def test_FieldVariable(self): self.assertRaises(TypeError, ph.FieldVariable, "test_funcs", [0, 0]) # list instead of tuple self.assertRaises(ValueError, ph.FieldVariable, "test_funcs", (3, 0)) # order too high self.assertRaises(ValueError, ph.FieldVariable, "test_funcs", (0, 3)) # order too high self.assertRaises(ValueError, ph.FieldVariable, "test_funcs", (2, 2)) # order too high self.assertRaises(ValueError, ph.FieldVariable, "test_funcs", (-1, 3)) # order negative # defaults a = ph.FieldVariable("test_funcs") self.assertEqual((0, 0), a.order) self.assertEqual( "test_funcs", a.data["weight_lbl"]) # default weight label is function label self.assertEqual(None, a.location) self.assertEqual(1, a.data["exponent"]) # default exponent is 1 b = ph.FieldVariable("test_funcs", order=(1, 1), location=7, weight_label="test_lbl", exponent=10) self.assertEqual((1, 1), b.order) self.assertEqual( "test_lbl", b.data["weight_lbl"]) # default weight label is function label self.assertEqual(7, b.location) self.assertEqual(10, b.data["exponent"])
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): 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 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])
]) # create test-functions nodes, fem_funcs = sh.cure_interval(sh.LagrangeFirstOrder, spatial_domain.bounds, node_count=len(spatial_domain)) # register functions re.register_base("adjoint_eig_funcs", adjoint_eig_funcs, overwrite=True) re.register_base("eig_funcs", eig_funcs, overwrite=True) re.register_base("eig_funcs_i", eig_funcs_i, overwrite=True) re.register_base("eig_funcs_ti", eig_funcs_ti, overwrite=True) re.register_base("fem_funcs", fem_funcs, overwrite=True) # original intermediate (_i), target intermediate (_ti) and fem field variable fem_field_variable = ph.FieldVariable("fem_funcs", location=l) field_variable_i = ph.FieldVariable("eig_funcs_i", weight_label="eig_funcs", location=l) d_field_variable_i = ph.SpatialDerivedFieldVariable("eig_funcs_i", 1, weight_label="eig_funcs", location=l) field_variable_ti = ph.FieldVariable("eig_funcs_ti", weight_label="eig_funcs", location=l) d_field_variable_ti = ph.SpatialDerivedFieldVariable("eig_funcs_ti", 1, weight_label="eig_funcs", location=l)
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, spatial_domain.bounds, norm_fac[i]) for i in range(n)]) re.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, spatial_domain.bounds, norm_fac_t[i]) for i in range(n)]) re.register_base("eig_funcs_t", eig_funcs_t, overwrite=True) # init controller x_at_1 = ph.FieldVariable("eig_funcs", location=1) xt_at_1 = ph.FieldVariable("eig_funcs_t", weight_label="eig_funcs", location=1) controller = ct.Controller(ct.ControlLaw([ph.ScalarTerm(x_at_1, 1), ph.ScalarTerm(xt_at_1, -1)])) # derive initial field variable x(z,0) and weights start_state = cr.Function(lambda z: init_profile) initial_weights = cr.project_on_base(start_state, eig_funcs) # init trajectory traj = tr.RadTrajectory(l, T, param_t, bound_cond_type, actuation_type) # input with feedback control_law = sim.SimulationInputSum([traj, controller]) # determine (A,B) with modal-transfomation A = np.diag(eig_values)
