The following cell implements a solution to a first-order linear model in the form
\begin{align} \tau\frac{dy}{dt} + y & = K u(t) \\ \end{align}where $\tau$ and $K$ are model parameters, and $u(t)$ is an external process input.
% matplotlib inline
from pyomo.environ import *
from pyomo.dae import *
import matplotlib.pyplot as plt
tf = 10
tau = 1
K = 5
# define u(t)
u = lambda t: 0 if t < 5 else 1
# create a model object
model = ConcreteModel()
# define the independent variable
model.t = ContinuousSet(bounds=(0, tf))
# define the dependent variables
model.y = Var(model.t)
model.dydt = DerivativeVar(model.y)
# fix the initial value of y
model.y[0].fix(0)
# define the differential equation as a constraint
model.ode = Constraint(model.t,
rule=lambda model, t: tau*model.dydt[t] + model.y[t] == K*u(t))
# transform dae model to discrete optimization problem
TransformationFactory('dae.finite_difference').apply_to(model, nfe=50, method='BACKWARD')
# solve the model
SolverFactory('ipopt').solve(model).write()
# access elements of a ContinuousSet object
t = [t for t in model.t]
# access elements of a Var object
y = [model.y[t]() for t in model.y]
plt.plot(t,y)
plt.xlabel('time / sec')
plt.ylabel('response')
plt.title('Response of a linear first-order ODE')
# ========================================================== # = Solver Results = # ========================================================== # ---------------------------------------------------------- # Problem Information # ---------------------------------------------------------- Problem: - Lower bound: -inf Upper bound: inf Number of objectives: 1 Number of constraints: 101 Number of variables: 101 Sense: unknown # ---------------------------------------------------------- # Solver Information # ---------------------------------------------------------- Solver: - Status: ok Message: Ipopt 3.12.8\x3a Optimal Solution Found Termination condition: optimal Id: 0 Error rc: 0 Time: 0.024537086486816406 # ---------------------------------------------------------- # Solution Information # ---------------------------------------------------------- Solution: - number of solutions: 0 number of solutions displayed: 0
Text(0.5,1,'Response of a linear first-order ODE')
