def test_variables(self):
N = 10
ocp = Ocp(t0=2 * pi, T=10)
p = ocp.parameter(grid='control')
v = ocp.variable(grid='control')
x = ocp.state()
ocp.set_der(x, 0)
ocp.subject_to(ocp.at_t0(x) == 0)
ts = linspace(0, 10, N)
ocp.add_objective(ocp.integral(sin(v - p)**2, grid='control'))
ocp.method(MultipleShooting(N=N))
ocp.solver('ipopt')
ocp.set_value(p, ts)
ocp.set_initial(v, ts)
sol = ocp.solve()
_, xs = sol.sample(v, grid='control')
assert_array_almost_equal(xs[:-1], ts)
ocp.set_initial(v, 0.1 + 2 * pi + ts)
sol = ocp.solve()
_, xs = sol.sample(v, grid='control')
assert_array_almost_equal(xs[:-1], 2 * pi + ts)
ocp.set_initial(v, 0.1 + ocp.t)
sol = ocp.solve()
_, xs = sol.sample(v, grid='control')
assert_array_almost_equal(xs[:-1], 2 * pi + ts)
ocp.set_initial(v, 0.1 + 2 * pi)
sol = ocp.solve()
_, xs = sol.sample(v, grid='control')
with self.assertRaises(AssertionError):
assert_array_almost_equal(xs[:-1], 2 * pi + ts)
def test_der(self):
T = 1
M = 1
b = 1
t0 = 0
x0 = 0
ocp = Ocp(t0=t0, T=T)
x = ocp.state()
u = ocp.control()
ocp.set_der(x, u)
y = 2 * x
ocp.subject_to(ocp.der(y) <= 2 * b)
ocp.subject_to(-2 * b <= ocp.der(y))
ocp.add_objective(ocp.at_tf(x))
ocp.subject_to(ocp.at_t0(x) == x0)
ocp.solver('ipopt')
ocp.method(MultipleShooting(N=4, M=M, intg='rk'))
sol = ocp.solve()
ts, xs = sol.sample(x, grid='control')
self.assertAlmostEqual(xs[0], x0, places=6)
self.assertAlmostEqual(xs[-1], x0 - b * T, places=6)
self.assertAlmostEqual(ts[0], t0)
self.assertAlmostEqual(ts[-1], t0 + T)
def test_integral(self):
t0 = 1.2
T = 5.7
ocp = Ocp(t0=t0, T=T)
x = ocp.state()
u = ocp.control()
ocp.set_der(x, u)
ocp.subject_to(ocp.at_t0(x) == 0)
ocp.subject_to(u <= 1)
f = ocp.integral(x * ocp.t)
ocp.add_objective(-f) # (t-t0)*t -> t^3/3-t^2/2*t0
ocp.solver('ipopt')
opts = {"abstol": 1e-8, "reltol": 1e-8, "quad_err_con": True}
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(f, grid='control')
I = lambda t: t**3 / 3 - t**2 / 2 * t0
x_ref = I(t0 + T) - I(t0)
assert_array_almost_equal(xs[-1], x_ref)
def test_param(self):
ocp = Ocp(T=1)
x = ocp.state()
u = ocp.control()
p = ocp.parameter()
ocp.set_der(x, u)
ocp.subject_to(u <= 1)
ocp.subject_to(-1 <= u)
ocp.add_objective(ocp.at_tf(x))
ocp.subject_to(ocp.at_t0(x) == p)
ocp.solver('ipopt')
ocp.method(MultipleShooting())
ocp.set_value(p, 0)
sol = ocp.solve()
ts, xs = sol.sample(x, grid='control')
self.assertAlmostEqual(xs[0], 0)
ocp.set_value(p, 1)
sol = ocp.solve()
ts, xs = sol.sample(x, grid='control')
self.assertAlmostEqual(xs[0], 1)
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 vdp(method,grid='control'):
ocp = Ocp(T=10)
# Define 2 states
x1 = ocp.state()
x2 = ocp.state()
# Define 1 control
u = ocp.control(order=0)
# Specify ODE
ocp.set_der(x1, (1 - x2**2) * x1 - x2 + u)
ocp.set_der(x2, x1)
# Lagrange objective
ocp.add_objective(ocp.integral(x1**2 + x2**2 + u**2))
# Path constraints
ocp.subject_to(-1 <= (u <= 1))
ocp.subject_to(x1 >= -0.25, grid=grid)
# Initial constraints
ocp.subject_to(ocp.at_t0(x1) == 0)
ocp.subject_to(ocp.at_t0(x2) == 1)
# Pick an NLP solver backend
ocp.solver('ipopt')
# Pick a solution method
ocp.method(method)
return (ocp, x1, x2, u)
def integrator_control_problem(T=1, u_max=1, x0=0, stage_method=None, t0=0):
if stage_method is None:
stage_method = MultipleShooting()
ocp = Ocp(t0=t0, T=T)
x = ocp.state()
u = ocp.control()
ocp.set_der(x, u)
ocp.subject_to(u <= u_max)
ocp.subject_to(-u_max <= u)
ocp.add_objective(ocp.at_tf(x))
if x0 is not None:
ocp.subject_to(ocp.at_t0(x) == x0)
ocp.solver('ipopt')
ocp.method(stage_method)
return (ocp, x, u)
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)
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)
T = 10 tgrid = np.linspace(0, T, N) y_ref_val = np.sin(tgrid) x0 = [0, 0] ocp = Ocp(t0=tgrid[0], T=tgrid[-1]) # Define states x = ocp.state(2) # Define controls u = ocp.control() # Specify ODE model_rhs = pendulum_ode(x, u, model_param) ocp.set_der(x, model_rhs) # Set initial conditions ocp.subject_to(ocp.at_t0(x) == x0) # Define reference y_ref = ocp.parameter(grid='control') ocp.set_value(y_ref, y_ref_val) # Define output correction beta = ocp.parameter(grid='control') # Define previous control u_prev = ocp.parameter(grid='control') # Set ILC objective
x = ocp.state() y = ocp.state() theta = ocp.state() v = ocp.control() w = ocp.control() #--------------------------path parameters s_obs = ocp.state() sdot_obs = ocp.control() #-----------------------------ODEs ocp.set_der(x, v * cos(theta)) ocp.set_der(y, v * sin(theta)) ocp.set_der(theta, w) ocp.set_der(s_obs, sdot_obs) #-------------------------------------------------------------------------------# # Solve the first iteration # #-------------------------------------------------------------------------------# #------------------------- Constraints on initial point X_0 = ocp.parameter(4) X = vertcat(x, y, theta, s_obs) ocp.subject_to(ocp.at_t0(X) == X_0)
#------------- Initialize OCP ocp = Ocp(T=N * dt) #------------------------- System model x = ocp.state() y = ocp.state() theta = ocp.state() v = ocp.control() w = ocp.control() #-----------------------------ODEs ocp.set_der(x, v * cos(theta)) ocp.set_der(y, v * sin(theta)) ocp.set_der(theta, w) #-------------------------------------------------------------------------------# # Solve the first iteration # #-------------------------------------------------------------------------------# #------------------------- Constraints on initial point X_0 = ocp.parameter(3) X = vertcat(x, y, theta) ocp.subject_to(ocp.at_t0(X) == X_0) current_X = vertcat(initial_pos_x, initial_pos_y, 0.0)
