def test_fem(self):
self.act_funcs = "fem_funcs"
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(self.l),
original_beta=self.beta_i,
target_beta=self.beta_ti,
trajectory=self.traj,
scale=self.transform_i(-self.l))
# determine (A,B) with modal-transfomation
rad_pde = ut.get_parabolic_robin_weak_form(self.act_funcs, self.act_funcs, controller, self.param,
self.dz.bounds)
cf = sim.parse_weak_formulation(rad_pde)
ss_weak = cf.convert_to_state_space()
# simulate
self.t, self.q = sim.simulate_state_space(ss_weak, np.zeros((len(self.fem_funcs))),
self.dt)
eval_d = sim.evaluate_approximation(self.act_funcs, self.q, self.t, self.dz)
x_0t = eval_d.output_data[:, 0]
yc, tc = tr.gevrey_tanh(self.T, 1)
x_0t_desired = np.interp(self.t, tc, yc[0, :])
self.assertLess(np.average((x_0t - x_0t_desired) ** 2), 1e-3)
# display results
if show_plots:
win1 = vis.PgAnimatedPlot([eval_d], title="Test")
win2 = vis.PgSurfacePlot(eval_d)
app.exec_()
def test_modal(self):
self.act_funcs = "eig_funcs"
a2, a1, a0, alpha, beta = self.param
controller = ut.get_parabolic_robin_backstepping_controller(state=self.x_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(self.l),
original_beta=self.beta_i,
target_beta=self.beta_ti,
trajectory=self.traj,
scale=self.transform_i(-self.l))
# determine (A,B) with modal-transfomation
A = np.diag(np.real(self.eig_val))
B = a2 * np.array([self.adjoint_eig_funcs[i](self.l) for i in range(self.n)])
ss_modal = sim.StateSpace(self.act_funcs, A, B, input_handle=controller)
# simulate
self.t, self.q = sim.simulate_state_space(ss_modal, np.zeros((len(self.adjoint_eig_funcs))),
self.dt)
eval_d = sim.evaluate_approximation(self.act_funcs, self.q, self.t, self.dz)
x_0t = eval_d.output_data[:, 0]
yc, tc = tr.gevrey_tanh(self.T, 1)
x_0t_desired = np.interp(self.t, tc, yc[0, :])
self.assertLess(np.average((x_0t - x_0t_desired) ** 2), 1e-2)
# display results
if show_plots:
win1 = vis.PgAnimatedPlot([eval_d], title="Test")
win2 = vis.PgSurfacePlot(eval_d)
app.exec_()
def test_modal(self):
self.act_funcs = "eig_funcs"
a2, a1, a0, alpha, beta = self.param
controller = ut.get_parabolic_robin_backstepping_controller(
state=self.x_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(self.l),
original_beta=self.beta_i,
target_beta=self.beta_ti,
trajectory=self.traj,
scale=self.transform_i(-self.l))
# determine (A,B) with modal-transfomation
A = np.diag(np.real(self.eig_val))
B = a2 * np.array(
[self.adjoint_eig_funcs[i](self.l) for i in range(self.n)])
ss_modal = sim.StateSpace(self.act_funcs,
A,
B,
input_handle=controller)
# simulate
self.t, self.q = sim.simulate_state_space(
ss_modal, np.zeros((len(self.adjoint_eig_funcs))), self.dt)
eval_d = sim.evaluate_approximation(self.act_funcs, self.q, self.t,
self.dz)
x_0t = eval_d.output_data[:, 0]
yc, tc = tr.gevrey_tanh(self.T, 1)
x_0t_desired = np.interp(self.t, tc, yc[0, :])
self.assertLess(np.average((x_0t - x_0t_desired)**2), 1e-2)
# display results
if show_plots:
win1 = vis.PgAnimatedPlot([eval_d], title="Test")
win2 = vis.PgSurfacePlot(eval_d)
app.exec_()
def test_fem(self):
self.act_funcs = "fem_funcs"
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(self.l),
original_beta=self.beta_i,
target_beta=self.beta_ti,