[ef.SecondOrderRobinEigenfunction(eig_freq_t[i], param_t, spatial_domain.bounds).scale(eig_funcs[i](0)) for i in range(n)]) # create fem test functions nodes, fem_funcs = sh.cure_interval(sh.LagrangeFirstOrder, spatial_domain.bounds, node_count=len(spatial_domain)) # register eigenfunctions re.register_base("eig_funcs", eig_funcs, overwrite=True) re.register_base("adjoint_eig_funcs", adjoint_eig_funcs, overwrite=True) re.register_base("eig_funcs_t", eig_funcs_t, overwrite=True) re.register_base("fem_funcs", fem_funcs, overwrite=True) # original () and target (_t) field variable fem_field_variable = ph.FieldVariable("fem_funcs", location=l) field_variable = ph.FieldVariable("eig_funcs", location=l) d_field_variable = ph.SpatialDerivedFieldVariable("eig_funcs", 1, 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) def transform_i(z): """ intermediate (_i) transformation at z=l """ return np.exp(a1 / 2 / a2 * z) # x_i = x * transform_i def transform_ti(z): """
def test_fem(self): # system/simulation parameters actuation_type = 'robin' bound_cond_type = 'robin' self.l = 1. spatial_disc = 30 self.dz = sim.Domain(bounds=(0, self.l), num=spatial_disc) self.T = 1. temporal_disc = 1e2 self.dt = sim.Domain(bounds=(0, self.T), num=temporal_disc) self.n = 12 # original system parameters a2 = 1.5 a1 = 2.5 a0 = 28 alpha = -2 beta = -3 self.param = [a2, a1, a0, alpha, beta] adjoint_param = ef.get_adjoint_rad_evp_param(self.param) # target system parameters (controller parameters) a1_t = -5 a0_t = -25 alpha_t = 3 beta_t = 2 self.param_t = [a2, a1_t, a0_t, alpha_t, beta_t] # actuation_type by b which is close to b_desired on a k times subdivided spatial domain b_desired = self.l / 2 k = 51 # = k1 + k2 k1, k2, self.b = ut.split_domain(k, b_desired, self.l, mode='coprime')[0:3] M = np.linalg.inv( ut.get_inn_domain_transformation_matrix(k1, k2, mode="2n")) # original intermediate ("_i") and traget intermediate ("_ti") system parameters _, _, a0_i, self.alpha_i, self.beta_i = ef.transform2intermediate( self.param) self.param_i = a2, 0, a0_i, self.alpha_i, self.beta_i _, _, a0_ti, self.alpha_ti, self.beta_ti = ef.transform2intermediate( self.param_t) self.param_ti = a2, 0, a0_ti, self.alpha_ti, self.beta_ti # create (not normalized) eigenfunctions eig_freq, self.eig_val = ef.compute_rad_robin_eigenfrequencies( self.param, self.l, self.n) init_eig_funcs = np.array([ ef.SecondOrderRobinEigenfunction(om, self.param, self.dz.bounds) for om in eig_freq ]) init_adjoint_eig_funcs = np.array([ ef.SecondOrderRobinEigenfunction(om, adjoint_param, self.dz.bounds) for om in eig_freq ]) # normalize eigenfunctions and adjoint eigenfunctions adjoint_and_eig_funcs = [ cr.normalize_function(init_eig_funcs[i], init_adjoint_eig_funcs[i]) for i in range(self.n) ] eig_funcs = np.array([f_tuple[0] for f_tuple in adjoint_and_eig_funcs]) self.adjoint_eig_funcs = np.array( [f_tuple[1] for f_tuple in adjoint_and_eig_funcs]) # eigenfunctions of the in-domain intermediate (_id) and the intermediate (_i) system eig_freq_i, eig_val_i = ef.compute_rad_robin_eigenfrequencies( self.param_i, self.l, self.n) self.assertTrue(all(np.isclose(eig_val_i, self.eig_val))) eig_funcs_id = np.array([ ef.SecondOrderRobinEigenfunction(eig_freq_i[i], self.param_i, self.dz.bounds, eig_funcs[i](0)) for