def test_derivative_elimination(self):
"""
Test that state derivatives are not eliminated even if causalized in torn blocks.
"""
cost_ref = 88.740458
u_norm_ref = 4.242559212
op_manual = self.op_der_loop_manual
op_automatic = self.op_der_loop_automatic
# Eliminate
blt_op_manual = BLTOptimizationProblem(op_manual)
blt_op_automatic = BLTOptimizationProblem(op_automatic)
# Check remaining variables
var_manual = sorted([
var.getName()
for var in blt_op_manual.getVariables(blt_op_manual.REAL_ALGEBRAIC)
if not var.isAlias()
])
var_automatic = sorted([
var.getName() for var in blt_op_automatic.getVariables(
blt_op_automatic.REAL_ALGEBRAIC) if not var.isAlias()
])
assert len(var_manual) == 2
assert len(var_automatic) == 1
# Optimize and check result
res_manual = blt_op_manual.optimize()
res_automatic = blt_op_automatic.optimize()
assert_results(res_manual, cost_ref, u_norm_ref, u_norm_rtol=1e-2)
assert_results(res_automatic, cost_ref, u_norm_ref, u_norm_rtol=1e-2)
def test_automatic_tearing(self):
"""
Test consistency between with and without automatic tearing.
"""
cost_ref = 1.06183273
u_norm_ref = 0.398219663
# Perform elimination
blt_op_automatic = BLTOptimizationProblem(self.op_illust_automatic)
blt_op_manual = BLTOptimizationProblem(self.op_illust_manual)
# Check remaining variables
var_automatic = sorted([
var.getName() for var in blt_op_automatic.getVariables(
blt_op_automatic.REAL_ALGEBRAIC) if not var.isAlias()
])
var_manual = sorted([
var.getName()
for var in blt_op_manual.getVariables(blt_op_manual.REAL_ALGEBRAIC)
if not var.isAlias()
])
assert len(var_automatic) == 3
assert len(var_manual) == 4
# Optimize and check result
res_automatic = blt_op_automatic.optimize()
res_manual = blt_op_manual.optimize()
assert_results(res_automatic, cost_ref, u_norm_ref, u_norm_rtol=1e-2)
assert_results(res_manual, cost_ref, u_norm_ref, u_norm_rtol=1e-2)
def test_closed_form(self):
"""
Test creation of closed form expressions when not using tearing.
"""
# Perform elimination
elim_opts = EliminationOptions()
elim_opts['inline_solved'] = True
elim_opts['closed_form'] = True
blt_op = BLTOptimizationProblem(self.op_illust_manual, elim_opts)
for res in blt_op.getDaeResidual():
if 'y1)*y2)*' in res.getRepresentation():
residual = res.getRepresentation()
N.testing.assert_string_equal(residual,
"SX(((((2*y1)*y2)*sqrt(y5))-sqrt(x1)))")
def test_ineliminable(self):
"""
Test ineliminable variables for both manual and automatic tearing.
"""
cost_ref = 1.08181340
u_norm_ref = 0.399868319
# Mark bounded varible as ineliminable
elim_opts = EliminationOptions()
elim_opts['ineliminable'] = ['y1']
# Manual tearing
op_manual = self.op_illust_manual_bound
op_manual.getVariable('y3').setTearing(True)
for eq in op_manual.getDaeEquations():
if 'y1)*y2)*y4)' in eq.getResidual().getRepresentation():
eq.setTearing(True)
blt_op_manual = BLTOptimizationProblem(op_manual, elim_opts)
# Automatic and manual tearing
op_hybrid = self.op_illust_automatic_bound
op_hybrid.getVariable('y1').setTearing(False)
for eq in op_hybrid.getDaeEquations():
if '(y1*y4)+sqrt(y3)' in eq.getResidual().getRepresentation():
eq.setTearing(False)
blt_op_hybrid = BLTOptimizationProblem(op_hybrid, elim_opts)
# Check remaining variables
var_manual = sorted([
var.getName()
for var in blt_op_manual.getVariables(blt_op_manual.REAL_ALGEBRAIC)
if not var.isAlias()
])
var_hybrid = sorted([
var.getName()
for var in blt_op_hybrid.getVariables(blt_op_hybrid.REAL_ALGEBRAIC)
if not var.isAlias()
])
assert len(var_manual) == 3
assert len(var_hybrid) == 3
# Optimize and check result
res_manual = blt_op_manual.optimize()
res_hybrid = blt_op_hybrid.optimize()
assert_results(res_manual, cost_ref, u_norm_ref, u_norm_rtol=1e-2)
assert_results(res_hybrid, cost_ref, u_norm_ref, u_norm_rtol=1e-2)
# Reset tearing choices
op_manual.getVariable('y3').setTearing(False)
for eq in op_manual.getDaeEquations():
if 'y1)*y2)*y4)' in eq.getResidual().getRepresentation():
eq.setTearing(False)
op_hybrid.getVariable('y1').setTearing(True)
for eq in op_hybrid.getDaeEquations():
if '(y1*y4)+sqrt(y3)' in eq.getResidual().getRepresentation():
eq.setTearing(True)
def test_linear_tearing(self):
"""
Test solution of torn linear blocks both symbolically with SX and numerically with MX.
