def NLAR_sim(params, S, S_lag):
'''simulate 1 step of a non-linear AR(2) model of speed
params = ps, pa, pt, ptv, std
'''
ps, pa, pt, ptv, std = params
A = (S-S_lag)/dt
Tnl = torque_law(S)/damp - S
Tnl_lag = torque_law(S_lag)/damp - S_lag
Tvar = (Tnl - Tnl_lag)/dt
innov = rnd.normal(scale=innov_std, size=S.shape)
# dynimical equation:
S_next = ps*S + pa*A + pt*Tnl + ptv*Tvar + innov
return S_next
# Remove extremal ticks:
ax1t.yaxis.get_major_locator().set_params(prune='both') # extremal speed ticks
ax2t.yaxis.get_major_locator().set_params(prune='both') # extremal power ticks
ax2t.xaxis.get_major_locator().set_params(prune='both') # extremal time ticks
ax2tz.xaxis.get_major_locator().set_params(prune='both') # extremal time ticks
### Phase portrait
fig = plt.figure('phase portrait')
ax = fig.add_subplot(111,
title=u'Phase portrait (Ω,a) of the SEAREV\n'
'with leveled viscous torque PTO',
xlabel='speed (rad/s)',
ylabel=u'accel (rad/s²)')
s = np.linspace(-1, 1, 500)
ax.plot(s, torque_law(s) / damp, 'r', label=u'T(Ω)/damp')
ax.plot(s, s, 'r--', label=u'Ω')
ax.plot(speed, accel, '-', lw=0.3)
ax.legend(loc='upper right')
# 3D phase portrait:
#mlab.plot3d(angle, speed, accel, tube_radius=None, opacity=0.3, color=(0,0,1))
# 3D auto-regression plot:
#from mayavi import mlab
#mlab.points3d(angle[:-1], speed[:-1], speed[1:], mode='point')
plt.show()
ar_coef, innov_var = ar_acf_fit(speed, 2, maxlags = ar_fitlags)
print('Speed AR(2) model from acf fit on %.0f s:' % (ar_fitlags*dt))
print(' * AR coef: '+ str(ar_coef))
print(' * innovation sd: {:g}'.format(np.sqrt(innov_var)))
# speed series simulation:
model = {'ar':ar_coef, 'ma':[]}
innov_gen = lambda size: np.random.normal(scale = np.sqrt(innov_var), size=size)
speed_sim = armaToolbox.arima_sim(model, n_pts, innov_gen = innov_gen)
accel_sim = np.zeros(n_pts)
accel_sim[:-1] = np.diff(speed_sim)/dt
power_sim = speed_sim*torque_law(speed_sim)/1e6
### State space stochastic model (VAR kind of) #################################
# state variables:
X = np.vstack((speed, accel))
p1, p2 = ar_coef
# Conversion of AR2 to state-space
T_AR2 = np.array([[p1+p2, -dt*p2],
[(p1+p2-1)/dt, -p2]])
T = T_AR2
fitlags = 200
def pred_objective(T):
'''mean square prediction error at horizon `fitlags`'''
T = T.reshape((2,2))
X_prev = X
# Remove extremal ticks:
ax1t.yaxis.get_major_locator().set_params(prune='both') # extremal speed ticks
ax2t.yaxis.get_major_locator().set_params(prune='both') # extremal power ticks
ax2t.xaxis.get_major_locator().set_params(prune='both') # extremal time ticks
ax2tz.xaxis.get_major_locator().set_params(prune='both') # extremal time ticks
### Phase portrait
fig = plt.figure('phase portrait')
ax = fig.add_subplot(111, title=u'Phase portrait (Ω,a) of the SEAREV\n'
'with leveled viscous torque PTO',
xlabel='speed (rad/s)', ylabel=u'accel (rad/s²)')
s = np.linspace(-1,1, 500)
ax.plot(s, torque_law(s)/damp, 'r', label=u'T(Ω)/damp')
ax.plot(s, s, 'r--', label=u'Ω')
ax.plot(speed, accel, '-', lw=0.3)
ax.legend(loc='upper right')
# 3D phase portrait:
#mlab.plot3d(angle, speed, accel, tube_radius=None, opacity=0.3, color=(0,0,1))
# 3D auto-regression plot:
#from mayavi import mlab
#mlab.points3d(angle[:-1], speed[:-1], speed[1:], mode='point')