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文章目录

• 前言
• 一、pandas是什么？
• 二、源码分析
• • 1.初始参数设定
  • 2.算法流程
• 总结

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前言

文章: Distributed model predictive control for heterogeneous vehicle platoons under unidirectional topologies
作者: Zheng, Yang, Shengbo Eben Li, Keqiang Li, Francesco Borrelli, and J. Karl Hedrick.
Journal: IEEE Transactions on Control Systems Technology 25, no. 3 (2017): 899-910.

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一、pandas是什么？

示例：pandas 是基于NumPy 的一种工具，该工具是为了解决数据分析任务而创建的。

二、源码分析

1.初始参数设定

文章中设置一辆领航车与七辆跟随车，跟随车的参数在文件PlatoonParameter.mat中。PlatoonParameter.mat中包含时间长度(10)、采样间隔(Time_step=0.1)、总步数(Num_step=100)、异构车辆的数量(7)、各个车的内部参数(质量、车轮半径、惯性滞后常数、摩擦系数、传动效率、空气阻力、物理约束等)、重力加速度等。
载入PlatoonParameter.mat：

clc;clear;close all;
load PlatoonParameter.mat  % This set of parameters was used in the paper

领航车辆参数初始化，理想车距设定为20，参数为位置、速度、加速度：

% Leading vehicle
d  = 20;                                % Desired spacing
a0 = zeros(Num_step,1); 
v0 = zeros(Num_step,1); 
x0 = zeros(Num_step,1);

跟随车辆初始化，参数为位置、速度、扭矩、理想扭矩以及性能指标和优化器停止标志：

%% Initial Virables 
Postion  = zeros(Num_step,Num_veh);     % postion of each vehicle;
Velocity = zeros(Num_step,Num_veh);     % velocity of each vehicle;
Torque   = zeros(Num_step,Num_veh);     % Braking or Tracking Torque of each vehicle;
U        = zeros(Num_step,Num_veh);     % Desired Braking or Tracking Torque of each vehicle;

Cost    = zeros(Num_step,Num_veh);      % Cost function
Exitflg = zeros(Num_step,Num_veh);      % Stop flag - solvers

领航车辆按预期的速度行驶，跟随车辆初始情况设定：

% Transient process of leader, which is given in advance
v0(1) = 20; 
a0(1/Tim_step+1:2/Tim_step) = 2; 
for i = 2:Num_step
    v0(i) = v0(i-1)+a0(i)*Tim_step; 
    x0(i) = x0(i-1)+v0(i)*Tim_step;    
end

% Zero initial error for the followers
for i = 1:Num_veh
    Postion(1,i)  = x0(1)-i*d;
    Velocity(1,i) = 20;             
    Torque(1,i)   = (Mass(i)*g*f + Ca(i)*Velocity(1,i)^2)*R(i)/Eta;
end

2.算法流程

本文采用分布式预测控制算法，预测时域Np = 20。对于每一步的滚动优化，基于上一时刻的预测进行当前的优化：

% Distributed MPC assumed state
Np = 20;                      % Predictive horizon
Pa = zeros(Np,Num_veh);       % Assumed postion of each vehicle;
Va = zeros(Np,Num_veh);       % Assumed velocity of each vehicle;
ua = zeros(Np,Num_veh);       % Assumed Braking or Tracking Torque input of each vehicle;

Pa_next = zeros(Np+1,Num_veh);  % 1(0): Assumed postion of each vehicle at the newt time step;
Va_next = zeros(Np+1,Num_veh);  % Assumed velocity of each vehicle at the newt time step;
ua_next = zeros(Np+1,Num_veh);  % Assumed Braking or Tracking Torque of each vehicle at the newt time step;

首个时刻初始化，第一个坐标代表时刻，第二个坐标代表车辆编号：

% Initialzie the assumed state for the first computation: constant speed
for i = 1:Num_veh
    ua(:,i) = Torque(1,i);
    Pa(1,i) = Postion(1,i);                % The first point should be interpreted as k = 0 (current state)
    Va(1,i) = Velocity(1,i);
    Ta(1,i) = Torque(1,i);
    for j = 1:Np
        [Pa(j+1,i),Va(j+1,i),Ta(j+1,i)] = VehicleDynamic(ua(j,i),Tim_step,Pa(j,i),Va(j,i),Ta(j,i),Mass(i),R(i),g,f,Eta,Ca(i),Tao(i));
    end    
end

tol_opt = 1e-5;
options = optimset('Display','off','TolFun', tol_opt, 'MaxIter', 2000,...
                'LargeScale', 'off', 'RelLineSrchBnd', [], 'RelLineSrchBndDuration', 1);