Numerical part of the test has been disabled, see #5208.
"""
cost_ref = 2.5843277
u_norm_ref = 0.647282415
# Select linear solver
nonlinear_opts = EliminationOptions()
nonlinear_opts['solve_blocks'] = False
symbolic_opts = EliminationOptions()
symbolic_opts['solve_blocks'] = True
symbolic_opts['solve_torn_linear_blocks'] = True
symbolic_opts['linear_solver'] = "symbolicqr"
numeric_opts = EliminationOptions()
numeric_opts['solve_blocks'] = True
numeric_opts['solve_torn_linear_blocks'] = True
numeric_opts['linear_solver'] = "lapackqr"
numeric_op_opts = {'expand_to_sx': 'no'}
# Eliminate
op = self.op_loop_automatic
blt_op_nonlinear = BLTOptimizationProblem(op, nonlinear_opts)
blt_op_symbolic = BLTOptimizationProblem(op, symbolic_opts)
#~ blt_op_numeric = BLTOptimizationProblem(op, numeric_opts)
# Check remaining variables
var_nonlinear = sorted([
var.getName() for var in blt_op_nonlinear.getVariables(
blt_op_nonlinear.REAL_ALGEBRAIC) if not var.isAlias()
])
var_symbolic = sorted([
var.getName() for var in blt_op_symbolic.getVariables(
blt_op_symbolic.REAL_ALGEBRAIC) if not var.isAlias()
])
#~ var_numeric = sorted([var.getName() for var in blt_op_numeric.getVariables(blt_op_numeric.REAL_ALGEBRAIC)
#~ if not var.isAlias()])
assert len(var_nonlinear) == 1
assert len(var_symbolic) == 0
#~ assert len(var_numeric) == 0
# Optimize and check result
res_nonlinear = blt_op_nonlinear.optimize()
res_symbolic = blt_op_symbolic.optimize()
#~ res_numeric = blt_op_numeric.optimize(options=numeric_op_opts)
assert_results(res_nonlinear, cost_ref, u_norm_ref, u_norm_rtol=1e-2)
assert_results(res_symbolic, cost_ref, u_norm_ref, u_norm_rtol=1e-2)
def test_constraint_and_objective(self):
"""
Test eliminating variables occurring in constraints and objective.
"""
cost_ref = 1.08181340
u_norm_ref = 0.399868319
# Mark bounded varible as ineliminable for reference
inelim_opts = EliminationOptions()
inelim_opts['ineliminable'] = ['y1']
elim_opts = EliminationOptions()
# Manual tearing
op = self.op_illust_manual_constraint
op.getVariable('y3').setTearing(True)
for eq in op.getDaeEquations():
if 'y1)*y2)*y4)' in eq.getResidual().getRepresentation():
eq.setTearing(True)
blt_op_inelim = BLTOptimizationProblem(op, inelim_opts)
blt_op_elim = BLTOptimizationProblem(op, elim_opts)
# Check remaining variables
var_inelim = sorted([
var.getName()
for var in blt_op_inelim.getVariables(blt_op_inelim.REAL_ALGEBRAIC)
if not var.isAlias()
])
var_elim = sorted([
var.getName()
for var in blt_op_elim.getVariables(blt_op_elim.REAL_ALGEBRAIC)
if not var.isAlias()
])
assert len(var_inelim) == 3
assert len(var_elim) == 2
# Optimize and check result
res_inelim = blt_op_inelim.optimize()
res_elim = blt_op_elim.optimize()
assert_results(res_inelim, cost_ref, u_norm_ref, u_norm_rtol=1e-2)
assert_results(res_elim, cost_ref, u_norm_ref, u_norm_rtol=1e-2)
# Reset tearing choices
op.getVariable('y3').setTearing(False)
for eq in op.getDaeEquations():
if 'y1)*y2)*y4)' in eq.getResidual().getRepresentation():
eq.setTearing(False)
def test_par_est(self):
"""
Test parameter estimation.