trajectory=self.traj,
scale=self.transform_i(-self.l))
# determine (A,B) with modal-transfomation
rad_pde = ut.get_parabolic_robin_weak_form(self.act_funcs,
self.act_funcs, controller,
self.param, self.dz.bounds)
cf = sim.parse_weak_formulation(rad_pde)
ss_weak = cf.convert_to_state_space()
# simulate
self.t, self.q = sim.simulate_state_space(
ss_weak, np.zeros((len(self.fem_funcs))), self.dt)
eval_d = sim.evaluate_approximation(self.act_funcs, self.q, self.t,
self.dz)
x_0t = eval_d.output_data[:, 0]
yc, tc = tr.gevrey_tanh(self.T, 1)
x_0t_desired = np.interp(self.t, tc, yc[0, :])
self.assertLess(np.average((x_0t - x_0t_desired)**2), 1e-3)
# display results
if show_plots:
win1 = vis.PgAnimatedPlot([eval_d], title="Test")
win2 = vis.PgSurfacePlot(eval_d)
app.exec_()
x_i_at_l = [ph.ScalarTerm(field_variable, transform_i_at_l)]
xd_i_at_l = [ph.ScalarTerm(d_field_variable, transform_i_at_l),
ph.ScalarTerm(field_variable, transform_i_at_l * a1_z(l) / 2 / a2)]
x_ti_at_l = [ph.ScalarTerm(field_variable_t, transform_ti_at_l)]
xd_ti_at_l = [ph.ScalarTerm(d_field_variable_t, transform_ti_at_l),
ph.ScalarTerm(field_variable_t, transform_ti_at_l * a1_t / 2 / a2)]
# discontinuous operator (Kx)(t) = int_kernel_zz(l)*x(l,t)
int_kernel_zz = alpha_ti - alpha_i + si.quad(lambda z: (a0_i(z) - a0_ti) / 2 / a2, 0, l)[0]
# init controller
controller = ut.get_parabolic_robin_backstepping_controller(state=x_fem_i_at_l,
approx_state=x_i_at_l,
d_approx_state=xd_i_at_l,
approx_target_state=x_ti_at_l,
d_approx_target_state=xd_ti_at_l,
integral_kernel_zz=int_kernel_zz,
original_beta=beta_i,
target_beta=beta_ti,
trajectory=traj,
scale=inv_transform_i_at_l)
rad_pde = ut.get_parabolic_robin_weak_form("fem_funcs", "fem_funcs", controller, param, spatial_domain.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(fem_funcs))), temporal_domain)
# pyqtgraph visualization
evald_x = sim.evaluate_approximation("fem_funcs", q, t, spatial_domain, name="x(z,t)")
win1 = vis.PgAnimatedPlot([evald_x], title="animation", dt=temporal_domain.step*.25)
sh_field_variable_id = ph.FieldVariable("sh_eig_funcs_id", weight_label="eig_funcs", location=l)
sh_x_fem_i_at_l = [ph.ScalarTerm(sh_fem_field_variable_i),
ph.ScalarTerm(field_variable_i),
ph.ScalarTerm(sh_field_variable_id, -1)]
# controller initialization
def int_kernel_zz(z):
return alpha_ti - alpha_i + (a0_i - a0_ti) / 2 / a2 * z
controller = ut.get_parabolic_robin_backstepping_controller(state=sh_x_fem_i_at_l,
approx_state=x_i_at_l,
d_approx_state=xd_i_at_l,
approx_target_state=x_ti_at_l,
d_approx_target_state=xd_ti_at_l,
integral_kernel_zz=int_kernel_zz(l),
original_beta=beta_i,
target_beta=beta_ti,
trajectory=traj,
scale=np.exp(-a1 / 2 / a2 * b))
# determine (A,B) with modal transformation
rad_pde = ut.get_parabolic_robin_weak_form("fem_funcs", "fem_funcs", controller, param, spatial_domain.bounds, b)
cf = sim.parse_weak_formulation(rad_pde)
ss_weak = cf.convert_to_state_space()
# simulate (t: time vector, q: weights matrix)
t, q = sim.simulate_state_space(ss_weak, init_profile * np.ones((len(fem_funcs))), temporal_domain)
# compute modal weights (for the intermediate system: evald_modal_xi)
mat = cr.calculate_base_transformation_matrix(fem_funcs, eig_funcs)
