def setUp(self):
self.u = CorrectInput()
# setup temp and spat domain
spat_domain = sim.Domain((0, 1), num=3)
nodes, ini_funcs = sf.cure_interval(sf.LagrangeFirstOrder,
spat_domain.bounds,
node_count=3)
register_base("init_funcs", ini_funcs, overwrite=True)
# enter string with mass equations for testing
int1 = ph.IntegralTerm(
ph.Product(ph.TemporalDerivedFieldVariable("init_funcs", 2),
ph.TestFunction("init_funcs")), spat_domain.bounds)
s1 = ph.ScalarTerm(
ph.Product(
ph.TemporalDerivedFieldVariable("init_funcs", 2, location=0),
ph.TestFunction("init_funcs", location=0)))
int2 = ph.IntegralTerm(
ph.Product(ph.SpatialDerivedFieldVariable("init_funcs", 1),
ph.TestFunction("init_funcs", order=1)),
spat_domain.bounds)
s2 = ph.ScalarTerm(
ph.Product(ph.Input(self.u),
ph.TestFunction("init_funcs", location=1)), -1)
string_pde = sim.WeakFormulation([int1, s1, int2, s2])
self.cf = sim.parse_weak_formulation(string_pde)
self.ic = np.zeros((3, 2))
def setUp(self):
self.u = CorrectInput()
# setup temp and spat domain
spat_domain = sim.Domain((0, 1), num=3)
nodes, ini_funcs = sf.cure_interval(sf.LagrangeFirstOrder, spat_domain.bounds, node_count=3)
register_base("init_funcs", ini_funcs, overwrite=True)
# enter string with mass equations for testing
int1 = ph.IntegralTerm(
ph.Product(ph.TemporalDerivedFieldVariable("init_funcs", 2),
ph.TestFunction("init_funcs")), spat_domain.bounds)
s1 = ph.ScalarTerm(
ph.Product(ph.TemporalDerivedFieldVariable("init_funcs", 2, location=0),
ph.TestFunction("init_funcs", location=0)))
int2 = ph.IntegralTerm(
ph.Product(ph.SpatialDerivedFieldVariable("init_funcs", 1),
ph.TestFunction("init_funcs", order=1)), spat_domain.bounds)
s2 = ph.ScalarTerm(
ph.Product(ph.Input(self.u), ph.TestFunction("init_funcs", location=1)), -1)
string_pde = sim.WeakFormulation([int1, s1, int2, s2])
self.cf = sim.parse_weak_formulation(string_pde)
self.ic = np.zeros((3, 2))
def _build_bases(self, n_modal, orig_params, tar_params, spatial_domain):
"""
Compute approximation bases for original and target system
"""
# eigenvalues and -vectors of the original system
orig_eig_values, orig_eig_vectors = \
pi.SecondOrderDirichletEigenfunction.cure_interval(spatial_domain,
param=orig_params,
n=n_modal)
norm_orig_eig_vectors = pi.normalize_base(orig_eig_vectors)
pi.register_base(self.orig_base_lbl, norm_orig_eig_vectors)
# eigenvectors of the target system (slightly different)
target_eig_freq = pi.SecondOrderDirichletEigenfunction.eigval_tf_eigfreq(
tar_params, eig_val=orig_eig_values)
orig_scale = np.array(
[vec(0) for vec in norm_orig_eig_vectors.derive(1)])
tar_scale = orig_scale / target_eig_freq
_, tar_eig_vectors = pi.SecondOrderDirichletEigenfunction.cure_interval(
spatial_domain,
param=tar_params,
eig_val=orig_eig_values,
scale=tar_scale)
pi.register_base(self.tar_base_lbl, tar_eig_vectors)
if self.verbose:
pi.visualize_functions(orig_eig_vectors)
pi.visualize_functions(norm_orig_eig_vectors)
pi.visualize_functions(tar_eig_vectors)
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 xrange(len(eig_freq))])
ss_modal = sim.StateSpace("eig_funcs", A, B)
# 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)), np.sort(np.linalg.eigvals(ss_modal.A)),
rtol=1e-05, atol=0.))
self.assertTrue(np.allclose(np.array([i[0] for i in ss_weak.B]), ss_modal.B))
# display results
if show_plots:
t, q = sim.simulate_state_space(ss_modal, u, initial_weights, dt)
eval_d = ut.evaluate_approximation("eig_funcs", q, t, dz, spat_order=1)
win1 = vis.PgAnimatedPlot([eval_d], title="Test")
win2 = vis.PgSurfacePlot(eval_d[0])
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 xrange(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 xrange(spatial_disc)])
ss_modal = sim.StateSpace("eig_funcs", A, B)
# 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)), np.sort(np.linalg.eigvals(ss_modal.A)),
rtol=3e-1, atol=0.))
self.assertTrue(np.allclose(np.array([i[0] for i in ss_weak.B]), ss_modal.B))
# display results
if show_plots:
t, q = sim.simulate_state_space(ss_modal, u, initial_weights, dt)
eval_d = ut.evaluate_approximation("eig_funcs", q, t, dz, spat_order=1)
win2 = vis.PgSurfacePlot(eval_d)
app.exec_()
def test_fem(self):
"""
use best documented fem case to test all steps in simulation process
"""
# enter string with mass equations
nodes, ini_funcs = pyinduct.shapefunctions.cure_interval(pyinduct.shapefunctions.LagrangeSecondOrder,
self.dz.bounds, node_count=10)
register_base("init_funcs", ini_funcs, overwrite=True)
int1 = ph.IntegralTerm(
ph.Product(ph.TemporalDerivedFieldVariable("init_funcs", 2),
ph.TestFunction("init_funcs")), self.dz.bounds, scale=self.params.sigma*self.params.tau**2)
s1 = ph.ScalarTerm(
ph.Product(ph.TemporalDerivedFieldVariable("init_funcs", 2, location=0),
ph.TestFunction("init_funcs", location=0)), scale=self.params.m)
int2 = ph.IntegralTerm(
ph.Product(ph.SpatialDerivedFieldVariable("init_funcs", 1),
ph.TestFunction("init_funcs", order=1)), self.dz.bounds, scale=self.params.sigma)
s2 = ph.ScalarTerm(
ph.Product(ph.Input(self.u), ph.TestFunction("init_funcs", location=1)), -self.params.sigma)
# derive sate-space system
string_pde = sim.WeakFormulation([int1, s1, int2, s2], name="fem_test")
self.cf = sim.parse_weak_formulation(string_pde)
ss = self.cf.convert_to_state_space()
# generate initial conditions for weights
q0 = np.array([cr.project_on_base(self.ic[idx], ini_funcs) for idx in range(2)]).flatten()
# simulate
t, q = sim.simulate_state_space(ss, self.cf.input_function, q0, self.dt)
# calculate result data
eval_data = []
for der_idx in range(2):
eval_data.append(
ut.evaluate_approximation("init_funcs", q[:, der_idx * ini_funcs.size:(der_idx + 1) * ini_funcs.size],