"""
cost_ref = 1.1109779e-2
u_norm_ref = 0.556018602
model = self.model_illust
op = self.op_illust_est_automatic
# Simulate with nominal parameters
n_meas = 16
sim_res = model.simulate(input=('u', lambda t: -0.2 + N.sin(t)),
final_time=4.,
options={
'ncp': n_meas,
'CVode_options': {
'rtol': 1e-10
}
})
# Assemble external data by adding noise to simulation
Q = N.diag([1., 1.])
N.random.seed(1)
t_meas = sim_res['time']
sigma = 0.05
x1_meas = sim_res['x1'] + sigma * N.random.randn(n_meas + 1)
x2_meas = sim_res['x2'] + sigma * N.random.randn(n_meas + 1)
u_meas = sim_res['u']
data_x1 = N.vstack([t_meas, x1_meas])
data_x2 = N.vstack([t_meas, x2_meas])
data_u1 = N.vstack([t_meas, u_meas])
quad_pen = OrderedDict()
quad_pen['x1'] = data_x1
quad_pen['x2'] = data_x2
eliminated = OrderedDict()
eliminated['u'] = data_u1
external_data = ExternalData(Q=Q,
quad_pen=quad_pen,
eliminated=eliminated)
# Eliminate
blt_op = BLTOptimizationProblem(op)
# Check remaining variables
alg_vars = sorted([
var.getName() for var in blt_op.getVariables(blt_op.REAL_ALGEBRAIC)
if not var.isAlias()
])
assert len(alg_vars) == 3
# Set up options
dae_opts = op.optimize_options()
dae_opts['init_traj'] = sim_res
dae_opts['nominal_traj'] = sim_res
dae_opts['external_data'] = external_data
blt_opts = blt_op.optimize_options()
blt_opts['init_traj'] = sim_res
blt_opts['nominal_traj'] = sim_res
blt_opts['external_data'] = external_data
# Optimize and check result
res_dae = op.optimize(options=dae_opts)
res_blt = blt_op.optimize(options=blt_opts)
assert_results(res_dae, cost_ref, u_norm_ref, u_norm_rtol=1e-2)
assert_results(res_blt, cost_ref, u_norm_ref, u_norm_rtol=1e-2)
N.testing.assert_allclose([res_dae['p1'][0], res_dae['p3'][0]],
[2.022765, 0.992965],
rtol=2e-3)
N.testing.assert_allclose([res_blt['p1'][0], res_blt['p3'][0]],
[2.022765, 0.992965],
rtol=2e-3)
def test_sparsity_preservation(self):
"""
Test sparsity preservation for both LMFI and Markowitz.
"""
cost_ref = 1.06183273
u_norm_ref = 0.398219663
# Specify sparsity preservation
elim_opts_lmfi = EliminationOptions()
elim_opts_lmfi['dense_tol'] = 2
elim_opts_lmfi['dense_measure'] = "lmfi"
elim_opts_mrkwtz = EliminationOptions()
elim_opts_mrkwtz['dense_tol'] = 4
elim_opts_mrkwtz['dense_measure'] = "Markowitz"
elim_opts_mrkwtz2 = EliminationOptions()
elim_opts_mrkwtz2['dense_tol'] = 2
elim_opts_mrkwtz2['dense_measure'] = "Markowitz"
# Manual tearing
op = self.op_illust_manual
op.getVariable('y3').setTearing(True)
for eq in op.getDaeEquations():
if 'y1)*y2)*y4)' in eq.getResidual().getRepresentation():
eq.setTearing(True)
blt_op_lmfi = BLTOptimizationProblem(op, elim_opts_lmfi)
blt_op_mrkwtz = BLTOptimizationProblem(op, elim_opts_mrkwtz)
blt_op_mrkwtz2 = BLTOptimizationProblem(op, elim_opts_mrkwtz2)
# Check remaining variables
var_lmfi = sorted([
var.getName()
for var in blt_op_lmfi.getVariables(blt_op_lmfi.REAL_ALGEBRAIC)
if not var.isAlias()
])
var_mrkwtz = sorted([
var.getName()
for var in blt_op_mrkwtz.getVariables(blt_op_mrkwtz.REAL_ALGEBRAIC)
if not var.isAlias()
])
var_mrkwtz2 = sorted([
var.getName() for var in blt_op_mrkwtz2.getVariables(
blt_op_mrkwtz.REAL_ALGEBRAIC) if not var.isAlias()
])
assert len(var_lmfi) == 3
assert len(var_mrkwtz) == 3
assert len(var_mrkwtz2) == 4
# Optimize and check result
res_lmfi = blt_op_lmfi.optimize()
res_mrkwtz = blt_op_mrkwtz.optimize()
res_mrkwtz2 = blt_op_mrkwtz2.optimize()
assert_results(res_lmfi, cost_ref, u_norm_ref, u_norm_rtol=1e-2)
assert_results(res_mrkwtz, cost_ref, u_norm_ref, u_norm_rtol=1e-2)
assert_results(res_mrkwtz2, cost_ref, u_norm_ref, u_norm_rtol=1e-2)
# Reset tearing choices
op.getVariable('y3').setTearing(False)
for eq in op.getDaeEquations():
if 'y1)*y2)*y4)' in eq.getResidual().getRepresentation():
eq.setTearing(False)
def test_hybrid_tearing(self):
"""
Test consistency between manual and automatic tearing as well as the combination of both.
