Y. Zhao and T. Wang, “Distributed control for high-speed trains movements,” 2017 29th Chinese Control And Decision Conference (CCDC), 2017, pp. 7591-7596, doi: 10.1109/CCDC.2017.7978561.

文章目录

Abstract
1 Introduction
2 Preliminaries
3 Distributed control for a single high-speed train
- 3.1 The model of a high-speed train
- 3.2 The distributed controller design
- 3.3 Example 1
4 Conclusion

Abstract

1 Introduction

2 Preliminaries

3 Distributed control for a single high-speed train

3.1 The model of a high-speed train

假设有 n1n_1n1 辆车，把这些车分为三类：首车，中间车，尾车。动力学模型分别如下：
{x˙i(t)=vi,i=1,2,⋯,n1m1v˙1(t)=u1−k(x1−x2−l)−(c0+c1v1+c2v12)m1miv˙i(t)=ui+k(xi−1−xi−l)−k(xi−xi+1−l)−(c0+c1vi+c2vi2)mi,i=2,3,⋯,n1−1mn1v˙n1(t)=un1+k(xn1−1−xn1−l)−(c0+c1vn1+c2vn12)mn1(1)\left\{\begin{aligned} \dot{x}_i(t) &= v_i, ~~~~ i = 1,2,\cdots, n_1 \\ m_1 \dot{v}_1(t) &= u_1 - k(x_1 - x_2 - l) - (c_0 + c_1 v_1 + c_2 v_1^2) m_1 \\ m_i \dot{v}_i(t) &= u_i + k(x_{i-1} - x_{i} - l) - k(x_{i} - x_{i+1} - l) - (c_0 + c_1 v_i + c_2 v_i^2) m_i, ~~~~ i=2,3,\cdots,n_1-1 \\ m_{n_1} \dot{v}_{n_1}(t) &= u_{n_1} + k(x_{n_1-1} - x_{n_1} - l) - (c_0 + c_1 v_{n_1} + c_2 v_{n_1}^2) m_{n_1} \end{aligned}\right. \tag{1}⎩⎪⎪⎪⎪⎨⎪⎪⎪⎪⎧x˙i(t)m1v˙1(t)miv˙i(t)mn1v˙n1(t)=vi, i=1,2,⋯,n1=u1−k(x1−x2−l)−(c0+c1v1+c2v12)m1=ui+k(xi−1−xi−l)−k(xi−xi+1−l)−(c0+c1vi+c2vi2)mi, i=2,3,⋯,n1−1=un1+k(xn1−1−xn1−l)−(c0+c1vn1+c2vn12)mn1(1)

符号表示为：
xix_ixi：表示位置，
mim_imi：表示质量，
lll：表示原始弹性长度。

当达到平衡状态后，用以下符号进行表示。

首先速度均达到一致，公式表示为
vˉ1=vˉ2=⋯=vˉn1=vr\bar{v}_1 = \bar{v}_2 = \cdots = \bar{v}_{n_1} = v_rvˉ1=vˉ2=⋯=vˉn1=vr

距离达到均等间隔，公式表示为
xˉi−1−xˉi=l\bar{x}_{i-1} - \bar{x}_i = lxˉi−1−xˉi=l

xˉi=vˉit\bar{x}_i = \bar{v}_i txˉi=vˉit

平衡状态时的控制输入为
uˉi=c0mi+c1mivr+c2mivr2(2)\bar{u}_i = c_0 m_i + c_1 m_i v_r + c_2 m_i v_r^2 \tag{2}uˉi=c0mi+c1mivr+c2mivr2(2)

这个公式相当于 Davis formula 和 Newton’s second law （uˉi=Ri/mi\bar{u}_i = R_i/m_iuˉi=Ri/mi）的变形。

接下来定义误差动力学方程。

令 x^i(t)=xi(t)−xˉi(t)\hat{x}_{i}(t) = {x}_{i}(t) - \bar{x}_{i}(t)x^i(t)=xi(t)−xˉi(t)，v^i(t)=vi(t)−vr(t)\hat{v}_{i}(t) = {v}_{i}(t) - {v}_{r}(t)v^i(t)=vi(t)−vr(t)，u^i(t)=ui(t)−uˉi(t)\hat{u}_{i}(t) = {u}_{i}(t) - \bar{u}_{i}(t)u^i(t)=ui(t)−uˉi(t)。整合公式（1）和（2）就可以得到如下误差动力学方程。

{x^˙i=v^i,i=1,2,⋯,n1miv^˙i=u^i+k∑j=1n1adij(x^j−x^i)−2c2mivrv^i−c1miv^i,i=2,3,⋯,n1−1(3)\left\{\begin{aligned} \dot{\hat{x}}_i &= \hat{v}_i, ~~~~ i = 1,2,\cdots, n_1 \\ m_i \dot{\hat{v}}_i &= \hat{u}_i + k \sum_{j=1}^{n_1} a_{dij} (\hat{x}_{j} - \hat{x}_{i}) - 2 c_2 m_i v_r \hat{v}_i - c_1 m_i \hat{v}_i, ~~~~ i=2,3,\cdots,n_1-1 \\ \end{aligned}\right. \tag{3}⎩⎪⎪⎨⎪⎪⎧x^˙imiv^˙i=v^i, i=1,2,⋯,n1=u^i+kj=1∑n1adij(x^j−x^i)−2c2mivrv^i−c1miv^i, i=2,3,⋯,n1−1(3)