t, self.dz))
eval_data[-1].name = "{0}{1}".format(self.cf.name, "_"+"".join(["d" for x in range(der_idx)])
+ "t" if der_idx > 0 else "")
# display results
if show_plots:
win = vis.PgAnimatedPlot(eval_data[:2], title="fem 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))
# TODO dump in pyinduct/tests/ressources
file_path = os.sep.join(["resources", "test_data.res"])
if not os.path.isdir("resources"):
os.makedirs("resources")
with open(file_path, "w") as f:
f.write(dumps(eval_data))
def setUp(self):
self.u = np.sin
self.input = ph.Input(self.u) # control input
nodes, self.ini_funcs = pyinduct.shapefunctions.cure_interval(pyinduct.shapefunctions.LagrangeFirstOrder, (0, 1), node_count=3)
register_base("ini_funcs", self.ini_funcs, overwrite=True)
self.phi = ph.TestFunction("ini_funcs") # eigenfunction or something else
self.dphi = ph.TestFunction("ini_funcs", order=1) # eigenfunction or something else
self.dphi_at1 = ph.TestFunction("ini_funcs", order=1, location=1) # eigenfunction or something else
self.field_var = ph.FieldVariable("ini_funcs")
self.field_var_at1 = ph.FieldVariable("ini_funcs", location=1)
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 = pyinduct.shapefunctions.cure_interval(pyinduct.shapefunctions.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):
self.node_cnt = 5
self.time_step = 1e-1
self.dates = np.arange(0, 10, self.time_step)
self.spat_int = (0, 1)
self.nodes = np.linspace(self.spat_int[0], self.spat_int[1], self.node_cnt)
# create initial functions
self.nodes, self.funcs = sh.cure_interval(sh.LagrangeFirstOrder, self.spat_int, node_count=self.node_cnt)
register_base("approx_funcs", self.funcs, overwrite=True)
# create a slow rising, nearly horizontal line
self.weights = np.array(list(range(self.node_cnt*self.dates.size))).reshape((self.dates.size, len(self.nodes)))
def setUp(self):
interval = (0, 10)
node_cnt = 11
self.nodes, self.initial_functions = shapefunctions.cure_interval(shapefunctions.LagrangeFirstOrder, interval,
node_count=node_cnt)
register_base("ini_funcs", self.initial_functions, overwrite=True)
# "real" functions
self.z_values = np.linspace(interval[0], interval[1], 100*node_cnt) # because we are smarter
self.funcs = [core.Function(lambda x: 2),
core.Function(lambda x: 2*x),
core.Function(lambda x: x**2),
core.Function(lambda x: np.sin(x))
]
self.real_values = [func(self.z_values) for func in self.funcs]
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):
self.u = np.sin
self.input = ph.Input(self.u) # control input
nodes, self.ini_funcs = cure_interval(LagrangeFirstOrder, (0, 1),
node_count=3)
register_base("ini_funcs", self.ini_funcs, overwrite=True)
self.phi = ph.TestFunction(
"ini_funcs") # eigenfunction or something else
self.dphi = ph.TestFunction("ini_funcs",
order=1) # eigenfunction or something else
self.dphi_at1 = ph.TestFunction(
"ini_funcs", order=1,
location=1) # eigenfunction or something else
self.field_var = ph.FieldVariable("ini_funcs")
self.field_var_at1 = ph.FieldVariable("ini_funcs", location=1)
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 = 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):
self.node_cnt = 5
self.time_step = 1e-1
self.dates = np.arange(0, 10, self.time_step)
self.spat_int = (0, 1)
self.nodes = np.linspace(self.spat_int[0], self.spat_int[1],
self.node_cnt)
# create initial functions
self.nodes, self.funcs = sh.cure_interval(sh.LagrangeFirstOrder,
self.spat_int,
node_count=self.node_cnt)
register_base("approx_funcs", self.funcs, overwrite=True)
# create a slow rising, nearly horizontal line
self.weights = np.array(list(range(
self.node_cnt * self.dates.size))).reshape(
(self.dates.size, len(self.nodes)))
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 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 shape_generator(self, cls, der_order):
"""
verify the correct connection with visual feedback
"""
dz = pi.Domain((0, 1), step=.001)
dt = pi.Domain((0, 0), num=1)
nodes, funcs = pi.cure_interval(cls, dz.bounds, node_count=11)
pi.register_base("test", funcs, overwrite=True)
# approx_func = pi.Function(np.cos, domain=dz.bounds,
# derivative_handles=[lambda z: -np.sin(z), lambda z: -np.cos(z)])
approx_func = pi.Function(lambda z: np.sin(3*z), domain=dz.bounds,
derivative_handles=[lambda z: 3*np.cos(3*z), lambda z: -9*np.sin(3*z)])
weights = approx_func(nodes)
hull = pi.evaluate_approximation("test", np.atleast_2d(weights),
temp_domain=dt, spat_domain=dz, spat_order=der_order)
if show_plots:
# plot shapefunctions
c_map = pi.visualization.create_colormap(len(funcs))
pw = pg.plot(title="{}-Test".format(cls.__name__))
pw.addLegend()
pw.showGrid(x=True, y=True, alpha=0.5)
[pw.addItem(pg.PlotDataItem(np.array(dz),
weights[idx]*func.derive(der_order)(dz),
pen=pg.mkPen(color=c_map[idx]),
name="{}.{}".format(cls.__name__, idx)))
for idx, func in enumerate(funcs)]
# plot hull curve
pw.addItem(pg.PlotDataItem(np.array(hull.input_data[1]), hull.output_data[0, :],
pen=pg.mkPen(width=2), name="hull-curve"))
# plot original function
pw.addItem(pg.PlotDataItem(np.array(dz), approx_func.derive(der_order)(dz),
pen=pg.mkPen(color="m", width=2, style=pg.QtCore.Qt.DashLine), name="original"))
pg.QtCore.QCoreApplication.instance().exec_()
return np.sum(np.abs(hull.output_data[0, :] - approx_func.derive(der_order)(dz)))
def __init__(self, params, n_modal, spat_dom):
a2 = params[0]
z_start, z_end = spat_dom.bounds
self.base_lbl = "eigen_vectors"
# eigenvalues and -vectors of the system system
eig_values, eig_vectors = \
pi.SecondOrderDirichletEigenfunction.cure_interval(spat_dom,
param=params,