"""
cost_ref = 1.06183273
u_norm_ref = 0.398219663
# Automatic tearing
blt_op_automatic = BLTOptimizationProblem(self.op_illust_automatic)
# Manual tearing
op_manual = self.op_illust_manual
op_manual.getVariable('y3').setTearing(True)
for eq in op_manual.getDaeEquations():
if 'y1)*y2)*y4)' in eq.getResidual().getRepresentation():
eq.setTearing(True)
blt_op_manual = BLTOptimizationProblem(op_manual)
# Automatic and manual tearing
op_hybrid = self.op_illust_automatic
op_hybrid.getVariable('y1').setTearing(False)
for eq in op_hybrid.getDaeEquations():
if '(y1*y4)+sqrt(y3)' in eq.getResidual().getRepresentation():
eq.setTearing(False)
blt_op_hybrid = BLTOptimizationProblem(op_hybrid)
# Check remaining variables
var_automatic = sorted([
var.getName() for var in blt_op_automatic.getVariables(
blt_op_automatic.REAL_ALGEBRAIC) if not var.isAlias()
])
var_manual = sorted([
var.getName()
for var in blt_op_manual.getVariables(blt_op_manual.REAL_ALGEBRAIC)
if not var.isAlias()
])
var_hybrid = sorted([
var.getName()
for var in blt_op_hybrid.getVariables(blt_op_hybrid.REAL_ALGEBRAIC)
if not var.isAlias()
])
assert len(var_automatic) == 3
assert len(var_manual) == 2
assert len(var_hybrid) == 2
# Optimize and check result
res_automatic = blt_op_automatic.optimize()
res_manual = blt_op_manual.optimize()
res_hybrid = blt_op_hybrid.optimize()
assert_results(res_automatic, cost_ref, u_norm_ref, u_norm_rtol=1e-2)
assert_results(res_manual, cost_ref, u_norm_ref, u_norm_rtol=1e-2)
assert_results(res_hybrid, cost_ref, u_norm_ref, u_norm_rtol=1e-2)
# Reset tearing choices
op_manual.getVariable('y3').setTearing(False)
for eq in op_manual.getDaeEquations():
if 'y1)*y2)*y4)' in eq.getResidual().getRepresentation():
eq.setTearing(False)
op_hybrid.getVariable('y1').setTearing(True)
for eq in op_hybrid.getDaeEquations():
if '(y1*y4)+sqrt(y3)' in eq.getResidual().getRepresentation():
eq.setTearing(True)
def run_demo(with_plots=True):
"""
This example solves the optimization problem from pyjmi.examples.ccpp using the Python-based symbolic elimination.
