文章目录
- 2 Preliminaries and Problem Statement
- 3 Nonlinear Consensus Algorithm
- 4 Convergence Analysis
- 5 Numerical Experiments
2 Preliminaries and Problem Statement
2.1 Graph Theory
2.2 Mathematical Preliminaries
2.3 Consensus Problem Statement
系统采用的是二阶积分器模型：
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\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 h c h_c hc 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
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\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} (v_j(t-\tau_{ij}(t)) - v_i(t)) \\ &+ \green{\gamma} (x_j(t-\tau_{ij}(t)) - x_i(t)) \\ &+ \green{v_L} (t-\tau_{iL}(t)) \tau_{ij}(t) - r_{i,j}) ] \\ &+ \red{k_{i,L}} [\beta (v_L(t-\tau_{iL}(t)) - v_i(t)) \\ &+ \gamma (x_L(t-\tau_{iL}(t)) + v_L(t-\tau_{iL}) \tau_{iL}(t) - x_i(t) - r_{i,L})] \end{aligned}
ui(t)=j=1∑nai,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)]
4 Convergence Analysis
4.1 Convergence Analysis
4.2 Convergence Speed Analysis
5 Numerical Experiments
Nodes:
- one leader
- nine follower
Three conditions:
- no time delays
- heterogeneous time delays
- homogeneous time delays
5.1 Simulation Setting
Sampling interval Δ t = 0.01 s \Delta t = 0.01s Δt=0.01s
The initial positions are x ( 0 ) = [ 0 10 20.5 31.5 43 55 67.5 80.5 94 108 ] T x(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)=[01020.531.5435567.580.594108]T m on a lane.
The initial velocities are set as = [ 7 7 7 7 7 7 7 7 7 7 ] T = \left[\begin{matrix} 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 \end{matrix}\right]^\text{T} =[7777777777]T m/s.
The heterogeneous time delays are selected as τ = [ 0 0.15 0.18 0.19 0.20 0.21 0.22 0.23 0.27 0.30 ] 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=[00.150.180.190.200.210.220.230.270.30]T.
The homogeneous time delays are set as 0.20 s 0.20s 0.20s.
The desired gaps between the followers and the leader are set as [ 45 40 35 30 25 20 15 10 5 ] T \left[\begin{matrix} 45 & 40 & 35 & 30 & 25 & 20 & 15 & 10 & 5 \end{matrix}\right]^\text{T} [45403530252015105]T.
relevant parameters: α = 3.5 s − 1 \alpha = 3.5 s^{-1} α=3.5s−1