从第二个时刻开始，对每一步进行优化，程序中?代表车辆编号：

    Vehicle_Type = [Mass(?),R(?),g,f,Eta,Ca(?),Tao(?)];                 % the vehicle parameters ： Mass,R,g,f,Eta,Ca,Tao, 
    X0 = [Postion(i-1,?),Velocity(i-1,?),Torque(i-1,?)];                % the vehicle variable in the last time
    Pd = zeros(Np+1,1);  Vd = zeros(Np+1,1);                      
    Xdes = [Pd,Vd];  
    Xa = [Pa(:,?),Va(:,?)];                                         
    Xnfa = [Pa(:,?) - d, Va(:,?)];                                         
   
    u0 = ua(:,?);   
    A = [];b = []; Aeq = []; beq = [];                                   
    lb = Torquebound(2,1)*ones(Np,1); ub = Torquebound(2,2)*ones(Np,1);         
    Pnp = Xnfa(end,1); Vnp = Xnfa(end,2);  
    Xend(i,?) = Pnp; Vend(i,?) = Vnp; Tnp = (Ca(?)*Vnp.^2 + Mass(?)*g*f)/Eta*R(?);
    % MPC - subporblem for vehicle ?
    [u, Cost(i,?), Exitflg(i,?), output] = fmincon(@(u) Costfunction( Np, Tim_step, X0 ,u, Vehicle_Type,Q,Xdes,R,F,Xa,G,Xnfa), ...
        u0, A, b, Aeq, beq, lb, ub, @(u) Nonlinearconstraints(Np, Tim_step, X0, u, Vehicle_Type,Pnp,Vnp,Tnp),options); 

将序列u的第一个值作用到系统中，更新状态：

    % state involves one step
    U(i,?) = u(1);
    [Postion(i,?),Velocity(i,?),Torque(i,?)] = VehicleDynamic(U(i,?),Tim_step,Postion(i-1,?),Velocity(i-1,?),Torque(i-1,?),Mass(?),R(?),g,f,Eta,Ca(?),Tao(?));  

根据对优化问题求解得到的u更新下个时刻的控制估计ua，ua前Np-1个值对应于u的后Np-1个结果；对于ua的最后一个元素由预期设定的方式决定，即与当前时刻的终端的最优速度 v ∗ ( N p ) v^*(Np) v∗(Np)有关；得到ua的控制序列之后，可以对下个时刻的Pa和Va的预测序列。

    % Update assumed state
    Temp = zeros(Np+1,3);
    Temp(1,:) = [Postion(i,?),Velocity(i,?),Torque(i,?)]; 
    ua(1:Np-1,?) = u(2:Np);
    for j = 1:Np-1
        [Temp(j+1,1),Temp(j+1,2),Temp(j+1,3)] = VehicleDynamic(ua(j,?),Tim_step,Temp(j,1),Temp(j,2),Temp(j,3),Mass(?),R(?),g,f,Eta,Ca(?),Tao(?));
    end 
    ua(Np,?) = (Ca(?)*Temp(Np,2).^2 + Mass(?)*g*f)/Eta*R(?);
    [Temp(Np+1,1),Temp(Np+1,2),Temp(Np+1,3)] = VehicleDynamic(ua(Np,?),Tim_step,Temp(Np,1),Temp(Np,2),Temp(Np,3),Mass(?),R(?),g,f,Eta,Ca(?),Tao(?));
    Pa_next(:,2) = Temp(:,1);
    Va_next(:,2) = Temp(:,2);
     %% Update assumed data
    Pa = Pa_next;
    Va = Va_next;

其中，fmincon()用于寻找非线性约束最小值，将序列u的第一个值作用到系统中，更新状态；同时

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总结

提示：这里对文章进行总结：
例如：以上就是今天要讲的内容，本文仅仅简单介绍了pandas的使用，而pandas提供了大量能使我们快速便捷地处理数据的函数和方法。