"""
# Load initial gues
init_path = os.path.join(get_files_path(), "ccpp_init.txt")
init_res = LocalDAECollocationAlgResult(
result_data=ResultDymolaTextual(init_path))
# Compile model
class_name = "CombinedCycleStartup.Startup6"
file_paths = (os.path.join(get_files_path(), "CombinedCycle.mo"),
os.path.join(get_files_path(), "CombinedCycleStartup.mop"))
compiler_options = {'equation_sorting': True, 'automatic_tearing': True}
op = transfer_optimization_problem("CombinedCycleStartup.Startup6",
file_paths, compiler_options)
# Set elimination options
elim_opts = EliminationOptions()
elim_opts['ineliminable'] = ['plant.sigma']
if with_plots:
elim_opts['draw_blt'] = True
# Eliminate algebraic variables
op = BLTOptimizationProblem(op, elim_opts)
# Set collocation options
opt_opts = op.optimize_options()
opt_opts['init_traj'] = init_res
opt_opts['nominal_traj'] = init_res
# Solve the optimal control problem
opt_res = op.optimize(options=opt_opts)
# Extract variable profiles
opt_plant_p = opt_res['plant.p']
opt_plant_sigma = opt_res['plant.sigma']
opt_plant_load = opt_res['plant.load']
opt_time = opt_res['time']
# Plot the optimized trajectories
if with_plots:
plt.close(2)
plt.figure(2)
plt.subplot(3, 1, 1)
plt.plot(opt_time, opt_plant_p * 1e-6)
plt.ylabel('evaporator pressure [MPa]')
plt.grid(True)
plt.title('Optimized trajectories')
plt.subplot(3, 1, 2)
plt.plot(opt_time, opt_plant_sigma * 1e-6)
plt.grid(True)
plt.ylabel('turbine thermal stress [MPa]')
plt.subplot(3, 1, 3)
plt.plot(opt_time, opt_plant_load)
plt.grid(True)
plt.ylabel('input load [1]')
plt.xlabel('time [s]')
# Verify solution for testing purposes
try:
import casadi
except:
pass
else:
cost = float(opt_res.solver.solver_object.getOutput('f'))
N.testing.assert_allclose(cost, 17492.465548193624, rtol=1e-5)
def run_demo(with_plots=True):
"""
Demonstrate symbolic elimination.
"""
# Compile and load optimization problem
file_path = os.path.join(get_files_path(), "JMExamples_opt.mop")
compiler_options = {'equation_sorting': True, 'automatic_tearing': True}
op = transfer_optimization_problem("JMExamples_opt.EliminationExample",
file_path, compiler_options)
# Set up and perform elimination
elim_opts = EliminationOptions()
elim_opts['ineliminable'] = [
'y1'
] # Provide list of variable names to not eliminate
# Variables with any of the following properties are recommended to mark as ineliminable:
# Potentially active bounds
# Occurring in the objective or constraints
# Numerically unstable pivots (difficult to predict)
if with_plots:
elim_opts['draw_blt'] = True
elim_opts['draw_blt_strings'] = True
op = BLTOptimizationProblem(op, elim_opts)
# Optimize and extract solution
res = op.optimize()
x1 = res['x1']
x2 = res['x2']
y1 = res['y1']
u = res['u']
time = res['time']
# Plot
if with_plots:
plt.figure(1)
plt.clf()
plt.subplot(4, 1, 1)
plt.plot(time, x1)
plt.grid()
plt.ylabel('x1')
plt.subplot(4, 1, 2)
plt.plot(time, x2)
plt.grid()
plt.ylabel('x2')
plt.subplot(4, 1, 3)
plt.plot(time, y1)
plt.grid()
plt.ylabel('y1')
plt.subplot(4, 1, 4)
plt.plot(time, u)
plt.grid()
plt.ylabel('u')
plt.xlabel('time')
plt.show()
# Verify solution for testing purposes
try:
import casadi
except:
pass
else:
cost = float(res.solver.solver_object.getOutput('f'))
N.testing.assert_allclose(cost, 1.0818134, rtol=1e-4)
def run_demo(with_plots=True, use_ma57=True):
"""
This example is based on a single-track model of a car with tire dynamics.
The optimization problem is to minimize the duration of a 90-degree turn
with actuators on the steer angle and front and rear wheel torques.
This example also demonstrates the usage of blocking factors, to enforce
piecewise constant inputs.
This example needs one of the linear solvers MA27 or MA57 to work.
The precense of MA27 or MA57 is not detected in the example, so if only
MA57 is present, then True must be passed in the use_ma57 argument.
