def setUp(self):
self.input = ph.Input(np.sin)
phi = cr.Function(np.sin)
psi = cr.Function(np.cos)
self.t_funcs = np.array([phi, psi])
register_base("funcs", self.t_funcs, overwrite=True)
self.test_funcs = ph.TestFunction("funcs")
self.s_funcs = np.array([cr.Function(self.scale)])[[0, 0]]
register_base("scale_funcs", self.s_funcs, overwrite=True)
self.scale_funcs = ph.ScalarFunction("scale_funcs")
nodes, self.ini_funcs = cure_interval(LagrangeFirstOrder, (0, 1),
node_count=2)
register_base("prod_ini_funcs", self.ini_funcs, overwrite=True)
self.field_var = ph.FieldVariable("prod_ini_funcs")
self.field_var_dz = ph.SpatialDerivedFieldVariable("prod_ini_funcs", 1)
def test_ScalarTerm(self):
self.assertRaises(TypeError, ph.ScalarTerm, 7) # factor is number
self.assertRaises(TypeError, ph.ScalarTerm,
cr.Function(np.sin)) # factor is Function
ph.ScalarTerm(self.input)
self.assertRaises(ValueError, ph.ScalarTerm,
self.test_func) # integration has to be done
t1 = ph.ScalarTerm(self.xdz_at1)
self.assertEqual(t1.scale, 1.0) # default scale
# check if automated evaluation works
self.assertTrue(np.allclose(t1.arg.args[0].data, np.array([-1, 1])))
def setUp(self):
z_start = 0
z_end = 1
z_step = 0.1
self.dz = sim.Domain(bounds=(z_start, z_end), num=9)
t_start = 0
t_end = 10
t_step = 0.01
self.dt = sim.Domain(bounds=(t_start, t_end), step=t_step)
self.params = ut.Parameters
self.params.node_distance = 0.1
self.params.m = 1.0
self.params.order = 8
self.params.sigma = 1
self.params.tau = 1
self.y_end = 10
self.u = tr.FlatString(0, self.y_end, z_start, z_end, 0, 5,
self.params)
def x(z, t):
"""
initial conditions for testing
"""
return 0
def x_dt(z, t):
"""
initial conditions for testing
"""
return 0
# initial conditions
self.ic = np.array([
cr.Function(lambda z: x(z, 0)), # x(z, 0)
cr.Function(lambda z: x_dt(z, 0)), # dx_dt(z, 0)
])
def setUp(self):
self.input = ph.Input(np.sin)
self.phi = np.array([cr.Function(lambda x: 2 * x)])
register_base("phi", self.phi, overwrite=True)
self.test_func = ph.TestFunction("phi")
nodes, self.ini_funcs = cure_interval(LagrangeFirstOrder, (0, 1),
node_count=2)
register_base("ini_funcs", self.ini_funcs, overwrite=True)
self.xdt = ph.TemporalDerivedFieldVariable("ini_funcs", order=1)
self.xdz_at1 = ph.SpatialDerivedFieldVariable("ini_funcs",
order=1,
location=1)
self.prod = ph.Product(self.input, self.xdt)
def setUp(self):
interval = (0, 1)
nodes, funcs = sf.cure_interval(sf.LagrangeFirstOrder, interval, 3)
register_base("funcs", funcs, overwrite=True)
x = ph.FieldVariable("funcs")
x_dt = ph.TemporalDerivedFieldVariable("funcs", 1)
x_dz = ph.SpatialDerivedFieldVariable("funcs", 1)
register_base("scal_func", cr.Function(np.exp), overwrite=True)
exp = ph.ScalarFunction("scal_func")
alpha = 2
self.term1 = ph.IntegralTerm(x_dt, interval, 1 + alpha)
self.term2 = ph.IntegralTerm(x_dz, interval, 2)
self.term3 = ph.IntegralTerm(ph.Product(x, exp), interval)
self.weight_label = "funcs"
self.weights = np.hstack([1, 1, 1, 2, 2, 2])
def setUp(self):
interval = (0, 1)
