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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 *
# In this example, we fit a nonlinear model to measurements
#
# This example uses more advanced constructs than the vdp* examples:
# Since the number of control intervals is potentially very large here,
# we use memory-efficient Map and MapAccum, in combination with
# codegeneration.
#
# We will be working with a 2-norm objective:
# || y_measured - y_simulated ||_2^2
#
# This form is well-suited for the Gauss-Newton Hessian approximation.
############ SETTINGS #####################
N = 10000 # Number of samples
fs = 610.1 # Sampling frequency [hz]
param_truth = DM([5.625e-6,2.3e-4,1,4.69])
param_guess = DM([5,2,1,5])
scale = vertcat(1e-6,1e-4,1,1)
############ MODELING #####################
y = MX.sym('y')
dy = MX.sym('dy')
u = MX.sym('u')
states = vertcat(y,dy);
controls = u;
M = MX.sym("x")
c = MX.sym("c")
k = MX.sym("k")
k_NL = MX.sym("k_NL")
params = vertcat(M,c,k,k_NL)
rhs = vertcat(dy , (u-k_NL*y**3-k*y-c*dy)/M)
# Form an ode function
ode = Function('ode',[states,controls,params],[rhs])
############ Creating a simulator ##########
N_steps_per_sample = 10
dt = 1/fs/N_steps_per_sample
# Build an integrator for this system: Runge Kutta 4 integrator
k1 = ode(states,controls,params)
k2 = ode(states+dt/2.0*k1,controls,params)
k3 = ode(states+dt/2.0*k2,controls,params)
k4 = ode(states+dt*k3,controls,params)
states_final = states+dt/6.0*(k1+2*k2+2*k3+k4)
# Create a function that simulates one step propagation in a sample
one_step = Function('one_step',[states, controls, params],[states_final]);
X = states;
for i in range(N_steps_per_sample):
X = one_step(X, controls, params)
# Create a function that simulates all step propagation on a sample
one_sample = Function('one_sample',[states, controls, params], [X])
############ Simulating the system ##########
all_samples = one_sample.mapaccum("all_samples", N)
# Choose an excitation signal
numpy.random.seed(0)
u_data = DM(0.1*numpy.random.random(N))
x0 = DM([0,0])
X_measured = all_samples(x0, u_data, repmat(param_truth,1,N))
y_data = X_measured[0,:].T
# You may add some noise here
#y_data+= 0.001*numpy.random.random(N)
# When noise is absent, the fit will be perfect.
# Use just-in-time compilation to speed up the evaluation
if Importer.has_plugin('clang'):
with_jit = True
compiler = 'clang'
elif Importer.has_plugin('shell'):
with_jit = True
compiler = 'shell'
else:
print("WARNING; running without jit. This may result in very slow evaluation times")
with_jit = False
compiler = ''
############ Create a Gauss-Newton solver ##########
def gauss_newton(e,nlp,V):
J = jacobian(e,V)
H = triu(mtimes(J.T, J))
sigma = MX.sym("sigma")
hessLag = Function('nlp_hess_l',{'x':V,'lam_f':sigma, 'hess_gamma_x_x':sigma*H},
['x','p','lam_f','lam_g'], ['hess_gamma_x_x'],
dict(jit=with_jit, compiler=compiler))
return nlpsol("solver","ipopt", nlp, dict(hess_lag=hessLag, jit=with_jit, compiler=compiler))
############ Identifying the simulated system: single shooting strategy ##########
# Note, it is in general a good idea to scale your decision variables such
# that they are in the order of ~0.1..100
X_symbolic = all_samples(x0, u_data, repmat(params*scale,1,N))
e = y_data-X_symbolic[0,:].T;
nlp = {'x':params, 'f':0.5*dot(e,e)}
solver = gauss_newton(e,nlp, params)
sol = solver(x0=param_guess)
print(sol["x"]*scale)
assert(norm_inf(sol["x"]*scale-param_truth)<1e-8)
############ Identifying the simulated system: multiple shooting strategy ##########
# All states become decision variables
X = MX.sym("X", 2, N)
Xn = one_sample.map(N, 'openmp')(X, u_data.T, repmat(params*scale,1,N))
gaps = Xn[:,:-1]-X[:,1:]
e = y_data-Xn[0,:].T;
V = veccat(params, X)
nlp = {'x':V, 'f':0.5*dot(e,e),'g': vec(gaps)}
# Multipleshooting allows for careful initialization
yd = np.diff(y_data,axis=0)*fs
X_guess = horzcat(y_data , vertcat(yd,yd[-1])).T;
x0 = veccat(param_guess,X_guess)
solver = gauss_newton(e,nlp, V)
sol = solver(x0=x0,lbg=0,ubg=0)
print(sol["x"][:4]*scale)
assert(norm_inf(sol["x"][:4]*scale-param_truth)<1e-8)
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