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GP_NMPC_batch.py
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# import necessary packages
import numpy as np
from os import getcwd
import pickle
from scipy.spatial import distance
from Problem_definition import *
from scipy.optimize import minimize
import math as math
from pylab import *
from scipy.io import savemat
from scipy.spatial.distance import cdist
from casadi import *
from matplotlib.colors import LinearSegmentedColormap
from scipy.spatial.distance import cdist
import numpy.random as random
from scipy.stats import qmc
from pyDOE import *
from scipy.stats import beta
import sys
import os
class GP_batch:
def __init__(self):
# Variable definitions
self.nk, self.ndat0, self.tf, self.x0, self.backoff_repeats,\
self.MC_n_iter, self.backoff_MC, self.learning, self.Lsolver, self.c_code\
,self.state_dep, self.multi_hyper, self.filter_par = specifications()
self.xd, self.xa, self.u, self.ODEeq, self.Aeq, self.Obj_M, self.Obj_L,\
self.R, self.u_min, self.u_max, self.states, self.algebraics, self.inputs,\
self.ngp, self.gpfcn, self.pgp, self.Sigma_v, self.nd, self.na, self.nu,\
self.path, self.Sigma_w, self.eps, self.Sigma_w0 = DAE_system()
self.deltat = self.tf/self.nk
# Internal function calls
self.covSEfcn = self.covSEard()
self.Xdat, self.Ydat = self.generate_data()
self.Xnorm, self.Ynorm, self.stdX, self.stdY, self.meanX, self.meanY \
= self.normalize_data()
self.hypopt, self.invKopt = self.determine_hyperparameters()
self.meanfcn, self.varfcn, self.meanfcn2, self.varfcnsd\
= self.GP_predictor()
self.Xdat, self.Ydat = self.GP_datareduction()
self.Xnorm, self.Ynorm, self.stdX, self.stdY, self.meanX, self.meanY \
= self.normalize_data()
self.hypopt, self.invKopt = self.determine_hyperparameters()
self.meanfcn, self.varfcn, self.meanfcn2, self.varfcnsd\
= self.GP_predictor()
def model_fcn(self):
xd, xa, u, xu, ODEeq, Aeq = self.xd, self.xa, self.u, self.xu, self.ODEeq, self.Aeq
t = SX.sym("t")
p_s = SX.sym("p_s")
xddot = SX.sym("xddot",self.nd)
res = []
for i in range(self.nd):
res = vertcat(res,ODEeq[i]*p_s - xddot[i])
for i in range(self.na):
res = vertcat(res,Aeq[i])
ffcn = Function('ffcn', [t,xddot,xd,xa,xu,u,p_s],[res])
return ffcn
def simulator(self,xd_previous,uNMPC,t0,tf):
''' Simulates the Dynamic "real" system given an initial state, t0, tf '''
xd, xa, u, ODEeq, Aeq = self.xd, self.xa, self.u, self.ODEeq, self.Aeq
ODE = []
for i in range(self.nd):
ODE = vertcat(ODE,substitute(ODEeq[i],u,SX(DM(uNMPC))))
A = []
for i in range(self.na):
A = vertcat(A,substitute(Aeq[i],u,SX(DM(uNMPC))))
dae = {'x':xd, 'z':xa, 'ode':ODE, 'alg':A}
I = integrator('I', 'idas', dae, {'t0':t0, 'tf':tf, 'abstol':1e-12, \
'reltol':1e-12})
res = I(x0=xd_previous)
xd_current = np.array(res['xf'])
xa_current = np.array(res['zf'])
return xd_current, xa_current
def generate_data(self):
''' Generates data to train initial GP '''
ndat0, nu = 300, self.nu
nd, x0, deltat = self.nd, self.x0, self.deltat
Sigma_v, tf = self.Sigma_v, self.tf
u_min, u_max = self.u_min, self.u_max
nk, simulator = self.nk, self.simulator
Xdat = np.zeros((ndat0,nd+nu))
Ydat = np.zeros((ndat0,nd))
nruns = int(ndat0*deltat/tf)
sampler_Sobol1 = qmc.Sobol(d=nu, scramble=True)
udat1 = sampler_Sobol1.random(ndat0)
x_min, x_max = np.array([0.,50.,0.]), np.array([20.,800.,0.18])
sampler_Sobol2 = qmc.Sobol(d=nu+nd, scramble=True)
Xdat = sampler_Sobol2.random(ndat0)
original = False
if original:
i = 0
for j in range(nruns*100):
xd_current = x0
t0, tf = 0., 0.
