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+https://github.com/OpenMDAO/dymos.git
Online documentation is available at https://openmdao.github.io/dymos/
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_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
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.
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.
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.