-
Notifications
You must be signed in to change notification settings - Fork 3
/
activeInferencePID.py
610 lines (487 loc) · 31.6 KB
/
activeInferencePID.py
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
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""
Created on Sun Mar 4 15:52:21 2018
Active inference version of a PID controller. Example built on cruise control
problem from Astrom and Murray (2010), pp 65-69.
In this specific example, only Proportional and Integral terms are used,
since standard cruise control problems do not usually adopt the D-term.
@author: manuelbaltieri
"""
import numpy as np
import matplotlib.pyplot as plt
plt.close('all')
large_value = np.exp(50)
### define font size for plots ###
#
SMALL_SIZE = 16
MEDIUM_SIZE = 20
BIGGER_SIZE = 22
plt.rc('font', size=MEDIUM_SIZE) # controls default text sizes
plt.rc('axes', titlesize=MEDIUM_SIZE) # fontsize of the axes title
plt.rc('axes', labelsize=SMALL_SIZE) # fontsize of the x and y labels
plt.rc('xtick', labelsize=SMALL_SIZE) # fontsize of the tick labels
plt.rc('ytick', labelsize=SMALL_SIZE) # fontsize of the tick labels
plt.rc('legend', fontsize=SMALL_SIZE) # legend fontsize
plt.rc('figure', titlesize=BIGGER_SIZE) # fontsize of the figure title
#
dt = .05 # integration step
### generative model constants ###
#
alpha = 100000. # drift in Generative Model
gamma = 1 # drift in OU process (if you want to simulate coloured noise)
obs_states = 1
hidden_states = 1 # x, in Friston's work
hidden_causes = 1 # v, in Friston's work
##states = obs_states + hidden_states
temp_orders_states = 3 # generalised coordinates for hidden states x
temp_orders_causes = 3 # generalised coordinates for hidden causes v
#
### cruise control problem from Astrom and Murray (2010), pp 65-69
# environment parameters
ga = 9.81 # gravitational acceleration
theta = 4. # hill angle
C_r = .01 # rolling friction coefficient
C_d = .32 # drag coefficient
rho = 1.3 # air density
A = 2.4 # frontal area agent
# car's parameters
#m = 1000 # car mass (book example)
m = 100 # car mass
T_m = 190 # maximum torque
omega_m = 420 # engine speed to reach T_m
r_g = 12 # = gear ration/wheel radius,
# a1 = 40, a2 = 25, a3 = 16, a4 = 12, a5 = 10
beta = .4 # motor constant
#
### FUNCTIONS ###
## motor action ##
def sigmoid(x): # limit the motor control (not used in the paper)
return np.tanh(x)
return 1 / (1+np.exp(-x))
def dsigmoid(x): # derivative of the above function to improve active inference (not used in the paper)
return sigmoid(x) * (1 - sigmoid(x))
## cruise control problem ##
def force_gravitation(theta):
return m * ga * np.sin(theta)
def force_friction(v):
return m * ga * C_r * np.sign(v)
def force_drag(v):
return .5 * rho * C_d * A * v**2
def force_disturbance(v, theta):
return force_gravitation(theta) + force_friction(v) + force_drag(v)
def force_drive(v, u):
return r_g * u * torque(v)
def torque(v):
return T_m * (1 - beta * (omega(v) / omega_m)**2)
def omega(v):
return r_g * v
## free energy functions ##
# generative process
# x: hidden states
# v: hidden causes
# a: action
# w: fluctuations in process dynamics
def g(x, v):
return x
def f(x, v, a):
return (force_drive(x, v + a) - force_disturbance(x, theta)) / m
# generative model
def g_gm(x, v):
return g(x, v)
def f_gm(x, v):
# a = 0.0, no action in generative model
return f(x, v, 0.0)
def getObservation(x, v, a, w):
# w = 0.0, no added fluctuations in the car dynamics
x[:, 1:] = f(x[:, :-1], v, a)# + w
x[:, 0] += dt * x[:, 1]
return g(x[:, :-1], v)
# main function, PID control
def pidControl(simulation, T, dt, switch_condition_time, learning, limit_case, mu_gamma_z_input, mu_gamma_w_input):
iterations = int(T / dt)
### car variables ###
x = np.zeros((hidden_states, temp_orders_states)) # hidden states