nodes, funcs = sf.cure_interval(sf.LagrangeFirstOrder, interval, 3)
register_base("funcs", funcs, overwrite=True)
x_at1 = ph.FieldVariable("funcs", location=1)
x_dt_at1 = ph.TemporalDerivedFieldVariable("funcs", 1, location=1)
x_dz_at0 = ph.SpatialDerivedFieldVariable("funcs", 1, location=0)
exp_func = cr.Function(np.exp)
register_base("exp_func", exp_func, overwrite=True)
exp_at1 = ph.ScalarFunction("exp_func", location=1)
alpha = 2
self.term1 = ph.ScalarTerm(x_dt_at1, 1 + alpha)
self.term2 = ph.ScalarTerm(x_dz_at0, 2)
self.term3 = ph.ScalarTerm(ph.Product(x_at1, exp_at1))
self.weight_label = "funcs"
self.weights = np.array([1, 1, 1, 2, 2, 2])
def test_IntegralTerm(self):
self.assertRaises(TypeError, ph.IntegralTerm, 7,
(0, 1)) # integrand is number
self.assertRaises(TypeError, ph.IntegralTerm, cr.Function(np.sin),
(0, 1)) # integrand is Function
self.assertRaises(ValueError, ph.IntegralTerm, self.xdz_at1,
(0, 1)) # nothing left after evaluation
self.assertRaises(TypeError, ph.IntegralTerm, self.xdt,
[0, 1]) # limits is list
ph.IntegralTerm(self.test_func, (0, 1)) # integrand is Placeholder
self.assertRaises(ValueError, ph.IntegralTerm, self.input,
(0, 1)) # nothing to do
ph.IntegralTerm(self.xdt, (0, 1)) # integrand is Placeholder
ph.IntegralTerm(self.prod, (0, 1)) # integrand is Product
t1 = ph.IntegralTerm(self.xdt, (0, 1))
self.assertEqual(t1.scale, 1.0) # default scale
self.assertEqual(t1.arg.args[0],
self.xdt) # automated product creation
self.assertEqual(t1.limits, (0, 1))
# eigenfunctions target system
eig_freq_t = np.sqrt(-eig_values.astype(complex))
norm_fac_t = norm_fac * eig_freq / eig_freq_t
eig_funcs_t = np.asarray(
[ef.SecondOrderDirichletEigenfunction(eig_freq_t[i], param_t, spatial_domain.bounds, norm_fac_t[i]) for i in
range(n)])
re.register_base("eig_funcs_t", eig_funcs_t, overwrite=True)
# init controller
x_at_1 = ph.FieldVariable("eig_funcs", location=1)
xt_at_1 = ph.FieldVariable("eig_funcs_t", weight_label="eig_funcs", location=1)
controller = ct.Controller(ct.ControlLaw([ph.ScalarTerm(x_at_1, 1), ph.ScalarTerm(xt_at_1, -1)]))
# derive initial field variable x(z,0) and weights
start_state = cr.Function(lambda z: init_profile)
initial_weights = cr.project_on_base(start_state, eig_funcs)
# init trajectory
traj = tr.RadTrajectory(l, T, param_t, bound_cond_type, actuation_type)
# input with feedback
control_law = sim.SimulationInputSum([traj, controller])
# determine (A,B) with modal-transfomation
A = np.diag(eig_values)
B = -a2 * np.array([eig_funcs[i].derive()(l) for i in range(n)])
ss = sim.StateSpace("eig_funcs", A, B, input_handle=control_law)
# evaluate desired output data
z_d = np.linspace(0, l, len(spatial_domain))
def setUp(self):
self.psi = cr.Function(np.sin)
register_base("funcs", self.psi, overwrite=True)
self.funcs = ph.TestFunction("funcs")
def test_it(self):
# system/simulation parameters
actuation_type = 'robin'
bound_cond_type = 'robin'
self.l = 1.
spatial_disc = 10
self.dz = sim.Domain(bounds=(0, self.l), num=spatial_disc)
self.T = 1.
