FF8C00}{r_{i,L}} = i \cdot (l_c + h_c)ri,L​=i⋅(lc​+hc​) : 节点 iii 与 leader 的期望距离
ri,j=riL−rjL=(i−j)(lc+hc)\textcolor{#FF8C00}{r_{i,j}} = r_{iL} - r_{jL} = (i-j)(l_c + h_c)ri,j​=riL​−rjL​=(i−j)(lc​+hc​) : 节点 iii 与 jjj 的期望距离
τij(t)\tau_{ij}(t)τij​(t) : 信息由 jjj 到 iii 的传输时滞
τiL(t)\tau_{iL}(t)τiL​(t) : 信息由 leader 到 iii 的传输时滞

Remark 1

vL(t−τiL(t))τij(t)\textcolor{#9932CC}{v_L(t-\tau_{iL}(t)) \tau_{ij}(t)}vL​(t−τiL​(t))τij​(t) : 间隔补偿项
(13c) denotes the position difference between vehicles iii and jjj with respect to the desired gap ri,jr_{i,j}ri,j​

vL(t−τiL(t))τiL(t)\textcolor{#9932CC}{v_L(t-\tau_{iL}(t)) \tau_{iL}(t)}vL​(t−τiL​(t))τiL​(t) : 间隔补偿项

具体原文中的 Remark 1 如下

代入数值并展开（因为节点 1 只受到 leader 的影响，因此只有协议的第二项）：
u1(t)=β(vL(t−τ1L(t))−v1(t))+γ(xL(t−τ1L(t))+vL(t−τ1L)τ1L(t)−x1(t)−r1,L)u_1(t) = \green{\beta} ( ~ \blue{v_L(t-\tau_{1L}(t))} - v_1(t) ~ ) +\green{\gamma} ( ~ \blue{x_L(t-\tau_{1L}(t))} + \textcolor{#9932CC}{v_L(t-\tau_{1L}) \tau_{1L}(t)} - x_1(t) - \textcolor{#FF8C00}{r_{1,L}} ~ )u1​(t)=β( vL​(t−τ1L​(t))−v1​(t) )+γ( xL​(t−τ1L​(t))+vL​(t−τ1L​)τ1L​(t)−x1​(t)−r1,L​ )u2(t)=1[α(V2(h2,1(t))−v2(t))+β(v1(t−τ21(t))−v2(t))+γ(x1(t−τ21(t))−x2(t))+vL(t−τ2L(t))τ21(t)−r2,1)]+1[β(vL(t−τ2L(t))−v2(t))+γ(xL(t−τ2L(t))+vL(t−τ2L)τ2L(t)−x2(t)−r2,L)]\begin{aligned} u_2(t) =& \red{1} [ \green{\alpha} (~V_2(h_{2,1}(t)) - v_2(t)~) \\ & + \green{\beta} (~\blue{v_1(t-\tau_{21}(t))} - v_2(t)~) \\ & + \green{\gamma} (~\blue{x_1(t-\tau_{21}(t))} - x_2(t)~) \\ & + \textcolor{#9932CC}{v_L (t-\tau_{2L}(t)) \tau_{21}(t)} - \textcolor{#FF8C00}{r_{2,1}}) ~] \\ +& \red{1} [ ~ \green{\beta} ( ~ \blue{v_L(t-\tau_{2L}(t))} - v_2(t) ~ ) \\ & +\green{\gamma} ( ~ \blue{x_L(t-\tau_{2L}(t))} + \textcolor{#9932CC}{v_L(t-\tau_{2L}) \tau_{2L}(t)} - x_2(t) - \textcolor{#FF8C00}{r_{2,L}} ~ ) ~ ] \end{aligned}u2​(t)=+​1[α( V2​(h2,1​(t))−v2​(t) )+β( v1​(t−τ21​(t))−v2​(t) )+γ( x1​(t−τ21​(t))−x2​(t) )+vL​(t−τ2L​(t))τ21​(t)−r2,1​) ]1[ β( vL​(t−τ2L​(t))−v2​(t) )+γ( xL​(t−τ2L​(t))+vL​(t−τ2L​)τ2L​(t)−x2​(t)−r2,L​ ) ]​u3(t)=⋯u_3(t) =\cdots u3​(t)=⋯⋮\vdots ⋮