ph.ScalarTerm(field_variable_i),
ph.ScalarTerm(sh_field_variable_id, -1)
]
# controller initialization
def int_kernel_zz(z):
return alpha_ti - alpha_i + (a0_i - a0_ti) / 2 / a2 * z
controller = ut.get_parabolic_robin_backstepping_controller(
state=sh_x_fem_i_at_l,
approx_state=x_i_at_l,
d_approx_state=xd_i_at_l,
approx_target_state=x_ti_at_l,
d_approx_target_state=xd_ti_at_l,
integral_kernel_zz=int_kernel_zz(l),
original_beta=beta_i,
target_beta=beta_ti,
trajectory=traj,
scale=np.exp(-a1 / 2 / a2 * b))
# determine (A,B) with modal transformation
rad_pde = ut.get_parabolic_robin_weak_form("fem_funcs", "fem_funcs",
controller, param,
spatial_domain.bounds, b)
cf = sim.parse_weak_formulation(rad_pde)
ss_weak = cf.convert_to_state_space()
# simulate (t: time vector, q: weights matrix)
t, q = sim.simulate_state_space(ss_weak,
def test_fem(self):
# system/simulation parameters
actuation_type = 'robin'
bound_cond_type = 'robin'
self.l = 1.
spatial_disc = 30
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 = 12
# original system parameters
a2 = 1.5
a1 = 2.5
a0 = 28
alpha = -2
beta = -3
self.param = [a2, a1, a0, alpha, beta]
adjoint_param = ef.get_adjoint_rad_evp_param(self.param)
# 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]
# actuation_type by b which is close to b_desired on a k times subdivided spatial domain
b_desired = self.l / 2
k = 51 # = k1 + k2
k1, k2, self.b = ut.split_domain(k, b_desired, self.l, mode='coprime')[0:3]
M = np.linalg.inv(ut.get_inn_domain_transformation_matrix(k1, k2, mode="2n"))
# original intermediate ("_i") and traget intermediate ("_ti") system parameters
_, _, a0_i, self.alpha_i, self.beta_i = ef.transform2intermediate(self.param)
self.param_i = a2, 0, a0_i, self.alpha_i, self.beta_i
_, _, a0_ti, self.alpha_ti, self.beta_ti = ef.transform2intermediate(self.param_t)
self.param_ti = a2, 0, a0_ti, self.alpha_ti, self.beta_ti
# create (not normalized) eigenfunctions
eig_freq, self.eig_val = ef.compute_rad_robin_eigenfrequencies(self.param, self.l, self.n)
init_eig_funcs = np.array([ef.SecondOrderRobinEigenfunction(om, self.param, self.dz.bounds) for om in eig_freq])
init_adjoint_eig_funcs = np.array(
[ef.SecondOrderRobinEigenfunction(om, adjoint_param, self.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(self.n)]
eig_funcs = np.array([f_tuple[0] for f_tuple in adjoint_and_eig_funcs])
self.adjoint_eig_funcs = np.array([f_tuple[1] for f_tuple in adjoint_and_eig_funcs])
# eigenfunctions of the in-domain intermediate (_id) and the intermediate (_i) system
eig_freq_i, eig_val_i = ef.compute_rad_robin_eigenfrequencies(self.param_i, self.l, self.n)
self.assertTrue(all(np.isclose(eig_val_i, self.eig_val)))
eig_funcs_id = np.array([ef.SecondOrderRobinEigenfunction(eig_freq_i[i],
self.param_i,
self.dz.bounds,
eig_funcs[i](0))
for i in range(self.n)])
eig_funcs_i = np.array([ef.SecondOrderRobinEigenfunction(eig_freq_i[i],
self.param_i,
self.dz.bounds,
eig_funcs[i](0) * eig_funcs_id[i](self.l) /
eig_funcs_id[i](self.b))
for i in range(self.n)])
# eigenfunctions from target system ("_ti")
eig_freq_ti = np.sqrt((a0_ti - self.eig_val) / a2)
eig_funcs_ti = np.array([ef.SecondOrderRobinEigenfunction(eig_freq_ti[i],
self.param_ti,
self.dz.bounds,
eig_funcs_i[i](0))