temporal_disc = 1e2
self.dt = sim.Domain(bounds=(0, self.T), num=temporal_disc)
self.n = 10
# original system parameters
a2 = 1.5
a1_z = cr.Function(lambda z: 1, derivative_handles=[lambda z: 0])
a0_z = lambda z: 3
alpha = -2
beta = -3
self.param = [a2, a1_z, a0_z, alpha, beta]
# target system parameters (controller parameters)
a1_t = -5
a0_t = -25
alpha_t = 3
beta_t = 2
self.param_t = [a2, a1_t, a0_t, alpha_t, beta_t]
adjoint_param_t = ef.get_adjoint_rad_evp_param(self.param_t)
# original intermediate ("_i") and traget intermediate ("_ti") system parameters
_, _, a0_i, alpha_i, beta_i = ef.transform2intermediate(self.param,
d_end=self.l)
self.param_i = a2, 0, a0_i, alpha_i, beta_i
_, _, a0_ti, alpha_ti, beta_ti = ef.transform2intermediate(
self.param_t)
self.param_ti = a2, 0, a0_ti, alpha_ti, beta_ti
# create (not normalized) target (_t) eigenfunctions
eig_freq_t, self.eig_val_t = ef.compute_rad_robin_eigenfrequencies(
self.param_t, self.l, self.n)
init_eig_funcs_t = np.array([
ef.SecondOrderRobinEigenfunction(om, self.param_t, self.dz.bounds)
for om in eig_freq_t
])
init_adjoint_eig_funcs_t = np.array([
ef.SecondOrderRobinEigenfunction(om, adjoint_param_t,
self.dz.bounds)
for om in eig_freq_t
])
# normalize eigenfunctions and adjoint eigenfunctions
adjoint_and_eig_funcs_t = [
cr.normalize_function(init_eig_funcs_t[i],
init_adjoint_eig_funcs_t[i])
for i in range(self.n)
]
eig_funcs_t = np.array(
[f_tuple[0] for f_tuple in adjoint_and_eig_funcs_t])
self.adjoint_eig_funcs_t = np.array(
[f_tuple[1] for f_tuple in adjoint_and_eig_funcs_t])
# # transformed original eigenfunctions
self.eig_funcs = np.array([
ef.TransformedSecondOrderEigenfunction(
self.eig_val_t[i],
[eig_funcs_t[i](0), alpha * eig_funcs_t[i](0), 0, 0],
[a2, a1_z, a0_z], np.linspace(0, self.l, 1e4))
for i in range(self.n)
])
# create testfunctions
nodes, self.fem_funcs = sf.cure_interval(sf.LagrangeFirstOrder,
self.dz.bounds,
node_count=self.n)
# register functions
register_base("eig_funcs_t", eig_funcs_t, overwrite=True)
register_base("adjoint_eig_funcs_t",
self.adjoint_eig_funcs_t,
overwrite=True)
register_base("eig_funcs", self.eig_funcs, overwrite=True)
register_base("fem_funcs", self.fem_funcs, overwrite=True)
# init trajectory
self.traj = tr.RadTrajectory(self.l, self.T, self.param_ti,
bound_cond_type, actuation_type)
# original () and target (_t) field variable
fem_field_variable = ph.FieldVariable("fem_funcs", location=self.l)
field_variable_t = ph.FieldVariable("eig_funcs_t",
weight_label="eig_funcs",
location=self.l)
d_field_variable_t = ph.SpatialDerivedFieldVariable(
"eig_funcs_t", 1, weight_label="eig_funcs", location=self.l)
field_variable = ph.FieldVariable("eig_funcs", location=self.l)
d_field_variable = ph.SpatialDerivedFieldVariable("eig_funcs",
1,
location=self.l)
# intermediate (_i) and target intermediate (_ti) transformations by z=l
# x_i = x * transform_i_at_l
self.transform_i_at_l = np.exp(
integrate.quad(lambda z: a1_z(z) / 2 / a2, 0, self.l)[0])
# x = x_i * inv_transform_i_at_l
self.inv_transform_i_at_l = np.exp(
-integrate.quad(lambda z: a1_z(z) / 2 / a2, 0, self.l)[0])
# x_ti = x_t * transform_ti_at_l
self.transform_ti_at_l = np.exp(a1_t / 2 / a2 * self.l)