i in range(self.n) ]) eig_funcs_i = np.array([ ef.SecondOrderRobinEigenfunction( eig_freq_i[i], self.param_i, self.dz.bounds, eig_funcs[i](0) * eig_funcs_id[i](self.l) / eig_funcs_id[i](self.b)) for i in range(self.n) ]) # eigenfunctions from target system ("_ti") eig_freq_ti = np.sqrt((a0_ti - self.eig_val) / a2) eig_funcs_ti = np.array([ ef.SecondOrderRobinEigenfunction(eig_freq_ti[i], self.param_ti, self.dz.bounds, eig_funcs_i[i](0)) for i in range(self.n) ]) # create testfunctions nodes, self.fem_funcs = sf.cure_interval(sf.LagrangeFirstOrder, self.dz.bounds, node_count=self.n) # register eigenfunctions # register_functions("eig_funcs", eig_funcs, overwrite=True) register_base("adjoint_eig_funcs", self.adjoint_eig_funcs, overwrite=True) register_base("eig_funcs", eig_funcs, overwrite=True) register_base("eig_funcs_i", eig_funcs_i, overwrite=True) register_base("eig_funcs_ti", eig_funcs_ti, overwrite=True) register_base("fem_funcs", self.fem_funcs, overwrite=True) # init trajectory self.traj = tr.RadTrajectory(self.l, self.T, self.param_ti, bound_cond_type, actuation_type) # original () and target (_t) field variable fem_field_variable = ph.FieldVariable("fem_funcs", location=self.l) field_variable_i = ph.FieldVariable("eig_funcs_i", weight_label="eig_funcs", location=self.l) d_field_variable_i = ph.SpatialDerivedFieldVariable( "eig_funcs_i", 1, weight_label="eig_funcs", location=self.l) field_variable_ti = ph.FieldVariable("eig_funcs_ti", weight_label="eig_funcs", location=self.l) d_field_variable_ti = ph.SpatialDerivedFieldVariable( "eig_funcs_ti", 1, weight_label="eig_funcs", location=self.l) # intermediate (_i) and target intermediate (_ti) field variable (list of scalar terms = sum of scalar terms) self.x_fem_i_at_l = [ph.ScalarTerm(fem_field_variable)] self.x_i_at_l = [ph.ScalarTerm(field_variable_i)] self.xd_i_at_l = [ph.ScalarTerm(d_field_variable_i)] self.x_ti_at_l = [ph.ScalarTerm(field_variable_ti)] self.xd_ti_at_l = [ph.ScalarTerm(d_field_variable_ti)] # shift transformation shifted_fem_funcs_i = np.array([ ef.FiniteTransformFunction( func, M, self.b, self.l, scale_func=lambda z: np.exp(a1 / 2 / a2 * z)) for func in self.fem_funcs ]) shifted_eig_funcs_id = np.array([ ef.FiniteTransformFunction(func, M, self.b, self.l) for func in eig_funcs_id ]) register_base("sh_fem_funcs_i", shifted_fem_funcs_i, overwrite=True) register_base("sh_eig_funcs_id", shifted_eig_funcs_id, overwrite=True) sh_fem_field_variable_i = ph.FieldVariable("sh_fem_funcs_i", weight_label="fem_funcs", location=self.l) sh_field_variable_id = ph.FieldVariable("sh_eig_funcs_id", weight_label="eig_funcs", location=self.l) self.sh_x_fem_i_at_l = [ ph.ScalarTerm(sh_fem_field_variable_i), ph.ScalarTerm(field_variable_i), ph.ScalarTerm(sh_field_variable_id, -1) ] # discontinuous operator (Kx)(t) = int_kernel_zz(l)*x(l,t) self.int_kernel_zz = lambda z: self.alpha_ti - self.alpha_i + ( a0_i - a0_ti) / 2 / a2 * z a2, a1, _, _, _ = self.param controller = ut.get_parabolic_robin_backstepping_controller( state=self.sh_x_fem_i_at_l, approx_state=self.x_i_at_l, d_approx_state=self.xd_i_at_l, approx_target_state=self.x_ti_at_l, d_approx_target_state=self.xd_ti_at_l, integral_kernel_zz=self.int_kernel_zz(self.l), original_beta=self.beta_i, target_beta=self.beta_ti, trajectory=self.traj, scale=np.exp(-a1 / 2 / a2 * self.b)) # determine (A,B) with modal-transfomation rad_pde = ut.get_parabolic_robin_weak_form("fem_funcs", "fem_funcs", controller, self.param, self.dz.bounds, self.b) cf = sim.parse_weak_formulation(rad_pde) ss_weak = cf.convert_to_state_space() # simulate t, q = sim.simulate_state_space(ss_weak, np.zeros( (len(self.fem_funcs))), self.dt) # weights of the intermediate system mat = cr.calculate_base_transformation_matrix(self.fem_funcs, eig_funcs) q_i = np.zeros((q.shape[0], len(eig_funcs_i))) for i in range(q.shape[0]): q_i[i, :] = np.dot(q[i, :], np.transpose(mat)) eval_i = sim.evaluate_approximation("eig_funcs_i", q_i, t, self.dz) x_0t = eval_i.output_data[:, 0] yc, tc = tr.gevrey_tanh(self.T, 1) x_0t_desired = np.interp(t, tc, yc[0, :]) self.assertLess(np.average((x_0t - x_0t_desired)**2), 1e-2) # display results if show_plots: eval_d = sim.evaluate_approximation("fem_funcs", q, t, self.dz) win1 = vis.PgSurfacePlot(eval_i) win2 = vis.PgSurfacePlot(eval_d) app.exec_()
def test_it(self): # system/simulation parameters actuation_type = 'robin' bound_cond_type = 'robin' self.l = 1. spatial_disc = 10 self.dz = sim.Domain(bounds=(0, self.l), num=spatial_disc) self.T = 1. temporal_disc = 1e2 self.dt = sim.Domain(bounds=(0, self.T), num=temporal_disc) self.n = 10 # original system parameters a2 = 1.5 a1_z = cr.Function(lambda z: 1, derivative_handles=[lambda z: 0]) a0_z = lambda z: 3 alpha = -2 beta = -3 self.param = [a2, a1_z, a0_z, alpha, beta] # target system parameters (controller parameters) a1_t = -5 a0_t = -25 alpha_t = 3 beta_t = 2 self.param_t = [a2, a1_t, a0_t, alpha_t, beta_t] adjoint_param_t = ef.get_adjoint_rad_evp_param(self.param_t) # original intermediate ("_i") and traget intermediate ("_ti") system parameters _, _, a0_i, alpha_i, beta_i = ef.transform2intermediate(self.param, d_end=self.l) self.param_i = a2, 0, a0_i, alpha_i, beta_i _, _, a0_ti, alpha_ti, beta_ti = ef.transform2intermediate( self.param_t) self.param_ti = a2, 0, a0_ti, alpha_ti, beta_ti # create (not normalized) target (_t) eigenfunctions eig_freq_t, self.eig_val_t = ef.compute_rad_robin_eigenfrequencies( self.param_t, self.l, self.n) init_eig_funcs_t = np.array([ ef.SecondOrderRobinEigenfunction(om, self.param_t, self.dz.bounds) for om in eig_freq_t ]) init_adjoint_eig_funcs_t = np.array([ ef.SecondOrderRobinEigenfunction(om, adjoint_param_t, self.dz.bounds) for om in eig_freq_t ]) # normalize eigenfunctions and adjoint eigenfunctions adjoint_and_eig_funcs_t = [ cr.normalize_function(init_eig_funcs_t[i], init_adjoint_eig_funcs_t[i]) for i in range(self.n) ] eig_funcs_t = np.array( [f_tuple[0] for f_tuple in adjoint_and_eig_funcs_t]) self.adjoint_eig_funcs_t = np.array( [f_tuple[1] for f_tuple in adjoint_and_eig_funcs_t]) # # transformed original eigenfunctions self.eig_funcs = np.array([ ef.TransformedSecondOrderEigenfunction( self.eig_val_t[i], [eig_funcs_t[i](0), alpha * eig_funcs_t[i](0), 0, 0], [a2, a1_z, a0_z], np.linspace(0, self.l, 1e4)) for i in range(self.n) ]) # create testfunctions nodes, self.fem_funcs = sf.cure_interval(sf.LagrangeFirstOrder, self.dz.bounds, node_count=self.n) # register