https://github.com/Pyomo/pyomo/blob/master/examples/dae/Path_Constraint.py
#
# Pyomo: Python Optimization Modeling Objects
# Copyright 2017 National Technology and Engineering Solutions of Sandia, LLC
# Under the terms of Contract DE-NA0003525 with National Technology and
# Engineering Solutions of Sandia, LLC, the U.S. Government retains certain
# rights in this software.
# This software is distributed under the 3-clause BSD License.
# Sample Problem 3: Inequality State Path Constraint
# (Ex 4 from Dynopt Guide)
#
# min x3(tf)
# s.t. X1_dot = X2 X1(0) = 0
# X2_dot = -X2+u X2(0) = -1
# X3_dot = X1^2+x2^2+0.005*u^2 X3(0) = 0
# X2-8*(t-0.5)^2+0.5 <= 0
# tf = 1
#
from pyomo.environ import *
from pyomo.dae import *
m = ConcreteModel()
m.tf = Param(initialize=1)
m.t = ContinuousSet(bounds=(0,m.tf))
m.u = Var(m.t, initialize=0)
m.x1 = Var(m.t)
m.x2 = Var(m.t)
m.x3 = Var(m.t)
m.dx1 = DerivativeVar(m.x1, wrt=m.t)
m.dx2 = DerivativeVar(m.x2, wrt=m.t)
m.dx3 = DerivativeVar(m.x3, wrt=m.t)
m.obj = Objective(expr=m.x3[m.tf])
def _x1dot(m, t):
if t == 0:
return Constraint.Skip
return m.dx1[t] == m.x2[t]
m.x1dotcon = Constraint(m.t, rule=_x1dot)
def _x2dot(m, t):
if t == 0:
return Constraint.Skip
return m.dx2[t] == -m.x2[t]+m.u[t]
m.x2dotcon = Constraint(m.t, rule=_x2dot)
def _x3dot(m, t):
if t == 0:
return Constraint.Skip
return m.dx3[t] == m.x1[t]**2+m.x2[t]**2+0.005*m.u[t]**2
m.x3dotcon = Constraint(m.t, rule=_x3dot)
def _con(m, t):
return m.x2[t]-8*(t-0.5)**2+0.5 <= 0
m.con = Constraint(m.t, rule=_con)
def _init(m):
yield m.x1[0] == 0
yield m.x2[0] == -1
yield m.x3[0] == 0
m.init_conditions = ConstraintList(rule=_init)
# transform and solve
TransformationFactory('dae.collocation').apply_to(m, wrt=m.t, nfe=3, ncp=12, method='BACKWARD')
SolverFactory('ipopt').solve(m).write()
# ========================================================== # = Solver Results = # ========================================================== # ---------------------------------------------------------- # Problem Information # ---------------------------------------------------------- Problem: - Lower bound: -inf Upper bound: inf Number of objectives: 1 Number of constraints: 256 Number of variables: 255 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: 0.08427214622497559 # ---------------------------------------------------------- # Solution Information # ---------------------------------------------------------- Solution: - number of solutions: 0 number of solutions displayed: 0
%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
t = [t for t in m.t]
u = [m.u[t]() for t in t]
x1 = [m.x1[t]() for t in m.x1]
x2 = [m.x2[t]() for t in m.x2]
x3 = [m.x3[t]() for t in m.x3]
plt.figure(figsize=(10,12))
plt.subplot(4,1,1)
plt.plot(t, x1)
#X2-8*(t-0.5)^2+0.5 <= 0
plt.subplot(4,1,2)
plt.plot(t, x2)
plt.plot(t, 8*(np.array(t)-0.5)**2 - 0.5)
plt.subplot(4,1,3)
plt.plot(t, x3)
plt.subplot(4,1,4)
plt.plot(t, u)
[<matplotlib.lines.Line2D at 0x1233bfd30>]
from pyomo.environ import *
from pyomo.dae import *
m = ConcreteModel()
m.tf = Param(initialize=1)
m.t = ContinuousSet(bounds=(0, m.tf))
m.u = Var(m.t, initialize=0)
m.x1 = Var(m.t)
m.x2 = Var(m.t)
m.x3 = Var(m.t)
m.dx1 = DerivativeVar(m.x1)
m.dx2 = DerivativeVar(m.x2)
m.dx3 = DerivativeVar(m.x3)
m.obj = Objective(expr=m.x3[m.tf])
m.x1dotcon = Constraint(m.t, rule=lambda m, t: m.dx1[t] == m.x2[t])
m.x2dotcon = Constraint(m.t, rule=lambda m, t: m.dx2[t] == -m.x2[t] + m.u[t])
m.x3dotcon = Constraint(m.t, rule=lambda m, t: m.dx3[t] == m.x1[t]**2+m.x2[t]**2+0.005*m.u[t]**2)
m.con = Constraint(m.t, rule=lambda m, t: m.x2[t]-8*(t-0.5)**2+0.5 <= 0)
