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mpc_sim_gurobi.m
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mpc_sim_gurobi.m
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clc
clear variables
close all
disp_curr_exec_time = true;
save_data = true;
save_name = 'mpc_sim_gurobi_1.mat'; path = 'sim_data/';
%% Plant data
nx = 12;
nu = 4;
Ts = 20e-03; % Discrete time step
T_int = 1e-03; % Plant model integration time step
%% MPC data
% Horizons
Np = 40;
Nc = 10;
Tp = 20e-03; % Prediction time step
% Weighting matrices
Q = diag([1e02, 1e02, 1e03, 1e-03, 1e-03, 1e02, 1e-03, 1e-03, 1e-03, 1e-03, 1e-03, 1e-03]);
R = diag([1e-04, 1e-04, 1e-04, 1e-04]);
Rd = diag([1e01, 1e01, 1e01, 1e01]);
P = 1e01*Q;
S = 1e06*eye(nx);
% Bounds
u_lb = [0, -2, -2, -2]';
u_ub = [15, 2, 2, 2]';
x_lb = [-1.5, -0.5, 0, deg2rad(-75), deg2rad(-75), deg2rad(-180), -inf, -inf, -inf, -inf, -inf, -inf]';
x_ub = [2.5, 2.5, 4, deg2rad(75), deg2rad(75), deg2rad(180), inf, inf, inf, inf, inf, inf]';
%% MPC formulation
% ===== Cost function =====
% Q_bar
cell_tmp = repmat({Q},Np-1,1); cell_tmp{Np} = P;
Q_bar = blkdiag(cell_tmp{:});
% R_bar, T
cell_tmp = repmat({R},Np,1);
R_bar_tilde = blkdiag(cell_tmp{:});
mat_tmp_1 = repmat(eye(nu),floor(Np/Nc),1);
mat_tmp_2 = repmat(eye(nu),floor(Np/Nc)+mod(Np,Nc),1);
cell_tmp = [repmat({mat_tmp_1},Nc-1,1); {mat_tmp_2}];
T = blkdiag(cell_tmp{:});
R_bar = T'*R_bar_tilde*T;
% R_delta_bar
mat_tmp = 2*eye(Np) + diag(-1*ones(Np-1,1),1) + diag(-1*ones(Np-1,1),-1);
mat_tmp(1,1) = 1; mat_tmp(Np,Np) = 1;
Rd_bar_tilde = kron(mat_tmp,Rd);
Rd_bar = T'*Rd_bar_tilde*T;
% H, f
H = blkdiag(Q_bar, R_bar + Rd_bar, S);
c = @(X_r) [-Q_bar'*X_r; zeros(nu*Nc,1); zeros(nx,1)];
% ===== Equality constraints (prediction model) =====
% Ap
mat_tmp = diag(ones(Np-1,1),-1);
Ap = @(A) kron(mat_tmp,A);
% Bp
mat_tmp = diag(ones(Np,1));
Bp = @(B) kron(mat_tmp,B)*T;
% cp
cp = @(x0, A, c) [A*x0 + c; repmat(c,Np-1,1)];
% A_eq, b_eq
cell_tmp = repmat({eye(nx)},Np,1);
A_eq = @(A, B) [blkdiag(cell_tmp{:}) - Ap(A), -Bp(B), zeros(nx*Np,nx)];
b_eq = @(x0, A, c) cp(x0, A, c);
% ===== Inequality constraints (bounds) =====
% Input
Eu = [eye(nu*Nc); -eye(nu*Nc)];
hu = [repmat(u_ub,Nc,1); repmat(-u_lb,Nc,1)];
% Slack
Ee = -eye(nx);
he = zeros(nx,1);
% State
Ex = [eye(nx*Np); -eye(nx*Np)];
Te = repmat(eye(nx),Np,1); Eep = [Te; Te];
hx = [repmat(x_ub,floor(0.5*Np),1); repmat(inf*ones(nx,1),Np-floor(0.5*Np),1);...
