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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 casadi import *
from numpy import *
import matplotlib.pyplot as plt
# Excercise 1, chapter 10 from Larry Biegler's book
print("program started")
# Test with different number of elements
for N in range(1,11):
print("N = ", N)
# Degree of interpolating polynomial
K = 2
# Legrandre roots
tau_root = [0., 0.211325, 0.788675]
# Radau roots (K=3)
#tau_root = [0, 0.155051, 0.644949, 1]
# Time
t = SX.sym("t")
# Differential equation
z = SX.sym("z")
F = Function("dz_dt", [z],[z*z - 2*z + 1])
z0 = -3
# Analytic solution
z_analytic = Function("z_analytic", [t], [(4*t-3)/(3*t+1)])
# Collocation point
tau = SX.sym("tau")
# Step size
h = 1.0/N
# Get the coefficients of the continuity and collocation equations
D = DM.zeros(K+1)
C = DM.zeros(K+1,K+1)
for j in range(K+1):
# Lagrange polynomial
L = 1
for k in range(K+1):
if(k != j):
L *= (tau-tau_root[k])/(tau_root[j]-tau_root[k])
# Evaluate at end for coefficients of continuity equation
lfcn = Function("lfcn", [tau],[L])
D[j] = lfcn(1.)
# Differentiate and evaluate at collocation points
tfcn = Function("tfcn", [tau],[tangent(L,tau)])
for k in range(K+1): C[j,k] = tfcn(tau_root[k])
print("C = ", C)
print("D = ", D)
# Collocated states
Z = SX.sym("Z",N,K+1)
# Construct the NLP
x = vec(Z.T)
g = []
for i in range(N):
for k in range(1,K+1):
# Add collocation equations to NLP
rhs = 0
for j in range(K+1):
rhs += Z[i,j]*C[j,k]
FF = F(Z[i,k])
g.append(h*FF-rhs)
# Add continuity equation to NLP
rhs = 0
for j in range(K+1):
rhs += D[j]*Z[i,j]
if(i<N-1):
g.append(Z[i+1,0] - rhs)
g = vertcat(*g)
print("g = ", g)
# NLP
nlp = {'x':x, 'f':x[0]**2, 'g':g}
## ----
## SOLVE THE NLP
## ----
# NLP solver options
opts = {"ipopt.tol" : 1e-10}
# Allocate an NLP solver and buffer
solver = nlpsol("solver", "ipopt", nlp, opts)
arg = {}
# Initial condition
arg["x0"] = x.nnz() * [0]
# Bounds on x
lbx = x.nnz()*[-100]
ubx = x.nnz()*[100]
lbx[0] = ubx[0] = z0
arg["lbx"] = lbx
arg["ubx"] = ubx
# Bounds on the constraints
arg["lbg"] = 0
arg["ubg"] = 0
# Solve the problem
res = solver(**arg)
## Print the time points
t_opt = N*(K+1) * [0]
for i in range(N):
for j in range(K+1):
t_opt[j + (K+1)*i] = h*(i + tau_root[j])
print("time points: ", t_opt)
# Print the optimal cost
print("optimal cost: ", float(res["f"]))
# Print the optimal solution
xopt = res["x"].nonzeros()
print("optimal solution: ", xopt)
# plot to screen
plt.plot(t_opt,xopt)
# show the plots
plt.show()
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