先分析没有时滞时，协议可以简化为
u1(t)=β(vL(t−τ1L(t))−v1(t))+γ(xL(t−τ1L(t))+vL(t−τ1L)τ1L(t)−x1(t)−r1,L)=β(vL(t)−v1(t))+γ(xL(t)−x1(t)−r1,L)\begin{aligned} u_1(t) =& \green{\beta} ( ~ \blue{v_L(t-\tau_{1L}(t))} - v_1(t) ~ ) +\green{\gamma} ( ~ \blue{x_L(t-\tau_{1L}(t))} + \textcolor{#9932CC}{v_L(t-\tau_{1L}) \tau_{1L}(t)} - x_1(t) - \textcolor{#FF8C00}{r_{1,L}} ~ ) \\ =& \green{\beta} ( ~ \blue{v_L(t)} - v_1(t) ~ ) +\green{\gamma} ( ~ \blue{x_L(t)} - x_1(t) - \textcolor{#FF8C00}{r_{1,L}} ~ ) \end{aligned}u1​(t)==​β( vL​(t−τ1L​(t))−v1​(t) )+γ( xL​(t−τ1L​(t))+vL​(t−τ1L​)τ1L​(t)−x1​(t)−r1,L​ )β( vL​(t)−v1​(t) )+γ( xL​(t)−x1​(t)−r1,L​ )​u2(t)=α(V2(h2,1(t))−v2(t))+β(v1(t−τ21(t))−v2(t))+γ(x1(t−τ21(t))−x2(t))+vL(t−τ2L(t))τ21(t)−r2,1)+β(vL(t−τ2L(t))−v2(t))+γ(xL(t−τ2L(t))+vL(t−τ2L)τ2L(t)−x2(t)−r2,L)=α(V2(h2,1(t))−v2(t))+β(v1(t)−v2(t))+γ(x1(t)−x2(t))−r2,1)+β(vL(t)−v2(t))+γ(xL(t)−x2(t)−r2,L)\begin{aligned} u_2(t) =& \green{\alpha} (~V_2(h_{2,1}(t)) - v_2(t)~) + \green{\beta} (~\blue{v_1(t-\tau_{21}(t))} - v_2(t)~) + \green{\gamma} (~\blue{x_1(t-\tau_{21}(t))} - x_2(t)~) + \textcolor{#9932CC}{v_L (t-\tau_{2L}(t)) \tau_{21}(t)} - \textcolor{#FF8C00}{r_{2,1}}) \\ &+ \green{\beta} ( ~ \blue{v_L(t-\tau_{2L}(t))} - v_2(t) ~ ) +\green{\gamma} ( ~ \blue{x_L(t-\tau_{2L}(t))} + \textcolor{#9932CC}{v_L(t-\tau_{2L}) \tau_{2L}(t)} - x_2(t) - \textcolor{#FF8C00}{r_{2,L}} ~ ) \\ =& \green{\alpha} (~V_2(h_{2,1}(t)) - v_2(t)~) +\green{\beta} (~\blue{v_1(t)} - v_2(t)~) +\green{\gamma} (~\blue{x_1(t)} - x_2(t)~) -\textcolor{#FF8C00}{r_{2,1}}) \\ &+\green{\beta} ( ~ \blue{v_L(t)} - v_2(t) ~ ) +\green{\gamma} ( ~ \blue{x_L(t)} - x_2(t) - \textcolor{#FF8C00}{r_{2,L}} ~ ) \end{aligned}u2​(t)==​α( V2​(h2,1​(t))−v2​(t) )+β( v1​(t−τ21​(t))−v2​(t) )+γ( x1​(t−τ21​(t))−x2​(t) )+vL​(t−τ2L​(t))τ21​(t)−r2,1​)+β( vL​(t−τ2L​(t))−v2​(t) )+γ( xL​(t−τ2L​(t))+vL​(t−τ2L​)τ2L​(t)−x2​(t)−r2,L​ )α( V2​(h2,1​(t))−v2​(t) )+β( v1​(t)−v2​(t) )+γ( x1​(t)−x2​(t) )−r2,1​)+β( vL​(t)−v2​(t) )+γ( xL​(t)−x2​(t)−r2,L​ )​u3(t)=α(V3(h3,2(t))−v3(t))+β(v2(t)−v3(t))+γ(x2(t)−x3(t))−r3,2)+β(vL(t)−v3(t))+γ(xL(t)−x3(t)−r3,L)\begin{aligned} u_3(t) =& \green{\alpha} (~V_3(h_{3,2}(t)) - v_3(t)~) +\green{\beta} (~\blue{v_2(t)} - v_3(t)~) +\green{\gamma} (~\blue{x_2(t)} - x_3(t)~) -\textcolor{#FF8C00}{r_{3,2}}) \\ &+\green{\beta} ( ~ \blue{v_L(t)} - v_3(t) ~ ) +\green{\gamma} ( ~ \blue{x_L(t)} - x_3(t) - \textcolor{#FF8C00}{r_{3,L}} ~ ) \end{aligned}u3​(t)=​α( V3​(h3,2​(t))−v3​(t) )+β( v2​(t)−v3​(t) )+γ( x2​(t)−x3​(t) )−r3,2​)+β( vL​(t)−v3​(t) )+γ( xL​(t)−x3​(t)−r3,L​ )​u4(t)=⋯u_4(t) = \cdotsu4​(t)=⋯⋮\vdots⋮