repmat(-x_lb,floor(0.5*Np),1); repmat(inf*ones(nx,1),Np-floor(0.5*Np),1)];
% A_ineq, B_ineq
A_ineq = [Ex, zeros(2*nx*Np, nu*Nc), -Eep;
zeros(2*nu*Nc, nx*Np), Eu, zeros(2*nu*Nc,nx);
zeros(nx, nx*Np), zeros(nx, nu*Nc), Ee];
b_ineq = [hx; hu; he];
% Remove unbounded ineq. constraints
ind_inf = (b_ineq == inf);
b_ineq(ind_inf) = []; A_ineq(ind_inf,:) = [];
%% MPC simulation
T = 5;
N = floor(T/Ts);
N_sim = N;
t = linspace(0, N_sim*Ts, N_sim+1);
% Reference traj. (lemniscate)
traj = @(t) [1 + 2*cos(sqrt(2)*2*pi*t);
1 + 2*sin(sqrt(2)*2*pi*t).*cos(sqrt(2)*2*pi*t);
2 + 0*t;
0*t;
0*t;
pi/4 + 0*t;
zeros(nx-6,1)*t];
x_r_traj = traj((0:Ts:T)/T);
x_mpc = zeros(nx,N_sim+1);
u_mpc = zeros(nu,N_sim);
prim_res = zeros(N_sim,1);
dual_res = zeros(N_sim,1);
% Initial state
x0 = [0, 0, 0, 0, 0, pi/4, 0, 0, 0, 0, 0, 0]';
u0 = zeros(nu,1); % Initial input guess
% ===== MPC simulation start =====
exec_time = []; n_iter = [];
x_mpc(:,1) = x0;
fprintf('MPC simulation (Gurobi)...\n');
for k=1:1:N
% Reference traj. over pred. horizon
if N - k >= Np - 1
X_r = x_r_traj(:,k:1:k+Np-1);
X_r = reshape(X_r, [nx*Np, 1]);
else
vec_tmp_1 = x_r_traj(:,k:1:end);
vec_tmp_2 = repmat(x_r_traj(:,end), 1, Np - size(x_r_traj(:,k:1:end),2));
X_r = [vec_tmp_1, vec_tmp_2];
X_r = reshape(X_r, [nx*Np, 1]);
end
% Linearize pred. model
if k == 1
J_f_x = num_jacobian(@(x)plant_dt(x,u0,Tp), x_mpc(:,k));
J_f_u = num_jacobian(@(u)plant_dt(x_mpc(:,k),u,Tp), u0);
A = J_f_x;
B = J_f_u;
b = plant_dt(x_mpc(:,k),u0,Tp) - A*x_mpc(:,k) - B*u0;
else
J_f_x = num_jacobian(@(x)plant_dt(x,u_mpc(:,k-1),Tp), x_mpc(:,k));
J_f_u = num_jacobian(@(u)plant_dt(x_mpc(:,k),u,Tp), u_mpc(:,k-1));
A = J_f_x;
B = J_f_u;
b = plant_dt(x_mpc(:,k),u_mpc(:,k-1),Tp) - A*x_mpc(:,k) - B*u_mpc(:,k-1);
end
% QP-MPC matrices
H_mpc = H;
c_mpc = 2*c(X_r);
A_eq_mpc = A_eq(A,B);
b_eq_mpc = b_eq(x_mpc(:,k), A, b);
A_ineq_mpc = A_ineq;
b_ineq_mpc = b_ineq;
% Primal-dual infeas. functions
prim_inf_fun = @(y) max( norm(A_eq_mpc*y-b_eq_mpc,inf), norm(max(A_ineq_mpc*y-b_ineq_mpc,0),inf) ); % Primal infeas.
dual_inf_fun = @(y,mu,lambda) norm(H_mpc*y + c_mpc - A_eq_mpc'*mu - A_ineq_mpc'*lambda,inf); % Dual infeas.
options.OutputFlag = 0;
options.FeasibilityTol = 1e-06; % Primal infeas. tol.
options.OptimalityTol = 1e-06; % Dual infeas. tol.
