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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 *
import numpy as N
import matplotlib.pyplot as plt
'''
Demonstration on how to construct a fixed-step implicit Runge-Kutta integrator
@author: Joel Andersson, KU Leuven 2013
'''
# End time
tf = 10.0
# Dimensions
nx = 3
np = 1
# Declare variables
x = SX.sym('x', nx) # state
p = SX.sym('u', np) # control
# ODE right hand side function
ode = vertcat((1 - x[1]*x[1])*x[0] - x[1] + p, \
x[0], \
x[0]*x[0] + x[1]*x[1] + p*p)
dae = {'x':x, 'p':p, 'ode':ode}
f = Function('f', [x, p], [ode])
# Number of finite elements
n = 100
# Size of the finite elements
h = tf/n
# Degree of interpolating polynomial
d = 4
# Choose collocation points
tau_root = [0] + collocation_points(d, 'legendre')
# Coefficients of the collocation equation
C = N.zeros((d+1,d+1))
# Coefficients of the continuity equation
D = N.zeros(d+1)
# Dimensionless time inside one control interval
tau = SX.sym('tau')
# For all collocation points
for j in range(d+1):
# Construct Lagrange polynomials to get the polynomial basis at the collocation point
L = 1
for r in range(d+1):
if r != j:
L *= (tau-tau_root[r])/(tau_root[j]-tau_root[r])
# Evaluate the polynomial at the final time to get the coefficients of the continuity equation
lfcn = Function('lfcn', [tau], [L])
D[j] = lfcn(1.0)
# Evaluate the time derivative of the polynomial at all collocation points to get the coefficients of the continuity equation
tfcn = Function('tfcn', [tau], [tangent(L,tau)])
for r in range(d+1): C[j,r] = tfcn(tau_root[r])
# Total number of variables for one finite element
X0 = MX.sym('X0',nx)
P = MX.sym('P',np)
V = MX.sym('V',d*nx)
# Get the state at each collocation point
X = [X0] + vertsplit(V,[r*nx for r in range(d+1)])
# Get the collocation quations (that define V)
V_eq = []
for j in range(1,d+1):
# Expression for the state derivative at the collocation point
xp_j = 0
for r in range (d+1):
xp_j += C[r,j]*X[r]
# Append collocation equations
f_j = f(X[j],P)
V_eq.append(h*f_j - xp_j)
# Concatenate constraints
V_eq = vertcat(*V_eq)
# Root-finding function, implicitly defines V as a function of X0 and P
vfcn = Function('vfcn', [V, X0, P], [V_eq])
# Convert to SX to decrease overhead
vfcn_sx = vfcn.expand()
# Create a implicit function instance to solve the system of equations
ifcn = rootfinder('ifcn', 'newton', vfcn_sx)
V = ifcn(MX(),X0,P)
X = [X0 if r==0 else V[(r-1)*nx:r*nx] for r in range(d+1)]
# Get an expression for the state at the end of the finie element
XF = 0
for r in range(d+1):
XF += D[r]*X[r]
# Get the discrete time dynamics
F = Function('F', [X0,P],[XF])
# Do this iteratively for all finite elements
X = X0
for i in range(n):
X = F(X,P)
# Fixed-step integrator
irk_integrator = Function('irk_integrator', {'x0':X0, 'p':P, 'xf':X},
integrator_in(), integrator_out())
# Create a convensional integrator for reference
ref_integrator = integrator('ref_integrator', 'cvodes', dae, 0, tf)
# Test values
x0_val = N.array([0,1,0])
p_val = 0.2
# Make sure that both integrators give consistent results
for F in (irk_integrator,ref_integrator):
print('-------')
print('Testing ' + F.name())
print('-------')
# Generate a new function that calculates forward and reverse directional derivatives
dF = F.factory('dF', ['x0', 'p', 'fwd:x0', 'fwd:p', 'adj:xf'],
['xf', 'fwd:xf', 'adj:x0', 'adj:p']);
arg = {}
# Pass arguments
arg['x0'] = x0_val
arg['p'] = p_val
# Forward sensitivity analysis, first direction: seed p and x0[0]
arg['fwd_x0'] = [1,0,0]
arg['fwd_p'] = 1
# Adjoint sensitivity analysis, seed xf[2]
arg['adj_xf'] = [0,0,1]
# Integrate
res = dF(**arg)
# Get the nondifferentiated results
print('%30s = %s' % ('xf', res['xf']))
# Get the forward sensitivities
print('%30s = %s' % ('d(xf)/d(p)+d(xf)/d(x0[0])', res['fwd_xf']))
# Get the adjoint sensitivities
print('%30s = %s' % ('d(xf[2])/d(x0)', res['adj_x0']))
print('%30s = %s' % ('d(xf[2])/d(p)', res['adj_p']))
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