for k in range(nk):
if i >= ndat0:
break
udat = u_min + udat1[i,:]*(u_max-u_min)
tf += deltat
xtemp = np.array(vertcat(DM(xd_current),udat)).flatten()
xd_current, xa_current = simulator(xd_current,DM(udat),t0,tf)
xd_current = xd_current.flatten()
if xd_current[1] < 900.:
Ydat[i,:] = np.array(xd_current).flatten() +\
np.random.multivariate_normal(np.zeros(nd),Sigma_v)
Xdat[i,:] = xtemp
i += 1
t0 += deltat
else:
for i in range(ndat0):
xdat = x_min + Xdat[i,:nd]*(x_max-x_min)
udat = u_min + Xdat[i,nd:]*(u_max-u_min)
Xdat[i,:] = np.hstack((xdat,udat)).flatten()
xd_current, xa_current = simulator(DM(xdat),DM(udat),0.,deltat)
xd_current = xd_current.flatten()
Ydat[i,:] = np.array(xd_current).flatten() +\
np.random.multivariate_normal(np.zeros(nd),Sigma_v)
return Xdat, Ydat
def GP_datareduction(self):
Xnorm, ndat0 = self.Xnorm, self.ndat0
Xdat, Ydat = self.Xdat, self.Ydat
varfcnsd = self.varfcnsd
sf2opt = np.exp(2.*self.hypopt[self.nd+self.nu,:])
for i in range(300):
distmat = np.zeros((300-i))
for j in range(300-i):
distmat[j] = sum1(varfcnsd(Xnorm[j,:])/sf2opt)
Xnorm2D = Xnorm[j,:].reshape(1,self.nd+self.nu)
indexsort = np.argsort(distmat)
Xnorm = Xnorm[indexsort[1:],:]
Xdat = Xdat[indexsort[1:],:]
Ydat = Ydat[indexsort[1:],:]
if Xnorm.shape[0]==ndat0:
break
return Xdat, Ydat
def normalize_data(self):
''' Routine that outputs normalization utililies '''
nX, nd = self.nd + self.nu, self.nd
Xdat, Ydat = self.Xdat, self.Ydat
ndat = Xdat.shape[0]
Xnorm, Ynorm = np.zeros((ndat,nX)), np.zeros((ndat,nd))
stdX , stdY = np.std(Xdat,0) , np.std(Ydat,0)
meanX, meanY = np.mean(Xdat,0), np.mean(Ydat,0)
for i in range(ndat):
Xnorm[i,:] = (Xdat[i,:] - meanX)/stdX
for i in range(ndat):
Ynorm[i,:] = (Ydat[i,:] - meanY)/stdY
return Xnorm, Ynorm, stdX, stdY, meanX, meanY
def initialize_back_offs(self):
''' Routine that initializes the backoffs '''
nk, backoff_repeats, ngp = self.nk, self.backoff_repeats, self.ngp
Conp_back_off = np.zeros((ngp, nk))
ALL_Conp_back_off = np.zeros((ngp, nk, backoff_repeats))
return Conp_back_off, ALL_Conp_back_off
def determine_hyperparameters(self):
nd, nu = self.nd, self.nu
Xnorm, Ynorm, nX = self.Xnorm, self.Ynorm, nd + nu
ndat = Xnorm.shape[0]
lb = np.array([-3.]*(nX+1) + [-8.])
ub = np.array([3.]*(nX+1) + [ 4.])
bounds = np.hstack((lb.reshape(nX+2,1),ub.reshape(nX+2,1)))
multi_start = self.multi_hyper
sampler_Sobol3 = qmc.Sobol(d=nX+2, scramble=True)
multi_startvec = sampler_Sobol3.random(multi_start)
options = {'disp':False,'maxiter':10000}
hypopt = np.zeros((nX+2,nd))
localsol = [0.]*multi_start
localval = np.zeros((multi_start))
invKopt = []
for i in range(nd):
for j in range(multi_start):
self.print2(['multi_start hyper parameter optimization iteration = ',j,' state = ',i])
hyp_init = lb + (ub-lb)*multi_startvec[j,:]
res = minimize(self.negative_loglikelihood,hyp_init,args=(Xnorm,Ynorm[:,i])\
,method='SLSQP',options=options,bounds=bounds,tol=1e-12)
localsol[j] = res.x
localval[j] = res.fun
minindex = np.argmin(localval)
hypopt[:,i] = localsol[minindex]
ellopt = np.exp(2.*hypopt[:nX,i])
sf2opt = np.exp(2.*hypopt[nX,i])
sn2opt = np.exp(2.*hypopt[nX+1,i]) + 1e-6
cov_mat = self.calc_cov_matrix(Xnorm,ellopt,sf2opt) + sn2opt*np.eye(ndat)
invKopt += [np.linalg.solve(cov_mat,np.eye(ndat))]
return hypopt, invKopt
def Online_MatrixInv(self,Conv_inv,xnorm,sf2,sn2,Xsample,ell):
# array manipulation
calc_cov_sample = self.calc_cov_sample
k = calc_cov_sample(xnorm,Xsample,ell,sf2)
A22 = np.array([sn2]); A22 = A22.reshape((1,1))
A12 = k.reshape((k.shape[0], 1))
A21 = k.reshape((1,k.shape[0]))
I = Conv_inv
II = np.matmul(A21,I)
III = np.matmul(I,A12)
IV = np.matmul(A21,III)
V = IV - A22
VI = 1./V
C12 = III * VI
C21 = VI * II
VII = np.matmul(III,C21)
C11 = I - VII
C22 = -VI
C = np.block([[C11,C12],[C21,C22]])
return C
def GP_predictor(self):
nd, invKopt, hypopt = self.nd, self.invKopt, self.hypopt
Ynorm, Xnorm = SX(DM(self.Ynorm)), SX(DM(self.Xnorm))
ndat = Xnorm.shape[0]
nX, covSEfcn, nk = self.nd + self.nu, self.covSEfcn, self.nk
stdX, stdY, meanX, meanY = SX(self.stdX),SX(self.stdY),SX(self.meanX),SX(self.meanY)