model.Q = sparse(H_mpc);
model.obj = c_mpc;
model.A = sparse(A_eq_mpc);
model.rhs = b_eq_mpc;
model.sense = '=';
model.modelsense = 'min';
model.QC = sparse(A_ineq_mpc);
model.QCrhs = b_ineq_mpc;
model.QCname = 'inequality';
model.lb = [repmat(x_lb,floor(0.5*Np),1); repmat(-inf*ones(nx,1),Np-floor(0.5*Np),1); repmat(u_lb,Nc,1); zeros(nx,1)];
model.ub = [repmat(x_ub,floor(0.5*Np),1); repmat(inf*ones(nx,1),Np-floor(0.5*Np),1); repmat(u_ub,Nc,1); inf*ones(nx,1)];
tim1 = tic;
sol = gurobi(model, options);
curr_exec_time = toc(tim1);
curr_n_iter = sol.baritercount;
exec_time = [exec_time, curr_exec_time];
n_iter = [n_iter, curr_n_iter];
if disp_curr_exec_time == true
fprintf('MPC step solved (%d / %d) (time: %.4f ms, iterations: %d)\n',...
k, N, curr_exec_time*1e03, curr_n_iter)
end
y_opt = sol.x;
x_opt = y_opt(1:1:nx*Np);
x_opt = reshape(x_opt, [nx,Np]);
u_opt = y_opt(nx*Np+1:1:nx*Np+nu*Nc);
u_opt = reshape(u_opt, [nu,Nc]);
e_opt = y_opt(nx*Np+nu*Nc+1:1:end);
u_mpc(:,k) = u_opt(:,1);
% Plant model integration
x_tmp = x_mpc(:,k);
for i = 1:1:floor(Ts/T_int)
x_tmp = plant_dt(x_tmp,u_mpc(:,k),T_int);
end
x_mpc(:,k+1) = x_tmp;
end
fprintf('\n===== MPC (Gurobi) =====\n');
fprintf('Exec. time: [%.4f, %.4f] ms; mean = %.4f ms\n', min(exec_time)*1e03, max(exec_time)*1e03, mean(exec_time)*1e03)
fprintf('Iterations: [%d, %d]; mean = %d\n', min(n_iter), max(n_iter), round(mean(n_iter)))
if save_data == true
save(strcat(path,save_name), 't', 'x_mpc', 'u_mpc', 'x_r_traj', 'x_lb', 'x_ub', 'u_lb', 'u_ub', ...
'exec_time', 'n_iter');
end
%% Plots
% - Plot 1: 3D trajectory
f1 = figure(1); set(f1,'WindowStyle','docked');
tiledlayout(1,1,'tilespacing','none','padding','none')
nexttile, hold on
plot3(x_r_traj(1,:), x_r_traj(2,:), x_r_traj(3,:), 'k-', 'linewidth', 0.5)
plot3(x_mpc(1,:), x_mpc(2,:), x_mpc(3,:), 'r-', 'linewidth', 1)
patch([2.5 2.5 2.5 2.5], [-0.5 2.5 2.5 -0.5], [0 0 4 4], 'k','facealpha',0.2)
hold off, grid on, axis equal
set(gca,'TickLabelInterpreter','latex','fontsize',13)
xlim([-1.5,3.5]), ylim([-0.5,2.5]), zlim([0, 4])
xlabel('$x$ [m]','interpreter','latex')
ylabel('$y$ [m]','interpreter','latex')
zlabel('$z$ [m]','interpreter','latex')
view(-35,50)
% - Plot 2: Orientation
f2 = figure(2); set(f2,'WindowStyle','docked');
tiledlayout(3,1,'tilespacing','compact','padding','none')
% -- 1:
nexttile, hold on
plot(t, rad2deg(x_mpc(4,:)), 'r-', 'linewidth', 1)
yline(rad2deg(x_lb(4)),'k--', 'linewidth', 0.5)
yline(rad2deg(x_ub(4)),'k--', 'linewidth', 0.5)
hold off, grid on, grid(gca,'minor')
set(gca,'TickLabelInterpreter','latex','fontsize',14)
xlim([0,5]), ylim([1.05*rad2deg(x_lb(4)),1.05*rad2deg(x_ub(4))])
ylabel('$\phi$ [deg]','interpreter','latex')
% -- 2:
nexttile, hold on
plot(t, rad2deg(x_mpc(5,:)), 'r-', 'linewidth', 1)
yline(rad2deg(x_lb(5)),'k--', 'linewidth', 0.5)
