Dymos: Open Source Optimization of Dynamic Multidisciplinary Systems

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.

Installation

pip install git+https://github.com/OpenMDAO/dymos.git

Documentation

Online documentation is available at https://openmdao.github.io/dymos/

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 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_time(units='s')
@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
        self.add_input('v',
                       val=np.zeros(nn),
                       desc='velocity',
                       units='m/s')

        self.add_input('g',
                       val=9.80665*np.ones(nn),
                       desc='gravitational acceleration',
                       units='m/s/s')

        self.add_input('theta',
                       val=np.zeros(nn),
                       desc='angle of wire',
                       units='rad')

        self.add_output('xdot',
                        val=np.zeros(nn),
                        desc='velocity component in x',
                        units='m/s')

        self.add_output('ydot',
                        val=np.zeros(nn),
                        desc='velocity component in y',
                        units='m/s')

        self.add_output('vdot',
                        val=np.zeros(nn),
                        desc='acceleration magnitude',
                        units='m/s**2')

        self.add_output('check',
                        val=np.zeros(nn),
                        desc='A check on the solution: v/sin(theta) = constant',
                        units='m/s')

        # 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 ScipyODEIntegrator provides an OpenMDAO group which simulates the ODE system it is given. This explicit integration capability can be used to check solutions of the implicit collocation techniques or to generate an initial guess for state-time histories of the implicit collocation.

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 GLM)
• Equations of motion
• Boundary and path constraints

Each dymos Phase is ultimately just an OpenMDAO Group that can exist in a problem along with numerous other groups.