再分析具有不同时滞时，协议为（注意因为 leader 速度是个常量，因此不受时滞影响，这一点在文章中也有体现）
u1(t)=β(vL(t−τ1L(t))−v1(t))+γ(xL(t−τ1L(t))+vL(t−τ1L)τ1L(t)−x1(t)−r1,L)=β(vL(t)−v1(t))+γ(xL(t−τ1L(t))+vL(t)τ1L(t)−x1(t)−r1,L)\begin{aligned} u_1(t) &= \green{\beta} ( ~ \blue{v_L(t-\tau_{1L}(t))} - v_1(t) ~ ) +\green{\gamma} ( ~ \blue{x_L(t-\tau_{1L}(t))} + \textcolor{#9932CC}{v_L(t-\tau_{1L}) \tau_{1L}(t)} - x_1(t) - \textcolor{#FF8C00}{r_{1,L}} ~ ) \\ &= \green{\beta} ( ~ \blue{v_L(t)} - v_1(t) ~ ) +\green{\gamma} ( ~ \blue{x_L(t-\tau_{1L}(t))} + \textcolor{#9932CC}{v_L(t) \tau_{1L}(t)} - x_1(t) - \textcolor{#FF8C00}{r_{1,L}} ~ ) \end{aligned}u1​(t)​=β( vL​(t−τ1L​(t))−v1​(t) )+γ( xL​(t−τ1L​(t))+vL​(t−τ1L​)τ1L​(t)−x1​(t)−r1,L​ )=β( vL​(t)−v1​(t) )+γ( xL​(t−τ1L​(t))+vL​(t)τ1L​(t)−x1​(t)−r1,L​ )​

