def l2_cost_function_noinit(Q, n_states, n_uncert, p):
"""
The expectation cost function is of the form
E(int_{0}^{t} x^Top Q x dt) = int_{0}^{t} X^Top Q_det X dt
"""
xi_symbols = [
symbols('xi' + str(i)) for i in range(1, n_uncert - n_states + 1)
]
xi = Matrix(xi_symbols)
if n_uncert - n_states == 1:
xi = Matrix([symbols('xi')])
Hp = Matrix(GaussHermitePC(n_uncert - n_states, p))
Phi = Matrix(np.kron(np.identity(n_states), Hp.T))
Q_dummy = Phi.T * Q * Phi
n_nodesQuadrature = 20
[weight, node] = UnivariateGaussHermiteQuadrature(n_nodesQuadrature)
# computing the Expectation with respect to the uncertain parameters
for xi_i in xi:
Q_dummy = integrate_gauss(weight, node, Q_dummy, xi_i)
#
Q_det = Q_dummy
#
return np.round(np.array(Q_det, dtype=float), 5)
def quadratic_chance_constraint_noinit(A, n_states, n_uncert, p):
"""
Pr{x^Top A x geq c} \leq eps
Converts to SOCP
"""
xi_symbols = [
symbols('xi' + str(i)) for i in range(1, n_uncert - n_states + 1)
]
xi = Matrix(xi_symbols)
if n_uncert - n_states == 1:
xi = Matrix([symbols('xi')])
Hp = Matrix(GaussHermitePC(n_uncert - n_states, p))
l = Hp.shape[0]
H_dummy = zeros(l, 1)
for i in range(1, l):
H_dummy[i] = Hp[i]
HH_dummy = H_dummy * H_dummy.T
n_nodesQuadrature = 20
[weight, node] = UnivariateGaussHermiteQuadrature(n_nodesQuadrature)
# computing the Expectation with respect to the uncertain parameters
for xi_i in xi:
HH_dummy = integrate_gauss(weight, node, HH_dummy, xi_i)
A_det = np.kron(A, HH_dummy)
return np.round(np.array(A_det, dtype=float), 5)
def initial_conditions(x, x_mean,x_var, num_uncertainty,p):
n_states = int(x.shape[0])
n_uncert = int(num_uncertainty)
# n_uncert-n_states,n_uncert
xi_symbols = [symbols('xi'+str(i)) for i in range(1,n_uncert+1)]
xi = Matrix(xi_symbols)
if n_uncert-n_states == 1:
xi = Matrix([symbols('xi')])
print(xi)
x_subs= zeros(n_states,1)
print(x_subs)
for i in range(n_states):
x_subs[i,0] = x_mean[i] + np.sqrt(x_var[i])*xi[i+int(n_uncert-n_states)]
print(x_subs)
#rescaled Hermite polynomials
Hp = GaussHermitePC(n_uncert,p)
#print(Hp)
# Kronecker Product
Phi = Matrix(np.kron(np.identity(n_states),Hp.T))
left = Phi.T*Phi
right = Phi.T*x_subs
n_nodesQuadrature = 20
[weight,node] = UnivariateGaussHermiteQuadrature(n_nodesQuadrature)
left_int = left
right_int = right
for xi_i in xi:
left_int = integrate_gauss(weight,node,left_int, xi_i)
right_int = integrate_gauss(weight,node,right_int, xi_i)
X_init = np.asarray(left_int.inv()*right_int)
return np.round(np.array(X_init,dtype=float),5)
def linear_chance_constraint_matrices_noinit(n_states, n_uncert, p):
xi_symbols = [
symbols('xi' + str(i)) for i in range(1, n_uncert - n_states + 1)
]
xi = Matrix(xi_symbols)
if n_uncert - n_states == 1:
xi = Matrix([symbols('xi')])
Hp = Matrix(GaussHermitePC(n_uncert - n_states, p))
l = Hp.shape[0]
M = zeros(1, l)
M[0] = 1
H_dummy = zeros(l - 1, 1)
for i in range(1, l):
H_dummy[i - 1] = Hp[i]
HH_dummy = H_dummy * H_dummy.T
n_nodesQuadrature = 20
[weight, node] = UnivariateGaussHermiteQuadrature(n_nodesQuadrature)
# computing the Expectation with respect to the uncertain parameters
for xi_i in xi:
HH_dummy = integrate_gauss(weight, node, HH_dummy, xi_i)
#
EHH = sqrtm(np.array(HH_dummy, float))
H = zeros(l, l)
for i in range(1, l):
for j in range(1, l):
H[i, j] = EHH[i - 1, j - 1]
N = np.kron(np.ones((n_states, n_states)), H)
return M, N
def make_deterministic_sym(f, g, x, u, theta, num_uncertainty, p, init_uncertain = False):
"""
Converts stochasitic dynamics
