def test_time_dep_ode(self):
t0 = 1.2
T = 5.7
ocp = Ocp(t0=t0, T=5.7)
x = ocp.state()
ocp.set_der(x, ocp.t**2)
ocp.subject_to(ocp.at_t0(x) == 0)
tf = t0 + T
x_ref = tf**3 / 3 - t0**3 / 3
ocp.solver('ipopt')
opts = {"abstol": 1e-9, "reltol": 1e-9}
for method in [
MultipleShooting(intg='rk'),
MultipleShooting(intg='cvodes', intg_options=opts),
MultipleShooting(intg='idas', intg_options=opts),
DirectCollocation()
]:
ocp.method(method)
sol = ocp.solve()
ts, xs = sol.sample(x, grid='control')
x_ref = ts**3 / 3 - t0**3 / 3
assert_array_almost_equal(xs, x_ref)
def test_basic_t0_free(self):
xf = 2
t0 = 0
for T in [2]:
for x0 in [0, 1]:
for b in [1, 2]:
for method in [
MultipleShooting(N=4, intg='rk'),
MultipleShooting(N=4, intg='cvodes'),
MultipleShooting(N=4, intg='idas'),
DirectCollocation(N=4)
]:
ocp = Ocp(t0=FreeTime(2), T=T)
x = ocp.state()
u = ocp.control()
ocp.set_der(x, u)
ocp.subject_to(u <= b)
ocp.subject_to(-b <= u)
ocp.add_objective(ocp.tf)
ocp.subject_to(ocp.at_t0(x) == x0)
ocp.subject_to(ocp.at_tf(x) == xf)
ocp.subject_to(ocp.t0 >= 0)
ocp.solver('ipopt')
ocp.method(method)
sol = ocp.solve()
ts, xs = sol.sample(x, grid='control')
self.assertAlmostEqual(xs[0], x0, places=6)
self.assertAlmostEqual(xs[-1], xf, places=6)
self.assertAlmostEqual(ts[0], t0)
self.assertAlmostEqual(ts[-1], t0 + T)
def bang_bang_problem(stage_method):
ocp = Ocp(T=FreeTime(1))
p = ocp.state()
v = ocp.state()
u = ocp.control()
ocp.set_der(p, v)
ocp.set_der(v, u)
ocp.subject_to(u <= 1)
ocp.subject_to(-1 <= u)
ocp.add_objective(ocp.T)
ocp.subject_to(ocp.at_t0(p) == 0)
ocp.subject_to(ocp.at_t0(v) == 0)
ocp.subject_to(ocp.at_tf(p) == 1)
ocp.subject_to(ocp.at_tf(v) == 0)
ocp.solver('ipopt')
ocp.method(stage_method)
return (ocp, ocp.solve(), p, v, u)
# After bounce 1
stage2, p2, v2 = create_bouncing_ball_stage(ocp)
ocp.subject_to(stage2.at_t0(v2) == -0.9 * stage1.at_tf(v1))
ocp.subject_to(stage2.at_t0(p2) == stage1.at_tf(p1))
ocp.subject_to(stage2.t0 == stage1.tf)
ocp.subject_to(stage2.at_tf(p2) == 0)
# After bounce 2
stage3, p3, v3 = create_bouncing_ball_stage(ocp)
ocp.subject_to(stage3.at_t0(v3) == -0.9 * stage2.at_tf(v2))
ocp.subject_to(stage3.at_t0(p3) == stage2.at_tf(p2))
ocp.subject_to(stage3.t0 == stage2.tf)
ocp.subject_to(stage3.at_tf(v3) == 0)
ocp.subject_to(stage3.at_tf(p3) == 0.5) # Stop at a half meter!
ocp.solver('ipopt')
# Solve
sol = ocp.solve()
# Plot the 3 bounces
plt.figure()
ts1, ps1 = sol(stage1).sample(p1, grid='integrator')
ts2, ps2 = sol(stage2).sample(p2, grid='integrator')
ts3, ps3 = sol(stage3).sample(p3, grid='integrator')
plt.plot(ts1, ps1)
plt.plot(ts2, ps2)
plt.plot(ts3, ps3)
plt.show(block=True)
def test_dae_casadi(self):
# cross check with dae_colloation
xref = 0.1 # chariot reference
l = 1. #- -> crane, + -> pendulum
m = 1.