u2(t)=α(V2(h2,1(t))−v2(t))+β(v1(t−τ21(t))−v2(t))+γ(x1(t−τ21(t))−x2(t))+vL(t−τ2L(t))τ21(t)−r2,1)+β(vL(t−τ2L(t))−v2(t))+γ(xL(t−τ2L(t))+vL(t−τ2L)τ2L(t)−x2(t)−r2,L)=α(V2(h2,1(t))−v2(t))+β(v1(t−τ21(t))−v2(t))+γ(x1(t−τ21(t))−x2(t))+vL(t)τ21(t)−r2,1)+β(vL(t)−v2(t))+γ(xL(t−τ2L(t))+vL(t)τ2L(t)−x2(t)−r2,L)\begin{aligned} u_2(t) =& \green{\alpha} (~V_2(h_{2,1}(t)) - v_2(t)~) + \green{\beta} (~\blue{v_1(t-\tau_{21}(t))} - v_2(t)~) + \green{\gamma} (~\blue{x_1(t-\tau_{21}(t))} - x_2(t)~) + \textcolor{#9932CC}{v_L (t-\tau_{2L}(t)) \tau_{21}(t)} - \textcolor{#FF8C00}{r_{2,1}}) \\ &+ \green{\beta} ( ~ \blue{v_L(t-\tau_{2L}(t))} - v_2(t) ~ ) +\green{\gamma} ( ~ \blue{x_L(t-\tau_{2L}(t))} + \textcolor{#9932CC}{v_L(t-\tau_{2L}) \tau_{2L}(t)} - x_2(t) - \textcolor{#FF8C00}{r_{2,L}} ~ ) \\ =& \green{\alpha} (~V_2(h_{2,1}(t)) - v_2(t)~) + \green{\beta} (~\blue{v_1(t-\tau_{21}(t))} - v_2(t)~) + \green{\gamma} (~\blue{x_1(t-\tau_{21}(t))} - x_2(t)~) + \textcolor{#9932CC}{v_L (t) \tau_{21}(t)} - \textcolor{#FF8C00}{r_{2,1}}) \\ &+ \green{\beta} ( ~ \blue{v_L(t)} - v_2(t) ~ ) +\green{\gamma} ( ~ \blue{x_L(t-\tau_{2L}(t))} + \textcolor{#9932CC}{v_L(t) \tau_{2L}(t)} - x_2(t) - \textcolor{#FF8C00}{r_{2,L}} ~ ) \end{aligned}u2​(t)==​α( V2​(h2,1​(t))−v2​(t) )+β( v1​(t−τ21​(t))−v2​(t) )+γ( x1​(t−τ21​(t))−x2​(t) )+vL​(t−τ2L​(t))τ21​(t)−r2,1​)+β( vL​(t−τ2L​(t))−v2​(t) )+γ( xL​(t−τ2L​(t))+vL​(t−τ2L​)τ2L​(t)−x2​(t)−r2,L​ )α( V2​(h2,1​(t))−v2​(t) )+β( v1​(t−τ21​(t))−v2​(t) )+γ( x1​(t−τ21​(t))−x2​(t) )+vL​(t)τ21​(t)−r2,1​)+β( vL​(t)−v2​(t) )+γ( xL​(t−τ2L​(t))+vL​(t)τ2L​(t)−x2​(t)−r2,L​ )​

u3(t)=⋯u_3(t) =\cdots u3​(t)=⋯⋮\vdots ⋮

剩下一个具有相同时滞的，就比较简单了
u1(t)=β(vL(t−τ1L(t))−v1(t))+γ(xL(t−τ1L(t))+vL(t−τ1L)τ1L(t)−x1(t)−r1,L)=β(vL(t)−v1(t))+γ(xL(t−τ)+vL(t)τ−x1(t)−r1,L)\begin{aligned} u_1(t) &= \green{\beta} ( ~ \blue{v_L(t-\tau_{1L}(t))} - v_1(t) ~ ) +\green{\gamma} ( ~ \blue{x_L(t-\tau_{1L}(t))} + \textcolor{#9932CC}{v_L(t-\tau_{1L}) \tau_{1L}(t)} - x_1(t) - \textcolor{#FF8C00}{r_{1,L}} ~ ) \\ &= \green{\beta} ( ~ \blue{v_L(t)} - v_1(t) ~ ) +\green{\gamma} ( ~ \blue{x_L(t-\tau)} + \textcolor{#9932CC}{v_L(t) \tau} - x_1(t) - \textcolor{#FF8C00}{r_{1,L}} ~ ) \end{aligned}u1​(t)​=β( vL​(t−τ1L​(t))−v1​(t) )+γ( xL​(t−τ1L​(t))+vL​(t−τ1L​)τ1L​(t)−x1​(t)−r1,L​ )=β( vL​(t)−v1​(t) )+γ( xL​(t−τ)+vL​(t)τ−x1​(t)−r1,L​ )​