functions register_base("eig_funcs_t", eig_funcs_t, overwrite=True) register_base("adjoint_eig_funcs_t", self.adjoint_eig_funcs_t, overwrite=True) register_base("eig_funcs", self.eig_funcs, overwrite=True) register_base("fem_funcs", self.fem_funcs, overwrite=True) # init trajectory self.traj = tr.RadTrajectory(self.l, self.T, self.param_ti, bound_cond_type, actuation_type) # original () and target (_t) field variable fem_field_variable = ph.FieldVariable("fem_funcs", location=self.l) field_variable_t = ph.FieldVariable("eig_funcs_t", weight_label="eig_funcs", location=self.l) d_field_variable_t = ph.SpatialDerivedFieldVariable( "eig_funcs_t", 1, weight_label="eig_funcs", location=self.l) field_variable = ph.FieldVariable("eig_funcs", location=self.l) d_field_variable = ph.SpatialDerivedFieldVariable("eig_funcs", 1, location=self.l) # intermediate (_i) and target intermediate (_ti) transformations by z=l # x_i = x * transform_i_at_l self.transform_i_at_l = np.exp( integrate.quad(lambda z: a1_z(z) / 2 / a2, 0, self.l)[0]) # x = x_i * inv_transform_i_at_l self.inv_transform_i_at_l = np.exp( -integrate.quad(lambda z: a1_z(z) / 2 / a2, 0, self.l)[0]) # x_ti = x_t * transform_ti_at_l self.transform_ti_at_l = np.exp(a1_t / 2 / a2 * self.l) # intermediate (_i) and target intermediate (_ti) field variable (list of scalar terms = sum of scalar terms) self.x_fem_i_at_l = [ ph.ScalarTerm(fem_field_variable, self.transform_i_at_l) ] self.x_i_at_l = [ph.ScalarTerm(field_variable, self.transform_i_at_l)] self.xd_i_at_l = [ ph.ScalarTerm(d_field_variable, self.transform_i_at_l), ph.ScalarTerm(field_variable, self.transform_i_at_l * a1_z(self.l) / 2 / a2) ] self.x_ti_at_l = [ ph.ScalarTerm(field_variable_t, self.transform_ti_at_l) ] self.xd_ti_at_l = [ ph.ScalarTerm(d_field_variable_t, self.transform_ti_at_l), ph.ScalarTerm(field_variable_t, self.transform_ti_at_l * a1_t / 2 / a2) ] # discontinuous operator (Kx)(t) = int_kernel_zz(l)*x(l,t) self.int_kernel_zz = alpha_ti - alpha_i + integrate.quad( lambda z: (a0_i(z) - a0_ti) / 2 / a2, 0, self.l)[0] controller = ut.get_parabolic_robin_backstepping_controller( state=self.x_fem_i_at_l, approx_state=self.x_i_at_l, d_approx_state=self.xd_i_at_l, approx_target_state=self.x_ti_at_l, d_approx_target_state=self.xd_ti_at_l, integral_kernel_zz=self.int_kernel_zz, original_beta=beta_i, target_beta=beta_ti, trajectory=self.traj, scale=self.inv_transform_i_at_l) rad_pde = ut.get_parabolic_robin_weak_form("fem_funcs", "fem_funcs", controller, self.param, self.dz.bounds) cf = sim.parse_weak_formulation(rad_pde) ss_weak = cf.convert_to_state_space() # simulate t, q = sim.simulate_state_space(ss_weak, np.zeros( (len(self.fem_funcs))), self.dt) eval_d = sim.evaluate_approximation("fem_funcs", q, t, self.dz) x_0t = eval_d.output_data[:, 0] yc, tc = tr.gevrey_tanh(self.T, 1) x_0t_desired = np.interp(t, tc, yc[0, :]) self.assertLess(np.average((x_0t - x_0t_desired)**2), 1e-4) # display results if show_plots: win1 = vis.PgAnimatedPlot([eval_d], title="Test") win2 = vis.PgSurfacePlot(eval_d) app.exec_()