x = SX.sym('x',nX)
xnorm = (x - meanX)/stdX
k = SX.zeros(ndat)
k2 = SX.zeros(ndat+nk)
mean = SX.zeros(nd)
mean2 = SX.zeros(nd)
var = SX.zeros(nd)
Xnorm2 = SX.sym('Xnorm2',ndat+nk,nX)
invKY2 = SX.sym('invKY2',ndat+nk,nd)
for i in range(nd):
invK = SX(DM(invKopt[i]))
hyper = SX(DM(hypopt[:,i]))
ellopt, sf2opt = exp(2*hyper[:nX]), exp(2*hyper[nX])
for j in range(ndat):
k[j] = covSEfcn(xnorm,Xnorm[j,:],ellopt,sf2opt)
for j in range(ndat+nk):
k2[j] = covSEfcn(xnorm,Xnorm2[j,:],ellopt,sf2opt)
invKYnorm = mtimes(invK,Ynorm[:,i])
mean[i] = mtimes(k.T,invKYnorm)
mean2[i] = mtimes(k2.T,invKY2[:,i])
var[i] = sf2opt - mtimes(mtimes(k.T,invK),k)
meanfcn = Function('meanfcn',[x],[mean*stdY + meanY])
meanfcn2 = Function('meanfcn2',[x,Xnorm2,invKY2],[mean2*stdY + meanY])
varfcn = Function('varfcn',[x] ,[var*stdY**2])
varfcnsd = Function('varfcnsd',[x],[var])
return meanfcn, varfcn, meanfcn2, varfcnsd
def GP_predictor_np(self, x, invKsample, Xsample, Ysample):
nd, hypopt = self.nd, self.hypopt
nX = self.nd + self.nu
stdX, stdY, meanX, meanY = self.stdX, self.stdY, self.meanX, self.meanY
calc_cov_sample = self.calc_cov_sample
Sigma_w = self.Sigma_w
xnorm = (x - meanX)/stdX
mean = np.zeros(nd)
var = np.zeros(nd)
for i in range(nd):
invK = invKsample[i]
hyper = hypopt[:,i]
ellopt, sf2opt = np.exp(2*hyper[:nX]), np.exp(2*hyper[nX])
k = calc_cov_sample(xnorm,Xsample,ellopt,sf2opt)
mean[i] = np.matmul(np.matmul(k.T,invK),Ysample[:,i])
var[i] = sf2opt + Sigma_w[i,i]/stdY[i]**2 - np.matmul(np.matmul(k.T,invK),k)
mean_sample = mean*stdY + meanY
var_sample = var*stdY**2
return mean_sample, var_sample
def calc_cov_sample(self,xnorm,Xnorm,ell,sf2):
nd, nu = self.nd, self.nu
n, D = Xnorm.shape
dist = cdist(Xnorm, xnorm.reshape(1,nd+nu), 'seuclidean', V=ell)**2
cov_matrix = sf2 * np.exp(-.5*dist)
return cov_matrix
def covSEard(self):
nd, nu = self.nd, self.nu
ell = SX.sym('ell',nd+nu)
sf2 = SX.sym('sf2')
x, z = SX.sym('x',nd+nu), SX.sym('z',nd+nu)
dist = sum1((x - z)**2 / ell)
covSEfcn = Function('covSEfcn',[x,z,ell,sf2],[sf2*exp(-.5*dist)])
return covSEfcn
def calc_cov_matrix(self,Xnorm,ell,sf2):
dist = cdist(Xnorm,Xnorm,'seuclidean',V=ell)**2
cov_matrix = sf2*np.exp(-0.5*dist)
return cov_matrix
def negative_loglikelihood(self,hyper,X,Y):
n, nX = X.shape[0], X.shape[1]
ell = np.exp(2*hyper[:nX])
sf2 = np.exp(2*hyper[nX])
lik = np.exp(2*hyper[nX+1])
K = self.calc_cov_matrix(X,ell,sf2)
K = K + (lik+1e-8)*np.eye(n)
K = (K + K.T)*0.5
L = np.linalg.cholesky(K)
logdetK = 2 * np.sum(np.log(np.diag(L)))
invLY = np.linalg.solve(L,Y)
alpha = np.linalg.solve(L.T,invLY)
NLL = np.dot(Y.T,alpha) + logdetK
return NLL
def compute_Conp(self, x_opt, Conp_MC, bo_MC):
''' collect data computed by the MPC routine for path constraints '''
nk, gpfcn = self.nk, self.gpfcn
for step in range(nk):
Conp_MC[:,step,bo_MC] = np.array(DM(gpfcn(x_opt[:,step+1]))).flatten()
return Conp_MC
def collect_MC_data(self, U_data, Xd_data, Xa_data, Conp_data, Cont_data,\
t_data, x_opt, u_opt, un):
''' collect data computed by the MPC routine '''
nk = self.nk
Xd_data[:,:,un] = x_opt
U_data[:,:,un] = u_opt
for step in range(nk+1):
Conp_data[:,step,un] = np.array(DM(self.gpfcn(x_opt[:,step]))).flatten()
Cont_data[:,step,un] = np.array(DM(self.gtfcn(x_opt[:,step]))).flatten()
return U_data, Xd_data, Xa_data, Conp_data, Cont_data, t_data
def MPC_params(self, invK, Xmeasure, Ymeasure):
nk, ndat0, nd = self.nk, self.ndat0, self.nd
calc_cov_sample = self.calc_cov_sample
ndat = invK[0].shape[0]
nX = self.nd + self.nu
invK_MPC = np.zeros((nk+ndat0, nk+ndat0))
X_MPC = np.zeros((nk+ndat0, nX))
par = np.zeros(((ndat0+nk)**2)*nd + (nk+ndat0)*nX + (nk+ndat0)*(nd))
for ij in range(nd):
invK_MPC[:ndat,:ndat] = invK[ij]
par[((ndat0+nk)**2)*ij:((ndat0+nk)**2)*(ij+1)] =\
np.array(reshape(invK_MPC,((ndat0+nk)**2),1)).flatten()
X_MPC[:ndat,:] = Xmeasure
par[((ndat0+nk)**2)*nd: ((ndat0+nk)**2)*nd+(nk+ndat0)*nX] =\
np.array(reshape(X_MPC, (nk+ndat0)*nX, 1)).flatten()
y = Ymeasure[:,ij].flatten()
y = np.concatenate((y,np.zeros(nk+ndat0-ndat)))