v = np.zeros((hidden_causes, temp_orders_states - 1)) # hidden causes
y = np.zeros((obs_states, temp_orders_states - 1)) # observations (without noise)
# noise on sensory input (world - generative process)
# gamma_z = 0. * np.ones((obs_states, temp_orders_states)) # sensory log-precisions, correlated noise
# gamma_z[:,1] = gamma_z[:,0] - np.log(2 * gamma)
# gamma_z[:,2] = gamma_z[:,1] - np.log(2 * gamma)
gamma_z = 5. * np.ones((obs_states, temp_orders_states)) # sensory log-precisions, uncorrelated noise
pi_z = np.exp(gamma_z) * np.ones((obs_states, temp_orders_states))
sigma_z = 1 / (np.sqrt(pi_z))
z = np.zeros((iterations, obs_states, temp_orders_states))
for i in range(obs_states):
for j in range(temp_orders_states):
z[:, i, j] = sigma_z[i, j] * np.random.randn(1, iterations)
# noise on motion of hidden states (world - generative process)
gamma_w = 32 * np.ones((hidden_states, temp_orders_states)) # process log-precisions, uncorrelated noise (not used in the paper)
gamma_w[:,1] = gamma_w[:,0] - np.log(2 * gamma)
gamma_w[:,2] = gamma_w[:,1] - np.log(2 * gamma)
pi_w = np.exp(gamma_w) * np.ones((hidden_states, temp_orders_states))
sigma_w = 1 / (np.sqrt(pi_w))
w = np.zeros((iterations, hidden_states, temp_orders_states))
for i in range(hidden_states):
for j in range(temp_orders_states - 1):
w[:, i, j] = sigma_w[i, j] * np.random.randn(1, iterations)
### free energy variables ###
mu_x = 0.0001*np.random.randn(hidden_states, temp_orders_states) # expected hidden states
mu_v = np.zeros((hidden_causes, temp_orders_states)) # expected hidden causes (not used in the paper, see equations 20-21)
a = np.zeros((obs_states, temp_orders_states-1))
# priors
eta_x = np.zeros((hidden_causes, temp_orders_states - 1)) # priors on expected hidden states, entering as expected hidden causes
eta_gamma_z = np.zeros((obs_states, temp_orders_states - 1)) # priors on expected sensory precisions (not used in the paper), hyperpriors
eta_gamma_w = np.zeros((hidden_states, temp_orders_states - 1)) # priors on expected process precisions (not used in the paper), hyperpriors
# minimisation of variables and (hyper)parameters
dFdmu_x = np.zeros((hidden_states, temp_orders_states))
dFdmu_v = np.zeros((hidden_causes, temp_orders_states))
Dmu_x = np.zeros((hidden_states, temp_orders_states))
Dmu_v = np.zeros((hidden_causes, temp_orders_states))
dFdmu_gamma_z = np.zeros((hidden_causes, temp_orders_states))
phi_z = np.zeros((obs_states, temp_orders_states-1))
phi_w = np.zeros((hidden_states, temp_orders_states-1))
# learning rates (not used in the paper)
k_mu_x = 1 # learning rate perception
k_a = 1 # learning rate action
k_mu_gamma_z = 1 # learning rate attention (sensory precisions)
k_mu_gamma_w = 1 # learning rate attention (process precisions)
# damping terms for hyperparameters optimisation
kappa_z = 5 # damping on sensory precisions minimisation
kappa_w = 10 # damping on process precisions minimisation
# agent's estimates of the noise (agent - generative model)
mu_gamma_z = mu_gamma_z_input * np.ones((obs_states, temp_orders_states - 1)) # sensory log-precisions, correlated noise
mu_gamma_z[0, 1] = mu_gamma_z[0, 0] - np.log(2 * gamma)
mu_pi_z = np.exp(mu_gamma_z) * np.ones((obs_states, temp_orders_states - 1))
mu_gamma_w = mu_gamma_w_input * np.ones((hidden_states, temp_orders_states - 1)) # process log-precisions, correlated noise
mu_gamma_w[0, 1] = mu_gamma_w[0, 0] - np.log(2)
mu_pi_w = np.exp(mu_gamma_w) * np.ones((hidden_states, temp_orders_states - 1))
# hyperpriors' precisions (not used in the paper, left for future work)
mu_gamma_gamma_z = - large_value * np.ones((obs_states, temp_orders_states - 1))
mu_p_gamma_z = np.exp(mu_gamma_gamma_z) * np.ones((obs_states, temp_orders_states - 1))