n=n_modal)
# pi.visualize_functions(orig_eig_vectors)
norm_eig_vectors = pi.normalize_base(eig_vectors)
# pi.visualize_functions(normalized_eig_vectors)
pi.register_base(self.base_lbl, norm_eig_vectors)
self.a_mat = np.diag(np.real_if_close(eig_values))
b_mat = -a2 * np.array(
[eig_vect.derive()(z_end) for eig_vect in norm_eig_vectors])
self.b_mat = np.reshape(b_mat, (b_mat.size, 1))
def setUp(self):
# enter string with mass equations
self.u = cr.Function(lambda x: 0)
interval = (0, 1)
nodes, ini_funcs = pyinduct.shapefunctions.cure_interval(pyinduct.shapefunctions.LagrangeFirstOrder, interval, node_count=3)
register_base("init_funcs", ini_funcs, overwrite=True)
int1 = ph.IntegralTerm(
ph.Product(ph.TemporalDerivedFieldVariable("init_funcs", 2),
ph.TestFunction("init_funcs")), interval)
s1 = ph.ScalarTerm(
ph.Product(ph.TemporalDerivedFieldVariable("init_funcs", 2, location=0),
ph.TestFunction("init_funcs", location=0)))
int2 = ph.IntegralTerm(
ph.Product(ph.SpatialDerivedFieldVariable("init_funcs", 1),
ph.TestFunction("init_funcs", order=1)), interval)
s2 = ph.ScalarTerm(
ph.Product(ph.Input(self.u), ph.TestFunction("init_funcs", location=1)), -1)
string_pde = sim.WeakFormulation([int1, s1, int2, s2])
self.cf = sim.parse_weak_formulation(string_pde)
def setUp(self):
# real function
self.z_values = np.linspace(0, 1, 1000)
self.real_func = core.Function(lambda x: x)
self.real_func_handle = np.vectorize(self.real_func)
# approximation by lag1st
self.nodes, self.src_test_funcs = shapefunctions.cure_interval(shapefunctions.LagrangeFirstOrder, (0, 1),
node_count=2)
register_base("test_funcs", self.src_test_funcs, overwrite=True)
self.src_weights = core.project_on_base(self.real_func, self.src_test_funcs)
self.assertTrue(np.allclose(self.src_weights, [0, 1])) # just to be sure
self.src_approx_handle = core.back_project_from_base(self.src_weights, self.src_test_funcs)
# approximation by sin(w*x)
def trig_factory(freq):
def func(x):
return np.sin(freq*x)
return func
self.trig_test_funcs = np.array([core.Function(trig_factory(w), domain=(0, 1)) for w in range(1, 3)])
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 = pyinduct.shapefunctions.cure_interval(pyinduct.shapefunctions.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 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_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 setUp(self):
# 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
# a1_t = a1; a0_t = a0; alpha_t = alpha; beta_t = beta
self.param_t = [a2, a1_t, a0_t, alpha_t, beta_t]
# original intermediate ("_i") and target 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
# 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
# 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 from target system ("_t")
eig_freq_t = np.sqrt(-a1_t ** 2 / 4 / a2 ** 2 + (a0_t - self.eig_val) / a2)
eig_funcs_t = np.array([ef.SecondOrderRobinEigenfunction(eig_freq_t[i],
self.param_t, self.dz.bounds).scale(eig_funcs[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_base("eig_funcs", eig_funcs, overwrite=True)
register_base("adjoint_eig_funcs", self.adjoint_eig_funcs, overwrite=True)
register_base("eig_funcs_t", eig_funcs_t, 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 = ph.FieldVariable("eig_funcs", location=self.l)
d_field_variable = ph.SpatialDerivedFieldVariable("eig_funcs", 1, 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)
# intermediate (_i) and target intermediate (_ti) transformations by z=l
self.transform_i = lambda z: np.exp(a1 / 2 / a2 * z) # x_i = x * transform_i
self.transform_ti = lambda z: np.exp(a1_t / 2 / a2 * z) # x_ti = x_t * transform_ti
# 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(self.l))]
self.x_i_at_l = [ph.ScalarTerm(field_variable, self.transform_i(self.l))]
self.xd_i_at_l = [ph.ScalarTerm(d_field_variable, self.transform_i(self.l)),
ph.ScalarTerm(field_variable, self.transform_i(self.l) * a1 / 2 / a2)]
self.x_ti_at_l = [ph.ScalarTerm(field_variable_t, self.transform_ti(self.l))]
self.xd_ti_at_l = [ph.ScalarTerm(d_field_variable_t, self.transform_ti(self.l)),
ph.ScalarTerm(field_variable_t, self.transform_ti(self.l) * a1_t / 2 / a2)]
# 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
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):
# 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 setUp(self):
# 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
# a1_t = a1; a0_t = a0; alpha_t = alpha; beta_t = beta
self.param_t = [a2, a1_t, a0_t, alpha_t, beta_t]
# original intermediate ("_i") and target 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
# 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
# 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 from target system ("_t")
eig_freq_t = np.sqrt(-a1_t**2 / 4 / a2**2 + (a0_t - self.eig_val) / a2)
eig_funcs_t = np.array([
ef.SecondOrderRobinEigenfunction(eig_freq_t[i], self.param_t,
self.dz.bounds).scale(
eig_funcs[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_base("eig_funcs", eig_funcs, overwrite=True)
register_base("adjoint_eig_funcs",
self.adjoint_eig_funcs,
overwrite=True)
register_base("eig_funcs_t", eig_funcs_t, 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 = ph.FieldVariable("eig_funcs", location=self.l)
d_field_variable = ph.SpatialDerivedFieldVariable("eig_funcs",
1,
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)
# intermediate (_i) and target intermediate (_ti) transformations by z=l
self.transform_i = lambda z: np.exp(a1 / 2 / a2 * z
) # x_i = x * transform_i
self.transform_ti = lambda z: np.exp(a1_t / 2 / a2 * z
) # x_ti = x_t * transform_ti
