Is there a way to implement equations as Dymos path constraints? - optimization

Question

For example, if I have a function h_max(mach) and I want the altitude to always respect this predefined altitude-mach relationship throughout the flight enveloppe, how could I impliment this?
I have tried calculating the limit quantity (in this case, h_max) as its own state and then calculating another state as h_max-h and then constraining that through a path constraint to being greater than 0. This type of approach has worked, but involved two explicit components, a group and alot of extra coding just to get a constraint working. I was wondering if there was a better way?
Thanks so much in advance.

Answer 1

The next version of Dymos, 1.7.0 will be released soon and will support this.
In the mean time, you can install the latest developmental version of Dymos directly from github to have access to this capability:
python -m pip install git+https://github.com/OpenMDAO/dymos.git
Then, you can define boundary and path constraints with an equation. Note the equation must have an equals sign in it, and then lower, upper, or equals will apply to the result of the equation.
In reality, dymos is just inserting an OpenMDAO ExecComp for you under the hood, so the one caveat to this is that your expression must be compatible with complex-step differentiation.
Here's an example of the brachistochrone that uses constraint expressions to set the final y value to a specific value while satisfying a path constraint defined with a second equation.
import openmdao.api as om
import dymos as dm
from dymos.examples.plotting import plot_results
from dymos.examples.brachistochrone import BrachistochroneODE
import matplotlib.pyplot as plt
#
# Initialize the Problem and the optimization driver
#
p = om.Problem(model=om.Group())
p.driver = om.ScipyOptimizeDriver()
p.driver.declare_coloring()
#
# Create a trajectory and add a phase to it
#
traj = p.model.add_subsystem('traj', dm.Trajectory())
phase = traj.add_phase('phase0',
dm.Phase(ode_class=BrachistochroneODE,
transcription=dm.GaussLobatto(num_segments=10)))
#
# Set the variables
#
phase.set_time_options(fix_initial=True, duration_bounds=(.5, 10))
phase.add_state('x', fix_initial=True, fix_final=True)
phase.add_state('y', fix_initial=True, fix_final=False)
phase.add_state('v', fix_initial=True, fix_final=False)
phase.add_control('theta', continuity=True, rate_continuity=True,
units='deg', lower=0.01, upper=179.9)
phase.add_parameter('g', units='m/s**2', val=9.80665)
Y_FINAL = 5.0
Y_MIN = 5.0
phase.add_boundary_constraint(f'bcf_y = y - {Y_FINAL}', loc='final', equals=0.0)
phase.add_path_constraint(f'path_y = y - {Y_MIN}', lower=0.0)
#
# Minimize time at the end of the phase
#
phase.add_objective('time', loc='final', scaler=10)
p.model.linear_solver = om.DirectSolver()
#
# Setup the Problem
#
p.setup()
#
# Set the initial values
#
p['traj.phase0.t_initial'] = 0.0
p['traj.phase0.t_duration'] = 2.0
p.set_val('traj.phase0.states:x', phase.interp('x', ys=[0, 10]))
p.set_val('traj.phase0.states:y', phase.interp('y', ys=[10, 5]))
p.set_val('traj.phase0.states:v', phase.interp('v', ys=[0, 9.9]))
p.set_val('traj.phase0.controls:theta', phase.interp('theta', ys=[5, 100.5]))
#
# Solve for the optimal trajectory
#
dm.run_problem(p)
# Check the results
print('final time')
print(p.get_val('traj.phase0.timeseries.time')[-1])
p.list_problem_vars()
Note the constraints from the list_problem_vars() call that come from timeseries_exec_comp - this is the OpenMDAO ExecComp that Dymos automatically inserts for you.
--- Constraint Report [traj] ---
--- phase0 ---
[final] 0.0000e+00 == bcf_y [None]
[path] 0.0000e+00 <= path_y [None]
/usr/local/lib/python3.8/dist-packages/openmdao/recorders/sqlite_recorder.py:227: UserWarning:The existing case recorder file, dymos_solution.db, is being overwritten.
Model viewer data has already been recorded for Driver.
Full total jacobian was computed 3 times, taking 0.057485 seconds.
Total jacobian shape: (71, 51)
Jacobian shape: (71, 51) (12.51% nonzero)
FWD solves: 12 REV solves: 0
Total colors vs. total size: 12 vs 51 (76.5% improvement)
Sparsity computed using tolerance: 1e-25
Time to compute sparsity: 0.057485 sec.
Time to compute coloring: 0.054118 sec.
Memory to compute coloring: 0.000000 MB.
/usr/local/lib/python3.8/dist-packages/openmdao/core/total_jac.py:1585: DerivativesWarning:Constraints or objectives [('traj.phases.phase0.timeseries.timeseries_exec_comp.path_y', inds=[(0, 0)])] cannot be impacted by the design variables of the problem.
Optimization terminated successfully (Exit mode 0)
Current function value: [18.02999766]
Iterations: 14
Function evaluations: 14
Gradient evaluations: 14
Optimization Complete
-----------------------------------
final time
[1.80299977]
----------------
Design Variables
----------------
name val size indices
-------------------------- -------------- ---- ---------------------------------------------
traj.phase0.t_duration [1.80299977] 1 None
traj.phase0.states:x |12.14992234| 9 [1 2 3 4 5 6 7 8 9]
traj.phase0.states:y |22.69124774| 10 [ 1 2 3 4 5 6 7 8 9 10]
traj.phase0.states:v |24.46289861| 10 [ 1 2 3 4 5 6 7 8 9 10]
traj.phase0.controls:theta |266.48489386| 21 [ 0 1 2 3 4 5 ... 4 15 16 17 18 19 20]
-----------
Constraints
-----------
name val size indices alias
----------------------------------------------------------- ------------- ---- --------------------------------------------- ----------------------------------------------------
timeseries.timeseries_exec_comp.bcf_y [0.] 1 [29] traj.phases.phase0->final_boundary_constraint->bcf_y
timeseries.timeseries_exec_comp.path_y |15.73297378| 30 [ 0 1 2 3 4 5 ... 3 24 25 26 27 28 29] traj.phases.phase0->path_constraint->path_y
traj.phase0.collocation_constraint.defects:x |6e-08| 10 None None
traj.phase0.collocation_constraint.defects:y |7e-08| 10 None None
traj.phase0.collocation_constraint.defects:v |3e-08| 10 None None
traj.phase0.continuity_comp.defect_control_rates:theta_rate |0.0| 9 None None
----------
Objectives
----------
name val size indices
------------- ------------- ---- -------
traj.phase0.t [18.02999766] 1 -1