m.x1dotcon[0].deactivate()
m.x2dotcon[0].deactivate()
m.x3dotcon[0].deactivate()
m.ic = ConstraintList()
m.ic.add(m.x1[0] == 0)
m.ic.add(m.x2[0] == -1)
m.ic.add(m.x3[0] == 0)
<pyomo.core.base.constraint._GeneralConstraintData at 0x11591ec48>
# transform and solve
TransformationFactory('dae.collocation').apply_to(m, wrt=m.t, nfe=3, ncp=12, method='BACKWARD')
SolverFactory('ipopt').solve(m).write()
%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
t = [t for t in m.t]
u = [m.u[t]() for t in t]
x1 = [m.x1[t]() for t in m.x1]
x2 = [m.x2[t]() for t in m.x2]
x3 = [m.x3[t]() for t in m.x3]
plt.figure(figsize=(10,12))
plt.subplot(4,1,1)
plt.plot(t, x1)
#X2-8*(t-0.5)^2+0.5 <= 0
plt.subplot(4,1,2)
plt.plot(t, x2)
plt.plot(t, 8*(np.array(t)-0.5)**2 - 0.5)
plt.subplot(4,1,3)
plt.plot(t, x3)
plt.subplot(4,1,4)
plt.plot(t, u)
# ========================================================== # = Solver Results = # ========================================================== # ---------------------------------------------------------- # Problem Information # ---------------------------------------------------------- Problem: - Lower bound: -inf Upper bound: inf Number of objectives: 1 Number of constraints: 256 Number of variables: 255 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: 0.1155850887298584 # ---------------------------------------------------------- # Solution Information # ---------------------------------------------------------- Solution: - number of solutions: 0 number of solutions displayed: 0
[<matplotlib.lines.Line2D at 0x115d9b390>]
from pyomo.environ import *
from pyomo.dae import *
m = ConcreteModel()
m.tf = Param(initialize=1)
m.t = ContinuousSet(bounds=(0, m.tf))
m.u = Var(m.t, initialize=0)
m.N = [1, 2, 3]
m.x = Var(m.N, m.t)
m.dx = DerivativeVar(m.x)
m.obj = Objective(expr=m.x[3,m.tf])
def ode(m, n, t):
if t==0:
return Constraint.Skip
odes = {
1: m.dx[1,t] == m.x[2,t],
2: m.dx[2,t] == -m.x[2,t] + m.u[t],
3: m.dx[3,t] == m.x[1,t]**2 + m.x[2,t]**2 + 0.005*m.u[t]**2
}
return odes[n]
m.ode = Constraint(m.N, m.t, rule=ode)
m.con = Constraint(m.t, rule=lambda m, t: m.x[2,t]-8*(t-0.5)**2+0.5 <= 0)
m.ic = ConstraintList()
m.ic.add(m.x[1,0] == 0)
m.ic.add(m.x[2,0] == -1)
m.ic.add(m.x[3,0] == 0)
<pyomo.core.base.constraint._GeneralConstraintData at 0x116778408>
# transform and solve
TransformationFactory('dae.collocation').apply_to(m, wrt=m.t, nfe=3, ncp=12, method='BACKWARD')
SolverFactory('ipopt').solve(m).write()
%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
t = [t for t in m.t]
u = [m.u[t]() for t in t]
x1 = [m.x[1,t]() for t in t]
x2 = [m.x[2,t]() for t in t]
x3 = [m.x[3,t]() for t in t]
plt.figure(figsize=(10,12))
plt.subplot(4,1,1)
plt.plot(t, x1)
plt.subplot(4,1,2)
plt.plot(t, x2)
plt.plot(t, 8*(np.array(t)-0.5)**2 - 0.5)
plt.subplot(4,1,3)
plt.plot(t, x3)
plt.subplot(4,1,4)
plt.plot(t, u)
# ========================================================== # = Solver Results = # ========================================================== # ---------------------------------------------------------- # Problem Information # ---------------------------------------------------------- Problem: - Lower bound: -inf Upper bound: inf Number of objectives: 1 Number of constraints: 256 Number of variables: 255 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: 0.09073829650878906 # ---------------------------------------------------------- # Solution Information # ---------------------------------------------------------- Solution: - number of solutions: 0 number of solutions displayed: 0
[<matplotlib.lines.Line2D at 0x11676d898>]