mu_gamma_gamma_w = - large_value * np.ones((hidden_states, temp_orders_states - 1))
mu_p_gamma_w = np.exp(mu_gamma_gamma_w) * np.ones((hidden_states, temp_orders_states - 1))
# history, here for reference, only some variable are used
x_history = np.zeros((iterations, hidden_states, temp_orders_states))
y_history = np.zeros((iterations, obs_states, temp_orders_states - 1))
v_history = np.zeros((iterations, hidden_causes, temp_orders_states - 1))
psi_history = np.zeros((iterations, obs_states, temp_orders_states - 1))
mu_x_history = np.zeros((iterations, hidden_states, temp_orders_states))
eta_x_history = np.zeros((iterations, hidden_causes, temp_orders_states - 1))
eta_gamma_z_history = np.zeros((iterations, obs_states, temp_orders_states - 1))
eta_gamma_w_history = np.zeros((iterations, hidden_causes, temp_orders_states - 1))
a_history = np.zeros((iterations, temp_orders_states - 1))
mu_gamma_z_history = np.zeros((iterations, temp_orders_states-1))
mu_gamma_w_history = np.zeros((iterations, temp_orders_states-1))
mu_pi_z_history = np.zeros((iterations, temp_orders_states-1))
mu_pi_w_history = np.zeros((iterations, temp_orders_states-1))
dFdmu_gamma_z_history = np.zeros((iterations, temp_orders_states-1))
dFdmu_gamma_w_history = np.zeros((iterations, temp_orders_states-1))
xi_z_history = np.zeros((iterations, obs_states, temp_orders_states - 1))
xi_w_history = np.zeros((iterations, hidden_states, temp_orders_states - 1))
kappa_z_history = np.zeros((iterations,1))
kappa_w_history = np.zeros((iterations,1))
gamma_z_history = np.zeros((iterations, temp_orders_states))
gamma_w_history = np.zeros((iterations, temp_orders_states))
### initialisation ###
# hyperpriors (not used in the paper, left for reference)
eta_gamma_z[0, 0] = 3.
eta_gamma_z[0, 1] = 1.
eta_gamma_w[0, 0] = - 18.
eta_gamma_w[0, 1] = - 18.
desired_velocity = 10.
eta_x[0, 0] = desired_velocity
### main loop ###
for i in range(iterations - 1):
print(i)
# re-encode precisions after hyperparameters update
mu_pi_z = np.exp(mu_gamma_z) * np.ones((obs_states, temp_orders_states - 1))
mu_pi_w = np.exp(mu_gamma_w) * np.ones((hidden_states, temp_orders_states - 1))
mu_p_gamma_z = np.exp(mu_gamma_gamma_z) * np.ones((obs_states, temp_orders_states - 1))
mu_p_gamma_w = np.exp(mu_gamma_gamma_w) * np.ones((obs_states, temp_orders_states - 1))
# include an external disturbance to test integral term
if (simulation == 0) or (simulation == 1):
if (i > iterations/2):
v[0,0] = 3.0
# test 2DOF
if (simulation == 2):
if (i > iterations/2):
eta_x[0, 0] = desired_velocity - 3
# Analytical noise, for one extra level of generalised cooordinates, this is equivalent to an ornstein-uhlenbeck process
# dw2 = - gamma * w[i, 0, 1] + w[i, 0, 2] / np.sqrt(dt)
# dz2 = - gamma * z[i, 0, 1] + z[i, 0, 2] / np.sqrt(dt)
#
# w[i+1, 0, 1] = w[i, 0, 1] + dt * dw2
# z[i+1, 0, 1] = z[i, 0, 1] + dt * dz2
#
# dw = - gamma * w[i, 0, 0] + w[i, 0, 1]
# dz = - gamma * z[i, 0, 0] + z[i, 0, 1]
#
# w[i+1, 0, 0] = w[i, 0, 0] + dt * dw
# z[i+1, 0, 0] = z[i, 0, 0] + dt * dz
y = getObservation(x, v, a, w[i, 0, :-1])
psi = y + z[i, 0, :-1]
### minimise free energy ###
# perception
Dmu_x[0, :-1] = mu_x[0, 1:]
# dFdmu_x[0, :-1] = np.array([mu_pi_z * - (psi - mu_x[0, :-1]) + mu_pi_w * alpha * (mu_x[0, 1:] + alpha * (mu_x[0, :-1] - eta_x))]) # if the noise is not white
if limit_case != 1:
dFdmu_x[0, :-1] = np.array([mu_pi_z * - (y + z[i, 0, :-1]/np.sqrt(dt) - mu_x[0, :-1]) + mu_pi_w * alpha * (mu_x[0, 1:] + alpha * (mu_x[0, :-1] - eta_x))])
else:
# or use the line below to simulate indipendence of set-point adaptation
# with respect to precisions of measurement noise
dFdmu_x[0, :-1] = np.array([mu_pi_w * alpha * (mu_x[0, 1:] + alpha * (mu_x[0, :-1] - eta_x))])