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): # original system parameters a2 = 1.5 a1 = 2.5 a0 = 28 alpha = -2 beta = -3 param = [a2, a1, a0, alpha, beta] adjoint_param = ef.get_adjoint_rad_evp_param(param) # target system parameters (controller parameters) a1_t = -5 a0_t = -25 alpha_t = 3 beta_t = 2 # a1_t = a1; a0_t = a0; alpha_t = alpha; beta_t = beta param_t = [a2, a1_t, a0_t, alpha_t, beta_t] # original intermediate ("_i") and traget intermediate ("_ti") system parameters _, _, a0_i, alpha_i, beta_i = ef.transform2intermediate(param) _, _, a0_ti, alpha_ti, beta_ti = ef.transform2intermediate(param_t) # system/simulation parameters actuation_type = 'robin' bound_cond_type = 'robin' self.l = 1. spatial_disc = 10 dz = sim.Domain(bounds=(0, self.l), num=spatial_disc) T = 1. temporal_disc = 1e2 dt = sim.Domain(bounds=(0, T), num=temporal_disc) n = 10 # create (not normalized) eigenfunctions eig_freq, eig_val = ef.compute_rad_robin_eigenfrequencies( param, self.l, n) init_eig_funcs = np.array([ ef.SecondOrderRobinEigenfunction(om, param, dz.bounds) for om in eig_freq ]) init_adjoint_eig_funcs = np.array([ ef.SecondOrderRobinEigenfunction(om, adjoint_param, dz.bounds) for om in eig_freq ]) # normalize eigenfunctions and adjoint eigenfunctions adjoint_and_eig_funcs = [ cr.normalize_function(init_eig_funcs[i], init_adjoint_eig_funcs[i]) for i in range(n) ] eig_funcs = np.array([f_tuple[0] for f_tuple in adjoint_and_eig_funcs]) adjoint_eig_funcs = np.array( [f_tuple[1] for f_tuple in adjoint_and_eig_funcs]) # eigenfunctions from target system ("_t") eig_freq_t = np.sqrt(-a1_t**2 / 4 / a2**2 + (a0_t - eig_val) / a2) eig_funcs_t = np.array([ ef.SecondOrderRobinEigenfunction(eig_freq_t[i], param_t, dz.bounds).scale(eig_funcs[i](0)) for i in range(n) ]) # register eigenfunctions register_base("eig_funcs", eig_funcs, overwrite=True) register_base("adjoint_eig_funcs", adjoint_eig_funcs, overwrite=True) register_base("eig_funcs_t", eig_funcs_t, overwrite=True) # derive initial field variable x(z,0) and weights start_state = cr.Function(lambda z: 0., domain=(0, self.l)) initial_weights = cr.project_on_base(start_state, adjoint_eig_funcs) # controller initialization x_at_l = ph.FieldVariable("eig_funcs", location=self.l) xd_at_l = ph.SpatialDerivedFieldVariable("eig_funcs", 1, location=self.l) x_t_at_l = ph.FieldVariable("eig_funcs_t", weight_label="eig_funcs", location=self.l) xd_t_at_l = ph.SpatialDerivedFieldVariable("eig_funcs_t", 1, weight_label="eig_funcs", location=self.l) combined_transform = lambda z: np.exp((a1_t - a1) / 2 / a2 * z) int_kernel_zz = lambda z: alpha_ti - alpha_i + (a0_i - a0_ti ) / 2 / a2 * z controller = ct.Controller( ct.ControlLaw([ ph.ScalarTerm(x_at_l, (beta_i - beta_ti - int_kernel_zz(self.l))), ph.ScalarTerm(x_t_at_l, -beta_ti * combined_transform(self.l)), ph.ScalarTerm(x_at_l, beta_ti), ph.ScalarTerm(xd_t_at_l, -combined_transform(self.l)), ph.ScalarTerm(x_t_at_l, -a1_t / 2 / a2 * combined_transform(self.l)), ph.ScalarTerm(xd_at_l, 1), ph.ScalarTerm(x_at_l, a1 / 2 / a2 + int_kernel_zz(self.l)) ])) # init trajectory traj = tr.RadTrajectory(self.l, T, param_t, bound_cond_type, actuation_type) traj.scale = combined_transform(self.l) # input with feedback control_law = sim.SimulationInputSum([traj, controller]) # control_law = sim.simInputSum([traj]) # determine (A,B) with