xdot = f(x,u) + g(x,u, theta)
to deterministic approximation, with certain bounds
# Uses SYmbolic Integration
Parametersyp
----------
f : sympy expression in x and u
the initial deterministic component
g : sympy expression in x, u, and theta
the stochastic component
mu : sympy constant/number
the mean of theta
sigma : sympy constant/number
the variance of theta
x : sympy symbol
the state
u : sympy symbol
the control
theta : sympy symbol
the normally distributed noise
p : order of the polynomial expansion (user design choice)
num_uncertainty: total uncertainty = num_states+num_uncertainparam_dyn
Returns
-------
sympy expression in x and u
the deterministic dynamics
"""
n_states = x.shape[0]
n_uncert = num_uncertainty
dt_symbol = symbols('dt')
if init_uncertain == True:
n_range = n_uncert
else:
n_range = n_uncert - n_states
#===============================================================================#
xi_symbols = [symbols('xi'+str(i)) for i in range(1,n_range+1)]
xi = Matrix(xi_symbols)
#print('xi symbols')
#print(xi)
#Express theta_i in terms of unit gaussian xi_i
if n_range==1:
xi = Matrix([symbols('xi')])
theta_subs= zeros(1,n_uncert-n_states)
for i in range(n_uncert-n_states):
theta_subs[i] = sqrt(dt_symbol)*xi[i]
#print(theta_subs)
for j in range(n_uncert-n_states):
g = g.subs({theta[j]:theta_subs[j]})
#rescaled Hermite polynomials
Hp = GaussHermitePC(n_range,p)
print(Hp)
#expansions of states using the Hermite polynomials
## X in the paper
l = Hp.shape[0]
xc_vals_dummy = zeros(l,n_states)
for i in range(0,n_states):
xc_vals_dummy[:,i] = Matrix([symbols('x'+str(i)+str(j)) for j in range(0,l)])
states = xc_vals_dummy.T.reshape(n_states*l,1)
# print(states)
controls = u.reshape(u.shape[0],1)
# print(controls)
## input to the dynamics
# input_dyn = list(states.col_join(controls).col_join(Matrix([[dt_symbol]])))
# print(input_dyn)
# ## input to the dynamics
# input_dyn = list(states.col_join(controls).col_join(Matrix([[dt_symbol]])))
# following for uncontrolled system
input_dyn = list(states.col_join(Matrix([[dt_symbol]])))
print(input_dyn)
# print(input_dyn.reshape(int(n_states*l+u.shape[0]+1),0))
# Generating the Symbolic Polynomial Chaos Expansion with Coefficients
xct = zeros(n_states,1)
for i in range(0,n_states):
xc_vals = [symbols('x'+str(i)+str(j)) for j in range(0,l)]
xct_dummy = Matrix(xc_vals)
xct[i] = xct_dummy.T*Hp
#print(xc_vals)
for i in range(n_states):
f = f.subs({x[i]:xct[i]})
g = g.subs({x[i]:xct[i]})
# Dynamics Interms of the coefficients and the germ $\xi$
dyn = (f*dt_symbol + g) # g includes \sqrt
# print(dyn)
# Kronecker Product
# Phi = Matrix(np.kron(np.identity(n_states),Hp.T))
norm_hp = zeros(l,1)
for jj in range(l):
norm_hp[jj,0] = Hp[jj]*Hp[jj]
for xi_i in xi:
norm_hp[jj,0] = integrate(gaussian_density(xi_i)*norm_hp[jj,0], (xi_i,-oo,oo))
# print(xi_i)
print(jj)
print(Hp)
print(norm_hp)
print('Computing the projected dynamics\n')
[weight,node] = UnivariateGaussHermiteQuadrature(10)
dyn_det = zeros(l*n_states,1)
print(l*n_states)
for ii in range(n_states):
for jj in range(l):
dyn_det[ii*l + jj] = Hp[jj]*dyn[ii]
for xi_i in xi:
# dyn_det[ii*l + jj] = integrate(gaussian_density(xi_i)*dyn_det[ii*l + jj], (xi_i,-oo,oo))
dyn_det[ii*l + jj,:] = integrate_gauss(weight,node,dyn_det[ii*l + jj,:],xi_i)
# print(xi_i)
dyn_det[ii*l + jj,:] = dyn_det[ii*l + jj,:]/norm_hp[jj,0]
# print('xx')
print('dxdt[%d]'%(ii*l + jj))
print('xx =',dyn_det[ii*l + jj,:][0])
# print(states)
print('Lambdifying the file.')
