MPC模型预测控制（四）-MATLAB跟踪圆

参考https://github.com/Janani-Mohan

%% YALMIP : Circular Trajectory Tracking using MPC 
clc;
clear;
close all;
yalmip('clear')

%% MPC Parameters definition
% Model Parameters
params.Ts = 0.01;   % Sampling time 
params.nstates = 3; % No of States in the system [x, y,theta]
params.ninputs = 1; % No of Inputs in the system [delta_f], steering angle
params.l_f = 0.17; % Distance from the CG to the front wheel
params.l_r = 0.16; % Distance from the CG to the rear wheel
params.l_q = params.l_r / (params.l_r + params.l_f);

% Control Parameters
params.Np = 10;   % The horizon 30
params.Q = [1 0 0; 0 1 0; 0 0 1]; % The Q matrix for running cost for error in trajectory tracking
params.Q_N = [1 0 0; 0 1 0; 0 0 1];% The Q matrix for terminal cost for error in trajectory tracking
params.R = 1; %  The R matrix for running cost for input

% Initial conditions for the state of the vehicle       
params.x0 = 1.5;  %1.5
params.y0 = 0;  
params.v0 = 1; 
params.psi0 = pi/2; 

% Number of iterations
params.N = 500;%1000

% Every point on the circle is a reference to be followed by the vehicle
% x0 and y0 are the coordinates of the circle's center,r is its radius.
trajectory_params.x0 = 0;
trajectory_params.y0 = 0;
trajectory_params.r = 1.5;

%% Simulation environment
% Initialization
z(1,:)  = [params.x0, params.y0, params.psi0]; % State representation
u(1) = 0; % Control Input

% Generate the trajectory to be followed
traj = trajectory_circular(trajectory_params);

% Obtain the symbolic form of the matrices A, B
syms xkn ykn psikn xk yk psik Ts lr lf delta
[symbolic_linear_A, symbolic_linear_B] = Symbolic_Kinematic_Model(params);

k = 0;
while (k < params.N)
  k = k+1;  
%   fprintf("Iteration number = %d ",k);
  
  % The distances of all the points in the trajectory to the vehicle
  d = sqrt((traj(:,1)-z(k,1)).^2 + (traj(:,2)-z(k,2)).^2);
  
  % The index of the point that is the closest to the vehicle
  [~, idx] = min(d);
  
  % The first reference point for x, y, theta
  z_ref(k,:) = traj(idx, :);   
  
  refs = zeros(params.Np+1, 3);
  
  for i = 1:params.Np+1

       theta = z_ref(k,3) +(params.v0 / trajectory_params.r) * params.Ts * i;
        x = trajectory_params.x0 + trajectory_params.r * cos(theta - pi/2);
        y = trajectory_params.y0 + trajectory_params.r * sin(theta - pi/2);

    
    refs(i,:) = [x, y, theta];
  end
%   plot(refs(:,1),refs(:,2));
% The first case
if (k == 1)
  params.A = subs(symbolic_linear_A, [Ts lr lf psik delta], [params.Ts params.l_r params.l_f z(k,3) 0]);
  params.B = subs(symbolic_linear_B, [Ts lr lf psik delta], [params.Ts params.l_r params.l_f z(k,3) 0]);
  params.A = double(params.A);
  params.B = double(params.B);
end
% The remaining cases
if (k > 1)
  % Completing the circle
  if (z_ref(k,3) < z_ref(k-1,3))    
    refs(:,3) = refs(:,3) + 2*pi;
    z_ref(k,3) = z_ref(k,3) + 2*pi;
    traj(idx,3) = traj(idx,3) + 2*pi;
  end
  params.A = subs(symbolic_linear_A, [Ts lr lf psik delta], [params.Ts params.l_r params.l_f z(k,3) u(k-1)]);
  params.B = subs(symbolic_linear_B, [Ts lr lf psik delta], [params.Ts params.l_r params.l_f z(k,3) u(k-1)]);
  params.A = double(params.A);
  params.B = double(params.B);
end

% Ensure stability : Ricatti Equation Solver is used to ensure stability 
% finding stable Q matrix
  [params.Qf,~,~,err] = dare(params.A, params.B, params.Q, params.R);
  if (err == -1 || err == -2)
    params.Qf = params.Q;
  end
 yalmip('clear')

  % The predicted state and control variables
  z_mpc = sdpvar(params.Np+1, params.nstates);
  u_mpc = sdpvar(params.Np, params.ninputs);

  % Reset constraints and objective function
  constraints = [];
  J = 0;

  for i = 1:params.Np
    
    if i < params.Np
      J = J + (z_mpc(i,:)-refs(i,:)) * params.Qf * (z_mpc(i,:)-refs(i,:))' +  ...
        u_mpc(i,:) * params.R * u_mpc(i,:)';


      % Model constraints
      constraints = [constraints, ...
        z_mpc(i+1,:)' == params.A * z_mpc(i,:)' + params.B * u_mpc(i,:)'];

      % Input constraints
      constraints = [constraints, ...
        -pi/4 <= u_mpc(i) <= pi/4];
    else
      J = J + ...
        (z_mpc(i,:)-refs(i,:)) * params.Q_N * (z_mpc(i,:)-refs(i,:))' +  ...
        u_mpc(i,:) * params.R * u_mpc(i,:)';