par[((ndat0+nk)**2)*nd+(nk+ndat0)*nX+(nk+ndat0)*(ij) :\
((ndat0+nk)**2)*nd+(nk+ndat0)*nX+(nk+ndat0)*(ij+1)] = y
return par
def OCP_step_GP(self, x_opt, Ufcn_, res, u_opt, step, xd_current, invKsample,
Xsample, Ysample, sf2, ell):
Sigma_v, nd = self.Sigma_v, self.nd
GP_predictor_np = self.GP_predictor_np
meanY, stdY = self.meanY, self.stdY
meanX, stdX, ngp = self.meanX, self.stdX, self.ngp
Online_MatrixInv = self.Online_MatrixInv
u_ = np.array(Ufcn_(np.array(res["x"][:,0])))
u_opt[:,step] = u_[:,0]
xnew_measured = xd_current
xnew_measured = np.concatenate((xnew_measured.reshape(nd,1),u_), axis=None)
xnew = np.concatenate((xd_current.reshape(nd,1),u_), axis=None)
xd_Mean, xd_Sigma = GP_predictor_np(xnew, invKsample, Xsample, Ysample)
xd_Sigma = xd_Sigma*np.eye(nd)
xd_current = (xd_Mean
+ np.random.multivariate_normal(np.zeros(nd),xd_Sigma))
xd_measured = xd_current
x_opt[:,step+1] = xd_current[:]
xd_current_norm = (xd_current - meanY)/stdY
xnew_norm = (xnew - meanX)/stdX
for ij in range(nd):
invKsample[ij]= Online_MatrixInv(invKsample[ij],xnew_norm,sf2[ij],1e-6,Xsample,ell[ij])
Xsample = np.vstack((Xsample,xnew_norm))
Ysample = np.vstack((Ysample,xd_current_norm))
return Xsample, Ysample, invKsample, x_opt, xd_current, xnew_measured, xd_measured, u_opt
def OCP_step_Plant(self, x_opt, Ufcn_, res, u_opt, step, xd_current, tfi, t0is, MC_i):
Sigma_v, nd, simulator = self.Sigma_v, self.nd, self.simulator
Sigma_w = self.Sigma_w
u_ = np.array(Ufcn_(np.array(res["x"][:,0])))
u_opt[:,step,MC_i] = u_[:,0]
xnew_measured = xd_current
xnew_measured = np.concatenate((xnew_measured.reshape(nd,1),u_), axis=None)
xd_current, _ = simulator(xd_current,u_,t0is,tfi)
xd_current = xd_current.flatten() + np.random.multivariate_normal(np.zeros(nd),Sigma_w).flatten()
xd_measured = xd_current
x_opt[:,step+1] = xd_current[:]
return x_opt, xd_current, xnew_measured, xd_measured, u_opt
def Compute_beta(self,ngp,Conp_back_off,Conp_MC):
path = self.path
S = Conp_MC.shape[2]
alpha = 0.01 # confidence interval of the cdf
scaling = [800.,0.01,200.]
bj = 0.
for j in range(S):
ai = 0.
for i in range(ngp):
if path[i]:
ai += np.sum(Conp_MC[i,:,j]/scaling[i] >= 1e-4)
else:
ai += np.sum(Conp_MC[i,-1,j]/scaling[i] >= 1e-4)
bj += ai > 0.
beta_cor = 1. - bj/S
beta_ = 1. - beta.ppf(1. - alpha, S+1-beta_cor*S, beta_cor*S)
return beta_
def update_backoff(self,ngp,Conp_nominal0,Conp_MC0,backoff_factor):
pgp, nk = np.float(self.pgp), self.nk
Conp_nominal0 = np.reshape(Conp_nominal0, (ngp, nk) ,order='F')
F_inv = np.percentile(Conp_MC0, (1.-np.float(pgp))*100., axis=2)
Conp_back_off = (F_inv - Conp_nominal0)*backoff_factor
return Conp_back_off
def load_varsopt(self, MC_i, step, args):
if MC_i != 0:
try:
with open("varsopt_dir/varsopt" + str(step)+".pkl", 'rb') as a_file:
args[step]["x0"] = pickle.load(a_file)
except:
self.print2(["error loading, step = ",step])
return
def save_varsopt(self, step, res):
try:
with open("varsopt_dir/varsopt" + str(step)+".pkl", 'wb') as a_file:
pickle.dump(np.array(res["x"]), a_file)
except:
self.print2(["error saving, step = ",step])
return
def plot_results_Plant(self, Xd_plant2, u_opt2, Conp_plant2, Eobj_GP_MC, PorGP, folder):
''' Plot results and save to files '''
nd, nu = self.nd, self.nu
ngp, MC_n_iter = self.ngp, self.MC_n_iter
nk, tf = self.nk, self.tf
t_X = np.linspace(0, tf, nk+1, endpoint=True)
for j in range(nd):
plt.figure()
for i in range(MC_n_iter):
plt.plot(t_X,list(Xd_plant2[j,:,i]),'-')
plt.ylabel('plant x_'+str(j))
plt.xlabel('time')
#plt.xlim([0,np.ndarray.max(t_pasts[-1,:])])
plt.savefig(folder+'/'+'x_'+str(j)+PorGP+
'_.png', dpi=150)
plt.close()
for j in range(nu):
plt.figure()
for i in range(MC_n_iter):
plt.step(t_X[:-1],list(u_opt2[j,:,i]),'-')
plt.ylabel('plant control u_'+str(j))
plt.xlabel('time')
#plt.xlim([0,np.ndarray.max(t_pasts[-1,:])])
plt.savefig(folder+'/'+'u_'+str(j)+PorGP+
'.png', dpi=150)
plt.close()
Conp_data_mean = np.mean(Conp_plant2, axis=2)
Conp_data_std = np.std(Conp_plant2, axis=2)
for j in range(ngp):
plt.figure()
for i in range(MC_n_iter):