# intermediate (_i) and target intermediate (_ti) field variable (list of scalar terms = sum of scalar terms)
self.x_fem_i_at_l = [
ph.ScalarTerm(fem_field_variable, self.transform_i_at_l)
]
self.x_i_at_l = [ph.ScalarTerm(field_variable, self.transform_i_at_l)]
self.xd_i_at_l = [
ph.ScalarTerm(d_field_variable, self.transform_i_at_l),
ph.ScalarTerm(field_variable,
self.transform_i_at_l * a1_z(self.l) / 2 / a2)
]
self.x_ti_at_l = [
ph.ScalarTerm(field_variable_t, self.transform_ti_at_l)
]
self.xd_ti_at_l = [
ph.ScalarTerm(d_field_variable_t, self.transform_ti_at_l),
ph.ScalarTerm(field_variable_t,
self.transform_ti_at_l * a1_t / 2 / a2)
]
# discontinuous operator (Kx)(t) = int_kernel_zz(l)*x(l,t)
self.int_kernel_zz = alpha_ti - alpha_i + integrate.quad(
lambda z: (a0_i(z) - a0_ti) / 2 / a2, 0, self.l)[0]
controller = ut.get_parabolic_robin_backstepping_controller(
state=self.x_fem_i_at_l,
approx_state=self.x_i_at_l,
d_approx_state=self.xd_i_at_l,
approx_target_state=self.x_ti_at_l,
d_approx_target_state=self.xd_ti_at_l,
integral_kernel_zz=self.int_kernel_zz,
original_beta=beta_i,
target_beta=beta_ti,
trajectory=self.traj,
scale=self.inv_transform_i_at_l)
rad_pde = ut.get_parabolic_robin_weak_form("fem_funcs", "fem_funcs",
controller, self.param,
self.dz.bounds)
cf = sim.parse_weak_formulation(rad_pde)
ss_weak = cf.convert_to_state_space()
# simulate
t, q = sim.simulate_state_space(ss_weak, np.zeros(
(len(self.fem_funcs))), self.dt)
eval_d = sim.evaluate_approximation("fem_funcs", q, t, self.dz)
x_0t = eval_d.output_data[:, 0]
yc, tc = tr.gevrey_tanh(self.T, 1)
x_0t_desired = np.interp(t, tc, yc[0, :])
self.assertLess(np.average((x_0t - x_0t_desired)**2), 1e-4)
# display results
if show_plots:
win1 = vis.PgAnimatedPlot([eval_d], title="Test")
win2 = vis.PgSurfacePlot(eval_d)
app.exec_()
def test_it(self):
# original system parameters
a2 = 1.5
a1 = 2.5
a0 = 28
alpha = -2
beta = -3
param = [a2, a1, a0, alpha, beta]
adjoint_param = ef.get_adjoint_rad_evp_param(param)
# target system parameters (controller parameters)
a1_t = -5
a0_t = -25
alpha_t = 3
beta_t = 2
# a1_t = a1; a0_t = a0; alpha_t = alpha; beta_t = beta
param_t = [a2, a1_t, a0_t, alpha_t, beta_t]
# original intermediate ("_i") and traget intermediate ("_ti") system parameters
_, _, a0_i, alpha_i, beta_i = ef.transform2intermediate(param)
_, _, a0_ti, alpha_ti, beta_ti = ef.transform2intermediate(param_t)
# system/simulation parameters
actuation_type = 'robin'
bound_cond_type = 'robin'
self.l = 1.
spatial_disc = 10
dz = sim.Domain(bounds=(0, self.l), num=spatial_disc)
T = 1.
temporal_disc = 1e2
dt = sim.Domain(bounds=(0, T), num=temporal_disc)
n = 10
# create (not normalized) eigenfunctions
eig_freq, eig_val = ef.compute_rad_robin_eigenfrequencies(
param, self.l, n)
init_eig_funcs = np.array([
ef.SecondOrderRobinEigenfunction(om, param, dz.bounds)
for om in eig_freq
])
init_adjoint_eig_funcs = np.array([
ef.SecondOrderRobinEigenfunction(om, adjoint_param, dz.bounds)
for om in eig_freq
])
# normalize eigenfunctions and adjoint eigenfunctions
adjoint_and_eig_funcs = [
cr.normalize_function(init_eig_funcs[i], init_adjoint_eig_funcs[i])