# action
# dFdy = mu_pi_z * (psi - mu_x[0, :-1]) # if the noise is not white
dFdy = mu_pi_z * (y + z[i, 0, :-1]/np.sqrt(dt) - mu_x[0, :-1])
dyda = np.ones((obs_states, temp_orders_states-1))
dFda = np.zeros((obs_states, temp_orders_states-1))
dFda[0, 0] = np.sum(dFdy * dyda)
# attention
# dFdmu_gamma_z = .5 * (mu_pi_z * (psi - mu_x[0, :-1])**2 - 1) + mu_p_gamma_z * (mu_gamma_z - eta_gamma_z) # if the noise is not white
dFdmu_gamma_z = .5 * (mu_pi_z * (y**2 + z[i, 0, :-1]**2 + mu_x[0, :-1]**2 + 2*y*z[i, 0, :-1]/np.sqrt(dt) - 2*mu_x[0, :-1]*z[i, 0, :-1]/np.sqrt(dt) - 2*y*mu_x[0, :-1]) - 1) + mu_p_gamma_z * (mu_gamma_z - eta_gamma_z)
dFdmu_gamma_w = .5 * (mu_pi_w * (mu_x[0, 1:] + alpha * (mu_x[0, :-1] - eta_x))**2 - 1) + mu_p_gamma_w * (mu_gamma_w - eta_gamma_w)
## update equations ##
mu_x += dt * (Dmu_x - k_mu_x * dFdmu_x)
a += dt * - k_a * dFda
# only used for hyperparameters
phi_z += dt * (- dFdmu_gamma_z - kappa_z * phi_z)
phi_w += dt * (- dFdmu_gamma_w - kappa_w * phi_w)
# test conditions for hyperparameters optimisation
if simulation == 1 and (i > switch_condition_time/dt):
if (i < iterations/2):
mu_gamma_z += dt * k_mu_gamma_z * phi_z
if simulation == 3 and (i > switch_condition_time/dt):
if learning == 1:
mu_gamma_z += dt * k_mu_gamma_z * phi_z
else:
if i < iterations/2:
mu_gamma_z += dt * k_mu_gamma_z * phi_z
if i == iterations/2:
mu_gamma_gamma_z = -111 * np.ones((obs_states, temp_orders_states - 1)) # uncomment or comment to get a prior or just follow the changing measurement noise
gamma_z = 2. * np.ones((obs_states, temp_orders_states)) # one of the sensors "breaks"
pi_z = np.exp(gamma_z) * np.ones((obs_states, temp_orders_states))
sigma_z = 1 / (np.sqrt(pi_z))
z = np.zeros((iterations, obs_states, temp_orders_states))
for j in range(obs_states):
for k in range(temp_orders_states):
z[:, j, k] = sigma_z[j, k] * np.random.randn(1, iterations)
# save history
y_history[i, :] = y
psi_history[i, :] = psi
mu_x_history[i, :, :] = mu_x
eta_x_history[i] = eta_x
eta_gamma_z_history[i] = eta_gamma_z
eta_gamma_w_history[i] = eta_gamma_w
a_history[i] = a
v_history[i] = v
mu_gamma_z_history[i] = mu_gamma_z
mu_gamma_w_history[i] = mu_gamma_w
gamma_z_history[i] = gamma_z
gamma_w_history[i] = gamma_w
return psi_history, y_history, mu_x_history, eta_x_history, eta_gamma_z_history, eta_gamma_w_history, a_history, v_history, gamma_z_history, mu_gamma_z_history, gamma_w_history, mu_gamma_w_history
### Simulations ###
# 0: PID control as active inference (Fig 2)
# 1: PID parameters tuning (Fig 4)
# 2: Active inference PID control with 2DOF, load disturbance response affected by pi_z + set point response affected by pi_w (Fig 3)
# 3: Summary statistics of IAE with/without parameters tuning (Fig 5)
# 4: Summary statistics of variance with continual/interrupted adaptation (Fig 6)
simulation = 4
learning = 0 # learning precisions, 0 off, 1 on
limit_case = 0 # simulation 3 requires to explicitly implement equation 24,
# since numerical approximation prevent the assumptions
# to be in place for varying precisions in simulation 3, 0 implicit, 1 explicit
if simulation == 0:
T = 300
switch_condition_time = 0 # time to start optimising precisions (not used in simulation 0)
psi_history, y_history, mu_x_history, eta_x_history, eta_gamma_z_history, eta_gamma_w_history, a_history, v_history, gamma_z_history, mu_gamma_z_history, gamma_w_history, mu_gamma_w_history = pidControl(simulation, T, dt, switch_condition_time, learning, limit_case, -3.,-20)
elif simulation == 1:
T = 300
switch_condition_time = T/10 # time to start optimising precisions
psi_history, y_history, mu_x_history, eta_x_history, eta_gamma_z_history, eta_gamma_w_history, a_history, v_history, gamma_z_history, mu_gamma_z_history, gamma_w_history, mu_gamma_w_history = pidControl(simulation, T, dt, switch_condition_time, learning, limit_case, -3.,-20)
