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Merge pull request #199 from moorepants/old-example
Add Vyasarayani example to the gallery, but left multi option version in examples dir.
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theory | ||
examples/index.rst | ||
examples | ||
api | ||
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.. include:: ../README.rst | ||
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""" | ||
Multi-minimum Parameter Identification | ||
====================================== | ||
This example is taken from the following paper: | ||
Vyasarayani, Chandrika P., Thomas Uchida, Ashwin Carvalho, and John McPhee. | ||
"Parameter Identification in Dynamic Systems Using the Homotopy Optimization | ||
Approach". Multibody System Dynamics 26, no. 4 (2011): 411-24. | ||
In Section 3.1 there is a simple example of a single pendulum parameter | ||
identification that has many local minima. | ||
For the following differential equations that describe a single pendulum acting | ||
under the influence of gravity, the goals is to identify the parameter p given | ||
noisy measurements of the angle, y1. | ||
:: | ||
-- -- -- -- | ||
| y1' | | y2 | | ||
y' = f(y, t) = | | = | | | ||
| y2' | | -p*sin(y1) | | ||
-- -- -- -- | ||
""" | ||
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import numpy as np | ||
import sympy as sm | ||
from scipy.integrate import odeint | ||
import matplotlib.pyplot as plt | ||
from opty import Problem | ||
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# %% | ||
# Specify the symbolic equations of motion. | ||
p, t = sm.symbols('p, t') | ||
y1, y2 = [f(t) for f in sm.symbols('y1, y2', cls=sm.Function)] | ||
y = sm.Matrix([y1, y2]) | ||
f = sm.Matrix([y2, -p*sm.sin(y1)]) | ||
eom = y.diff(t) - f | ||
sm.pprint(eom) | ||
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# %% | ||
# Generate some data by integrating the equations of motion. | ||
duration = 50.0 | ||
num_nodes = 5000 | ||
interval = duration/(num_nodes - 1) | ||
time = np.linspace(0.0, duration, num=num_nodes) | ||
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p_val = 10.0 | ||
y0 = [np.pi/6.0, 0.0] | ||
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def eval_f(y, t, p): | ||
return np.array([y[1], -p*np.sin(y[0])]) | ||
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y_meas = odeint(eval_f, y0, time, args=(p_val,)) | ||
y1_meas = y_meas[:, 0] | ||
y2_meas = y_meas[:, 1] | ||
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# %% | ||
# Add measurement noise. | ||
y1_meas += np.random.normal(scale=0.05, size=y1_meas.shape) | ||
y2_meas += np.random.normal(scale=0.1, size=y2_meas.shape) | ||
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# %% | ||
# Setup the optimization problem to minimize the error in the simulated angle | ||
# and the measured angle. The midpoint integration method is preferable to the | ||
# backward Euler method because no artificial damping is introduced. | ||
def obj(free): | ||
"""Minimize the error in the angle, y1.""" | ||
return interval*np.sum((y1_meas - free[:num_nodes])**2) | ||
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def obj_grad(free): | ||
grad = np.zeros_like(free) | ||
grad[:num_nodes] = 2.0*interval*(free[:num_nodes] - y1_meas) | ||
return grad | ||
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prob = Problem(obj, obj_grad, eom, (y1, y2), num_nodes, interval, | ||
time_symbol=t, integration_method='midpoint') | ||
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num_states = len(y) | ||
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# %% | ||
# Give noisy measurements as the initial state guess and a random positive | ||
# values as the parameter guess. | ||
initial_guess = np.hstack((y1_meas, y2_meas, 100.0*np.random.random(1))) | ||
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# %% | ||
# Find the optimal solution. | ||
solution, info = prob.solve(initial_guess) | ||
p_sol = solution[-1] | ||
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# %% | ||
# Print the result. | ||
known_msg = "Known value of p = {}".format(p_val) | ||
guess_msg = "Initial guess for p = {}".format(initial_guess[-1]) | ||
identified_msg = "Identified value of p = {}".format(p_sol) | ||
divider = '='*max(len(known_msg), len(identified_msg)) | ||
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print(divider) | ||
print(known_msg) | ||
print(guess_msg) | ||
print(identified_msg) | ||
print(divider) | ||
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# %% | ||
# Simulate with the identified parameter. | ||
y_sim = odeint(eval_f, y0, time, args=(p_sol,)) | ||
y1_sim = y_sim[:, 0] | ||
y2_sim = y_sim[:, 1] | ||
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# %% | ||
# Plot results | ||
fig_y1, axes_y1 = plt.subplots(3, 1, layout='constrained') | ||
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legend = ['measured', 'initial guess', 'direct collocation solution', | ||
'identified simulated'] | ||
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axes_y1[0].plot(time, y1_meas, '.k', | ||
time, initial_guess[:num_nodes], '.b', | ||
time, solution[:num_nodes], '.r', | ||
time, y1_sim, 'g') | ||
axes_y1[0].set_xlabel('Time [s]') | ||
axes_y1[0].set_ylabel('y1 [rad]') | ||
axes_y1[0].legend(legend) | ||
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axes_y1[1].set_title('Initial Guess Constraint Violations') | ||
axes_y1[1].plot(prob.con(initial_guess)[:num_nodes - 1]) | ||
axes_y1[2].set_title('Solution Constraint Violations') | ||
axes_y1[2].plot(prob.con(solution)[:num_nodes - 1]) | ||
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# %% | ||
fig_y2, axes_y2 = plt.subplots(3, 1, layout='constrained') | ||
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axes_y2[0].plot(time, y2_meas, '.k', | ||
time, initial_guess[num_nodes:-1], '.b', | ||
time, solution[num_nodes:-1], '.r', | ||
time, y2_sim, 'g') | ||
axes_y2[0].set_xlabel('Time [s]') | ||
axes_y2[0].set_ylabel('y2 [rad]') | ||
axes_y2[0].legend(legend) | ||
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axes_y2[1].set_title('Initial Guess Constraint Violations') | ||
axes_y2[1].plot(prob.con(initial_guess)[num_nodes - 1:]) | ||
axes_y2[2].set_title('Solution Constraint Violations') | ||
axes_y2[2].plot(prob.con(solution)[num_nodes - 1:]) | ||
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plt.show() |