# 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(self.l))
]
self.x_i_at_l = [
ph.ScalarTerm(field_variable, self.transform_i(self.l))
]
self.xd_i_at_l = [
ph.ScalarTerm(d_field_variable, self.transform_i(self.l)),
ph.ScalarTerm(field_variable,
self.transform_i(self.l) * a1 / 2 / a2)
]
self.x_ti_at_l = [
ph.ScalarTerm(field_variable_t, self.transform_ti(self.l))
]
self.xd_ti_at_l = [
ph.ScalarTerm(d_field_variable_t, self.transform_ti(self.l)),
ph.ScalarTerm(field_variable_t,
self.transform_ti(self.l) * a1_t / 2 / a2)
]
# 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
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_()
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_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 setUp(self):
# scalars
self.scalars = ph.Scalars(np.vstack(list(range(3))))
# inputs
self.u = np.sin
self.input = ph.Input(self.u)
self.input_squared = ph.Input(self.u, exponent=2)
nodes, self.ini_funcs = sf.cure_interval(sf.LagrangeFirstOrder, (0, 1),
node_count=3)
# TestFunctions
register_base("ini_funcs", self.ini_funcs, overwrite=True)
self.phi = ph.TestFunction("ini_funcs")
self.phi_at0 = ph.TestFunction("ini_funcs", location=0)
self.phi_at1 = ph.TestFunction("ini_funcs", location=1)
self.dphi = ph.TestFunction("ini_funcs", order=1)
self.dphi_at1 = ph.TestFunction("ini_funcs", order=1, location=1)
# FieldVars
self.field_var = ph.FieldVariable("ini_funcs")
self.field_var_squared = ph.FieldVariable("ini_funcs", exponent=2)
self.odd_weight_field_var = ph.FieldVariable(
"ini_funcs", weight_label="special_weights")
self.field_var_at1 = ph.FieldVariable("ini_funcs", location=1)
self.field_var_at1_squared = ph.FieldVariable("ini_funcs",
location=1,
exponent=2)
self.field_var_dz = ph.SpatialDerivedFieldVariable("ini_funcs", 1)
self.field_var_dz_at1 = ph.SpatialDerivedFieldVariable("ini_funcs",
1,
location=1)
self.field_var_ddt = ph.TemporalDerivedFieldVariable("ini_funcs", 2)
self.field_var_ddt_at0 = ph.TemporalDerivedFieldVariable("ini_funcs",
2,
location=0)
self.field_var_ddt_at1 = ph.TemporalDerivedFieldVariable("ini_funcs",
2,
location=1)
# create all possible kinds of input variables
self.input_term1 = ph.ScalarTerm(ph.Product(self.phi_at1, self.input))
self.input_term1_swapped = ph.ScalarTerm(
ph.Product(self.input, self.phi_at1))
self.input_term1_squared = ph.ScalarTerm(
ph.Product(self.input_squared, self.phi_at1))
self.input_term2 = ph.ScalarTerm(ph.Product(self.dphi_at1, self.input))
self.func_term = ph.ScalarTerm(self.phi_at1)
# same goes for field variables
self.field_term_at1 = ph.ScalarTerm(self.field_var_at1)
self.field_term_at1_squared = ph.ScalarTerm(self.field_var_at1_squared)
self.field_term_dz_at1 = ph.ScalarTerm(self.field_var_dz_at1)
self.field_term_ddt_at1 = ph.ScalarTerm(self.field_var_ddt_at1)
self.field_int = ph.IntegralTerm(self.field_var, (0, 1))
self.field_squared_int = ph.IntegralTerm(self.field_var_squared,
(0, 1))
self.field_dz_int = ph.IntegralTerm(self.field_var_dz, (0, 1))
self.field_ddt_int = ph.IntegralTerm(self.field_var_ddt, (0, 1))
self.prod_term_fs_at1 = ph.ScalarTerm(
ph.Product(self.field_var_at1, self.scalars))
self.prod_int_fs = ph.IntegralTerm(
ph.Product(self.field_var, self.scalars), (0, 1))
self.prod_int_f_f = ph.IntegralTerm(
ph.Product(self.field_var, self.phi), (0, 1))
self.prod_int_f_squared_f = ph.IntegralTerm(
ph.Product(self.field_var_squared, self.phi), (0, 1))
self.prod_int_f_f_swapped = ph.IntegralTerm(
ph.Product(self.phi, self.field_var), (0, 1))
self.prod_int_f_at1_f = ph.IntegralTerm(
ph.Product(self.field_var_at1, self.phi), (0, 1))
self.prod_int_f_at1_squared_f = ph.IntegralTerm(
ph.Product(self.field_var_at1_squared, self.phi), (0, 1))
self.prod_int_f_f_at1 = ph.IntegralTerm(
ph.Product(self.field_var, self.phi_at1), (0, 1))
self.prod_int_f_squared_f_at1 = ph.IntegralTerm(
ph.Product(self.field_var_squared, self.phi_at1), (0, 1))
self.prod_term_f_at1_f_at1 = ph.ScalarTerm(
ph.Product(self.field_var_at1, self.phi_at1))
self.prod_term_f_at1_squared_f_at1 = ph.ScalarTerm(
ph.Product(self.field_var_at1_squared, self.phi_at1))
self.prod_int_fddt_f = ph.IntegralTerm(
ph.Product(self.field_var_ddt, self.phi), (0, 1))
self.prod_term_fddt_at0_f_at0 = ph.ScalarTerm(
ph.Product(self.field_var_ddt_at0, self.phi_at0))
self.prod_term_f_at1_dphi_at1 = ph.ScalarTerm(
ph.Product(self.field_var_at1, self.dphi_at1))
self.temp_int = ph.IntegralTerm(
ph.Product(self.field_var_ddt, self.phi), (0, 1))
self.spat_int = ph.IntegralTerm(
ph.Product(self.field_var_dz, self.dphi), (0, 1))
self.spat_int_asymmetric = ph.IntegralTerm(
ph.Product(self.field_var_dz, self.phi), (0, 1))
self.alternating_weights_term = ph.IntegralTerm(
self.odd_weight_field_var, (0, 1))
def setUp(self):
nodes, ini_funcs = cure_interval(LagrangeFirstOrder, (0, 1),
node_count=2)
register_base("test_funcs", ini_funcs, overwrite=True)
def setUp(self):
nodes, ini_funcs = cure_interval(LagrangeFirstOrder, (0, 1), node_count=2)
register_base("test_funcs", ini_funcs, overwrite=True)
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_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 setUp(self):
self.psi = cr.Function(np.sin)
register_base("funcs", self.psi, overwrite=True)
self.funcs = ph.TestFunction("funcs")
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 shape_generator(self, cls, der_order):
"""
verify the correct connection with visual feedback
"""
dz = pi.Domain((0, 1), step=.001)
dt = pi.Domain((0, 0), num=1)
nodes, funcs = pi.cure_interval(cls, dz.bounds, node_count=11)
pi.register_base("test", funcs, overwrite=True)