for i in range(n)
]
eig_funcs = np.array([f_tuple[0] for f_tuple in adjoint_and_eig_funcs])
adjoint_eig_funcs = np.array(
[f_tuple[1] for f_tuple in adjoint_and_eig_funcs])
# eigenfunctions from target system ("_t")
eig_freq_t = np.sqrt(-a1_t**2 / 4 / a2**2 + (a0_t - eig_val) / a2)
eig_funcs_t = np.array([
ef.SecondOrderRobinEigenfunction(eig_freq_t[i], param_t,
dz.bounds).scale(eig_funcs[i](0))
for i in range(n)
])
# register eigenfunctions
register_base("eig_funcs", eig_funcs, overwrite=True)
register_base("adjoint_eig_funcs", adjoint_eig_funcs, overwrite=True)
register_base("eig_funcs_t", eig_funcs_t, overwrite=True)
# derive initial field variable x(z,0) and weights
start_state = cr.Function(lambda z: 0., domain=(0, self.l))
initial_weights = cr.project_on_base(start_state, adjoint_eig_funcs)
# controller initialization
x_at_l = ph.FieldVariable("eig_funcs", location=self.l)
xd_at_l = ph.SpatialDerivedFieldVariable("eig_funcs",
1,
location=self.l)
x_t_at_l = ph.FieldVariable("eig_funcs_t",
weight_label="eig_funcs",
location=self.l)
xd_t_at_l = ph.SpatialDerivedFieldVariable("eig_funcs_t",
1,
weight_label="eig_funcs",
location=self.l)
combined_transform = lambda z: np.exp((a1_t - a1) / 2 / a2 * z)
int_kernel_zz = lambda z: alpha_ti - alpha_i + (a0_i - a0_ti
) / 2 / a2 * z
controller = ct.Controller(
ct.ControlLaw([
ph.ScalarTerm(x_at_l,
(beta_i - beta_ti - int_kernel_zz(self.l))),
ph.ScalarTerm(x_t_at_l, -beta_ti * combined_transform(self.l)),
ph.ScalarTerm(x_at_l, beta_ti),
ph.ScalarTerm(xd_t_at_l, -combined_transform(self.l)),
ph.ScalarTerm(x_t_at_l,
-a1_t / 2 / a2 * combined_transform(self.l)),
ph.ScalarTerm(xd_at_l, 1),
ph.ScalarTerm(x_at_l, a1 / 2 / a2 + int_kernel_zz(self.l))
]))
# init trajectory
traj = tr.RadTrajectory(self.l, T, param_t, bound_cond_type,
actuation_type)
traj.scale = combined_transform(self.l)
# input with feedback
control_law = sim.SimulationInputSum([traj, controller])
# control_law = sim.simInputSum([traj])
# determine (A,B) with modal-transformation
A = np.diag(np.real(eig_val))
B = a2 * np.array(
[adjoint_eig_funcs[i](self.l) for i in range(len(eig_freq))])
ss_modal = sim.StateSpace("eig_funcs", A, B, input_handle=control_law)
# simulate
t, q = sim.simulate_state_space(ss_modal, initial_weights, dt)
eval_d = sim.evaluate_approximation("eig_funcs", q, t, dz)
x_0t = eval_d.output_data[:, 0]
yc, tc = tr.gevrey_tanh(T, 1)
x_0t_desired = np.interp(t, tc, yc[0, :])
self.assertLess(np.average((x_0t - x_0t_desired)**2), 1e-4)
# display results
if show_plots:
win1 = vis.PgAnimatedPlot([eval_d], title="Test")
win2 = vis.PgSurfacePlot(eval_d)
app.exec_()
def test_it(self):
# original system parameters
a2 = 1
a1 = 0 # attention: only a2 = 1., a1 =0 supported in this test case
a0 = 0
param = [a2, a1, a0, None, None]
# target system parameters (controller parameters)
a1_t = 0
a0_t = 0 # attention: only a2 = 1., a1 =0 and a0 =0 supported in this test case
param_t = [a2, a1_t, a0_t, None, None]
# system/simulation parameters
actuation_type = 'dirichlet'
bound_cond_type = 'dirichlet'
l = 1.
spatial_disc = 10
dz = sim.Domain(bounds=(0, l), num=spatial_disc)
T = 1.