M = 1.
g = 9.81
ocp = Ocp(T=5)
x = ocp.state()
y = ocp.state()
w = ocp.state()
dx = ocp.state()
dy = ocp.state()
dw = ocp.state()
xa = ocp.algebraic()
u = ocp.control()
ocp.set_der(x, dx)
ocp.set_der(y, dy)
ocp.set_der(w, dw)
ddx = (w - x) * xa / m
ddy = g - y * xa / m
ddw = ((x - w) * xa - u) / M
ocp.set_der(dx, ddx)
ocp.set_der(dy, ddy)
ocp.set_der(dw, ddw)
ocp.add_alg((x - w) * (ddx - ddw) + y * ddy + dy * dy + (dx - dw)**2)
ocp.add_objective(
ocp.at_tf((x - xref) * (x - xref) + (w - xref) * (w - xref) +
dx * dx + dy * dy))
ocp.add_objective(
ocp.integral((x - xref) * (x - xref) + (w - xref) * (w - xref)))
ocp.subject_to(-2 <= (u <= 2))
ocp.subject_to(ocp.at_t0(x) == 0)
ocp.subject_to(ocp.at_t0(y) == l)
ocp.subject_to(ocp.at_t0(w) == 0)
ocp.subject_to(ocp.at_t0(dx) == 0)
ocp.subject_to(ocp.at_t0(dy) == 0)
ocp.subject_to(ocp.at_t0(dw) == 0)
#ocp.subject_to(xa>=0,grid='integrator_roots')
ocp.set_initial(y, l)
ocp.set_initial(xa, 9.81)
# Pick an NLP solver backend
# NOTE: default scaling strategy of MUMPS leads to a singular matrix error
ocp.solver(
'ipopt', {
"ipopt.linear_solver": "mumps",
"ipopt.mumps_scaling": 0,
"ipopt.tol": 1e-12
})
# Pick a solution method
method = DirectCollocation(N=50)
ocp.method(method)
# Solve
sol = ocp.solve()
assert_array_almost_equal(
sol.sample(xa, grid='integrator', refine=1)[1][0],
9.81011622448889)
assert_array_almost_equal(
sol.sample(xa, grid='integrator', refine=1)[1][1],
9.865726317147214)
assert_array_almost_equal(
sol.sample(xa, grid='integrator')[1][0], 9.81011622448889)
assert_array_almost_equal(
sol.sample(xa, grid='integrator')[1][1], 9.865726317147214)
assert_array_almost_equal(
sol.sample(xa, grid='control')[1][0], 9.81011622448889)
assert_array_almost_equal(
sol.sample(xa, grid='control')[1][1], 9.865726317147214)
def test_stage_cloning_t0_T(self):
for t0_stage, t0_sol_stage in [(None, 0), (-1, -1),
(FreeTime(-1), -1)]:
for T_stage, T_sol_stage in [(None, 2), (2, 2), (FreeTime(1), 2)]:
kwargs = {}
if t0_stage is not None:
kwargs["t0"] = t0_stage
if T_stage is not None:
kwargs["T"] = T_stage
stage = Stage(**kwargs)
p = stage.state()
v = stage.state()
u = stage.control()
stage.set_der(p, v)
stage.set_der(v, u)
stage.subject_to(u <= 1)
stage.subject_to(-1 <= u)
stage.add_objective(stage.tf)
stage.subject_to(stage.at_t0(p) == 0)
stage.subject_to(stage.at_t0(v) == 0)
stage.subject_to(stage.at_tf(p) == 1)
stage.subject_to(stage.at_tf(v) == 0)
stage.method(MultipleShooting(N=2))
for t0, t0_sol in ([] if t0_stage is None else [
(None, t0_sol_stage)
]) + [(-1, -1), (FreeTime(-1), -1)]:
for T, T_sol in ([] if T_stage is None else [
(None, T_sol_stage)
]) + [(2, 2), (FreeTime(1), 2)]:
ocp = Ocp()
kwargs = {}
if t0 is not None:
kwargs["t0"] = t0
if T is not None:
kwargs["T"] = T
mystage = ocp.stage(stage, **kwargs)
if mystage.is_free_starttime():
ocp.subject_to(mystage.t0 >= t0_sol)
ocp.solver('ipopt')
sol = ocp.solve()
tolerance = 1e-6
ts, ps = sol(mystage).sample(p,
grid='integrator',
refine=10)
ps_ref = np.hstack(
((0.5 * np.linspace(0, 1, 10 + 1)**2)[:-1],
np.linspace(0.5, 1.5, 10 + 1) -
0.5 * np.linspace(0, 1, 10 + 1)**2))
np.testing.assert_allclose(ps, ps_ref, atol=tolerance)
ts_ref = t0_sol + np.linspace(0, 2, 10 * 2 + 1)
ts, vs = sol(mystage).sample(v,
grid='integrator',
refine=10)
np.testing.assert_allclose(ts, ts_ref, atol=tolerance)
vs_ref = np.hstack((np.linspace(0, 1, 10 + 1)[:-1],
np.linspace(1, 0, 10 + 1)))
np.testing.assert_allclose(vs, vs_ref, atol=tolerance)
u_ref = np.array([1.0] * 10 + [-1.0] * 11)
ts, us = sol(mystage).sample(u,
grid='integrator',
refine=10)
np.testing.assert_allclose(us, u_ref, atol=tolerance)