u2(t)=α(V2(h2,1(t))−v2(t))+β(v1(t−τ21(t))−v2(t))+γ(x1(t−τ21(t))−x2(t))+vL(t−τ2L(t))τ21(t)−r2,1)+β(vL(t−τ2L(t))−v2(t))+γ(xL(t−τ2L(t))+vL(t−τ2L)τ2L(t)−x2(t)−r2,L)=α(V2(h2,1(t))−v2(t))+β(v1(t−τ)−v2(t))+γ(x1(t−τ)−x2(t))+vL(t)τ−r2,1)+β(vL(t)−v2(t))+γ(xL(t−τ)+vL(t)τ−x2(t)−r2,L)\begin{aligned} u_2(t) =& \green{\alpha} (~V_2(h_{2,1}(t)) - v_2(t)~) + \green{\beta} (~\blue{v_1(t-\tau_{21}(t))} - v_2(t)~) + \green{\gamma} (~\blue{x_1(t-\tau_{21}(t))} - x_2(t)~) + \textcolor{#9932CC}{v_L (t-\tau_{2L}(t)) \tau_{21}(t)} - \textcolor{#FF8C00}{r_{2,1}}) \\ &+ \green{\beta} ( ~ \blue{v_L(t-\tau_{2L}(t))} - v_2(t) ~ ) +\green{\gamma} ( ~ \blue{x_L(t-\tau_{2L}(t))} + \textcolor{#9932CC}{v_L(t-\tau_{2L}) \tau_{2L}(t)} - x_2(t) - \textcolor{#FF8C00}{r_{2,L}} ~ ) \\ =& \green{\alpha} (~V_2(h_{2,1}(t)) - v_2(t)~) + \green{\beta} (~\blue{v_1(t-\tau)} - v_2(t)~) + \green{\gamma} (~\blue{x_1(t-\tau)} - x_2(t)~) + \textcolor{#9932CC}{v_L (t) \tau} - \textcolor{#FF8C00}{r_{2,1}}) \\ &+ \green{\beta} ( ~ \blue{v_L(t)} - v_2(t) ~ ) +\green{\gamma} ( ~ \blue{x_L(t-\tau)} + \textcolor{#9932CC}{v_L(t) \tau} - x_2(t) - \textcolor{#FF8C00}{r_{2,L}} ~ ) \end{aligned}u2​(t)==​α( V2​(h2,1​(t))−v2​(t) )+β( v1​(t−τ21​(t))−v2​(t) )+γ( x1​(t−τ21​(t))−x2​(t) )+vL​(t−τ2L​(t))τ21​(t)−r2,1​)+β( vL​(t−τ2L​(t))−v2​(t) )+γ( xL​(t−τ2L​(t))+vL​(t−τ2L​)τ2L​(t)−x2​(t)−r2,L​ )α( V2​(h2,1​(t))−v2​(t) )+β( v1​(t−τ)−v2​(t) )+γ( x1​(t−τ)−x2​(t) )+vL​(t)τ−r2,1​)+β( vL​(t)−v2​(t) )+γ( xL​(t−τ)+vL​(t)τ−x2​(t)−r2,L​ )​

u3(t)=⋯u_3(t) =\cdots u3​(t)=⋯⋮\vdots ⋮

接下来继续往下看。
Vi(hi,j(t))=V1+V2tanh(C1(hi,j(t))−C2)(14)V_i(h_{i,j}(t)) = V_1 + V_2 ~ \text{tanh}(C_1 (h_{i,j}(t)) - C_2) \tag{14}Vi​(hi,j​(t))=V1​+V2​ tanh(C1​(hi,j​(t))−C2​)(14)

hi,j(t)=(xj(t)−xi(t)−(i−j)lc)/(i−j)(15)h_{i,j}(t) = (x_j(t) - x_i(t) - (i-j)l_c) / (i-j) \tag{15}hi,j​(t)=(xj​(t)−xi​(t)−(i−j)lc​)/(i−j)(15)