# approx_func = pi.Function(np.cos, domain=dz.bounds,
# derivative_handles=[lambda z: -np.sin(z), lambda z: -np.cos(z)])
approx_func = pi.Function(lambda z: np.sin(3 * z),
domain=dz.bounds,
derivative_handles=[
lambda z: 3 * np.cos(3 * z),
lambda z: -9 * np.sin(3 * z)
])
weights = approx_func(nodes)
hull = pi.evaluate_approximation("test",
np.atleast_2d(weights),
temp_domain=dt,
spat_domain=dz,
spat_order=der_order)
if show_plots:
# plot shapefunctions
c_map = pi.visualization.create_colormap(len(funcs))
pw = pg.plot(title="{}-Test".format(cls.__name__))
pw.addLegend()
pw.showGrid(x=True, y=True, alpha=0.5)
[
pw.addItem(
pg.PlotDataItem(np.array(dz),
weights[idx] * func.derive(der_order)(dz),
pen=pg.mkPen(color=c_map[idx]),
name="{}.{}".format(cls.__name__, idx)))
for idx, func in enumerate(funcs)
]
# plot hull curve
pw.addItem(
pg.PlotDataItem(np.array(hull.input_data[1]),
hull.output_data[0, :],
pen=pg.mkPen(width=2),
name="hull-curve"))
# plot original function
pw.addItem(
pg.PlotDataItem(np.array(dz),
approx_func.derive(der_order)(dz),
pen=pg.mkPen(color="m",
width=2,
style=pg.QtCore.Qt.DashLine),
name="original"))
pg.QtCore.QCoreApplication.instance().exec_()
return np.sum(
np.abs(hull.output_data[0, :] - approx_func.derive(der_order)(dz)))
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
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_fem(self):
"""
use best documented fem case to test all steps in simulation process
"""
# enter string with mass equations
# nodes, ini_funcs = sf.cure_interval(sf.LagrangeFirstOrder,
nodes, ini_funcs = sf.cure_interval(sf.LagrangeSecondOrder,
self.dz.bounds,
node_count=11)
register_base("init_funcs", ini_funcs, overwrite=True)
int1 = ph.IntegralTerm(ph.Product(
ph.TemporalDerivedFieldVariable("init_funcs", 2),
ph.TestFunction("init_funcs")),
self.dz.bounds,
scale=self.params.sigma * self.params.tau**2)
s1 = ph.ScalarTerm(ph.Product(
ph.TemporalDerivedFieldVariable("init_funcs", 2, location=0),
ph.TestFunction("init_funcs", location=0)),
scale=self.params.m)
int2 = ph.IntegralTerm(ph.Product(
ph.SpatialDerivedFieldVariable("init_funcs", 1),
ph.TestFunction("init_funcs", order=1)),
self.dz.bounds,
scale=self.params.sigma)
s2 = ph.ScalarTerm(
ph.Product(ph.Input(self.u),
ph.TestFunction("init_funcs", location=1)),
-self.params.sigma)
# derive sate-space system
string_pde = sim.WeakFormulation([int1, s1, int2, s2], name="fem_test")
self.cf = sim.parse_weak_formulation(string_pde)
ss = self.cf.convert_to_state_space()
# generate initial conditions for weights
q0 = np.array([
cr.project_on_base(self.ic[idx], ini_funcs) for idx in range(2)
]).flatten()
# simulate
t, q = sim.simulate_state_space(ss, q0, self.dt)
# calculate result data
eval_data = []
for der_idx in range(2):
eval_data.append(
sim.evaluate_approximation(
"init_funcs",
q[:,
der_idx * ini_funcs.size:(der_idx + 1) * ini_funcs.size],
t, self.dz))
eval_data[-1].name = "{0}{1}".format(
self.cf.name, "_" + "".join(["d" for x in range(der_idx)]) +
"t" if der_idx > 0 else "")
# display results
if show_plots:
win = vis.PgAnimatedPlot(eval_data[:2],
title="fem 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))
# save some test data for later use
root_dir = os.getcwd()
if root_dir.split(os.sep)[-1] == "tests":
res_dir = os.sep.join([os.getcwd(), "resources"])
else:
res_dir = os.sep.join([os.getcwd(), "tests", "resources"])
if not os.path.isdir(res_dir):
os.makedirs(res_dir)
file_path = os.sep.join([res_dir, "test_data.res"])
with open(file_path, "w+b") as f:
dump(eval_data, f)
def _build_system(self):
# initial and test functions
nodes = pi.Domain(self.bounds, num=self.approx_cnt)
full_fem_base = pi.LagrangeFirstOrder.cure_interval(nodes)
act_fem_base = pi.Base(full_fem_base[-1])
not_act_fem_base = pi.Base(full_fem_base[1:-1])
pi.register_base("act_base", act_fem_base)
pi.register_base("sim_base", not_act_fem_base)
pi.register_base(self.base_lbl, full_fem_base)
a2, a1, a0, _, _ = self.params
# weak form
x = pi.FieldVariable("sim_base")
x_dt = x.derive(temp_order=1)
x_dz = x.derive(spat_order=1)
phi = pi.TestFunction("sim_base")
phi_dz = phi.derive(1)
act_phi = pi.ScalarFunction("act_base")
act_phi_dz = act_phi.derive(1)
u = pi.ConstantTrajectory(0) # dummy input
weak_form = pi.WeakFormulation(
[
# ... of the homogeneous part of the system
pi.IntegralTerm(pi.Product(x_dt, phi), limits=self.bounds),
pi.IntegralTerm(
pi.Product(x_dz, phi_dz), limits=self.bounds, scale=a2),
pi.IntegralTerm(
pi.Product(x_dz, phi), limits=self.bounds, scale=-a1),
pi.IntegralTerm(
pi.Product(x, phi), limits=self.bounds, scale=-a0),
# ... of the inhomogeneous part of the system
pi.IntegralTerm(pi.Product(pi.Product(act_phi, phi),
pi.Input(u, order=1)),
limits=self.bounds),
pi.IntegralTerm(pi.Product(pi.Product(act_phi_dz, phi_dz),
pi.Input(u)),
limits=self.bounds,
scale=a2),
pi.IntegralTerm(pi.Product(pi.Product(act_phi_dz, phi),
pi.Input(u)),
limits=self.bounds,
scale=-a1),
pi.IntegralTerm(pi.Product(pi.Product(act_phi, phi),
pi.Input(u)),
limits=self.bounds,
scale=-a0)
],
name="main_system")
# system matrices \dot x = A x + b0 u + b1 \dot u
cf = pi.parse_weak_formulation(weak_form)
ss = pi.create_state_space(cf)
a_mat = ss.A[1]
b0 = ss.B[0][1]
self.b1 = ss.B[1][1]
# Idea: \bar x = \tilde A x + b1 u
self.a_tilde = np.diag(np.ones(a_mat.shape[0]), 0)
self.a_tilde_inv = np.linalg.inv(self.a_tilde)
# Yields: \dot \bar x = \bar A \bar x + \bar b u
self.a_bar = (self.a_tilde @ a_mat) @ self.a_tilde_inv
self.b_bar = self.a_tilde @ (a_mat @ self.b1) + b0
def test_it(self):
param = [2., -1.5, -3., 2., .5]
a2, a1, a0, alpha, beta = param
l = 1.
spatial_disc = 10
dz = sim.Domain(bounds=(0, l), num=spatial_disc)
T = 1.