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 eigenfunctions
# register_functions("eig_funcs", eig_funcs, overwrite=True)
register_base("adjoint_eig_funcs", self.adjoint_eig_funcs, overwrite=True)
register_base("eig_funcs", eig_funcs, overwrite=True)
register_base("eig_funcs_i", eig_funcs_i, overwrite=True)
register_base("eig_funcs_ti", eig_funcs_ti, 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_i = ph.FieldVariable("eig_funcs_i", weight_label="eig_funcs", location=self.l)
d_field_variable_i = ph.SpatialDerivedFieldVariable("eig_funcs_i", 1, weight_label="eig_funcs", location=self.l)
field_variable_ti = ph.FieldVariable("eig_funcs_ti", weight_label="eig_funcs", location=self.l)
d_field_variable_ti = ph.SpatialDerivedFieldVariable("eig_funcs_ti", 1, weight_label="eig_funcs",
location=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.x_i_at_l = [ph.ScalarTerm(field_variable_i)]
self.xd_i_at_l = [ph.ScalarTerm(d_field_variable_i)]
self.x_ti_at_l = [ph.ScalarTerm(field_variable_ti)]
self.xd_ti_at_l = [ph.ScalarTerm(d_field_variable_ti)]
# shift transformation
shifted_fem_funcs_i = np.array(
[ef.FiniteTransformFunction(func, M, self.b, self.l, scale_func=lambda z: np.exp(a1 / 2 / a2 * z))
for func in self.fem_funcs])
shifted_eig_funcs_id = np.array([ef.FiniteTransformFunction(func, M, self.b, self.l) for func in eig_funcs_id])
register_base("sh_fem_funcs_i", shifted_fem_funcs_i, overwrite=True)
register_base("sh_eig_funcs_id", shifted_eig_funcs_id, overwrite=True)
sh_fem_field_variable_i = ph.FieldVariable("sh_fem_funcs_i", weight_label="fem_funcs", location=self.l)
sh_field_variable_id = ph.FieldVariable("sh_eig_funcs_id", weight_label="eig_funcs", location=self.l)
self.sh_x_fem_i_at_l = [ph.ScalarTerm(sh_fem_field_variable_i),
ph.ScalarTerm(field_variable_i),
ph.ScalarTerm(sh_field_variable_id, -1)]
# discontinuous operator (Kx)(t) = int_kernel_zz(l)*x(l,t)
self.int_kernel_zz = lambda z: self.alpha_ti - self.alpha_i + (a0_i - a0_ti) / 2 / a2 * z
a2, a1, _, _, _ = self.param
controller = ut.get_parabolic_robin_backstepping_controller(state=self.sh_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(self.l),
original_beta=self.beta_i,
target_beta=self.beta_ti,
trajectory=self.traj,
scale=np.exp(-a1 / 2 / a2 * self.b))
# determine (A,B) with modal-transfomation
rad_pde = ut.get_parabolic_robin_weak_form("fem_funcs", "fem_funcs", controller, self.param, self.dz.bounds,
self.b)
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)
# weights of the intermediate system
mat = cr.calculate_base_transformation_matrix(self.fem_funcs, eig_funcs)
q_i = np.zeros((q.shape[0], len(eig_funcs_i)))
for i in range(q.shape[0]):
q_i[i, :] = np.dot(q[i, :], np.transpose(mat))
eval_i = sim.evaluate_approximation("eig_funcs_i", q_i, t, self.dz)
x_0t = eval_i.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-2)
# display results
if show_plots:
eval_d = sim.evaluate_approximation("fem_funcs", q, t, self.dz)
win1 = vis.PgSurfacePlot(eval_i)
win2 = vis.PgSurfacePlot(eval_d)
app.exec_()
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_()
ph.ScalarTerm(field_variable_t, transform_ti(l) * a1_t / 2 / a2)]
# discontinuous operator (Kx)(t) = int_kernel_zz(l)*x(l,t)
def int_kernel_zz(z):