temporal_disc = 1e2
dt = sim.Domain(bounds=(0, T), num=temporal_disc)
n = 10
# eigenvalues /-functions original system
eig_freq = np.array([(i + 1) * np.pi / l for i in range(n)])
eig_values = a0 - a2 * eig_freq**2 - a1**2 / 4. / a2
norm_fac = np.ones(eig_freq.shape) * np.sqrt(2)
eig_funcs = np.asarray([
ef.SecondOrderDirichletEigenfunction(eig_freq[i], param, dz.bounds,
norm_fac[i]) for i in range(n)
])
register_base("eig_funcs", eig_funcs, overwrite=True)
# eigenfunctions target system
eig_freq_t = np.sqrt(-eig_values.astype(complex))
norm_fac_t = norm_fac * eig_freq / eig_freq_t
eig_funcs_t = np.asarray([
ef.SecondOrderDirichletEigenfunction(eig_freq_t[i], param_t,
dz.bounds, norm_fac_t[i])
for i in range(n)
])
register_base("eig_funcs_t", eig_funcs_t, overwrite=True)
# derive initial field variable x(z,0) and weights
start_state = cr.Function(lambda z: 0., domain=(0, l))
initial_weights = cr.project_on_base(start_state, eig_funcs)
# init trajectory / input of target system
traj = tr.RadTrajectory(l, T, param_t, bound_cond_type, actuation_type)
# init controller
x_at_1 = ph.FieldVariable("eig_funcs", location=1)
xt_at_1 = ph.FieldVariable("eig_funcs_t",
weight_label="eig_funcs",
location=1)
controller = ct.Controller(
ct.ControlLaw(
[ph.ScalarTerm(x_at_1, 1),
ph.ScalarTerm(xt_at_1, -1)]))
# input with feedback
control_law = sim.SimulationInputSum([traj, controller])
# determine (A,B) with modal-transfomation
A = np.diag(eig_values)
B = -a2 * np.array([eig_funcs[i].derive()(l) for i in range(n)])
ss = sim.StateSpace("eig_funcs", A, B, input_handle=control_law)
# simulate
t, q = sim.simulate_state_space(ss, initial_weights, dt)
eval_d = sim.evaluate_approximation("eig_funcs", q, t, dz)
x_0t = eval_d.output_data[:, 0]
yc, tc = tr.gevrey_tanh(T, 1)
x_0t_desired = np.interp(t, tc, yc[0, :])
self.assertLess(np.average((x_0t - x_0t_desired)**2), 0.5)
# display results
if show_plots:
eval_d = sim.evaluate_approximation("eig_funcs", q, t, dz)
win2 = vis.PgSurfacePlot(eval_d)
app.exec_()
def test_it(self):
actuation_type = 'dirichlet'
bound_cond_type = 'dirichlet'
param = [1., -2., -1., None, None]
adjoint_param = ef.get_adjoint_rad_evp_param(param)
a2, a1, a0, _, _ = param
l = 1.
spatial_disc = 10
dz = sim.Domain(bounds=(0, l), num=spatial_disc)
T = 1.
temporal_disc = 1e2
dt = sim.Domain(bounds=(0, T), num=temporal_disc)
omega = np.array([(i + 1) * np.pi / l for i in range(spatial_disc)])
eig_values = a0 - a2 * omega**2 - a1**2 / 4. / a2
norm_fak = np.ones(omega.shape) * np.sqrt(2)
eig_funcs = np.array([
ef.SecondOrderDirichletEigenfunction(omega[i], param, dz.bounds,
norm_fak[i])
for i in range(spatial_disc)
])
register_base("eig_funcs", eig_funcs, overwrite=True)
adjoint_eig_funcs = np.array([
ef.SecondOrderDirichletEigenfunction(omega[i], adjoint_param,
dz.bounds, norm_fak[i])
for i in range(spatial_disc)
])
register_base("adjoint_eig_funcs", adjoint_eig_funcs, overwrite=True)
# derive initial field variable x(z,0) and weights
start_state = cr.Function(lambda z: 0., domain=(0, l))
initial_weights = cr.project_on_base(start_state, adjoint_eig_funcs)
# init trajectory
u = tr.RadTrajectory(l, T, param, bound_cond_type, actuation_type)
# determine (A,B) with weak-formulation (pyinduct)
# derive sate-space system
rad_pde = ut.get_parabolic_dirichlet_weak_form("eig_funcs",
"adjoint_eig_funcs", u,
param, dz.bounds)
cf = sim.parse_weak_formulation(rad_pde)
ss_weak = cf.convert_to_state_space()
# determine (A,B) with modal-transfomation
A = np.diag(eig_values)
B = -a2 * np.array(
[adjoint_eig_funcs[i].derive()(l) for i in range(spatial_disc)])
ss_modal = sim.StateSpace("eig_funcs", A, B, input_handle=u)
# TODO: resolve the big tolerance (rtol=3e-01) between ss_modal.A and ss_weak.A
# check if ss_modal.(A,B) is close to ss_weak.(A,B)
self.assertTrue(
np.allclose(np.sort(np.linalg.eigvals(ss_weak.A[1])),
np.sort(np.linalg.eigvals(ss_modal.A[1])),
rtol=3e-1,
atol=0.))