plt.plot(t_X,list(Conp_plant2[j,:,i]),'--', color='grey')
plt.plot(t_X,list((Conp_data_mean+Conp_data_std)[j,:]),'-', color='black')
plt.plot(t_X,list((Conp_data_mean-Conp_data_std)[j,:]),'-', color='black')
plt.plot(t_X,[0.0 for i in range(len(Conp_data_mean[j,:]))],'-.', color='black')
plt.ylabel(PorGP+'_path_constraint_'+str(j))
plt.xlabel('time')
#plt.xlim([0,np.ndarray.max(t_pasts[-1,:])])
plt.savefig(folder+'/'+PorGP+'path constraint '
+str(j)+'.png', dpi=150)
plt.close()
plt.figure()
plt.plot(list(Eobj_GP_MC),'--')
plt.ylabel('objective')
plt.xlabel('iterations')
#plt.xlim([0,np.ndarray.max(t_pasts[-1,:])])
plt.savefig(folder+'/'+'objective in back-offs.png', dpi=150)
plt.close()
return
def plot_results_GP(self, Xd_MC, Conp_MC, Ud_MC, Conp_nominal, PorGP, folder, ALL_Conp_back_off):
''' Plot results and save to files '''
nd, nu = self.nd, self.nu
ngp, backoff_repeats = self.ngp, self.backoff_repeats
nk, tf, backoff_MC = self.nk, self.tf, self.backoff_MC
t_X = np.linspace(0, tf, nk+1, endpoint=True)
t_U = np.linspace(0, tf, nk, endpoint=False)
for j in range(nd):
for k in range(backoff_repeats):
plt.figure()
for i in range(backoff_MC):
plt.plot(t_X,list(Xd_MC[j,:,i,k]),'-')
plt.ylabel(PorGP+' repeat '+str(k)+' x_'+str(j))
plt.xlabel('time')
#plt.xlim([0,np.ndarray.max(t_pasts[-1,:])])
plt.savefig(folder+'/'+' repeat '+str(k)+'x_'+str(j)+PorGP+
'.png', dpi=150)
plt.close()
for j in range(nu):
for k in range(backoff_repeats):
plt.figure()
for i in range(backoff_MC):
plt.step(t_X[:-1],list(Ud_MC[j,:,i,k]),'-')
plt.ylabel(PorGP+' repeat '+str(k)+' u_'+str(j))
plt.xlabel('time')
#plt.xlim([0,np.ndarray.max(t_pasts[-1,:])])
plt.savefig(folder+'/'+' repeat '+str(k)+'u_'+str(j)+PorGP+
'.png', dpi=150)
plt.close()
Conp_data_mean = np.mean(Conp_MC, axis=2)
Conp_data_std = np.std(Conp_MC, axis=2)
''' plot of last iteration of the back-off '''
for j in range(ngp):
plt.figure()
for i in range(backoff_MC):
plt.plot(t_X[1:],list(Conp_MC[j,:,i]),'--', color='grey')
plt.plot(t_X[1:],list(Conp_nominal[j,:]),'-', color='black')
plt.plot(t_X[1:],[0.0 for i in range(len(Conp_data_mean[j,:]))],'-.', color='black')
plt.ylabel(PorGP+'constraint '+str(j))
plt.xlabel('time')
#plt.xlim([0,np.ndarray.max(t_pasts[-1,:])])
plt.savefig(folder+'/'+PorGP+
' path constraint '+str(j)+'.png', dpi=150)
plt.close()
c_ = [(backoff_repeats - float(i))/backoff_repeats for i in range(backoff_repeats)]
for j in range(ngp):
plt.figure()
for i in range(backoff_repeats):
plt.plot(t_X[1:],list(ALL_Conp_back_off[j,:,i]),'--', color=str(c_[i]))
plt.ylabel(PorGP+'back-off '+str(j))
plt.xlabel('time')
#plt.xlim([0,np.ndarray.max(t_pasts[-1,:])])
plt.savefig(folder+'/'+PorGP+
' ALL path constraint '+str(j)+'.png', dpi=150)
plt.close()
return
def save_to_file(self, input_list, ref, folder):
names_l = ['Xd_MC', 'Conp_MC', 'Ud_MC', 'ALL_Conp_back_off', 'Xd_plant2', 'u_opt2',
'Conp_plant2', 'Eobj_GP_MC','Conp_nominal', 'betamat', 'backoff_factormat']
for li in range(len(input_list)):
with open('PlotFiles/'+folder+'/'+str(names_l[li])+ref+"_file"+".pkl", 'wb') as a_file:
pickle.dump(input_list[li], a_file)
def save_to_file_nom(self, input_list, ref, folder):
names_l = ['Xd_plant_nom', 'u_opt_nom', 'Conp_plant_nom']
for li in range(len(input_list)):
with open('PlotFiles/'+folder+'/'+str(names_l[li])+ref+"_file"+".pkl", 'wb') as a_file:
pickle.dump(input_list[li], a_file)
def GP_nominal_backoff(self,nk_sh,precompile,c_code):
''' Optimization of nominal (mean) GP incorporating backoff constraints '''
meanfcn2, nd, nu = self.meanfcn2, self.nd, self.nu
Xdat, u_min, u_max = self.Xdat, self.u_min, self.u_max
gpfcn, ngp = self.gpfcn, self.ngp
Obj_M, Obj_L, nX = self.Obj_M, self.Obj_L, (self.nd + self.nu)
ndat0, path = self.ndat0, self.path
Lsolver, nk = self.Lsolver, self.nk
state_dep, hypopt = self.state_dep, self.hypopt
varfcnsd, R = self.varfcnsd, SX(DM(self.R))
# Define optimization variables
par = MX.sym("par",nd+ngp*nk + ((ndat0+nk)**2)*nd\
+ nX*(ndat0+nk) + (nk+ndat0)*nd + nu)
varsopt = MX.sym("varsopt",nd*nk_sh+nu*nk_sh)