elif simulation == 2:
T = 100
switch_condition_time = 0 # time to start optimising precisions (not used in simulation 0)
limit_case = 1
simulation = 0
psi_history, y_history, mu_x_history, eta_x_history, eta_gamma_z_history, eta_gamma_w_history, a_history, v_history, gamma_z_history, mu_gamma_z_history, gamma_w_history, mu_gamma_w_history = pidControl(simulation, T, dt, switch_condition_time, learning, limit_case, 1.,-24)
psi_history2, y_history, mu_x_history, eta_x_history, eta_gamma_z_history, eta_gamma_w_history, a_history2, v_history, gamma_z_history, mu_gamma_z_history, gamma_w_history, mu_gamma_w_history2 = pidControl(simulation, T, dt, switch_condition_time, learning, limit_case, 1.,-22)
psi_history3, y_history, mu_x_history, eta_x_history, eta_gamma_z_history, eta_gamma_w_history, a_history3, v_history, gamma_z_history, mu_gamma_z_history, gamma_w_history, mu_gamma_w_history3 = pidControl(simulation, T, dt, switch_condition_time, learning, limit_case, 1.,-20)
simulation = 2
psi_history4, y_history, mu_x_history, eta_x_history, eta_gamma_z_history, eta_gamma_w_history, a_history4, v_history, gamma_z_history, mu_gamma_z_history, gamma_w_history, mu_gamma_w_history4 = pidControl(simulation, T, dt, switch_condition_time, learning, limit_case, 1.,-24)
psi_history5, y_history, mu_x_history, eta_x_history, eta_gamma_z_history, eta_gamma_w_history, a_history5, v_history, gamma_z_history, mu_gamma_z_history, gamma_w_history, mu_gamma_w_history5 = pidControl(simulation, T, dt, switch_condition_time, learning, limit_case, 1.,-22)
psi_history6, y_history, mu_x_history, eta_x_history, eta_gamma_z_history, eta_gamma_w_history, a_history6, v_history, gamma_z_history, mu_gamma_z_history, gamma_w_history, mu_gamma_w_history6 = pidControl(simulation, T, dt, switch_condition_time, learning, limit_case, 1.,-20)
elif simulation == 3:
T = 300
iterations = int(T / dt)
simulations_number = 20
switch_condition_time = T/10 # time to start optimising precisions
variance_before = np.zeros((simulations_number, 2))
variance_after = np.zeros((simulations_number, 2))
for i in range(simulations_number):
learning = 0
psi_history, y_history, mu_x_history, eta_x_history, eta_gamma_z_history, eta_gamma_w_history, a_history, v_history, gamma_z_history, mu_gamma_z_history, gamma_w_history, mu_gamma_w_history = pidControl(simulation, T, dt, switch_condition_time, learning, limit_case, -2.,-20)
variance_before[i, 0] = np.var(y_history[int(T/dt/4):int(T/dt/2)-1, 0, 0])
variance_after[i, 0] = np.var(y_history[int(T/dt/4*3):int(T/dt)-1, 0, 0])
learning = 1
psi_history2, y_history2, mu_x_history, eta_x_history, eta_gamma_z_history, eta_gamma_w_history, a_history2, v_history, gamma_z_history, mu_gamma_z_history2, gamma_w_history, mu_gamma_w_history = pidControl(simulation, T, dt, switch_condition_time, learning, limit_case, -2.,-20)
variance_before[i, 1] = np.var(y_history2[int(T/dt/4):int(T/dt/2)-1, 0, 0])
variance_after[i, 1] = np.var(y_history2[int(T/dt/4*3):int(T/dt)-1, 0, 0])
elif simulation == 4:
T = 300
iterations = int(T / dt)
simulations_number = 20
desired_velocity = 10
epsilon = 0.5
simulation = 0
y_history_stats = np.zeros((simulations_number, iterations, obs_states, temp_orders_states - 1))
y_history_stats2 = np.zeros((simulations_number, iterations, obs_states, temp_orders_states - 1))
tauNoAdaptation = np.zeros(simulations_number,)
tauAdaptation = np.zeros(simulations_number,)
iaeNoAdaptation = np.zeros(simulations_number,)
iaeAdaptation = np.zeros(simulations_number,)
for i in range(simulations_number):
random_sensory_precision = np.random.rand()*2 - 3.