self.assertTrue(
np.allclose(np.array([i[0] for i in ss_weak.B[1]]), ss_modal.B[1]))
# display results
if show_plots:
t, q = sim.simulate_state_space(ss_modal, initial_weights, dt)
eval_d = sim.evaluate_approximation("eig_funcs",
q,
t,
dz,
spat_order=1)
win2 = vis.PgSurfacePlot(eval_d)
app.exec_()
def test_modal(self):
order = 8
def char_eq(w):
return w * (np.sin(w) + self.params.m * w * np.cos(w))
def phi_k_factory(freq, derivative_order=0):
def eig_func(z):
return np.cos(
freq * z) - self.params.m * freq * np.sin(freq * z)
def eig_func_dz(z):
return -freq * (np.sin(freq * z) +
self.params.m * freq * np.cos(freq * z))
def eig_func_ddz(z):
return freq**2 * (-np.cos(freq * z) +
self.params.m * freq * np.sin(freq * z))
if derivative_order == 0:
return eig_func
elif derivative_order == 1:
return eig_func_dz
elif derivative_order == 2:
return eig_func_ddz
else:
raise ValueError
# create eigenfunctions
eig_frequencies = ut.find_roots(char_eq,
n_roots=order,
grid=np.arange(0, 1e3, 2),
rtol=-2)
print("eigenfrequencies:")
print(eig_frequencies)
# create eigen function vectors
class SWMFunctionVector(cr.ComposedFunctionVector):
"""
String With Mass Function Vector, necessary due to manipulated scalar product
"""
@property
def func(self):
return self.members["funcs"][0]
@property
def scalar(self):
return self.members["scalars"][0]
eig_vectors = []
for n in range(order):
eig_vectors.append(
SWMFunctionVector(
cr.Function(phi_k_factory(eig_frequencies[n]),
derivative_handles=[
phi_k_factory(eig_frequencies[n],
der_order)
for der_order in range(1, 3)
],
domain=self.dz.bounds,
nonzero=self.dz.bounds),
phi_k_factory(eig_frequencies[n])(0)))
# normalize eigen vectors
norm_eig_vectors = [cr.normalize_function(vec) for vec in eig_vectors]
norm_eig_funcs = np.array([vec.func for vec in norm_eig_vectors])
register_base("norm_eig_funcs", norm_eig_funcs, overwrite=True)
norm_eig_funcs[0](1)
# debug print eigenfunctions
if 0:
func_vals = []
for vec in eig_vectors:
func_vals.append(np.vectorize(vec.func)(self.dz))
norm_func_vals = []
for func in norm_eig_funcs:
norm_func_vals.append(np.vectorize(func)(self.dz))
clrs = ["r", "g", "b", "c", "m", "y", "k", "w"]
for n in range(1, order + 1, len(clrs)):
pw_phin_k = pg.plot(title="phin_k for k in [{0}, {1}]".format(
n, min(n + len(clrs), order)))
for k in range(len(clrs)):
if k + n > order:
break
pw_phin_k.plot(x=np.array(self.dz),
y=norm_func_vals[n + k - 1],
pen=clrs[k])
app.exec_()
# create terms of weak formulation
terms = [
ph.IntegralTerm(ph.Product(
ph.FieldVariable("norm_eig_funcs", order=(2, 0)),
ph.TestFunction("norm_eig_funcs")),
self.dz.bounds,
scale=-1),
ph.ScalarTerm(ph.Product(
ph.FieldVariable("norm_eig_funcs", order=(2, 0), location=0),
ph.TestFunction("norm_eig_funcs", location=0)),
scale=-1),
ph.ScalarTerm(
ph.Product(ph.Input(self.u),
ph.TestFunction("norm_eig_funcs", location=1))),
ph.ScalarTerm(ph.Product(
ph.FieldVariable("norm_eig_funcs", location=1),
ph.TestFunction("norm_eig_funcs", order=1, location=1)),
scale=-1),
ph.ScalarTerm(
ph.Product(
ph.FieldVariable("norm_eig_funcs", location=0),
ph.TestFunction("norm_eig_funcs", order=1, location=0))),
ph.IntegralTerm(
ph.Product(ph.FieldVariable("norm_eig_funcs"),
ph.TestFunction("norm_eig_funcs", order=2)),
self.dz.bounds)
]
modal_pde = sim.WeakFormulation(terms, name="swm_lib-modal")
eval_data = sim.simulate_system(modal_pde,
self.ic,
self.dt,
self.dz,
der_orders=(2, 0))
# display results
if show_plots:
win = vis.PgAnimatedPlot(eval_data[0:2],
title="modal approx and derivative")
win2 = vis.PgSurfacePlot(eval_data[0])
app.exec_()
# test for correct transition
self.assertTrue(
np.isclose(eval_data[0].output_data[-1, 0], self.y_end, atol=1e-3))
def test_it(self):
actuation_type = 'robin'
bound_cond_type = 'robin'
param = [2., 1.5, -3., -1., -.5]
adjoint_param = ef.get_adjoint_rad_evp_param(param)
a2, a1, a0, alpha, beta = param
l = 1.