return alpha_ti - alpha_i + (a0_i - a0_ti) / 2 / a2 * z
# init trajectory
traj = tr.RadTrajectory(l, T, param_ti, bound_cond_type, actuation_type)
# controller initialization
controller = ut.get_parabolic_robin_backstepping_controller(state=x_i_at_l,
approx_state=x_i_at_l,
d_approx_state=xd_i_at_l,
approx_target_state=x_ti_at_l,
d_approx_target_state=xd_ti_at_l,
integral_kernel_zz=int_kernel_zz(l),
original_beta=beta_i,
target_beta=beta_ti,
trajectory=traj,
scale=transform_i(-l))
# determine (A,B)
rad_pde = ut.get_parabolic_robin_weak_form("fem_funcs", "fem_funcs", controller, param, spatial_domain.bounds)
cf = sim.parse_weak_formulation(rad_pde)
ss_weak = cf.convert_to_state_space()
# simulate
t, q = sim.simulate_state_space(ss_weak, init_profile * np.ones(n_fem), temporal_domain)
# evaluate desired output data
y_d, t_d = tr.gevrey_tanh(T, 80)
C = tr.coefficient_recursion(y_d, alpha * y_d, param)
def test_fem(self):
# system/simulation parameters
actuation_type = 'robin'
bound_cond_type = 'robin'
self.l = 1.
spatial_disc = 30
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 = 12
# original system parameters
a2 = 1.5
a1 = 2.5
a0 = 28
alpha = -2
beta = -3
self.param = [a2, a1, a0, alpha, beta]
adjoint_param = ef.get_adjoint_rad_evp_param(self.param)
# 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]
# actuation_type by b which is close to b_desired on a k times subdivided spatial domain
b_desired = self.l / 2
k = 51 # = k1 + k2
k1, k2, self.b = ut.split_domain(k, b_desired, self.l,
mode='coprime')[0:3]
M = np.linalg.inv(
ut.get_inn_domain_transformation_matrix(k1, k2, mode="2n"))
# original intermediate ("_i") and traget intermediate ("_ti") system parameters
_, _, a0_i, self.alpha_i, self.beta_i = ef.transform2intermediate(
self.param)
self.param_i = a2, 0, a0_i, self.alpha_i, self.beta_i
_, _, a0_ti, self.alpha_ti, self.beta_ti = ef.transform2intermediate(
self.param_t)
self.param_ti = a2, 0, a0_ti, self.alpha_ti, self.beta_ti
# create (not normalized) eigenfunctions
eig_freq, self.eig_val = ef.compute_rad_robin_eigenfrequencies(
self.param, self.l, self.n)
init_eig_funcs = np.array([
ef.SecondOrderRobinEigenfunction(om, self.param, self.dz.bounds)
for om in eig_freq
])
init_adjoint_eig_funcs = np.array([
ef.SecondOrderRobinEigenfunction(om, adjoint_param, self.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(self.n)
]
eig_funcs = np.array([f_tuple[0] for f_tuple in adjoint_and_eig_funcs])
self.adjoint_eig_funcs = np.array(
[f_tuple[1] for f_tuple in adjoint_and_eig_funcs])
# eigenfunctions of the in-domain intermediate (_id) and the intermediate (_i) system
eig_freq_i, eig_val_i = ef.compute_rad_robin_eigenfrequencies(
self.param_i, self.l, self.n)
self.assertTrue(all(np.isclose(eig_val_i, self.eig_val)))
eig_funcs_id = np.array([
ef.SecondOrderRobinEigenfunction(eig_freq_i[i], self.param_i,
self.dz.bounds, eig_funcs[i](0))
for i in range(self.n)
])
eig_funcs_i = np.array([
ef.SecondOrderRobinEigenfunction(
eig_freq_i[i], self.param_i, self.dz.bounds, eig_funcs[i](0) *
eig_funcs_id[i](self.l) / eig_funcs_id[i](self.b))
for i in range(self.n)
])
# eigenfunctions from target system ("_ti")
eig_freq_ti = np.sqrt((a0_ti - self.eig_val) / a2)