代入文章中的仿真部分的参数，我们可以得到：

h2,1=(x1(t)−x2(t)−(2−1)lc)/(2−1)h_{2,1} = ( x_1(t)-x_2(t) - (2-1) l_c ) / (2-1) h2,1​=(x1​(t)−x2​(t)−(2−1)lc​)/(2−1)V2(h2,1(t))=V1+V2∗tanh(C1∗h2,1(t)−C2)V_2(\red{h_{2,1}(t)}) = V_1 + V_2 * \text{tanh}(C_1 * \red{h_{2,1}(t)} - C_2)V2​(h2,1​(t))=V1​+V2​∗tanh(C1​∗h2,1​(t)−C2​)

h3,2=(x2(t)−x3(t)−(3−2)lc)/(3−2)h_{3,2} = ( x_2(t)-x_3(t) - (3-2) l_c ) / (3-2) h3,2​=(x2​(t)−x3​(t)−(3−2)lc​)/(3−2)V3(h3,2(t))=V1+V2∗tanh(C1∗h3,2(t)−C2)V_3(\red{h_{3,2}(t)}) = V_1 + V_2 * \text{tanh}(C_1 * \red{h_{3,2}(t)} - C_2)V3​(h3,2​(t))=V1​+V2​∗tanh(C1​∗h3,2​(t)−C2​)

4 Convergence Analysis

4.1 Convergence Analysis

4.2 Convergence Speed Analysis

5 Numerical Experiments

Nodes:

  1. one leader
  2. nine follower

Three conditions:

  1. no time delays
  2. heterogeneous time delays
  3. homogeneous time delays

5.1 Simulation Setting

Sampling interval Δt=0.01s\Delta t = 0.01sΔt=0.01s

The initial positions are x(0)=[01020.531.5435567.580.594108]Tx(0) = \left[\begin{matrix} 0 & 10 & 20.5 & 31.5 & 43 & 55 & 67.5 & 80.5 & 94 & 108 \end{matrix}\right]^\text{T}x(0)=[0​10​20.5​31.5​43​55​67.5​80.5​94​108​]T m on a lane.

The initial velocities are set as =[7777777777]T= \left[\begin{matrix} 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 \end{matrix}\right]^\text{T}=[7​7​7​7​7​7​7​7​7​7​]T m/s.

The heterogeneous time delays are selected as τ=[00.150.180.190.200.210.220.230.270.30]T\tau = \left[\begin{matrix} 0 & 0.15 & 0.18 & 0.19 & 0.20 & 0.21 & 0.22 & 0.23 & 0.27 & 0.30 \end{matrix}\right]^\text{T}τ=[0​0.15​0.18​0.19​0.20​0.21​0.22​0.23​0.27​0.30​]T.

The homogeneous time delays are set as 0.20s0.20s0.20s.

The desired gaps between the followers and the leader are set as [45403530252015105]T\left[\begin{matrix} 45 & 40 & 35 & 30 & 25 & 20 & 15 & 10 & 5 \end{matrix}\right]^\text{T}[45​40​35​30​25​20​15​10​5​]T.

relevant parameters: α=3.5s−1\alpha = 3.5 s^{-1}α=3.5s−1

5.2 Discussion of Results

5.3 Comparison to Existing Approaches

ZGC Codes

zgc.1 Leader’s states

zgc.2 Leader and some followers’ states

zgc.3 Leader and all followers’ states (no time delays)

zgc.4 Leader and all followers’ states (heterogeneous time delays)

zgc.5 Leader and all followers’ states (homogeneous time delays)

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