spatial_disc = 10
dz = sim.Domain(bounds=(0, l), num=spatial_disc)
T = 1.
temporal_disc = 1e2
dt = sim.Domain(bounds=(0, T), num=temporal_disc)
n = 10
eig_freq, eig_val = ef.compute_rad_robin_eigenfrequencies(param, l, n)
init_eig_funcs = np.array([
ef.SecondOrderRobinEigenfunction(om, param, dz.bounds)
for om in eig_freq
])
init_adjoint_eig_funcs = np.array([
ef.SecondOrderRobinEigenfunction(om, adjoint_param, dz.bounds)
for om in eig_freq
])
# normalize eigenfunctions and adjoint eigenfunctions
adjoint_and_eig_funcs = [
cr.normalize_function(init_eig_funcs[i], init_adjoint_eig_funcs[i])
for i in range(n)
]
eig_funcs = np.array([f_tuple[0] for f_tuple in adjoint_and_eig_funcs])
adjoint_eig_funcs = np.array(
[f_tuple[1] for f_tuple in adjoint_and_eig_funcs])
# register eigenfunctions
register_base("eig_funcs", eig_funcs, overwrite=True)
register_base("adjoint_eig_funcs", adjoint_eig_funcs, overwrite=True)
# derive initial field variable x(z,0) and weights
start_state = cr.Function(lambda z: 0., domain=(0, l))
initial_weights = cr.project_on_base(start_state, adjoint_eig_funcs)
# init trajectory
u = tr.RadTrajectory(l, T, param, bound_cond_type, actuation_type)
# determine (A,B) with weak-formulation (pyinduct)
rad_pde = ut.get_parabolic_robin_weak_form("eig_funcs",
"adjoint_eig_funcs", u,
param, dz.bounds)
cf = sim.parse_weak_formulation(rad_pde)
ss_weak = cf.convert_to_state_space()
# determine (A,B) with modal-transfomation
A = np.diag(np.real_if_close(eig_val))
B = a2 * np.array(
[adjoint_eig_funcs[i](l) for i in range(len(eig_freq))])
ss_modal = sim.StateSpace("eig_funcs", A, B, input_handle=u)
# check if ss_modal.(A,B) is close to ss_weak.(A,B)
self.assertTrue(
np.allclose(np.sort(np.linalg.eigvals(ss_weak.A[1])),
np.sort(np.linalg.eigvals(ss_modal.A[1])),
rtol=1e-05,
atol=0.))
self.assertTrue(
np.allclose(np.array([i[0] for i in ss_weak.B[1]]), ss_modal.B[1]))
# display results
if show_plots:
t, q = sim.simulate_state_space(ss_modal, initial_weights, dt)
eval_d = sim.evaluate_approximation("eig_funcs",
q,
t,
dz,
spat_order=1)
win1 = vis.PgAnimatedPlot([eval_d], title="Test")
win2 = vis.PgSurfacePlot(eval_d)
app.exec_()
from pyinduct import eigenfunctions as ef
from pyinduct import simulation as sim
from pyinduct import visualization as vis
# system/simulation parameters
actuation_type = 'robin'
bound_cond_type = 'robin'
l = 1.
T = 1
spatial_domain = sim.Domain(bounds=(0, l), num=30)
temporal_domain = sim.Domain(bounds=(0, T), num=1e2)
n = 10
# original system parameters
a2 = .5
a1_z = cr.Function(lambda z: 0.1 * np.exp(4 * z),
derivative_handles=[lambda z: 0.4 * np.exp(4 * z)])
a0_z = lambda z: 1 + 10 * z + 2 * np.sin(4 * np.pi / l * z)
alpha = -1
beta = -1
param = [a2, a1_z, a0_z, alpha, beta]
# target system parameters (controller parameters)
a1_t = -0
a0_t = -6
alpha_t = 3
beta_t = 3
param_t = [a2, a1_t, a0_t, alpha_t, beta_t]
adjoint_param_t = ef.get_adjoint_rad_evp_param(param_t)
# original intermediate ("_i") and target intermediate ("_ti") system parameters
_, _, a0_i, alpha_i, beta_i = ef.transform2intermediate(param, d_end=l)