符号表示为：
adij>0a_{dij} > 0adij>0：表示 i,ji,ji,j 是一组，否则为 adij=0a_{dij} = 0adij=0。

3.2 The distributed controller design

设计控制律：
u^i=ui(t)−uˉi=u1i(t)+u2i(t)+u3i(t)+u4i(t)=−kddi(xi−xˉi)−miδidvi(vi(t)−vr)+miδi∑j=in1avij(vj(t)−vi(t))−mi∑j=1n1adij∇xiUij(4)\begin{aligned} \hat{u}_i &= u_i(t) - \bar{u}_i \\ &= u_{1i}(t) + u_{2i}(t) + u_{3i}(t) + u_{4i}(t) \\ &= -k d_{di} (x_i - \bar{x}_i) \\ &- m_i \delta_i d_{vi} (v_i(t) - v_r) \\ &+ m_i \delta_i \sum_{j=i}^{n_1} a_{vij} (v_j(t) - v_i(t)) \\ &- m_i \sum_{j=1}^{n_1} a_{dij} \nabla_{xi} U_{ij} \end{aligned} \tag{4}u^i=ui(t)−uˉi=u1i(t)+u2i(t)+u3i(t)+u4i(t)=−kddi(xi−xˉi)−miδidvi(vi(t)−vr)+miδij=i∑n1avij(vj(t)−vi(t))−mij=1∑n1adij∇xiUij(4)

控制律分别由以下四部分组成：

u1i(t)=−kddi(xi−xˉi)u_{1i}(t) = -k d_{di} (x_i - \bar{x}_i)u1i(t)=−kddi(xi−xˉi)

u2i(t)=−miδidvi(vi(t)−vr)u_{2i}(t) = - m_i \delta_i d_{vi} (v_i(t) - v_r)u2i(t)=−miδidvi(vi(t)−vr)

u3i(t)=+miδi∑j=in1avij(vj(t)−vi(t))u_{3i}(t) = + m_i \delta_i \sum_{j=i}^{n_1} a_{vij} (v_j(t) - v_i(t))u3i(t)=+miδij=i∑n1avij(vj(t)−vi(t))

u4i(t)=−mi∑j=1n1adij∇xiUiju_{4i}(t) = - m_i \sum_{j=1}^{n_1} a_{dij} \nabla_{xi} U_{ij}u4i(t)=−mij=1∑n1adij∇xiUij

3.3 Example 1

4 Conclusion

【Paper】2017_Distributed control for high-speed trains movements相关推荐

【Paper】2019_Consensus Control of Multiple AUVs Recovery System Under Switching Topologies and Time D
Zhang W, Zeng J, Yan Z, et al. Consensus control of multiple AUVs recovery system under switching to ...
【Paper】2021_Consensus Control of Leader-Following Multi-Agent Systems in Directed Topology
参考文献格式: Qinglai Wei, Xin Wang, Xiangnan Zhong and Naiqi Wu, "Consensus Control of Leader-Follow ...
【Paper】2007_Consensus control for a class of networks of dynamic agents 二阶静态一致性
文章目录 3. Control Protocol and Network Dynamics 4. Network with Fixed Topology 定理 1 Remark 1 Lemma 2 证 ...
【Paper】2013_Cooperative control of multi-agent systems 二阶动态一致性
Lewis F L, Zhang H, Hengster-Movric K, et al. Cooperative control of multi-agent systems: optimal an ...
【Paper】2006_Time-Optimal Control of a Hovering Quad-Rotor Helicopter
Lai L C, Yang C C, Wu C J. Time-optimal control of a hovering quad-rotor helicopter[J]. Journal of I ...
【Paper】2017_水下潜航器编队海洋勘测的协调控制方法研究
友情链接:[paper]2019_Consensus Control of Multiple AUVs Recovery System Under Switching Topologies and T ...
【Paper】2015_El H_Decentralized Control Architecture for UAV-UGV Cooperation
Decentralized Control Architecture for UAV-UGV Cooperation 1 Introduction 2 Problem Statement and Ar ...
【Paper】2021_Distributed Consensus Tracking of Networked Agent Systems Under Denial-of-Service Attack
Y. Wan, G. Wen, X. Yu and T. Huang, "Distributed Consensus Tracking of Networked Agent Systems ...
【Paper】2003_Consensus Problems in Networks of Agents with Switching Topology and Time-Delays
此篇文章主要在于仿真代码,关于文献的解释请参考 [Paper]2003_Murr_Consensus Problems in Networks of Agents with Switching Top ...

【Paper】2017_Distributed control for high-speed trains movements