temporal_disc = 2e2
dt = sim.Domain(bounds=(0, T), num=temporal_disc)
# create test functions
nodes_1, ini_funcs_1 = pyinduct.shapefunctions.cure_interval(pyinduct.shapefunctions.LagrangeFirstOrder,
dz.bounds,
node_count=spatial_disc)
register_base("init_funcs_1", ini_funcs_1, overwrite=True)
nodes_2, ini_funcs_2 = pyinduct.shapefunctions.cure_interval(pyinduct.shapefunctions.LagrangeSecondOrder,
dz.bounds,
node_count=spatial_disc)
register_base("init_funcs_2", ini_funcs_2, overwrite=True)
def test_dd():
# trajectory
bound_cond_type = 'dirichlet'
actuation_type = 'dirichlet'
u = tr.RadTrajectory(l, T, param, bound_cond_type, actuation_type)
# derive state-space system
rad_pde = ut.get_parabolic_dirichlet_weak_form("init_funcs_2", "init_funcs_2", u, param, dz.bounds)
cf = sim.parse_weak_formulation(rad_pde)
ss = cf.convert_to_state_space()
# simulate system
t, q = sim.simulate_state_space(ss, cf.input_function, np.zeros(ini_funcs_2.shape), dt)
return t, q
def test_rd():
# trajectory
bound_cond_type = 'robin'
actuation_type = 'dirichlet'
u = tr.RadTrajectory(l, T, param, bound_cond_type, actuation_type)
# integral terms
int1 = ph.IntegralTerm(ph.Product(ph.TemporalDerivedFieldVariable("init_funcs_2", order=1),
ph.TestFunction("init_funcs_2", order=0)), dz.bounds)
int2 = ph.IntegralTerm(ph.Product(ph.SpatialDerivedFieldVariable("init_funcs_2", order=0),
ph.TestFunction("init_funcs_2", order=2)), dz.bounds, -a2)
int3 = ph.IntegralTerm(ph.Product(ph.SpatialDerivedFieldVariable("init_funcs_2", order=1),
ph.TestFunction("init_funcs_2", order=0)), dz.bounds, -a1)
int4 = ph.IntegralTerm(ph.Product(ph.SpatialDerivedFieldVariable("init_funcs_2", order=0),
ph.TestFunction("init_funcs_2", order=0)), dz.bounds, -a0)
# scalar terms from int 2
s1 = ph.ScalarTerm(ph.Product(ph.SpatialDerivedFieldVariable("init_funcs_2", order=1, location=l),
ph.TestFunction("init_funcs_2", order=0, location=l)), -a2)
s2 = ph.ScalarTerm(ph.Product(ph.SpatialDerivedFieldVariable("init_funcs_2", order=0, location=0),
ph.TestFunction("init_funcs_2", order=0, location=0)), a2*alpha)
s3 = ph.ScalarTerm(ph.Product(ph.SpatialDerivedFieldVariable("init_funcs_2", order=0, location=0),
ph.TestFunction("init_funcs_2", order=1, location=0)), -a2)
s4 = ph.ScalarTerm(ph.Product(ph.Input(u),
ph.TestFunction("init_funcs_2", order=1, location=l)), a2)
# derive state-space system
rad_pde = sim.WeakFormulation([int1, int2, int3, int4, s1, s2, s3, s4])
cf = sim.parse_weak_formulation(rad_pde)
ss = cf.convert_to_state_space()
# simulate system
t, q = sim.simulate_state_space(ss, cf.input_function, np.zeros(ini_funcs_2.shape), dt)
return t, q
def test_dr():
# trajectory
bound_cond_type = 'dirichlet'
actuation_type = 'robin'
u = tr.RadTrajectory(l, T, param, bound_cond_type, actuation_type)
# integral terms
int1 = ph.IntegralTerm(ph.Product(ph.TemporalDerivedFieldVariable("init_funcs_1", order=1),
ph.TestFunction("init_funcs_1", order=0)), dz.bounds)
int2 = ph.IntegralTerm(ph.Product(ph.SpatialDerivedFieldVariable("init_funcs_1", order=1),
ph.TestFunction("init_funcs_1", order=1)), dz.bounds, a2)
int3 = ph.IntegralTerm(ph.Product(ph.SpatialDerivedFieldVariable("init_funcs_1", order=0),
ph.TestFunction("init_funcs_1", order=1)), dz.bounds, a1)
int4 = ph.IntegralTerm(ph.Product(ph.SpatialDerivedFieldVariable("init_funcs_1", order=0),
ph.TestFunction("init_funcs_1", order=0)), dz.bounds, -a0)
# scalar terms from int 2
s1 = ph.ScalarTerm(ph.Product(ph.SpatialDerivedFieldVariable("init_funcs_1", order=0, location=l),
ph.TestFunction("init_funcs_1", order=0, location=l)), -a1)
s2 = ph.ScalarTerm(ph.Product(ph.SpatialDerivedFieldVariable("init_funcs_1", order=0, location=l),
ph.TestFunction("init_funcs_1", order=0, location=l)), a2*beta)
s3 = ph.ScalarTerm(ph.Product(ph.SpatialDerivedFieldVariable("init_funcs_1", order=1, location=0),
ph.TestFunction("init_funcs_1", order=0, location=0)), a2)
s4 = ph.ScalarTerm(ph.Product(ph.Input(u),
ph.TestFunction("init_funcs_1", order=0, location=l)), -a2)
# derive state-space system
rad_pde = sim.WeakFormulation([int1, int2, int3, int4, s1, s2, s3, s4])
cf = sim.parse_weak_formulation(rad_pde)
ss = cf.convert_to_state_space()
# simulate system
t, q = sim.simulate_state_space(ss, cf.input_function, np.zeros(ini_funcs_1.shape), dt)
# check if (x'(0,t_end) - 1.) < 0.1
self.assertLess(np.abs(ini_funcs_1[0].derive(1)(sys.float_info.min) * (q[-1, 0] - q[-1, 1])) - 1, 0.1)
return t, q
def test_rr():
# trajectory
bound_cond_type = 'robin'
actuation_type = 'robin'
u = tr.RadTrajectory(l, T, param, bound_cond_type, actuation_type)
# derive state-space system
rad_pde = ut.get_parabolic_robin_weak_form("init_funcs_1", "init_funcs_1", u, param, dz.bounds)
cf = sim.parse_weak_formulation(rad_pde)
ss = cf.convert_to_state_space()
# simulate system
t, q = sim.simulate_state_space(ss, cf.input_function, np.zeros(ini_funcs_1.shape), dt)
# check if (x(0,t_end) - 1.) < 0.1
self.assertLess(np.abs(ini_funcs_1[0].derive(0)(0) * q[-1, 0]) - 1, 0.1)
return t, q
t, q = test_dr()
_, _ = test_rr()
# TODO: fit LagrangeSecondOrder to test_dd and test_rd
# t, q = test_dd()
# t, q = test_rd()
# display results
if show_plots:
eval_d = ut.evaluate_approximation("init_funcs_1", q, t, dz, spat_order=1)
win1 = vis.PgAnimatedPlot([eval_d], title="Test")
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 setUp(self):
# scalars
self.scalars = ph.Scalars(np.vstack(range(3)))
# inputs
self.u = np.sin
self.input = ph.Input(self.u) # control input