dyn_det_file = lambdify(((input_dyn)),dyn_det)
# dyn_det_file = theano_function(input_dyn,Matrix(dyn_det),on_unused_input='ignore')
# print(dyn_det_file)
return Matrix(dyn_det), dyn_det_file, input_dyn
def make_deterministic_sym(f, g, mu, sigma, x, u, theta, num_uncertainty, p):
"""
Converts stochasitic dynamics
xdot = f(x,u) + g(x,u, theta)
to deterministic approximation, with certain bounds
# Uses SYmbolic Integration
Parametersyp
----------
f : sympy expression in x and u
the initial deterministic component
g : sympy expression in x, u, and theta
the stochastic component
mu : sympy constant/number
the mean of theta
sigma : sympy constant/number
the variance of theta
x : sympy symbol
the state
u : sympy symbol
the control
theta : sympy symbol
the normally distributed noise
p : order of the polynomial expansion (user design choice)
num_uncertainty: total uncertainty = num_states+num_uncertainparam_dyn
Returns
-------
sympy expression in x and u
the deterministic dynamics
"""
n_states = x.shape[0]
n_uncert = num_uncertainty
#===============================================================================#
xi_symbols = [symbols('xi' + str(i)) for i in range(1, n_uncert + 1)]
xi = Matrix(xi_symbols)
#print('xi symbols')
#print(xi)
#Express theta_i in terms of unit gaussian xi_i
theta_subs = zeros(1, n_uncert - n_states)
for i in range(n_uncert - n_states):
theta_subs[i] = mu[i] + sqrt(sigma[i]) * xi[i]
#print(theta_subs)
for j in range(n_uncert - n_states):
g = g.subs({theta[j]: theta_subs[j]})
#rescaled Hermite polynomials
Hp = GaussHermitePC(n_uncert, p)
print(Hp)
#expansions of states using the Hermite polynomials
## X in the paper
l = Hp.shape[0]
# Generating the Symbolic Polynomial Chaos Expansion with Coefficients
xct = zeros(n_states, 1)
for i in range(0, n_states):
xc_vals = [symbols('x' + str(i) + str(j)) for j in range(0, l)]
xct_dummy = Matrix(xc_vals)
xct[i] = xct_dummy.T * Hp
#print(xc_vals)
for i in range(n_states):
f = f.subs({x[i]: xct[i]})
g = g.subs({x[i]: xct[i]})
# Dynamics Interms of the coefficients and the germ $\xi$
dyn = (f + g)
# print(dyn)
# Kronecker Product
Phi = Matrix(np.kron(np.identity(n_states), Hp.T))
left = Phi.T * Phi
right = Phi.T * dyn
left_int = left
right_int = right
# print(right)
print('Working on computing projections.\n')
left_dummy = left.shape[0]
[weight, node] = UnivariateGaussHermiteQuadrature(10)
for numl in range(0, left_dummy):
for xi_i in xi:
# left_int[numl,:] = integrate(gaussian_density(xi_i)*left_int[numl,:], (xi_i,-oo,oo))
left_int[numl, :] = integrate_gauss(weight, node,
left_int[numl, :], xi_i)
#print(xi_i)
print(numl)
print('Finished computing the integrals on left hand side. \n')
right_dummy = right.shape[0]
for numr in range(0, right_dummy):
for xi_i in xi:
# right_int[numr] = integrate(gaussian_density(xi_i)*right_int[numr], (xi_i,-oo,oo))
right_int[numr, :] = integrate_gauss(weight, node,
right_int[numr, :], xi_i)
print(xi_i)
print(numr)
# print("LEFT")
# print(N(left_int,5))
# print("RIGHT")
# print(right_int)
# quit()
print('Computing the projected dynamics\n')
dyn_det = left_int.inv() * right_int
xc_vals_dummy = zeros(l, n_states)
for i in range(0, n_states):
xc_vals_dummy[:, i] = Matrix(
[symbols('x' + str(i) + str(j)) for j in range(0, l)])
states = xc_vals_dummy.T.reshape(n_states * l, 1)
# print(states)
print('Lambdifying the file.')
dyn_det_file = lambdify((states), dyn_det, 'numpy')
# print(dyn_det_file)
return Matrix(dyn_det), dyn_det_file