      % Model constraints
      constraints = [constraints, ...
        z_mpc(i+1,:)' == params.A * z_mpc(i,:)' + params.B * u_mpc(i,:)'];

      % Input constraints
      constraints = [constraints, ...
        -pi/4 <= u_mpc(i) <= pi/4];
    end
  end

  
  %  terminal cost
  J = J + (z_mpc(params.Np+1, :) - refs(params.Np+1,:)) * params.Qf * (z_mpc(params.Np+1, :)-refs(params.Np+1,:))';

  % terminal constraints
    constraints = [constraints, ...
       z_mpc(params.Np+1, 3) - refs(params.Np+1, 3) == 0];
        
  assign(z_mpc(1,:), z(k,:));

  % Options
  ops = sdpsettings('solver', 'quadprog');

  % Optimize
  optimize([constraints, z_mpc(1,:) == z(k,:)], J, ops);

  % Take the first predicted input 
  u(k) = value(u_mpc(1));

  % simulate the vehicle
  z(k+1,:) = vehicle_model(z(k,:), u(k), params);
  plot(traj(:,1), traj(:,2))
  hold on
  plot(z(2:end,1), z(2:end,2));
  plot(refs(:,1), refs(:,2), '*', 'Color','b')
  plot(z(k,1), z(k,2), '*', 'Color', 'r')
  axis equal
  axis([-2 2 -2 2])
  drawnow
  title('Trajectory Tracking')
  hold off

end
% Plot state deviations and inputs
figure
subplot(2,2,1)
plot(traj(:,1), traj(:,2),'-*r')
hold on
plot(z(:,1), z(:,2))
title('Trajectory Tracking')
legend ('Given Trajectory','My Trajectory','location','southeast');
axis equal
hold off

subplot(2,2,2)
plot((z(1:end-1, 1) - z_ref(:,1)).^2 + (z(1:end-1, 2) - z_ref(:,2)).^2)
xlabel('No of iterations');
ylabel('Deviation from the reference trajectory');
title('Distance deviation')


subplot(2,2,3)
plot((z_ref(:,3) - z(1:end-1,3))*180/pi)
xlabel('No of iterations');
ylabel('Deviation from the reference trajectory');
title('Orientation deviation')

figure
plot(u * 180 / pi)
xlabel('No of iterations');
ylabel('Variation in steering angle');
title('Input Steering Angle')
% 
% %% Trajectory Generator
% 
% %Creates the Trajectory to Follow
% function traj = trajectory_circular(trajectory_params)
% % Center of the circular trajectory 
%   x0 = trajectory_params.x0;
%   y0 = trajectory_params.y0;
%   r = trajectory_params.r;
% 
% % Creates the circular trajectory and corresponding angle
%   for i = 0:1:360
%       x = x0 + r*cos(pi * i / 180);
%       y = y0 + r*sin(pi * i / 180);
%       theta = pi - atan2(x - x0, y - y0);
% 
%       traj(i+1, 1) = x;
%       traj(i+1, 2) = y;
%       traj(i+1, 3) = theta;
%   end
%   
% end
% 
% %% Kinematic Bicycle Model
% 
% function [A, B] = Symbolic_Kinematic_Model(params)
% 
%   syms xkn ykn psikn xk yk psik Ts lr lf delta
%   
%   xkn = xk + Ts*params.v0*cos(psik + atan((lr/(lr+lf))*tan(delta)));
%   ykn = yk + Ts*params.v0*sin(psik + atan((lr/(lr+lf))*tan(delta)));
%   psikn = psik + Ts*(params.v0/lr)*sin(atan((lr/(lr+lf))*tan(delta)));
% 
%   A(1,1) = diff(xkn, xk);
%   A(1,2) = diff(xkn, yk);
%   A(1,3) = diff(xkn, psik);
% 
%   A(2,1) = diff(ykn, xk);
%   A(2,2) = diff(ykn, yk);
%   A(2,3) = diff(ykn, psik);
% 
%   A(3,1) = diff(psikn, xk);
%   A(3,2) = diff(psikn, yk);
%   A(3,3) = diff(psikn, psik);
% 
%   B(1,1) = diff(xkn, delta);
%   B(2,1) = diff(ykn, delta);
%   B(3,1) = diff(psikn, delta);
% 
% end
% 
% %% Vehicle Model
% function z_new = vehicle_model(z_curr, u_curr, params)
%     
%   delta = u_curr;
%   
%   % vehicle dynamics equations
%   z_new(1) = z_curr(1) + (params.v0*cos(z_curr(3)+atan((params.l_r/(params.l_r+params.l_f))*tan(delta))))*params.Ts;
%   z_new(2) = z_curr(2) + (params.v0*sin(z_curr(3)+atan((params.l_r/(params.l_r+params.l_f))*tan(delta))))*params.Ts;
%   z_new(3) = z_curr(3) + (params.v0 / params.l_r * sin(atan((params.l_r/(params.l_r+params.l_f))*tan(delta))))*params.Ts;
% 
% end

注释掉的有三个函数：轨迹函数，运动自行车模型函数，实际的运动结果函数。需要放到同文件夹里生成.m的函数。这里没有一一复制。这个需要安装yawmip，免费的。参考链接：https://blog.csdn.net/XIXI_0921/article/details/47981869

希望有志之士能把这个函数改为跟踪8字形。我试了下没成功，用NMPC的程序倒是能跟踪上。大家有问题就交流。对程序里有不理解的也可以评论问我，我会做相应解答。