yline(rad2deg(x_ub(5)),'k--', 'linewidth', 0.5)
hold off, grid on, grid(gca,'minor')
set(gca,'TickLabelInterpreter','latex','fontsize',14)
xlim([0,5]), ylim([1.1*rad2deg(x_lb(5)),1.1*rad2deg(x_ub(5))])
ylabel('$\theta$ [deg]','interpreter','latex')
% -- 3:
nexttile, hold on
plot(t, rad2deg(x_mpc(6,:)), 'r-', 'linewidth', 1)
yline(rad2deg(x_lb(6)),'k--', 'linewidth', 0.5)
yline(rad2deg(x_ub(6)),'k--', 'linewidth', 0.5)
hold off, grid on, grid(gca,'minor')
set(gca,'TickLabelInterpreter','latex','fontsize',14)
xlim([0,5]), ylim([1.1*rad2deg(x_lb(6)),1.1*rad2deg(x_ub(6))])
xlabel('Simulation time [s]','interpreter','latex')
ylabel('$\psi$ [deg]','interpreter','latex')
% - Plot 3: Control inputs
f3 = figure(3); set(f3,'WindowStyle','docked');
tiledlayout(2,2,'tilespacing','compact','padding','none')
% -- 1:
nexttile, hold on
plot(t(1:1:end-1), u_mpc(1,:), 'r-', 'linewidth', 1);
yline(u_lb(1),'k--', 'linewidth', 0.5);
yline(u_ub(1),'k--', 'linewidth', 0.5)
hold off, grid on, grid(gca,'minor')
set(gca,'TickLabelInterpreter','latex','fontsize',14)
xlim([0,5]), ylim([-1,1.1*u_ub(1)])
ylabel('$f$ [N]','interpreter','latex')
% -- 2:
nexttile, hold on
plot(t(1:1:end-1), u_mpc(2,:), 'r-', 'linewidth', 1)
yline(u_lb(2),'k--', 'linewidth', 0.5)
yline(u_ub(2),'k--', 'linewidth', 0.5)
hold off, grid on, grid(gca,'minor')
set(gca,'TickLabelInterpreter','latex','fontsize',14)
xlim([0,5]), ylim([1.1*u_lb(2),1.1*u_ub(2)])
ylabel('$\tau_x$ [$\mathrm{N \, m}$]','interpreter','latex')
% -- 3:
nexttile, hold on
plot(t(1:1:end-1), u_mpc(3,:), 'r-', 'linewidth', 1)
yline(u_lb(3),'k--', 'linewidth', 0.5)
yline(u_ub(3),'k--', 'linewidth', 0.5)
hold off, grid on, grid(gca,'minor')
set(gca,'TickLabelInterpreter','latex','fontsize',14)
xlim([0,5]), ylim([1.1*u_lb(3),1.1*u_ub(3)])
xlabel('Simulation time [s]','interpreter','latex')
ylabel('$\tau_y$ [$\mathrm{N \, m}$]','interpreter','latex')
% -- 4:
nexttile, hold on
plot(t(1:1:end-1), u_mpc(4,:), 'r-', 'linewidth', 1)
yline(u_lb(4),'k--', 'linewidth', 0.5)
yline(u_ub(4),'k--', 'linewidth', 0.5)
hold off, grid on, grid(gca,'minor')
set(gca,'TickLabelInterpreter','latex','fontsize',14)
xlim([0,5]), ylim([1.05*u_lb(4),1.05*u_ub(4)])
xlabel('Simulation time [s]','interpreter','latex')
ylabel('$\tau_z$ [$\mathrm{N \, m}$]','interpreter','latex')
% - Plot 4: Execution time
f4 = figure(4); set(f4,'WindowStyle','docked');
tiledlayout(1,1,'tilespacing','none','padding','none')
nexttile, hold on
plot(t(1:1:end-1), 1e03*exec_time, 'r-', 'linewidth', 1);
hold off, grid on, grid(gca,'minor')
set(gca,'TickLabelInterpreter','latex','fontsize',14)
xlabel('Simulation time [s]','interpreter','latex')
ylabel('Execution time [ms]','interpreter','latex')
% ========================================
function xs = sat(x,bounds)
xs = min(max(x,bounds(:,1)),bounds(:,2));
end