# TestFunctions
nodes, self.ini_funcs = pyinduct.shapefunctions.cure_interval(pyinduct.shapefunctions.LagrangeFirstOrder,
(0, 1), node_count=3)
register_base("ini_funcs", self.ini_funcs, overwrite=True)
self.phi = ph.TestFunction("ini_funcs") # eigenfunction or something else
self.phi_at0 = ph.TestFunction("ini_funcs", location=0) # eigenfunction or something else
self.phi_at1 = ph.TestFunction("ini_funcs", location=1) # eigenfunction or something else
self.dphi = ph.TestFunction("ini_funcs", order=1) # eigenfunction or something else
self.dphi_at1 = ph.TestFunction("ini_funcs", order=1, location=1) # eigenfunction or something else
# FieldVars
self.field_var = ph.FieldVariable("ini_funcs")
self.odd_weight_field_var = ph.FieldVariable("ini_funcs", weight_label="special_weights")
self.field_var_at1 = ph.FieldVariable("ini_funcs", location=1)
self.field_var_dz = ph.SpatialDerivedFieldVariable("ini_funcs", 1)
self.field_var_dz_at1 = ph.SpatialDerivedFieldVariable("ini_funcs", 1, location=1)
self.field_var_ddt = ph.TemporalDerivedFieldVariable("ini_funcs", 2)
self.field_var_ddt_at0 = ph.TemporalDerivedFieldVariable("ini_funcs", 2, location=0)
self.field_var_ddt_at1 = ph.TemporalDerivedFieldVariable("ini_funcs", 2, location=1)
# create all possible kinds of input variables
self.input_term1 = ph.ScalarTerm(ph.Product(self.phi_at1, self.input))
self.input_term1_swapped = ph.ScalarTerm(ph.Product(self.input, self.phi_at1))
self.input_term2 = ph.ScalarTerm(ph.Product(self.dphi_at1, self.input))
self.func_term = ph.ScalarTerm(self.phi_at1)
self.field_term_at1 = ph.ScalarTerm(self.field_var_at1)
self.field_term_dz_at1 = ph.ScalarTerm(self.field_var_dz_at1)
self.field_term_ddt_at1 = ph.ScalarTerm(self.field_var_ddt_at1)
self.field_int = ph.IntegralTerm(self.field_var, (0, 1))
self.field_dz_int = ph.IntegralTerm(self.field_var_dz, (0, 1))
self.field_ddt_int = ph.IntegralTerm(self.field_var_ddt, (0, 1))
self.prod_term_fs_at1 = ph.ScalarTerm(
ph.Product(self.field_var_at1, self.scalars))
self.prod_int_fs = ph.IntegralTerm(ph.Product(self.field_var, self.scalars), (0, 1))
self.prod_int_f_f = ph.IntegralTerm(ph.Product(self.field_var, self.phi), (0, 1))
self.prod_int_f_f_swapped = ph.IntegralTerm(ph.Product(self.phi, self.field_var), (0, 1))
self.prod_int_f_at1_f = ph.IntegralTerm(
ph.Product(self.field_var_at1, self.phi), (0, 1))
self.prod_int_f_f_at1 = ph.IntegralTerm(
ph.Product(self.field_var, self.phi_at1), (0, 1))
self.prod_term_f_at1_f_at1 = ph.ScalarTerm(
ph.Product(self.field_var_at1, self.phi_at1))
self.prod_int_fddt_f = ph.IntegralTerm(
ph.Product(self.field_var_ddt, self.phi), (0, 1))
self.prod_term_fddt_at0_f_at0 = ph.ScalarTerm(
ph.Product(self.field_var_ddt_at0, self.phi_at0))
self.prod_term_f_at1_dphi_at1 = ph.ScalarTerm(
ph.Product(self.field_var_at1, self.dphi_at1))
self.temp_int = ph.IntegralTerm(ph.Product(self.field_var_ddt, self.phi), (0, 1))
self.spat_int = ph.IntegralTerm(ph.Product(self.field_var_dz, self.dphi), (0, 1))
self.spat_int_asymmetric = ph.IntegralTerm(
ph.Product(self.field_var_dz, self.phi), (0, 1))
self.alternating_weights_term = ph.IntegralTerm(self.odd_weight_field_var, (0, 1))
def test_it(self):
param = [2., -1.5, -3., 2., .5]
a2, a1, a0, alpha, beta = param
l = 1.
spatial_disc = 11
dz = sim.Domain(bounds=(0, l), num=spatial_disc)
T = 1.
temporal_disc = 2e2
dt = sim.Domain(bounds=(0, T), num=temporal_disc)
# create test functions
nodes_1, ini_funcs_1 = sf.cure_interval(sf.LagrangeFirstOrder,
dz.bounds,
node_count=spatial_disc)
register_base("init_funcs_1", ini_funcs_1, overwrite=True)
nodes_2, ini_funcs_2 = sf.cure_interval(sf.LagrangeSecondOrder,
dz.bounds,
node_count=spatial_disc)
register_base("init_funcs_2", ini_funcs_2, overwrite=True)
def test_dd():
# trajectory
bound_cond_type = 'dirichlet'
actuation_type = 'dirichlet'
u = tr.RadTrajectory(l, T, param, bound_cond_type, actuation_type)
# derive state-space system
rad_pde = ut.get_parabolic_dirichlet_weak_form(
"init_funcs_2", "init_funcs_2", u, param, dz.bounds)
cf = sim.parse_weak_formulation(rad_pde)
ss = cf.convert_to_state_space()
# simulate system
t, q = sim.simulate_state_space(ss, np.zeros(ini_funcs_2.shape),
dt)
return t, q
def test_rd():
# trajectory
bound_cond_type = 'robin'
actuation_type = 'dirichlet'
u = tr.RadTrajectory(l, T, param, bound_cond_type, actuation_type)
# integral terms
int1 = ph.IntegralTerm(
ph.Product(
ph.TemporalDerivedFieldVariable("init_funcs_2", order=1),
ph.TestFunction("init_funcs_2", order=0)), dz.bounds)
int2 = ph.IntegralTerm(
ph.Product(
ph.SpatialDerivedFieldVariable("init_funcs_2", order=0),
ph.TestFunction("init_funcs_2", order=2)), dz.bounds, -a2)
int3 = ph.IntegralTerm(
ph.Product(
ph.SpatialDerivedFieldVariable("init_funcs_2", order=1),
ph.TestFunction("init_funcs_2", order=0)), dz.bounds, -a1)
int4 = ph.IntegralTerm(
ph.Product(
ph.SpatialDerivedFieldVariable("init_funcs_2", order=0),
ph.TestFunction("init_funcs_2", order=0)), dz.bounds, -a0)
# scalar terms from int 2
s1 = ph.ScalarTerm(
ph.Product(
ph.SpatialDerivedFieldVariable("init_funcs_2",
order=1,
location=l),
ph.TestFunction("init_funcs_2", order=0, location=l)), -a2)
s2 = ph.ScalarTerm(
ph.Product(
ph.SpatialDerivedFieldVariable("init_funcs_2",
order=0,
location=0),
ph.TestFunction("init_funcs_2", order=0, location=0)),
a2 * alpha)
s3 = ph.ScalarTerm(
ph.Product(
ph.SpatialDerivedFieldVariable("init_funcs_2",
order=0,
location=0),
ph.TestFunction("init_funcs_2", order=1, location=0)), -a2)
s4 = ph.ScalarTerm(
ph.Product(
ph.Input(u),
ph.TestFunction("init_funcs_2", order=1, location=l)), a2)
