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#
# MIT No Attribution
#
# Copyright (C) 2010-2023 Joel Andersson, Joris Gillis, Moritz Diehl, KU Leuven.
#
# Permission is hereby granted, free of charge, to any person obtaining a copy of this
# software and associated documentation files (the "Software"), to deal in the Software
# without restriction, including without limitation the rights to use, copy, modify,
# merge, publish, distribute, sublicense, and/or sell copies of the Software, and to
# permit persons to whom the Software is furnished to do so.
#
# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED,
# INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A
# PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
# HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION
# OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE
# SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
#
# -*- coding: utf-8 -*-
from __future__ import print_function
from casadi import *
from copy import deepcopy
print('Testing sensitivity analysis in CasADi')
# All ODE and DAE integrators to be tested
DAE_integrators = ['idas','collocation']
ODE_integrators = ['cvodes','rk'] + DAE_integrators
for Integrators in (ODE_integrators,DAE_integrators):
if Integrators==ODE_integrators: # rocket example
print('******')
print('Testing ODE example')
# Time
t = SX.sym('t')
# Parameter
u = SX.sym('u')
# Differential states
s = SX.sym('s'); v = SX.sym('v'); m = SX.sym('m')
x = vertcat(s,v,m)
# Constants
alpha = 0.05 # friction
beta = 0.1 # fuel consumption rate
# Differential equation
ode = vertcat(
v,
(u-alpha*v*v)/m,
-beta*u*u)
# Quadrature
quad = v**3 + ((3-sin(t)) - u)**2
# DAE callback function
dae = {'t':t, 'x':x, 'p':u, 'ode':ode, 'quad':quad}
# Time length
tf = 0.5
# Initial position
x0 = [0.,0.,1.]
# Parameter
u0 = 0.4
else: # Simple DAE example
print('******')
print('Testing DAE example')
# Differential state
x = SX.sym('x')
# Algebraic variable
z = SX.sym('z')
# Parameter
u = SX.sym('u')
# Differential equation
ode = -x + 0.5*x*x + u + 0.5*z
# Algebraic constraint
alg = z + exp(z) - 1.0 + x
# Quadrature
quad = x*x + 3.0*u*u
# DAE callback function
dae = {'x':x, 'z':z, 'p':u, 'ode':ode, 'alg':alg, 'quad':quad}
# End time
tf = 5.
# Initial position
x0 = 1.
# Parameter
u0 = 0.4
# Integrator
for MyIntegrator in Integrators:
print('========')
print('Integrator: ', MyIntegrator)
print('========')
# Integrator options
opts = {}
if MyIntegrator=='collocation':
opts['rootfinder'] = 'kinsol'
# Integrator
I = integrator('I', MyIntegrator, dae, 0, tf, opts)
# Integrate to get results
res = I(x0=x0, p=u0)
xf = res['xf']
qf = res['qf']
print('%50s' % 'Unperturbed solution:', 'xf = ', xf, ', qf = ', qf)
# Perturb solution to get a finite difference approximation
h = 0.001
res = I(x0=x0, p=u0+h)
fd_xf = (res['xf']-xf)/h
fd_qf = (res['qf']-qf)/h
print('%50s' % 'Finite difference approximation:', 'd(xf)/d(p) = ', fd_xf, ', d(qf)/d(p) = ', fd_qf)
# Calculate once, forward
I_fwd = I.factory('I_fwd', ['x0', 'z0', 'p', 'fwd:p'], ['fwd:xf', 'fwd:qf'])
res = I_fwd(x0=x0, p=u0, fwd_p=1)
fwd_xf = res['fwd_xf']
fwd_qf = res['fwd_qf']
print('%50s' % 'Forward sensitivities:', 'd(xf)/d(p) = ', fwd_xf, ', d(qf)/d(p) = ', fwd_qf)
# Calculate once, adjoint
I_adj = I.factory('I_adj', ['x0', 'z0', 'p', 'adj:qf'], ['adj:x0', 'adj:p'])
res = I_adj(x0=x0, p=u0, adj_qf=1)
adj_x0 = res['adj_x0']
adj_p = res['adj_p']
print('%50s' % 'Adjoint sensitivities:', 'd(qf)/d(x0) = ', adj_x0, ', d(qf)/d(p) = ', adj_p)
# Perturb adjoint solution to get a finite difference approximation of the second order sensitivities
res = I_adj(x0=x0, p=u0+h, adj_qf=1)
fd_adj_x0 = (res['adj_x0']-adj_x0)/h
fd_adj_p = (res['adj_p']-adj_p)/h
print('%50s' % 'FD of adjoint sensitivities:', 'd2(qf)/d(x0)d(p) = ', fd_adj_x0, ', d2(qf)/d(p)d(p) = ', fd_adj_p)
# Forward over adjoint to get the second order sensitivities
I_foa = I_adj.factory('I_foa', ['x0', 'z0', 'p', 'adj_qf', 'fwd:p'], ['fwd:adj_x0', 'fwd:adj_p'])
res = I_foa(x0=x0, p=u0, adj_qf=1, fwd_p=1)
fwd_adj_x0 = res['fwd_adj_x0']
fwd_adj_p = res['fwd_adj_p']
print('%50s' % 'Forward over adjoint sensitivities:', 'd2(qf)/d(x0)d(p) = ', fwd_adj_x0, ', d2(qf)/d(p)d(p) = ', fwd_adj_p)
# Adjoint over adjoint to get the second order sensitivities
I_aoa = I_adj.factory('I_aoa', ['x0', 'z0', 'p', 'adj_qf', 'adj:adj_p'], ['adj:x0', 'adj:p'])
res = I_aoa(x0=x0, p=u0, adj_qf=1, adj_adj_p=1)
adj_x0 = res['adj_x0']
adj_p = res['adj_p']
print('%50s' % 'Adjoint over adjoint sensitivities:', 'd2(qf)/d(x0)d(p) = ', adj_x0, ', d2(qf)/d(p)d(p) = ', adj_p)
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