Betts 10 58

examples-gallery/plot_betts_10_58.py

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+"""
+Heat Equation
+=============
+
+This is example 10.58 from Betts' book "Practical Methods for Optimal Control
+Using Nonlinear Programming", 3rd edition, chapter 10: Test Problems.
+It deals with the 'discretization' of a PDE.
+
+**States**
+
+- :math:`y_0, .....y_{10}, w` : state variables
+
+**Specifieds**
+
+- :math:`v, q_{00}, q_{11}` : control variables
+
+"""
+
+import numpy as np
+import sympy as sm
+import sympy.physics.mechanics as me
+from opty.direct_collocation import Problem
+
+# %%
+# Equations of Motion.
+#---------------------
+t = me.dynamicsymbols._t
+T = sm.symbols('T', cls=sm.Function)
+
+q = list(me.dynamicsymbols(f'q:{10}'))
+w = me.dynamicsymbols('w')
+v, q00, q11 = me.dynamicsymbols('v q00 q11')
+
+# %%
+# parameters fom the example
+qa = 0.2
+gamma = 0.04
+h = 10.0
+delta = 1/9
+
+# %%
+# In addition to the 11 DEs for the 11 state variables, the book gives two
+# algebraic equations. As opty needs exactly one equation per state variable,
+# I solve the AEs for a state variable and insert into the DEs.
+
+q01 = sm.solve(sm.Matrix([h*(q[0] - w) - 1/(2*delta)*(q[1] - q00)]), q[0])[q[0]]
+q81 = sm.solve(sm.Matrix([1/(2*delta)*(q11 - q[-1])]), q[-1])[q[-1]]
+
+# %%
+uq = [q[i].diff(t) for i in range(10)]
+
+eom = sm.Matrix([
+    -uq[0] + 1/delta**2 * (q[1] - 2*q01 + q00),
+    -uq[1] + 1/delta**2 * (q[2] - 2*q[1] + q01),
+    *[-uq[i] - 1/delta**2 * (q[i+1] - 2*q[i] + q[i-1]) for i in range(2, 9)],
+    -uq[-1] + 1/delta**2 * (q11 - 2*q[-1] + q81),
+    -w.diff(t) + 1/gamma*(v - w),
+])
+
+# %%
+# Optimization
+# ------------
+# I put the optimization into a function,because this makes it a bit easier
+# to change parameters.
+
+def solve_optimization(nodes, tf, iterations=1):
+    t0, tf = 0.0, tf
+    num_nodes = nodes
+    interval_value = (tf - t0)/(num_nodes - 1)
+
+    state_symbols = q + [w]
+    specified_symbols = (v, q00, q11)
+
+    # Specify the objective function and form the gradient.
+
+    #obj_func = 1/(2*delta) * (qa - q[0](tf))**2
+    #   + 1/(2*delta) * (q[9](tf) - qa)**2
+    #   + sum([1/(2*delta) * (q[i](tf) - qa)**2 for i in range(1, 9)])
+    # this is the objective function to be minimized.
+
+    def obj(free):
+        value1 = 1/(2*delta) * (qa - free[num_nodes-1])**2
+        value2 = 1/(2*delta) * (qa - free[10*num_nodes-1])**2
+        value3 = 1/delta * np.sum([(qa - free[(i+1)*num_nodes-1])**2 for i
+                in range(1, 9)])
+        return value1 + value2 + value3
+
+    def obj_grad(free):
+        grad = np.zeros_like(free)
+        grad[num_nodes-1] = 2*1/(2*delta) * (qa - free[num_nodes-1])
+        grad[10*num_nodes-1] = 2*1/(2*delta) * (qa - free[10*num_nodes-1])
+        for i in range(1, 9):
+            grad[(i+1)*num_nodes-1] = 2*1/delta * (qa - free[(i+1)*num_nodes-1])
+        return -grad
+
+    # Specify the symbolic instance constraints, as per the example
+    instance_constraints = (
+        *[q[i].func(t0) - 0 for i in range(10)],
+        w.func(t0) - 0,
+    )
+
+    bounds = {v: (0, 1)}
+    # Create the optimization problem and set any options.
+    prob = Problem(obj,
+                obj_grad,
+                eom,
+                state_symbols,
+                num_nodes,
+                interval_value,
+                instance_constraints=instance_constraints,
+                bounds=bounds,
+    )
+
+    prob.add_option('max_iter', 3000 )
+
+    # Give some rough estimates for the trajectories.
+    initial_guess = np.zeros(prob.num_free)
+
+    # Find the optimal solution.
+    for _ in range(iterations):
+        solution, info = prob.solve(initial_guess)
+        initial_guess = solution
+        print(info['status_msg'])
+        print(f'Objective value achieved: {info['obj_val']:.4e}, ' +
+            f'as per the book it is {3.879*1.e-5} \n')
+
+    return prob, solution, info
+
+# %%
+# As per the example final time tf = 0.2
+tf = 0.2
+num_nodes = 101
+prob, solution, info = solve_optimization(num_nodes, tf, iterations=3)
+
+# %%
+# 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()
+
+# %%
+# sphinx_gallery_thumbnail_number = 2
+