random_process_precision = np.random.rand()*2 - 22.
simulation = 0
switch_condition_time = 0
psi_history, y_history_stats[i,:,:,:], mu_x_history, eta_x_history, eta_gamma_z_history, eta_gamma_w_history, a_history, v_history, gamma_z_history, mu_gamma_z_history, gamma_w_history, mu_gamma_w_history = pidControl(simulation, T, dt, switch_condition_time, learning, limit_case, random_sensory_precision, random_process_precision)
foo = np.where((y_history_stats[i,int(T/2/dt)+3:,0,0] <= desired_velocity + epsilon) & (y_history_stats[i,int(T/2/dt)+3:,0,0] >= desired_velocity - epsilon))[0]
# consider 3*dt of distance to avoid t = tau
tauNoAdaptation[i] = foo[0] # first zero crossing after load disturbance
simulation = 1
switch_condition_time = T/10
psi_history, y_history_stats2[i,:,:,:], mu_x_history, eta_x_history, eta_gamma_z_history, eta_gamma_w_history, a_history, v_history, gamma_z_history, mu_gamma_z_history, gamma_w_history, mu_gamma_w_history = pidControl(simulation, T, dt, switch_condition_time, learning, limit_case, random_sensory_precision, random_process_precision)
foo2 = np.where((y_history_stats2[i,int(T/2/dt)+3:,0,0] <= desired_velocity + epsilon) & (y_history_stats2[i,int(T/2/dt)+3:,0,0] >= desired_velocity - epsilon))[0]
# consider 3*dt of distance to avoid t = tau
tauAdaptation[i] = foo2[0] # first zero crossing after load disturbance
iaeNoAdaptation[i] = np.sum(np.absolute(y_history_stats[i,int(T/2/dt):int(T/2/dt+3+tauNoAdaptation[i]), 0, 0]))
iaeAdaptation[i] = np.sum(np.absolute(y_history_stats2[i, int(T/2/dt):int(T/2/dt+3+tauAdaptation[i]), 0, 0]))
simulation = 4 # to plot the right figure
### FIGURES ###
if simulation < 3:
plt.figure(figsize=(9, 6))
plt.xlim((0., T))
if simulation == 0 or simulation == 1:
plt.plot(np.arange(0, T-dt, dt), psi_history[:-1,0,0], 'b', linewidth=1, label='Sensed velocity, $\psi$')
plt.plot(np.arange(0, T-dt, dt), mu_x_history[:-1,0,0], 'r', linewidth=1, label='Expec. of velocity, $\mu_x$')
#plt.plot(np.arange(0, T-dt, dt), eta_x_history[:-1,0,0], 'k--', linewidth=1, label='Desired velocity, $\eta_x$')
elif simulation == 2:
x_min = 0
x_max = 20
plt.xlim((x_min, x_max))
plt.plot(np.arange(x_min, x_max, dt), psi_history[int((x_min+40)/dt):int((x_max+40)/dt),0,0], 'b', linewidth=1, label='Sensed velocity, $\psi_1$; $\pi_w = exp($'+str(mu_gamma_w_history[1,0])+')')
plt.plot(np.arange(x_min, x_max, dt), psi_history2[int((x_min+40)/dt):int((x_max+40)/dt),0,0], 'r', linewidth=1, label='Sensed velocity, $\psi_1$; $\pi_w = exp($'+str(mu_gamma_w_history2[1,0])+')')
plt.plot(np.arange(x_min, x_max, dt), psi_history3[int((x_min+40)/dt):int((x_max+40)/dt),0,0], 'g', linewidth=1, label='Sensed velocity, $\psi_1$; $\pi_w = exp($'+str(mu_gamma_w_history[31,0])+')')
plt.title('Car velocity')
plt.xlabel('Time ($s$)')
plt.ylabel('Velocity ($km/h$)')
plt.legend(loc=1)
if simulation == 0:
plt.text(T+20, eta_x_history[-2,0,0], "$\eta_x$", size=20, rotation=0., ha="center", va="center", bbox=dict(boxstyle="larrow", fc="w", ec="0.5", alpha=0.9))
plt.ylim((-10., 50.))