vars_lb = np.zeros(nd*nk_sh+nu*nk_sh)
vars_ub = np.zeros(nd*nk_sh+nu*nk_sh)
vars_init = np.zeros(nd*nk_sh+nu*nk_sh)
XD = np.resize(np.array([],dtype=MX),(nk_sh+1))
U = np.resize(np.array([],dtype=MX),(nk_sh))
offset = 0
XD[0] = par[:nd]
Conp_back_off = reshape(par[nd:nd + ngp*nk],ngp,nk)
parmean = par[nd+ngp*nk : nd+ngp*nk + ((ndat0+nk)**2)*nd
+ nX*(ndat0+nk) + (nk+ndat0)*nd]
invKpar = parmean[:((ndat0+nk)**2)*nd]
Xnormpar = parmean[((ndat0+nk)**2)*nd : ((ndat0+nk)**2)*nd + nX*(ndat0+nk)]
Ypar = parmean[((ndat0+nk)**2)*nd + nX*(ndat0+nk) :\
((ndat0+nk)**2)*nd + nX*(ndat0+nk) + (nk+ndat0)*nd]
Xnorm2 = reshape(Xnormpar,ndat0+nk,nX)
u0 = par[nd+ngp*nk + ((ndat0+nk)**2)*nd\
+ nX*(ndat0+nk) + (nk+ndat0)*nd:\
nd+ngp*nk + ((ndat0+nk)**2)*nd\
+ nX*(ndat0+nk) + (nk+ndat0)*nd +nu]
Ya = SX.sym('Ya',nk+ndat0)
invKa = SX.sym('invKa',nk+ndat0,nk+ndat0)
invKYfcn = Function('invKYfcn',[invKa,Ya],[mtimes(invKa,Ya)])
invKY2 = MX.zeros(ndat0+nk,nd)
for i in range(nd):
invK2 = reshape(invKpar[i*((ndat0+nk)**2):\
(i+1)*((ndat0+nk)**2)],ndat0+nk,ndat0+nk)
Y2 = Ypar[i*(ndat0+nk) : (i+1)*(ndat0+nk)]
invKY2[:,i] = invKYfcn(invK2,Y2)
for i in range(nk_sh):
XD[i+1] = varsopt[offset:offset+nd]
vars_lb[offset:offset+nd] = np.ones(nd)*-inf
vars_ub[offset:offset+nd] = np.ones(nd)*inf
vars_init[offset:offset+nd] = np.array(Xdat[i+(nk-nk_sh),:nd])
offset += nd
U[i] = varsopt[offset:offset+nu]
vars_lb[offset:offset+nu] = u_min
vars_ub[offset:offset+nu] = u_max
vars_init[offset:offset+nu] = Xdat[i+(nk-nk_sh),nd:]
offset += nu
# Define constraints
g = []
lbg = []
ubg = []
for i in range(nk_sh):
g += [meanfcn2(vertcat(XD[i],U[i]),Xnorm2,invKY2) - XD[i+1]]
lbg.append(np.zeros(nd))
ubg.append(np.zeros(nd))
for i in range(nk_sh):
for j in range(ngp):
if path[j]:
g += [gpfcn(XD[i+1])[j] + Conp_back_off[j,(nk-nk_sh) + i]]
lbg.append(np.ones(1)*-inf)
ubg.append(np.zeros(1))
else:
if i == (nk_sh-1):
g += [gpfcn(XD[i+1])[j] + Conp_back_off[j,(nk-nk_sh) + i]]
lbg.append(np.ones(1)*-inf)
ubg.append(np.zeros(1))
# Define objective
Obj = 0
if state_dep:
sf2 = exp(2*hypopt[nd+nu,:])
beta = 15.
Obj += beta*(sum1(varfcnsd(vertcat(XD[0],U[0]))/sf2))
for i in range(nk_sh):
Obj += Obj_L(XD[i+1],U[i])
Obj += Obj_M(XD[nk_sh])
# Control penality
u1 = SX.sym('u1',nu)
u2 = SX.sym('u2',nu)
dufcn = Function('dufcn',[u1,u2],[mtimes(mtimes(transpose(u2-u1),R),u2-u1)])
deltau = MX.zeros(1)
for k in range(nk_sh-1):
if k == 0:
deltau += dufcn(u0,U[k])
else:
deltau += dufcn(U[k],U[k+1])
Obj += deltau
# Define NLP
opts = {}
opts["expand"] = True
opts["ipopt.print_level"] = 0
opts["ipopt.max_iter"] = 1000
opts["ipopt.tol"] = 1e-8
opts["ipopt.linear_solver"] = Lsolver
opts["calc_lam_p"] = False
opts["calc_multipliers"] = False
nlp = {'x':varsopt,'p':par,'f':Obj,'g':vertcat(*g)}
soname = 'nlp' + str(nk_sh) + '.so'
if (precompile or (not c_code)):
GPNMPC = nlpsol("solver","ipopt",nlp,opts)
else:
GPNMPC = nlpsol("solver","ipopt",soname)
Ufcn = Function('Ufcn',[varsopt],[U[0]])
args = {}
args["lbx"] = vars_lb
args["ubx"] = vars_ub
args["x0"] = vars_init
args["lbg"] = np.concatenate(lbg)
args["ubg"] = np.concatenate(ubg)
return GPNMPC, Ufcn, args
def GP_backoff_computation(self, backoff_repeats, GPNMPC2,
Ufcn2, args2,Conp_back_off, ALL_Conp_back_off):
''' Compute the backoffs '''
backoff_MC = self.backoff_MC #MC iterations to get average violation
backoff_repeats = self.backoff_repeats
MPC_run_scenarioGP = self.MPC_run_scenarioGP
nd, ngp, nk, nu = self.nd, self.ngp, self.nk, self.nu
compute_Conp = self.compute_Conp
update_backoff = self.update_backoff
MPC_GP_nominal = self.MPC_GP_nominal
hypopt = self.hypopt
Compute_beta = self.Compute_beta
sf2 = [0]*nd; sn2 = [0]*nd; ell = [0]*nd;
for ii in range(nd):
sf2[ii] = np.exp(2*hypopt[nd+nu,ii])
sn2[ii] = np.exp(2*hypopt[nd+nu+1,ii])
ell[ii] = np.exp(2*hypopt[:nd+nu,ii])
''' Initialize data collectors '''
Xd_MC = np.zeros((nd, nk+1, backoff_MC, backoff_repeats))
Ud_MC = np.zeros((nu, nk, backoff_MC, backoff_repeats))
Eobj_GP_MC = np.zeros((backoff_repeats))
''' back-off iterations '''
backoff_factor_a = 0.
backoff_factor_b = 4.
