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Open Source Optimization of Dynamic Multidisciplinary Systems
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robfalck Merge develop -> 0.14.0 (#211)
* Decorator deprecation (#208)

* Revert "Changed input promotion to avoid warnings in latest OpenMDAO version. (#193)"

This reverts commit 8d763d9.

* moves connections from trajectory to sub phases to configure in preparation for some trajectory parameter and linkage updates.

* removed ode decorators from brachistochrone example

* more brachistochrone examples changed to not use decorators

* finished converting brachistochrone examples to the decorator-free usage

* aircraft example converted to decoratorless format

* more conversion to undecorated odes.  found a coloring bug in the two burn orbit raise linkages test.

* examples have all decorated ODEs removed.  decorators have been deprecated.

* deprecated set_state_options in favor of add_state.

* fixed deprecation of set_state_options

* cleaned up orbit raise tests with SLSQP

* more issues with slsqp and the two burn orbit raise

* updated PETSc version for Python 2.7 checking

* matplotlib issues during documentation build

* Bump version: 0.13.0 → 0.14.0 (#210)
Latest commit ae13bde Aug 14, 2019

Dymos: Open Source Optimization of Dynamic Multidisciplinary Systems

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Dymos is a framework for the simulation and optimization of dynamical systems within the OpenMDAO Multidisciplinary Analysis and Optimization environment. Dymos leverages implicit and explicit simulation techniques to simulate generic dynamic systems of arbitary complexity.

The software has two primary objectives:

  • Provide a generic ODE integration interface that allows for the analysis of dynamical systems.
  • Allow the user to solve optimal control problems involving dynamical multidisciplinary systems.


pip install git+


Online documentation is available at

Defining Ordinary Differential Equations

The first step in simulating or optimizing a dynamical system is to define the ordinary differential equations to be integrated. The user first builds an OpenMDAO model which has outputs that provide the rates of the state variables. This model can be an OpenMDAO model of arbitrary complexity, including nested groups and components, layers of nonlinear solvers, etc.

Next we can wrap our system with decorators that provide information regarding the states to be integrated, which sources in the model provide their rates, and where any externally provided parameters should be connected. When used in an optimal control context, these external parameters may serve as controls.

import numpy as np
from openmdao.api import ExplicitComponent

from dymos import declare_time, declare_state, declare_parameter

@declare_state('x', rate_source='xdot', units='m')
@declare_state('y', rate_source='ydot', units='m')
@declare_state('v', rate_source='vdot', targets=['v'], units='m/s')
@declare_parameter('theta', targets=['theta'])
@declare_parameter('g', units='m/s**2', targets=['g'])
class BrachistochroneEOM(ExplicitComponent):

    def initialize(self):
        self.metadata.declare('num_nodes', types=int)

    def setup(self):
        nn = self.metadata['num_nodes']

        # Inputs

                       desc='gravitational acceleration',

                       desc='angle of wire',

                        desc='velocity component in x',

                        desc='velocity component in y',

                        desc='acceleration magnitude',

                        desc='A check on the solution: v/sin(theta) = constant',

        # Setup partials
        arange = np.arange(self.metadata['num_nodes'])

        self.declare_partials(of='vdot', wrt='g', rows=arange, cols=arange, val=1.0)
        self.declare_partials(of='vdot', wrt='theta', rows=arange, cols=arange, val=1.0)

        self.declare_partials(of='xdot', wrt='v', rows=arange, cols=arange, val=1.0)
        self.declare_partials(of='xdot', wrt='theta', rows=arange, cols=arange, val=1.0)

        self.declare_partials(of='ydot', wrt='v', rows=arange, cols=arange, val=1.0)
        self.declare_partials(of='ydot', wrt='theta', rows=arange, cols=arange, val=1.0)

        self.declare_partials(of='check', wrt='v', rows=arange, cols=arange, val=1.0)
        self.declare_partials(of='check', wrt='theta', rows=arange, cols=arange, val=1.0)

    def compute(self, inputs, outputs):
        theta = inputs['theta']
        cos_theta = np.cos(theta)
        sin_theta = np.sin(theta)
        g = inputs['g']
        v = inputs['v']

        outputs['vdot'] = g*cos_theta
        outputs['xdot'] = v*sin_theta
        outputs['ydot'] = -v*cos_theta
        outputs['check'] = v/sin_theta

    def compute_partials(self, inputs, jacobian):
        theta = inputs['theta']
        cos_theta = np.cos(theta)
        sin_theta = np.sin(theta)
        g = inputs['g']
        v = inputs['v']

        jacobian['vdot', 'g'] = cos_theta
        jacobian['vdot', 'theta'] = -g*sin_theta

        jacobian['xdot', 'v'] = sin_theta
        jacobian['xdot', 'theta'] = v*cos_theta

        jacobian['ydot', 'v'] = -cos_theta
        jacobian['ydot', 'theta'] = v*sin_theta

        jacobian['check', 'v'] = 1/sin_theta
        jacobian['check', 'theta'] = -v*cos_theta/sin_theta**2

Integrating Ordinary Differential Equations

Dymos's RungeKutta and solver-based pseudspectral transcriptions provide the ability to numerically integrate the ODE system it is given. Used in an optimal control context, these provide a shooting method in which each iteration provides a physically viable trajectory.

Pseudospectral Methods

dymos currently supports the Radau Pseudospectral Method and high-order Gauss-Lobatto transcriptions. These implicit techniques rely on the optimizer to impose "defect" constraints which enforce the physical accuracy of the resulting trajectories. To verify the physical accuracy of the solutions, Dymos can explicitly integrate them using variable-step methods.

Solving Optimal Control Problems

dymos uses the concept of phases to support optimal control of dynamical systems. Users connect one or more phases to construct trajectories. Each phase can have its own:

  • Optimal Control Transcription (Gauss-Lobatto, Radau Pseudospectral, or RungeKutta)
  • Equations of motion
  • Boundary and path constraints

dymos Phases and Trajectories are ultimately just OpenMDAO Groups that can exist in a problem along with numerous other models, allowing for the simultaneous optimization of systems and dynamics.

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