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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.
#
from casadi import *
T = 10. # Time horizon
N = 20 # number of control intervals
# Declare model variables
x1 = MX.sym('x1')
x2 = MX.sym('x2')
x = vertcat(x1, x2)
u = MX.sym('u')
# Model equations
xdot = vertcat((1-x2**2)*x1 - x2 + u, x1)
# Objective term
L = x1**2 + x2**2 + u**2
# Formulate discrete time dynamics
if False:
# CVODES from the SUNDIALS suite
dae = {'x':x, 'p':u, 'ode':xdot, 'quad':L}
F = integrator('F', 'cvodes', dae, 0, T/N)
else:
# Fixed step Runge-Kutta 4 integrator
M = 4 # RK4 steps per interval
DT = T/N/M
f = Function('f', [x, u], [xdot, L])
X0 = MX.sym('X0', 2)
U = MX.sym('U')
X = X0
Q = 0
for j in range(M):
k1, k1_q = f(X, U)
k2, k2_q = f(X + DT/2 * k1, U)
k3, k3_q = f(X + DT/2 * k2, U)
k4, k4_q = f(X + DT * k3, U)
X=X+DT/6*(k1 +2*k2 +2*k3 +k4)
Q = Q + DT/6*(k1_q + 2*k2_q + 2*k3_q + k4_q)
F = Function('F', [X0, U], [X, Q],['x0','p'],['xf','qf'])
# Evaluate at a test point
Fk = F(x0=[0.2,0.3],p=0.4)
print(Fk['xf'])
print(Fk['qf'])
# Start with an empty NLP
w=[]
w0 = []
lbw = []
ubw = []
J = 0
g=[]
lbg = []
ubg = []
# "Lift" initial conditions
Xk = MX.sym('X0', 2)
w += [Xk]
lbw += [0, 1]
ubw += [0, 1]
w0 += [0, 1]
# Formulate the NLP
for k in range(N):
# New NLP variable for the control
Uk = MX.sym('U_' + str(k))
w += [Uk]
lbw += [-1]
ubw += [1]
w0 += [0]
# Integrate till the end of the interval
Fk = F(x0=Xk, p=Uk)
Xk_end = Fk['xf']
J=J+Fk['qf']
# New NLP variable for state at end of interval
Xk = MX.sym('X_' + str(k+1), 2)
w += [Xk]
lbw += [-0.25, -inf]
ubw += [ inf, inf]
w0 += [0, 0]
# Add equality constraint
g += [Xk_end-Xk]
lbg += [0, 0]
ubg += [0, 0]
# Create an NLP solver
prob = {'f': J, 'x': vertcat(*w), 'g': vertcat(*g)}
solver = nlpsol('solver', 'ipopt', prob);
# Solve the NLP
sol = solver(x0=w0, lbx=lbw, ubx=ubw, lbg=lbg, ubg=ubg)
w_opt = sol['x'].full().flatten()
# Plot the solution
x1_opt = w_opt[0::3]
x2_opt = w_opt[1::3]
u_opt = w_opt[2::3]
tgrid = [T/N*k for k in range(N+1)]
import matplotlib.pyplot as plt
plt.figure(1)
plt.clf()
plt.plot(tgrid, x1_opt, '--')
plt.plot(tgrid, x2_opt, '-')
plt.step(tgrid, vertcat(DM.nan(1), u_opt), '-.')
plt.xlabel('t')
plt.legend(['x1','x2','u'])
plt.grid()
plt.show()
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