"""
# Set up optimization
mop_path = os.path.join(get_files_path(), "vehicle_turn.mop")
op = transfer_optimization_problem('Turn', mop_path)
opts = op.optimize_options()
opts['IPOPT_options']['linear_solver'] = "ma57" if use_ma57 else "ma27"
opts['IPOPT_options']['tol'] = 1e-9
opts['n_e'] = 60
# Set blocking factors
factors = {
'delta_u': opts['n_e'] / 2 * [2],
'Twf_u': opts['n_e'] / 4 * [4],
'Twr_u': opts['n_e'] / 4 * [4]
}
rad2deg = 180. / (2 * N.pi)
du_bounds = {'delta_u': 2. / rad2deg}
bf = BlockingFactors(factors, du_bounds=du_bounds)
opts['blocking_factors'] = bf
# Use Dymola simulation result as initial guess
init_path = os.path.join(get_files_path(), "vehicle_turn_dymola.txt")
init_guess = ResultDymolaTextual(init_path)
opts['init_traj'] = init_guess
# Symbolic elimination
op = BLTOptimizationProblem(op)
# Solve optimization problem
res = op.optimize(options=opts)
# Extract solution
time = res['time']
X = res['car.X']
Y = res['car.Y']
delta = res['delta_u']
Twf = res['Twf_u']
Twr = res['Twr_u']
Ri = op.get('Ri')
Ro = op.get('Ro')
# Verify result
N.testing.assert_allclose(time[-1], 4.0118, rtol=5e-3)
# Plot solution
if with_plots:
# Plot road
plt.close(1)
plt.figure(1)
plt.plot(X, Y, 'b')
xi = N.linspace(0., Ri, 100)
xo = N.linspace(0., Ro, 100)
yi = (Ri**8 - xi**8)**(1. / 8.)
yo = (Ro**8 - xo**8)**(1. / 8.)
plt.plot(xi, yi, 'r--')
plt.plot(xo, yo, 'r--')
plt.xlabel('X [m]')
plt.ylabel('Y [m]')
plt.legend(['position', 'road'], loc=3)
# Plot inputs
plt.close(2)
plt.figure(2)
plt.plot(time, delta * rad2deg, drawstyle='steps-post')
plt.plot(time, Twf * 1e-3, drawstyle='steps-post')
plt.plot(time, Twr * 1e-3, drawstyle='steps-post')
plt.xlabel('time [s]')
plt.legend(['delta [deg]', 'Twf [kN]', 'Twr [kN]'], loc=4)
plt.show()
def run_demo(with_plots=True):
"""
This example is based on the multibody mechanics double pendulum example from the Modelica Standard Library (MSL).
The MSL example has been modified by adding a torque on the first revolute joint of the pendulum as a top-level
input. The considered optimization problem is to invert both pendulum bodies with bounded torque.
This example needs linear solver MA27 to work.
"""
# Simulate system with linear state feedback to generate initial guess
file_paths = (os.path.join(get_files_path(), "DoublePendulum.mo"),
os.path.join(get_files_path(), "DoublePendulum.mop"))
comp_opts = {'inline_functions': 'all', 'dynamic_states': False,
'expose_temp_vars_in_fmu': True, 'equation_sorting': True, 'automatic_tearing': True}
init_fmu = load_fmu(compile_fmu("DoublePendulum.Feedback", file_paths, compiler_options=comp_opts))
init_res = init_fmu.simulate(final_time=3., options={'CVode_options': {'rtol': 1e-10}})
# Set up optimization
op = transfer_optimization_problem('Opt', file_paths, compiler_options=comp_opts)
opts = op.optimize_options()
opts['IPOPT_options']['linear_solver'] = "ma27"
opts['n_e'] = 100
opts['init_traj'] = init_res
opts['nominal_traj'] = init_res
# Symbolic elimination
op = BLTOptimizationProblem(op)
# Solve optimization problem
res = op.optimize(options=opts)
# Extract solution
time = res['time']
phi1 = res['pendulum.revolute1.phi']
phi2 = res['pendulum.revolute2.phi']
u = res['u']
# Verify solution for testing purposes
try:
import casadi
except:
pass
else:
cost = float(res.solver.solver_object.output(casadi.NLP_SOLVER_F))
N.testing.assert_allclose(cost, 9.632883808252522, rtol=5e-3)
# Plot solution
if with_plots:
plt.close(1)
plt.figure(1)
plt.subplot(2, 1, 1)
plt.plot(time, phi1, 'b')
plt.plot(time, phi2, 'r')
plt.legend(['$\phi_1$', '$\phi_2$'])
plt.ylabel('$\phi$')
plt.xlabel('$t$')
plt.grid()
plt.subplot(2, 1, 2)
plt.plot(time, u)
plt.ylabel('$u$')
plt.xlabel('$t$')
plt.grid()
plt.show()