plt.savefig("figures/activeInferencePID_a.pdf")
elif simulation == 1:
plt.text(T+20, eta_x_history[-2,0,0], "$\eta_x$", size=20, rotation=0., ha="center", va="center", bbox=dict(boxstyle="larrow", fc="w", ec="0.5", alpha=0.9))
plt.axvline(x=switch_condition_time, linewidth=3, color='k', linestyle='-.')
plt.ylim((-10., 50.))
plt.savefig("figures/activeInferencePIDTuning_a.pdf")
elif simulation == 2:
plt.text(x_max+1.5, eta_x_history[-2,0,0], "$\eta_x$", size=20, rotation=0., ha="center", va="center", bbox=dict(boxstyle="larrow", fc="w", ec="0.5", alpha=0.9))
plt.ylim((0., 15.))
plt.title('Car velocity - Load disturbance')
plt.legend(loc=4)
plt.annotate("new ext. input",
xy=((x_min+x_max)/2, 9.), xycoords='data',
xytext=((x_min+x_max)/2-2.85, 6.), textcoords='data',
arrowprops=dict(arrowstyle="simple",
connectionstyle="arc3"))
plt.savefig("figures/activeInferencePIDLoad.pdf")
if simulation == 2:
plt.figure(figsize=(9, 6))
x_min = 0
x_max = 20
plt.xlim((x_min, x_max))
plt.plot(np.arange(x_min, x_max, dt), psi_history4[int((x_min+40)/dt):int((x_max+40)/dt),0,0], 'b', linewidth=1, label='Sensed velocity, $\psi_1$; $\pi_w = exp($'+str(mu_gamma_w_history4[1,0])+')')
plt.plot(np.arange(x_min, x_max, dt), psi_history5[int((x_min+40)/dt):int((x_max+40)/dt),0,0], 'r', linewidth=1, label='Sensed velocity, $\psi_1$; $\pi_w = exp($'+str(mu_gamma_w_history5[1,0])+')')
plt.plot(np.arange(x_min, x_max, dt), psi_history6[int((x_min+40)/dt):int((x_max+40)/dt),0,0], 'g', linewidth=1, label='Sensed velocity, $\psi_1$; $\pi_w = exp($'+str(mu_gamma_w_history6[1,0])+')')
plt.title('Car velocity')
plt.xlabel('Time ($s$)')
plt.ylabel('Velocity ($km/h$)')
plt.legend(loc=1)
plt.text(x_max+1.5, eta_x_history[-2,0,0], "$\eta_x$", size=20, rotation=0., ha="center", va="center", bbox=dict(boxstyle="larrow", fc="w", ec="0.5", alpha=0.9))
plt.ylim((0., 20.))
plt.title('Car velocity - Set-point change')
plt.legend(loc=4)
plt.annotate("new target",
xy=((x_min+x_max)/2, 9.), xycoords='data',
xytext=((x_min+x_max)/2-2.15, 6.), textcoords='data',
arrowprops=dict(arrowstyle="simple",
connectionstyle="arc3"))
plt.savefig("figures/activeInferencePIDSetPoint.pdf")
if simulation == 0 or simulation == 1:
plt.figure(figsize=(9, 6))
plt.xlim((0., T))
plt.plot(np.arange(0, T-dt, dt), psi_history[:-1,0,1], 'b', linewidth=1, label='Sensed acceleration, $\psi\'$')
plt.plot(np.arange(0, T-dt, dt), mu_x_history[:-1,0,1], 'r', linewidth=1, label='Expec. of acceleration, $\mu_x\'$')
#plt.plot(np.arange(0, T-dt, dt), eta_x_history[:-1,0,1], 'g', label='Desired acceleration, $\eta_x\'$')
plt.title('Car acceleration')
plt.xlabel('Time ($s$)')
plt.ylabel('Acceleration ($km/h^2$)')
plt.legend(loc=1)
if simulation == 0:
plt.text(T+22, eta_x_history[-2,0,1], "$\eta'_x$", size=20, rotation=0., ha="center", va="center", bbox=dict(boxstyle="larrow", fc="w", ec="0.5", alpha=0.9))
plt.ylim((-70., 100.))