modal-transformation A = np.diag(np.real(eig_val)) B = a2 * np.array( [adjoint_eig_funcs[i](self.l) for i in range(len(eig_freq))]) ss_modal = sim.StateSpace("eig_funcs", A, B, input_handle=control_law) # simulate t, q = sim.simulate_state_space(ss_modal, initial_weights, dt) eval_d = sim.evaluate_approximation("eig_funcs", q, t, dz) x_0t = eval_d.output_data[:, 0] yc, tc = tr.gevrey_tanh(T, 1) x_0t_desired = np.interp(t, tc, yc[0, :]) self.assertLess(np.average((x_0t - x_0t_desired)**2), 1e-4) # display results if show_plots: win1 = vis.PgAnimatedPlot([eval_d], title="Test") win2 = vis.PgSurfacePlot(eval_d) app.exec_()
def test_it(self): # original system parameters a2 = 1 a1 = 0 # attention: only a2 = 1., a1 =0 supported in this test case a0 = 0 param = [a2, a1, a0, None, None] # target system parameters (controller parameters) a1_t = 0 a0_t = 0 # attention: only a2 = 1., a1 =0 and a0 =0 supported in this test case param_t = [a2, a1_t, a0_t, None, None] # system/simulation parameters actuation_type = 'dirichlet' bound_cond_type = 'dirichlet' l = 1. spatial_disc = 10 dz = sim.Domain(bounds=(0, l), num=spatial_disc) T = 1. temporal_disc = 1e2 dt = sim.Domain(bounds=(0, T), num=temporal_disc) n = 10 # eigenvalues /-functions original system eig_freq = np.array([(i + 1) * np.pi / l for i in range(n)]) eig_values = a0 - a2 * eig_freq**2 - a1**2 / 4. / a2 norm_fac = np.ones(eig_freq.shape) * np.sqrt(2) eig_funcs = np.asarray([ ef.SecondOrderDirichletEigenfunction(eig_freq[i], param, dz.bounds, norm_fac[i]) for i in range(n) ]) register_base("eig_funcs", eig_funcs, overwrite=True) # eigenfunctions target system eig_freq_t = np.sqrt(-eig_values.astype(complex)) norm_fac_t = norm_fac * eig_freq / eig_freq_t eig_funcs_t = np.asarray([ ef.SecondOrderDirichletEigenfunction(eig_freq_t[i], param_t, dz.bounds, norm_fac_t[i]) for i in range(n) ]) register_base("eig_funcs_t", eig_funcs_t, overwrite=True) # derive initial field variable x(z,0) and weights start_state = cr.Function(lambda z: 0., domain=(0, l)) initial_weights = cr.project_on_base(start_state, eig_funcs) # init trajectory / input of target system traj = tr.RadTrajectory(l, T, param_t, bound_cond_type, actuation_type) # init controller x_at_1 = ph.FieldVariable("eig_funcs", location=1) xt_at_1 = ph.FieldVariable("eig_funcs_t", weight_label="eig_funcs", location=1) controller = ct.Controller( ct.ControlLaw( [ph.ScalarTerm(x_at_1, 1), ph.ScalarTerm(xt_at_1, -1)])) # input with feedback control_law = sim.SimulationInputSum([traj, controller]) # determine (A,B) with modal-transfomation A = np.diag(eig_values) B = -a2 * np.array([eig_funcs[i].derive()(l) for i in range(n)]) ss = sim.StateSpace("eig_funcs", A, B, input_handle=control_law) # simulate t, q = sim.simulate_state_space(ss, initial_weights, dt) eval_d = sim.evaluate_approximation("eig_funcs", q, t, dz) x_0t = eval_d.output_data[:, 0] yc, tc = tr.gevrey_tanh(T, 1) x_0t_desired = np.interp(t, tc, yc[0, :]) self.assertLess(np.average((x_0t - x_0t_desired)**2), 0.5) # display results if show_plots: eval_d = sim.evaluate_approximation("eig_funcs", q, t, dz) win2 = vis.PgSurfacePlot(eval_d) app.exec_()
def test_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))