Conp_MC0 = np.zeros((ngp, nk, backoff_MC))
Conp_pred0 = np.zeros((ngp, nk, backoff_MC))
betamat = np.zeros((backoff_repeats))
backoff_factormat = np.zeros((backoff_repeats))
for un in range(backoff_repeats):
Conp_MC = np.zeros((ngp, nk, backoff_MC))
Conp_pred = np.zeros((ngp, nk, backoff_MC))
obj_MC = np.zeros((backoff_MC))
if un != 0:
backoff_factor_c = (backoff_factor_a + backoff_factor_b)/2.
backoff_factor = backoff_factor_c
else:
backoff_factor = backoff_factor_a
if un != 0:
Conp_back_off = \
update_backoff(ngp,Conp_nominal0,Conp_MC0,backoff_factor)
for bo_MC in range(backoff_MC): # MC iterations to get average violation
''' Solve MPC '''
x_opt, u_opt, obj_f, self.par, x_pred = MPC_run_scenarioGP(
bo_MC, un, GPNMPC2,Ufcn2, args2,sf2, ell, Conp_back_off, sn2)
obj_MC[bo_MC] = obj_f
if un == 0:
''' Compute path constraints '''
Conp_MC0 = compute_Conp(x_opt , Conp_MC0 , bo_MC)
Conp_pred0 = compute_Conp(x_pred, Conp_pred0, bo_MC)
Conp_MC = compute_Conp(x_opt , Conp_MC , bo_MC)
Conp_pred = compute_Conp(x_pred, Conp_pred, bo_MC)
''' Collect states last iteration MC '''
Xd_MC[:,:,bo_MC, un] = x_opt
Ud_MC[:,:,bo_MC, un] = u_opt
self.print2(['Conp_back_off = ',Conp_back_off])
''' Nominal path constraints '''
if un == 0:
x_nominal0, u_nominal0 = \
MPC_GP_nominal(GPNMPC2, Ufcn2, args2, sf2, ell, Conp_back_off)
Conp_nominal0 = np.zeros((ngp,nk,1))
Conp_nominal0 = compute_Conp(x_nominal0,Conp_nominal0,0)
x_nominal, u_nominal = \
MPC_GP_nominal(GPNMPC2, Ufcn2, args2, sf2, ell, Conp_back_off)
Conp_nominal = np.zeros((ngp,nk,1))
Conp_nominal = compute_Conp(x_nominal,Conp_nominal,0)
beta_ = Compute_beta(ngp,Conp_back_off,Conp_MC)
''' collect results '''
ALL_Conp_back_off[:,:,un] = Conp_back_off
Eobj_GP_MC[un] = np.mean(obj_MC, axis=0)
betamat[un] = beta_
backoff_factormat[un] = backoff_factor
if un != 0:
beta_c = beta_ - (1.-self.eps)
else:
beta_a = beta_ - (1.-self.eps)
if un != 0:
if np.sign(beta_c) == np.sign(beta_a):
backoff_factor_a = backoff_factor_c
beta_a = beta_c
else:
backoff_factor_b = backoff_factor_c
beta_b = beta_c
return Xd_MC, Conp_MC, Eobj_GP_MC, Conp_back_off, Ud_MC,\
ALL_Conp_back_off, Conp_nominal, Conp_pred, backoff_factor,\
beta_, backoff_factor_a, backoff_factor_b, beta_a, beta_c, betamat, backoff_factormat
def MPC_GP_nominal(self, GPNMPC2, Ufcn2, args2, sf2, ell, Conp_back_off):
''' simulates the nominal MPC '''
nd, invK, nu = self.nd, self.invKopt[:], self.nu
xd_current, nk = self.x0, self.nk
GP_predictor_np = self.GP_predictor_np
ngp,MPC_params = self.ngp, self.MPC_params
load_varsopt = self.load_varsopt
save_varsopt = self.save_varsopt
Xnorm, Ynorm = self.Xnorm[:], self.Ynorm[:]
meanfcn2 = self.meanfcn2
u_min = self.u_min
x_nominal = np.zeros((nd,nk+1)); u_nominal = np.zeros((nu,nk))
x_nominal[:,0] = xd_current[:]
u_ = u_min
for step in range(nk):
self.print2(['Computing nominal MPC, step = ',step])
load_varsopt(1,step,args2)
p_backoff = Conp_back_off.reshape((ngp*nk) ,order='F')
if self.learning:
par = MPC_params(invK,Xnorm[:],Ynorm[:])
else:
par = MPC_params(self.invKopt[:],self.Xnorm[:],self.Ynorm[:])
args2[step]["p"] = np.concatenate((xd_current,p_backoff,par,u_.flatten())) # define current value of states
res = GPNMPC2[step](**args2[step]) # solve nominal model (mean GP)
self.print2(['solver status = ',GPNMPC2[step].stats()['return_status']])
save_varsopt(step, res)
u_ = np.array(Ufcn2[step](np.array(res["x"][:,0])))
u_nominal[:,step] = u_[:,0]
xnew = np.concatenate((xd_current.reshape(nd,1),u_), axis=None)
xd_Mean, _ = GP_predictor_np(xnew, invK, Xnorm, Ynorm)
xd_current = (xd_Mean)
x_nominal[:,step+1] = xd_current[:]
return x_nominal, u_nominal
def MPC_run_scenarioGP(self, bo_MC, un, GPNMPC2, Ufcn2, args2, sf2, ell, Conp_back_off,sn2):
'''
Used to calculate backoffs
Simulates the MPC routine by a data driven approch using the GP from the
scenario perspective and lears on the new data that becomes available.
