-
Notifications
You must be signed in to change notification settings - Fork 0
/
mpc_sim_qpalm.m
419 lines (299 loc) · 10.6 KB
/
mpc_sim_qpalm.m
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
clc
clear variables
close all
import qpalm.qpalm
disp_curr_exec_time = true;
save_data = true;
save_name = 'mpc_sim_qpalm_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 (QPALM)...\n');
for k=1:1:N_sim
% 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 = 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.
prob = qpalm;
options = prob.default_settings();
options.eps_abs = 1e-06; % Abs. tol.
options.eps_rel = 1e-06; % Rel. tol.
options.eps_prim_inf = 1e-06; % Prim. infeas. tol.
options.eps_dual_inf = 1e-06; % Dual infeas. tol.
prob.setup(H_mpc, c_mpc, [A_eq_mpc; A_ineq_mpc], ...
[b_eq_mpc; -inf*ones(length(b_ineq_mpc),1)], [b_eq_mpc;
b_ineq_mpc], options);
tim1 = tic;
sol = prob.solve();
curr_exec_time = toc(tim1);
curr_n_iter = sol.info.iter;
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;
% Evaluate optimal primal-dual residuals
mu_opt = -sol.y(1:1:nx*Np);
lambda_opt = -sol.y(nx*Np+1:1:end);
prim_res(k) = prim_inf_fun(y_opt);
dual_res(k) = dual_inf_fun(y_opt, mu_opt, lambda_opt);
end
fprintf('\n===== MPC (QPALM) =====\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', ...
'prim_res', 'dual_res');
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')
% - Plot 5: Optimal primal-dual residuals
f5 = figure(5); set(f5,'WindowStyle','normal');
tiledlayout(1,1,'tilespacing','none','padding','none')
nexttile
semilogy(t(1:end-1), prim_res, 'r-', 'linewidth', 1), hold on
semilogy(t(1:end-1), dual_res, 'r:', 'linewidth', 1), hold off
grid on, grid(gca,'minor')
set(gca,'TickLabelInterpreter','latex','fontsize',14)
xlabel('Simulation time [s]','interpreter','latex')
ylabel('Residuals','interpreter','latex')
legend('Primal residual','Dual residual','interpreter','latex','fontsize',12)
% ========================================
function xs = sat(x,bounds)
xs = min(max(x,bounds(:,1)),bounds(:,2));
end