本文主要是介绍MPC模型预测控制（四）-MATLAB跟踪圆，希望对大家解决编程问题提供一定的参考价值，需要的开发者们随着小编来一起学习吧！
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 vehicled = 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, thetaz_ref(k,:) = traj(idx, :); refs = zeros(params.Np+1, 3);for i = 1:params.Np+1theta = 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 circleif (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;endparams.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;endyalmip('clear')% The predicted state and control variablesz_mpc = sdpvar(params.Np+1, params.nstates);u_mpc = sdpvar(params.Np, params.ninputs);% Reset constraints and objective functionconstraints = [];J = 0;for i = 1:params.Npif i < params.NpJ = J + (z_mpc(i,:)-refs(i,:)) * params.Qf * (z_mpc(i,:)-refs(i,:))' + ...u_mpc(i,:) * params.R * u_mpc(i,:)';% Model constraintsconstraints = [constraints, ...z_mpc(i+1,:)' == params.A * z_mpc(i,:)' + params.B * u_mpc(i,:)'];% Input constraintsconstraints = [constraints, ...-pi/4 <= u_mpc(i) <= pi/4];elseJ = J + ...(z_mpc(i,:)-refs(i,:)) * params.Q_N * (z_mpc(i,:)-refs(i,:))' + ...u_mpc(i,:) * params.R * u_mpc(i,:)';% Model constraintsconstraints = [constraints, ...z_mpc(i+1,:)' == params.A * z_mpc(i,:)' + params.B * u_mpc(i,:)'];% Input constraintsconstraints = [constraints, ...-pi/4 <= u_mpc(i) <= pi/4];endend% terminal costJ = J + (z_mpc(params.Np+1, :) - refs(params.Np+1,:)) * params.Qf * (z_mpc(params.Np+1, :)-refs(params.Np+1,:))';% terminal constraintsconstraints = [constraints, ...z_mpc(params.Np+1, 3) - refs(params.Np+1, 3) == 0];assign(z_mpc(1,:), z(k,:));% Optionsops = sdpsettings('solver', 'quadprog');% Optimizeoptimize([constraints, z_mpc(1,:) == z(k,:)], J, ops);% Take the first predicted input u(k) = value(u_mpc(1));% simulate the vehiclez(k+1,:) = vehicle_model(z(k,:), u(k), params);plot(traj(:,1), traj(:,2))hold onplot(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 equalaxis([-2 2 -2 2])drawnowtitle('Trajectory Tracking')hold offend
% 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 offsubplot(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。
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