Betts 10 1

examples-gallery/plot_betts_10_1.py

@@ -0,0 +1,165 @@
+"""
+Alp Rider
+=========
+
+This is example 10.1 from Betts' book "Practical Methods for Optimal Control
+Using NonlinearProgramming", 3rd edition, Chapter 10: Test Problems.
+More details are in sectionn 4.11.7 of the book.
+
+The main issue here is an inequality constraint for the state variables:
+:math:`y_1^2 + y_2^2 + y_3^2 + y_4^2 \\geq function(time, parameters)`
+
+As presently *opty* does not support inequality constraints, I introduced a new
+state variable `fact` to take care of the inequality. The inequality is then
+reformulated as
+:math:`y_1^2 + y_2^2 + y_3^2 + y_4^2 = fact \\cdot function(time, parameters)`.
+
+and I restrict fact :math:`\\geq 1.0`.
+
+**States**
+
+- :math:`y_1, .., y_4` : state variables as per the book
+- :math:`fact`: additional state variable to take care of the inequality
+
+**Specifieds**
+
+- :math:`u_1, u_2` : control variables
+
+Note: I simply copied the equations of motion, the bounds and the constants
+from the book. I do not know their meaning.
+
+"""
+
+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
+
+# %%
+# Equations of Motion
+# -------------------
+#
+t = me.dynamicsymbols._t
+y1, y2, y3, y4 = me.dynamicsymbols('y1 y2 y3 y4')
+fact = me.dynamicsymbols('fact')
+u1, u2 = me.dynamicsymbols('u1 u2')
+
+# %%
+# As the time occurs explicitly in the equations of motion, I use a function
+# to represent it symbolically. This will be a known_trajectory_map in the
+# optimization problem.
+
+T = sm.symbols('T', cls=sm.Function)
+
+# %%
+def p(t, a, b):
+    return sm.exp(-b*(t - a)**2)
+
+rhs = (3.0*(p(T(t), 3, 12) + p(T(t), 6, 10) + p(T(t), 10, 6)) +
+                8.0*p(T(t), 15, 4) + 0.01)
+
+eom = sm.Matrix([
+                -y1.diff(t) - 10*y1 + u1 + u2,
+                -y2.diff(t) - 2*y2 + u1 - 2*u2,
+                -y3.diff(t) - 3*y3 + 5*y4 + u1 - u2,
+                -y4.diff(t) + 5*y3 - 3*y4 + u1 + 3*u2,
+                y1**2 + y2**2 + y3**2 + y4**2 - fact*rhs,
+])
+sm.pprint(eom)
+
+# %%
+# Set up and Solve the Optimization Problem
+# -----------------------------------------
+
+t0, tf = 0.0, 20.0
+num_nodes = 601
+interval_value = (tf - t0)/(num_nodes - 1)
+times = np.linspace(t0, tf, num_nodes)
+
+state_symbols = (y1, y2, y3, y4, fact)
+specified_symbols = (u1, u2)
+integration_method = 'backward euler'
+
+# Specify the objective function and form the gradient.
+obj_func = sm.Integral(100*(y1**2 + y2**2 + y3**2 + y4**2)
+            + 0.01*(u1**2 + u2**2), t)
+
+obj, obj_grad = create_objective_function(
+        obj_func,
+        state_symbols,
+        specified_symbols,
+        tuple(),
+        num_nodes,
+        node_time_interval=interval_value,
+        integration_method=integration_method,
+)
+
+# Specify the symbolic instance constraints, as per the example.
+instance_constraints = (
+        y1.func(t0) - 2.0,
+        y2.func(t0) - 1.0,
+        y3.func(t0) - 2.0,
+        y4.func(t0) - 1.0,
+        y1.func(tf) - 2.0,
+        y2.func(tf) - 3.0,
+        y3.func(tf) - 1.0,
+        y4.func(tf) + 2.0,
+)
+
+bounds = {fact: (1.0 , 10000 )}
+
+# %%
+# Create the optimization problem and set any options.
+prob = Problem(
+        obj,
+        obj_grad,
+        eom,
+        state_symbols,
+        num_nodes,
+        interval_value,
+        instance_constraints=instance_constraints,
+        known_trajectory_map={T(t): times},
+        bounds=bounds,
+        integration_method=integration_method,
+)
+
+#prob.add_option('nlp_scaling_method', 'gradient-based')
+prob.add_option('max_iter', 7000)
+
+# %%
+# Give some rough estimates for the trajectories.
+# In order to speed up the optimization, I used the solution from a previous
+# run as the initial guess.
+initial_guess = np.zeros(prob.num_free)
+initial_guess = np.load('betts_10_1_opty_solution.npy')
+
+# %%
+# Find the optimal solution.
+for _ in range(1):
+    solution, info = prob.solve(initial_guess)
+    initial_guess = solution
+    print(info['status_msg'])
+    print(f'Objective value achieved: {info['obj_val']:.4f}, as per the book ' +
+        f'it is {2030.85609}, so the difference is: '
+        f'{(info['obj_val'] - 2030.85609)/2030.85609*100:.3f} % ')
+
+# %%
+# 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()
+
+# %%
+# Verify that the inequality is always kept, i.e. that fact >= 1.0 always.
+print(f'minimum value of fact = {np.min(solution[4*num_nodes: 5*num_nodes]):.4e}')
+
+# %%
+# sphinx_gallery_thumbnail_number = 2
+