plt.savefig("figures/activeInferencePID_b.pdf")
elif simulation == 1:
plt.text(T+22, eta_x_history[-2,0,1], "$\eta'_x$", size=20, rotation=0., ha="center", va="center", bbox=dict(boxstyle="larrow", fc="w", ec="0.5", alpha=0.9))
plt.axvline(x=switch_condition_time, linewidth=3, color='k', linestyle='-.')
plt.ylim((-70., 100.))
plt.savefig("figures/activeInferencePIDTuning_b.pdf")
if simulation == 0 or simulation == 1:
plt.figure(figsize=(9, 6))
plt.xlim((0., T))
plt.title('Motor output')
plt.plot(np.arange(0, T-dt, dt), a_history[:-1,0], linewidth=1, label='Action, a')
plt.xlabel('Time ($s$)')
plt.ylabel('Acceleration ($km/h^2$)')
if simulation == 0:
plt.plot(np.arange(0, T-dt, dt), v_history[:-1,0,0], 'k', linewidth=1, label='Ext. input, v')
plt.legend(loc=1)
plt.ylim((-5., 5.))
plt.savefig("figures/activeInferencePID_c.pdf")
elif simulation == 1:
plt.axvline(x=switch_condition_time, linewidth=3, color='k', linestyle='-.')
plt.plot(np.arange(0, T-dt, dt), v_history[:-1,0,0], 'k', label='Ext. input, v')
plt.ylim((-5., 5.))
plt.legend(loc=1)
plt.savefig("figures/activeInferencePIDTuning_c.pdf")
if simulation == 1:
plt.figure(figsize=(9, 6))
plt.xlim((0., T))
plt.ylim((-6., 6.))
plt.title('Log-(sensory) precisions (= log-PI gains)')
plt.plot(np.arange(0, T-dt, dt), mu_gamma_z_history[:-1, 0], 'r', label='Expec. of log-precision, $\mu_{\gamma_z}$')
plt.plot(np.arange(0, T-dt, dt), gamma_z_history[:-1, 0], 'b')
plt.plot(np.arange(0, T-dt, dt), mu_gamma_z_history[:-1, 1], color='orange', label='Expec. of log-precision, $\mu_{\gamma_{z\'}}$')
plt.text(T+20, gamma_z_history[-2,0], "$\gamma_{z}$", size=20, rotation=0., ha="center", va="center", bbox=dict(boxstyle="larrow", fc="w", ec="0.5", alpha=0.9))
plt.xlabel('Time ($s$)')
plt.legend(loc=4)
plt.axvline(x=switch_condition_time, linewidth=3, color='k', linestyle='-.')
mid = int(T/2)
plt.axvline(x=mid, linewidth=3, color='k', linestyle='-')
plt.text(mid-60, gamma_z_history[-2,0]-1, "Adaptation", size=20, rotation=0., ha="center", va="center", bbox=dict(boxstyle="round", fc="w", ec="0.5", alpha=0.9))
plt.text(mid+75, gamma_z_history[-2,0]-1, "Control", size=20, rotation=0., ha="center", va="center", bbox=dict(boxstyle="round", fc="w", ec="0.5", alpha=0.9))
plt.savefig("figures/activeInferencePIDTuning_d.pdf")
if simulation == 3:
plt.figure(figsize=(10, 6))
plt.boxplot([variance_after[:, 0], variance_after[:, 1]])
plt.xticks([1, 2], ['Adaptation interrupted', 'Continual adaptation'])
plt.ylabel('Variance (a.u.)')
plt.savefig("figures/activeInferencePIDVaryingNoise.pdf")
if simulation == 4:
plt.figure(figsize=(10, 6))
plt.boxplot([iaeNoAdaptation, iaeAdaptation])
plt.xticks([1, 2], ['No adaptation', 'Adaptation'])
plt.ylabel('IAE (a.u.)')
plt.savefig("figures/activeInferencePIDiae.pdf")