eig_funcs_ti = np.array([
ef.SecondOrderRobinEigenfunction(eig_freq_ti[i], self.param_ti,
self.dz.bounds, eig_funcs_i[i](0))
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 eigenfunctions
# register_functions("eig_funcs", eig_funcs, overwrite=True)
register_base("adjoint_eig_funcs",
self.adjoint_eig_funcs,
overwrite=True)
register_base("eig_funcs", eig_funcs, overwrite=True)
register_base("eig_funcs_i", eig_funcs_i, overwrite=True)
register_base("eig_funcs_ti", eig_funcs_ti, 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_i = ph.FieldVariable("eig_funcs_i",
weight_label="eig_funcs",
location=self.l)
d_field_variable_i = ph.SpatialDerivedFieldVariable(
"eig_funcs_i", 1, weight_label="eig_funcs", location=self.l)
field_variable_ti = ph.FieldVariable("eig_funcs_ti",
weight_label="eig_funcs",
location=self.l)
d_field_variable_ti = ph.SpatialDerivedFieldVariable(
"eig_funcs_ti", 1, weight_label="eig_funcs", location=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.x_i_at_l = [ph.ScalarTerm(field_variable_i)]
self.xd_i_at_l = [ph.ScalarTerm(d_field_variable_i)]
self.x_ti_at_l = [ph.ScalarTerm(field_variable_ti)]
self.xd_ti_at_l = [ph.ScalarTerm(d_field_variable_ti)]
# shift transformation
shifted_fem_funcs_i = np.array([
ef.FiniteTransformFunction(
func,
M,
self.b,
self.l,
scale_func=lambda z: np.exp(a1 / 2 / a2 * z))
for func in self.fem_funcs
])
shifted_eig_funcs_id = np.array([
ef.FiniteTransformFunction(func, M, self.b, self.l)
for func in eig_funcs_id
])
register_base("sh_fem_funcs_i", shifted_fem_funcs_i, overwrite=True)
register_base("sh_eig_funcs_id", shifted_eig_funcs_id, overwrite=True)
sh_fem_field_variable_i = ph.FieldVariable("sh_fem_funcs_i",
weight_label="fem_funcs",
location=self.l)
sh_field_variable_id = ph.FieldVariable("sh_eig_funcs_id",
weight_label="eig_funcs",
location=self.l)
self.sh_x_fem_i_at_l = [
ph.ScalarTerm(sh_fem_field_variable_i),
ph.ScalarTerm(field_variable_i),
ph.ScalarTerm(sh_field_variable_id, -1)
]
# discontinuous operator (Kx)(t) = int_kernel_zz(l)*x(l,t)
self.int_kernel_zz = lambda z: self.alpha_ti - self.alpha_i + (
a0_i - a0_ti) / 2 / a2 * z
a2, a1, _, _, _ = self.param
controller = ut.get_parabolic_robin_backstepping_controller(
state=self.sh_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(self.l),
original_beta=self.beta_i,
target_beta=self.beta_ti,
trajectory=self.traj,
scale=np.exp(-a1 / 2 / a2 * self.b))
# determine (A,B) with modal-transfomation
rad_pde = ut.get_parabolic_robin_weak_form("fem_funcs", "fem_funcs",
controller, self.param,
self.dz.bounds, self.b)
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)
# weights of the intermediate system
mat = cr.calculate_base_transformation_matrix(self.fem_funcs,
eig_funcs)
q_i = np.zeros((q.shape[0], len(eig_funcs_i)))
for i in range(q.shape[0]):
q_i[i, :] = np.dot(q[i, :], np.transpose(mat))
eval_i = sim.evaluate_approximation("eig_funcs_i", q_i, t, self.dz)
x_0t = eval_i.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-2)
# display results
if show_plots:
eval_d = sim.evaluate_approximation("fem_funcs", q, t, self.dz)
win1 = vis.PgSurfacePlot(eval_i)
win2 = vis.PgSurfacePlot(eval_d)
app.exec_()
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_()