Betts 10 144 145

examples-gallery/plot_betts_10_144_145.py

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+"""
+Van der Pol Oscillator
+======================
+
+https://en.wikipedia.org/wiki/Van_der_Pol_oscillator
+
+These are example 10.144 / 145 from Betts' book "Practical Methods for Optimal Control
+Using NonlinearProgramming", 3rd edition, Chapter 10: Test Problems.
+It is described in more detail in section 4.14. example 4.11 of the book.
+
+"""
+import numpy as np
+import sympy as sm
+import sympy.physics.mechanics as me
+from opty.direct_collocation import Problem
+from opty.utils import create_objective_function
+import matplotlib.pyplot as plt
+
+# %%
+# First version of the problem (10.144).
+# --------------------------------------
+# **States**
+# :math:`y_1, y_2` : state variables
+#
+# **Controls**
+# :math:`u` : control variables
+#
+# Equations of Motion.
+# --------------------
+t = me.dynamicsymbols._t
+
+y1, y2 = me.dynamicsymbols('y1, y2')
+u = me.dynamicsymbols('u')
+
+eom = sm.Matrix([
+    -y1.diff() + y2,
+    -y2.diff(t) + (1 - y1**2)*y2 - y1 + u,
+])
+sm.pprint(eom)
+
+# %%
+# Define and Solve the Optimization Problem.
+# ------------------------------------------
+num_nodes = 2001
+iterations = 1000
+
+t0, tf = 0.0, 5.0
+interval_value = (tf - t0)/(num_nodes - 1)
+
+state_symbols = (y1, y2)
+unkonwn_input_trajectories = (u, )
+
+objective = sm.integrate(u**2 + y1**2 + y2**2, t)
+obj, obj_grad = create_objective_function(objective,
+                state_symbols,
+                unkonwn_input_trajectories,
+                tuple(),
+                num_nodes,
+                interval_value
+)
+
+instance_constraints = (
+        y1.func(t0) - 1.0,
+        y2.func(t0),
+)
+
+bounds = {
+        y2: (-0.4, np.inf),
+}
+
+prob = Problem(
+        obj,
+        obj_grad,
+        eom,
+        state_symbols,
+        num_nodes,
+        interval_value,
+        instance_constraints= instance_constraints,
+        bounds=bounds,
+)
+
+# %%
+# Solve the optimization problem.
+# Give some rough estimates for the trajectories.
+
+initial_guess = np.ones(prob.num_free)
+
+# Find the optimal solution.
+for i in range(1):
+    solution, info = prob.solve(initial_guess)
+    initial_guess = solution
+    print(info['status_msg'])
+    print(f'Objectve is: {info['obj_val']:.8f}, ' +
+          f'as per the book it is {2.95369916}, so the deviation is: ' +
+        f'{(info['obj_val'] -2.95369916) /2.95369916*100 :.5e} %')
+solution1 = solution
+# %%
+# Plot the optimal state and input trajectories.
+prob.plot_trajectories(solution)
+
+# %%
+# Plot the constraint violations.
+prob.plot_constraint_violations(solution)
+
+# %%
+# Plot the objective function as a function of optimizer iteration.
+prob.plot_objective_value()
+
+# %%
+# Second version of the problem (10.145).
+# ---------------------------------------
+# This is example 10.145 from Betts' book "Practical Methods for Optimal Control
+# Using NonlinearProgramming", 3rd edition, Chapter 10: Test Problems.
+# It is same as problem 10.144 but a bit reformulated.
+# It has two control variables and one additional algebraic equation of motion.
+# As opty needs as many state variables as equations of motion, I simply call
+# the additional control variable v a state variable.
+# It is described in more detail in section 4.14, example 4.11 of the book.
+#
+# This formulation seems to be more accurate compared to the above, when
+# considering the violations of the constraints.
+#
+# **States**
+# :math:`y_1, y_2, v` : state variables
+#
+# **Controls**
+# :math:`u` : control variables
+#
+# Equations of Motion.
+# --------------------
+y1, y2, v = me.dynamicsymbols('y1, y2, v')
+u = me.dynamicsymbols('u')
+
+eom = sm.Matrix([
+    -y1.diff() + y2,
+    -y2.diff(t) + v - y1 + u,
+    v - (1-y1**2)*y2,
+])
+sm.pprint(eom)
+
+# %%
+# Define and Solve the Optimization Problem.
+# ------------------------------------------
+state_symbols = (y1, y2, v)
+unkonwn_input_trajectories = (u,)
+
+objective = sm.integrate(u**2 + y1**2 + y2**2, t)
+obj, obj_grad = create_objective_function(objective,
+                state_symbols,
+                unkonwn_input_trajectories,
+                tuple(),
+                num_nodes,
+                interval_value
+)
+
+instance_constraints = (
+        y1.func(t0) - 1.0,
+        y2.func(t0),
+)
+
+bounds = {
+        y2: (-0.4, np.inf),
+}
+
+prob = Problem(
+        obj,
+        obj_grad,
+        eom,
+        state_symbols,
+        num_nodes,
+        interval_value,
+        instance_constraints= instance_constraints,
+        bounds=bounds,
+)
+
+# %%
+# Solve the optimization problem.
+# Give some rough estimates for the trajectories.
+
+initial_guess = np.ones(prob.num_free)
+
+# Find the optimal solution.
+for i in range(1):
+    solution, info = prob.solve(initial_guess)
+    initial_guess = solution
+    print(info['status_msg'])
+    print(f'Objectve is: {info['obj_val']:.8f}, ' +
+          f'as per the book it is {2.95369916}, so the deviation is: ' +
+        f'{(info['obj_val'] -2.95369916) /2.95369916*100 :.5e} %')
+# %%
+# Plot the optimal state and input trajectories.
+prob.plot_trajectories(solution)
+
+# %%
+# Plot the constraint violations.
+prob.plot_constraint_violations(solution)
+
+# %%
+# Plot the objective function as a function of optimizer iteration.
+prob.plot_objective_value()
+
+# %%
+# Plot the Difference between the two Solutions.
+# ----------------------------------------------
+diffy1 = solution1[: num_nodes] - solution[: num_nodes]
+diffy2 = solution1[num_nodes: 2*num_nodes] - solution[num_nodes: 2*num_nodes]
+diffu = solution1[2*num_nodes:] - solution[3*num_nodes:]
+times = np.linspace(t0, tf, num_nodes)
+
+fig, ax = plt.subplots(3, 1, figsize=(7, 4), sharex=True, constrained_layout=True)
+ax[0].plot(times, diffy1, label='Delta y1')
+ax[0].legend()
+ax[1].plot(times, diffy2, label='Delta y2')
+ax[1].legend()
+ax[2].plot(times, diffu, label='Delta u')
+ax[2].legend()
+ax[2].set_xlabel('Time')
+ax[0].set_title('Differences between the two solutions')
+avoid_printing = True
+# sphinx_gallery_thumbnail_number = 2