It also compilates data given a control sequence.
'''
nd, invKsample, invKMPC = self.nd, self.invKopt[:], self.invKopt[:]
Xsample, Ysample = self.Xnorm[:], self.Ynorm[:]
Xmeasure, Ymeasure = self.Xnorm[:], self.Ynorm[:]
xd_current, nk, ngp = self.x0, self.nk, self.ngp
GP_predictor_np, Sigma_v = self.GP_predictor_np, self.Sigma_v
meanY, stdY, meanX, stdX = self.meanY, self.stdY, self.meanX, self.stdX
Online_MatrixInv, OCP_step_GP = self.Online_MatrixInv, self.OCP_step_GP
MPC_params, load_varsopt = self.MPC_params, self.load_varsopt
save_varsopt = self.save_varsopt
u_min = self.u_min
Sigma_w0 = self.Sigma_w0
Sigma_w = self.Sigma_w
meanfcn2 = self.meanfcn2
ndat0 = self.ndat0
''' initialize data collectors '''
xd_current = xd_current + (np.random.multivariate_normal(np.zeros(nd),Sigma_w0))
xd_current = xd_current.clip(min=0)
x_opt = np.zeros((nd,nk+1));
u_opt = np.zeros((self.nu,nk))
x_opt[:,0] = xd_current[:]
u_ = u_min
x_pred = np.zeros((nd,nk+1));
x_pred[:,0] = 0.
invKYMPC = np.zeros((ndat0+nk, nd))
XMPC = np.zeros((ndat0+nk, nd+self.nu))
for step in range(nk):
self.print2(['Computing back-offs Back_off_iter = ',un,' step = ',step,' MC_iter = ', bo_MC])
load_varsopt(un, step, args2)
''' Solves open-loop optimization '''
p_backoff = Conp_back_off.reshape((ngp*nk) ,order='F')
if self.learning:
par = MPC_params(invKMPC,Xmeasure,Ymeasure)
else:
par = MPC_params(self.invKopt[:],self.Xnorm[:],self.Ynorm[:])
args2[step]["p"] = np.concatenate((xd_current,p_backoff,par,u_.flatten()))
res = GPNMPC2[step](**args2[step])
self.print2(['solver status = ',GPNMPC2[step].stats()['return_status']])
save_varsopt(step, res)
''' calculate u(k+1), x(k+1), xm(k+1) and invK(k+1), X(k+1),Y(k+1) '''
Xsample, Ysample, invKsample, x_opt, xd_current, xnew_measured, xd_measured, u_opt = \
OCP_step_GP(x_opt, Ufcn2[step], res, u_opt, step, xd_current,
invKsample,Xsample, Ysample, sf2, ell)
u_ = u_opt[:,step]
''' calculate differences '''
for ij in range(nd):
invKYMPC[:step-nk,ij] = np.array(DM(mtimes(invKMPC[ij],Ymeasure[:,ij]))).flatten()
XMPC[:step-nk,:] = Xmeasure
x_in = np.concatenate((x_opt[:,step],u_opt[:,step]))
x_pred[:,step+1] = np.array(DM(meanfcn2(x_in,XMPC,invKYMPC))).flatten()
''' update invK, X and Y '''
xd_measured_norm = (xd_measured - meanY)/stdY
xnewmeasured_norm = (xnew_measured - meanX)/stdX
for ij in range(nd):
invKMPC[ij] = Online_MatrixInv(invKMPC[ij],xnewmeasured_norm,
sf2[ij],Sigma_w[ij,ij]/(stdY[ij])**2,Xmeasure,ell[ij])
Xmeasure = np.vstack((Xmeasure,xnewmeasured_norm))
Ymeasure = np.vstack((Ymeasure,xd_measured_norm))
if step == (nk-1):
E_obj_data = res["f"]
return x_opt, u_opt, E_obj_data, par, x_pred
def MC_MPC_plant(self, GPNMPC2, Ufcn2, args2, Conp_back_off):
''' Do a multirun MC MPC on the plant include GP learning '''
x0, MC_n_iter, nd, nk = self.x0, self.MC_n_iter, self.nd, self.nk
nu, deltat, Sigma_v = self.nu, self.deltat, self.Sigma_v
meanY, stdY = self.meanY, self.stdY
meanX, stdX, simulator= self.meanX, self.stdX,self.simulator
Online_MatrixInv = self.Online_MatrixInv
gpfcn, ngp, Sigma_v = self.gpfcn, self.ngp, self.Sigma_v
OCP_step_Plant = self.OCP_step_Plant
load_varsopt = self.load_varsopt
save_varsopt = self.save_varsopt
nu, nd = self.nu, self.nd
u_min = self.u_min
hypopt = self.hypopt
Sigma_w0 = self.Sigma_w0
Sigma_w = self.Sigma_w
sf2 = [0]*nd; sn2 = [0]*nd; ell = [0]*nd;
for ii in range(nd):
sf2[ii] = np.exp(2*hypopt[nd+nu,ii])
sn2[ii] = np.exp(2*hypopt[nd+nu+1,ii])
ell[ii] = np.exp(2*hypopt[:nd+nu,ii])
''' initlialize data collectors '''
Xd_plant = np.zeros((nd, nk+1, MC_n_iter))
Conp_plant = np.zeros((ngp,nk+1, MC_n_iter))
u_opt = np.zeros((nu, nk, MC_n_iter))
u_ = u_min
''' each MC run '''
for MC_i in range(MC_n_iter):