# derive state-space system
rad_pde = sim.WeakFormulation(
[int1, int2, int3, int4, s1, s2, s3, s4])
cf = sim.parse_weak_formulation(rad_pde)
ss = cf.convert_to_state_space()
# simulate system
t, q = sim.simulate_state_space(ss, np.zeros(ini_funcs_2.shape),
dt)
return t, q
def test_dr():
# trajectory
bound_cond_type = 'dirichlet'
actuation_type = 'robin'
u = tr.RadTrajectory(l, T, param, bound_cond_type, actuation_type)
# integral terms
int1 = ph.IntegralTerm(
ph.Product(
ph.TemporalDerivedFieldVariable("init_funcs_1", order=1),
ph.TestFunction("init_funcs_1", order=0)), dz.bounds)
int2 = ph.IntegralTerm(
ph.Product(
ph.SpatialDerivedFieldVariable("init_funcs_1", order=1),
ph.TestFunction("init_funcs_1", order=1)), dz.bounds, a2)
int3 = ph.IntegralTerm(
ph.Product(
ph.SpatialDerivedFieldVariable("init_funcs_1", order=0),
ph.TestFunction("init_funcs_1", order=1)), dz.bounds, a1)
int4 = ph.IntegralTerm(
ph.Product(
ph.SpatialDerivedFieldVariable("init_funcs_1", order=0),
ph.TestFunction("init_funcs_1", order=0)), dz.bounds, -a0)
# scalar terms from int 2
s1 = ph.ScalarTerm(
ph.Product(
ph.SpatialDerivedFieldVariable("init_funcs_1",
order=0,
location=l),
ph.TestFunction("init_funcs_1", order=0, location=l)), -a1)
s2 = ph.ScalarTerm(
ph.Product(
ph.SpatialDerivedFieldVariable("init_funcs_1",
order=0,
location=l),
ph.TestFunction("init_funcs_1", order=0, location=l)),
a2 * beta)
s3 = ph.ScalarTerm(
ph.Product(
ph.SpatialDerivedFieldVariable("init_funcs_1",
order=1,
location=0),
ph.TestFunction("init_funcs_1", order=0, location=0)), a2)
s4 = ph.ScalarTerm(
ph.Product(
ph.Input(u),
ph.TestFunction("init_funcs_1", order=0, location=l)), -a2)
# derive state-space system
rad_pde = sim.WeakFormulation(
[int1, int2, int3, int4, s1, s2, s3, s4])
cf = sim.parse_weak_formulation(rad_pde)
ss = cf.convert_to_state_space()
# simulate system
t, q = sim.simulate_state_space(ss, np.zeros(ini_funcs_1.shape),
dt)
# check if (x'(0,t_end) - 1.) < 0.1
self.assertLess(
np.abs(ini_funcs_1[0].derive(1)(sys.float_info.min) *
(q[-1, 0] - q[-1, 1])) - 1, 0.1)
return t, q
def test_rr():
# trajectory
bound_cond_type = 'robin'
actuation_type = 'robin'
u = tr.RadTrajectory(l, T, param, bound_cond_type, actuation_type)
# derive state-space system
rad_pde = ut.get_parabolic_robin_weak_form("init_funcs_1",
"init_funcs_1", u,
param, dz.bounds)
cf = sim.parse_weak_formulation(rad_pde)
ss = cf.convert_to_state_space()
# simulate system
t, q = sim.simulate_state_space(ss, np.zeros(ini_funcs_1.shape),
dt)
# check if (x(0,t_end) - 1.) < 0.1
self.assertLess(
np.abs(ini_funcs_1[0].derive(0)(0) * q[-1, 0]) - 1, 0.1)
return t, q
t, q = test_dr()
_, _ = test_rr()
# TODO: fit LagrangeSecondOrder to test_dd and test_rd
# t, q = test_dd()
# t, q = test_rd()
# display results
if show_plots:
eval_d = sim.evaluate_approximation("init_funcs_1",
q,
t,
dz,
spat_order=1)
win1 = vis.PgAnimatedPlot([eval_d], title="Test")
win2 = vis.PgSurfacePlot(eval_d)
app.exec_()
adjoint_eig_funcs_t = np.array([f_tuple[1] for f_tuple in adjoint_and_eig_funcs_t])
# transformed original eigenfunctions
eig_funcs = np.array([ef.TransformedSecondOrderEigenfunction(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, l, 1e4))
for i in range(n)])
# create testfunctions
nodes, fem_funcs = sh.cure_interval(sh.LagrangeFirstOrder,
spatial_domain.bounds,
node_count=len(spatial_domain))
# register functions
register_base("eig_funcs_t", eig_funcs_t, overwrite=True)
register_base("adjoint_eig_funcs_t", adjoint_eig_funcs_t, overwrite=True)
register_base("eig_funcs", eig_funcs, overwrite=True)
register_base("fem_funcs", fem_funcs, overwrite=True)
# init trajectory
traj = tr.RadTrajectory(l, T, param_ti, bound_cond_type, actuation_type)
# original () and target (_t) field variable
fem_field_variable = ph.FieldVariable("fem_funcs", location=l)
field_variable_t = ph.FieldVariable("eig_funcs_t", weight_label="eig_funcs", location=l)
d_field_variable_t = ph.SpatialDerivedFieldVariable("eig_funcs_t", 1, weight_label="eig_funcs", location=l)
field_variable = ph.FieldVariable("eig_funcs", location=l)
d_field_variable = ph.SpatialDerivedFieldVariable("eig_funcs", 1, location=l)
# intermediate (_i) and target intermediate (_ti) transformations by z=l
def setUp(self):
nodes, ini_funcs = pyinduct.shapefunctions.cure_interval(pyinduct.shapefunctions.LagrangeFirstOrder, (0, 1), node_count=2)
register_base("test_funcs", ini_funcs, overwrite=True)
[f_tuple[1] for f_tuple in adjoint_and_eig_funcs_t])
# transformed original eigenfunctions
eig_funcs = np.array([
ef.TransformedSecondOrderEigenfunction(
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, l, 1e4)) for i in range(n)
])
# create testfunctions
nodes, fem_funcs = sh.cure_interval(sh.LagrangeFirstOrder,
spatial_domain.bounds,
node_count=len(spatial_domain))
# register functions
register_base("eig_funcs_t", eig_funcs_t, overwrite=True)
register_base("adjoint_eig_funcs_t", adjoint_eig_funcs_t, overwrite=True)
register_base("eig_funcs", eig_funcs, overwrite=True)
register_base("fem_funcs", fem_funcs, overwrite=True)
# init trajectory
traj = tr.RadTrajectory(l, T, param_ti, bound_cond_type, actuation_type)
# original () and target (_t) field variable
fem_field_variable = ph.FieldVariable("fem_funcs", location=l)
field_variable_t = ph.FieldVariable("eig_funcs_t",
weight_label="eig_funcs",
location=l)
d_field_variable_t = ph.SpatialDerivedFieldVariable("eig_funcs_t",
1,
weight_label="eig_funcs",
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_()