#
#     This file is part of CasADi.
#
#     CasADi -- A symbolic framework for dynamic optimization.
#     Copyright (C) 2010-2023 Joel Andersson, Joris Gillis, Moritz Diehl,
#                             KU Leuven. All rights reserved.
#     Copyright (C) 2011-2014 Greg Horn
#
#     CasADi is free software; you can redistribute it and/or
#     modify it under the terms of the GNU Lesser General Public
#     License as published by the Free Software Foundation; either
#     version 3 of the License, or (at your option) any later version.
#
#     CasADi is distributed in the hope that it will be useful,
#     but WITHOUT ANY WARRANTY; without even the implied warranty of
#     MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
#     Lesser General Public License for more details.
#
#     You should have received a copy of the GNU Lesser General Public
#     License along with CasADi; if not, write to the Free Software
#     Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA

#
# IDAS integrator
# =====================
#
# We solve a system
#   $\dot{x}(t)=f(x(t),y(t),t)$
#   $0=g(x(t),y(t),t)$

from casadi import *
from numpy import *
from pylab import *

# We solve the following simple dae system that describes
# the dynamics of a pendulum:
# x' = u, y' = v, u' = lambda * x, v' =lambda * y - g
#   s.t. x^2+y^2 = L
#
# We retain g and L as parameters
# http://en.wikipedia.org/wiki/Differential_algebraic_equation#Examples

L = SX.sym("L")
g = SX.sym("g")

# differential states

x=SX.sym("x")
y=SX.sym("y")
u=SX.sym("u")
v=SX.sym("v")

# algebraic states

lambd=SX.sym("lambda")

# All states and parameters

x_all = vertcat(x,u,y,v)
z_all = lambd
p_all = vertcat(L,g)

# the initial state of the pendulum

P_ = [5,10] # parameters
X_ = [3,-1.0/3,4,1.0/4] # differential states
XDOT_ = [-1.0/3,1147.0/240,1.0/4,-653.0/180] # state derivatives
Z_ = [1147.0/720] # algebraic state

# We construct the DAE system

ode = vertcat(u,lambd*x,v,lambd*y+g)
alg = x**2+y**2-L**2
dae = {'x':x_all, 'z':z_all, 'p':p_all, 'ode':ode, 'alg':alg}
f = Function('f', [x_all, z_all, p_all], [ode, alg], ['x', 'z', 'p'], ['ode', 'alg'])

# Let's check we have consistent initial conditions:

res = f(p=P_, x=X_, z=Z_)
print(res['ode']) # This should be same as XDOT_
print(res['alg']) # This should be all zeros


# Let's check our jacobian $\frac{dg}{dy}$:

j = jacobian(alg,lambd)
print(j)

# Note that the jacobian is not invertible: it is not of DAE-index 1
#
# This system is not solvable with idas, because it is of DAE-index 3.
# It is impossible to lambda from the last element of the residual.

# We create a DAE system solver

I = integrator('I', 'idas', dae, {'calc_ic':False, 'init_xdot':XDOT_})

# This system is not solvable with idas, because it is of DAE-index 3.
# It is impossible obtain lambda from the last element of the residual.

try:
  I(p=P_, x0=X_, z0=Z_)
except Exception as e:
  print(e)

# We construct a reworked version od the DAE (index reduced), now it is DAE-index 1

ode = vertcat(u,lambd*x)
alg = vertcat(x**2+y**2-L**2, u*x+v*y,u**2-g*y+v**2+L**2*lambd)
x_all = vertcat(x,u)
z_all = vertcat(y,v,lambd)
dae = {'x':x_all, 'z':z_all, 'p':p_all, 'ode':ode, 'alg':alg}
f = Function('f', [x_all, z_all, p_all], [ode, alg], ['x', 'z', 'p'], ['ode', 'alg'])

# the initial state of the pendulum

P_ = [5,10] # parameters
X_ = [3,-1.0/3] # differential states
XDOT_ = [-1.0/3,1147.0/240] # state derivatives
Z_ = [4,1.0/4,1147.0/720] # algebraic state

# Let's check we have consistent initial conditions:

res = f(p=P_, x=X_, z=Z_)
print(res['ode']) # This should be the same as XDOT_
print(res['alg']) # This should be all zeros

# Let's check our jacobian:

J = f.factory('J', f.name_in(), ['jac:alg:z'])
res = J(p=P_, x=X_, z=Z_)
print(array(res["jac_alg_z"]))

# $\frac{dg}{dy}$ is invertible this time.

# We create a DAE system solver

I = integrator('I', 'idas', dae, 0, 1, {'init_xdot':XDOT_})
res = I(p=P_, x0=X_, z0=Z_)
print(res['xf'])

# Possible problems
# ==================

# If you would initialize with:

P_ = [5,10] # parameters
X_ = [5,0]  # states

# You will get an error:

try:
  I(p=P_, x0=X_, z0=Z_)
except Exception as e:
  print(e)

# Although this initialisation is consistent,
# it coincides with a singular point.
